Commit 0fe7a2d4fff9dc12418eef835bac61d086cb573a
2003-06-08T20:09:10
added libtommath-0.20
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diff --git a/bn.pdf b/bn.pdf
index 37866c8..bdbc819 100644
Binary files a/bn.pdf and b/bn.pdf differ
diff --git a/bn.tex b/bn.tex
index a69c13c..3c07991 100644
--- a/bn.tex
+++ b/bn.tex
@@ -1,7 +1,7 @@
\documentclass[]{article}
\begin{document}
-\title{LibTomMath v0.19 \\ A Free Multiple Precision Integer Library \\ http://math.libtomcrypt.org }
+\title{LibTomMath v0.20 \\ A Free Multiple Precision Integer Library \\ http://math.libtomcrypt.org }
\author{Tom St Denis \\ tomstdenis@iahu.ca}
\maketitle
\newpage
diff --git a/bn_mp_exptmod_fast.c b/bn_mp_exptmod_fast.c
index de42ff8..54de53d 100644
--- a/bn_mp_exptmod_fast.c
+++ b/bn_mp_exptmod_fast.c
@@ -80,7 +80,6 @@ mp_exptmod_fast (mp_int * G, mp_int * X, mp_int * P, mp_int * Y, int redmode)
if (((P->used * 2 + 1) < MP_WARRAY) &&
P->used < (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) {
redux = fast_mp_montgomery_reduce;
-
} else {
/* use slower baselien method */
redux = mp_montgomery_reduce;
diff --git a/changes.txt b/changes.txt
index f874a2b..5756d6a 100644
--- a/changes.txt
+++ b/changes.txt
@@ -1,3 +1,7 @@
+June 8th, 2003
+v0.20 -- Removed the book from the package. Added the TDCAL license document.
+ -- This release is officially pure-bred TDCAL again [last officially TDCAL based release was v0.16]
+
June 6th, 2003
v0.19 -- Fixed a bug in mp_montgomery_reduce() which was introduced when I tweaked mp_rshd() in the previous release.
Essentially the digits were not trimmed before the compare which cause a subtraction to occur all the time.
diff --git a/etc/2kprime.1 b/etc/2kprime.1
index eb12565..e1384db 100644
--- a/etc/2kprime.1
+++ b/etc/2kprime.1
@@ -1,2 +1 @@
-256-bits (k = 36113) = 115792089237316195423570985008687907853269984665640564039457584007913129603823
-512-bits (k = 38117) = 13407807929942597099574024998205846127479365820592393377723561443721764030073546976801874298166903427690031858186486050853753882811946569946433649006045979
+259-bits (k = 17745) = 926336713898529563388567880069503262826159877325124512315660672063305037101743
diff --git a/etc/2kprime.c b/etc/2kprime.c
index 47b0e1d..4f1d4bb 100644
--- a/etc/2kprime.c
+++ b/etc/2kprime.c
@@ -7,7 +7,7 @@ int sizes[] = {256, 512, 768, 1024, 1536, 2048, 3072, 4096};
int main(void)
{
char buf[2000];
- int x, y, t;
+ int x, y;
mp_int q, p;
FILE *out;
clock_t t1;
diff --git a/makefile b/makefile
index 15972af..a835a2e 100644
--- a/makefile
+++ b/makefile
@@ -1,6 +1,6 @@
CFLAGS += -I./ -Wall -W -Wshadow -O3 -fomit-frame-pointer -funroll-loops
-VERSION=0.19
+VERSION=0.20
default: libtommath.a
@@ -103,5 +103,6 @@ clean:
zipup: clean manual poster
perl gen.pl ; mv mpi.c pre_gen/ ; \
cd .. ; rm -rf ltm* libtommath-$(VERSION) ; mkdir libtommath-$(VERSION) ; \
- cp -R ./libtommath/* ./libtommath-$(VERSION)/ ; tar -c libtommath-$(VERSION)/* > ltm-$(VERSION).tar ; \
+ cp -R ./libtommath/* ./libtommath-$(VERSION)/ ; cp tdcal.pdf ./libtommath-$(VERSION)/ ; cd ./libtommath-$(VERSION) ; rm -f tommath.src tommath.tex tommath.out ; cd pics ; rm -f * ; cd .. ; cd .. ; ls ; \
+ tar -c libtommath-$(VERSION)/* > ltm-$(VERSION).tar ; \
bzip2 -9vv ltm-$(VERSION).tar ; zip -9 -r ltm-$(VERSION).zip libtommath-$(VERSION)/*
diff --git a/pics/expt_state.sxd b/pics/expt_state.sxd
deleted file mode 100644
index 6518404..0000000
Binary files a/pics/expt_state.sxd and /dev/null differ
diff --git a/pics/expt_state.tif b/pics/expt_state.tif
deleted file mode 100644
index cb06e8e..0000000
Binary files a/pics/expt_state.tif and /dev/null differ
diff --git a/pics/makefile b/pics/makefile
deleted file mode 100644
index 302adec..0000000
--- a/pics/makefile
+++ /dev/null
@@ -1,23 +0,0 @@
-# makes the images... yeah
-
-default: pses
-
-
-sliding_window.ps: sliding_window.tif
- tiff2ps -c -e sliding_window.tif > sliding_window.ps
-
-expt_state.ps: expt_state.tif
- tiff2ps -c -e expt_state.tif > expt_state.ps
-
-sliding_window.pdf: sliding_window.ps
- epstopdf sliding_window.ps
-
-expt_state.pdf: expt_state.ps
- epstopdf expt_state.ps
-
-pses: sliding_window.ps expt_state.ps
-pdfes: sliding_window.pdf expt_state.pdf
-
-clean:
- rm -rf *.ps *.pdf .xvpics
-
\ No newline at end of file
diff --git a/pics/sliding_window.TIF b/pics/sliding_window.TIF
deleted file mode 100644
index bb4cb96..0000000
Binary files a/pics/sliding_window.TIF and /dev/null differ
diff --git a/pics/sliding_window.sxd b/pics/sliding_window.sxd
deleted file mode 100644
index 91e7c0d..0000000
Binary files a/pics/sliding_window.sxd and /dev/null differ
diff --git a/poster.pdf b/poster.pdf
index a1ecd06..629336c 100644
Binary files a/poster.pdf and b/poster.pdf differ
diff --git a/pre_gen/mpi.c b/pre_gen/mpi.c
index e5b9347..c728269 100644
--- a/pre_gen/mpi.c
+++ b/pre_gen/mpi.c
@@ -2155,7 +2155,6 @@ mp_exptmod_fast (mp_int * G, mp_int * X, mp_int * P, mp_int * Y, int redmode)
if (((P->used * 2 + 1) < MP_WARRAY) &&
P->used < (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) {
redux = fast_mp_montgomery_reduce;
-
} else {
/* use slower baselien method */
redux = mp_montgomery_reduce;
diff --git a/tdcal.pdf b/tdcal.pdf
new file mode 100644
index 0000000..1566a9d
Binary files /dev/null and b/tdcal.pdf differ
diff --git a/tommath.out b/tommath.out
deleted file mode 100644
index fb54c12..0000000
--- a/tommath.out
+++ /dev/null
@@ -1,143 +0,0 @@
-\BOOKMARK [0][-]{chapter.1}{Introduction}{}
-\BOOKMARK [1][-]{section.1.1}{Multiple Precision Arithmetic}{chapter.1}
-\BOOKMARK [2][-]{subsection.1.1.1}{The Need for Multiple Precision Arithmetic}{section.1.1}
-\BOOKMARK [2][-]{subsection.1.1.2}{Multiple Precision Arithmetic}{section.1.1}
-\BOOKMARK [2][-]{subsection.1.1.3}{Benefits of Multiple Precision Arithmetic}{section.1.1}
-\BOOKMARK [2][-]{subsection.1.1.4}{Basis of Operations}{section.1.1}
-\BOOKMARK [1][-]{section.1.2}{Purpose of This Text}{chapter.1}
-\BOOKMARK [1][-]{section.1.3}{Discussion and Notation}{chapter.1}
-\BOOKMARK [2][-]{subsection.1.3.1}{Notation}{section.1.3}
-\BOOKMARK [2][-]{subsection.1.3.2}{Work Effort}{section.1.3}
-\BOOKMARK [1][-]{section.1.4}{Exercises}{chapter.1}
-\BOOKMARK [0][-]{chapter.2}{Introduction to LibTomMath}{}
-\BOOKMARK [1][-]{section.2.1}{What is LibTomMath?}{chapter.2}
-\BOOKMARK [1][-]{section.2.2}{Goals of LibTomMath}{chapter.2}
-\BOOKMARK [1][-]{section.2.3}{Choice of LibTomMath}{chapter.2}
-\BOOKMARK [2][-]{subsection.2.3.1}{Code Base}{section.2.3}
-\BOOKMARK [2][-]{subsection.2.3.2}{API Simplicity}{section.2.3}
-\BOOKMARK [2][-]{subsection.2.3.3}{Optimizations}{section.2.3}
-\BOOKMARK [2][-]{subsection.2.3.4}{Portability and Stability}{section.2.3}
-\BOOKMARK [2][-]{subsection.2.3.5}{Choice}{section.2.3}
-\BOOKMARK [0][-]{chapter.3}{Getting Started}{}
-\BOOKMARK [1][-]{section.3.1}{Library Basics}{chapter.3}
-\BOOKMARK [1][-]{section.3.2}{What is a Multiple Precision Integer?}{chapter.3}
-\BOOKMARK [2][-]{subsection.3.2.1}{The mp\137int structure}{section.3.2}
-\BOOKMARK [1][-]{section.3.3}{Argument Passing}{chapter.3}
-\BOOKMARK [1][-]{section.3.4}{Return Values}{chapter.3}
-\BOOKMARK [1][-]{section.3.5}{Initialization and Clearing}{chapter.3}
-\BOOKMARK [2][-]{subsection.3.5.1}{Initializing an mp\137int}{section.3.5}
-\BOOKMARK [2][-]{subsection.3.5.2}{Clearing an mp\137int}{section.3.5}
-\BOOKMARK [1][-]{section.3.6}{Other Initialization Routines}{chapter.3}
-\BOOKMARK [2][-]{subsection.3.6.1}{Initializing Variable Sized mp\137int Structures}{section.3.6}
-\BOOKMARK [2][-]{subsection.3.6.2}{Creating a Clone}{section.3.6}
-\BOOKMARK [2][-]{subsection.3.6.3}{Multiple Integer Initializations And Clearings}{section.3.6}
-\BOOKMARK [1][-]{section.3.7}{Maintenance}{chapter.3}
-\BOOKMARK [2][-]{subsection.3.7.1}{Augmenting Integer Precision}{section.3.7}
-\BOOKMARK [2][-]{subsection.3.7.2}{Clamping Excess Digits}{section.3.7}
-\BOOKMARK [0][-]{chapter.4}{Basic Operations}{}
-\BOOKMARK [1][-]{section.4.1}{Copying an Integer}{chapter.4}
-\BOOKMARK [1][-]{section.4.2}{Zeroing an Integer}{chapter.4}
-\BOOKMARK [1][-]{section.4.3}{Sign Manipulation}{chapter.4}
-\BOOKMARK [2][-]{subsection.4.3.1}{Absolute Value}{section.4.3}
-\BOOKMARK [2][-]{subsection.4.3.2}{Integer Negation}{section.4.3}
-\BOOKMARK [1][-]{section.4.4}{Small Constants}{chapter.4}
-\BOOKMARK [2][-]{subsection.4.4.1}{Setting Small Constants}{section.4.4}
-\BOOKMARK [2][-]{subsection.4.4.2}{Setting Large Constants}{section.4.4}
-\BOOKMARK [1][-]{section.4.5}{Comparisons}{chapter.4}
-\BOOKMARK [2][-]{subsection.4.5.1}{Unsigned Comparisions}{section.4.5}
-\BOOKMARK [2][-]{subsection.4.5.2}{Signed Comparisons}{section.4.5}
-\BOOKMARK [0][-]{chapter.5}{Basic Arithmetic}{}
-\BOOKMARK [1][-]{section.5.1}{Building Blocks}{chapter.5}
-\BOOKMARK [1][-]{section.5.2}{Addition and Subtraction}{chapter.5}
-\BOOKMARK [2][-]{subsection.5.2.1}{Low Level Addition}{section.5.2}
-\BOOKMARK [2][-]{subsection.5.2.2}{Low Level Subtraction}{section.5.2}
-\BOOKMARK [2][-]{subsection.5.2.3}{High Level Addition}{section.5.2}
-\BOOKMARK [2][-]{subsection.5.2.4}{High Level Subtraction}{section.5.2}
-\BOOKMARK [1][-]{section.5.3}{Bit and Digit Shifting}{chapter.5}
-\BOOKMARK [2][-]{subsection.5.3.1}{Multiplication by Two}{section.5.3}
-\BOOKMARK [2][-]{subsection.5.3.2}{Division by Two}{section.5.3}
-\BOOKMARK [1][-]{section.5.4}{Polynomial Basis Operations}{chapter.5}
-\BOOKMARK [2][-]{subsection.5.4.1}{Multiplication by x}{section.5.4}
-\BOOKMARK [2][-]{subsection.5.4.2}{Division by x}{section.5.4}
-\BOOKMARK [1][-]{section.5.5}{Powers of Two}{chapter.5}
-\BOOKMARK [2][-]{subsection.5.5.1}{Multiplication by Power of Two}{section.5.5}
-\BOOKMARK [2][-]{subsection.5.5.2}{Division by Power of Two}{section.5.5}
-\BOOKMARK [2][-]{subsection.5.5.3}{Remainder of Division by Power of Two}{section.5.5}
-\BOOKMARK [0][-]{chapter.6}{Multiplication and Squaring}{}
-\BOOKMARK [1][-]{section.6.1}{The Multipliers}{chapter.6}
-\BOOKMARK [1][-]{section.6.2}{Multiplication}{chapter.6}
-\BOOKMARK [2][-]{subsection.6.2.1}{The Baseline Multiplication}{section.6.2}
-\BOOKMARK [2][-]{subsection.6.2.2}{Faster Multiplication by the ``Comba'' Method}{section.6.2}
-\BOOKMARK [2][-]{subsection.6.2.3}{Polynomial Basis Multiplication}{section.6.2}
-\BOOKMARK [2][-]{subsection.6.2.4}{Karatsuba Multiplication}{section.6.2}
-\BOOKMARK [2][-]{subsection.6.2.5}{Toom-Cook 3-Way Multiplication}{section.6.2}
-\BOOKMARK [2][-]{subsection.6.2.6}{Signed Multiplication}{section.6.2}
-\BOOKMARK [1][-]{section.6.3}{Squaring}{chapter.6}
-\BOOKMARK [2][-]{subsection.6.3.1}{The Baseline Squaring Algorithm}{section.6.3}
-\BOOKMARK [2][-]{subsection.6.3.2}{Faster Squaring by the ``Comba'' Method}{section.6.3}
-\BOOKMARK [2][-]{subsection.6.3.3}{Polynomial Basis Squaring}{section.6.3}
-\BOOKMARK [2][-]{subsection.6.3.4}{Karatsuba Squaring}{section.6.3}
-\BOOKMARK [2][-]{subsection.6.3.5}{Toom-Cook Squaring}{section.6.3}
-\BOOKMARK [2][-]{subsection.6.3.6}{High Level Squaring}{section.6.3}
-\BOOKMARK [0][-]{chapter.7}{Modular Reduction}{}
-\BOOKMARK [1][-]{section.7.1}{Basics of Modular Reduction}{chapter.7}
-\BOOKMARK [1][-]{section.7.2}{The Barrett Reduction}{chapter.7}
-\BOOKMARK [2][-]{subsection.7.2.1}{Fixed Point Arithmetic}{section.7.2}
-\BOOKMARK [2][-]{subsection.7.2.2}{Choosing a Radix Point}{section.7.2}
-\BOOKMARK [2][-]{subsection.7.2.3}{Trimming the Quotient}{section.7.2}
-\BOOKMARK [2][-]{subsection.7.2.4}{Trimming the Residue}{section.7.2}
-\BOOKMARK [2][-]{subsection.7.2.5}{The Barrett Algorithm}{section.7.2}
-\BOOKMARK [2][-]{subsection.7.2.6}{The Barrett Setup Algorithm}{section.7.2}
-\BOOKMARK [1][-]{section.7.3}{The Montgomery Reduction}{chapter.7}
-\BOOKMARK [2][-]{subsection.7.3.1}{Digit Based Montgomery Reduction}{section.7.3}
-\BOOKMARK [2][-]{subsection.7.3.2}{Baseline Montgomery Reduction}{section.7.3}
-\BOOKMARK [2][-]{subsection.7.3.3}{Faster ``Comba'' Montgomery Reduction}{section.7.3}
-\BOOKMARK [2][-]{subsection.7.3.4}{Montgomery Setup}{section.7.3}
-\BOOKMARK [1][-]{section.7.4}{The Diminished Radix Algorithm}{chapter.7}
-\BOOKMARK [2][-]{subsection.7.4.1}{Choice of Moduli}{section.7.4}
-\BOOKMARK [2][-]{subsection.7.4.2}{Choice of k}{section.7.4}
-\BOOKMARK [2][-]{subsection.7.4.3}{Restricted Diminished Radix Reduction}{section.7.4}
-\BOOKMARK [2][-]{subsection.7.4.4}{Unrestricted Diminished Radix Reduction}{section.7.4}
-\BOOKMARK [1][-]{section.7.5}{Algorithm Comparison}{chapter.7}
-\BOOKMARK [0][-]{chapter.8}{Exponentiation}{}
-\BOOKMARK [1][-]{section.8.1}{Exponentiation Basics}{chapter.8}
-\BOOKMARK [2][-]{subsection.8.1.1}{Single Digit Exponentiation}{section.8.1}
-\BOOKMARK [1][-]{section.8.2}{k-ary Exponentiation}{chapter.8}
-\BOOKMARK [2][-]{subsection.8.2.1}{Optimal Values of k}{section.8.2}
-\BOOKMARK [2][-]{subsection.8.2.2}{Sliding-Window Exponentiation}{section.8.2}
-\BOOKMARK [1][-]{section.8.3}{Modular Exponentiation}{chapter.8}
-\BOOKMARK [2][-]{subsection.8.3.1}{Barrett Modular Exponentiation}{section.8.3}
-\BOOKMARK [1][-]{section.8.4}{Quick Power of Two}{chapter.8}
-\BOOKMARK [0][-]{chapter.9}{Higher Level Algorithms}{}
-\BOOKMARK [1][-]{section.9.1}{Integer Division with Remainder}{chapter.9}
-\BOOKMARK [1][-]{section.9.2}{Single Digit Helpers}{chapter.9}
-\BOOKMARK [2][-]{subsection.9.2.1}{Single Digit Addition}{section.9.2}
-\BOOKMARK [2][-]{subsection.9.2.2}{Single Digit Subtraction}{section.9.2}
-\BOOKMARK [2][-]{subsection.9.2.3}{Single Digit Multiplication}{section.9.2}
-\BOOKMARK [2][-]{subsection.9.2.4}{Single Digit Division}{section.9.2}
-\BOOKMARK [2][-]{subsection.9.2.5}{Single Digit Modulo}{section.9.2}
-\BOOKMARK [2][-]{subsection.9.2.6}{Single Digit Root Extraction}{section.9.2}
-\BOOKMARK [1][-]{section.9.3}{Random Number Generation}{chapter.9}
-\BOOKMARK [1][-]{section.9.4}{Formatted Output}{chapter.9}
-\BOOKMARK [2][-]{subsection.9.4.1}{Getting The Output Size}{section.9.4}
-\BOOKMARK [2][-]{subsection.9.4.2}{Generating Radix-n Output}{section.9.4}
-\BOOKMARK [2][-]{subsection.9.4.3}{Reading Radix-n Input}{section.9.4}
-\BOOKMARK [1][-]{section.9.5}{Unformatted Output}{chapter.9}
-\BOOKMARK [2][-]{subsection.9.5.1}{Getting The Output Size}{section.9.5}
-\BOOKMARK [2][-]{subsection.9.5.2}{Generating Output}{section.9.5}
-\BOOKMARK [2][-]{subsection.9.5.3}{Reading Input}{section.9.5}
-\BOOKMARK [0][-]{chapter.10}{Number Theoretic Algorithms}{}
-\BOOKMARK [1][-]{section.10.1}{Greatest Common Divisor}{chapter.10}
-\BOOKMARK [1][-]{section.10.2}{Least Common Multiple}{chapter.10}
-\BOOKMARK [1][-]{section.10.3}{Jacobi Symbol Computation}{chapter.10}
-\BOOKMARK [1][-]{section.10.4}{Modular Inverse}{chapter.10}
-\BOOKMARK [2][-]{subsection.10.4.1}{General Case}{section.10.4}
-\BOOKMARK [2][-]{subsection.10.4.2}{Odd Moduli}{section.10.4}
-\BOOKMARK [1][-]{section.10.5}{Primality Tests}{chapter.10}
-\BOOKMARK [2][-]{subsection.10.5.1}{Trial Division}{section.10.5}
-\BOOKMARK [2][-]{subsection.10.5.2}{The Fermat Test}{section.10.5}
-\BOOKMARK [2][-]{subsection.10.5.3}{The Miller-Rabin Test}{section.10.5}
-\BOOKMARK [2][-]{subsection.10.5.4}{Primality Test in a Bottle}{section.10.5}
-\BOOKMARK [2][-]{subsection.10.5.5}{The Next Prime}{section.10.5}
-\BOOKMARK [1][-]{section.10.6}{Root Extraction}{chapter.10}
-\BOOKMARK [0][-]{appendix*.16}{Appendix}{}
diff --git a/tommath.src b/tommath.src
deleted file mode 100644
index fd11871..0000000
--- a/tommath.src
+++ /dev/null
@@ -1,4675 +0,0 @@
-\documentclass[b5paper]{book}
-\usepackage{hyperref}
-\usepackage{makeidx}
-\usepackage{amssymb}
-\usepackage{color}
-\usepackage{alltt}
-\usepackage{graphicx}
-\usepackage{layout}
-\def\union{\cup}
-\def\intersect{\cap}
-\def\getsrandom{\stackrel{\rm R}{\gets}}
-\def\cross{\times}
-\def\cat{\hspace{0.5em} \| \hspace{0.5em}}
-\def\catn{$\|$}
-\def\divides{\hspace{0.3em} | \hspace{0.3em}}
-\def\nequiv{\not\equiv}
-\def\approx{\raisebox{0.2ex}{\mbox{\small $\sim$}}}
-\def\lcm{{\rm lcm}}
-\def\gcd{{\rm gcd}}
-\def\log{{\rm log}}
-\def\ord{{\rm ord}}
-\def\abs{{\mathit abs}}
-\def\rep{{\mathit rep}}
-\def\mod{{\mathit\ mod\ }}
-\renewcommand{\pmod}[1]{\ ({\rm mod\ }{#1})}
-\newcommand{\floor}[1]{\left\lfloor{#1}\right\rfloor}
-\newcommand{\ceil}[1]{\left\lceil{#1}\right\rceil}
-\def\Or{{\rm\ or\ }}
-\def\And{{\rm\ and\ }}
-\def\iff{\hspace{1em}\Longleftrightarrow\hspace{1em}}
-\def\implies{\Rightarrow}
-\def\undefined{{\rm ``undefined"}}
-\def\Proof{\vspace{1ex}\noindent {\bf Proof:}\hspace{1em}}
-\let\oldphi\phi
-\def\phi{\varphi}
-\def\Pr{{\rm Pr}}
-\newcommand{\str}[1]{{\mathbf{#1}}}
-\def\F{{\mathbb F}}
-\def\N{{\mathbb N}}
-\def\Z{{\mathbb Z}}
-\def\R{{\mathbb R}}
-\def\C{{\mathbb C}}
-\def\Q{{\mathbb Q}}
-\definecolor{DGray}{gray}{0.5}
-\newcommand{\emailaddr}[1]{\mbox{$<${#1}$>$}}
-\def\twiddle{\raisebox{0.3ex}{\mbox{\tiny $\sim$}}}
-\def\gap{\vspace{0.5ex}}
-\makeindex
-\begin{document}
-\frontmatter
-\pagestyle{empty}
-\title{Multiple-Precision Integer Arithmetic, \\ A Case Study Involving the LibTomMath Project \\ - DRAFT - }
-\author{\mbox{
-%\begin{small}
-\begin{tabular}{c}
-Tom St Denis \\
-Algonquin College \\
-\\
-Mads Rasmussen \\
-Open Communications Security \\
-\\
-Greg Rose \\
-QUALCOMM Australia \\
-\end{tabular}
-%\end{small}
-}
-}
-\maketitle
-This text in its entirety is copyright \copyright{}2003 by Tom St Denis. It may not be redistributed
-electronically or otherwise without the sole permission of the author. The text is freely redistributable as long as
-it is packaged along with the LibTomMath library in a non-commercial project. Contact the
-author for other redistribution rights.
-
-This text corresponds to the v0.17 release of the LibTomMath project.
-
-\begin{alltt}
-Tom St Denis
-111 Banning Rd
-Ottawa, Ontario
-K2L 1C3
-Canada
-
-Phone: 1-613-836-3160
-Email: tomstdenis@iahu.ca
-\end{alltt}
-
-This text is formatted to the international B5 paper size of 176mm wide by 250mm tall using the \LaTeX{}
-{\em book} macro package and the Perl {\em booker} package.
-
-\tableofcontents
-\listoffigures
-\chapter*{Preface}
-Blah.
-
-\mainmatter
-\pagestyle{headings}
-\chapter{Introduction}
-\section{Multiple Precision Arithmetic}
-\subsection{The Need for Multiple Precision Arithmetic}
-The most prevalent use for multiple precision arithmetic (\textit{often referred to as bignum math}) is within public
-key cryptography. Algorithms such as RSA, Diffie-Hellman and Elliptic Curve Cryptography require large integers in order to
-resist known cryptanalytic attacks. Typical modern programming languages such as C and Java only provide small
-single-precision data types which are incapable of precisely representing integers which are often hundreds of bits long.
-
-For example, consider multiplying $1,234,567$ by $9,876,543$ in C with an ``unsigned long'' data type. With an
-x86 machine the result is $4,136,875,833$ while the true result is $12,193,254,061,881$. The original inputs
-were approximately $21$ and $24$ bits respectively. If the C language cannot multiply two relatively small values
-together precisely how does anyone expect it to multiply two values that are considerably larger?
-
-Most advancements in fast multiple precision arithmetic stem from the desire for faster cryptographic primitives. However, cryptography
-is not the only field of study that can benefit from fast large integer routines. Another auxiliary use for multiple precision integers is
-high precision floating point data types. The basic IEEE standard floating point type is made up of an integer mantissa $q$ and an exponent $e$.
-Numbers are given in the form $n = q \cdot b^e$ where $b = 2$ is specified. Since IEEE is meant to be implemented in
-hardware the precision of the mantissa is often fairly small (\textit{23, 48 and 64 bits}). Since the mantissa is merely an
-integer a large multiple precision integer could be used. In effect very high precision floating point arithmetic
-could be performed. This would be useful where scientific applications must minimize the total output error over long simulations.
-
-\subsection{Multiple Precision Arithmetic}
-\index{multiple precision}
-Multiple precision arithmetic attempts to the solve the shortcomings of single precision data types such as those from
-the C and Java programming languages. In essence multiple precision arithmetic is a set of operations that can be
-performed on members of an algebraic group whose precision is not fixed. The algorithms when implemented to be multiple
-precision can allow a developer to work with any practical precision required.
-
-Typically the arithmetic over the ring of integers denoted by $\Z$ is performed by routines that are collectively and
-casually referred to as ``bignum'' routines. However, it is possible to have rings of polynomials as well typically
-denoted by $\Z/p\Z \left [ X \right ]$ which could have variable precision (\textit{or degree}). This text will
-discuss implementation of the former, however implementing polynomial basis routines should be relatively easy after reading this text.
-
-\subsection{Benefits of Multiple Precision Arithmetic}
-\index{precision} \index{accuracy}
-Precision of the real value to a given precision is defined loosely as the proximity of the real value to a given representation.
-Accuracy is defined as the reproducibility of the result. For example, the calculation $1/3 = 0.25$ is imprecise but can be accurate provided
-it is reproducible.
-
-The benefit of multiple precision representations over single precision representations is that
-often no precision is lost while representing the result of an operation which requires excess precision. For example,
-the multiplication of two $n$-bit integers requires at least $2n$ bits to represent the result. A multiple precision
-system would augment the precision of the destination to accomodate the result while a single precision system would
-truncate excess bits to maintain a fixed level of precision.
-
-Multiple precision representations allow for the precision to be very high (\textit{if not exacting}) but at a cost of
-modest computer resources. The only reasonable case where a multiple precision system will lose precision is when
-emulating a floating point data type. However, with multiple precision integer arithmetic no precision is lost.
-
-\subsection{Basis of Operations}
-At the heart of all multiple precision integer operations are the ``long-hand'' algorithms we all learned as children
-in grade school. For example, to multiply $1,234$ by $981$ the student is not taught to memorize the times table for
-$1,234$, instead they are taught how to long-multiply. That is to multiply each column using simple single digit
-multiplications, line up the partial results, and add the resulting products by column. The representation that most
-are familiar with is known as decimal or formally as radix-10. A radix-$n$ representation simply means there are
-$n$ possible values per digit. For example, binary would be a radix-2 representation.
-
-In essence computer based multiple precision arithmetic is very much the same. The most notable difference is the usage
-of a binary friendly radix. That is to use a radix of the form $2^k$ where $k$ is typically the size of a machine
-register. Also occasionally more optimal algorithms are used to perform certain operations such as multiplication and
-squaring instead of traditional long-hand algorithms.
-
-\section{Purpose of This Text}
-The purpose of this text is to instruct the reader regarding how to implement multiple precision algorithms. That is
-to not only explain the core theoretical algorithms but also the various ``house keeping'' tasks that are neglected by
-authors of other texts on the subject. Texts such as \cite[HAC]{HAC} and \cite{TAOCPV2} give considerably detailed
-explanations of the theoretical aspects of the algorithms and very little regarding the practical aspects.
-
-How an algorithm is explained and how it is actually implemented are two very different
-realities. For example, algorithm 14.7 on page 594 of HAC lists a relatively simple algorithm for performing multiple
-precision integer addition. However, what the description lacks is any discussion concerning the fact that the two
-integer inputs may be of differing magnitudes. Similarly the division routine (\textit{Algorithm 14.20, pp. 598})
-does not discuss how to handle sign or handle the dividend's decreasing magnitude in the main loop (\textit{Step \#3}).
-
-As well as the numerous practical oversights both of the texts do not discuss several key optimal algorithms required
-such as ``Comba'' and Karatsuba multipliers and fast modular inversion. These optimal algorithms are vital to achieve
-any form of useful performance in non-trivial applications.
-
-To solve this problem the focus of this text is on the practical aspects of implementing the algorithms that
-constitute a multiple precision integer package with light discussions on the theoretical aspects. As a case
-study the ``LibTomMath''\footnote{Available freely at http://math.libtomcrypt.org} package is used to demonstrate
-algorithms with implementations that have been field tested and work very well.
-
-\section{Discussion and Notation}
-\subsection{Notation}
-A multiple precision integer of $n$-digits shall be denoted as $x = (x_n ... x_1 x_0)_{ \beta }$ to be the
-multiple precision notation for the integer $x \equiv \sum_{i=0}^{n} x_i\beta^i$. The elements of the array $x$ are
-said to be the radix $\beta$ digits of the integer. For example, $x = (1,2,3)_{10}$ would represent the
-integer $1\cdot 10^2 + 2\cdot10^1 + 3\cdot10^0 = 123$.
-
-A ``mp\_int'' shall refer to a composite structure which contains the digits of the integer as well as auxilary data
-required to manipulate the data. These additional members are discussed in ~BASICOP~. For the purposes of this text
-a ``multiple precision integer'' and a ``mp\_int'' are synonymous.
-
-\index{single-precision} \index{double-precision} \index{mp\_digit} \index{mp\_word}
-For the purposes of this text a single-precision variable must be able to represent integers in the range $0 \le x < 2 \beta$ while
-a double-precision variable must be able to represent integers in the range $0 \le x < 2 \beta^2$. Within the source code that will be
-presented the data type \textbf{mp\_digit} will represent a single-precision type while \textbf{mp\_word} will represent a
-double-precision type. In several algorithms (\textit{notably the Comba routines}) temporary results
-will be stored in a double-precision arrays. For the purposes of this text $x_j$ will refer to the
-$j$'th digit of a single-precision array and $\hat x_j$ will refer to the $j$'th digit of a double-precision
-array.
-
-The $\lfloor \mbox{ } \rfloor$ brackets represent a value truncated and rounded down to the nearest integer. The $\lceil \mbox{ } \rceil$ brackets
-represent a value truncated and rounded up to the nearest integer. Typically when the $/$ division symbol is used the intention is to perform an integer
-division. For example, $5/2 = 2$ which will often be written as $\lfloor 5/2 \rfloor = 2$ for clarity. When a value is presented as a fraction
-such as $5 \over 2$ a real value division is implied.
-
-\subsection{Work Effort}
-\index{big-O}
-To measure the efficiency of various algorithms a modified big-O notation is used. In this system all
-single precision operations are considered to have the same cost\footnote{Except where explicitly noted.}.
-That is a single precision addition, multiplication and division are assumed to take the same time to
-complete. While this is generally not true in practice it will simplify the discussions considerably.
-
-Some algorithms have slight advantages over others which is why some constants will not be removed in
-the notation. For example, a normal multiplication requires $O(n^2)$ work while a squaring requires
-$O({{n^2 + n}\over 2})$ work. In standard big-O notation these would be said to be equivalent. However, in the
-context of the this text the magnitude of the inputs will not approach an infinite size. This means the conventional limit
-notation wisdom does not apply to the cancellation of constants.
-
-Throughout the discussions various ``work levels'' will be discussed. These levels are the $O(1)$,
-$O(n)$, $O(n^2)$, ..., $O(n^k)$ work efforts. For example, operations at the $O(n^k)$ ``level'' are said to be
-executed more frequently than operations at the $O(n^m)$ ``level'' when $k > m$. Obviously most optimizations will pay
-off the most at the higher levels since they represent the bulk of the effort required.
-
-\section{Exercises}
-Within the more advanced chapters a section will be set aside to give the reader some challenging exercises. These exercises are not
-designed to be prize winning problems, but to be thought provoking. Wherever possible the problems are forward minded stating
-problems that will be answered in subsequent chapters. The reader is encouraged to finish the exercises as they appear to get a
-better understanding of the subject material.
-
-Similar to the exercises of \cite{TAOCPV2} as explained on pp.\textit{ix} these exercises are given a scoring system. However, unlike
-\cite{TAOCPV2} the problems do not get nearly as hard as often. The scoring of these exercises ranges from one (\textit{the easiest}) to
-five (\textit{the hardest}). The following table sumarizes the scoring.
-
-\vspace{5mm}
-\begin{tabular}{cl}
-$\left [ 1 \right ]$ & An easy problem that should only take the reader a manner of \\
- & minutes to solve. Usually does not involve much computer time. \\
- & \\
-$\left [ 2 \right ]$ & An easy problem that involves a marginal amount of computer \\
- & time usage. Usually requires a program to be written to \\
- & solve the problem. \\
- & \\
-$\left [ 3 \right ]$ & A moderately hard problem that requires a non-trivial amount \\
- & of work. Usually involves trivial research and development of \\
- & new theory from the perspective of a student. \\
- & \\
-$\left [ 4 \right ]$ & A moderately hard problem that involves a non-trivial amount \\
- & of work and research. The solution to which will demonstrate \\
- & a higher mastery of the subject matter. \\
- & \\
-$\left [ 5 \right ]$ & A hard problem that involves concepts that are non-trivial. \\
- & Solutions to these problems will demonstrate a complete mastery \\
- & of the given subject. \\
- & \\
-\end{tabular}
-
-Essentially problems at the first level are meant to be simple questions that the reader can answer quickly without programming a solution or
-devising new theory. These problems are quick tests to see if the material is understood. Problems at the second level are also
-designed to be easy but will require a program or algorithm to be implemented to arrive at the answer.
-
-Problems at the third level are meant to be a bit more difficult. Often the answer is fairly obvious but arriving at an exacting solution
-requires some thought and skill. These problems will almost always involve devising a new algorithm or implementing a variation of
-another algorithm.
-
-Problems at the fourth level are meant to be even more difficult as well as involve some research. The reader will most likely not know
-the answer right away nor will this text provide the exact details of the answer (\textit{or at least not until a subsequent chapter}). Problems
-at the fifth level are meant to be the hardest problems relative to all the other problems in the chapter. People who can correctly
-answer fifth level problems have a mastery of the subject matter at hand.
-
-Often problems will be tied together. The purpose of this is to start a chain of thought that will be discussed in future chapters. The reader
-is encouraged to answer the follow-up problems and try to draw the relevence of problems.
-
-\chapter{Introduction to LibTomMath}
-
-\section{What is LibTomMath?}
-LibTomMath is a free and open source multiple precision library written in portable ISO C source code. By portable it is
-meant that the library does not contain any code that is computer platform dependent or otherwise problematic to use on any
-given platform. The library has been successfully tested under numerous operating systems including Solaris, MacOS, Windows,
-Linux, PalmOS and on standalone hardware such as the Gameboy Advance. The library is designed to contain enough
-functionality to be able to develop applications such as public key cryptosystems.
-
-\section{Goals of LibTomMath}
-
-Even though the library is written entirely in portable ISO C considerable care has been taken to
-optimize the algorithm implementations within the library. Specifically the code has been written to work well with
-the GNU C Compiler (\textit{GCC}) on both x86 and ARMv4 processors. Wherever possible highly efficient
-algorithms (\textit{such as Karatsuba multiplication, sliding window exponentiation and Montgomery reduction}) have
-been provided to make the library as efficient as possible. Even with the optimal and sometimes specialized
-algorithms that have been included the Application Programing Interface (\textit{API}) has been kept as simple as possible.
-Often generic place holder routines will make use of specialized algorithms automatically without the developer's
-attention. One such example is the generic multiplication algorithm \textbf{mp\_mul()} which will automatically use
-Karatsuba multiplication if the inputs are of a specific size.
-
-Making LibTomMath as efficient as possible is not the only goal of the LibTomMath project. Ideally the library should
-be source compatible with another popular library which makes it more attractive for developers to use. In this case the
-MPI library was used as a API template for all the basic functions.
-
-The project is also meant to act as a learning tool for students. The logic being that no easy-to-follow ``bignum''
-library exists which can be used to teach computer science students how to perform fast and reliable multiple precision
-arithmetic. To this end the source code has been given quite a few comments and algorithm discussion points. Often routines have
-more comments than lines of code.
-
-\section{Choice of LibTomMath}
-LibTomMath was chosen as the case study of this text not only because the author of both projects is one and the same but
-for more worthy reasons. Other libraries such as GMP, MPI, LIP and OpenSSL have multiple precision
-integer arithmetic routines but would not be ideal for this text for reasons as will be explained in the
-following sub-sections.
-
-\subsection{Code Base}
-The LibTomMath code base is all portable ISO C source code. This means that there are no platform dependent conditional
-segments of code littered throughout the source. This clean and uncluttered approach to the library means that a
-developer can more readily ascertain the true intent of a given section of source code without trying to keep track of
-what conditional code will be used.
-
-The code base of LibTomMath is also well organized. Each function is in its own separate source code file
-which allows the reader to find a given function very fast. When compiled with GCC for the x86 processor the entire
-library is a mere 87,760 bytes (\textit{$116,182$ bytes for ARMv4 processors}). This includes every single function
-LibTomMath provides from basic arithmetic to various number theoretic functions such as modular exponentiation, various
-reduction algorithms and Jacobi symbol computation.
-
-By comparison MPI which has fewer functions than LibTomMath compiled with the same conditions is 45,429 bytes
-(\textit{$54,536$ for ARMv4}). GMP which has rather large collection of functions with the default configuration on an
-x86 Athlon is 2,950,688 bytes. Note that while LibTomMath has fewer functions than GMP it has been used as the sole basis
-for several public key cryptosystems without having to seek additional outside functions to supplement the library.
-
-\subsection{API Simplicity}
-LibTomMath is designed after the MPI library and shares the API design. Quite often programs that use MPI will build
-with LibTomMath without change. The function names are relatively straight forward as to what they perform. Almost all of the
-functions except for a few minor exceptions which as will be discussed are for good reasons share the same parameter passing
-convention. The learning curve is fairly shallow with the API provided which is an extremely valuable benefit for the
-student and developer alike.
-
-The LIP library is an example of a library with an API that is awkward to work with. LIP uses function names that are often ``compressed'' to
-illegible short hand. LibTomMath does not share this fault.
-
-\subsection{Optimizations}
-While LibTomMath is certainly not the fastest library (\textit{GMP often beats LibTomMath by a factor of two}) it does
-feature a set of optimal algorithms for tasks ranging from modular reduction to squaring. GMP and LIP also feature
-such optimizations while MPI only uses baseline algorithms with no optimizations.
-
-LibTomMath is almost always an order of magnitude faster than the MPI library at computationally expensive tasks such as modular
-exponentiation. In the grand scheme of ``bignum'' libraries LibTomMath is faster than the average library and usually
-slower than the best libraries such as GMP and OpenSSL by a small factor.
-
-\subsection{Portability and Stability}
-LibTomMath will build ``out of the box'' on any platform equipped with a modern version of the GNU C Compiler
-(\textit{GCC}). This means that without changes the library will build without configuration or setting up any
-variables. LIP and MPI will build ``out of the box'' as well but have numerous known bugs. Most notably the author of
-MPI is not working on his library anymore.
-
-GMP requires a configuration script to run and will not build out of the box. GMP and LibTomMath are still in active
-development and are very stable across a variety of platforms.
-
-\subsection{Choice}
-LibTomMath is a relatively compact, well documented, highly optimized and portable library which seems only natural for
-the case study of this text. Various source files from the LibTomMath project will be included within the text. However, the
-reader is encouraged to download their own copy of the library to actually be able to work with the library.
-
-\chapter{Getting Started}
-MARK,BASICOP
-\section{Library Basics}
-To begin the design of a multiple precision integer library a primitive data type and a series of primitive algorithms must be established. A data
-type that will hold the information required to maintain a multiple precision integer must be designed. With this basic data type of a series
-of low level algorithms for initializing, clearing, growing and optimizing multiple precision integers can be developed to form the basis of
-the entire library of algorithms.
-
-\section{What is a Multiple Precision Integer?}
-Recall that most programming languages (\textit{in particular C}) only have fixed precision data types that on their own cannot be used
-to represent values larger than their precision alone will allow. The purpose of multiple precision algorithms is to use these fixed precision
-data types to create multiple precision integers which may represent values that are much larger.
-
-As a well known analogy, school children are taught how to form numbers larger than nine by prepending more radix ten digits. In the decimal system
-the largest value is only $9$ since the digits may only have values from $0$ to $9$. However, by concatenating digits together larger numbers
-may be represented. Computer based multiple precision arithmetic is essentially the same concept except with a different radix.
-
-What most people probably do not think about explicitly are the various other attributes that describe a multiple precision integer. For example,
-the integer $154_{10}$ has two immediately obvious properties. First, the integer is positive, that is the sign of this particular integer
-is positive as oppose to negative. Second, the integer has three digits in its representation. There is an additional property that the integer
-posesses that does not concern pencil-and-paper arithmetic. The third property is how many digits are allowed for the integer.
-
-The human analogy of this third property is ensuring there is enough space on the paper to right the integer. Computers must maintain a
-strict control on memory usage with respect to the digits of a multiple precision integer. These three properties make up what is known
-as a multiple precision integer or mp\_int for short.
-
-\subsection{The mp\_int structure}
-The mp\_int structure is the ISO C based manifestation of what represents a multiple precision integer. The ISO C standard does not provide for
-any such data type but it does provide for making composite data types known as structures. The following is the structure definition
-used within LibTomMath.
-
-\index{mp\_int}
-\begin{verbatim}
-typedef struct {
- int used, alloc, sign;
- mp_digit *dp;
-} mp_int;
-\end{verbatim}
-
-The mp\_int structure can be broken down as follows.
-
-\begin{enumerate}
-\item The \textbf{used} parameter denotes how many digits of the array \textbf{dp} contain the digits used to represent
-a given integer. The \textbf{used} count must not exceed the \textbf{alloc} count.
-
-\item The array \textbf{dp} holds the digits that represent the given integer. It is padded with $\textbf{alloc} - \textbf{used}$ zero
-digits.
-
-\item The \textbf{alloc} parameter denotes how
-many digits are available in the array to use by functions before it has to increase in size. When the \textbf{used} count
-of a result would exceed the \textbf{alloc} count all of the algorithms will automatically increase the size of the
-array to accommodate the precision of the result.
-
-\item The \textbf{sign} parameter denotes the sign as either zero/positive (\textbf{MP\_ZPOS}) or negative (\textbf{MP\_NEG}).
-\end{enumerate}
-
-\section{Argument Passing}
-A convention of argument passing must be adopted early on in the development of any library. Making the function prototypes
-consistent will help eliminate many headaches in the future as the library grows to significant complexity. In LibTomMath the multiple precision
-integer functions accept parameters from left to right as pointers to mp\_int structures. That means that the source operands are
-placed on the left and the destination on the right. Consider the following examples.
-
-\begin{verbatim}
- mp_mul(&a, &b, &c); /* c = a * b */
- mp_add(&a, &b, &a); /* a = a + b */
- mp_sqr(&a, &b); /* b = a * a */
-\end{verbatim}
-
-The left to right order is a fairly natural way to implement the functions since it lets the developer read aloud the
-functions and make sense of them. For example, the first function would read ``multiply a and b and store in c''.
-
-Certain libraries (\textit{LIP by Lenstra for instance}) accept parameters the other way around. That is the destination
-on the left and arguments on the right. In truth it is entirely a matter of preference. In the case of LibTomMath the
-convention from the MPI library has been adopted.
-
-Another very useful design consideration is whether to allow argument sources to also be a destination. For example, the
-second example (\textit{mp\_add}) adds $a$ to $b$ and stores in $a$. This is an important feature to implement since it
-allows the higher up functions to cut down on the number of variables. However, to implement this feature specific
-care has to be given to ensure the destination is not modified before the source is fully read.
-
-\section{Return Values}
-A well implemented library, no matter what its purpose, should trap as many runtime errors as possible and return them to the
-caller. By catching runtime errors a library can be guaranteed to prevent undefined behaviour. In a multiple precision
-library the only errors that can occur occur are related to inappropriate inputs (\textit{division by zero for instance}) or
-memory allocation errors.
-
-In LibTomMath any function that can cause a runtime error will return an error as an \textbf{int} data type with one of the
-following values.
-
-\index{MP\_OKAY} \index{MP\_VAL} \index{MP\_MEM}
-\begin{center}
-\begin{tabular}{|l|l|}
-\hline \textbf{Value} & \textbf{Meaning} \\
-\hline \textbf{MP\_OKAY} & The function was successful \\
-\hline \textbf{MP\_VAL} & One of the input value(s) was invalid \\
-\hline \textbf{MP\_MEM} & The function ran out of heap memory \\
-\hline
-\end{tabular}
-\end{center}
-
-When an error is detected within a function it should free any memory it allocated and return as soon as possible. The goal
-is to leave the system in the same state the system was when the function was called. Error checking with this style of API is fairly simple.
-
-\begin{verbatim}
- int err;
- if ((err = mp_add(&a, &b, &c)) != MP_OKAY) {
- printf("Error: %d\n", err);
- exit(EXIT_FAILURE);
- }
-\end{verbatim}
-
-The GMP library uses C style \textit{signals} to flag errors which is of questionable use. Not all errors are fatal
-and it was not deemed ideal by the author of LibTomMath to force developers to have signal handlers for such cases.
-
-\section{Initialization and Clearing}
-The logical starting point when actually writing multiple precision integer functions is the initialization and
-clearing of the integers. These two functions will be used by far the most throughout the algorithms whenever
-temporary integers are required.
-
-Given the basic mp\_int structure an initialization routine must first allocate memory to hold the digits of
-the integer. Often it is optimal to allocate a sufficiently large pre-set number of digits even considering
-the initial integer will represent zero. If only a single digit were allocated quite a few re-allocations
-would occur for the majority of inputs. There is a tradeoff between how many default digits to allocate
-and how many re-allocations are tolerable.
-
-If the memory for the digits has been successfully allocated then the rest of the members of the structure must
-be initialized. Since the initial state is to represent a zero integer the digits allocated must all be zeroed. The
-\textbf{used} count set to zero and \textbf{sign} set to \textbf{MP\_ZPOS}.
-
-\subsection{Initializing an mp\_int}
-To initialize an mp\_int the mp\_init algorithm shall be used. The purpose of this algorithm is to allocate
-the memory required and initialize the integer to a default representation of zero.
-
-\begin{figure}[here]
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_init}. \\
-\textbf{Input}. An mp\_int $a$ \\
-\textbf{Output}. Allocate memory for the digits and set to a zero state. \\
-\hline \\
-1. Allocate memory for \textbf{MP\_PREC} digits. \\
-2. If the allocation failed then return(\textit{MP\_MEM}) \\
-3. for $n$ from $0$ to $MP\_PREC - 1$ do \\
-\hspace{3mm}3.1 $a_n \leftarrow 0$\\
-4. $a.sign \leftarrow MP\_ZPOS$\\
-5. $a.used \leftarrow 0$\\
-6. $a.alloc \leftarrow MP\_PREC$\\
-7. Return(\textit{MP\_OKAY})\\
-\hline
-\end{tabular}
-\end{center}
-\caption{Algorithm mp\_init}
-\end{figure}
-
-\textbf{Algorithm mp\_init.}
-The \textbf{MP\_PREC} variable is a simple constant used to dictate minimal precision of allocated integers. It is ideally at least equal to $32$ but
-can be any reasonable power of two. Steps one and two allocate the memory and account for it. If the allocation fails the algorithm returns
-immediately to signal the failure. Step three will ensure that all the digits are in the default state of zero. Finally steps
-four through six set the default settings of the \textbf{sign}, \textbf{used} and \textbf{alloc} members of the mp\_int structure.
-
-EXAM,bn_mp_init.c
-
-The \textbf{OPT\_CAST} type cast on line @22,OPT_CAST@ is designed to allow C++ compilers to build the code out of
-the box. Microsoft C V5.00 is known to cause problems without the cast. Also note that if the memory
-allocation fails the other members of the mp\_int will be in an undefined state. The code from
-line @29,a->used@ to line @31,a->sign@ sets the default state for a mp\_int which is zero, positive and no used digits.
-
-\subsection{Clearing an mp\_int}
-When an mp\_int is no longer required the memory allocated for it can be cleared from the heap with
-the mp\_clear algorithm.
-
-\begin{figure}[here]
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_clear}. \\
-\textbf{Input}. An mp\_int $a$ \\
-\textbf{Output}. The memory for $a$ is cleared. \\
-\hline \\
-1. If $a$ has been previously freed then return(\textit{MP\_OKAY}). \\
-2. Free the digits of $a$ and mark $a$ as freed. \\
-3. $a.used \leftarrow 0$ \\
-4. $a.alloc \leftarrow 0$ \\
-5. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\caption{Algorithm mp\_clear}
-\end{figure}
-
-\textbf{Algorithm mp\_clear.}
-In steps one and two the memory for the digits are only free'd if they had not been previously released before.
-This is more of concern for the implementation since it is used to prevent ``double-free'' errors. It also helps catch
-code errors where mp\_ints are used after being cleared. Similarly steps three and four set the
-\textbf{used} and \textbf{alloc} to known values which would be easy to spot during debugging. For example, if an mp\_int is expected
-to be non-zero and its \textbf{used} member is observed to be zero (\textit{due to being cleared}) then an obvious bug in the code has been
-spotted.
-
-EXAM,bn_mp_clear.c
-
-The \textbf{if} statement on line @21,a->dp != NULL@ prevents the heap from being corrupted if a user double-frees an
-mp\_int. For example, a trivial case of this bug would be as follows.
-
-\begin{verbatim}
-mp_int a;
-mp_init(&a);
-mp_clear(&a);
-mp_clear(&a);
-\end{verbatim}
-
-Without that check the code would try to free the memory allocated for the digits twice which will cause most standard C
-libraries to cause a fault. Also by setting the pointer to \textbf{NULL} it helps debug code that may inadvertently
-free the mp\_int before it is truly not needed. The allocated digits are set to zero before being freed on line @24,memset@.
-This is ideal for cryptographic situations where the mp\_int is a secret parameter.
-
-The following snippet is an example of using both the init and clear functions.
-
-\begin{small}
-\begin{verbatim}
-#include <tommath.h>
-#include <stdio.h>
-#include <stdlib.h>
-int main(void)
-{
- mp_int num;
- int err;
-
- /* init the bignum */
- if ((err = mp_init(&num)) != MP_OKAY) {
- printf("Error: %d\n", err);
- return EXIT_FAILURE;
- }
-
- /* do work with it ... */
-
- /* clear up */
- mp_clear(&num);
-
- return EXIT_SUCCESS;
-}
-\end{verbatim}
-\end{small}
-
-\section{Other Initialization Routines}
-
-It is often helpful to have specialized initialization algorithms to simplify the design of other algorithms. For example, an
-initialization followed by a copy is a common operation when temporary copies of integers are required. It is quite
-beneficial to have a series of simple helper functions available.
-
-\subsection{Initializing Variable Sized mp\_int Structures}
-Occasionally the number of digits required will be known in advance of an initialization. In these
-cases the mp\_init\_size algorithm can be of use. The purpose of this algorithm is similar to mp\_init except that
-it will allocate \textit{at least} a specified number of digits. This is ideal to prevent re-allocations when the
-input size is known.
-
-\newpage\begin{figure}[here]
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_init\_size}. \\
-\textbf{Input}. An mp\_int $a$ and the requested number of digits $b$\\
-\textbf{Output}. $a$ is initialized to hold at least $b$ digits. \\
-\hline \\
-1. $u \leftarrow b\mbox{ (mod }MP\_PREC\mbox{)}$ \\
-2. $v \leftarrow b + 2 \cdot MP\_PREC - u$ \\
-3. Allocate $v$ digits. \\
-4. If the allocation failed then return(\textit{MP\_MEM}). \\
-5. for $n$ from $0$ to $v - 1$ do \\
-\hspace{3mm}5.1 $a_n \leftarrow 0$ \\
-6. $a.sign \leftarrow MP\_ZPOS$\\
-7. $a.used \leftarrow 0$\\
-8. $a.alloc \leftarrow v$\\
-9. Return(\textit{MP\_OKAY})\\
-\hline
-\end{tabular}
-\end{center}
-\caption{Algorithm mp\_init\_size}
-\end{figure}
-
-\textbf{Algorithm mp\_init\_size.}
-The value of $v$ is calculated to be at least the requested amount of digits $b$ plus additional padding. The padding is calculated
-to be at least \textbf{MP\_PREC} digits plus enough digits to make the digit count a multiple of \textbf{MP\_PREC}. This padding is used to
-prevent trivial allocations from becoming a bottleneck in the rest of the algorithms that depend on this.
-
-EXAM,bn_mp_init_size.c
-
-Line @23,MP_PREC@ will ensure that the number of digits actually allocated is padded up to the next multiple of
-\textbf{MP\_PREC} plus an additional \textbf{MP\_PREC}. This ensures that the number of allocated digit is
-always greater than the amount requested. As a result it prevents many trivial memory allocations. The value of
-\textbf{MP\_PREC} is defined in ``tommath.h'' and must be a power of two.
-
-\subsection{Creating a Clone}
-Another common sequence of operations is to make a local temporary copy of an argument. To initialize then copy a mp\_int will be known as
-creating a clone. This is useful within functions that need to modify an integer argument but do not wish to actually modify the original copy.
-The mp\_init\_copy algorithm will perform this very task.
-
-\begin{figure}[here]
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_init\_copy}. \\
-\textbf{Input}. An mp\_int $a$ and $b$\\
-\textbf{Output}. $a$ is initialized to be a copy of $b$. \\
-\hline \\
-1. Init $a$. (\textit{mp\_init}) \\
-2. If the init of $a$ was unsuccessful return(\textit{MP\_MEM}) \\
-3. Copy $b$ to $a$. (\textit{mp\_copy}) \\
-4. Return the status of the copy operation. \\
-\hline
-\end{tabular}
-\end{center}
-\caption{Algorithm mp\_init\_copy}
-\end{figure}
-
-\textbf{Algorithm mp\_init\_copy.}
-This algorithm will initialize a mp\_int variable and copy another previously initialized mp\_int variable into it. The algorithm will
-detect when the initialization fails and returns the error to the calling algorithm. As such this algorithm will perform two operations
-in one step.
-
-EXAM,bn_mp_init_copy.c
-
-This will initialize \textbf{a} and make it a verbatim copy of the contents of \textbf{b}. Note that
-\textbf{a} will have its own memory allocated which means that \textbf{b} may be cleared after the call
-and \textbf{a} will be left intact.
-
-\subsection{Multiple Integer Initializations And Clearings}
-Occasionally a function will require a series of mp\_int data types to be made available. The mp\_init\_multi algorithm
-is provided to simplify such cases. The purpose of this algorithm is to initialize a variable length array of mp\_int
-structures at once. As a result algorithms that require multiple integers only has to use
-one algorithm to initialize all the mp\_int variables.
-
-\begin{figure}[here]
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_init\_multi}. \\
-\textbf{Input}. Variable length array of mp\_int variables of length $k$. \\
-\textbf{Output}. The array is initialized such that each each mp\_int is ready to use. \\
-\hline \\
-1. for $n$ from 0 to $k - 1$ do \\
-\hspace{+3mm}1.1. Initialize the $n$'th mp\_int (\textit{mp\_init}) \\
-\hspace{+3mm}1.2. If initialization failed then do \\
-\hspace{+6mm}1.2.1. for $j$ from $0$ to $n$ do \\
-\hspace{+9mm}1.2.1.1. Free the $j$'th mp\_int (\textit{mp\_clear}) \\
-\hspace{+6mm}1.2.2. Return(\textit{MP\_MEM}) \\
-2. Return(\textit{MP\_OKAY}) \\
-\hline
-\end{tabular}
-\end{center}
-\caption{Algorithm mp\_init\_multi}
-\end{figure}
-
-\textbf{Algorithm mp\_init\_multi.}
-The algorithm will initialize the array of mp\_int variables one at a time. As soon as an runtime error is detected (\textit{step 1.2}) all of
-the previously initialized variables are cleared. The goal is an ``all or nothing'' initialization which allows for quick recovery from runtime
-errors.
-
-Similarly to clear a variable length array of mp\_int structures the mp\_clear\_multi algorithm will be used.
-
-Consider the following snippet which demonstrates how to use both routines.
-\begin{small}
-\begin{verbatim}
-#include <tommath.h>
-#include <stdio.h>
-#include <stdlib.h>
-int main(void)
-{
- mp_int num1, num2, num3;
- int err;
-
- if ((err = mp_init_multi(&num1, &num2, &num3, NULL)) !- MP_OKAY) {
- printf("Error: %d\n", err);
- return EXIT_FAILURE;
- }
-
- /* at this point num1/num2/num3 are ready */
-
- /* free them */
- mp_clear_multi(&num1, &num2, &num3, NULL);
-
- return EXIT_SUCCESS;
-}
-\end{verbatim}
-\end{small}
-
-Note how both lists are terminated with the \textbf{NULL} variable. This indicates to the algorithms to stop fetching parameters off
-of the stack. If it is not present the functions will most likely cause a segmentation fault.
-
-EXAM,bn_mp_multi.c
-
-Both routines are implemented in the same source file since they are typically used in conjunction with each other.
-
-\section{Maintenance}
-A small useful collection of mp\_int maintenance functions will also prove useful.
-
-\subsection{Augmenting Integer Precision}
-When storing a value in an mp\_int sufficient digits must be available to accomodate the entire value without
-loss of precision. Quite often the size of the array given by the \textbf{alloc} member is large enough to simply
-increase the \textbf{used} digit count. However, when the size of the array is too small it must be re-sized
-appropriately to accomodate the result. The mp\_grow algorithm will provide this functionality.
-
-\begin{figure}[here]
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_grow}. \\
-\textbf{Input}. An mp\_int $a$ and an integer $b$. \\
-\textbf{Output}. $a$ is expanded to accomodate $b$ digits. \\
-\hline \\
-1. if $a.alloc \ge b$ then return(\textit{MP\_OKAY}) \\
-2. $u \leftarrow b\mbox{ (mod }MP\_PREC\mbox{)}$ \\
-3. $v \leftarrow b + 2 \cdot MP\_PREC - u$ \\
-4. Re-Allocate the array of digits $a$ to size $v$ \\
-5. If the allocation failed then return(\textit{MP\_MEM}). \\
-6. for n from a.alloc to $v - 1$ do \\
-\hspace{+3mm}6.1 $a_n \leftarrow 0$ \\
-7. $a.alloc \leftarrow v$ \\
-8. Return(\textit{MP\_OKAY}) \\
-\hline
-\end{tabular}
-\end{center}
-\caption{Algorithm mp\_grow}
-\end{figure}
-
-\textbf{Algorithm mp\_grow.}
-Step one will prevent a re-allocation from being performed if it was not required. This is useful to prevent mp\_ints
-from growing excessively in code that erroneously calls mp\_grow. Similar to mp\_init\_size the requested digit count
-is padded to provide more digits than requested.
-
-In step four it is assumed that the reallocation leaves the lower $a.alloc$ digits intact. This is much akin to how the
-\textit{realloc} function from the standard C library works. Since the newly allocated digits are assumed to contain
-undefined values they are also initially zeroed.
-
-EXAM,bn_mp_grow.c
-
-The first step is to see if we actually need to perform a re-allocation at all. This is tested for on line
-@24,a->alloc < size@. Similar to mp\_init\_size the same code on line @26,MP_PREC - 1@ was used to resize the
-digits requested. A simple for loop from line @34,a->alloc@ to line @38,}@ will zero all digits that were above the
-old \textbf{alloc} limit to make sure the integer is in a known state.
-
-\subsection{Clamping Excess Digits}
-When a function anticipates a result will be $n$ digits it is simpler to assume this is true within the body of
-the function. For example, a multiplication of a $i$ digit number by a $j$ digit produces a result of at most
-$i + j$ digits. It is entirely possible that the result is $i + j - 1$ though, with no final carry into the last
-position. However, suppose the destination had to be first expanded (\textit{via mp\_grow}) to accomodate $i + j - 1$
-digits than further expanded to accomodate the final carry. That would be a considerable waste of time since heap
-operations are relatively slow.
-
-The ideal solution is to always assume the result is $i + j$ and fix up the \textbf{used} count after the function
-terminates. This way a single heap operation (\textit{at most}) is required. However, if the result was not checked
-there would be an excess high order zero digit.
-
-For example, suppose the product of two integers was $x_n = (0x_{n-1}x_{n-2}...x_0)_{\beta}$. The leading zero digit
-will not contribute to the precision of the result. In fact, through subsequent operations more leading zero digits would
-accumulate to the point the size of the integer would be prohibitive. As a result even though the precision is very
-low the representation is excessively large.
-
-The mp\_clamp algorithm is designed to solve this very problem. It will trim leading zeros by decrementing the
-\textbf{used} count until a non-zero leading digit is found. Also in this system, zero is considered to be a positive
-number which means that if the \textbf{used} count is decremented to zero the sign must be set to \textbf{MP\_ZPOS}.
-
-\begin{figure}[here]
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_clamp}. \\
-\textbf{Input}. An mp\_int $a$ \\
-\textbf{Output}. Any excess leading zero digits of $a$ are removed \\
-\hline \\
-1. while $a.used > 0$ and $a_{a.used - 1} = 0$ do \\
-\hspace{+3mm}1.1 $a.used \leftarrow a.used - 1$ \\
-2. if $a.used = 0$ then do \\
-\hspace{+3mm}2.1 $a.sign \leftarrow MP\_ZPOS$ \\
-\hline \\
-\end{tabular}
-\end{center}
-\caption{Algorithm mp\_clamp}
-\end{figure}
-
-\textbf{Algorithm mp\_clamp.}
-As can be expected this algorithm is very simple. The loop on step one is expected to iterate only once or twice at
-the most. For example, this will happen in cases where there is not a carry to fill the last position. Step two fixes the sign for
-when all of the digits are zero to ensure that the mp\_int is valid at all times.
-
-EXAM,bn_mp_clamp.c
-
-Note on line @27,while@ how to test for the \textbf{used} count is made on the left of the \&\& operator. In the C programming
-language the terms to \&\& are evaluated left to right with a boolean short-circuit if any condition fails. This is
-important since if the \textbf{used} is zero the test on the right would fetch below the array. That is obviously
-undesirable. The parenthesis on line @28,a->used@ is used to make sure the \textbf{used} count is decremented and not
-the pointer ``a''.
-
-\section*{Exercises}
-\begin{tabular}{cl}
-$\left [ 1 \right ]$ & Discuss the relevance of the \textbf{used} member of the mp\_int structure. \\
- & \\
-$\left [ 1 \right ]$ & Discuss the consequences of not using padding when performing allocations. \\
- & \\
-$\left [ 2 \right ]$ & Estimate an ideal value for \textbf{MP\_PREC} when performing 1024-bit RSA \\
- & encryption when $\beta = 2^{28}$. \\
- & \\
-$\left [ 1 \right ]$ & Discuss the relevance of the algorithm mp\_clamp. What does it prevent? \\
- & \\
-$\left [ 1 \right ]$ & Give an example of when the algorithm mp\_init\_copy might be useful. \\
- & \\
-\end{tabular}
-
-
-\chapter{Basic Operations}
-\section{Copying an Integer}
-After the various house-keeping routines are in place, simple algorithms can be designed to take advantage of them. Being able
-to make a verbatim copy of an integer is a very useful function to have. To copy an integer the mp\_copy algorithm will be used.
-
-\newpage\begin{figure}[here]
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_copy}. \\
-\textbf{Input}. An mp\_int $a$ and $b$. \\
-\textbf{Output}. Store a copy of $a$ in $b$. \\
-\hline \\
-1. Check if $a$ and $b$ point to the same location in memory. \\
-2. If true then return(\textit{MP\_OKAY}). \\
-3. If $b.alloc < a.used$ then grow $b$ to $a.used$ digits. (\textit{mp\_grow}) \\
-4. If failed to grow then return(\textit{MP\_MEM}). \\
-5. for $n$ from 0 to $a.used - 1$ do \\
-\hspace{3mm}5.1 $b_{n} \leftarrow a_{n}$ \\
-6. if $a.used < b.used - 1$ then \\
-\hspace{3mm}6.1. for $n$ from $a.used$ to $b.used - 1$ do \\
-\hspace{6mm}6.1.1 $b_{n} \leftarrow 0$ \\
-7. $b.used \leftarrow a.used$ \\
-8. $b.sign \leftarrow a.sign$ \\
-9. return(\textit{MP\_OKAY}) \\
-\hline
-\end{tabular}
-\end{center}
-\caption{Algorithm mp\_copy}
-\end{figure}
-
-\textbf{Algorithm mp\_copy.}
-Step 1 and 2 make sure that the two mp\_ints are unique. This allows the user to call the copy function with
-potentially the same input and not waste time. Step 3 and 4 ensure that the destination is large enough to
-hold a copy of the input $a$. Note that the \textbf{used} member of $b$ may be smaller than the \textbf{used}
-member of $a$ but a memory re-allocation is only required if the \textbf{alloc} member of $b$ is smaller. This
-prevents trivial memory reallocations.
-
-Step 5 copies the digits from $a$ to $b$ while step 6 ensures that if initially $\vert b \vert > \vert a \vert$,
-the more significant digits of $b$ will be zeroed. Finally steps 7 and 8 copies the \textbf{used} and \textbf{sign} members over
-which completes the copy operation.
-
-EXAM,bn_mp_copy.c
-
-Source lines @23,if dst ==@-@31,}@ do the initial house keeping. That is to see if the input is unique and if so to
-make sure there is enough room. If not enough space is available it returns the error and leaves the destination variable
-intact.
-
-The inner loop of the copy operation is contained between lines @34,{@ and @50,}@. Many LibTomMath routines are designed with this source code style
-in mind, making aliases to shorten lengthy pointers (\textit{see line @38,->@ and @39,->@}) for rapid use. Also the
-use of nested braces creates a simple way to denote various portions of code that reside on various work levels. Here, the copy loop is at the
-$O(n)$ level.
-
-\section{Zeroing an Integer}
-Reseting an mp\_int to the default state is a common step in many algorithms. The mp\_zero algorithm will be the algorithm used to
-perform this task.
-
-\begin{figure}[here]
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_zero}. \\
-\textbf{Input}. An mp\_int $a$ \\
-\textbf{Output}. Zero the contents of $a$ \\
-\hline \\
-1. $a.used \leftarrow 0$ \\
-2. $a.sign \leftarrow$ MP\_ZPOS \\
-3. for $n$ from 0 to $a.alloc - 1$ do \\
-\hspace{3mm}3.1 $a_n \leftarrow 0$ \\
-\hline
-\end{tabular}
-\end{center}
-\caption{Algorithm mp\_zero}
-\end{figure}
-
-\textbf{Algorithm mp\_zero.}
-This algorithm simply resets a mp\_int to the default state.
-
-EXAM,bn_mp_zero.c
-
-After the function is completed, all of the digits are zeroed, the \textbf{used} count is zeroed and the
-\textbf{sign} variable is set to \textbf{MP\_ZPOS}.
-
-\section{Sign Manipulation}
-\subsection{Absolute Value}
-With the mp\_int representation of an integer, calculating the absolute value is trivial. The mp\_abs algorithm will compute
-the absolute value of an mp\_int.
-
-\begin{figure}[here]
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_abs}. \\
-\textbf{Input}. An mp\_int $a$ \\
-\textbf{Output}. Computes $b = \vert a \vert$ \\
-\hline \\
-1. Copy $a$ to $b$. (\textit{mp\_copy}) \\
-2. If the copy failed return(\textit{MP\_MEM}). \\
-3. $b.sign \leftarrow MP\_ZPOS$ \\
-4. Return(\textit{MP\_OKAY}) \\
-\hline
-\end{tabular}
-\end{center}
-\caption{Algorithm mp\_abs}
-\end{figure}
-
-\textbf{Algorithm mp\_abs.}
-This algorithm computes the absolute of an mp\_int input. As can be expected the algorithm is very trivial.
-
-EXAM,bn_mp_abs.c
-
-\subsection{Integer Negation}
-With the mp\_int representation of an integer, calculating the negation is also trivial. The mp\_neg algorithm will compute
-the negative of an mp\_int input.
-
-\newpage\begin{figure}[here]
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_neg}. \\
-\textbf{Input}. An mp\_int $a$ \\
-\textbf{Output}. Computes $b = -a$ \\
-\hline \\
-1. Copy $a$ to $b$. (\textit{mp\_copy}) \\
-2. If the copy failed return(\textit{MP\_MEM}). \\
-3. If $a.sign = MP\_ZPOS$ then do \\
-\hspace{3mm}3.1 $b.sign = MP\_NEG$. \\
-4. else do \\
-\hspace{3mm}4.1 $b.sign = MP\_ZPOS$. \\
-5. Return(\textit{MP\_OKAY}) \\
-\hline
-\end{tabular}
-\end{center}
-\caption{Algorithm mp\_neg}
-\end{figure}
-
-\textbf{Algorithm mp\_neg.}
-This algorithm computes the negation of an input.
-
-EXAM,bn_mp_neg.c
-
-\section{Small Constants}
-\subsection{Setting Small Constants}
-Often a mp\_int must be set to a relatively small value such as $1$ or $2$. For these cases the mp\_set algorithm is useful.
-
-\newpage\begin{figure}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_set}. \\
-\textbf{Input}. An mp\_int $a$ and a digit $b$ \\
-\textbf{Output}. Make $a$ equivalent to $b$ \\
-\hline \\
-1. Zero $a$ (\textit{mp\_zero}). \\
-2. $a_0 \leftarrow b \mbox{ (mod }\beta\mbox{)}$ \\
-3. $a.used \leftarrow \left \lbrace \begin{array}{ll}
- 1 & \mbox{if }a_0 > 0 \\
- 0 & \mbox{if }a_0 = 0
- \end{array} \right .$ \\
-\hline
-\end{tabular}
-\end{center}
-\caption{Algorithm mp\_set}
-\end{figure}
-
-\textbf{Algorithm mp\_set.}
-This algorithm sets a mp\_int to a small single digit value. Step number 1 ensures that the integer is reset to the default state. The
-single digit is set (\textit{modulo $\beta$}) and the \textbf{used} count is adjusted accordingly.
-
-EXAM,bn_mp_set.c
-
-Line @21,mp_zero@ calls mp\_zero() to clear the mp\_int and reset the sign. Line @22,MP_MASK@ copies the digit
-into the least significant location. Note the usage of a new constant \textbf{MP\_MASK}. This constant is used to quickly
-reduce an integer modulo $\beta$. Since $\beta$ is of the form $2^k$ for any suitable $k$ it suffices to perform a binary AND with
-$MP\_MASK = 2^k - 1$ to perform the reduction. Finally line @23,a->used@ will set the \textbf{used} member with respect to the
-digit actually set. This function will always make the integer positive.
-
-One important limitation of this function is that it will only set one digit. The size of a digit is not fixed, meaning source that uses
-this function should take that into account. Meaning that only trivially small constants can be set using this function.
-
-\subsection{Setting Large Constants}
-To overcome the limitations of the mp\_set algorithm the mp\_set\_int algorithm is provided. It accepts a ``long''
-data type as input and will always treat it as a 32-bit integer.
-
-\begin{figure}[here]
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_set\_int}. \\
-\textbf{Input}. An mp\_int $a$ and a ``long'' integer $b$ \\
-\textbf{Output}. Make $a$ equivalent to $b$ \\
-\hline \\
-1. Zero $a$ (\textit{mp\_zero}) \\
-2. for $n$ from 0 to 7 do \\
-\hspace{3mm}2.1 $a \leftarrow a \cdot 16$ (\textit{mp\_mul2d}) \\
-\hspace{3mm}2.2 $u \leftarrow \lfloor b / 2^{4(7 - n)} \rfloor \mbox{ (mod }16\mbox{)}$\\
-\hspace{3mm}2.3 $a_0 \leftarrow a_0 + u$ \\
-\hspace{3mm}2.4 $a.used \leftarrow a.used + 1$ \\
-3. Clamp excess used digits (\textit{mp\_clamp}) \\
-\hline
-\end{tabular}
-\end{center}
-\caption{Algorithm mp\_set\_int}
-\end{figure}
-
-\textbf{Algorithm mp\_set\_int.}
-The algorithm performs eight iterations of a simple loop where in each iteration four bits from the source are added to the
-mp\_int. Step 2.1 will multiply the current result by sixteen making room for four more bits in the less significant positions. In step 2.2 the
-next four bits from the source are extracted and are added to the mp\_int. The \textbf{used} digit count is
-incremented to reflect the addition. The \textbf{used} digit counter is incremented since if any of the leading digits were zero the mp\_int would have
-zero digits used and the newly added four bits would be ignored.
-
-Excess zero digits are trimmed in steps 2.1 and 3 by using higher level algorithms mp\_mul2d and mp\_clamp.
-
-EXAM,bn_mp_set_int.c
-
-This function sets four bits of the number at a time to handle all practical \textbf{DIGIT\_BIT} sizes. The weird
-addition on line @38,a->used@ ensures that the newly added in bits are added to the number of digits. While it may not
-seem obvious as to why the digit counter does not grow exceedingly large it is because of the shift on line @27,mp_mul_2d@
-as well as the call to mp\_clamp() on line @40,mp_clamp@. Both functions will clamp excess leading digits which keeps
-the number of used digits low.
-
-\section{Comparisons}
-\subsection{Unsigned Comparisions}
-Comparing a multiple precision integer is performed with the exact same algorithm used to compare two decimal numbers. For example,
-to compare $1,234$ to $1,264$ the digits are extracted by their positions. That is we compare $1 \cdot 10^3 + 2 \cdot 10^2 + 3 \cdot 10^1 + 4 \cdot 10^0$
-to $1 \cdot 10^3 + 2 \cdot 10^2 + 6 \cdot 10^1 + 4 \cdot 10^0$ by comparing single digits at a time starting with the highest magnitude
-positions. If any leading digit of one integer is greater than a digit in the same position of another integer then obviously it must be greater.
-
-The first comparision routine that will be developed is the unsigned magnitude compare which will perform a comparison based on the digits of two
-mp\_int variables alone. It will ignore the sign of the two inputs. Such a function is useful when an absolute comparison is required or if the
-signs are known to agree in advance.
-
-To facilitate working with the results of the comparison functions three constants are required.
-
-\begin{figure}[here]
-\begin{center}
-\begin{tabular}{|r|l|}
-\hline \textbf{Constant} & \textbf{Meaning} \\
-\hline \textbf{MP\_GT} & Greater Than \\
-\hline \textbf{MP\_EQ} & Equal To \\
-\hline \textbf{MP\_LT} & Less Than \\
-\hline
-\end{tabular}
-\end{center}
-\caption{Comparison Return Codes}
-\end{figure}
-
-\begin{figure}[here]
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_cmp\_mag}. \\
-\textbf{Input}. Two mp\_ints $a$ and $b$. \\
-\textbf{Output}. Unsigned comparison results ($a$ to the left of $b$). \\
-\hline \\
-1. If $a.used > b.used$ then return(\textit{MP\_GT}) \\
-2. If $a.used < b.used$ then return(\textit{MP\_LT}) \\
-3. for n from $a.used - 1$ to 0 do \\
-\hspace{+3mm}3.1 if $a_n > b_n$ then return(\textit{MP\_GT}) \\
-\hspace{+3mm}3.2 if $a_n < b_n$ then return(\textit{MP\_LT}) \\
-4. Return(\textit{MP\_EQ}) \\
-\hline
-\end{tabular}
-\end{center}
-\caption{Algorithm mp\_cmp\_mag}
-\end{figure}
-
-\textbf{Algorithm mp\_cmp\_mag.}
-By saying ``$a$ to the left of $b$'' it is meant that the comparison is with respect to $a$, that is if $a$ is greater than $b$ it will return
-\textbf{MP\_GT} and similar with respect to when $a = b$ and $a < b$. The first two steps compare the number of digits used in both $a$ and $b$.
-Obviously if the digit counts differ there would be an imaginary zero digit in the smaller number where the leading digit of the larger number is.
-If both have the same number of digits than the actual digits themselves must be compared starting at the leading digit.
-
-By step three both inputs must have the same number of digits so its safe to start from either $a.used - 1$ or $b.used - 1$ and count down to
-the zero'th digit. If after all of the digits have been compared, no difference is found, the algorithm returns \textbf{MP\_EQ}.
-
-EXAM,bn_mp_cmp_mag.c
-
-The two if statements on lines @24,if@ and @28,if@ compare the number of digits in the two inputs. These two are performed before all of the digits
-are compared since it is a very cheap test to perform and can potentially save considerable time. The implementation given is also not valid
-without those two statements. $b.alloc$ may be smaller than $a.used$, meaning that undefined values will be read from $b$ past the end of the
-array of digits.
-
-\subsection{Signed Comparisons}
-Comparing with sign considerations is also fairly critical in several routines (\textit{division for example}). Based on an unsigned magnitude
-comparison a trivial signed comparison algorithm can be written.
-
-\begin{figure}[here]
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_cmp}. \\
-\textbf{Input}. Two mp\_ints $a$ and $b$ \\
-\textbf{Output}. Signed Comparison Results ($a$ to the left of $b$) \\
-\hline \\
-1. if $a.sign = MP\_NEG$ and $b.sign = MP\_ZPOS$ then return(\textit{MP\_LT}) \\
-2. if $a.sign = MP\_ZPOS$ and $b.sign = MP\_NEG$ then return(\textit{MP\_GT}) \\
-3. if $a.sign = MP\_NEG$ then \\
-\hspace{+3mm}3.1 Return the unsigned comparison of $b$ and $a$ (\textit{mp\_cmp\_mag}) \\
-4 Otherwise \\
-\hspace{+3mm}4.1 Return the unsigned comparison of $a$ and $b$ \\
-\hline
-\end{tabular}
-\end{center}
-\caption{Algorithm mp\_cmp}
-\end{figure}
-
-\textbf{Algorithm mp\_cmp.}
-The first two steps compare the signs of the two inputs. If the signs do not agree then it can return right away with the appropriate
-comparison code. When the signs are equal the digits of the inputs must be compared to determine the correct result. In step
-three the unsigned comparision flips the order of the arguments since they are both negative. For instance, if $-a > -b$ then
-$\vert a \vert < \vert b \vert$. Step number four will compare the two when they are both positive.
-
-EXAM,bn_mp_cmp.c
-
-The two if statements on lines @22,if@ and @26,if@ perform the initial sign comparison. If the signs are not the equal then which ever
-has the positive sign is larger. At line @30,if@, the inputs are compared based on magnitudes. If the signs were both negative then
-the unsigned comparison is performed in the opposite direction (\textit{line @31,mp_cmp_mag@}). Otherwise, the signs are assumed to
-be both positive and a forward direction unsigned comparison is performed.
-
-\section*{Exercises}
-\begin{tabular}{cl}
-$\left [ 2 \right ]$ & Modify algorithm mp\_set\_int to accept as input a variable length array of bits. \\
- & \\
-$\left [ 3 \right ]$ & Give the probability that algorithm mp\_cmp\_mag will have to compare $k$ digits \\
- & of two random digits (of equal magnitude) before a difference is found. \\
- & \\
-$\left [ 1 \right ]$ & Suggest a simple method to speed up the implementation of mp\_cmp\_mag based \\
- & on the observations made in the previous problem. \\
- &
-\end{tabular}
-
-\chapter{Basic Arithmetic}
-\section{Building Blocks}
-At this point algorithms for initialization, clearing, zeroing, copying, comparing and setting small constants have been
-established. The next logical set of algorithms to develop are addition, subtraction and digit shifting algorithms. These
-algorithms make use of the lower level algorithms and are the cruicial building block for the multiplication algorithms. It is very important
-that these algorithms are highly optimized. On their own they are simple $O(n)$ algorithms but they can be called from higher level algorithms
-which easily places them at $O(n^2)$ or even $O(n^3)$ work levels.
-
-MARK,SHIFTS
-All nine algorithms within this chapter make use of the logical bit shift operations denoted by $<<$ and $>>$ for left and right
-logical shifts respectively. A logical shift is analogous to sliding the decimal point of radix-10 representations. For example, the real
-number $0.9345$ is equivalent to $93.45\%$ which is found by sliding the the decimal two places to the right (\textit{multiplying by $10^2$}).
-Mathematically a logical shift is equivalent to a division or multiplication by a power of two.
-For example, $a << k = a \cdot 2^k$ while $a >> k = \lfloor a/2^k \rfloor$.
-
-One significant difference between a logical shift and the way decimals are shifted is that digits below the zero'th position are removed
-from the number. For example, consider $1101_2 >> 1$ using decimal notation this would produce $110.1_2$. However, with a logical shift the
-result is $110_2$.
-
-\section{Addition and Subtraction}
-In normal fixed precision arithmetic negative numbers are easily represented by subtraction from the modulus. For example, with 32-bit integers
-$a - b\mbox{ (mod }2^{32}\mbox{)}$ is the same as $a + (2^{32} - b) \mbox{ (mod }2^{32}\mbox{)}$ since $2^{32} \equiv 0 \mbox{ (mod }2^{32}\mbox{)}$.
-As a result subtraction can be performed with a trivial series of logical operations and an addition.
-
-However, in multiple precision arithmetic negative numbers are not represented in the same way. Instead a sign flag is used to keep track of the
-sign of the integer. As a result signed addition and subtraction are actually implemented as conditional usage of lower level addition or
-subtraction algorithms with the sign fixed up appropriately.
-
-The lower level algorithms will add or subtract integers without regard to the sign flag. That is they will add or subtract the magnitude of
-the integers respectively.
-
-\subsection{Low Level Addition}
-An unsigned addition of multiple precision integers is performed with the same long-hand algorithm used to add decimal numbers. That is to add the
-trailing digits first and propagate the resulting carry upwards. Since this is a lower level algorithm the name will have a ``s\_'' prefix.
-Historically that convention stems from the MPI library where ``s\_'' stood for static functions that were hidden from the developer entirely.
-
-\newpage
-\begin{figure}[!here]
-\begin{center}
-\begin{small}
-\begin{tabular}{l}
-\hline Algorithm \textbf{s\_mp\_add}. \\
-\textbf{Input}. Two mp\_ints $a$ and $b$ \\
-\textbf{Output}. The unsigned addition $c = \vert a \vert + \vert b \vert$. \\
-\hline \\
-1. if $a.used > b.used$ then \\
-\hspace{+3mm}1.1 $min \leftarrow b.used$ \\
-\hspace{+3mm}1.2 $max \leftarrow a.used$ \\
-\hspace{+3mm}1.3 $x \leftarrow a$ \\
-2. else \\
-\hspace{+3mm}2.1 $min \leftarrow a.used$ \\
-\hspace{+3mm}2.2 $max \leftarrow b.used$ \\
-\hspace{+3mm}2.3 $x \leftarrow b$ \\
-3. If $c.alloc < max + 1$ then grow $c$ to hold at least $max + 1$ digits (\textit{mp\_grow}) \\
-4. If failed to grow $c$ return(\textit{MP\_MEM}) \\
-5. $oldused \leftarrow c.used$ \\
-6. $c.used \leftarrow max + 1$ \\
-7. $u \leftarrow 0$ \\
-8. for $n$ from $0$ to $min - 1$ do \\
-\hspace{+3mm}8.1 $c_n \leftarrow a_n + b_n + u$ \\
-\hspace{+3mm}8.2 $u \leftarrow c_n >> lg(\beta)$ \\
-\hspace{+3mm}8.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\
-9. if $min \ne max$ then do \\
-\hspace{+3mm}9.1 for $n$ from $min$ to $max - 1$ do \\
-\hspace{+6mm}9.1.1 $c_n \leftarrow x_n + u$ \\
-\hspace{+6mm}9.1.2 $u \leftarrow c_n >> lg(\beta)$ \\
-\hspace{+6mm}9.1.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\
-10. $c_{max} \leftarrow u$ \\
-11. if $olduse > max$ then \\
-\hspace{+3mm}11.1 for $n$ from $max + 1$ to $olduse - 1$ do \\
-\hspace{+6mm}11.1.1 $c_n \leftarrow 0$ \\
-12. Clamp excess digits in $c$. (\textit{mp\_clamp}) \\
-13. Return(\textit{MP\_OKAY}) \\
-\hline
-\end{tabular}
-\end{small}
-\end{center}
-\caption{Algorithm s\_mp\_add}
-\end{figure}
-
-\textbf{Algorithm s\_mp\_add.}
-This algorithm is loosely based on algorithm 14.7 of HAC \cite[pp. 594]{HAC} but has been extended to allow the inputs to have different magnitudes.
-Coincidentally the description of algorithm A in Knuth \cite[pp. 266]{TAOCPV2} shares the same deficiency as the algorithm from \cite{HAC}. Even the
-MIX pseudo machine code presented by Knuth \cite[pp. 266-267]{TAOCPV2} is incapable of handling inputs which are of different magnitudes.
-
-Steps 1 and 2 will sort the two inputs based on their \textbf{used} digit count. This allows the inputs to have varying magnitudes which not
-only makes it more efficient than the trivial algorithm presented in the references but more flexible. The variable $min$ is given the lowest
-digit count while $max$ is given the highest digit count. If both inputs have the same \textbf{used} digit count both $min$ and $max$ are
-set to the same value. The variable $x$ is an \textit{alias} for the largest input and not meant to be a copy of it. After the inputs are sorted,
-steps 3 and 4 will ensure that the destination $c$ can accommodate the result. The old \textbf{used} count from $c$ is copied to
-$oldused$ so that excess digits can be cleared later, and the new \textbf{used} count is set to $max+1$, so that a carry from the most significant
-word can be handled.
-
-At step 7 the carry variable $u$ is set to zero and the first part of the addition loop can begin. The first step of the loop (\textit{8.1}) adds
-digits from the two inputs together along with the carry variable $u$. The following step extracts the carry bit by shifting the result of the
-preceding step right by $lg(\beta)$ positions. The shift to extract the carry is similar to how carry extraction works with decimal addition.
-
-Consider adding $77$ to $65$, the first addition of the first column is $7 + 5$ which produces the result $12$. The trailing digit of the result
-is $2 \equiv 12 \mbox{ (mod }10\mbox{)}$ and the carry is found by dividing (\textit{and ignoring the remainder}) $12$ by the radix or in this case $10$. The
-division and multiplication of $10$ is simply a logical right or left shift, respectively, of the digits. In otherwords the carry can be extracted
-by shifting one digit to the right.
-
-Note that $lg()$ is simply the base two logarithm such that $lg(2^k) = k$. This implies that $lg(\beta)$ is the number of bits in a radix-$\beta$
-digit. Therefore, a logical shift right of the summand by $lg(\beta)$ will extract the carry. The final step of the loop reduces the digit
-modulo the radix $\beta$ to ensure it is in range.
-
-After step 8 the smallest input (\textit{or both if they are the same magnitude}) has been exhausted. Step 9 decides whether
-the inputs were of equal magnitude. If not than another loop similar to that in step 8, must be executed. The loop at step
-number 9.1 differs from the previous loop since it only adds the mp\_int $x$ along with the carry.
-
-Step 10 finishes the addition phase by copying the final carry to the highest location in the result $c_{max}$. Step 11 ensures that
-leading digits that were originally present in $c$ are cleared. Finally excess leading digits are clamped and the algorithm returns success.
-
-EXAM,bn_s_mp_add.c
-
-Lines @27,if@ to @35,}@ perform the initial sorting of the inputs and determine the $min$ and $max$ variables. Note that $x$ is a pointer to a
-mp\_int assigned to the largest input, in effect it is a local alias. Lines @37,init@ to @42,}@ ensure that the destination is grown to
-accomodate the result of the addition.
-
-Similar to the implementation of mp\_copy this function uses the braced code and local aliases coding style. The three aliases that are on
-lines @56,tmpa@, @59,tmpb@ and @62,tmpc@ represent the two inputs and destination variables respectively. These aliases are used to ensure the
-compiler does not have to dereference $a$, $b$ or $c$ (respectively) to access the digits of the respective mp\_int.
-
-The initial carry $u$ is cleared on line @65,u = 0@, note that $u$ is of type mp\_digit which ensures type compatibility within the
-implementation. The initial addition loop begins on line @66,for@ and ends on line @75,}@. Similarly the conditional addition loop
-begins on line @81,for@ and ends on line @90,}@. The addition is finished with the final carry being stored in $tmpc$ on line @94,tmpc++@.
-Note the ``++'' operator on the same line. After line @94,tmpc++@ $tmpc$ will point to the $c.used$'th digit of the mp\_int $c$. This is useful
-for the next loop on lines @97,for@ to @99,}@ which set any old upper digits to zero.
-
-\subsection{Low Level Subtraction}
-The low level unsigned subtraction algorithm is very similar to the low level unsigned addition algorithm. The principle difference is that the
-unsigned subtraction algorithm requires the result to be positive. That is when computing $a - b$ the condition $\vert a \vert \ge \vert b\vert$ must
-be met for this algorithm to function properly. Keep in mind this low level algorithm is not meant to be used in higher level algorithms directly.
-This algorithm as will be shown can be used to create functional signed addition and subtraction algorithms.
-
-MARK,GAMMA
-
-For this algorithm a new variable is required to make the description simpler. Recall from section 1.3.1 that a mp\_digit must be able to represent
-the range $0 \le x < 2\beta$ for the algorithms to work correctly. However, it is allowable that a mp\_digit represent a larger range of values. For
-this algorithm we will assume that the variable $\gamma$ represents the number of bits available in a
-mp\_digit (\textit{this implies $2^{\gamma} > \beta$}).
-
-For example, the default for LibTomMath is to use a ``unsigned long'' for the mp\_digit ``type'' while $\beta = 2^{28}$. In ISO C an ``unsigned long''
-data type must be able to represent $0 \le x < 2^{32}$ meaning that in this case $\gamma = 32$.
-
-\newpage\begin{figure}[!here]
-\begin{center}
-\begin{small}
-\begin{tabular}{l}
-\hline Algorithm \textbf{s\_mp\_sub}. \\
-\textbf{Input}. Two mp\_ints $a$ and $b$ ($\vert a \vert \ge \vert b \vert$) \\
-\textbf{Output}. The unsigned subtraction $c = \vert a \vert - \vert b \vert$. \\
-\hline \\
-1. $min \leftarrow b.used$ \\
-2. $max \leftarrow a.used$ \\
-3. If $c.alloc < max$ then grow $c$ to hold at least $max$ digits. (\textit{mp\_grow}) \\
-4. If the reallocation failed return(\textit{MP\_MEM}). \\
-5. $oldused \leftarrow c.used$ \\
-6. $c.used \leftarrow max$ \\
-7. $u \leftarrow 0$ \\
-8. for $n$ from $0$ to $min - 1$ do \\
-\hspace{3mm}8.1 $c_n \leftarrow a_n - b_n - u$ \\
-\hspace{3mm}8.2 $u \leftarrow c_n >> (\gamma - 1)$ \\
-\hspace{3mm}8.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\
-9. if $min < max$ then do \\
-\hspace{3mm}9.1 for $n$ from $min$ to $max - 1$ do \\
-\hspace{6mm}9.1.1 $c_n \leftarrow a_n - u$ \\
-\hspace{6mm}9.1.2 $u \leftarrow c_n >> (\gamma - 1)$ \\
-\hspace{6mm}9.1.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\
-10. if $oldused > max$ then do \\
-\hspace{3mm}10.1 for $n$ from $max$ to $oldused - 1$ do \\
-\hspace{6mm}10.1.1 $c_n \leftarrow 0$ \\
-11. Clamp excess digits of $c$. (\textit{mp\_clamp}). \\
-12. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{small}
-\end{center}
-\caption{Algorithm s\_mp\_sub}
-\end{figure}
-
-\textbf{Algorithm s\_mp\_sub.}
-This algorithm performs the unsigned subtraction of two mp\_int variables under the restriction that the result must be positive. That is when
-passing variables $a$ and $b$ the condition that $\vert a \vert \ge \vert b \vert$ must be met for the algorithm to function correctly. This
-algorithm is loosely based on algorithm 14.9 \cite[pp. 595]{HAC} and is similar to algorithm S in \cite[pp. 267]{TAOCPV2} as well. As was the case
-of the algorithm s\_mp\_add both other references lack discussion concerning various practical details such as when the inputs differ in magnitude.
-
-The initial sorting of the inputs is trivial in this algorithm since $a$ is guaranteed to have at least the same magnitude of $b$. Steps 1 and 2
-set the $min$ and $max$ variables. Unlike the addition routine there is guaranteed to be no carry which means that the final result can be at
-most $max$ digits in length as opposed to $max + 1$. Similar to the addition algorithm the \textbf{used} count of $c$ is copied locally and
-set to the maximal count for the operation.
-
-The subtraction loop that begins on step 8 is essentially the same as the addition loop of algorithm s\_mp\_add except single precision
-subtraction is used instead. Note the use of the $\gamma$ variable to extract the carry (\textit{also known as the borrow}) within the subtraction
-loops. Under the assumption that two's complement single precision arithmetic is used this will successfully extract the desired carry.
-
-For example, consider subtracting $0101_2$ from $0100_2$ where $\gamma = 4$ and $\beta = 2$. The least significant bit will force a carry upwards to
-the third bit which will be set to zero after the borrow. After the very first bit has been subtracted $4 - 1 \equiv 0011_2$ will remain, When the
-third bit of $0101_2$ is subtracted from the result it will cause another carry. In this case though the carry will be forced to propagate all the
-way to the most significant bit.
-
-Recall that $\beta < 2^{\gamma}$. This means that if a carry does occur just before the $lg(\beta)$'th bit it will propagate all the way to the most
-significant bit. Thus, the high order bits of the mp\_digit that are not part of the actual digit will either be all zero, or all one. All that
-is needed is a single zero or one bit for the carry. Therefore a single logical shift right by $\gamma - 1$ positions is sufficient to extract the
-carry. This method of carry extraction may seem awkward but the reason for it becomes apparent when the implementation is discussed.
-
-If $b$ has a smaller magnitude than $a$ then step 9 will force the carry and copy operation to propagate through the larger input $a$ into $c$. Step
-10 will ensure that any leading digits of $c$ above the $max$'th position are zeroed.
-
-EXAM,bn_s_mp_sub.c
-
-Line @24,min@ and @25,max@ perform the initial hardcoded sorting of the inputs. In reality the $min$ and $max$ variables are only aliases and are only
-used to make the source code easier to read. Again the pointer alias optimization is used within this algorithm. Lines @42,tmpa@, @43,tmpb@ and @44,tmpc@ initialize the aliases for
-$a$, $b$ and $c$ respectively.
-
-The first subtraction loop occurs on lines @47,u = 0@ through @61,}@. The theory behind the subtraction loop is exactly the same as that for
-the addition loop. As remarked earlier there is an implementation reason for using the ``awkward'' method of extracting the carry
-(\textit{see line @57, >>@}). The traditional method for extracting the carry would be to shift by $lg(\beta)$ positions and logically AND
-the least significant bit. The AND operation is required because all of the bits above the $\lg(\beta)$'th bit will be set to one after a carry
-occurs from subtraction. This carry extraction requires two relatively cheap operations to extract the carry. The other method is to simply
-shift the most significant bit to the least significant bit thus extracting the carry with a single cheap operation. This optimization only works on
-twos compliment machines which is a safe assumption to make.
-
-If $a$ has a larger magnitude than $b$ an additional loop (\textit{see lines @64,for@ through @73,}@}) is required to propagate the carry through
-$a$ and copy the result to $c$.
-
-\subsection{High Level Addition}
-Now that both lower level addition and subtraction algorithms have been established an effective high level signed addition algorithm can be
-established. This high level addition algorithm will be what other algorithms and developers will use to perform addition of mp\_int data
-types.
-
-Recall from section 5.2 that an mp\_int represents an integer with an unsigned mantissa (\textit{the array of digits}) and a \textbf{sign}
-flag. A high level addition is actually performed as a series of eight separate cases which can be optimized down to three unique cases.
-
-\begin{figure}[!here]
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_add}. \\
-\textbf{Input}. Two mp\_ints $a$ and $b$ \\
-\textbf{Output}. The signed addition $c = a + b$. \\
-\hline \\
-1. if $a.sign = b.sign$ then do \\
-\hspace{3mm}1.1 $c.sign \leftarrow a.sign$ \\
-\hspace{3mm}1.2 $c \leftarrow \vert a \vert + \vert b \vert$ (\textit{s\_mp\_add})\\
-2. else do \\
-\hspace{3mm}2.1 if $\vert a \vert < \vert b \vert$ then do (\textit{mp\_cmp\_mag}) \\
-\hspace{6mm}2.1.1 $c.sign \leftarrow b.sign$ \\
-\hspace{6mm}2.1.2 $c \leftarrow \vert b \vert - \vert a \vert$ (\textit{s\_mp\_sub}) \\
-\hspace{3mm}2.2 else do \\
-\hspace{6mm}2.2.1 $c.sign \leftarrow a.sign$ \\
-\hspace{6mm}2.2.2 $c \leftarrow \vert a \vert - \vert b \vert$ \\
-3. If any of the lower level operations failed return(\textit{MP\_MEM}) \\
-4. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\caption{Algorithm mp\_add}
-\end{figure}
-
-\textbf{Algorithm mp\_add.}
-This algorithm performs the signed addition of two mp\_int variables. There is no reference algorithm to draw upon from either \cite{TAOCPV2} or
-\cite{HAC} since they both only provide unsigned operations. The algorithm is fairly straightforward but restricted since subtraction can only
-produce positive results.
-
-\begin{figure}[here]
-\begin{small}
-\begin{center}
-\begin{tabular}{|c|c|c|c|c|}
-\hline \textbf{Sign of $a$} & \textbf{Sign of $b$} & \textbf{$\vert a \vert > \vert b \vert $} & \textbf{Unsigned Operation} & \textbf{Result Sign Flag} \\
-\hline $+$ & $+$ & Yes & $c = a + b$ & $a.sign$ \\
-\hline $+$ & $+$ & No & $c = a + b$ & $a.sign$ \\
-\hline $-$ & $-$ & Yes & $c = a + b$ & $a.sign$ \\
-\hline $-$ & $-$ & No & $c = a + b$ & $a.sign$ \\
-\hline &&&&\\
-
-\hline $+$ & $-$ & No & $c = b - a$ & $b.sign$ \\
-\hline $-$ & $+$ & No & $c = b - a$ & $b.sign$ \\
-
-\hline &&&&\\
-
-\hline $+$ & $-$ & Yes & $c = a - b$ & $a.sign$ \\
-\hline $-$ & $+$ & Yes & $c = a - b$ & $a.sign$ \\
-
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Addition Guide Chart}
-\label{fig:AddChart}
-\end{figure}
-
-Figure~\ref{fig:AddChart} lists all of the eight possible input combinations and is sorted to show that only three specific cases need to be handled. The
-return code of the unsigned operations at step 1.2, 2.1.2 and 2.2.2 are forwarded to step 3 to check for errors. This simplifies the description
-of the algorithm considerably and best follows how the implementation actually was achieved.
-
-Also note how the \textbf{sign} is set before the unsigned addition or subtraction is performed. Recall from the descriptions of algorithms
-s\_mp\_add and s\_mp\_sub that the mp\_clamp function is used at the end to trim excess digits. The mp\_clamp algorithm will set the \textbf{sign}
-to \textbf{MP\_ZPOS} when the \textbf{used} digit count reaches zero.
-
-For example, consider performing $-a + a$ with algorithm mp\_add. By the description of the algorithm the sign is set to \textbf{MP\_NEG} which would
-produce a result of $-0$. However, since the sign is set first then the unsigned addition is performed the subsequent usage of algorithm mp\_clamp
-within algorithm s\_mp\_add will force $-0$ to become $0$.
-
-EXAM,bn_mp_add.c
-
-The source code follows the algorithm fairly closely. The most notable new source code addition is the usage of the $res$ integer variable which
-is used to pass result of the unsigned operations forward. Unlike in the algorithm, the variable $res$ is merely returned as is without
-explicitly checking it and returning the constant \textbf{MP\_OKAY}. The observation is this algorithm will succeed or fail only if the lower
-level functions do so. Returning their return code is sufficient.
-
-\subsection{High Level Subtraction}
-The high level signed subtraction algorithm is essentially the same as the high level signed addition algorithm.
-
-\newpage\begin{figure}[!here]
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_sub}. \\
-\textbf{Input}. Two mp\_ints $a$ and $b$ \\
-\textbf{Output}. The signed subtraction $c = a - b$. \\
-\hline \\
-1. if $a.sign \ne b.sign$ then do \\
-\hspace{3mm}1.1 $c.sign \leftarrow a.sign$ \\
-\hspace{3mm}1.2 $c \leftarrow \vert a \vert + \vert b \vert$ (\textit{s\_mp\_add}) \\
-2. else do \\
-\hspace{3mm}2.1 if $\vert a \vert \ge \vert b \vert$ then do (\textit{mp\_cmp\_mag}) \\
-\hspace{6mm}2.1.1 $c.sign \leftarrow a.sign$ \\
-\hspace{6mm}2.1.2 $c \leftarrow \vert a \vert - \vert b \vert$ (\textit{s\_mp\_sub}) \\
-\hspace{3mm}2.2 else do \\
-\hspace{6mm}2.2.1 $c.sign \leftarrow \left \lbrace \begin{array}{ll}
- MP\_ZPOS & \mbox{if }a.sign = MP\_NEG \\
- MP\_NEG & \mbox{otherwise} \\
- \end{array} \right .$ \\
-\hspace{6mm}2.2.2 $c \leftarrow \vert b \vert - \vert a \vert$ \\
-3. If any of the lower level operations failed return(\textit{MP\_MEM}). \\
-4. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\caption{Algorithm mp\_sub}
-\end{figure}
-
-\textbf{Algorithm mp\_sub.}
-This algorithm performs the signed subtraction of two inputs. Similar to algorithm mp\_add there is no reference in either \cite{TAOCPV2} or
-\cite{HAC}. Also this algorithm is restricted by algorithm s\_mp\_sub. The following chart lists the eight possible inputs and
-the operations required.
-
-\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{|c|c|c|c|c|}
-\hline \textbf{Sign of $a$} & \textbf{Sign of $b$} & \textbf{$\vert a \vert \ge \vert b \vert $} & \textbf{Unsigned Operation} & \textbf{Result Sign Flag} \\
-\hline $+$ & $-$ & Yes & $c = a + b$ & $a.sign$ \\
-\hline $+$ & $-$ & No & $c = a + b$ & $a.sign$ \\
-\hline $-$ & $+$ & Yes & $c = a + b$ & $a.sign$ \\
-\hline $-$ & $+$ & No & $c = a + b$ & $a.sign$ \\
-\hline &&&& \\
-\hline $+$ & $+$ & Yes & $c = a - b$ & $a.sign$ \\
-\hline $-$ & $-$ & Yes & $c = a - b$ & $a.sign$ \\
-\hline &&&& \\
-\hline $+$ & $+$ & No & $c = b - a$ & $\mbox{opposite of }a.sign$ \\
-\hline $-$ & $-$ & No & $c = b - a$ & $\mbox{opposite of }a.sign$ \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Subtraction Guide Chart}
-\end{figure}
-
-Similar to the case of algorithm mp\_add the \textbf{sign} is set first before the unsigned addition or subtraction. That is to prevent the
-algorithm from producing $-a - -a = -0$ as a result.
-
-EXAM,bn_mp_sub.c
-
-Much like the implementation of algorithm mp\_add the variable $res$ is used to catch the return code of the unsigned addition or subtraction operations
-and forward it to the end of the function. On line @38, != MP_LT@ the ``not equal to'' \textbf{MP\_LT} expression is used to emulate a
-``greater than or equal to'' comparison.
-
-\section{Bit and Digit Shifting}
-MARK,POLY
-It is quite common to think of a multiple precision integer as a polynomial in $x$, that is $y = f(\beta)$ where $f(x) = \sum_{i=0}^{n-1} a_i x^i$.
-This notation arises within discussion of Montgomery and Diminished Radix Reduction as well as Karatsuba multiplication and squaring.
-
-In order to facilitate operations on polynomials in $x$ as above a series of simple ``digit'' algorithms have to be established. That is to shift
-the digits left or right as well to shift individual bits of the digits left and right. It is important to note that not all ``shift'' operations
-are on radix-$\beta$ digits.
-
-\subsection{Multiplication by Two}
-
-In a binary system where the radix is a power of two multiplication by two not only arises often in other algorithms it is a fairly efficient
-operation to perform. A single precision logical shift left is sufficient to multiply a single digit by two.
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_mul\_2}. \\
-\textbf{Input}. One mp\_int $a$ \\
-\textbf{Output}. $b = 2a$. \\
-\hline \\
-1. If $b.alloc < a.used + 1$ then grow $b$ to hold $a.used + 1$ digits. (\textit{mp\_grow}) \\
-2. If the reallocation failed return(\textit{MP\_MEM}). \\
-3. $oldused \leftarrow b.used$ \\
-4. $b.used \leftarrow a.used$ \\
-5. $r \leftarrow 0$ \\
-6. for $n$ from 0 to $a.used - 1$ do \\
-\hspace{3mm}6.1 $rr \leftarrow a_n >> (lg(\beta) - 1)$ \\
-\hspace{3mm}6.2 $b_n \leftarrow (a_n << 1) + r \mbox{ (mod }\beta\mbox{)}$ \\
-\hspace{3mm}6.3 $r \leftarrow rr$ \\
-7. If $r \ne 0$ then do \\
-\hspace{3mm}7.1 $b_{n + 1} \leftarrow r$ \\
-\hspace{3mm}7.2 $b.used \leftarrow b.used + 1$ \\
-8. If $b.used < oldused - 1$ then do \\
-\hspace{3mm}8.1 for $n$ from $b.used$ to $oldused - 1$ do \\
-\hspace{6mm}8.1.1 $b_n \leftarrow 0$ \\
-9. $b.sign \leftarrow a.sign$ \\
-10. Return(\textit{MP\_OKAY}).\\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_mul\_2}
-\end{figure}
-
-\textbf{Algorithm mp\_mul\_2.}
-This algorithm will quickly multiply a mp\_int by two provided $\beta$ is a power of two. Neither \cite{TAOCPV2} nor \cite{HAC} describe such
-an algorithm despite the fact it arises often in other algorithms. The algorithm is setup much like the lower level algorithm s\_mp\_add since
-it is for all intents and purposes equivalent to the operation $b = \vert a \vert + \vert a \vert$.
-
-Step 1 and 2 grow the input as required to accomodate the maximum number of \textbf{used} digits in the result. The initial \textbf{used} count
-is set to $a.used$ at step 4. Only if there is a final carry will the \textbf{used} count require adjustment.
-
-Step 6 is an optimization implementation of the addition loop for this specific case. That is since the two values being added together
-are the same there is no need to perform two reads from the digits of $a$. Step 6.1 performs a single precision shift on the current digit $a_n$ to
-obtain what will be the carry for the next iteration. Step 6.2 calculates the $n$'th digit of the result as single precision shift of $a_n$ plus
-the previous carry. Recall from ~SHIFTS~ that $a_n << 1$ is equivalent to $a_n \cdot 2$. An iteration of the addition loop is finished with
-forwarding the carry to the next iteration.
-
-Step 7 takes care of any final carry by setting the $a.used$'th digit of the result to the carry and augmenting the \textbf{used} count of $b$.
-Step 8 clears any leading digits of $b$ in case it originally had a larger magnitude than $a$.
-
-EXAM,bn_mp_mul_2.c
-
-This implementation is essentially an optimized implementation of s\_mp\_add for the case of doubling an input. The only noteworthy difference
-is the use of the logical shift operator on line @52,<<@ to perform a single precision doubling.
-
-\subsection{Division by Two}
-A division by two can just as easily be accomplished with a logical shift right as multiplication by two can be with a logical shift left.
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_div\_2}. \\
-\textbf{Input}. One mp\_int $a$ \\
-\textbf{Output}. $b = a/2$. \\
-\hline \\
-1. If $b.alloc < a.used$ then grow $b$ to hold $a.used$ digits. (\textit{mp\_grow}) \\
-2. If the reallocation failed return(\textit{MP\_MEM}). \\
-3. $oldused \leftarrow b.used$ \\
-4. $b.used \leftarrow a.used$ \\
-5. $r \leftarrow 0$ \\
-6. for $n$ from $b.used - 1$ to $0$ do \\
-\hspace{3mm}6.1 $rr \leftarrow a_n \mbox{ (mod }2\mbox{)}$\\
-\hspace{3mm}6.2 $b_n \leftarrow (a_n >> 1) + (r << (lg(\beta) - 1)) \mbox{ (mod }\beta\mbox{)}$ \\
-\hspace{3mm}6.3 $r \leftarrow rr$ \\
-7. If $b.used < oldused - 1$ then do \\
-\hspace{3mm}7.1 for $n$ from $b.used$ to $oldused - 1$ do \\
-\hspace{6mm}7.1.1 $b_n \leftarrow 0$ \\
-8. $b.sign \leftarrow a.sign$ \\
-9. Clamp excess digits of $b$. (\textit{mp\_clamp}) \\
-10. Return(\textit{MP\_OKAY}).\\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_div\_2}
-\end{figure}
-
-\textbf{Algorithm mp\_div\_2.}
-This algorithm will divide an mp\_int by two using logical shifts to the right. Like mp\_mul\_2 it uses a modified low level addition
-core as the basis of the algorithm. Unlike mp\_mul\_2 the shift operations work from the leading digit to the trailing digit. The algorithm
-could be written to work from the trailing digit to the leading digit however, it would have to stop one short of $a.used - 1$ digits to prevent
-reading past the end of the array of digits.
-
-Essentially the loop at step 6 is similar to that of mp\_mul\_2 except the logical shifts go in the opposite direction and the carry is at the
-least significant bit not the most significant bit.
-
-EXAM,bn_mp_div_2.c
-
-\section{Polynomial Basis Operations}
-Recall from ~POLY~ that any integer can be represented as a polynomial in $x$ as $y = f(\beta)$. Such a representation is also known as
-the polynomial basis \cite[pp. 48]{ROSE}. Given such a notation a multiplication or division by $x$ amounts to shifting whole digits a single
-place. The need for such operations arises in several other higher level algorithms such as Barrett and Montgomery reduction, integer
-division and Karatsuba multiplication.
-
-Converting from an array of digits to polynomial basis is very simple. Consider the integer $y \equiv (a_2, a_1, a_0)_{\beta}$ and recall that
-$y = \sum_{i=0}^{2} a_i \beta^i$. Simply replace $\beta$ with $x$ and the expression is in polynomial basis. For example, $f(x) = 8x + 9$ is the
-polynomial basis representation for $89$ using radix ten. That is, $f(10) = 8(10) + 9 = 89$.
-
-\subsection{Multiplication by $x$}
-
-Given a polynomial in $x$ such as $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_0$ multiplying by $x$ amounts to shifting the coefficients up one
-degree. In this case $f(x) \cdot x = a_n x^{n+1} + a_{n-1} x^n + ... + a_0 x$. From a scalar basis point of view multiplying by $x$ is equivalent to
-multiplying by the integer $\beta$.
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_lshd}. \\
-\textbf{Input}. One mp\_int $a$ and an integer $b$ \\
-\textbf{Output}. $a \leftarrow a \cdot \beta^b$ (equivalent to multiplication by $x^b$). \\
-\hline \\
-1. If $b \le 0$ then return(\textit{MP\_OKAY}). \\
-2. If $a.alloc < a.used + b$ then grow $a$ to at least $a.used + b$ digits. (\textit{mp\_grow}). \\
-3. If the reallocation failed return(\textit{MP\_MEM}). \\
-4. $a.used \leftarrow a.used + b$ \\
-5. $i \leftarrow a.used - 1$ \\
-6. $j \leftarrow a.used - 1 - b$ \\
-7. for $n$ from $a.used - 1$ to $b$ do \\
-\hspace{3mm}7.1 $a_{i} \leftarrow a_{j}$ \\
-\hspace{3mm}7.2 $i \leftarrow i - 1$ \\
-\hspace{3mm}7.3 $j \leftarrow j - 1$ \\
-8. for $n$ from 0 to $b - 1$ do \\
-\hspace{3mm}8.1 $a_n \leftarrow 0$ \\
-9. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_lshd}
-\end{figure}
-
-\textbf{Algorithm mp\_lshd.}
-This algorithm multiplies an mp\_int by the $b$'th power of $x$. This is equivalent to multiplying by $\beta^b$. The algorithm differs
-from the other algorithms presented so far as it performs the operation in place instead storing the result in a separate location. The
-motivation behind this change is due to the way this function is typically used. Algorithms such as mp\_add store the result in an optionally
-different third mp\_int because the original inputs are often still required. Algorithm mp\_lshd (\textit{and similarly algorithm mp\_rshd}) is
-typically used on values where the original value is no longer required. The algorithm will return success immediately if
-$b \le 0$ since the rest of algorithm is only valid when $b > 0$.
-
-First the destination $a$ is grown as required to accomodate the result. The counters $i$ and $j$ are used to form a \textit{sliding window} over
-the digits of $a$ of length $b$. The head of the sliding window is at $i$ (\textit{the leading digit}) and the tail at $j$ (\textit{the trailing digit}).
-The loop on step 7 copies the digit from the tail to the head. In each iteration the window is moved down one digit. The last loop on
-step 8 sets the lower $b$ digits to zero.
-
-\newpage
-FIGU,sliding_window,Sliding Window Movement
-
-EXAM,bn_mp_lshd.c
-
-The if statement on line @24,if@ ensures that the $b$ variable is greater than zero. The \textbf{used} count is incremented by $b$ before
-the copy loop begins. This elminates the need for an additional variable in the for loop. The variable $top$ on line @42,top@ is an alias
-for the leading digit while $bottom$ on line @45,bottom@ is an alias for the trailing edge. The aliases form a window of exactly $b$ digits
-over the input.
-
-\subsection{Division by $x$}
-
-Division by powers of $x$ is easily achieved by shifting the digits right and removing any that will end up to the right of the zero'th digit.
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_rshd}. \\
-\textbf{Input}. One mp\_int $a$ and an integer $b$ \\
-\textbf{Output}. $a \leftarrow a / \beta^b$ (Divide by $x^b$). \\
-\hline \\
-1. If $b \le 0$ then return. \\
-2. If $a.used \le b$ then do \\
-\hspace{3mm}2.1 Zero $a$. (\textit{mp\_zero}). \\
-\hspace{3mm}2.2 Return. \\
-3. $i \leftarrow 0$ \\
-4. $j \leftarrow b$ \\
-5. for $n$ from 0 to $a.used - b - 1$ do \\
-\hspace{3mm}5.1 $a_i \leftarrow a_j$ \\
-\hspace{3mm}5.2 $i \leftarrow i + 1$ \\
-\hspace{3mm}5.3 $j \leftarrow j + 1$ \\
-6. for $n$ from $a.used - b$ to $a.used - 1$ do \\
-\hspace{3mm}6.1 $a_n \leftarrow 0$ \\
-7. $a.used \leftarrow a.used - b$ \\
-8. Return. \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_rshd}
-\end{figure}
-
-\textbf{Algorithm mp\_rshd.}
-This algorithm divides the input in place by the $b$'th power of $x$. It is analogous to dividing by a $\beta^b$ but much quicker since
-it does not require single precision division. This algorithm does not actually return an error code as it cannot fail.
-
-If the input $b$ is less than one the algorithm quickly returns without performing any work. If the \textbf{used} count is less than or equal
-to the shift count $b$ then it will simply zero the input and return.
-
-After the trivial cases of inputs have been handled the sliding window is setup. Much like the case of algorithm mp\_lshd a sliding window that
-is $b$ digits wide is used to copy the digits. Unlike mp\_lshd the window slides in the opposite direction from the trailing to the leading digit.
-Also the digits are copied from the leading to the trailing edge.
-
-Once the window copy is complete the upper digits must be zeroed and the \textbf{used} count decremented.
-
-EXAM,bn_mp_rshd.c
-
-The only noteworthy element of this routine is the lack of a return type.
-
--- Will update later to give it a return type...Tom
-
-\section{Powers of Two}
-
-Now that algorithms for moving single bits as well as whole digits exist algorithms for moving the ``in between'' distances are required. For
-example, to quickly multiply by $2^k$ for any $k$ without using a full multiplier algorithm would prove useful. Instead of performing single
-shifts $k$ times to achieve a multiplication by $2^{\pm k}$ a mixture of whole digit shifting and partial digit shifting is employed.
-
-\subsection{Multiplication by Power of Two}
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_mul\_2d}. \\
-\textbf{Input}. One mp\_int $a$ and an integer $b$ \\
-\textbf{Output}. $c \leftarrow a \cdot 2^b$. \\
-\hline \\
-1. $c \leftarrow a$. (\textit{mp\_copy}) \\
-2. If $c.alloc < c.used + \lfloor b / lg(\beta) \rfloor + 2$ then grow $c$ accordingly. \\
-3. If the reallocation failed return(\textit{MP\_MEM}). \\
-4. If $b \ge lg(\beta)$ then \\
-\hspace{3mm}4.1 $c \leftarrow c \cdot \beta^{\lfloor b / lg(\beta) \rfloor}$ (\textit{mp\_lshd}). \\
-\hspace{3mm}4.2 If step 4.1 failed return(\textit{MP\_MEM}). \\
-5. $d \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\
-6. If $d \ne 0$ then do \\
-\hspace{3mm}6.1 $mask \leftarrow 2^d$ \\
-\hspace{3mm}6.2 $r \leftarrow 0$ \\
-\hspace{3mm}6.3 for $n$ from $0$ to $c.used - 1$ do \\
-\hspace{6mm}6.3.1 $rr \leftarrow c_n >> (lg(\beta) - d) \mbox{ (mod }mask\mbox{)}$ \\
-\hspace{6mm}6.3.2 $c_n \leftarrow (c_n << d) + r \mbox{ (mod }\beta\mbox{)}$ \\
-\hspace{6mm}6.3.3 $r \leftarrow rr$ \\
-\hspace{3mm}6.4 If $r > 0$ then do \\
-\hspace{6mm}6.4.1 $c_{c.used} \leftarrow r$ \\
-\hspace{6mm}6.4.2 $c.used \leftarrow c.used + 1$ \\
-7. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_mul\_2d}
-\end{figure}
-
-\textbf{Algorithm mp\_mul\_2d.}
-This algorithm multiplies $a$ by $2^b$ and stores the result in $c$. The algorithm uses algorithm mp\_lshd and a derivative of algorithm mp\_mul\_2 to
-quickly compute the product.
-
-First the algorithm will multiply $a$ by $x^{\lfloor b / lg(\beta) \rfloor}$ which will ensure that the remainder multiplicand is less than
-$\beta$. For example, if $b = 37$ and $\beta = 2^{28}$ then this step will multiply by $x$ leaving a multiplication by $2^{37 - 28} = 2^{9}$
-left.
-
-After the digits have been shifted appropriately at most $lg(\beta) - 1$ shifts are left to perform. Step 5 calculates the number of remaining shifts
-required. If it is non-zero a modified shift loop is used to calculate the remaining product.
-Essentially the loop is a generic version of algorith mp\_mul2 designed to handle any shift count in the range $1 \le x < lg(\beta)$. The $mask$
-variable is used to extract the upper $d$ bits to form the carry for the next iteration.
-
-This algorithm is loosely measured as a $O(2n)$ algorithm which means that if the input is $n$-digits that it takes $2n$ ``time'' to
-complete. It is possible to optimize this algorithm down to a $O(n)$ algorithm at a cost of making the algorithm slightly harder to follow.
-
-EXAM,bn_mp_mul_2d.c
-
-Notes to be revised when code is updated. -- Tom
-
-\subsection{Division by Power of Two}
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_div\_2d}. \\
-\textbf{Input}. One mp\_int $a$ and an integer $b$ \\
-\textbf{Output}. $c \leftarrow \lfloor a / 2^b \rfloor, d \leftarrow a \mbox{ (mod }2^b\mbox{)}$. \\
-\hline \\
-1. If $b \le 0$ then do \\
-\hspace{3mm}1.1 $c \leftarrow a$ (\textit{mp\_copy}) \\
-\hspace{3mm}1.2 $d \leftarrow 0$ (\textit{mp\_zero}) \\
-\hspace{3mm}1.3 Return(\textit{MP\_OKAY}). \\
-2. $c \leftarrow a$ \\
-3. $d \leftarrow a \mbox{ (mod }2^b\mbox{)}$ (\textit{mp\_mod\_2d}) \\
-4. If $b \ge lg(\beta)$ then do \\
-\hspace{3mm}4.1 $c \leftarrow \lfloor c/\beta^{\lfloor b/lg(\beta) \rfloor} \rfloor$ (\textit{mp\_rshd}). \\
-5. $k \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\
-6. If $k \ne 0$ then do \\
-\hspace{3mm}6.1 $mask \leftarrow 2^k$ \\
-\hspace{3mm}6.2 $r \leftarrow 0$ \\
-\hspace{3mm}6.3 for $n$ from $c.used - 1$ to $0$ do \\
-\hspace{6mm}6.3.1 $rr \leftarrow c_n \mbox{ (mod }mask\mbox{)}$ \\
-\hspace{6mm}6.3.2 $c_n \leftarrow (c_n >> k) + (r << (lg(\beta) - k))$ \\
-\hspace{6mm}6.3.3 $r \leftarrow rr$ \\
-7. Clamp excess digits of $c$. (\textit{mp\_clamp}) \\
-8. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_div\_2d}
-\end{figure}
-
-\textbf{Algorithm mp\_div\_2d.}
-This algorithm will divide an input $a$ by $2^b$ and produce the quotient and remainder. The algorithm is designed much like algorithm
-mp\_mul\_2d by first using whole digit shifts then single precision shifts. This algorithm will also produce the remainder of the division
-by using algorithm mp\_mod\_2d.
-
-EXAM,bn_mp_div_2d.c
-
-The implementation of algorithm mp\_div\_2d is slightly different than the algorithm specifies. The remainder $d$ may be optionally
-ignored by passing \textbf{NULL} as the pointer to the mp\_int variable. The temporary mp\_int variable $t$ is used to hold the
-result of the remainder operation until the end. This allows $d$ and $a$ to represent the same mp\_int without modifying $a$ before
-the quotient is obtained.
-
-The remainder of the source code is essentially the same as the source code for mp\_mul\_2d. (-- Fix this paragraph up later, Tom).
-
-\subsection{Remainder of Division by Power of Two}
-
-The last algorithm in the series of polynomial basis power of two algorithms is calculating the remainder of division by $2^b$. This
-algorithm benefits from the fact that in twos complement arithmetic $a \mbox{ (mod }2^b\mbox{)}$ is the same as $a$ AND $2^b - 1$.
-
-\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_mod\_2d}. \\
-\textbf{Input}. One mp\_int $a$ and an integer $b$ \\
-\textbf{Output}. $c \leftarrow a \mbox{ (mod }2^b\mbox{)}$. \\
-\hline \\
-1. If $b \le 0$ then do \\
-\hspace{3mm}1.1 $c \leftarrow 0$ (\textit{mp\_zero}) \\
-\hspace{3mm}1.2 Return(\textit{MP\_OKAY}). \\
-2. If $b > a.used \cdot lg(\beta)$ then do \\
-\hspace{3mm}2.1 $c \leftarrow a$ (\textit{mp\_copy}) \\
-\hspace{3mm}2.2 Return the result of step 2.1. \\
-3. $c \leftarrow a$ \\
-4. If step 3 failed return(\textit{MP\_MEM}). \\
-5. for $n$ from $\lceil b / lg(\beta) \rceil$ to $c.used$ do \\
-\hspace{3mm}5.1 $c_n \leftarrow 0$ \\
-6. $k \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\
-7. $c_{\lfloor b / lg(\beta) \rfloor} \leftarrow c_{\lfloor b / lg(\beta) \rfloor} \mbox{ (mod }2^{k}\mbox{)}$. \\
-8. Clamp excess digits of $c$. (\textit{mp\_clamp}) \\
-9. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_mod\_2d}
-\end{figure}
-
-\textbf{Algorithm mp\_mod\_2d.}
-This algorithm will quickly calculate the value of $a \mbox{ (mod }2^b\mbox{)}$. First if $b$ is less than or equal to zero the
-result is set to zero. If $b$ is greater than the number of bits in $a$ then it simply copies $a$ to $c$ and returns. Otherwise, $a$
-is copied to $b$, leading digits are removed and the remaining leading digit is trimed to the exact bit count.
-
-EXAM,bn_mp_mod_2d.c
-
--- Add comments later, Tom.
-
-\section*{Exercises}
-\begin{tabular}{cl}
-$\left [ 3 \right ] $ & Devise an algorithm that performs $a \cdot 2^b$ for generic values of $b$ \\
- & in $O(n)$ time. \\
- &\\
-$\left [ 3 \right ] $ & Devise an efficient algorithm to multiply by small low hamming \\
- & weight values such as $3$, $5$ and $9$. Extend it to handle all values \\
- & upto $64$ with a hamming weight less than three. \\
- &\\
-$\left [ 2 \right ] $ & Modify the preceding algorithm to handle values of the form \\
- & $2^k - 1$ as well. \\
- &\\
-$\left [ 3 \right ] $ & Using only algorithms mp\_mul\_2, mp\_div\_2 and mp\_add create an \\
- & algorithm to multiply two integers in roughly $O(2n^2)$ time for \\
- & any $n$-bit input. Note that the time of addition is ignored in the \\
- & calculation. \\
- & \\
-$\left [ 5 \right ] $ & Improve the previous algorithm to have a working time of at most \\
- & $O \left (2^{(k-1)}n + \left ({2n^2 \over k} \right ) \right )$ for an appropriate choice of $k$. Again ignore \\
- & the cost of addition. \\
- & \\
-$\left [ 2 \right ] $ & Devise a chart to find optimal values of $k$ for the previous problem \\
- & for $n = 64 \ldots 1024$ in steps of $64$. \\
- & \\
-$\left [ 2 \right ] $ & Using only algorithms mp\_abs and mp\_sub devise another method for \\
- & calculating the result of a signed comparison. \\
- &
-\end{tabular}
-
-\chapter{Multiplication and Squaring}
-\section{The Multipliers}
-For most number theoretic problems including certain public key cryptographic algorithms, the ``multipliers'' form the most important subset of
-algorithms of any multiple precision integer package. The set of multiplier algorithms include integer multiplication, squaring and modular reduction
-where in each of the algorithms single precision multiplication is the dominant operation performed. This chapter will discuss integer multiplication
-and squaring, leaving modular reductions for the subsequent chapter.
-
-The importance of the multiplier algorithms is for the most part driven by the fact that certain popular public key algorithms are based on modular
-exponentiation, that is computing $d \equiv a^b \mbox{ (mod }c\mbox{)}$ for some arbitrary choice of $a$, $b$, $c$ and $d$. During a modular
-exponentiation the majority\footnote{Roughly speaking a modular exponentiation will spend about 40\% of the time performing modular reductions,
-35\% of the time performing squaring and 25\% of the time performing multiplications.} of the processor time is spent performing single precision
-multiplications.
-
-For centuries general purpose multiplication has required a lengthly $O(n^2)$ process, whereby each digit of one multiplicand has to be multiplied
-against every digit of the other multiplicand. Traditional long-hand multiplication is based on this process; while the techniques can differ the
-overall algorithm used is essentially the same. Only ``recently'' have faster algorithms been studied. First Karatsuba multiplication was discovered in
-1962. This algorithm can multiply two numbers with considerably fewer single precision multiplications when compared to the long-hand approach.
-This technique led to the discovery of polynomial basis algorithms (\textit{good reference?}) and subquently Fourier Transform based solutions.
-
-\section{Multiplication}
-\subsection{The Baseline Multiplication}
-\index{baseline multiplication}
-Computing the product of two integers in software can be achieved using a trivial adaptation of the standard $O(n^2)$ long-hand multiplication
-algorithm that school children are taught. The algorithm is considered an $O(n^2)$ algoritn since for two $n$-digit inputs $n^2$ single precision
-multiplications are required. More specifically for a $m$ and $n$ digit input $m \cdot n$ single precision multiplications are required. To
-simplify most discussions, it will be assumed that the inputs have comparable number of digits.
-
-The ``baseline multiplication'' algorithm is designed to act as the ``catch-all'' algorithm, only to be used when the faster algorithms cannot be
-used. This algorithm does not use any particularly interesting optimizations and should ideally be avoided if possible. One important
-facet of this algorithm, is that it has been modified to only produce a certain amount of output digits as resolution. The importance of this
-modification will become evident during the discussion of Barrett modular reduction. Recall that for a $n$ and $m$ digit input the product
-will be at most $n + m$ digits. Therefore, this algorithm can be reduced to a full multiplier by having it produce $n + m$ digits of the product.
-
-Recall from ~GAMMA~ the definition of $\gamma$ as the number of bits in the type \textbf{mp\_digit}. We shall now extend the variable set to
-include $\alpha$ which shall represent the number of bits in the type \textbf{mp\_word}. This implies that $2^{\alpha} > 2 \cdot \beta^2$. The
-constant $\delta = 2^{\alpha - 2lg(\beta)}$ will represent the maximal weight of any column in a product (\textit{see ~COMBA~ for more information}).
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{s\_mp\_mul\_digs}. \\
-\textbf{Input}. mp\_int $a$, mp\_int $b$ and an integer $digs$ \\
-\textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert \mbox{ (mod }\beta^{digs}\mbox{)}$. \\
-\hline \\
-1. If min$(a.used, b.used) < \delta$ then do \\
-\hspace{3mm}1.1 Calculate $c = \vert a \vert \cdot \vert b \vert$ by the Comba method (\textit{see algorithm~\ref{fig:COMBAMULT}}). \\
-\hspace{3mm}1.2 Return the result of step 1.1 \\
-\\
-Allocate and initialize a temporary mp\_int. \\
-2. Init $t$ to be of size $digs$ \\
-3. If step 2 failed return(\textit{MP\_MEM}). \\
-4. $t.used \leftarrow digs$ \\
-\\
-Compute the product. \\
-5. for $ix$ from $0$ to $a.used - 1$ do \\
-\hspace{3mm}5.1 $u \leftarrow 0$ \\
-\hspace{3mm}5.2 $pb \leftarrow \mbox{min}(b.used, digs - ix)$ \\
-\hspace{3mm}5.3 If $pb < 1$ then goto step 6. \\
-\hspace{3mm}5.4 for $iy$ from $0$ to $pb - 1$ do \\
-\hspace{6mm}5.4.1 $\hat r \leftarrow t_{iy + ix} + a_{ix} \cdot b_{iy} + u$ \\
-\hspace{6mm}5.4.2 $t_{iy + ix} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\
-\hspace{6mm}5.4.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\
-\hspace{3mm}5.5 if $ix + pb < digs$ then do \\
-\hspace{6mm}5.5.1 $t_{ix + pb} \leftarrow u$ \\
-6. Clamp excess digits of $t$. \\
-7. Swap $c$ with $t$ \\
-8. Clear $t$ \\
-9. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm s\_mp\_mul\_digs}
-\end{figure}
-
-\textbf{Algorithm s\_mp\_mul\_digs.}
-This algorithm computes the unsigned product of two inputs $a$ and $b$, limited to an output precision of $digs$ digits. While it may seem
-a bit awkward to modify the function from its simple $O(n^2)$ description, the usefulness of partial multipliers will arise in a subsequent
-algorithm. The algorithm is loosely based on algorithm 14.12 from \cite[pp. 595]{HAC} and is similar to Algorithm M of Knuth \cite[pp. 268]{TAOCPV2}.
-Algorithm s\_mp\_mul\_digs differs from these cited references since it can produce a variable output precision regardless of the precision of the
-inputs.
-
-The first thing this algorithm checks for is whether a Comba multiplier can be used instead. If the minimum digit count of either
-input is less than $\delta$, then the Comba method may be used instead. After the Comba method is ruled out, the baseline algorithm begins. A
-temporary mp\_int variable $t$ is used to hold the intermediate result of the product. This allows the algorithm to be used to
-compute products when either $a = c$ or $b = c$ without overwriting the inputs.
-
-All of step 5 is the infamous $O(n^2)$ multiplication loop slightly modified to only produce upto $digs$ digits of output. The $pb$ variable
-is given the count of digits to read from $b$ inside the nested loop. If $pb \le 1$ then no more output digits can be produced and the algorithm
-will exit the loop. The best way to think of the loops are as a series of $pb \times 1$ multiplications. That is, in each pass of the
-innermost loop $a_{ix}$ is multiplied against $b$ and the result is added (\textit{with an appropriate shift}) to $t$.
-
-For example, consider multiplying $576$ by $241$. That is equivalent to computing $10^0(1)(576) + 10^1(4)(576) + 10^2(2)(576)$ which is best
-visualized in the following table.
-
-\begin{figure}[here]
-\begin{center}
-\begin{tabular}{|c|c|c|c|c|c|l|}
-\hline && & 5 & 7 & 6 & \\
-\hline $\times$&& & 2 & 4 & 1 & \\
-\hline &&&&&&\\
- && & 5 & 7 & 6 & $10^0(1)(576)$ \\
- &2 & 3 & 6 & 1 & 6 & $10^1(4)(576) + 10^0(1)(576)$ \\
- 1 & 3 & 8 & 8 & 1 & 6 & $10^2(2)(576) + 10^1(4)(576) + 10^0(1)(576)$ \\
-\hline
-\end{tabular}
-\end{center}
-\caption{Long-Hand Multiplication Diagram}
-\end{figure}
-
-Each row of the product is added to the result after being shifted to the left (\textit{multiplied by a power of the radix}) by the appropriate
-count. That is in pass $ix$ of the inner loop the product is added starting at the $ix$'th digit of the reult.
-
-Step 5.4.1 introduces the hat symbol (\textit{e.g. $\hat r$}) which represents a double precision variable. The multiplication on that step
-is assumed to be a double wide output single precision multiplication. That is, two single precision variables are multiplied to produce a
-double precision result. The step is somewhat optimized from a long-hand multiplication algorithm because the carry from the addition in step
-5.4.1 is propagated through the nested loop. If the carry was not propagated immediately it would overflow the single precision digit
-$t_{ix+iy}$ and the result would be lost.
-
-At step 5.5 the nested loop is finished and any carry that was left over should be forwarded. The carry does not have to be added to the $ix+pb$'th
-digit since that digit is assumed to be zero at this point. However, if $ix + pb \ge digs$ the carry is not set as it would make the result
-exceed the precision requested.
-
-EXAM,bn_s_mp_mul_digs.c
-
-Lines @31,if@ to @35,}@ determine if the Comba method can be used first. The conditions for using the Comba routine are that min$(a.used, b.used) < \delta$ and
-the number of digits of output is less than \textbf{MP\_WARRAY}. This new constant is used to control
-the stack usage in the Comba routines. By default it is set to $\delta$ but can be reduced when memory is at a premium.
-
-Of particular importance is the calculation of the $ix+iy$'th column on lines @64,mp_word@, @65,mp_word@ and @66,mp_word@. Note how all of the
-variables are cast to the type \textbf{mp\_word}, which is also the type of variable $\hat r$. That is to ensure that double precision operations
-are used instead of single precision. The multiplication on line @65,) * (@ makes use of a specific GCC optimizer behaviour. On the outset it looks like
-the compiler will have to use a double precision multiplication to produce the result required. Such an operation would be horribly slow on most
-processors and drag this to a crawl. However, GCC is smart enough to realize that double wide output single precision multipliers can be used. For
-example, the instruction ``MUL'' on the x86 processor can multiply two 32-bit values and produce a 64-bit result.
-
-\subsection{Faster Multiplication by the ``Comba'' Method}
-MARK,COMBA
-
-One of the huge drawbacks of the ``baseline'' algorithms is that at the $O(n^2)$ level the carry must be computed and propagated upwards. This
-makes the nested loop very sequential and hard to unroll and implement in parallel. The ``Comba'' \cite{COMBA} method is named after little known
-(\textit{in cryptographic venues}) Paul G. Comba who described a method of implementing fast multipliers that do not require nested
-carry fixup operations. As an interesting aside it seems that Paul Barrett describes a similar technique in
-his 1986 paper \cite{BARRETT} written five years before.
-
-At the heart of the Comba technique is once again the long-hand algorithm. Except in this case a slight twist is placed on how
-the columns of the result are produced. In the standard long-hand algorithm rows of products are produced then added together to form the
-final result. In the baseline algorithm the columns are added together after each iteration to get the result instantaneously.
-
-In the Comba algorithm the columns of the result are produced entirely independently of each other. That is at the $O(n^2)$ level a
-simple multiplication and addition step is performed. The carries of the columns are propagated after the nested loop to reduce the amount
-of work requiored. Succintly the first step of the algorithm is to compute the product vector $\vec x$ as follows.
-
-\begin{equation}
-\vec x_n = \sum_{i+j = n} a_ib_j, \forall n \in \lbrace 0, 1, 2, \ldots, i + j \rbrace
-\end{equation}
-
-Where $\vec x_n$ is the $n'th$ column of the output vector. Consider the following example which computes the vector $\vec x$ for the multiplication
-of $576$ and $241$.
-
-\newpage\begin{figure}[here]
-\begin{small}
-\begin{center}
-\begin{tabular}{|c|c|c|c|c|c|}
- \hline & & 5 & 7 & 6 & First Input\\
- \hline $\times$ & & 2 & 4 & 1 & Second Input\\
-\hline & & $1 \cdot 5 = 5$ & $1 \cdot 7 = 7$ & $1 \cdot 6 = 6$ & First pass \\
- & $4 \cdot 5 = 20$ & $4 \cdot 7+5=33$ & $4 \cdot 6+7=31$ & 6 & Second pass \\
- $2 \cdot 5 = 10$ & $2 \cdot 7 + 20 = 34$ & $2 \cdot 6+33=45$ & 31 & 6 & Third pass \\
-\hline 10 & 34 & 45 & 31 & 6 & Final Result \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Comba Multiplication Diagram}
-\end{figure}
-
-At this point the vector $x = \left < 10, 34, 45, 31, 6 \right >$ is the result of the first step of the Comba multipler.
-Now the columns must be fixed by propagating the carry upwards. The resultant vector will have one extra dimension over the input vector which is
-congruent to adding a leading zero digit.
-
-\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{Comba Fixup}. \\
-\textbf{Input}. Vector $\vec x$ of dimension $k$ \\
-\textbf{Output}. Vector $\vec x$ such that the carries have been propagated. \\
-\hline \\
-1. for $n$ from $0$ to $k - 1$ do \\
-\hspace{3mm}1.1 $\vec x_{n+1} \leftarrow \vec x_{n+1} + \lfloor \vec x_{n}/\beta \rfloor$ \\
-\hspace{3mm}1.2 $\vec x_{n} \leftarrow \vec x_{n} \mbox{ (mod }\beta\mbox{)}$ \\
-2. Return($\vec x$). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm Comba Fixup}
-\end{figure}
-
-With that algorithm and $k = 5$ and $\beta = 10$ the following vector is produced $\vec x= \left < 1, 3, 8, 8, 1, 6 \right >$. In this case
-$241 \cdot 576$ is in fact $138816$ and the procedure succeeded. If the algorithm is correct and as will be demonstrated shortly more
-efficient than the baseline algorithm why not simply always use this algorithm?
-
-\subsubsection{Column Weight.}
-At the nested $O(n^2)$ level the Comba method adds the product of two single precision variables to each column of the output
-independently. A serious obstacle is if the carry is lost, due to lack of precision before the algorithm has a chance to fix
-the carries. For example, in the multiplication of two three-digit numbers the third column of output will be the sum of
-three single precision multiplications. If the precision of the accumulator for the output digits is less then $3 \cdot (\beta - 1)^2$ then
-an overflow can occur and the carry information will be lost. For any $m$ and $n$ digit inputs the maximum weight of any column is
-min$(m, n)$ which is fairly obvious.
-
-The maximum number of terms in any column of a product is known as the ``column weight'' and strictly governs when the algorithm can be used. Recall
-from earlier that a double precision type has $\alpha$ bits of resolution and a single precision digit has $lg(\beta)$ bits of precision. Given these
-two quantities we must not violate the following
-
-\begin{equation}
-k \cdot \left (\beta - 1 \right )^2 < 2^{\alpha}
-\end{equation}
-
-Which reduces to
-
-\begin{equation}
-k \cdot \left ( \beta^2 - 2\beta + 1 \right ) < 2^{\alpha}
-\end{equation}
-
-Let $\rho = lg(\beta)$ represent the number of bits in a single precision digit. By further re-arrangement of the equation the final solution is
-found.
-
-\begin{equation}
-k < {{2^{\alpha}} \over {\left (2^{2\rho} - 2^{\rho + 1} + 1 \right )}}
-\end{equation}
-
-The defaults for LibTomMath are $\beta = 2^{28}$ and $\alpha = 2^{64}$ which means that $k$ is bounded by $k < 257$. In this configuration
-the smaller input may not have more than $256$ digits if the Comba method is to be used. This is quite satisfactory for most applications since
-$256$ digits would allow for numbers in the range of $0 \le x < 2^{7168}$ which, is much larger than most public key cryptographic algorithms require.
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{fast\_s\_mp\_mul\_digs}. \\
-\textbf{Input}. mp\_int $a$, mp\_int $b$ and an integer $digs$ \\
-\textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert \mbox{ (mod }\beta^{digs}\mbox{)}$. \\
-\hline \\
-Place an array of \textbf{MP\_WARRAY} double precision digits named $\hat W$ on the stack. \\
-1. If $c.alloc < digs$ then grow $c$ to $digs$ digits. (\textit{mp\_grow}) \\
-2. If step 1 failed return(\textit{MP\_MEM}).\\
-\\
-Zero the temporary array $\hat W$. \\
-3. for $n$ from $0$ to $digs - 1$ do \\
-\hspace{3mm}3.1 $\hat W_n \leftarrow 0$ \\
-\\
-Compute the columns. \\
-4. for $ix$ from $0$ to $a.used - 1$ do \\
-\hspace{3mm}4.1 $pb \leftarrow \mbox{min}(b.used, digs - ix)$ \\
-\hspace{3mm}4.2 If $pb < 1$ then goto step 5. \\
-\hspace{3mm}4.3 for $iy$ from $0$ to $pb - 1$ do \\
-\hspace{6mm}4.3.1 $\hat W_{ix+iy} \leftarrow \hat W_{ix+iy} + a_{ix}b_{iy}$ \\
-\\
-Propagate the carries upwards. \\
-5. $oldused \leftarrow c.used$ \\
-6. $c.used \leftarrow digs$ \\
-7. If $digs > 1$ then do \\
-\hspace{3mm}7.1. for $ix$ from $1$ to $digs - 1$ do \\
-\hspace{6mm}7.1.1 $\hat W_{ix} \leftarrow \hat W_{ix} + \lfloor \hat W_{ix-1} / \beta \rfloor$ \\
-\hspace{6mm}7.1.2 $c_{ix - 1} \leftarrow \hat W_{ix - 1} \mbox{ (mod }\beta\mbox{)}$ \\
-8. else do \\
-\hspace{3mm}8.1 $ix \leftarrow 0$ \\
-9. $c_{ix} \leftarrow \hat W_{ix} \mbox{ (mod }\beta\mbox{)}$ \\
-\\
-Zero excess digits. \\
-10. If $digs < oldused$ then do \\
-\hspace{3mm}10.1 for $n$ from $digs$ to $oldused - 1$ do \\
-\hspace{6mm}10.1.1 $c_n \leftarrow 0$ \\
-11. Clamp excessive digits of $c$. (\textit{mp\_clamp}) \\
-12. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm fast\_s\_mp\_mul\_digs}
-\label{fig:COMBAMULT}
-\end{figure}
-
-\textbf{Algorithm fast\_s\_mp\_mul\_digs.}
-This algorithm performs the unsigned multiplication of $a$ and $b$ using the Comba method limited to $digs$ digits of precision. The algorithm
-essentially peforms the same calculation as algorithm s\_mp\_mul\_digs, just much faster.
-
-The array $\hat W$ is meant to be on the stack when the algorithm is used. The size of the array does not change which is ideal. Note also that
-unlike algorithm s\_mp\_mul\_digs no temporary mp\_int is required since the result is calculated directly in $\hat W$.
-
-The $O(n^2)$ loop on step four is where the Comba method's advantages begin to show through in comparison to the baseline algorithm. The lack of
-a carry variable or propagation in this loop allows the loop to be performed with only single precision multiplication and additions. Now that each
-iteration of the inner loop can be performed independent of the others the inner loop can be performed with a high level of parallelism.
-
-To measure the benefits of the Comba method over the baseline method consider the number of operations that are required. If the
-cost in terms of time of a multiply and addition is $p$ and the cost of a carry propagation is $q$ then a baseline multiplication would require
-$O \left ((p + q)n^2 \right )$ time to multiply two $n$-digit numbers. The Comba method requires only $O(pn^2 + qn)$ time, however in practice,
-the speed increase is actually much more. With $O(n)$ space the algorithm can be reduced to $O(pn + qn)$ time by implementing the $n$ multiply
-and addition operations in the nested loop in parallel.
-
-EXAM,bn_fast_s_mp_mul_digs.c
-
-The memset on line @47,memset@ clears the initial $\hat W$ array to zero in a single step. Like the slower baseline multiplication
-implementation a series of aliases (\textit{lines @67, tmpx@, @70, tmpy@ and @75,_W@}) are used to simplify the inner $O(n^2)$ loop.
-In this case a new alias $\_\hat W$ has been added which refers to the double precision columns offset by $ix$ in each pass.
-
-The inner loop on lines @83,for@, @84,mp_word@ and @85,}@ is where the algorithm will spend the majority of the time, which is why it has been
-stripped to the bones of any extra baggage\footnote{Hence the pointer aliases.}. On x86 processors the multiplication and additions amount to at the
-very least five instructions (\textit{two loads, two additions, one multiply}) while on the ARMv4 processors they amount to only three
-(\textit{one load, one store, one multiply-add}). For both of the x86 and ARMv4 processors the GCC compiler performs a good job at unrolling the loop
-and scheduling the instructions so there are very few dependency stalls.
-
-In theory the difference between the baseline and comba algorithms is a mere $O(qn)$ time difference. However, in the $O(n^2)$ nested loop of the
-baseline method there are dependency stalls as the algorithm must wait for the multiplier to finish before propagating the carry to the next
-digit. As a result fewer of the often multiple execution units\footnote{The AMD Athlon has three execution units and the Intel P4 has four.} can
-be simultaneously used.
-
-\subsection{Polynomial Basis Multiplication}
-To break the $O(n^2)$ barrier in multiplication requires a completely different look at integer multiplication. In the following algorithms
-the use of polynomial basis representation for two integers $a$ and $b$ as $f(x) = \sum_{i=0}^{n} a_i x^i$ and
-$g(x) = \sum_{i=0}^{n} b_i x^i$ respectively, is required. In this system both $f(x)$ and $g(x)$ have $n + 1$ terms and are of the $n$'th degree.
-
-The product $a \cdot b \equiv f(x)g(x)$ is the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$. The coefficients $w_i$ will
-directly yield the desired product when $\beta$ is substituted for $x$. The direct solution to solve for the $2n + 1$ coefficients
-requires $O(n^2)$ time and would in practice be slower than the Comba technique.
-
-However, numerical analysis theory indicates that only $2n + 1$ distinct points in $W(x)$ are required to determine the values of the $2n + 1$ unknown
-coefficients. This means by finding $\zeta_y = W(y)$ for $2n + 1$ small values of $y$ the coefficients of $W(x)$ can be found with
-Gaussian elimination. This technique is also occasionally refered to as the \textit{interpolation technique} (\textit{references please...}) since in
-effect an interpolation based on $2n + 1$ points will yield a polynomial equivalent to $W(x)$.
-
-The coefficients of the polynomial $W(x)$ are unknown which makes finding $W(y)$ for any value of $y$ impossible. However, since
-$W(x) = f(x)g(x)$ the equivalent $\zeta_y = f(y) g(y)$ can be used in its place. The benefit of this technique stems from the
-fact that $f(y)$ and $g(y)$ are much smaller than either $a$ or $b$ respectively. As a result finding the $2n + 1$ relations required
-by multiplying $f(y)g(y)$ involves multiplying integers that are much smaller than either of the inputs.
-
-When picking points to gather relations there are always three obvious points to choose, $y = 0, 1$ and $ \infty$. The $\zeta_0$ term
-is simply the product $W(0) = w_0 = a_0 \cdot b_0$. The $\zeta_1$ term is the product
-$W(1) = \left (\sum_{i = 0}^{n} a_i \right ) \left (\sum_{i = 0}^{n} b_i \right )$. The third point $\zeta_{\infty}$ is less obvious but rather
-simple to explain. The $2n + 1$'th coefficient of $W(x)$ is numerically equivalent to the most significant column in an integer multiplication.
-The point at $\infty$ is used symbolically to represent the most significant column, that is $W(\infty) = w_{2n} = a_nb_n$. Note that the
-points at $y = 0$ and $\infty$ yield the coefficients $w_0$ and $w_{2n}$ directly.
-
-If more points are required they should be of small values and powers of two such as $2^q$ and the related \textit{mirror points}
-$\left (2^q \right )^{2n} \cdot \zeta_{2^{-q}}$ for small values of $q$. The term ``mirror point'' stems from the fact that
-$\left (2^q \right )^{2n} \cdot \zeta_{2^{-q}}$ can be calculated in the exact opposite fashion as $\zeta_{2^q}$. For
-example, when $n = 2$ and $q = 1$ then following two equations are equivalent to the point $\zeta_{2}$ and its mirror.
-
-\begin{eqnarray}
-\zeta_{2} = f(2)g(2) = (4a_2 + 2a_1 + a_0)(4b_2 + 2b_1 + b_0) \nonumber \\
-16 \cdot \zeta_{1 \over 2} = 4f({1\over 2}) \cdot 4g({1 \over 2}) = (a_2 + 2a_1 + 4a_0)(b_2 + 2b_1 + 4b_0)
-\end{eqnarray}
-
-Using such points will allow the values of $f(y)$ and $g(y)$ to be independently calculated using only left shifts. For example, when $n = 2$ the
-polynomial $f(2^q)$ is equal to $2^q((2^qa_2) + a_1) + a_0$. This technique of polynomial representation is known as Horner's method.
-
-As a general rule of the algorithm when the inputs are split into $n$ parts each there are $2n - 1$ multiplications. Each multiplication is of
-multiplicands that have $n$ times fewer digits than the inputs. The asymptotic running time of this algorithm is
-$O \left ( k^{lg_n(2n - 1)} \right )$ for $k$ digit inputs (\textit{assuming they have the same number of digits}). Figure~\ref{fig:exponent}
-summarizes the exponents for various values of $n$.
-
-\begin{figure}
-\begin{center}
-\begin{tabular}{|c|c|c|}
-\hline \textbf{Split into $n$ Parts} & \textbf{Exponent} & \textbf{Notes}\\
-\hline $2$ & $1.584962501$ & This is Karatsuba Multiplication. \\
-\hline $3$ & $1.464973520$ & This is Toom-Cook Multiplication. \\
-\hline $4$ & $1.403677461$ &\\
-\hline $5$ & $1.365212389$ &\\
-\hline $10$ & $1.278753601$ &\\
-\hline $100$ & $1.149426538$ &\\
-\hline $1000$ & $1.100270931$ &\\
-\hline $10000$ & $1.075252070$ &\\
-\hline
-\end{tabular}
-\end{center}
-\caption{Asymptotic Running Time of Polynomial Basis Multiplication}
-\label{fig:exponent}
-\end{figure}
-
-At first it may seem like a good idea to choose $n = 1000$ since the exponent is approximately $1.1$. However, the overhead
-of solving for the 2001 terms of $W(x)$ will certainly consume any savings the algorithm could offer for all but exceedingly large
-numbers.
-
-\subsubsection{Cutoff Point}
-The polynomial basis multiplication algorithms all require fewer single precision multiplications than a straight Comba approach. However,
-the algorithms incur an overhead (\textit{at the $O(n)$ work level}) since they require a system of equations to be solved. This makes the
-polynomial basis approach more costly to use with small inputs.
-
-Let $m$ represent the number of digits in the multiplicands (\textit{assume both multiplicands have the same number of digits}). There exists a
-point $y$ such that when $m < y$ the polynomial basis algorithms are more costly than Comba, when $m = y$ they are roughly the same cost and
-when $m > y$ the Comba methods are slower than the polynomial basis algorithms.
-
-The exact location of $y$ depends on several key architectural elements of the computer platform in question.
-
-\begin{enumerate}
-\item The ratio of clock cycles for single precision multiplication versus other simpler operations such as addition, shifting, etc. For example
-on the AMD Athlon the ratio is roughly $17 : 1$ while on the Intel P4 it is $29 : 1$. The higher the ratio in favour of multiplication the lower
-the cutoff point $y$ will be.
-
-\item The complexity of the linear system of equations (\textit{for the coefficients of $W(x)$}) is. Generally speaking as the number of splits
-grows the complexity grows substantially. Ideally solving the system will only involve addition, subtraction and shifting of integers. This
-directly reflects on the ratio previous mentioned.
-
-\item To a lesser extent memory bandwidth and function call overheads. Provided the values are in the processor cache this is less of an
-influence over the cutoff point.
-
-\end{enumerate}
-
-A clean cutoff point separation occurs when a point $y$ is found such that all of the cutoff point conditions are met. For example, if the point
-is too low then there will be values of $m$ such that $m > y$ and the Comba method is still faster. Finding the cutoff points is fairly simple when
-a high resolution timer is available.
-
-\subsection{Karatsuba Multiplication}
-Karatsuba \cite{KARA} multiplication when originally proposed in 1962 was among the first set of algorithms to break the $O(n^2)$ barrier for
-general purpose multiplication. Given two polynomial basis representations $f(x) = ax + b$ and $g(x) = cx + d$, Karatsuba proved with
-light algebra \cite{KARAP} that the following polynomial is equivalent to multiplication of the two integers the polynomials represent.
-
-\begin{equation}
-f(x) \cdot g(x) = acx^2 + ((a - b)(c - d) + ac + bd)x + bd
-\end{equation}
-
-Using the observation that $ac$ and $bd$ could be re-used only three half sized multiplications would be required to produce the product. Applying
-this algorithm recursively, the work factor becomes $O(n^{lg(3)})$ which is substantially better than the work factor $O(n^2)$ of the Comba technique. It turns
-out what Karatsuba did not know or at least did not publish was that this is simply polynomial basis multiplication with the points
-$\zeta_0$, $\zeta_{\infty}$ and $-\zeta_{-1}$. Consider the resultant system of equations.
-
-\begin{center}
-\begin{tabular}{rcrcrcrc}
-$\zeta_{0}$ & $=$ & & & & & $w_0$ \\
-$-\zeta_{-1}$ & $=$ & $-w_2$ & $+$ & $w_1$ & $-$ & $w_0$ \\
-$\zeta_{\infty}$ & $=$ & $w_2$ & & & & \\
-\end{tabular}
-\end{center}
-
-By adding the first and last equation to the equation in the middle the term $w_1$ can be isolated and all three coefficients solved for. The simplicity
-of this system of equations has made Karatsuba fairly popular. In fact the cutoff point is often fairly low\footnote{With LibTomMath 0.18 it is 70 and 109 digits for the Intel P4 and AMD Athlon respectively.}
-making it an ideal algorithm to speed up certain public key cryptosystems such as RSA and Diffie-Hellman. It is worth noting that the point
-$\zeta_1$ could be substituted for $-\zeta_{-1}$. In this case the first and third row are subtracted instead of added to the second row.
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_karatsuba\_mul}. \\
-\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\
-\textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert$ \\
-\hline \\
-1. Init the following mp\_int variables: $x0$, $x1$, $y0$, $y1$, $t1$, $x0y0$, $x1y1$.\\
-2. If step 2 failed then return(\textit{MP\_MEM}). \\
-\\
-Split the input. e.g. $a = x1 \cdot \beta^B + x0$ \\
-3. $B \leftarrow \mbox{min}(a.used, b.used)/2$ \\
-4. $x0 \leftarrow a \mbox{ (mod }\beta^B\mbox{)}$ (\textit{mp\_mod\_2d}) \\
-5. $y0 \leftarrow b \mbox{ (mod }\beta^B\mbox{)}$ \\
-6. $x1 \leftarrow \lfloor a / \beta^B \rfloor$ (\textit{mp\_rshd}) \\
-7. $y1 \leftarrow \lfloor b / \beta^B \rfloor$ \\
-\\
-Calculate the three products. \\
-8. $x0y0 \leftarrow x0 \cdot y0$ (\textit{mp\_mul}) \\
-9. $x1y1 \leftarrow x1 \cdot y1$ \\
-10. $t1 \leftarrow x1 - x0$ (\textit{mp\_sub}) \\
-11. $x0 \leftarrow y1 - y0$ \\
-12. $t1 \leftarrow t1 \cdot x0$ \\
-\\
-Calculate the middle term. \\
-13. $x0 \leftarrow x0y0 + x1y1$ \\
-14. $t1 \leftarrow x0 - t1$ \\
-\\
-Calculate the final product. \\
-15. $t1 \leftarrow t1 \cdot \beta^B$ (\textit{mp\_lshd}) \\
-16. $x1y1 \leftarrow x1y1 \cdot \beta^{2B}$ \\
-17. $t1 \leftarrow x0y0 + t1$ \\
-18. $c \leftarrow t1 + x1y1$ \\
-19. Clear all of the temporary variables. \\
-20. Return(\textit{MP\_OKAY}).\\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_karatsuba\_mul}
-\end{figure}
-
-\textbf{Algorithm mp\_karatsuba\_mul.}
-This algorithm computes the unsigned product of two inputs using the Karatsuba multiplication algorithm. It is loosely based on the description
-from Knuth \cite[pp. 294-295]{TAOCPV2}.
-
-\index{radix point}
-In order to split the two inputs into their respective halves, a suitable \textit{radix point} must be chosen. The radix point chosen must
-be used for both of the inputs meaning that it must be smaller than the smallest input. Step 3 chooses the radix point $B$ as half of the
-smallest input \textbf{used} count. After the radix point is chosen the inputs are split into lower and upper halves. Step 4 and 5
-compute the lower halves. Step 6 and 7 computer the upper halves.
-
-After the halves have been computed the three intermediate half-size products must be computed. Step 8 and 9 compute the trivial products
-$x0 \cdot y0$ and $x1 \cdot y1$. The mp\_int $x0$ is used as a temporary variable after $x1 - x0$ has been computed. By using $x0$ instead
-of an additional temporary variable, the algorithm can avoid an addition memory allocation operation.
-
-The remaining steps 13 through 18 compute the Karatsuba polynomial through a variety of digit shifting and addition operations.
-
-EXAM,bn_mp_karatsuba_mul.c
-
-The new coding element in this routine, not seen in previous routines, is the usage of goto statements. The conventional
-wisdom is that goto statements should be avoided. This is generally true, however when every single function call can fail, it makes sense
-to handle error recovery with a single piece of code. Lines @61,if@ to @75,if@ handle initializing all of the temporary variables
-required. Note how each of the if statements goes to a different label in case of failure. This allows the routine to correctly free only
-the temporaries that have been successfully allocated so far.
-
-The temporary variables are all initialized using the mp\_init\_size routine since they are expected to be large. This saves the
-additional reallocation that would have been necessary. Also $x0$, $x1$, $y0$ and $y1$ have to be able to hold at least their respective
-number of digits for the next section of code.
-
-The first algebraic portion of the algorithm is to split the two inputs into their halves. However, instead of using mp\_mod\_2d and mp\_rshd
-to extract the halves, the respective code has been placed inline within the body of the function. To initialize the halves, the \textbf{used} and
-\textbf{sign} members are copied first. The first for loop on line @98,for@ copies the lower halves. Since they are both the same magnitude it
-is simpler to calculate both lower halves in a single loop. The for loop on lines @104,for@ and @109,for@ calculate the upper halves $x1$ and
-$y1$ respectively.
-
-By inlining the calculation of the halves, the Karatsuba multiplier has a slightly lower overhead and can be used for smaller magnitude inputs.
-
-When line @152,err@ is reached, the algorithm has completed succesfully. The ``error status'' variable $err$ is set to \textbf{MP\_OKAY} so that
-the same code that handles errors can be used to clear the temporary variables and return.
-
-\subsection{Toom-Cook $3$-Way Multiplication}
-Toom-Cook $3$-Way \cite{TOOM} multiplication is essentially the polynomial basis algorithm for $n = 3$ except that the points are
-chosen such that $\zeta$ is easy to compute and the resulting system of equations easy to reduce. Here, the points $\zeta_{0}$,
-$16 \cdot \zeta_{1 \over 2}$, $\zeta_1$, $\zeta_2$ and $\zeta_{\infty}$ make up the five required points to solve for the coefficients
-of the $W(x)$.
-
-With the five relations that Toom-Cook specifies, the following system of equations is formed.
-
-\begin{center}
-\begin{tabular}{rcrcrcrcrcr}
-$\zeta_0$ & $=$ & $0w_4$ & $+$ & $0w_3$ & $+$ & $0w_2$ & $+$ & $0w_1$ & $+$ & $1w_0$ \\
-$16 \cdot \zeta_{1 \over 2}$ & $=$ & $1w_4$ & $+$ & $2w_3$ & $+$ & $4w_2$ & $+$ & $8w_1$ & $+$ & $16w_0$ \\
-$\zeta_1$ & $=$ & $1w_4$ & $+$ & $1w_3$ & $+$ & $1w_2$ & $+$ & $1w_1$ & $+$ & $1w_0$ \\
-$\zeta_2$ & $=$ & $16w_4$ & $+$ & $8w_3$ & $+$ & $4w_2$ & $+$ & $2w_1$ & $+$ & $1w_0$ \\
-$\zeta_{\infty}$ & $=$ & $1w_4$ & $+$ & $0w_3$ & $+$ & $0w_2$ & $+$ & $0w_1$ & $+$ & $0w_0$ \\
-\end{tabular}
-\end{center}
-
-A trivial solution to this matrix requires $12$ subtractions, two multiplications by a small power of two, two divisions by a small power
-of two, two divisions by three and one multiplication by three. All of these $19$ sub-operations require less than quadratic time, meaning that
-the algorithm can be faster than a baseline multiplication. However, the greater complexity of this algorithm places the cutoff point
-(\textbf{TOOM\_MUL\_CUTOFF}) where Toom-Cook becomes more efficient much higher than the Karatsuba cutoff point.
-
-\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_toom\_mul}. \\
-\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\
-\textbf{Output}. $c \leftarrow a \cdot b $ \\
-\hline \\
-Split $a$ and $b$ into three pieces. E.g. $a = a_2 \beta^{2k} + a_1 \beta^{k} + a_0$ \\
-1. $k \leftarrow \lfloor \mbox{min}(a.used, b.used) / 3 \rfloor$ \\
-2. $a_0 \leftarrow a \mbox{ (mod }\beta^{k}\mbox{)}$ \\
-3. $a_1 \leftarrow \lfloor a / \beta^k \rfloor$, $a_1 \leftarrow a_1 \mbox{ (mod }\beta^{k}\mbox{)}$ \\
-4. $a_2 \leftarrow \lfloor a / \beta^{2k} \rfloor$, $a_2 \leftarrow a_2 \mbox{ (mod }\beta^{k}\mbox{)}$ \\
-5. $b_0 \leftarrow a \mbox{ (mod }\beta^{k}\mbox{)}$ \\
-6. $b_1 \leftarrow \lfloor a / \beta^k \rfloor$, $b_1 \leftarrow b_1 \mbox{ (mod }\beta^{k}\mbox{)}$ \\
-7. $b_2 \leftarrow \lfloor a / \beta^{2k} \rfloor$, $b_2 \leftarrow b_2 \mbox{ (mod }\beta^{k}\mbox{)}$ \\
-\\
-Find the five equations for $w_0, w_1, ..., w_4$. \\
-8. $w_0 \leftarrow a_0 \cdot b_0$ \\
-9. $w_4 \leftarrow a_2 \cdot b_2$ \\
-10. $tmp_1 \leftarrow 2 \cdot a_0$, $tmp_1 \leftarrow a_1 + tmp_1$, $tmp_1 \leftarrow 2 \cdot tmp_1$, $tmp_1 \leftarrow tmp_1 + a_2$ \\
-11. $tmp_2 \leftarrow 2 \cdot b_0$, $tmp_2 \leftarrow b_1 + tmp_2$, $tmp_2 \leftarrow 2 \cdot tmp_2$, $tmp_2 \leftarrow tmp_2 + b_2$ \\
-12. $w_1 \leftarrow tmp_1 \cdot tmp_2$ \\
-13. $tmp_1 \leftarrow 2 \cdot a_2$, $tmp_1 \leftarrow a_1 + tmp_1$, $tmp_1 \leftarrow 2 \cdot tmp_1$, $tmp_1 \leftarrow tmp_1 + a_0$ \\
-14. $tmp_2 \leftarrow 2 \cdot b_2$, $tmp_2 \leftarrow b_1 + tmp_2$, $tmp_2 \leftarrow 2 \cdot tmp_2$, $tmp_2 \leftarrow tmp_2 + b_0$ \\
-15. $w_3 \leftarrow tmp_1 \cdot tmp_2$ \\
-16. $tmp_1 \leftarrow a_0 + a_1$, $tmp_1 \leftarrow tmp_1 + a_2$, $tmp_2 \leftarrow b_0 + b_1$, $tmp_2 \leftarrow tmp_2 + b_2$ \\
-17. $w_2 \leftarrow tmp_1 \cdot tmp_2$ \\
-\\
-Continued on the next page.\\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_toom\_mul}
-\end{figure}
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_toom\_mul} (continued). \\
-\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\
-\textbf{Output}. $c \leftarrow a \cdot b $ \\
-\hline \\
-Now solve the system of equations. \\
-18. $w_1 \leftarrow w_4 - w_1$, $w_3 \leftarrow w_3 - w_0$ \\
-19. $w_1 \leftarrow \lfloor w_1 / 2 \rfloor$, $w_3 \leftarrow \lfloor w_3 / 2 \rfloor$ \\
-20. $w_2 \leftarrow w_2 - w_0$, $w_2 \leftarrow w_2 - w_4$ \\
-21. $w_1 \leftarrow w_1 - w_2$, $w_3 \leftarrow w_3 - w_2$ \\
-22. $tmp_1 \leftarrow 8 \cdot w_0$, $w_1 \leftarrow w_1 - tmp_1$, $tmp_1 \leftarrow 8 \cdot w_4$, $w_3 \leftarrow w_3 - tmp_1$ \\
-23. $w_2 \leftarrow 3 \cdot w_2$, $w_2 \leftarrow w_2 - w_1$, $w_2 \leftarrow w_2 - w_3$ \\
-24. $w_1 \leftarrow w_1 - w_2$, $w_3 \leftarrow w_3 - w_2$ \\
-25. $w_1 \leftarrow \lfloor w_1 / 3 \rfloor, w_3 \leftarrow \lfloor w_3 / 3 \rfloor$ \\
-\\
-Now substitute $\beta^k$ for $x$ by shifting $w_0, w_1, ..., w_4$. \\
-26. for $n$ from $1$ to $4$ do \\
-\hspace{3mm}26.1 $w_n \leftarrow w_n \cdot \beta^{nk}$ \\
-27. $c \leftarrow w_0 + w_1$, $c \leftarrow c + w_2$, $c \leftarrow c + w_3$, $c \leftarrow c + w_4$ \\
-28. Return(\textit{MP\_OKAY}) \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_toom\_mul (continued)}
-\end{figure}
-
-\textbf{Algorithm mp\_toom\_mul.}
-This algorithm computes the product of two mp\_int variables $a$ and $b$ using the Toom-Cook approach. Compared to the Karatsuba multiplication, this
-algorithm has a lower asymptotic running time of approximately $O(n^{1.464})$ but at an obvious cost in overhead. In this
-description, several statements have been compounded to save space. The intention is that the statements are executed from left to right across
-any given step.
-
-The two inputs $a$ and $b$ are first split into three $k$-digit integers $a_0, a_1, a_2$ and $b_0, b_1, b_2$ respectively. From these smaller
-integers the coefficients of the polynomial basis representations $f(x)$ and $g(x)$ are known and can be used to find the relations required.
-
-The first two relations $w_0$ and $w_4$ are the points $\zeta_{0}$ and $\zeta_{\infty}$ respectively. The relation $w_1, w_2$ and $w_3$ correspond
-to the points $16 \cdot \zeta_{1 \over 2}, \zeta_{2}$ and $\zeta_{1}$ respectively. These are found using logical shifts to independently find
-$f(y)$ and $g(y)$ which significantly speeds up the algorithm.
-
-After the five relations $w_0, w_1, \ldots, w_4$ have been computed, the system they represent must be solved in order for the unknown coefficients
-$w_1, w_2$ and $w_3$ to be isolated. The steps 18 through 25 perform the system reduction required as previously described. Each step of
-the reduction represents the comparable matrix operation that would be performed had this been performed by pencil. For example, step 18 indicates
-that row $1$ must be subtracted from row $4$ and simultaneously row $0$ subtracted from row $3$.
-
-Once the coeffients have been isolated, the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$ is known. By substituting $\beta^{k}$ for $x$, the integer
-result $a \cdot b$ is produced.
-
-EXAM,bn_mp_toom_mul.c
-
--- Comments to be added during editing phase.
-
-\subsection{Signed Multiplication}
-Now that algorithms to handle multiplications of every useful dimensions have been developed, a rather simple finishing touch is required. So far all
-of the multiplication algorithms have been unsigned multiplications which leaves only a signed multiplication algorithm to be established.
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_mul}. \\
-\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\
-\textbf{Output}. $c \leftarrow a \cdot b$ \\
-\hline \\
-1. If $a.sign = b.sign$ then \\
-\hspace{3mm}1.1 $sign = MP\_ZPOS$ \\
-2. else \\
-\hspace{3mm}2.1 $sign = MP\_ZNEG$ \\
-3. If min$(a.used, b.used) \ge TOOM\_MUL\_CUTOFF$ then \\
-\hspace{3mm}3.1 $c \leftarrow a \cdot b$ using algorithm mp\_toom\_mul \\
-4. else if min$(a.used, b.used) \ge KARATSUBA\_MUL\_CUTOFF$ then \\
-\hspace{3mm}4.1 $c \leftarrow a \cdot b$ using algorithm mp\_karatsuba\_mul \\
-5. else \\
-\hspace{3mm}5.1 $digs \leftarrow a.used + b.used + 1$ \\
-\hspace{3mm}5.2 If $digs < MP\_ARRAY$ and min$(a.used, b.used) \le \delta$ then \\
-\hspace{6mm}5.2.1 $c \leftarrow a \cdot b \mbox{ (mod }\beta^{digs}\mbox{)}$ using algorithm fast\_s\_mp\_mul\_digs. \\
-\hspace{3mm}5.3 else \\
-\hspace{6mm}5.3.1 $c \leftarrow a \cdot b \mbox{ (mod }\beta^{digs}\mbox{)}$ using algorithm s\_mp\_mul\_digs. \\
-6. $c.sign \leftarrow sign$ \\
-7. Return the result of the unsigned multiplication performed. \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_mul}
-\end{figure}
-
-\textbf{Algorithm mp\_mul.}
-This algorithm performs the signed multiplication of two inputs. It will make use of any of the three unsigned multiplication algorithms
-available when the input is of appropriate size. The \textbf{sign} of the result is not set until the end of the algorithm since algorithm
-s\_mp\_mul\_digs will clear it.
-
-EXAM,bn_mp_mul.c
-
-The implementation is rather simplistic and is not particularly noteworthy. Line @22,?@ computes the sign of the result using the ``?''
-operator from the C programming language. Line @37,<<@ computes $\delta$ using the fact that $1 << k$ is equal to $2^k$.
-
-\section{Squaring}
-
-Squaring is a special case of multiplication where both multiplicands are equal. At first it may seem like there is no significant optimization
-available but in fact there is. Consider the multiplication of $576$ against $241$. In total there will be nine single precision multiplications
-performed which are $1\cdot 6$, $1 \cdot 7$, $1 \cdot 5$, $4 \cdot 6$, $4 \cdot 7$, $4 \cdot 5$, $2 \cdot 6$, $2 \cdot 7$ and $2 \cdot 5$. Now consider
-the multiplication of $123$ against $123$. The nine products are $3 \cdot 3$, $3 \cdot 2$, $3 \cdot 1$, $2 \cdot 3$, $2 \cdot 2$, $2 \cdot 1$,
-$1 \cdot 3$, $1 \cdot 2$ and $1 \cdot 1$. On closer inspection some of the products are equivalent. For example, $3 \cdot 2 = 2 \cdot 3$
-and $3 \cdot 1 = 1 \cdot 3$.
-
-For any $n$-digit input, there are ${{\left (n^2 + n \right)}\over 2}$ possible unique single precision multiplications required compared to the $n^2$
-required for multiplication. The following diagram gives an example of the operations required.
-
-\begin{figure}[here]
-\begin{center}
-\begin{tabular}{ccccc|c}
-&&1&2&3&\\
-$\times$ &&1&2&3&\\
-\hline && $3 \cdot 1$ & $3 \cdot 2$ & $3 \cdot 3$ & Row 0\\
- & $2 \cdot 1$ & $2 \cdot 2$ & $2 \cdot 3$ && Row 1 \\
- $1 \cdot 1$ & $1 \cdot 2$ & $1 \cdot 3$ &&& Row 2 \\
-\end{tabular}
-\end{center}
-\caption{Squaring Optimization Diagram}
-\end{figure}
-
-MARK,SQUARE
-Starting from zero and numbering the columns from right to left a very simple pattern becomes obvious. For the purposes of this discussion let $x$
-represent the number being squared. The first observation is that in row $k$ the $2k$'th column of the product has a $\left (x_k \right)^2$ term in it.
-
-The second observation is that every column $j$ in row $k$ where $j \ne 2k$ is part of a double product. Every non-square term of a column will
-appear twice hence the name ``double product''. Every odd column is made up entirely of double products. In fact every column is made up of double
-products and at most one square (\textit{see the exercise section}).
-
-The third and final observation is that for row $k$ the first unique non-square term, that is, one that hasn't already appeared in an earlier row,
-occurs at column $2k + 1$. For example, on row $1$ of the previous squaring, column one is part of the double product with column one from row zero.
-Column two of row one is a square and column three is the first unique column.
-
-\subsection{The Baseline Squaring Algorithm}
-The baseline squaring algorithm is meant to be a catch-all squaring algorithm. It will handle any of the input sizes that the faster routines
-will not handle.
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{s\_mp\_sqr}. \\
-\textbf{Input}. mp\_int $a$ \\
-\textbf{Output}. $b \leftarrow a^2$ \\
-\hline \\
-1. Init a temporary mp\_int of at least $2 \cdot a.used +1$ digits. (\textit{mp\_init\_size}) \\
-2. If step 1 failed return(\textit{MP\_MEM}) \\
-3. $t.used \leftarrow 2 \cdot a.used + 1$ \\
-4. For $ix$ from 0 to $a.used - 1$ do \\
-\hspace{3mm}Calculate the square. \\
-\hspace{3mm}4.1 $\hat r \leftarrow t_{2ix} + \left (a_{ix} \right )^2$ \\
-\hspace{3mm}4.2 $t_{2ix} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\
-\hspace{3mm}Calculate the double products after the square. \\
-\hspace{3mm}4.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\
-\hspace{3mm}4.4 For $iy$ from $ix + 1$ to $a.used - 1$ do \\
-\hspace{6mm}4.4.1 $\hat r \leftarrow 2 \cdot a_{ix}a_{iy} + t_{ix + iy} + u$ \\
-\hspace{6mm}4.4.2 $t_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\
-\hspace{6mm}4.4.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\
-\hspace{3mm}Set the last carry. \\
-\hspace{3mm}4.5 While $u > 0$ do \\
-\hspace{6mm}4.5.1 $iy \leftarrow iy + 1$ \\
-\hspace{6mm}4.5.2 $\hat r \leftarrow t_{ix + iy} + u$ \\
-\hspace{6mm}4.5.3 $t_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\
-\hspace{6mm}4.5.4 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\
-5. Clamp excess digits of $t$. (\textit{mp\_clamp}) \\
-6. Exchange $b$ and $t$. \\
-7. Clear $t$ (\textit{mp\_clear}) \\
-8. Return(\textit{MP\_OKAY}) \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm s\_mp\_sqr}
-\end{figure}
-
-\textbf{Algorithm s\_mp\_sqr.}
-This algorithm computes the square of an input using the three observations on squaring. It is based fairly faithfully on algorithm 14.16 of HAC
-\cite[pp.596-597]{HAC}. Similar to algorithm s\_mp\_mul\_digs, a temporary mp\_int is allocated to hold the result of the squaring. This allows the
-destination mp\_int to be the same as the source mp\_int.
-
-The outer loop of this algorithm begins on step 4. It is best to think of the outer loop as walking down the rows of the partial results, while
-the inner loop computes the columns of the partial result. Step 4.1 and 4.2 compute the square term for each row, and step 4.3 and 4.4 propagate
-the carry and compute the double products.
-
-The requirement that a mp\_word be able to represent the range $0 \le x < 2 \beta^2$ arises from this
-very algorithm. The product $a_{ix}a_{iy}$ will lie in the range $0 \le x \le \beta^2 - 2\beta + 1$ which is obviously less than $\beta^2$ meaning that
-when it is multiplied by two, it can be properly represented by a mp\_word.
-
-Similar to algorithm s\_mp\_mul\_digs, after every pass of the inner loop, the destination is correctly set to the sum of all of the partial
-results calculated so far. This involves expensive carry propagation which will be eliminated in the next algorithm.
-
-EXAM,bn_s_mp_sqr.c
-
-Inside the outer loop (\textit{see line @32,for@}) the square term is calculated on line @35,r =@. Line @42,>>@ extracts the carry from the square
-term. Aliases for $a_{ix}$ and $t_{ix+iy}$ are initialized on lines @45,tmpx@ and @48,tmpt@ respectively. The doubling is performed using two
-additions (\textit{see line @57,r + r@}) since it is usually faster than shifting,if not at least as fast.
-
-\subsection{Faster Squaring by the ``Comba'' Method}
-A major drawback to the baseline method is the requirement for single precision shifting inside the $O(n^2)$ nested loop. Squaring has an additional
-drawback that it must double the product inside the inner loop as well. As for multiplication, the Comba technique can be used to eliminate these
-performance hazards.
-
-The first obvious solution is to make an array of mp\_words which will hold all of the columns. This will indeed eliminate all of the carry
-propagation operations from the inner loop. However, the inner product must still be doubled $O(n^2)$ times. The solution stems from the simple fact
-that $2a + 2b + 2c = 2(a + b + c)$. That is the sum of all of the double products is equal to double the sum of all the products. For example,
-$ab + ba + ac + ca = 2ab + 2ac = 2(ab + ac)$.
-
-However, we cannot simply double all of the columns, since the squares appear only once per row. The most practical solution is to have two mp\_word
-arrays. One array will hold the squares and the other array will hold the double products. With both arrays the doubling and carry propagation can be
-moved to a $O(n)$ work level outside the $O(n^2)$ level.
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{fast\_s\_mp\_sqr}. \\
-\textbf{Input}. mp\_int $a$ \\
-\textbf{Output}. $b \leftarrow a^2$ \\
-\hline \\
-Place two arrays of \textbf{MP\_WARRAY} mp\_words named $\hat W$ and $\hat {X}$ on the stack. \\
-1. If $b.alloc < 2a.used + 1$ then grow $b$ to $2a.used + 1$ digits. (\textit{mp\_grow}). \\
-2. If step 1 failed return(\textit{MP\_MEM}). \\
-3. for $ix$ from $0$ to $2a.used + 1$ do \\
-\hspace{3mm}3.1 $\hat W_{ix} \leftarrow 0$ \\
-\hspace{3mm}3.2 $\hat {X}_{ix} \leftarrow 0$ \\
-4. for $ix$ from $0$ to $a.used - 1$ do \\
-\hspace{3mm}Compute the square.\\
-\hspace{3mm}4.1 $\hat {X}_{ix+ix} \leftarrow \left ( a_ix \right )^2$ \\
-\\
-\hspace{3mm}Compute the double products.\\
-\hspace{3mm}4.2 for $iy$ from $ix + 1$ to $a.used - 1$ do \\
-\hspace{6mm}4.2.1 $\hat W_{ix+iy} \leftarrow \hat W_{ix+iy} + a_{ix}a_{iy}$ \\
-5. $oldused \leftarrow b.used$ \\
-6. $b.used \leftarrow 2a.used + 1$ \\
-\\
-Double the products and propagate the carries simultaneously. \\
-7. $\hat W_0 \leftarrow 2 \hat W_0 + \hat {X}_0$ \\
-8. for $ix$ from $1$ to $2a.used$ do \\
-\hspace{3mm}8.1 $\hat W_{ix} \leftarrow 2 \hat W_{ix} + \hat {X}_{ix}$ \\
-\hspace{3mm}8.2 $\hat W_{ix} \leftarrow \hat W_{ix} + \lfloor \hat W_{ix - 1} / \beta \rfloor$ \\
-\hspace{3mm}8.3 $b_{ix-1} \leftarrow W_{ix-1} \mbox{ (mod }\beta\mbox{)}$ \\
-9. $b_{2a.used} \leftarrow \hat W_{2a.used} \mbox{ (mod }\beta\mbox{)}$ \\
-10. if $2a.used + 1 < oldused$ then do \\
-\hspace{3mm}10.1 for $ix$ from $2a.used + 1$ to $oldused$ do \\
-\hspace{6mm}10.1.1 $b_{ix} \leftarrow 0$ \\
-11. Clamp excess digits from $b$. (\textit{mp\_clamp}) \\
-12. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm fast\_s\_mp\_sqr}
-\end{figure}
-
-\textbf{Algorithm fast\_s\_mp\_sqr.}
-This algorithm computes the square of an input using the Comba technique. It is designed to be a replacement for algorithm s\_mp\_sqr when
-the number of input digits is less than \textbf{MP\_WARRAY} and less than $\delta \over 2$.
-
-This routine requires two arrays of mp\_words to be placed on the stack. The first array $\hat W$ will hold the double products and the second
-array $\hat X$ will hold the squares. Though only at most $MP\_WARRAY \over 2$ words of $\hat X$ are used, it has proven faster on most
-processors to simply make it a full size array.
-
-The loop on step 3 will zero the two arrays to prepare them for the squaring step. Step 4.1 computes the squares of the product. Note how
-it simply assigns the value into the $\hat X$ array. The nested loop on step 4.2 computes the doubles of the products. This loop
-computes the sum of the products for each column. They are not doubled until later.
-
-After the squaring loop, the products stored in $\hat W$ musted be doubled and the carries propagated forwards. It makes sense to do both
-operations at the same time. The expression $\hat W_{ix} \leftarrow 2 \hat W_{ix} + \hat {X}_{ix}$ computes the sum of the double product and the
-squares in place.
-
-EXAM,bn_fast_s_mp_sqr.c
-
--- Write something deep and insightful later, Tom.
-
-\subsection{Polynomial Basis Squaring}
-The same algorithm that performs optimal polynomial basis multiplication can be used to perform polynomial basis squaring. The minor exception
-is that $\zeta_y = f(y)g(y)$ is actually equivalent to $\zeta_y = f(y)^2$ since $f(y) = g(y)$. Instead of performing $2n + 1$
-multiplications to find the $\zeta$ relations, squaring operations are performed instead.
-
-\subsection{Karatsuba Squaring}
-Let $f(x) = ax + b$ represent the polynomial basis representation of a number to square.
-Let $h(x) = \left ( f(x) \right )^2$ represent the square of the polynomial. The Karatsuba equation can be modified to square a
-number with the following equation.
-
-\begin{equation}
-h(x) = a^2x^2 + \left (a^2 + b^2 - (a - b)^2 \right )x + b^2
-\end{equation}
-
-Upon closer inspection this equation only requires the calculation of three half-sized squares: $a^2$, $b^2$ and $(a - b)^2$. As in
-Karatsuba multiplication, this algorithm can be applied recursively on the input and will achieve an asymptotic running time of
-$O \left ( n^{lg(3)} \right )$.
-
-You might ask yourself, if the asymptotic time of Karatsuba squaring and multiplication is the same, why not simply use the multiplication algorithm
-instead? The answer to this arises from the cutoff point for squaring. As in multiplication there exists a cutoff point, at which the
-time required for a Comba based squaring and a Karatsuba based squaring meet. Due to the overhead inherent in the Karatsuba method, the cutoff
-point is fairly high. For example, on an AMD Athlon XP processor with $\beta = 2^{28}$, the cutoff point is around 127 digits.
-
-Consider squaring a 200 digit number with this technique. It will be split into two 100 digit halves which are subsequently squared.
-The 100 digit halves will not be squared using Karatsuba, but instead using the faster Comba based squaring algorithm. If Karatsuba multiplication
-were used instead, the 100 digit numbers would be squared with a slower Comba based multiplication.
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_karatsuba\_sqr}. \\
-\textbf{Input}. mp\_int $a$ \\
-\textbf{Output}. $b \leftarrow a^2$ \\
-\hline \\
-1. Initialize the following temporary mp\_ints: $x0$, $x1$, $t1$, $t2$, $x0x0$ and $x1x1$. \\
-2. If any of the initializations on step 1 failed return(\textit{MP\_MEM}). \\
-\\
-Split the input. e.g. $a = x1\beta^B + x0$ \\
-3. $B \leftarrow \lfloor a.used / 2 \rfloor$ \\
-4. $x0 \leftarrow a \mbox{ (mod }\beta^B\mbox{)}$ (\textit{mp\_mod\_2d}) \\
-5. $x1 \leftarrow \lfloor a / \beta^B \rfloor$ (\textit{mp\_lshd}) \\
-\\
-Calculate the three squares. \\
-6. $x0x0 \leftarrow x0^2$ (\textit{mp\_sqr}) \\
-7. $x1x1 \leftarrow x1^2$ \\
-8. $t1 \leftarrow x1 - x0$ (\textit{mp\_sub}) \\
-9. $t1 \leftarrow t1^2$ \\
-\\
-Compute the middle term. \\
-10. $t2 \leftarrow x0x0 + x1x1$ (\textit{s\_mp\_add}) \\
-11. $t1 \leftarrow t2 - t1$ \\
-\\
-Compute final product. \\
-12. $t1 \leftarrow t1\beta^B$ (\textit{mp\_lshd}) \\
-13. $x1x1 \leftarrow x1x1\beta^{2B}$ \\
-14. $t1 \leftarrow t1 + x0x0$ \\
-15. $b \leftarrow t1 + x1x1$ \\
-16. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_karatsuba\_sqr}
-\end{figure}
-
-\textbf{Algorithm mp\_karatsuba\_sqr.}
-This algorithm computes the square of an input $a$ using the Karatsuba technique. This algorithm is very similar to the Karatsuba based
-multiplication algorithm with the exception that the three half-size multiplications have been replaced with three half-size squarings.
-
-The radix point for squaring is simply placed exactly in the middle of the digits when the input has an odd number of digits, otherwise it is
-placed just below the middle. Step 3, 4 and 5 compute the two halves required using $B$
-as the radix point. The first two squares in steps 6 and 7 are rather straightforward while the last square is of a more compact form.
-
-By expanding $\left (x1 - x0 \right )^2$, the $x1^2$ and $x0^2$ terms in the middle disappear, that is $x1^2 + x0^2 - (x1 - x0)^2 = 2 \cdot x0 \cdot x1$.
-Now if $5n$ single precision additions and a squaring of $n$-digits is faster than multiplying two $n$-digit numbers and doubling then
-this method is faster. Assuming no further recursions occur, the difference can be estimated with the following inequality.
-
-Let $p$ represent the cost of a single precision addition and $q$ the cost of a single precision multiplication both in terms of time\footnote{Or
-machine clock cycles.}.
-
-\begin{equation}
-5pn +{{q(n^2 + n)} \over 2} \le pn + qn^2
-\end{equation}
-
-For example, on an AMD Athlon XP processor $p = {1 \over 3}$ and $q = 6$. This implies that the following inequality should hold.
-\begin{center}
-\begin{tabular}{rcl}
-${5n \over 3} + 3n^2 + 3n$ & $<$ & ${n \over 3} + 6n^2$ \\
-${5 \over 3} + 3n + 3$ & $<$ & ${1 \over 3} + 6n$ \\
-${13 \over 9}$ & $<$ & $n$ \\
-\end{tabular}
-\end{center}
-
-This results in a cutoff point around $n = 2$. As a consequence it is actually faster to compute the middle term the ``long way'' on processors
-where multiplication is substantially slower\footnote{On the Athlon there is a 1:17 ratio between clock cycles for addition and multiplication. On
-the Intel P4 processor this ratio is 1:29 making this method even more beneficial. The only common exception is the ARMv4 processor which has a
-ratio of 1:7. } than simpler operations such as addition.
-
-EXAM,bn_mp_karatsuba_sqr.c
-
-This implementation is largely based on the implementation of algorithm mp\_karatsuba\_mul. It uses the same inline style to copy and
-shift the input into the two halves. The loop from line @54,{@ to line @70,}@ has been modified since only one input exists. The \textbf{used}
-count of both $x0$ and $x1$ is fixed up and $x0$ is clamped before the calculations begin. At this point $x1$ and $x0$ are valid equivalents
-to the respective halves as if mp\_rshd and mp\_mod\_2d had been used.
-
-By inlining the copy and shift operations the cutoff point for Karatsuba multiplication can be lowered. On the Athlon the cutoff point
-is exactly at the point where Comba squaring can no longer be used (\textit{128 digits}). On slower processors such as the Intel P4
-it is actually below the Comba limit (\textit{at 110 digits}).
-
-This routine uses the same error trap coding style as mp\_karatsuba\_sqr. As the temporary variables are initialized errors are redirected to
-the error trap higher up. If the algorithm completes without error the error code is set to \textbf{MP\_OKAY} and mp\_clears are executed normally.
-
-\textit{Last paragraph sucks. re-write! -- Tom}
-
-\subsection{Toom-Cook Squaring}
-The Toom-Cook squaring algorithm mp\_toom\_sqr is heavily based on the algorithm mp\_toom\_mul with the exception that squarings are used
-instead of multiplication to find the five relations.. The reader is encouraged to read the description of the latter algorithm and try to
-derive their own Toom-Cook squaring algorithm.
-
-\subsection{High Level Squaring}
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_sqr}. \\
-\textbf{Input}. mp\_int $a$ \\
-\textbf{Output}. $b \leftarrow a^2$ \\
-\hline \\
-1. If $a.used \ge TOOM\_SQR\_CUTOFF$ then \\
-\hspace{3mm}1.1 $b \leftarrow a^2$ using algorithm mp\_toom\_sqr \\
-2. else if $a.used \ge KARATSUBA\_SQR\_CUTOFF$ then \\
-\hspace{3mm}2.1 $b \leftarrow a^2$ using algorithm mp\_karatsuba\_sqr \\
-3. else \\
-\hspace{3mm}3.1 $digs \leftarrow a.used + b.used + 1$ \\
-\hspace{3mm}3.2 If $digs < MP\_ARRAY$ and $a.used \le \delta$ then \\
-\hspace{6mm}3.2.1 $b \leftarrow a^2$ using algorithm fast\_s\_mp\_sqr. \\
-\hspace{3mm}3.3 else \\
-\hspace{6mm}3.3.1 $b \leftarrow a^2$ using algorithm s\_mp\_sqr. \\
-4. $b.sign \leftarrow MP\_ZPOS$ \\
-5. Return the result of the unsigned squaring performed. \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_sqr}
-\end{figure}
-
-\textbf{Algorithm mp\_sqr.}
-This algorithm computes the square of the input using one of four different algorithms. If the input is very large and has at least
-\textbf{TOOM\_SQR\_CUTOFF} or \textbf{KARATSUBA\_SQR\_CUTOFF} digits then either the Toom-Cook or the Karatsuba Squaring algorithm is used. If
-neither of the polynomial basis algorithms should be used then either the Comba or baseline algorithm is used.
-
-EXAM,bn_mp_sqr.c
-
-\section*{Exercises}
-\begin{tabular}{cl}
-$\left [ 3 \right ] $ & Devise an efficient algorithm for selection of the radix point to handle inputs \\
- & that have different number of digits in Karatsuba multiplication. \\
- & \\
-$\left [ 3 \right ] $ & In ~SQUARE~ the fact that every column of a squaring is made up \\
- & of double products and at most one square is stated. Prove this statement. \\
- & \\
-$\left [ 2 \right ] $ & In the Comba squaring algorithm half of the $\hat X$ variables are not used. \\
- & Revise algorithm fast\_s\_mp\_sqr to shrink the $\hat X$ array. \\
- & \\
-$\left [ 3 \right ] $ & Prove the equation for Karatsuba squaring. \\
- & \\
-$\left [ 1 \right ] $ & Prove that Karatsuba squaring requires $O \left (n^{lg(3)} \right )$ time. \\
- & \\
-$\left [ 2 \right ] $ & Determine the minimal ratio between addition and multiplication clock cycles \\
- & required for equation $6.7$ to be true. \\
- & \\
-\end{tabular}
-
-\chapter{Modular Reduction}
-MARK,REDUCTION
-\section{Basics of Modular Reduction}
-\index{modular residue}
-Modular reduction is an operation that arises quite often within public key cryptography algorithms and various number theoretic algorithms,
-such as factoring. Modular reduction algorithms are the third class of algorithms of the ``multipliers'' set. A number $a$ is said to be reduced
-modulo another number $b$ by finding the remainder of the division $a/b$.
-
-Modular reduction is equivalent to solving for $r$ in the following equation. $a = bq + r$ where $q = \lfloor a/b \rfloor$. The result
-$r$ is said to be ``congruent to $a$ modulo $b$'' which is also written as $r \equiv a \mbox{ (mod }b\mbox{)}$. In other vernacular $r$ is known as the
-``modular residue'' which leads to ``quadratic residue''\footnote{That's fancy talk for $b \equiv a^2 \mbox{ (mod }p\mbox{)}$.} and
-other forms of residues.
-
-\index{modulus}
-Modular reductions are normally used to form finite groups such as fields and rings. For example, in the RSA public key algorithm \cite{RSAPAPER}
-two private primes $p$ and $q$ are chosen which when multiplied $n = pq$ forms a composite modulus. When operations such as multiplication and
-squaring are performed on units of the ring $\Z_n$ a finite multiplicative sub-group is formed.
-
-Modular reductions have a variety of other useful properties. For example, a number $x$ is a square if and only if it is a quadratic
-residue modulo a prime. With a finite set of primes $B = \left < p_0, p_1, \ldots, p_n \right >$ a quick test for whether $x$ is square or not can
-be performed\footnote{Provided none of the primes from $B$ divide $x$.}. Consider the figure~\ref{fig:QR} with the candiate $x = 955621$ a simple
-set of modular reductions modulo $3, 5, \ldots, 11$ may detect whether $x$ is a square or not. In this case $955621 \equiv 7 \mbox{ (mod }11\mbox{)}$
-and since $7$ is not a quadratic residue modulo $11$ the number $955621$ is not a square.
-
-\begin{figure}
-\begin{center}
-\begin{tabular}{|c|l|}
-\hline \textbf{Prime} & \textbf{Quadratic Residues} \\
-\hline $3$ & $1$ \\
-\hline $5$ & $1, 4$ \\
-\hline $7$ & $1, 2, 4$ \\
-\hline $11$ & $1, 3, 4, 5, 9$ \\
-\hline
-\end{tabular}
-\end{center}
-\caption{Quadratic Residues for primes less than $13$}
-\label{fig:QR}
-\end{figure}
-
-The most common usage for performance driven modular reductions is in modular exponentiation algorithms. That is to compute
-$d = a^b \mbox{ (mod }c\mbox{)}$ as fast as possible. As will be discussed in the subsequent chapter there exists fast algorithms for computing
-modular exponentiations without having to perform (\textit{in this example}) $b - 1$ multiplications. These algorithms will produce partial
-results in the range $0 \le x < c^2$ which can be taken advantage of to create several efficient algorithms.
-
-\section{The Barrett Reduction}
-The Barrett reduction algorithm \cite{BARRETT} was inspired by fast division algorithms which multiply by the reciprocal to emulate
-division. Barretts observation was that the residue $c$ of $a$ modulo $b$ is equal to
-
-\begin{equation}
-c = a - b \cdot \lfloor a/b \rfloor
-\end{equation}
-
-Since algorithms such as modular exponentiation would be using the same modulus extensively, typical DSP intuition would indicate the next step
-would be to replace $a/b$ by a multiplication by the reciprocal. However, DSP intuition on its own will not work as these numbers are considerably
-larger than the precision of common DSP floating point data types. It would take another common optimization to optimize the algorithm.
-
-\subsection{Fixed Point Arithmetic}
-The trick used to optimize the above equation is based on a technique of emulating floating point data types with fixed precision integers. Fixed
-point arithmetic would vastly popularlize the ``3d-shooter'' genre of games in the mid 1990s when floating point units were fairly slow. The idea behind
-fixed point arithmetic is to take a normal $k$-bit integer data type and break it into $p$-bit integer and a $q$-bit fraction part
-(\textit{where $p+q = k$}).
-
-In this system a $k$-bit integer $n$ would actually represent $n/2^q$. For example, with $q = 4$ the integer $n = 37$ would actually represent the
-value $2.3125$. To multiply two fixed point numbers the integers are multiplied using traditional arithmetic and subsequently normalized. For example,
-with $q = 4$ to multiply the integers $9$ and $5$ they must be converted to fixed point first by multiplying by $2^q$. Let $a = 9(2^q)$
-represent the fixed point representation of $9$ and $b = 5(2^q)$ represent the fixed point representation of $5$. The product $ab$ is equal to
-$45(2^{2q})$ which when normalized produces $45(2^q)$.
-
-Using fixed point arithmetic division can be easily achieved by multiplying by the reciprocal. If $2^q$ is equivalent to one than $2^q/b$ is
-equivalent to $1/b$ using real arithmetic. Using this fact dividing an integer $a$ by another integer $b$ can be achieved with the following
-expression.
-
-\begin{equation}
-\lfloor (a \cdot (\lfloor 2^q / b \rfloor))/2^q \rfloor
-\end{equation}
-
-The precision of the division is proportional to the value of $q$. If the divisor $b$ is used frequently as is the case with
-modular exponentiation pre-computing $2^q/b$ will allow a division to be performed with a multiplication and a right shift. Both operations
-are considerably faster than division on most processors.
-
-Consider dividing $19$ by $5$. The correct result is $\lfloor 19/5 \rfloor = 3$. With $q = 3$ the reciprocal is $\lfloor 2^q/5 \rfloor = 1$ which
-leads to a product of $19$ which when divided by $2^q$ produces $2$. However, with $q = 4$ the reciprocal is $\lfloor 2^q/5 \rfloor = 3$ and
-the result of the emulated division is $\lfloor 3 \cdot 19 / 2^q \rfloor = 3$ which is correct.
-
-Plugging this form of divison into the original equation the following modular residue equation arises.
-
-\begin{equation}
-c = a - b \cdot \lfloor (a \cdot (\lfloor 2^q / b \rfloor))/2^q \rfloor
-\end{equation}
-
-Using the notation from \cite{BARRETT} the value of $\lfloor 2^q / b \rfloor$ will be represented by the $\mu$ symbol. Using the $\mu$
-variable also helps re-inforce the idea that it is meant to be computed once and re-used.
-
-\begin{equation}
-c = a - b \cdot \lfloor (a \cdot \mu)/2^q \rfloor
-\end{equation}
-
-Provided that $2^q > b^2$ this algorithm will produce a quotient that is either exactly correct or off by a value of one. Let $n$ represent
-the number of digits in $b$. This algorithm requires approximately $2n^2$ single precision multiplications to produce the quotient and
-another $n^2$ single precision multiplications to find the residue. In total $3n^2$ single precision multiplications are required to
-reduce the number.
-
-For example, if $b = 1179677$ and $q = 41$ ($2^q > b^2$), then the reciprocal $\mu$ is equal to $\lfloor 2^q / b \rfloor = 1864089$. Consider reducing
-$a = 180388626447$ modulo $b$ using the above reduction equation. The quotient using the new formula is $\lfloor (a \cdot \mu) / 2^q \rfloor = 152913$.
-By subtracting $152913b$ from $a$ the correct residue $a \equiv 677346 \mbox{ (mod }b\mbox{)}$ is found.
-
-\subsection{Choosing a Radix Point}
-Using the fixed point representation a modular reduction can be performed with $3n^2$ single precision multiplications. If that were the best
-that could be achieved a full division might as well be used in its place. The key to optimizing the reduction is to reduce the precision of
-the initial multiplication that finds the quotient.
-
-Let $a$ represent the number of which the residue is sought. Let $b$ represent the modulus used to find the residue. Let $m$ represent
-the number of digits in $b$. For the purposes of this discussion we will assume that the number of digits in $a$ is $2m$. Dividing $a$ by
-$b$ is the same as dividing a $2m$ digit integer by a $m$ digit integer. Digits below the $m - 1$'th digit of $a$ will contribute at most a value
-of $1$ to the quotient because $\beta^k < b$ for any $0 \le k \le m - 1$.
-
-Since those digits do not contribute much to the quotient the observation is that they might as well be zero. However, if the digits
-``might as well be zero'' they might as well not be there in the first place. Let $q_0 = \lfloor a/\beta^{m-1} \rfloor$ represent the input
-with the zeroes trimmed. Now the modular reduction is trimmed to the almost equivalent equation
-
-\begin{equation}
-c = a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor
-\end{equation}
-
-Note that the original divisor $2^q$ has been replaced with $\beta^{m+1}$. Also note that the exponent on the divisor when added to the amount $q_0$
-was shifted by equals $2m$. If the optimization had not been performed the divisor would have the exponent $2m$ so in the end the exponents
-do ``add up''. Using the above equation the quotient $\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ can be off from the true quotient by at most
-two implying that $0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$. By first subtracting $b$ times the quotient and then
-conditionally subtracting $b$ once or twice the residue is found.
-
-The quotient is now found using $(m + 1)(m) = m^2 + m$ single precision multiplications and the residue with an additional $m^2$ single
-precision multiplications. In total $2m^2 + m$ single precision multiplications are required which is considerably faster than the original
-attempt.
-
-For example, let $\beta = 10$ represent the radix of the digits. Let $b = 9999$ represent the modulus which implies $m = 4$. Let $a = 99929878$
-represent the value of which the residue is desired. In this case $q = 8$ since $10^7 < 9999^2$ meaning that $\mu = \lfloor \beta^{q}/b \rfloor = 10001$.
-With the new observation the multiplicand for the quotient is equal to $q_0 = \lfloor a / \beta^{m - 1} \rfloor = 99929$. The quotient is then
-$\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor = 9993$. Subtracting $9993b$ from $a$ and the correct residue $a \equiv 9871 \mbox{ (mod }b\mbox{)}$
-is found.
-
-\subsection{Trimming the Quotient}
-So far the reduction algorithm has been optimized from $3m^2$ single precision multiplications down to $2m^2 + m$ single precision multiplications. As
-it stands now the algorithm is already fairly fast compared to a full integer division algorithm. However, there is still room for
-optimization.
-
-After the first multiplication inside the quotient ($q_0 \cdot \mu$) the value is shifted right by $m + 1$ places effectively nullifying the lower
-half of the product. It would be nice to be able to remove those digits from the product to effectively cut down the number of single precision
-multiplications. If the number of digits in the modulus $m$ is far less than $\beta$ a full product is not required for the algorithm to work properly.
-In fact the lower $m - 2$ digits will not affect the upper half of the product at all and do not need to be computed.
-
-The value of $\mu$ is a $m$-digit number and $q_0$ is a $m + 1$ digit number. Using a full multiplier $(m + 1)(m) = m^2 + m$ single precision
-multiplications would be required. Using a multiplier that will only produce digits at and above the $m - 1$'th digit reduces the number
-of single precision multiplications to ${m^2 + m} \over 2$ single precision multiplications.
-
-\subsection{Trimming the Residue}
-After the quotient has been calculated it is used to reduce the input. As previously noted the algorithm is not exact and it can be off by a small
-multiple of the modulus, that is $0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$. If $b$ is $m$ digits than the
-result of reduction equation is a value of at most $m + 1$ digits (\textit{provided $3 < \beta$}) implying that the upper $m - 1$ digits are
-implicitly zero.
-
-The next optimization arises from this very fact. Instead of computing $b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ using a full
-$O(m^2)$ multiplication algorithm only the lower $m+1$ digits of the product have to be computed. Similarly the value of $a$ can
-be reduced modulo $\beta^{m+1}$ before the multiple of $b$ is subtracted which simplifes the subtraction as well. A multiplication that produces
-only the lower $m+1$ digits requires ${m^2 + 3m - 2} \over 2$ single precision multiplications.
-
-With both optimizations in place the algorithm is the algorithm Barrett proposed. It requires $m^2 + 2m - 1$ single precision multiplications which
-is considerably faster than the straightforward $3m^2$ method.
-
-\subsection{The Barrett Algorithm}
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_reduce}. \\
-\textbf{Input}. mp\_int $a$, mp\_int $b$ and $\mu = \lfloor \beta^{2m}/b \rfloor$ $(0 \le a < b^2, b > 1)$ \\
-\textbf{Output}. $c \leftarrow a \mbox{ (mod }b\mbox{)}$ \\
-\hline \\
-Let $m$ represent the number of digits in $b$. \\
-1. Make a copy of $a$ and store it in $q$. (\textit{mp\_init\_copy}) \\
-2. $q \leftarrow \lfloor q / \beta^{m - 1} \rfloor$ (\textit{mp\_rshd}) \\
-\\
-Produce the quotient. \\
-3. $q \leftarrow q \cdot \mu$ (\textit{note: only produce digits at or above $m-1$}) \\
-4. $q \leftarrow \lfloor q / \beta^{m + 1} \rfloor$ \\
-\\
-Subtract the multiple of modulus from the input. \\
-5. $c \leftarrow a \mbox{ (mod }\beta^{m+1}\mbox{)}$ (\textit{mp\_mod\_2d}) \\
-6. $q \leftarrow q \cdot b \mbox{ (mod }\beta^{m+1}\mbox{)}$ (\textit{s\_mp\_mul\_digs}) \\
-7. $c \leftarrow c - q$ (\textit{mp\_sub}) \\
-\\
-Add $\beta^{m+1}$ if a carry occured. \\
-8. If $c < 0$ then (\textit{mp\_cmp\_d}) \\
-\hspace{3mm}8.1 $q \leftarrow 1$ (\textit{mp\_set}) \\
-\hspace{3mm}8.2 $q \leftarrow q \cdot \beta^{m+1}$ (\textit{mp\_lshd}) \\
-\hspace{3mm}8.3 $c \leftarrow c + q$ \\
-\\
-Now subtract the modulus if the residue is too large (e.g. quotient too small). \\
-9. While $c \ge b$ do (\textit{mp\_cmp}) \\
-\hspace{3mm}9.1 $c \leftarrow c - b$ \\
-10. Clear $q$. \\
-11. Return(\textit{MP\_OKAY}) \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_reduce}
-\end{figure}
-
-\textbf{Algorithm mp\_reduce.}
-This algorithm will reduce the input $a$ modulo $b$ in place using the Barrett algorithm. It is loosely based on algorithm 14.42 of HAC
-\cite[pp. 602]{HAC} which is based on the paper from Paul Barrett \cite{BARRETT}. The algorithm has several restrictions and assumptions which must be adhered to
-for the algorithm to work.
-
-First the modulus $b$ is assumed to be positive and greater than one. If the modulus were less than or equal to one than subtracting
-a multiple of it would either accomplish nothing or actually enlarge the input. The input $a$ must be in the range $0 \le a < b^2$ in order
-for the quotient to have enough precision. Technically the algorithm will still work if $a \ge b^2$ but it will take much longer to finish. The
-value of $\mu$ is passed as an argument to this algorithm and is assumed to be setup before the algorithm is used.
-
-Recall that the multiplication for the quotient on step 3 must only produce digits at or above the $m-1$'th position. An algorithm called
-$s\_mp\_mul\_high\_digs$ which has not been presented is used to accomplish this task. This optimal algorithm can only be used if the number
-of digits in $b$ is very much smaller than $\beta$.
-
-After the multiple of the modulus has been subtracted from $a$ the residue must be fixed up in case its negative. While it is known that
-$a \ge b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ only the lower $m+1$ digits are being used to compute the residue. In this case
-the invariant $\beta^{m+1}$ must be added to the residue to make it positive again.
-
-The while loop at step 9 will subtract $b$ until the residue is less than $b$. If the algorithm is performed correctly this step is only
-performed upto two times. However, if $a \ge b^2$ than it will iterate substantially more times than it should.
-
-EXAM,bn_mp_reduce.c
-
-The first multiplication that determines the quotient can be performed by only producing the digits from $m - 1$ and up. This essentially halves
-the number of single precision multiplications required. However, the optimization is only safe if $\beta$ is much larger than the number of digits
-in the modulus. In the source code this is evaluated on lines @36,if@ to @44,}@ where algorithm s\_mp\_mul\_high\_digs is used when it is
-safe to do so.
-
-\subsection{The Barrett Setup Algorithm}
-In order to use algorithm mp\_reduce the value of $\mu$ must be calculated in advance. Ideally this value should be computed once and stored for
-future use so that the Barrett algorithm can be used without delay.
-
-\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_reduce\_setup}. \\
-\textbf{Input}. mp\_int $a$ ($a > 1$) \\
-\textbf{Output}. $\mu \leftarrow \lfloor \beta^{2m}/a \rfloor$ \\
-\hline \\
-1. $\mu \leftarrow 2^{2 \cdot lg(\beta) \cdot m}$ (\textit{mp\_2expt}) \\
-2. $\mu \leftarrow \lfloor \mu / b \rfloor$ (\textit{mp\_div}) \\
-3. Return(\textit{MP\_OKAY}) \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_reduce\_setup}
-\end{figure}
-
-\textbf{Algorithm mp\_reduce\_setup.}
-This algorithm computes the reciprocal $\mu$ required for Barrett reduction. First $\beta^{2m}$ is calculated as $2^{2 \cdot lg(\beta) \cdot m}$ which
-is equivalent and much faster. The final value is computed by taking the integer quotient of $\lfloor \mu / b \rfloor$.
-
-EXAM,bn_mp_reduce_setup.c
-
-This simple routine calculates the reciprocal $\mu$ required by Barrett reduction. Note the extended usage of algorithm mp\_div where the variable
-which would received the remainder is passed as NULL. As will be discussed in ~DIVISION~ the division routine allows both the quotient and the
-remainder to be passed as NULL meaning to ignore the value.
-
-\section{The Montgomery Reduction}
-Montgomery reduction\footnote{Thanks to Niels Ferguson for his insightful explanation of the algorithm.} \cite{MONT} is by far the most interesting
-form of reduction in common use. It computes a modular residue which is not actually equal to the residue of the input yet instead equal to a
-residue times a constant. However, as perplexing as this may sound the algorithm is relatively simple and very efficient.
-
-Throughout this entire section the variable $n$ will represent the modulus used to form the residue. As will be discussed shortly the value of
-$n$ must be odd. The variable $x$ will represent the quantity of which the residue is sought. Similar to the Barrett algorithm the input
-is restricted to $0 \le x < n^2$. To begin the description some simple number theory facts must be established.
-
-\textbf{Fact 1.} Adding $n$ to $x$ does not change the residue since in effect it adds one to the quotient $\lfloor x / n \rfloor$. Another way
-to explain this is that $n$ (\textit{or multiples of $n$}) is congruent to zero modulo $n$. Adding zero will not change the value of the residue.
-
-\textbf{Fact 2.} If $x$ is even then performing a division by two in $\Z$ is congruent to $x \cdot 2^{-1} \mbox{ (mod }n\mbox{)}$. Actually
-this is an application of the fact that if $x$ is evenly divisible by any $k \in \Z$ then division in $\Z$ will be congruent to
-multiplication by $k^{-1}$ modulo $n$.
-
-From these two simple facts the following simple algorithm can be derived.
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{Montgomery Reduction}. \\
-\textbf{Input}. Integer $x$, $n$ and $k$ \\
-\textbf{Output}. $2^{-k}x \mbox{ (mod }n\mbox{)}$ \\
-\hline \\
-1. for $t$ from $1$ to $k$ do \\
-\hspace{3mm}1.1 If $x$ is odd then \\
-\hspace{6mm}1.1.1 $x \leftarrow x + n$ \\
-\hspace{3mm}1.2 $x \leftarrow x/2$ \\
-2. Return $x$. \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm Montgomery Reduction}
-\end{figure}
-
-The algorithm reduces the input one bit at a time using the two congruencies stated previously. Inside the loop $n$, which is odd, is
-added to $x$ if $x$ is odd. This forces $x$ to be even which allows the division by two in $\Z$ to be congruent to a modular division by two. Since
-$x$ is assumed to be initially much larger than $n$ the addition of $n$ will contribute an insignificant magnitude to $x$. Let $r$ represent the
-final result of the Montgomery algorithm. If $k > lg(n)$ and $0 \le x < n^2$ then the final result is limited to
-$0 \le r < \lfloor x/2^k \rfloor + n$. As a result at most a single subtraction is required to get the residue desired.
-
-\begin{figure}[here]
-\begin{small}
-\begin{center}
-\begin{tabular}{|c|l|}
-\hline \textbf{Step number ($t$)} & \textbf{Result ($x$)} \\
-\hline $1$ & $x + n = 5812$, $x/2 = 2906$ \\
-\hline $2$ & $x/2 = 1453$ \\
-\hline $3$ & $x + n = 1710$, $x/2 = 855$ \\
-\hline $4$ & $x + n = 1112$, $x/2 = 556$ \\
-\hline $5$ & $x/2 = 278$ \\
-\hline $6$ & $x/2 = 139$ \\
-\hline $7$ & $x + n = 396$, $x/2 = 198$ \\
-\hline $8$ & $x/2 = 99$ \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Example of Montgomery Reduction (I)}
-\label{fig:MONT1}
-\end{figure}
-
-Consider the example in figure~\ref{fig:MONT1} which reduces $x = 5555$ modulo $n = 257$ when $k = 8$. The final result $r = 99$ which is actually
-$2^{-8} \cdot 5555 \mbox{ (mod }257\mbox{)}$ can reveal the residue $x \equiv 158$ by multiplying by $2^8$ modulo $n$.
-
-Let $k = \lfloor lg(n) \rfloor + 1$ represent the number of bits in $n$. The current algorithm requires $2k^2$ single precision shifts
-and $k^2$ single precision additions. At this rate the algorithm is most certainly slower than Barrett reduction and not terribly useful.
-Fortunately there exists an alternative representation of the algorithm.
-
-\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{Montgomery Reduction} (modified I). \\
-\textbf{Input}. Integer $x$, $n$ and $k$ \\
-\textbf{Output}. $2^{-k}x \mbox{ (mod }n\mbox{)}$ \\
-\hline \\
-1. for $t$ from $0$ to $k - 1$ do \\
-\hspace{3mm}1.1 If the $t$'th bit of $x$ is one then \\
-\hspace{6mm}1.1.1 $x \leftarrow x + 2^tn$ \\
-2. Return $x/2^k$. \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm Montgomery Reduction (modified I)}
-\end{figure}
-
-This algorithm is equivalent since $2^tn$ is a multiple of $n$ and the lower $k$ bits of $x$ are zero by step 2. The number of single
-precision shifts has now been reduced from $2k^2$ to $k^2 + k$ which is only a small improvement.
-
-\begin{figure}[here]
-\begin{small}
-\begin{center}
-\begin{tabular}{|c|l|}
-\hline \textbf{Step number ($t$)} & \textbf{Result ($x$)} \\
-\hline $1$ & $x + 2^{0}n = 5812$ \\
-\hline $2$ & $5812$ \\
-\hline $3$ & $x + 2^{2}n = 6840$ \\
-\hline $4$ & $x + 2^{3}n = 8896$ \\
-\hline $5$ & $8896$ \\
-\hline $6$ & $8896$ \\
-\hline $7$ & $x + 2^{6}n = 25344$ \\
-\hline $8$ & $25344$ \\
-\hline -- & $x/2^k = 99$ \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Example of Montgomery Reduction (II)}
-\label{fig:MONT2}
-\end{figure}
-
-Figure~\ref{fig:MONT2} demonstrates the modified algorithm reducing $x = 4093$ modulo $n = 257$ with $k = 8$.
-With this algorithm a single shift right at the end is the only right shift required to reduce the input instead of $k$ right shifts inside the
-loop. Note that for the iterations $t = 2, 5, 6$ and $8$ where the result $x$ is not changed. In those iterations the $t$'th bit of $x$ is
-zero and the appropriate multiple of $n$ does not need to be added to force the $t$'th bit of the result to zero.
-
-\subsection{Digit Based Montgomery Reduction}
-Instead of computing the reduction on a bit-by-bit basis it is actually much faster to compute it on digit-by-digit basis. Consider the
-previous algorithm re-written to compute the Montgomery reduction in this new fashion.
-
-\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{Montgomery Reduction} (modified II). \\
-\textbf{Input}. Integer $x$, $n$ and $k$ \\
-\textbf{Output}. $\beta^{-k}x \mbox{ (mod }n\mbox{)}$ \\
-\hline \\
-1. for $t$ from $0$ to $k - 1$ do \\
-\hspace{3mm}1.1 $x \leftarrow x + \mu n \beta^t$ \\
-2. Return $x/\beta^k$. \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm Montgomery Reduction (modified II)}
-\end{figure}
-
-The value $\mu n \beta^t$ is a multiple of the modulus $n$ meaning that it will not change the residue. If the first digit of
-the value $\mu n \beta^t$ equals the negative (modulo $\beta$) of the $t$'th digit of $x$ then the addition will result in a zero digit. This
-problem breaks down to solving the following congruency.
-
-\begin{center}
-\begin{tabular}{rcl}
-$x_t + \mu n_0$ & $\equiv$ & $0 \mbox{ (mod }\beta\mbox{)}$ \\
-$\mu n_0$ & $\equiv$ & $-x_t \mbox{ (mod }\beta\mbox{)}$ \\
-$\mu$ & $\equiv$ & $-x_t/n_0 \mbox{ (mod }\beta\mbox{)}$ \\
-\end{tabular}
-\end{center}
-
-In each iteration of the loop on step 1 a new value of $\mu$ must be calculated. The value of $-1/n_0 \mbox{ (mod }\beta\mbox{)}$ is used
-extensively in this algorithm and should be precomputed. Let $\rho$ represent the negative of the modular inverse of $n_0$ modulo $\beta$.
-
-For example, let $\beta = 10$ represent the radix. Let $n = 17$ represent the modulus which implies $k = 2$ and $\rho \equiv 7$. Let $x = 33$
-represent the value to reduce.
-
-\newpage\begin{figure}
-\begin{center}
-\begin{tabular}{|c|c|c|}
-\hline \textbf{Step ($t$)} & \textbf{Value of $x$} & \textbf{Value of $\mu$} \\
-\hline -- & $33$ & --\\
-\hline $0$ & $33 + \mu n = 50$ & $1$ \\
-\hline $1$ & $50 + \mu n \beta = 900$ & $5$ \\
-\hline
-\end{tabular}
-\end{center}
-\caption{Example of Montgomery Reduction}
-\end{figure}
-
-The final result $900$ is then divided by $\beta^k$ to produce the final result $9$. The first observation is that $9 \nequiv x \mbox{ (mod }n\mbox{)}$
-which implies the result is not the modular residue of $x$ modulo $n$. However, recall that the residue is actually multiplied by $\beta^{-k}$ in
-the algorithm. To get the true residue the value must be multiplied by $\beta^k$. In this case $\beta^k \equiv 15 \mbox{ (mod }n\mbox{)}$ and
-the correct residue is $9 \cdot 15 \equiv 16 \mbox{ (mod }n\mbox{)}$.
-
-\subsection{Baseline Montgomery Reduction}
-The baseline Montgomery reduction algorithm will produce the residue for any size input. It is designed to be a catch-all algororithm for
-Montgomery reductions.
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_montgomery\_reduce}. \\
-\textbf{Input}. mp\_int $x$, mp\_int $n$ and a digit $\rho \equiv -1/n_0 \mbox{ (mod }n\mbox{)}$. \\
-\hspace{11.5mm}($0 \le x < n^2, n > 1, (n, \beta) = 1, \beta^k > n$) \\
-\textbf{Output}. $\beta^{-k}x \mbox{ (mod }n\mbox{)}$ \\
-\hline \\
-1. $digs \leftarrow 2n.used + 1$ \\
-2. If $digs < MP\_ARRAY$ and $m.used < \delta$ then \\
-\hspace{3mm}2.1 Use algorithm fast\_mp\_montgomery\_reduce instead. \\
-\\
-Setup $x$ for the reduction. \\
-3. If $x.alloc < digs$ then grow $x$ to $digs$ digits. \\
-4. $x.used \leftarrow digs$ \\
-\\
-Eliminate the lower $k$ digits. \\
-5. For $ix$ from $0$ to $k - 1$ do \\
-\hspace{3mm}5.1 $\mu \leftarrow x_{ix} \cdot \rho \mbox{ (mod }\beta\mbox{)}$ \\
-\hspace{3mm}5.2 $u \leftarrow 0$ \\
-\hspace{3mm}5.3 For $iy$ from $0$ to $k - 1$ do \\
-\hspace{6mm}5.3.1 $\hat r \leftarrow \mu n_{iy} + x_{ix + iy} + u$ \\
-\hspace{6mm}5.3.2 $x_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\
-\hspace{6mm}5.3.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\
-\hspace{3mm}5.4 While $u > 0$ do \\
-\hspace{6mm}5.4.1 $iy \leftarrow iy + 1$ \\
-\hspace{6mm}5.4.2 $x_{ix + iy} \leftarrow x_{ix + iy} + u$ \\
-\hspace{6mm}5.4.3 $u \leftarrow \lfloor x_{ix+iy} / \beta \rfloor$ \\
-\hspace{6mm}5.4.4 $x_{ix + iy} \leftarrow x_{ix+iy} \mbox{ (mod }\beta\mbox{)}$ \\
-\\
-Divide by $\beta^k$ and fix up as required. \\
-6. $x \leftarrow \lfloor x / \beta^k \rfloor$ \\
-7. If $x \ge n$ then \\
-\hspace{3mm}7.1 $x \leftarrow x - n$ \\
-8. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_montgomery\_reduce}
-\end{figure}
-
-\textbf{Algorithm mp\_montgomery\_reduce.}
-This algorithm reduces the input $x$ modulo $n$ in place using the Montgomery reduction algorithm. The algorithm is loosely based
-on algorithm 14.32 of \cite[pp.601]{HAC} except it merges the multiplication of $\mu n \beta^t$ with the addition in the inner loop. The
-restrictions on this algorithm are fairly easy to adapt to. First $0 \le x < n^2$ bounds the input to numbers in the same range as
-for the Barrett algorithm. Additionally $n > 1$ will ensure a modular inverse $\rho$ exists. $\rho$ must be calculated in
-advance of this algorithm. Finally the variable $k$ is fixed and a pseudonym for $n.used$.
-
-Step 2 decides whether a faster Montgomery algorithm can be used. It is based on the Comba technique meaning that there are limits on
-the size of the input. This algorithm is discussed in ~COMBARED~.
-
-Step 5 is the main reduction loop of the algorithm. The value of $\mu$ is calculated once per iteration in the outer loop. The inner loop
-calculates $x + \mu n \beta^{ix}$ by multiplying $\mu n$ and adding the result to $x$ shifted by $ix$ digits. Both the addition and
-multiplication are performed in the same loop to save time and memory. Step 5.4 will handle any additional carries that escape the inner loop.
-
-Using a quick inspection this algorithm requires $n$ single precision multiplications for the outer loop and $n^2$ single precision multiplications
-in the inner loop. In total $n^2 + n$ single precision multiplications which compares favourably to Barrett at $n^2 + 2n - 1$ single precision
-multiplications.
-
-EXAM,bn_mp_montgomery_reduce.c
-
-This is the baseline implementation of the Montgomery reduction algorithm. Lines @30,digs@ to @35,}@ determine if the Comba based
-routine can be used instead. Line @47,mu@ computes the value of $\mu$ for that particular iteration of the outer loop.
-
-The multiplication $\mu n \beta^{ix}$ is performed in one step in the inner loop. The alias $tmpx$ refers to the $ix$'th digit of $x$ and
-the alias $tmpn$ refers to the modulus $n$.
-
-\subsection{Faster ``Comba'' Montgomery Reduction}
-MARK,COMBARED
-
-The Montgomery reduction requires fewer single precision multiplications than a Barrett reduction, however it is much slower due to the serial
-nature of the inner loop. The Barrett reduction algorithm requires two slightly modified multipliers which can be implemented with the Comba
-technique. The Montgomery reduction algorithm cannot directly use the Comba technique to any significant advantage since the inner loop calculates
-a $k \times 1$ product $k$ times.
-
-The biggest obstacle is that at the $ix$'th iteration of the outer loop the value of $x_{ix}$ is required to calculate $\mu$. This means the
-carries from $0$ to $ix - 1$ must have been propagated upwards to form a valid $ix$'th digit. The solution as it turns out is very simple.
-Perform a Comba like multiplier and inside the outer loop just after the inner loop fix up the $ix + 1$'th digit by forwarding the carry.
-
-With this change in place the Montgomery reduction algorithm can be performed with a Comba style multiplication loop which substantially increases
-the speed of the algorithm.
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{fast\_mp\_montgomery\_reduce}. \\
-\textbf{Input}. mp\_int $x$, mp\_int $n$ and a digit $\rho \equiv -1/n_0 \mbox{ (mod }n\mbox{)}$. \\
-\hspace{11.5mm}($0 \le x < n^2, n > 1, (n, \beta) = 1, \beta^k > n$) \\
-\textbf{Output}. $\beta^{-k}x \mbox{ (mod }n\mbox{)}$ \\
-\hline \\
-Place an array of \textbf{MP\_WARRAY} mp\_word variables called $\hat W$ on the stack. \\
-1. if $x.alloc < n.used + 1$ then grow $x$ to $n.used + 1$ digits. \\
-Copy the digits of $x$ into the array $\hat W$ \\
-2. For $ix$ from $0$ to $x.used - 1$ do \\
-\hspace{3mm}2.1 $\hat W_{ix} \leftarrow x_{ix}$ \\
-3. For $ix$ from $x.used$ to $2n.used - 1$ do \\
-\hspace{3mm}3.1 $\hat W_{ix} \leftarrow 0$ \\
-Elimiate the lower $k$ digits. \\
-4. for $ix$ from $0$ to $n.used - 1$ do \\
-\hspace{3mm}4.1 $\mu \leftarrow \hat W_{ix} \cdot \rho \mbox{ (mod }\beta\mbox{)}$ \\
-\hspace{3mm}4.2 For $iy$ from $0$ to $n.used - 1$ do \\
-\hspace{6mm}4.2.1 $\hat W_{iy + ix} \leftarrow \hat W_{iy + ix} + \mu \cdot n_{iy}$ \\
-\hspace{3mm}4.3 $\hat W_{ix + 1} \leftarrow \hat W_{ix + 1} + \lfloor \hat W_{ix} / \beta \rfloor$ \\
-Propagate carries upwards. \\
-5. for $ix$ from $n.used$ to $2n.used + 1$ do \\
-\hspace{3mm}5.1 $\hat W_{ix + 1} \leftarrow \hat W_{ix + 1} + \lfloor \hat W_{ix} / \beta \rfloor$ \\
-Shift right and reduce modulo $\beta$ simultaneously. \\
-6. for $ix$ from $0$ to $n.used + 1$ do \\
-\hspace{3mm}6.1 $x_{ix} \leftarrow \hat W_{ix + n.used} \mbox{ (mod }\beta\mbox{)}$ \\
-Zero excess digits and fixup $x$. \\
-7. if $x.used > n.used + 1$ then do \\
-\hspace{3mm}7.1 for $ix$ from $n.used + 1$ to $x.used - 1$ do \\
-\hspace{6mm}7.1.1 $x_{ix} \leftarrow 0$ \\
-8. $x.used \leftarrow n.used + 1$ \\
-9. Clamp excessive digits of $x$. \\
-10. If $x \ge n$ then \\
-\hspace{3mm}10.1 $x \leftarrow x - n$ \\
-11. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm fast\_mp\_montgomery\_reduce}
-\end{figure}
-
-\textbf{Algorithm fast\_mp\_montgomery\_reduce.}
-This algorithm will compute the Montgomery reduction of $x$ modulo $n$ using the Comba technique. It is on most computer platforms significantly
-faster than algorithm mp\_montgomery\_reduce and algorithm mp\_reduce (\textit{Barrett reduction}). The algorithm has the same restrictions
-on the input as the baseline reduction algorithm. An additional two restrictions are imposed on this algorithm. The number of digits $k$ in the
-the modulus $n$ must not violate $MP\_WARRAY > 2k +1$ and $n < \delta$. When $\beta = 2^{28}$ this algorithm can be used to reduce modulo
-a modulus of at most $3,556$ bits in length.
-
-As in the other Comba reduction algorithms there is a $\hat W$ array which stores the columns of the product. It is initially filled with the
-contents of $x$ with the excess digits zeroed. The reduction loop is very similar the to the baseline loop at heart. The multiplication on step
-4.1 can be single precision only since $ab \mbox{ (mod }\beta\mbox{)} \equiv (a \mbox{ mod }\beta)(b \mbox{ mod }\beta)$. Some multipliers such
-as those on the ARM processors take a variable length time to complete depending on the number of bytes of result it must produce. By performing
-a single precision multiplication instead half the amount of time is spent.
-
-Also note that digit $\hat W_{ix}$ must have the carry from the $ix - 1$'th digit propagated upwards in order for this to work. That is what step
-4.3 will do. In effect over the $n.used$ iterations of the outer loop the $n.used$'th lower columns all have the their carries propagated forwards. Note
-how the upper bits of those same words are not reduced modulo $\beta$. This is because those values will be discarded shortly and there is no
-point.
-
-Step 5 will propgate the remainder of the carries upwards. On step 6 the columns are reduced modulo $\beta$ and shifted simultaneously as they are
-stored in the destination $x$.
-
-EXAM,bn_fast_mp_montgomery_reduce.c
-
-The $\hat W$ array is first filled with digits of $x$ on line @49,for@ then the rest of the digits are zeroed on line @54,for@. Both loops share
-the same alias variables to make the code easier to read.
-
-The value of $\mu$ is calculated in an interesting fashion. First the value $\hat W_{ix}$ is reduced modulo $\beta$ and cast to a mp\_digit. This
-forces the compiler to use a single precision multiplication and prevents any concerns about loss of precision. Line @101,>>@ fixes the carry
-for the next iteration of the loop by propagating the carry from $\hat W_{ix}$ to $\hat W_{ix+1}$.
-
-The for loop on line @113,for@ propagates the rest of the carries upwards through the columns. The for loop on line @126,for@ reduces the columns
-modulo $\beta$ and shifts them $k$ places at the same time. The alias $\_ \hat W$ actually refers to the array $\hat W$ starting at the $n.used$'th
-digit, that is $\_ \hat W_{t} = \hat W_{n.used + t}$.
-
-\subsection{Montgomery Setup}
-To calculate the variable $\rho$ a relatively simple algorithm will be required.
-
-\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_montgomery\_setup}. \\
-\textbf{Input}. mp\_int $n$ ($n > 1$ and $(n, 2) = 1$) \\
-\textbf{Output}. $\rho \equiv -1/n_0 \mbox{ (mod }\beta\mbox{)}$ \\
-\hline \\
-1. $b \leftarrow n_0$ \\
-2. If $b$ is even return(\textit{MP\_VAL}) \\
-3. $x \leftarrow ((b + 2) \mbox{ AND } 4) << 1) + b$ \\
-4. for $k$ from 0 to $3$ do \\
-\hspace{3mm}4.1 $x \leftarrow x \cdot (2 - bx)$ \\
-5. $\rho \leftarrow \beta - x \mbox{ (mod }\beta\mbox{)}$ \\
-6. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_montgomery\_setup}
-\end{figure}
-
-\textbf{Algorithm mp\_montgomery\_setup.}
-This algorithm will calculate the value of $\rho$ required within the Montgomery reduction algorithms. It uses a very interesting trick
-to calculate $1/n_0$ when $\beta$ is a power of two.
-
-EXAM,bn_mp_montgomery_setup.c
-
-This source code computes the value of $\rho$ required to perform Montgomery reduction. It has been modified to avoid performing excess
-multiplications when $\beta$ is not the default 28-bits.
-
-\section{The Diminished Radix Algorithm}
-The Diminished Radix method of modular reduction \cite{DRMET} is a fairly clever technique which can be more efficient than either the Barrett
-or Montgomery methods for certain forms of moduli. The technique is based on the following simple congruence.
-
-\begin{equation}
-(x \mbox{ mod } n) + k \lfloor x / n \rfloor \equiv x \mbox{ (mod }(n - k)\mbox{)}
-\end{equation}
-
-This observation was used in the MMB \cite{MMB} block cipher to create a diffusion primitive. It used the fact that if $n = 2^{31}$ and $k=1$ that
-then a x86 multiplier could produce the 62-bit product and use the ``shrd'' instruction to perform a double-precision right shift. The proof
-of the above equation is very simple. First write $x$ in the product form.
-
-\begin{equation}
-x = qn + r
-\end{equation}
-
-Now reduce both sides modulo $(n - k)$.
-
-\begin{equation}
-x \equiv qk + r \mbox{ (mod }(n-k)\mbox{)}
-\end{equation}
-
-The variable $n$ reduces modulo $n - k$ to $k$. By putting $q = \lfloor x/n \rfloor$ and $r = x \mbox{ mod } n$
-into the equation the original congruence is reproduced, thus concluding the proof. The following algorithm is based on this observation.
-
-\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{Diminished Radix Reduction}. \\
-\textbf{Input}. Integer $x$, $n$, $k$ \\
-\textbf{Output}. $x \mbox{ mod } (n - k)$ \\
-\hline \\
-1. $q \leftarrow \lfloor x / n \rfloor$ \\
-2. $q \leftarrow k \cdot q$ \\
-3. $x \leftarrow x \mbox{ (mod }n\mbox{)}$ \\
-4. $x \leftarrow x + q$ \\
-5. If $x \ge (n - k)$ then \\
-\hspace{3mm}5.1 $x \leftarrow x - (n - k)$ \\
-\hspace{3mm}5.2 Goto step 1. \\
-6. Return $x$ \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm Diminished Radix Reduction}
-\label{fig:DR}
-\end{figure}
-
-This algorithm will reduce $x$ modulo $n - k$ and return the residue. If $0 \le x < (n - k)^2$ then the algorithm will loop almost always
-once or twice and occasionally three times. For simplicity sake the value of $x$ is bounded by the following simple polynomial.
-
-\begin{equation}
-0 \le x < n^2 + k^2 - 2nk
-\end{equation}
-
-The true bound is $0 \le x < (n - k - 1)^2$ but this has quite a few more terms. The value of $q$ after step 1 is bounded by the following.
-
-\begin{equation}
-q < n - 2k - k^2/n
-\end{equation}
-
-Since $k^2$ is going to be considerably smaller than $n$ that term will always be zero. The value of $x$ after step 3 is bounded trivially as
-$0 \le x < n$. By step four the sum $x + q$ is bounded by
-
-\begin{equation}
-0 \le q + x < (k + 1)n - 2k^2 - 1
-\end{equation}
-
-With a second pass $q$ will be loosely bounded by $0 \le q < k^2$ after step 2 while $x$ will still be loosely bounded by $0 \le x < n$ after step 3. After the second pass it is highly unlike that the
-sum in step 4 will exceed $n - k$. In practice fewer than three passes of the algorithm are required to reduce virtually every input in the
-range $0 \le x < (n - k - 1)^2$.
-
-\begin{figure}
-\begin{small}
-\begin{center}
-\begin{tabular}{|l|}
-\hline
-$x = 123456789, n = 256, k = 3$ \\
-\hline $q \leftarrow \lfloor x/n \rfloor = 482253$ \\
-$q \leftarrow q*k = 1446759$ \\
-$x \leftarrow x \mbox{ mod } n = 21$ \\
-$x \leftarrow x + q = 1446780$ \\
-$x \leftarrow x - (n - k) = 1446527$ \\
-\hline
-$q \leftarrow \lfloor x/n \rfloor = 5650$ \\
-$q \leftarrow q*k = 16950$ \\
-$x \leftarrow x \mbox{ mod } n = 127$ \\
-$x \leftarrow x + q = 17077$ \\
-$x \leftarrow x - (n - k) = 16824$ \\
-\hline
-$q \leftarrow \lfloor x/n \rfloor = 65$ \\
-$q \leftarrow q*k = 195$ \\
-$x \leftarrow x \mbox{ mod } n = 184$ \\
-$x \leftarrow x + q = 379$ \\
-$x \leftarrow x - (n - k) = 126$ \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Example Diminished Radix Reduction}
-\label{fig:EXDR}
-\end{figure}
-
-Figure~\ref{fig:EXDR} demonstrates the reduction of $x = 123456789$ modulo $n - k = 253$ when $n = 256$ and $k = 3$. Note that even while $x$
-is considerably larger than $(n - k - 1)^2 = 63504$ the algorithm still converges on the modular residue exceedingly fast. In this case only
-three passes were required to find the residue $x \equiv 126$.
-
-
-\subsection{Choice of Moduli}
-On the surface this algorithm looks like a very expensive algorithm. It requires a couple of subtractions followed by multiplication and other
-modular reductions. The usefulness of this algorithm becomes exceedingly clear when an appropriate moduli is chosen.
-
-Division in general is a very expensive operation to perform. The one exception is when the division is by a power of the radix of representation used.
-Division by ten for example is simple for pencil and paper mathematics since it amounts to shifting the decimal place to the right. Similarly division
-by two (\textit{or powers of two}) is very simple for binary computers to perform. It would therefore seem logical to choose $n$ of the form $2^p$
-which would imply that $\lfloor x / n \rfloor$ is a simple shift of $x$ right $p$ bits.
-
-However, there is one operation related to division of power of twos that is even faster than this. If $n = \beta^p$ then the division may be
-performed by moving whole digits to the right $p$ places. In practice division by $\beta^p$ is much faster than division by $2^p$ for any $p$.
-Also with the choice of $n = \beta^p$ reducing $x$ modulo $n$ requires zeroing the digits above the $p-1$'th digit of $x$.
-
-Throughout the next section the term ``restricted modulus'' will refer to a modulus of the form $\beta^p - k$ where as the term ``unrestricted
-modulus'' will refer to a modulus of the form $2^p - k$. The word ``restricted'' in this case refers to the fact that it is based on the
-$2^p$ logic except $p$ must be a multiple of $lg(\beta)$.
-
-\subsection{Choice of $k$}
-Now that division and reduction (\textit{step 1 and 3 of figure~\ref{fig:DR}}) have been optimized to simple digit operations the multiplication by $k$
-in step 2 is the most expensive operation. Fortunately the choice of $k$ is not terribly limited. For all intents and purposes it might
-as well be a single digit. The smaller the value of $k$ is the faster the algorithm will be.
-
-\subsection{Restricted Diminished Radix Reduction}
-The restricted Diminished Radix algorithm can quickly reduce an input modulo a modulus of the form $n = \beta^p - k$. This algorithm can reduce
-an input $x$ within the range $0 \le x < n^2$ using only a couple passes of the algorithm demonstrated in figure~\ref{fig:DR}. The implementation
-of this algorithm has been optimized to avoid additional overhead associated with a division by $\beta^p$, the multiplication by $k$ or the addition
-of $x$ and $q$. The resulting algorithm is very efficient and can lead to substantial improvements over Barrett and Montgomery reduction when modular
-exponentiations are performed.
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_dr\_reduce}. \\
-\textbf{Input}. mp\_int $x$, $n$ and a mp\_digit $k = \beta - n_0$ \\
-\hspace{11.5mm}($0 \le x < n^2$, $n > 1$, $0 < k < \beta$) \\
-\textbf{Output}. $x \mbox{ mod } n$ \\
-\hline \\
-1. $m \leftarrow n.used$ \\
-2. If $x.alloc < 2m$ then grow $x$ to $2m$ digits. \\
-3. $\mu \leftarrow 0$ \\
-4. for $i$ from $0$ to $m - 1$ do \\
-\hspace{3mm}4.1 $\hat r \leftarrow k \cdot x_{m+i} + x_{i} + \mu$ \\
-\hspace{3mm}4.2 $x_{i} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\
-\hspace{3mm}4.3 $\mu \leftarrow \lfloor \hat r / \beta \rfloor$ \\
-5. $x_{m} \leftarrow \mu$ \\
-6. for $i$ from $m + 1$ to $x.used - 1$ do \\
-\hspace{3mm}6.1 $x_{i} \leftarrow 0$ \\
-7. Clamp excess digits of $x$. \\
-8. If $x \ge n$ then \\
-\hspace{3mm}8.1 $x \leftarrow x - n$ \\
-\hspace{3mm}8.2 Goto step 3. \\
-9. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_dr\_reduce}
-\end{figure}
-
-\textbf{Algorithm mp\_dr\_reduce.}
-This algorithm will perform the Dimished Radix reduction of $x$ modulo $n$. It has similar restrictions to that of the Barrett reduction
-with the addition that $n$ must be of the form $n = \beta^m - k$ where $0 < k <\beta$.
-
-This algorithm essentially implements the pseudo-code in figure~\ref{fig:DR} except with a slight optimization. The division by $\beta^m$, multiplication by $k$
-and addition of $x \mbox{ mod }\beta^m$ are all performed simultaneously inside the loop on step 4. The division by $\beta^m$ is emulated by accessing
-the term at the $m+i$'th position which is subsequently multiplied by $k$ and added to the term at the $i$'th position. After the loop the $m$'th
-digit is set to the carry and the upper digits are zeroed. Steps 5 and 6 emulate the reduction modulo $\beta^m$ that should have happend to
-$x$ before the addition of the multiple of the upper half.
-
-At step 8 if $x$ is still larger than $n$ another pass of the algorithm is required. First $n$ is subtracted from $x$ and then the algorithm resumes
-at step 3.
-
-EXAM,bn_mp_dr_reduce.c
-
-The first step is to grow $x$ as required to $2m$ digits since the reduction is performed in place on $x$. The label on line @49,top:@ is where
-the algorithm will resume if further reduction passes are required. In theory it could be placed at the top of the function however, the size of
-the modulus and question of whether $x$ is large enough are invariant after the first pass meaning that it would be a waste of time.
-
-The aliases $tmpx1$ and $tmpx2$ refer to the digits of $x$ where the latter is offset by $m$ digits. By reading digits from $x$ offset by $m$ digits
-a division by $\beta^m$ can be simulated virtually for free. The loop on line @61,for@ performs the bulk of the work (\textit{corresponds to step 4 of algorithm 7.11})
-in this algorithm.
-
-By line @68,mu@ the pointer $tmpx1$ points to the $m$'th digit of $x$ which is where the final carry will be placed. Similarly by line @71,for@ the
-same pointer will point to the $m+1$'th digit where the zeroes will be placed.
-
-Since the algorithm is only valid if both $x$ and $n$ are greater than zero an unsigned comparison suffices to determine if another pass is required.
-With the same logic at line @82,sub@ the value of $x$ is known to be greater than or equal to $n$ meaning that an unsigned subtraction can be used
-as well. Since the destination of the subtraction is the larger of the inputs the call to algorithm s\_mp\_sub cannot fail and the return code
-does not need to be checked.
-
-\subsubsection{Setup}
-To setup the restricted Diminished Radix algorithm the value $k = \beta - n_0$ is required. This algorithm is not really complicated but provided for
-completeness.
-
-\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_dr\_setup}. \\
-\textbf{Input}. mp\_int $n$ \\
-\textbf{Output}. $k = \beta - n_0$ \\
-\hline \\
-1. $k \leftarrow \beta - n_0$ \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_dr\_setup}
-\end{figure}
-
-EXAM,bn_mp_dr_setup.c
-
-\subsubsection{Modulus Detection}
-Another algorithm which will be useful is the ability to detect a restricted Diminished Radix modulus. An integer is said to be
-of restricted Diminished Radix form if all of the digits are equal to $\beta - 1$ except the trailing digit which may be any value.
-
-\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_dr\_is\_modulus}. \\
-\textbf{Input}. mp\_int $n$ \\
-\textbf{Output}. $1$ if $n$ is in D.R form, $0$ otherwise \\
-\hline
-1. If $n.used < 2$ then return($0$). \\
-2. for $ix$ from $1$ to $n.used - 1$ do \\
-\hspace{3mm}2.1 If $n_{ix} \ne \beta - 1$ return($0$). \\
-3. Return($1$). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_dr\_is\_modulus}
-\end{figure}
-
-\textbf{Algorithm mp\_dr\_is\_modulus.}
-This algorithm determines if a value is in Diminished Radix form. Step 1 rejects obvious cases where fewer than two digits are
-in the mp\_int. Step 2 tests all but the first digit to see if they are equal to $\beta - 1$. If the algorithm manages to get to
-step 3 then $n$ must of Diminished Radix form.
-
-EXAM,bn_mp_dr_is_modulus.c
-
-\subsection{Unrestricted Diminished Radix Reduction}
-The unrestricted Diminished Radix algorithm allows modular reductions to be performed when the modulus is of the form $2^p - k$. This algorithm
-is a straightforward adaptation of algorithm~\ref{fig:DR}.
-
-In general the restricted Diminished Radix reduction algorithm is much faster since it has considerably lower overhead. However, this new
-algorithm is much faster than either Montgomery or Barrett reduction when the moduli are of the appropriate form.
-
-\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_reduce\_2k}. \\
-\textbf{Input}. mp\_int $a$ and $n$. mp\_digit $k$ \\
-\hspace{11.5mm}($a \ge 0$, $n > 1$, $0 < k < \beta$, $n + k$ is a power of two) \\
-\textbf{Output}. $a \mbox{ (mod }n\mbox{)}$ \\
-\hline
-1. $p \leftarrow \lceil lg(n) \rceil$ (\textit{mp\_count\_bits}) \\
-2. While $a \ge n$ do \\
-\hspace{3mm}2.1 $q \leftarrow \lfloor a / 2^p \rfloor$ (\textit{mp\_div\_2d}) \\
-\hspace{3mm}2.2 $a \leftarrow a \mbox{ (mod }2^p\mbox{)}$ (\textit{mp\_mod\_2d}) \\
-\hspace{3mm}2.3 $q \leftarrow q \cdot k$ (\textit{mp\_mul\_d}) \\
-\hspace{3mm}2.4 $a \leftarrow a - q$ (\textit{s\_mp\_sub}) \\
-\hspace{3mm}2.5 If $a \ge n$ then do \\
-\hspace{6mm}2.5.1 $a \leftarrow a - n$ \\
-3. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_reduce\_2k}
-\end{figure}
-
-\textbf{Algorithm mp\_reduce\_2k.}
-This algorithm quickly reduces an input $a$ modulo an unrestricted Diminished Radix modulus $n$. Division by $2^p$ is emulated with a right
-shift which makes the algorithm fairly inexpensive to use.
-
-EXAM,bn_mp_reduce_2k.c
-
-The algorithm mp\_count\_bits calculates the number of bits in an mp\_int which is used to find the initial value of $p$. The call to mp\_div\_2d
-on line @31,mp_div_2d@ calculates both the quotient $q$ and the remainder $a$ required. By doing both in a single function call the code size
-is kept fairly small. The multiplication by $k$ is only performed if $k > 1$. This allows reductions modulo $2^p - 1$ to be performed without
-any multiplications.
-
-The unsigned s\_mp\_add, mp\_cmp\_mag and s\_mp\_sub are used in place of their full sign counterparts since the inputs are only valid if they are
-positive. By using the unsigned versions the overhead is kept to a minimum.
-
-\subsubsection{Unrestricted Setup}
-To setup this reduction algorithm the value of $k = 2^p - n$ is required.
-
-\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_reduce\_2k\_setup}. \\
-\textbf{Input}. mp\_int $n$ \\
-\textbf{Output}. $k = 2^p - n$ \\
-\hline
-1. $p \leftarrow \lceil lg(n) \rceil$ (\textit{mp\_count\_bits}) \\
-2. $x \leftarrow 2^p$ (\textit{mp\_2expt}) \\
-3. $x \leftarrow x - n$ (\textit{mp\_sub}) \\
-4. $k \leftarrow x_0$ \\
-5. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_reduce\_2k\_setup}
-\end{figure}
-
-\textbf{Algorithm mp\_reduce\_2k\_setup.}
-This algorithm computes the value of $k$ required for the algorithm mp\_reduce\_2k. By making a temporary variable $x$ equal to $2^p$ a subtraction
-is sufficient to solve for $k$. Alternatively if $n$ has more than one digit the value of $k$ is simply $\beta - n_0$.
-
-EXAM,bn_mp_reduce_2k_setup.c
-
-\subsubsection{Unrestricted Detection}
-An integer $n$ is a valid unrestricted Diminished Radix modulus if either of the following are true.
-
-\begin{enumerate}
-\item The number has only one digit.
-\item The number has more than one digit and every bit from the $\beta$'th to the most significant is one.
-\end{enumerate}
-
-If either condition is true than there is a power of two namely $2^p$ such that $0 < 2^p - n < \beta$. If the input is only
-one digit than it will always be of the correct form. Otherwise all of the bits above the first digit must be one. This arises from the fact
-that there will be value of $k$ that when added to the modulus causes a carry in the first digit which propagates all the way to the most
-significant bit. The resulting sum will be a power of two.
-
-\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_reduce\_is\_2k}. \\
-\textbf{Input}. mp\_int $n$ \\
-\textbf{Output}. $1$ if of proper form, $0$ otherwise \\
-\hline
-1. If $n.used = 0$ then return($0$). \\
-2. If $n.used = 1$ then return($1$). \\
-3. $p \leftarrow \rceil lg(n) \lceil$ (\textit{mp\_count\_bits}) \\
-4. for $x$ from $lg(\beta)$ to $p$ do \\
-\hspace{3mm}4.1 If the ($x \mbox{ mod }lg(\beta)$)'th bit of the $\lfloor x / lg(\beta) \rfloor$ of $n$ is zero then return($0$). \\
-5. Return($1$). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_reduce\_is\_2k}
-\end{figure}
-
-\textbf{Algorithm mp\_reduce\_is\_2k.}
-This algorithm quickly determines if a modulus is of the form required for algorithm mp\_reduce\_2k to function properly.
-
-EXAM,bn_mp_reduce_is_2k.c
-
-
-
-\section{Algorithm Comparison}
-So far three very different algorithms for modular reduction have been discussed. Each of the algorithms have their own strengths and weaknesses
-that makes having such a selection very useful. The following table sumarizes the three algorithms along with comparisons of work factors. Since
-all three algorithms have the restriction that $0 \le x < n^2$ and $n > 1$ those limitations are not included in the table.
-
-\begin{center}
-\begin{small}
-\begin{tabular}{|c|c|c|c|c|c|}
-\hline \textbf{Method} & \textbf{Work Required} & \textbf{Limitations} & \textbf{$m = 8$} & \textbf{$m = 32$} & \textbf{$m = 64$} \\
-\hline Barrett & $m^2 + 2m - 1$ & None & $79$ & $1087$ & $4223$ \\
-\hline Montgomery & $m^2 + m$ & $n$ must be odd & $72$ & $1056$ & $4160$ \\
-\hline D.R. & $2m$ & $n = \beta^m - k$ & $16$ & $64$ & $128$ \\
-\hline
-\end{tabular}
-\end{small}
-\end{center}
-
-In theory Montgomery and Barrett reductions would require roughly the same amount of time to complete. However, in practice since Montgomery
-reduction can be written as a single function with the Comba technique it is much faster. Barrett reduction suffers from the overhead of
-calling the half precision multipliers, addition and division by $\beta$ algorithms.
-
-For almost every cryptographic algorithm Montgomery reduction is the algorithm of choice. The one set of algorithms where Diminished Radix reduction truly
-shines are based on the discrete logarithm problem such as Diffie-Hellman \cite{DH} and ElGamal \cite{ELGAMAL}. In these algorithms
-primes of the form $\beta^m - k$ can be found and shared amongst users. These primes will allow the Diminished Radix algorithm to be used in
-modular exponentiation to greatly speed up the operation.
-
-
-
-\section*{Exercises}
-\begin{tabular}{cl}
-$\left [ 3 \right ]$ & Prove that the ``trick'' in algorithm mp\_montgomery\_setup actually \\
- & calculates the correct value of $\rho$. \\
- & \\
-$\left [ 2 \right ]$ & Devise an algorithm to reduce modulo $n + k$ for small $k$ quickly. \\
- & \\
-$\left [ 4 \right ]$ & Prove that the pseudo-code algorithm ``Diminished Radix Reduction'' \\
- & (\textit{figure~\ref{fig:DR}}) terminates. Also prove the probability that it will \\
- & terminate within $1 \le k \le 10$ iterations. \\
- & \\
-\end{tabular}
-
-
-\chapter{Exponentiation}
-Exponentiation is the operation of raising one variable to the power of another, for example, $a^b$. A variant of exponentiation, computed
-in a finite field or ring, is called modular exponentiation. This latter style of operation is typically used in public key
-cryptosystems such as RSA and Diffie-Hellman. The ability to quickly compute modular exponentiations is of great benefit to any
-such cryptosystem and many methods have been sought to speed it up.
-
-\section{Exponentiation Basics}
-A trivial algorithm would simply multiply $a$ against itself $b - 1$ times to compute the exponentiation desired. However, as $b$ grows in size
-the number of multiplications becomes prohibitive. Imagine what would happen if $b$ $\approx$ $2^{1024}$ as is the case when computing an RSA signature
-with a $1024$-bit key. Such a calculation could never be completed as it would take simply far too long.
-
-Fortunately there is a very simple algorithm based on the laws of exponents. Recall that $lg_a(a^b) = b$ and that $lg_a(a^ba^c) = b + c$ which
-are two trivial relationships between the base and the exponent. Let $b_i$ represent the $i$'th bit of $b$ starting from the least
-significant bit. If $b$ is a $k$-bit integer than the following equation is true.
-
-\begin{equation}
-a^b = \prod_{i=0}^{k-1} a^{2^i \cdot b_i}
-\end{equation}
-
-By taking the base $a$ logarithm of both sides of the equation the following equation is the result.
-
-\begin{equation}
-b = \sum_{i=0}^{k-1}2^i \cdot b_i
-\end{equation}
-
-The term $a^{2^i}$ can be found from the $i - 1$'th term by squaring the term since $\left ( a^{2^i} \right )^2$ is equal to
-$a^{2^{i+1}}$. This observation forms the basis of essentially all fast exponentiation algorithms. It requires $k$ squarings and on average
-$k \over 2$ multiplications to compute the result. This is indeed quite an improvement over simply multiplying by $a$ a total of $b-1$ times.
-
-While this current method is a considerable speed up there are further improvements to be made. For example, the $a^{2^i}$ term does not need to
-be computed in an auxilary variable. Consider the following equivalent algorithm.
-
-\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{Left to Right Exponentiation}. \\
-\textbf{Input}. Integer $a$, $b$ and $k$ \\
-\textbf{Output}. $c = a^b$ \\
-\hline \\
-1. $c \leftarrow 1$ \\
-2. for $i$ from $k - 1$ to $0$ do \\
-\hspace{3mm}2.1 $c \leftarrow c^2$ \\
-\hspace{3mm}2.2 $c \leftarrow c \cdot a^{b_i}$ \\
-3. Return $c$. \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Left to Right Exponentiation}
-\label{fig:LTOR}
-\end{figure}
-
-This algorithm starts from the most significant bit and works towards the least significant bit. When the $i$'th bit of $b$ is set $a$ is
-multiplied against the current product. In each iteration the product is squared which doubles the exponent of the individual terms of the
-product.
-
-For example, let $b = 101100_2 \equiv 44_{10}$. The following chart demonstrates the actions of the algorithm.
-
-\newpage\begin{figure}
-\begin{center}
-\begin{tabular}{|c|c|}
-\hline \textbf{Value of $i$} & \textbf{Value of $c$} \\
-\hline - & $1$ \\
-\hline $5$ & $a$ \\
-\hline $4$ & $a^2$ \\
-\hline $3$ & $a^4 \cdot a$ \\
-\hline $2$ & $a^8 \cdot a^2 \cdot a$ \\
-\hline $1$ & $a^{16} \cdot a^4 \cdot a^2$ \\
-\hline $0$ & $a^{32} \cdot a^8 \cdot a^4$ \\
-\hline
-\end{tabular}
-\end{center}
-\caption{Example of Left to Right Exponentiation}
-\end{figure}
-
-When the product $a^{32} \cdot a^8 \cdot a^4$ is simplified it is equal $a^{44}$ which is the desired exponentiation. This particular algorithm is
-called ``Left to Right'' because it reads the exponent in that order. All of the exponentiation algorithms that will be presented are of this nature.
-
-\subsection{Single Digit Exponentiation}
-The first algorithm in the series of exponentiation algorithms will be an unbounded algorithm where the exponent is a single digit. It is intended
-to be used when a small power of an input is required (\textit{e.g. $a^5$}). It is faster than simply multiplying $b - 1$ times for all values of
-$b$ that are greater than three.
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_expt\_d}. \\
-\textbf{Input}. mp\_int $a$ and mp\_digit $b$ \\
-\textbf{Output}. $c = a^b$ \\
-\hline \\
-1. $g \leftarrow a$ (\textit{mp\_init\_copy}) \\
-2. $c \leftarrow 1$ (\textit{mp\_set}) \\
-3. for $x$ from 1 to $lg(\beta)$ do \\
-\hspace{3mm}3.1 $c \leftarrow c^2$ (\textit{mp\_sqr}) \\
-\hspace{3mm}3.2 If $b$ AND $2^{lg(\beta) - 1} \ne 0$ then \\
-\hspace{6mm}3.2.1 $c \leftarrow c \cdot g$ (\textit{mp\_mul}) \\
-\hspace{3mm}3.3 $b \leftarrow b << 1$ \\
-4. Clear $g$. \\
-5. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_expt\_d}
-\end{figure}
-
-\textbf{Algorithm mp\_expt\_d.}
-This algorithm computes the value of $a$ raised to the power of a single digit $b$. It uses the left to right exponentiation algorithm to
-quickly compute the exponentiation. It is loosely based on algorithm 14.79 of HAC \cite[pp. 615]{HAC} with the difference that the
-exponent is a fixed width.
-
-A copy of $a$ is made first to allow destination variable $c$ be the same as the source variable $a$. The result is set to the initial value of
-$1$ in the subsequent step.
-
-Inside the loop the exponent is read from the most significant bit first down to the least significant bit. First $c$ is invariably squared
-on step 3.1. In the following step if the most significant bit of $b$ is one the copy of $a$ is multiplied against $c$. The value
-of $b$ is shifted left one bit to make the next bit down from the most signficant bit the new most significant bit. In effect each
-iteration of the loop moves the bits of the exponent $b$ upwards to the most significant location.
-
-EXAM,bn_mp_expt_d.c
-
--- Some note later.
-
-\section{$k$-ary Exponentiation}
-When calculating an exponentiation the most time consuming bottleneck is the multiplications which are in general a small factor
-slower than squaring. Recall from the previous algorithm that $b_{i}$ refers to the $i$'th bit of the exponent $b$. Suppose instead it referred to
-the $i$'th $k$-bit digit of the exponent of $b$. For $k = 1$ the definitions are synonymous and for $k > 1$ algorithm~\ref{fig:KARY}
-computes the same exponentiation. A group of $k$ bits from the exponent is called a \textit{window}. That is it is a small window on only a
-portion of the entire exponent. Consider the following modification to the basic left to right exponentiation algorithm.
-
-\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{$k$-ary Exponentiation}. \\
-\textbf{Input}. Integer $a$, $b$, $k$ and $t$ \\
-\textbf{Output}. $c = a^b$ \\
-\hline \\
-1. $c \leftarrow 1$ \\
-2. for $i$ from $t - 1$ to $0$ do \\
-\hspace{3mm}2.1 $c \leftarrow c^{2^k} $ \\
-\hspace{3mm}2.2 Extract the $i$'th $k$-bit word from $b$ and store it in $g$. \\
-\hspace{3mm}2.3 $c \leftarrow c \cdot a^g$ \\
-3. Return $c$. \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{$k$-ary Exponentiation}
-\label{fig:KARY}
-\end{figure}
-
-The squaring on step 2.1 can be calculated by squaring the value $c$ successively $k$ times. If the values of $a^g$ for $0 < g < 2^k$ have been
-precomputed this algorithm requires only $t$ multiplications and $tk$ squarings. The table can be generated with $2^{k - 1} - 1$ squarings and
-$2^{k - 1} + 1$ multiplications. This algorithm assumes that the number of bits in the exponent is evenly divisible by $k$.
-However, when it is not the remaining $0 < x \le k - 1$ bits can be handled with algorithm~\ref{fig:LTOR}.
-
-Suppose $k = 4$ and $t = 100$. This modified algorithm will require $109$ multiplications and $408$ squarings to compute the exponentiation. The
-original algorithm would on average have required $200$ multiplications and $400$ squrings to compute the same value. The total number of squarings
-has increased slightly but the number of multiplications has nearly halved.
-
-\subsection{Optimal Values of $k$}
-An optimal value of $k$ will minimize $2^{k} + \lceil n / k \rceil + n - 1$ for a fixed number of bits in the exponent $n$. The simplest
-approach is to brute force search amongst the values $k = 2, 3, \ldots, 8$ for the lowest result. Table~\ref{fig:OPTK} lists optimal values of $k$
-for various exponent sizes and compares the number of multiplication and squarings required against algorithm~\ref{fig:LTOR}.
-
-\begin{figure}[here]
-\begin{center}
-\begin{small}
-\begin{tabular}{|c|c|c|c|c|c|}
-\hline \textbf{Exponent (bits)} & \textbf{Optimal $k$} & \textbf{Work at $k$} & \textbf{Work with ~\ref{fig:LTOR}} \\
-\hline $16$ & $2$ & $27$ & $24$ \\
-\hline $32$ & $3$ & $49$ & $48$ \\
-\hline $64$ & $3$ & $92$ & $96$ \\
-\hline $128$ & $4$ & $175$ & $192$ \\
-\hline $256$ & $4$ & $335$ & $384$ \\
-\hline $512$ & $5$ & $645$ & $768$ \\
-\hline $1024$ & $6$ & $1257$ & $1536$ \\
-\hline $2048$ & $6$ & $2452$ & $3072$ \\
-\hline $4096$ & $7$ & $4808$ & $6144$ \\
-\hline
-\end{tabular}
-\end{small}
-\end{center}
-\caption{Optimal Values of $k$ for $k$-ary Exponentiation}
-\label{fig:OPTK}
-\end{figure}
-
-\subsection{Sliding-Window Exponentiation}
-A simple modification to the previous algorithm is only generate the upper half of the table in the range $2^{k-1} \le g < 2^k$. Essentially
-this is a table for all values of $g$ where the most significant bit of $g$ is a one. However, in order for this to be allowed in the
-algorithm values of $g$ in the range $0 \le g < 2^{k-1}$ must be avoided.
-
-Table~\ref{fig:OPTK2} lists optimal values of $k$ for various exponent sizes and compares the work required against algorithm~\ref{fig:KARY}.
-
-\begin{figure}[here]
-\begin{center}
-\begin{small}
-\begin{tabular}{|c|c|c|c|c|c|}
-\hline \textbf{Exponent (bits)} & \textbf{Optimal $k$} & \textbf{Work at $k$} & \textbf{Work with ~\ref{fig:KARY}} \\
-\hline $16$ & $3$ & $24$ & $27$ \\
-\hline $32$ & $3$ & $45$ & $49$ \\
-\hline $64$ & $4$ & $87$ & $92$ \\
-\hline $128$ & $4$ & $167$ & $175$ \\
-\hline $256$ & $5$ & $322$ & $335$ \\
-\hline $512$ & $6$ & $628$ & $645$ \\
-\hline $1024$ & $6$ & $1225$ & $1257$ \\
-\hline $2048$ & $7$ & $2403$ & $2452$ \\
-\hline $4096$ & $8$ & $4735$ & $4808$ \\
-\hline
-\end{tabular}
-\end{small}
-\end{center}
-\caption{Optimal Values of $k$ for Sliding Window Exponentiation}
-\label{fig:OPTK2}
-\end{figure}
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{Sliding Window $k$-ary Exponentiation}. \\
-\textbf{Input}. Integer $a$, $b$, $k$ and $t$ \\
-\textbf{Output}. $c = a^b$ \\
-\hline \\
-1. $c \leftarrow 1$ \\
-2. for $i$ from $t - 1$ to $0$ do \\
-\hspace{3mm}2.1 If the $i$'th bit of $b$ is a zero then \\
-\hspace{6mm}2.1.1 $c \leftarrow c^2$ \\
-\hspace{3mm}2.2 else do \\
-\hspace{6mm}2.2.1 $c \leftarrow c^{2^k}$ \\
-\hspace{6mm}2.2.2 Extract the $k$ bits from $(b_{i}b_{i-1}\ldots b_{i-(k-1)})$ and store it in $g$. \\
-\hspace{6mm}2.2.3 $c \leftarrow c \cdot a^g$ \\
-\hspace{6mm}2.2.4 $i \leftarrow i - k$ \\
-3. Return $c$. \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Sliding Window $k$-ary Exponentiation}
-\end{figure}
-
-Similar to the previous algorithm this algorithm must have a special handler when fewer than $k$ bits are left in the exponent. While this
-algorithm requires the same number of squarings it can potentially have fewer multiplications. The pre-computed table $a^g$ is also half
-the size as the previous table.
-
-Consider the exponent $b = 111101011001000_2 \equiv 31432_{10}$ with $k = 3$ using both algorithms. The first algorithm will divide the exponent up as
-the following five $3$-bit words $b \equiv \left ( 111, 101, 011, 001, 000 \right )_{2}$. The second algorithm will break the
-exponent as $b \equiv \left ( 111, 101, 0, 110, 0, 100, 0 \right )_{2}$. The single digit $0$ in the second representation are where
-a single squaring took place instead of a squaring and multiplication. In total the first method requires $10$ multiplications and $18$
-squarings. The second method requires $8$ multiplications and $18$ squarings.
-
-In general the sliding window method is never slower than the generic $k$-ary method and often it is slightly faster.
-
-\section{Modular Exponentiation}
-
-Modular exponentiation is essentially computing the power of a base within a finite field or ring. For example, computing
-$d \equiv a^b \mbox{ (mod }c\mbox{)}$ is a modular exponentiation. Instead of first computing $a^b$ and then reducing it
-modulo $c$ the intermediate result is reduced modulo $c$ after every squaring or multiplication operation.
-
-This guarantees that any intermediate result is bounded by $0 \le d \le c^2 - 2c + 1$ and can be reduced modulo $c$ quickly using
-one of the algorithms presented in ~REDUCTION~.
-
-Before the actual modular exponentiation algorithm can be written a wrapper algorithm must be written first. This algorithm
-will allow the exponent $b$ to be negative which is computed as $c \equiv \left (1 / a \right )^{\vert b \vert} \mbox{(mod }d\mbox{)}$. The
-value of $(1/a) \mbox{ mod }c$ is computed using the modular inverse (\textit{see ~MODINV~}). If no inverse exists the algorithm
-terminates with an error.
-
-\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_exptmod}. \\
-\textbf{Input}. mp\_int $a$, $b$ and $c$ \\
-\textbf{Output}. $y \equiv g^x \mbox{ (mod }p\mbox{)}$ \\
-\hline \\
-1. If $c.sign = MP\_NEG$ return(\textit{MP\_VAL}). \\
-2. If $b.sign = MP\_NEG$ then \\
-\hspace{3mm}2.1 $g' \leftarrow g^{-1} \mbox{ (mod }c\mbox{)}$ \\
-\hspace{3mm}2.2 $x' \leftarrow \vert x \vert$ \\
-\hspace{3mm}2.3 Compute $d \equiv g'^{x'} \mbox{ (mod }c\mbox{)}$ via recursion. \\
-3. if $p$ is odd \textbf{OR} $p$ is a D.R. modulus then \\
-\hspace{3mm}3.1 Compute $y \equiv g^{x} \mbox{ (mod }p\mbox{)}$ via algorithm mp\_exptmod\_fast. \\
-4. else \\
-\hspace{3mm}4.1 Compute $y \equiv g^{x} \mbox{ (mod }p\mbox{)}$ via algorithm s\_mp\_exptmod. \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_exptmod}
-\end{figure}
-
-\textbf{Algorithm mp\_exptmod.}
-The first algorithm which actually performs modular exponentiation is algorithm s\_mp\_exptmod. It is a sliding window $k$-ary algorithm
-which uses Barrett reduction to reduce the product modulo $p$. The second algorithm mp\_exptmod\_fast performs the same operation
-except it uses either Montgomery or Diminished Radix reduction. The two latter reduction algorithms are clumped in the same exponentiation
-algorithm since their arguments are essentially the same (\textit{two mp\_ints and one mp\_digit}).
-
-EXAM,bn_mp_exptmod.c
-
-In order to keep the algorithms in a known state the first step on line @29,if@ is to reject any negative modulus as input. If the exponent is
-negative the algorithm tries to perform a modular exponentiation with the modular inverse of the base $G$. The temporary variable $tmpG$ is assigned
-the modular inverse of $G$ and $tmpX$ is assigned the absolute value of $X$. The algorithm will recuse with these new values with a positive
-exponent.
-
-If the exponent is positive the algorithm resumes the exponentiation. Line @63,dr_@ determines if the modulus is of the restricted Diminished Radix
-form. If it is not line @65,reduce@ attempts to determine if it is of a unrestricted Diminished Radix form. The integer $dr$ will take on one
-of three values.
-
-\begin{enumerate}
-\item $dr = 0$ means that the modulus is not of either restricted or unrestricted Diminished Radix form.
-\item $dr = 1$ means that the modulus is of restricted Diminished Radix form.
-\item $dr = 2$ means that the modulus is of unrestricted Diminished Radix form.
-\end{enumerate}
-
-Line @69,if@ determines if the fast modular exponentiation algorithm can be used. It is allowed if $dr \ne 0$ or if the modulus is odd. Otherwise,
-the slower s\_mp\_exptmod algorithm is used which uses Barrett reduction.
-
-\subsection{Barrett Modular Exponentiation}
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{s\_mp\_exptmod}. \\
-\textbf{Input}. mp\_int $a$, $b$ and $c$ \\
-\textbf{Output}. $y \equiv g^x \mbox{ (mod }p\mbox{)}$ \\
-\hline \\
-1. $k \leftarrow lg(x)$ \\
-2. $winsize \leftarrow \left \lbrace \begin{array}{ll}
- 2 & \mbox{if }k \le 7 \\
- 3 & \mbox{if }7 < k \le 36 \\
- 4 & \mbox{if }36 < k \le 140 \\
- 5 & \mbox{if }140 < k \le 450 \\
- 6 & \mbox{if }450 < k \le 1303 \\
- 7 & \mbox{if }1303 < k \le 3529 \\
- 8 & \mbox{if }3529 < k \\
- \end{array} \right .$ \\
-3. Initialize $2^{winsize}$ mp\_ints in an array named $M$ and one mp\_int named $\mu$ \\
-4. Calculate the $\mu$ required for Barrett Reduction (\textit{mp\_reduce\_setup}). \\
-5. $M_1 \leftarrow g \mbox{ (mod }p\mbox{)}$ \\
-\\
-Setup the table of small powers of $g$. First find $g^{2^{winsize}}$ and then all multiples of it. \\
-6. $k \leftarrow 2^{winsize - 1}$ \\
-7. $M_{k} \leftarrow M_1$ \\
-8. for $ix$ from 0 to $winsize - 2$ do \\
-\hspace{3mm}8.1 $M_k \leftarrow \left ( M_k \right )^2$ (\textit{mp\_sqr}) \\
-\hspace{3mm}8.2 $M_k \leftarrow M_k \mbox{ (mod }p\mbox{)}$ (\textit{mp\_reduce}) \\
-9. for $ix$ from $2^{winsize - 1} + 1$ to $2^{winsize} - 1$ do \\
-\hspace{3mm}9.1 $M_{ix} \leftarrow M_{ix - 1} \cdot M_{1}$ (\textit{mp\_mul}) \\
-\hspace{3mm}9.2 $M_{ix} \leftarrow M_{ix} \mbox{ (mod }p\mbox{)}$ (\textit{mp\_reduce}) \\
-10. $res \leftarrow 1$ \\
-\\
-Start Sliding Window. \\
-11. $mode \leftarrow 0, bitcnt \leftarrow 1, buf \leftarrow 0, digidx \leftarrow x.used - 1, bitcpy \leftarrow 0, bitbuf \leftarrow 0$ \\
-12. Loop \\
-\hspace{3mm}12.1 $bitcnt \leftarrow bitcnt - 1$ \\
-\hspace{3mm}12.2 If $bitcnt = 0$ then do \\
-\hspace{6mm}12.2.1 If $digidx = -1$ goto step 13. \\
-\hspace{6mm}12.2.2 $buf \leftarrow x_{digidx}$ \\
-\hspace{6mm}12.2.3 $digidx \leftarrow digidx - 1$ \\
-\hspace{6mm}12.2.4 $bitcnt \leftarrow lg(\beta)$ \\
-Continued on next page. \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm s\_mp\_exptmod}
-\end{figure}
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{s\_mp\_exptmod} (\textit{continued}). \\
-\textbf{Input}. mp\_int $a$, $b$ and $c$ \\
-\textbf{Output}. $y \equiv g^x \mbox{ (mod }p\mbox{)}$ \\
-\hline \\
-\hspace{3mm}12.3 $y \leftarrow (buf >> (lg(\beta) - 1))$ AND $1$ \\
-\hspace{3mm}12.4 $buf \leftarrow buf << 1$ \\
-\hspace{3mm}12.5 if $mode = 0$ and $y = 0$ then goto step 12. \\
-\hspace{3mm}12.6 if $mode = 1$ and $y = 0$ then do \\
-\hspace{6mm}12.6.1 $res \leftarrow res^2$ \\
-\hspace{6mm}12.6.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\
-\hspace{6mm}12.6.3 Goto step 12. \\
-\hspace{3mm}12.7 $bitcpy \leftarrow bitcpy + 1$ \\
-\hspace{3mm}12.8 $bitbuf \leftarrow bitbuf + (y << (winsize - bitcpy))$ \\
-\hspace{3mm}12.9 $mode \leftarrow 2$ \\
-\hspace{3mm}12.10 If $bitcpy = winsize$ then do \\
-\hspace{6mm}Window is full so perform the squarings and single multiplication. \\
-\hspace{6mm}12.10.1 for $ix$ from $0$ to $winsize -1$ do \\
-\hspace{9mm}12.10.1.1 $res \leftarrow res^2$ \\
-\hspace{9mm}12.10.1.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\
-\hspace{6mm}12.10.2 $res \leftarrow res \cdot M_{bitbuf}$ \\
-\hspace{6mm}12.10.3 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\
-\hspace{6mm}Reset the window. \\
-\hspace{6mm}12.10.4 $bitcpy \leftarrow 0, bitbuf \leftarrow 0, mode \leftarrow 1$ \\
-\\
-No more windows left. Check for residual bits of exponent. \\
-13. If $mode = 2$ and $bitcpy > 0$ then do \\
-\hspace{3mm}13.1 for $ix$ form $0$ to $bitcpy - 1$ do \\
-\hspace{6mm}13.1.1 $res \leftarrow res^2$ \\
-\hspace{6mm}13.1.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\
-\hspace{6mm}13.1.3 $bitbuf \leftarrow bitbuf << 1$ \\
-\hspace{6mm}13.1.4 If $bitbuf$ AND $2^{winsize} \ne 0$ then do \\
-\hspace{9mm}13.1.4.1 $res \leftarrow res \cdot M_{1}$ \\
-\hspace{9mm}13.1.4.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\
-14. $y \leftarrow res$ \\
-15. Clear $res$, $mu$ and the $M$ array. \\
-16. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm s\_mp\_exptmod (continued)}
-\end{figure}
-
-\textbf{Algorithm s\_mp\_exptmod.}
-This algorithm computes the $x$'th power of $g$ modulo $p$ and stores the result in $y$. It takes advantage of the Barrett reduction
-algorithm to keep the product small throughout the algorithm.
-
-The first two steps determine the optimal window size based on the number of bits in the exponent. The larger the exponent the
-larger the window size becomes. After a window size $winsize$ has been chosen an array of $2^{winsize}$ mp\_int variables is allocated. This
-table will hold the values of $g^x \mbox{ (mod }p\mbox{)}$ for $2^{winsize - 1} \le x < 2^{winsize}$.
-
-After the table is allocated the first power of $g$ is found. Since $g \ge p$ is allowed it must be first reduced modulo $p$ to make
-the rest of the algorithm more efficient. The first element of the table at $2^{winsize - 1}$ is found by squaring $M_1$ successively $winsize - 2$
-times. The rest of the table elements are found by multiplying the previous element by $M_1$ modulo $p$.
-
-Now that the table is available the sliding window may begin. The following list describes the functions of all the variables in the window.
-\begin{enumerate}
-\item The variable $mode$ dictates how the bits of the exponent are interpreted.
-\begin{enumerate}
- \item When $mode = 0$ the bits are ignored since no non-zero bit of the exponent has been seen yet. For example, if the exponent were simply
- $1$ then there would be $lg(\beta) - 1$ zero bits before the first non-zero bit. In this case bits are ignored until a non-zero bit is found.
- \item When $mode = 1$ a non-zero bit has been seen before and a new $winsize$-bit window has not been formed yet. In this mode leading $0$ bits
- are read and a single squaring is performed. If a non-zero bit is read a new window is created.
- \item When $mode = 2$ the algorithm is in the middle of forming a window and new bits are appended to the window from the most significant bit
- downwards.
-\end{enumerate}
-\item The variable $bitcnt$ indicates how many bits are left in the current digit of the exponent left to be read. When it reaches zero a new digit
- is fetched from the exponent.
-\item The variable $buf$ holds the currently read digit of the exponent.
-\item The variable $digidx$ is an index into the exponents digits. It starts at the leading digit $x.used - 1$ and moves towards the trailing digit.
-\item The variable $bitcpy$ indicates how many bits are in the currently formed window. When it reaches $winsize$ the window is flushed and
- the appropriate operations performed.
-\item The variable $bitbuf$ holds the current bits of the window being formed.
-\end{enumerate}
-
-All of step 12 is the window processing loop. It will iterate while there are digits available form the exponent to read. The first step
-inside this loop is to extract a new digit if no more bits are available in the current digit. If there are no bits left a new digit is
-read and if there are no digits left than the loop terminates.
-
-After a digit is made available step 12.3 will extract the most significant bit of the current digit and move all other bits in the digit
-upwards. In effect the digit is read from most significant bit to least significant bit and since the digits are read from leading to
-trailing edges the entire exponent is read from most significant bit to least significant bit.
-
-At step 12.5 if the $mode$ and currently extracted bit $y$ are both zero the bit is ignored and the next bit is read. This prevents the
-algorithm from having to perform trivial squaring and reduction operations before the first non-zero bit is read. Step 12.6 and 12.7-10 handle
-the two cases of $mode = 1$ and $mode = 2$ respectively.
-
-FIGU,expt_state,Sliding Window State Diagram
-
-By step 13 there are no more digits left in the exponent. However, there may be partial bits in the window left. If $mode = 2$ then
-a Left-to-Right algorithm is used to process the remaining few bits.
-
-EXAM,bn_s_mp_exptmod.c
-
-Lines @26,if@ through @40,}@ determine the optimal window size based on the length of the exponent in bits. The window divisions are sorted
-from smallest to greatest so that in each \textbf{if} statement only one condition must be tested. For example, by the \textbf{if} statement
-on line @32,if@ the value of $x$ is already known to be greater than $140$.
-
-The conditional piece of code beginning on line @42,define@ allows the window size to be restricted to five bits. This logic is used to ensure
-the table of precomputed powers of $G$ remains relatively small.
-
-The for loop on line @49,for@ initializes the $M$ array while lines @59,mp_init@ and @62,mp_reduce@ compute the value of $\mu$ required for
-Barrett reduction.
-
--- More later.
-
-\section{Quick Power of Two}
-Calculating $b = 2^a$ can be performed much quicker than with any of the previous algorithms. Recall that a logical shift left $m << k$ is
-equivalent to $m \cdot 2^k$. By this logic when $m = 1$ a quick power of two can be achieved.
-
-\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_2expt}. \\
-\textbf{Input}. integer $b$ \\
-\textbf{Output}. $a \leftarrow 2^b$ \\
-\hline \\
-1. $a \leftarrow 0$ \\
-2. If $a.alloc < \lfloor b / lg(\beta) \rfloor + 1$ then grow $a$ appropriately. \\
-3. $a.used \leftarrow \lfloor b / lg(\beta) \rfloor + 1$ \\
-4. $a_{\lfloor b / lg(\beta) \rfloor} \leftarrow 1 << (b \mbox{ mod } lg(\beta))$ \\
-5. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_2expt}
-\end{figure}
-
-\textbf{Algorithm mp\_2expt.}
-
-EXAM,bn_mp_2expt.c
-
-\chapter{Higher Level Algorithms}
-
-This chapter discusses the various higher level algorithms that are required to complete a well rounded multiple precision integer package. These
-routines are less performance oriented than the algorithms of chapters five, six and seven but are no less important.
-
-The first section describes a method of integer division with remainder that is universally well known. It provides the signed division logic
-for the package. The subsequent section discusses a set of algorithms which allow a single digit to be the 2nd operand for a variety of operations.
-These algorithms serve mostly to simplify other algorithms where small constants are required. The last two sections discuss how to manipulate
-various representations of integers. For example, converting from an mp\_int to a string of character.
-
-\section{Integer Division with Remainder}
-MARK,DIVISION
-
-Integer division aside from modular exponentiation is most intensive algorithm to compute.
-
-
-\section{Single Digit Helpers}
-\subsection{Single Digit Addition}
-\subsection{Single Digit Subtraction}
-\subsection{Single Digit Multiplication}
-\subsection{Single Digit Division}
-\subsection{Single Digit Modulo}
-\subsection{Single Digit Root Extraction}
-\section{Random Number Generation}
-\section{Formatted Output}
-\subsection{Getting The Output Size}
-\subsection{Generating Radix-n Output}
-\subsection{Reading Radix-n Input}
-\section{Unformatted Output}
-\subsection{Getting The Output Size}
-\subsection{Generating Output}
-\subsection{Reading Input}
-
-\chapter{Number Theoretic Algorithms}
-\section{Greatest Common Divisor}
-\section{Least Common Multiple}
-\section{Jacobi Symbol Computation}
-\section{Modular Inverse}
-MARK,MODINV
-\subsection{General Case}
-\subsection{Odd Moduli}
-\section{Primality Tests}
-\subsection{Trial Division}
-\subsection{The Fermat Test}
-\subsection{The Miller-Rabin Test}
-\subsection{Primality Test in a Bottle}
-\subsection{The Next Prime}
-\section{Root Extraction}
-
-\backmatter
-\appendix
-\begin{thebibliography}{ABCDEF}
-\bibitem[1]{TAOCPV2}
-Donald Knuth, \textit{The Art of Computer Programming}, Third Edition, Volume Two, Seminumerical Algorithms, Addison-Wesley, 1998
-
-\bibitem[2]{HAC}
-A. Menezes, P. van Oorschot, S. Vanstone, \textit{Handbook of Applied Cryptography}, CRC Press, 1996
-
-\bibitem[3]{ROSE}
-Michael Rosing, \textit{Implementing Elliptic Curve Cryptography}, Manning Publications, 1999
-
-\bibitem[4]{COMBA}
-Paul G. Comba, \textit{Exponentiation Cryptosystems on the IBM PC}. IBM Systems Journal 29(4): 526-538 (1990)
-
-\bibitem[5]{KARA}
-A. Karatsuba, Doklay Akad. Nauk SSSR 145 (1962), pp.293-294
-
-\bibitem[6]{KARAP}
-Andre Weimerskirch and Christof Paar, \textit{Generalizations of the Karatsuba Algorithm for Polynomial Multiplication}, Submitted to Design, Codes and Cryptography, March 2002
-
-\bibitem[7]{BARRETT}
-Paul Barrett, \textit{Implementing the Rivest Shamir and Adleman Public Key Encryption Algorithm on a Standard Digital Signal Processor}, Advances in Cryptology, Crypto '86, Springer-Verlag.
-
-\bibitem[8]{MONT}
-P.L.Montgomery. \textit{Modular multiplication without trial division}. Mathematics of Computation, 44(170):519-521, April 1985.
-
-\bibitem[9]{DRMET}
-Chae Hoon Lim and Pil Joong Lee, \textit{Generating Efficient Primes for Discrete Log Cryptosystems}, POSTECH Information Research Laboratories
-
-\bibitem[10]{MMB}
-J. Daemen and R. Govaerts and J. Vandewalle, \textit{Block ciphers based on Modular Arithmetic}, State and {P}rogress in the {R}esearch of {C}ryptography, 1993, pp. 80-89
-
-\end{thebibliography}
-
-\input{tommath.ind}
-
-\chapter{Appendix}
-\subsection*{Appendix A -- Source Listing of tommath.h}
-
-The following is the source listing of the header file ``tommath.h'' for the LibTomMath project. It contains many of
-the definitions used throughout the code such as \textbf{mp\_int}, \textbf{MP\_PREC} and so on. The header is
-presented here for completeness.
-
-LIST,tommath.h
-
-\end{document}
\ No newline at end of file
diff --git a/tommath.tex b/tommath.tex
deleted file mode 100644
index cd2a97e..0000000
--- a/tommath.tex
+++ /dev/null
@@ -1,8141 +0,0 @@
-\documentclass[b5paper]{book}
-\usepackage{hyperref}
-\usepackage{makeidx}
-\usepackage{amssymb}
-\usepackage{color}
-\usepackage{alltt}
-\usepackage{graphicx}
-\usepackage{layout}
-\def\union{\cup}
-\def\intersect{\cap}
-\def\getsrandom{\stackrel{\rm R}{\gets}}
-\def\cross{\times}
-\def\cat{\hspace{0.5em} \| \hspace{0.5em}}
-\def\catn{$\|$}
-\def\divides{\hspace{0.3em} | \hspace{0.3em}}
-\def\nequiv{\not\equiv}
-\def\approx{\raisebox{0.2ex}{\mbox{\small $\sim$}}}
-\def\lcm{{\rm lcm}}
-\def\gcd{{\rm gcd}}
-\def\log{{\rm log}}
-\def\ord{{\rm ord}}
-\def\abs{{\mathit abs}}
-\def\rep{{\mathit rep}}
-\def\mod{{\mathit\ mod\ }}
-\renewcommand{\pmod}[1]{\ ({\rm mod\ }{#1})}
-\newcommand{\floor}[1]{\left\lfloor{#1}\right\rfloor}
-\newcommand{\ceil}[1]{\left\lceil{#1}\right\rceil}
-\def\Or{{\rm\ or\ }}
-\def\And{{\rm\ and\ }}
-\def\iff{\hspace{1em}\Longleftrightarrow\hspace{1em}}
-\def\implies{\Rightarrow}
-\def\undefined{{\rm ``undefined"}}
-\def\Proof{\vspace{1ex}\noindent {\bf Proof:}\hspace{1em}}
-\let\oldphi\phi
-\def\phi{\varphi}
-\def\Pr{{\rm Pr}}
-\newcommand{\str}[1]{{\mathbf{#1}}}
-\def\F{{\mathbb F}}
-\def\N{{\mathbb N}}
-\def\Z{{\mathbb Z}}
-\def\R{{\mathbb R}}
-\def\C{{\mathbb C}}
-\def\Q{{\mathbb Q}}
-\definecolor{DGray}{gray}{0.5}
-\newcommand{\emailaddr}[1]{\mbox{$<${#1}$>$}}
-\def\twiddle{\raisebox{0.3ex}{\mbox{\tiny $\sim$}}}
-\def\gap{\vspace{0.5ex}}
-\makeindex
-\begin{document}
-\frontmatter
-\pagestyle{empty}
-\title{Multiple-Precision Integer Arithmetic, \\ A Case Study Involving the LibTomMath Project \\ - DRAFT - }
-\author{\mbox{
-%\begin{small}
-\begin{tabular}{c}
-Tom St Denis \\
-Algonquin College \\
-\\
-Mads Rasmussen \\
-Open Communications Security \\
-\\
-Greg Rose \\
-QUALCOMM Australia \\
-\end{tabular}
-%\end{small}
-}
-}
-\maketitle
-This text in its entirety is copyright \copyright{}2003 by Tom St Denis. It may not be redistributed
-electronically or otherwise without the sole permission of the author. The text is freely redistributable as long as
-it is packaged along with the LibTomMath library in a non-commercial project. Contact the
-author for other redistribution rights.
-
-This text corresponds to the v0.17 release of the LibTomMath project.
-
-\begin{alltt}
-Tom St Denis
-111 Banning Rd
-Ottawa, Ontario
-K2L 1C3
-Canada
-
-Phone: 1-613-836-3160
-Email: tomstdenis@iahu.ca
-\end{alltt}
-
-This text is formatted to the international B5 paper size of 176mm wide by 250mm tall using the \LaTeX{}
-{\em book} macro package and the Perl {\em booker} package.
-
-\tableofcontents
-\listoffigures
-\chapter*{Preface}
-Blah.
-
-\mainmatter
-\pagestyle{headings}
-\chapter{Introduction}
-\section{Multiple Precision Arithmetic}
-\subsection{The Need for Multiple Precision Arithmetic}
-The most prevalent use for multiple precision arithmetic (\textit{often referred to as bignum math}) is within public
-key cryptography. Algorithms such as RSA, Diffie-Hellman and Elliptic Curve Cryptography require large integers in order to
-resist known cryptanalytic attacks. Typical modern programming languages such as C and Java only provide small
-single-precision data types which are incapable of precisely representing integers which are often hundreds of bits long.
-
-For example, consider multiplying $1,234,567$ by $9,876,543$ in C with an ``unsigned long'' data type. With an
-x86 machine the result is $4,136,875,833$ while the true result is $12,193,254,061,881$. The original inputs
-were approximately $21$ and $24$ bits respectively. If the C language cannot multiply two relatively small values
-together precisely how does anyone expect it to multiply two values that are considerably larger?
-
-Most advancements in fast multiple precision arithmetic stem from the desire for faster cryptographic primitives. However, cryptography
-is not the only field of study that can benefit from fast large integer routines. Another auxiliary use for multiple precision integers is
-high precision floating point data types. The basic IEEE standard floating point type is made up of an integer mantissa $q$ and an exponent $e$.
-Numbers are given in the form $n = q \cdot b^e$ where $b = 2$ is specified. Since IEEE is meant to be implemented in
-hardware the precision of the mantissa is often fairly small (\textit{23, 48 and 64 bits}). Since the mantissa is merely an
-integer a large multiple precision integer could be used. In effect very high precision floating point arithmetic
-could be performed. This would be useful where scientific applications must minimize the total output error over long simulations.
-
-\subsection{Multiple Precision Arithmetic}
-\index{multiple precision}
-Multiple precision arithmetic attempts to the solve the shortcomings of single precision data types such as those from
-the C and Java programming languages. In essence multiple precision arithmetic is a set of operations that can be
-performed on members of an algebraic group whose precision is not fixed. The algorithms when implemented to be multiple
-precision can allow a developer to work with any practical precision required.
-
-Typically the arithmetic over the ring of integers denoted by $\Z$ is performed by routines that are collectively and
-casually referred to as ``bignum'' routines. However, it is possible to have rings of polynomials as well typically
-denoted by $\Z/p\Z \left [ X \right ]$ which could have variable precision (\textit{or degree}). This text will
-discuss implementation of the former, however implementing polynomial basis routines should be relatively easy after reading this text.
-
-\subsection{Benefits of Multiple Precision Arithmetic}
-\index{precision} \index{accuracy}
-Precision of the real value to a given precision is defined loosely as the proximity of the real value to a given representation.
-Accuracy is defined as the reproducibility of the result. For example, the calculation $1/3 = 0.25$ is imprecise but can be accurate provided
-it is reproducible.
-
-The benefit of multiple precision representations over single precision representations is that
-often no precision is lost while representing the result of an operation which requires excess precision. For example,
-the multiplication of two $n$-bit integers requires at least $2n$ bits to represent the result. A multiple precision
-system would augment the precision of the destination to accomodate the result while a single precision system would
-truncate excess bits to maintain a fixed level of precision.
-
-Multiple precision representations allow for the precision to be very high (\textit{if not exacting}) but at a cost of
-modest computer resources. The only reasonable case where a multiple precision system will lose precision is when
-emulating a floating point data type. However, with multiple precision integer arithmetic no precision is lost.
-
-\subsection{Basis of Operations}
-At the heart of all multiple precision integer operations are the ``long-hand'' algorithms we all learned as children
-in grade school. For example, to multiply $1,234$ by $981$ the student is not taught to memorize the times table for
-$1,234$, instead they are taught how to long-multiply. That is to multiply each column using simple single digit
-multiplications, line up the partial results, and add the resulting products by column. The representation that most
-are familiar with is known as decimal or formally as radix-10. A radix-$n$ representation simply means there are
-$n$ possible values per digit. For example, binary would be a radix-2 representation.
-
-In essence computer based multiple precision arithmetic is very much the same. The most notable difference is the usage
-of a binary friendly radix. That is to use a radix of the form $2^k$ where $k$ is typically the size of a machine
-register. Also occasionally more optimal algorithms are used to perform certain operations such as multiplication and
-squaring instead of traditional long-hand algorithms.
-
-\section{Purpose of This Text}
-The purpose of this text is to instruct the reader regarding how to implement multiple precision algorithms. That is
-to not only explain the core theoretical algorithms but also the various ``house keeping'' tasks that are neglected by
-authors of other texts on the subject. Texts such as \cite[HAC]{HAC} and \cite{TAOCPV2} give considerably detailed
-explanations of the theoretical aspects of the algorithms and very little regarding the practical aspects.
-
-How an algorithm is explained and how it is actually implemented are two very different
-realities. For example, algorithm 14.7 on page 594 of HAC lists a relatively simple algorithm for performing multiple
-precision integer addition. However, what the description lacks is any discussion concerning the fact that the two
-integer inputs may be of differing magnitudes. Similarly the division routine (\textit{Algorithm 14.20, pp. 598})
-does not discuss how to handle sign or handle the dividend's decreasing magnitude in the main loop (\textit{Step \#3}).
-
-As well as the numerous practical oversights both of the texts do not discuss several key optimal algorithms required
-such as ``Comba'' and Karatsuba multipliers and fast modular inversion. These optimal algorithms are vital to achieve
-any form of useful performance in non-trivial applications.
-
-To solve this problem the focus of this text is on the practical aspects of implementing the algorithms that
-constitute a multiple precision integer package with light discussions on the theoretical aspects. As a case
-study the ``LibTomMath''\footnote{Available freely at http://math.libtomcrypt.org} package is used to demonstrate
-algorithms with implementations that have been field tested and work very well.
-
-\section{Discussion and Notation}
-\subsection{Notation}
-A multiple precision integer of $n$-digits shall be denoted as $x = (x_n ... x_1 x_0)_{ \beta }$ to be the
-multiple precision notation for the integer $x \equiv \sum_{i=0}^{n} x_i\beta^i$. The elements of the array $x$ are
-said to be the radix $\beta$ digits of the integer. For example, $x = (1,2,3)_{10}$ would represent the
-integer $1\cdot 10^2 + 2\cdot10^1 + 3\cdot10^0 = 123$.
-
-A ``mp\_int'' shall refer to a composite structure which contains the digits of the integer as well as auxilary data
-required to manipulate the data. These additional members are discussed in chapter three. For the purposes of this text
-a ``multiple precision integer'' and a ``mp\_int'' are synonymous.
-
-\index{single-precision} \index{double-precision} \index{mp\_digit} \index{mp\_word}
-For the purposes of this text a single-precision variable must be able to represent integers in the range $0 \le x < 2 \beta$ while
-a double-precision variable must be able to represent integers in the range $0 \le x < 2 \beta^2$. Within the source code that will be
-presented the data type \textbf{mp\_digit} will represent a single-precision type while \textbf{mp\_word} will represent a
-double-precision type. In several algorithms (\textit{notably the Comba routines}) temporary results
-will be stored in a double-precision arrays. For the purposes of this text $x_j$ will refer to the
-$j$'th digit of a single-precision array and $\hat x_j$ will refer to the $j$'th digit of a double-precision
-array.
-
-The $\lfloor \mbox{ } \rfloor$ brackets represent a value truncated and rounded down to the nearest integer. The $\lceil \mbox{ } \rceil$ brackets
-represent a value truncated and rounded up to the nearest integer. Typically when the $/$ division symbol is used the intention is to perform an integer
-division. For example, $5/2 = 2$ which will often be written as $\lfloor 5/2 \rfloor = 2$ for clarity. When a value is presented as a fraction
-such as $5 \over 2$ a real value division is implied.
-
-\subsection{Work Effort}
-\index{big-O}
-To measure the efficiency of various algorithms a modified big-O notation is used. In this system all
-single precision operations are considered to have the same cost\footnote{Except where explicitly noted.}.
-That is a single precision addition, multiplication and division are assumed to take the same time to
-complete. While this is generally not true in practice it will simplify the discussions considerably.
-
-Some algorithms have slight advantages over others which is why some constants will not be removed in
-the notation. For example, a normal multiplication requires $O(n^2)$ work while a squaring requires
-$O({{n^2 + n}\over 2})$ work. In standard big-O notation these would be said to be equivalent. However, in the
-context of the this text the magnitude of the inputs will not approach an infinite size. This means the conventional limit
-notation wisdom does not apply to the cancellation of constants.
-
-Throughout the discussions various ``work levels'' will be discussed. These levels are the $O(1)$,
-$O(n)$, $O(n^2)$, ..., $O(n^k)$ work efforts. For example, operations at the $O(n^k)$ ``level'' are said to be
-executed more frequently than operations at the $O(n^m)$ ``level'' when $k > m$. Obviously most optimizations will pay
-off the most at the higher levels since they represent the bulk of the effort required.
-
-\section{Exercises}
-Within the more advanced chapters a section will be set aside to give the reader some challenging exercises. These exercises are not
-designed to be prize winning problems, but to be thought provoking. Wherever possible the problems are forward minded stating
-problems that will be answered in subsequent chapters. The reader is encouraged to finish the exercises as they appear to get a
-better understanding of the subject material.
-
-Similar to the exercises of \cite{TAOCPV2} as explained on pp.\textit{ix} these exercises are given a scoring system. However, unlike
-\cite{TAOCPV2} the problems do not get nearly as hard as often. The scoring of these exercises ranges from one (\textit{the easiest}) to
-five (\textit{the hardest}). The following table sumarizes the scoring.
-
-\vspace{5mm}
-\begin{tabular}{cl}
-$\left [ 1 \right ]$ & An easy problem that should only take the reader a manner of \\
- & minutes to solve. Usually does not involve much computer time. \\
- & \\
-$\left [ 2 \right ]$ & An easy problem that involves a marginal amount of computer \\
- & time usage. Usually requires a program to be written to \\
- & solve the problem. \\
- & \\
-$\left [ 3 \right ]$ & A moderately hard problem that requires a non-trivial amount \\
- & of work. Usually involves trivial research and development of \\
- & new theory from the perspective of a student. \\
- & \\
-$\left [ 4 \right ]$ & A moderately hard problem that involves a non-trivial amount \\
- & of work and research. The solution to which will demonstrate \\
- & a higher mastery of the subject matter. \\
- & \\
-$\left [ 5 \right ]$ & A hard problem that involves concepts that are non-trivial. \\
- & Solutions to these problems will demonstrate a complete mastery \\
- & of the given subject. \\
- & \\
-\end{tabular}
-
-Essentially problems at the first level are meant to be simple questions that the reader can answer quickly without programming a solution or
-devising new theory. These problems are quick tests to see if the material is understood. Problems at the second level are also
-designed to be easy but will require a program or algorithm to be implemented to arrive at the answer.
-
-Problems at the third level are meant to be a bit more difficult. Often the answer is fairly obvious but arriving at an exacting solution
-requires some thought and skill. These problems will almost always involve devising a new algorithm or implementing a variation of
-another algorithm.
-
-Problems at the fourth level are meant to be even more difficult as well as involve some research. The reader will most likely not know
-the answer right away nor will this text provide the exact details of the answer (\textit{or at least not until a subsequent chapter}). Problems
-at the fifth level are meant to be the hardest problems relative to all the other problems in the chapter. People who can correctly
-answer fifth level problems have a mastery of the subject matter at hand.
-
-Often problems will be tied together. The purpose of this is to start a chain of thought that will be discussed in future chapters. The reader
-is encouraged to answer the follow-up problems and try to draw the relevence of problems.
-
-\chapter{Introduction to LibTomMath}
-
-\section{What is LibTomMath?}
-LibTomMath is a free and open source multiple precision library written in portable ISO C source code. By portable it is
-meant that the library does not contain any code that is computer platform dependent or otherwise problematic to use on any
-given platform. The library has been successfully tested under numerous operating systems including Solaris, MacOS, Windows,
-Linux, PalmOS and on standalone hardware such as the Gameboy Advance. The library is designed to contain enough
-functionality to be able to develop applications such as public key cryptosystems.
-
-\section{Goals of LibTomMath}
-
-Even though the library is written entirely in portable ISO C considerable care has been taken to
-optimize the algorithm implementations within the library. Specifically the code has been written to work well with
-the GNU C Compiler (\textit{GCC}) on both x86 and ARMv4 processors. Wherever possible highly efficient
-algorithms (\textit{such as Karatsuba multiplication, sliding window exponentiation and Montgomery reduction}) have
-been provided to make the library as efficient as possible. Even with the optimal and sometimes specialized
-algorithms that have been included the Application Programing Interface (\textit{API}) has been kept as simple as possible.
-Often generic place holder routines will make use of specialized algorithms automatically without the developer's
-attention. One such example is the generic multiplication algorithm \textbf{mp\_mul()} which will automatically use
-Karatsuba multiplication if the inputs are of a specific size.
-
-Making LibTomMath as efficient as possible is not the only goal of the LibTomMath project. Ideally the library should
-be source compatible with another popular library which makes it more attractive for developers to use. In this case the
-MPI library was used as a API template for all the basic functions.
-
-The project is also meant to act as a learning tool for students. The logic being that no easy-to-follow ``bignum''
-library exists which can be used to teach computer science students how to perform fast and reliable multiple precision
-arithmetic. To this end the source code has been given quite a few comments and algorithm discussion points. Often routines have
-more comments than lines of code.
-
-\section{Choice of LibTomMath}
-LibTomMath was chosen as the case study of this text not only because the author of both projects is one and the same but
-for more worthy reasons. Other libraries such as GMP, MPI, LIP and OpenSSL have multiple precision
-integer arithmetic routines but would not be ideal for this text for reasons as will be explained in the
-following sub-sections.
-
-\subsection{Code Base}
-The LibTomMath code base is all portable ISO C source code. This means that there are no platform dependent conditional
-segments of code littered throughout the source. This clean and uncluttered approach to the library means that a
-developer can more readily ascertain the true intent of a given section of source code without trying to keep track of
-what conditional code will be used.
-
-The code base of LibTomMath is also well organized. Each function is in its own separate source code file
-which allows the reader to find a given function very fast. When compiled with GCC for the x86 processor the entire
-library is a mere 87,760 bytes (\textit{$116,182$ bytes for ARMv4 processors}). This includes every single function
-LibTomMath provides from basic arithmetic to various number theoretic functions such as modular exponentiation, various
-reduction algorithms and Jacobi symbol computation.
-
-By comparison MPI which has fewer functions than LibTomMath compiled with the same conditions is 45,429 bytes
-(\textit{$54,536$ for ARMv4}). GMP which has rather large collection of functions with the default configuration on an
-x86 Athlon is 2,950,688 bytes. Note that while LibTomMath has fewer functions than GMP it has been used as the sole basis
-for several public key cryptosystems without having to seek additional outside functions to supplement the library.
-
-\subsection{API Simplicity}
-LibTomMath is designed after the MPI library and shares the API design. Quite often programs that use MPI will build
-with LibTomMath without change. The function names are relatively straight forward as to what they perform. Almost all of the
-functions except for a few minor exceptions which as will be discussed are for good reasons share the same parameter passing
-convention. The learning curve is fairly shallow with the API provided which is an extremely valuable benefit for the
-student and developer alike.
-
-The LIP library is an example of a library with an API that is awkward to work with. LIP uses function names that are often ``compressed'' to
-illegible short hand. LibTomMath does not share this fault.
-
-\subsection{Optimizations}
-While LibTomMath is certainly not the fastest library (\textit{GMP often beats LibTomMath by a factor of two}) it does
-feature a set of optimal algorithms for tasks ranging from modular reduction to squaring. GMP and LIP also feature
-such optimizations while MPI only uses baseline algorithms with no optimizations.
-
-LibTomMath is almost always an order of magnitude faster than the MPI library at computationally expensive tasks such as modular
-exponentiation. In the grand scheme of ``bignum'' libraries LibTomMath is faster than the average library and usually
-slower than the best libraries such as GMP and OpenSSL by a small factor.
-
-\subsection{Portability and Stability}
-LibTomMath will build ``out of the box'' on any platform equipped with a modern version of the GNU C Compiler
-(\textit{GCC}). This means that without changes the library will build without configuration or setting up any
-variables. LIP and MPI will build ``out of the box'' as well but have numerous known bugs. Most notably the author of
-MPI is not working on his library anymore.
-
-GMP requires a configuration script to run and will not build out of the box. GMP and LibTomMath are still in active
-development and are very stable across a variety of platforms.
-
-\subsection{Choice}
-LibTomMath is a relatively compact, well documented, highly optimized and portable library which seems only natural for
-the case study of this text. Various source files from the LibTomMath project will be included within the text. However, the
-reader is encouraged to download their own copy of the library to actually be able to work with the library.
-
-\chapter{Getting Started}
-\section{Library Basics}
-To begin the design of a multiple precision integer library a primitive data type and a series of primitive algorithms must be established. A data
-type that will hold the information required to maintain a multiple precision integer must be designed. With this basic data type of a series
-of low level algorithms for initializing, clearing, growing and optimizing multiple precision integers can be developed to form the basis of
-the entire library of algorithms.
-
-\section{What is a Multiple Precision Integer?}
-Recall that most programming languages (\textit{in particular C}) only have fixed precision data types that on their own cannot be used
-to represent values larger than their precision alone will allow. The purpose of multiple precision algorithms is to use these fixed precision
-data types to create multiple precision integers which may represent values that are much larger.
-
-As a well known analogy, school children are taught how to form numbers larger than nine by prepending more radix ten digits. In the decimal system
-the largest value is only $9$ since the digits may only have values from $0$ to $9$. However, by concatenating digits together larger numbers
-may be represented. Computer based multiple precision arithmetic is essentially the same concept except with a different radix.
-
-What most people probably do not think about explicitly are the various other attributes that describe a multiple precision integer. For example,
-the integer $154_{10}$ has two immediately obvious properties. First, the integer is positive, that is the sign of this particular integer
-is positive as oppose to negative. Second, the integer has three digits in its representation. There is an additional property that the integer
-posesses that does not concern pencil-and-paper arithmetic. The third property is how many digits are allowed for the integer.
-
-The human analogy of this third property is ensuring there is enough space on the paper to right the integer. Computers must maintain a
-strict control on memory usage with respect to the digits of a multiple precision integer. These three properties make up what is known
-as a multiple precision integer or mp\_int for short.
-
-\subsection{The mp\_int structure}
-The mp\_int structure is the ISO C based manifestation of what represents a multiple precision integer. The ISO C standard does not provide for
-any such data type but it does provide for making composite data types known as structures. The following is the structure definition
-used within LibTomMath.
-
-\index{mp\_int}
-\begin{verbatim}
-typedef struct {
- int used, alloc, sign;
- mp_digit *dp;
-} mp_int;
-\end{verbatim}
-
-The mp\_int structure can be broken down as follows.
-
-\begin{enumerate}
-\item The \textbf{used} parameter denotes how many digits of the array \textbf{dp} contain the digits used to represent
-a given integer. The \textbf{used} count must not exceed the \textbf{alloc} count.
-
-\item The array \textbf{dp} holds the digits that represent the given integer. It is padded with $\textbf{alloc} - \textbf{used}$ zero
-digits.
-
-\item The \textbf{alloc} parameter denotes how
-many digits are available in the array to use by functions before it has to increase in size. When the \textbf{used} count
-of a result would exceed the \textbf{alloc} count all of the algorithms will automatically increase the size of the
-array to accommodate the precision of the result.
-
-\item The \textbf{sign} parameter denotes the sign as either zero/positive (\textbf{MP\_ZPOS}) or negative (\textbf{MP\_NEG}).
-\end{enumerate}
-
-\section{Argument Passing}
-A convention of argument passing must be adopted early on in the development of any library. Making the function prototypes
-consistent will help eliminate many headaches in the future as the library grows to significant complexity. In LibTomMath the multiple precision
-integer functions accept parameters from left to right as pointers to mp\_int structures. That means that the source operands are
-placed on the left and the destination on the right. Consider the following examples.
-
-\begin{verbatim}
- mp_mul(&a, &b, &c); /* c = a * b */
- mp_add(&a, &b, &a); /* a = a + b */
- mp_sqr(&a, &b); /* b = a * a */
-\end{verbatim}
-
-The left to right order is a fairly natural way to implement the functions since it lets the developer read aloud the
-functions and make sense of them. For example, the first function would read ``multiply a and b and store in c''.
-
-Certain libraries (\textit{LIP by Lenstra for instance}) accept parameters the other way around. That is the destination
-on the left and arguments on the right. In truth it is entirely a matter of preference. In the case of LibTomMath the
-convention from the MPI library has been adopted.
-
-Another very useful design consideration is whether to allow argument sources to also be a destination. For example, the
-second example (\textit{mp\_add}) adds $a$ to $b$ and stores in $a$. This is an important feature to implement since it
-allows the higher up functions to cut down on the number of variables. However, to implement this feature specific
-care has to be given to ensure the destination is not modified before the source is fully read.
-
-\section{Return Values}
-A well implemented library, no matter what its purpose, should trap as many runtime errors as possible and return them to the
-caller. By catching runtime errors a library can be guaranteed to prevent undefined behaviour. In a multiple precision
-library the only errors that can occur occur are related to inappropriate inputs (\textit{division by zero for instance}) or
-memory allocation errors.
-
-In LibTomMath any function that can cause a runtime error will return an error as an \textbf{int} data type with one of the
-following values.
-
-\index{MP\_OKAY} \index{MP\_VAL} \index{MP\_MEM}
-\begin{center}
-\begin{tabular}{|l|l|}
-\hline \textbf{Value} & \textbf{Meaning} \\
-\hline \textbf{MP\_OKAY} & The function was successful \\
-\hline \textbf{MP\_VAL} & One of the input value(s) was invalid \\
-\hline \textbf{MP\_MEM} & The function ran out of heap memory \\
-\hline
-\end{tabular}
-\end{center}
-
-When an error is detected within a function it should free any memory it allocated and return as soon as possible. The goal
-is to leave the system in the same state the system was when the function was called. Error checking with this style of API is fairly simple.
-
-\begin{verbatim}
- int err;
- if ((err = mp_add(&a, &b, &c)) != MP_OKAY) {
- printf("Error: %d\n", err);
- exit(EXIT_FAILURE);
- }
-\end{verbatim}
-
-The GMP library uses C style \textit{signals} to flag errors which is of questionable use. Not all errors are fatal
-and it was not deemed ideal by the author of LibTomMath to force developers to have signal handlers for such cases.
-
-\section{Initialization and Clearing}
-The logical starting point when actually writing multiple precision integer functions is the initialization and
-clearing of the integers. These two functions will be used by far the most throughout the algorithms whenever
-temporary integers are required.
-
-Given the basic mp\_int structure an initialization routine must first allocate memory to hold the digits of
-the integer. Often it is optimal to allocate a sufficiently large pre-set number of digits even considering
-the initial integer will represent zero. If only a single digit were allocated quite a few re-allocations
-would occur for the majority of inputs. There is a tradeoff between how many default digits to allocate
-and how many re-allocations are tolerable.
-
-If the memory for the digits has been successfully allocated then the rest of the members of the structure must
-be initialized. Since the initial state is to represent a zero integer the digits allocated must all be zeroed. The
-\textbf{used} count set to zero and \textbf{sign} set to \textbf{MP\_ZPOS}.
-
-\subsection{Initializing an mp\_int}
-To initialize an mp\_int the mp\_init algorithm shall be used. The purpose of this algorithm is to allocate
-the memory required and initialize the integer to a default representation of zero.
-
-\begin{figure}[here]
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_init}. \\
-\textbf{Input}. An mp\_int $a$ \\
-\textbf{Output}. Allocate memory for the digits and set to a zero state. \\
-\hline \\
-1. Allocate memory for \textbf{MP\_PREC} digits. \\
-2. If the allocation failed then return(\textit{MP\_MEM}) \\
-3. for $n$ from $0$ to $MP\_PREC - 1$ do \\
-\hspace{3mm}3.1 $a_n \leftarrow 0$\\
-4. $a.sign \leftarrow MP\_ZPOS$\\
-5. $a.used \leftarrow 0$\\
-6. $a.alloc \leftarrow MP\_PREC$\\
-7. Return(\textit{MP\_OKAY})\\
-\hline
-\end{tabular}
-\end{center}
-\caption{Algorithm mp\_init}
-\end{figure}
-
-\textbf{Algorithm mp\_init.}
-The \textbf{MP\_PREC} variable is a simple constant used to dictate minimal precision of allocated integers. It is ideally at least equal to $32$ but
-can be any reasonable power of two. Steps one and two allocate the memory and account for it. If the allocation fails the algorithm returns
-immediately to signal the failure. Step three will ensure that all the digits are in the default state of zero. Finally steps
-four through six set the default settings of the \textbf{sign}, \textbf{used} and \textbf{alloc} members of the mp\_int structure.
-
-\index{bn\_mp\_init.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_init.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* init a new bigint */
-018 int
-019 mp_init (mp_int * a)
-020 \{
-021 /* allocate ram required and clear it */
-022 a->dp = OPT_CAST calloc (sizeof (mp_digit), MP_PREC);
-023 if (a->dp == NULL) \{
-024 return MP_MEM;
-025 \}
-026
-027 /* set the used to zero, allocated digits to the default precision
-028 * and sign to positive */
-029 a->used = 0;
-030 a->alloc = MP_PREC;
-031 a->sign = MP_ZPOS;
-032
-033 return MP_OKAY;
-034 \}
-\end{alltt}
-\end{small}
-
-The \textbf{OPT\_CAST} type cast on line 22 is designed to allow C++ compilers to build the code out of
-the box. Microsoft C V5.00 is known to cause problems without the cast. Also note that if the memory
-allocation fails the other members of the mp\_int will be in an undefined state. The code from
-line 29 to line 31 sets the default state for a mp\_int which is zero, positive and no used digits.
-
-\subsection{Clearing an mp\_int}
-When an mp\_int is no longer required the memory allocated for it can be cleared from the heap with
-the mp\_clear algorithm.
-
-\begin{figure}[here]
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_clear}. \\
-\textbf{Input}. An mp\_int $a$ \\
-\textbf{Output}. The memory for $a$ is cleared. \\
-\hline \\
-1. If $a$ has been previously freed then return(\textit{MP\_OKAY}). \\
-2. Free the digits of $a$ and mark $a$ as freed. \\
-3. $a.used \leftarrow 0$ \\
-4. $a.alloc \leftarrow 0$ \\
-5. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\caption{Algorithm mp\_clear}
-\end{figure}
-
-\textbf{Algorithm mp\_clear.}
-In steps one and two the memory for the digits are only free'd if they had not been previously released before.
-This is more of concern for the implementation since it is used to prevent ``double-free'' errors. It also helps catch
-code errors where mp\_ints are used after being cleared. Similarly steps three and four set the
-\textbf{used} and \textbf{alloc} to known values which would be easy to spot during debugging. For example, if an mp\_int is expected
-to be non-zero and its \textbf{used} member is observed to be zero (\textit{due to being cleared}) then an obvious bug in the code has been
-spotted.
-
-\index{bn\_mp\_clear.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_clear.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* clear one (frees) */
-018 void
-019 mp_clear (mp_int * a)
-020 \{
-021 if (a->dp != NULL) \{
-022
-023 /* first zero the digits */
-024 memset (a->dp, 0, sizeof (mp_digit) * a->used);
-025
-026 /* free ram */
-027 free (a->dp);
-028
-029 /* reset members to make debugging easier */
-030 a->dp = NULL;
-031 a->alloc = a->used = 0;
-032 \}
-033 \}
-\end{alltt}
-\end{small}
-
-The \textbf{if} statement on line 21 prevents the heap from being corrupted if a user double-frees an
-mp\_int. For example, a trivial case of this bug would be as follows.
-
-\begin{verbatim}
-mp_int a;
-mp_init(&a);
-mp_clear(&a);
-mp_clear(&a);
-\end{verbatim}
-
-Without that check the code would try to free the memory allocated for the digits twice which will cause most standard C
-libraries to cause a fault. Also by setting the pointer to \textbf{NULL} it helps debug code that may inadvertently
-free the mp\_int before it is truly not needed. The allocated digits are set to zero before being freed on line 24.
-This is ideal for cryptographic situations where the mp\_int is a secret parameter.
-
-The following snippet is an example of using both the init and clear functions.
-
-\begin{small}
-\begin{verbatim}
-#include <tommath.h>
-#include <stdio.h>
-#include <stdlib.h>
-int main(void)
-{
- mp_int num;
- int err;
-
- /* init the bignum */
- if ((err = mp_init(&num)) != MP_OKAY) {
- printf("Error: %d\n", err);
- return EXIT_FAILURE;
- }
-
- /* do work with it ... */
-
- /* clear up */
- mp_clear(&num);
-
- return EXIT_SUCCESS;
-}
-\end{verbatim}
-\end{small}
-
-\section{Other Initialization Routines}
-
-It is often helpful to have specialized initialization algorithms to simplify the design of other algorithms. For example, an
-initialization followed by a copy is a common operation when temporary copies of integers are required. It is quite
-beneficial to have a series of simple helper functions available.
-
-\subsection{Initializing Variable Sized mp\_int Structures}
-Occasionally the number of digits required will be known in advance of an initialization. In these
-cases the mp\_init\_size algorithm can be of use. The purpose of this algorithm is similar to mp\_init except that
-it will allocate \textit{at least} a specified number of digits. This is ideal to prevent re-allocations when the
-input size is known.
-
-\newpage\begin{figure}[here]
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_init\_size}. \\
-\textbf{Input}. An mp\_int $a$ and the requested number of digits $b$\\
-\textbf{Output}. $a$ is initialized to hold at least $b$ digits. \\
-\hline \\
-1. $u \leftarrow b\mbox{ (mod }MP\_PREC\mbox{)}$ \\
-2. $v \leftarrow b + 2 \cdot MP\_PREC - u$ \\
-3. Allocate $v$ digits. \\
-4. If the allocation failed then return(\textit{MP\_MEM}). \\
-5. for $n$ from $0$ to $v - 1$ do \\
-\hspace{3mm}5.1 $a_n \leftarrow 0$ \\
-6. $a.sign \leftarrow MP\_ZPOS$\\
-7. $a.used \leftarrow 0$\\
-8. $a.alloc \leftarrow v$\\
-9. Return(\textit{MP\_OKAY})\\
-\hline
-\end{tabular}
-\end{center}
-\caption{Algorithm mp\_init\_size}
-\end{figure}
-
-\textbf{Algorithm mp\_init\_size.}
-The value of $v$ is calculated to be at least the requested amount of digits $b$ plus additional padding. The padding is calculated
-to be at least \textbf{MP\_PREC} digits plus enough digits to make the digit count a multiple of \textbf{MP\_PREC}. This padding is used to
-prevent trivial allocations from becoming a bottleneck in the rest of the algorithms that depend on this.
-
-\index{bn\_mp\_init\_size.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_init\_size.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* init a mp_init and grow it to a given size */
-018 int
-019 mp_init_size (mp_int * a, int size)
-020 \{
-021
-022 /* pad size so there are always extra digits */
-023 size += (MP_PREC * 2) - (size & (MP_PREC - 1));
-024
-025 /* alloc mem */
-026 a->dp = OPT_CAST calloc (sizeof (mp_digit), size);
-027 if (a->dp == NULL) \{
-028 return MP_MEM;
-029 \}
-030 a->used = 0;
-031 a->alloc = size;
-032 a->sign = MP_ZPOS;
-033
-034 return MP_OKAY;
-035 \}
-\end{alltt}
-\end{small}
-
-Line 23 will ensure that the number of digits actually allocated is padded up to the next multiple of
-\textbf{MP\_PREC} plus an additional \textbf{MP\_PREC}. This ensures that the number of allocated digit is
-always greater than the amount requested. As a result it prevents many trivial memory allocations. The value of
-\textbf{MP\_PREC} is defined in ``tommath.h'' and must be a power of two.
-
-\subsection{Creating a Clone}
-Another common sequence of operations is to make a local temporary copy of an argument. To initialize then copy a mp\_int will be known as
-creating a clone. This is useful within functions that need to modify an integer argument but do not wish to actually modify the original copy.
-The mp\_init\_copy algorithm will perform this very task.
-
-\begin{figure}[here]
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_init\_copy}. \\
-\textbf{Input}. An mp\_int $a$ and $b$\\
-\textbf{Output}. $a$ is initialized to be a copy of $b$. \\
-\hline \\
-1. Init $a$. (\textit{mp\_init}) \\
-2. If the init of $a$ was unsuccessful return(\textit{MP\_MEM}) \\
-3. Copy $b$ to $a$. (\textit{mp\_copy}) \\
-4. Return the status of the copy operation. \\
-\hline
-\end{tabular}
-\end{center}
-\caption{Algorithm mp\_init\_copy}
-\end{figure}
-
-\textbf{Algorithm mp\_init\_copy.}
-This algorithm will initialize a mp\_int variable and copy another previously initialized mp\_int variable into it. The algorithm will
-detect when the initialization fails and returns the error to the calling algorithm. As such this algorithm will perform two operations
-in one step.
-
-\index{bn\_mp\_init\_copy.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_init\_copy.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* creates "a" then copies b into it */
-018 int
-019 mp_init_copy (mp_int * a, mp_int * b)
-020 \{
-021 int res;
-022
-023 if ((res = mp_init (a)) != MP_OKAY) \{
-024 return res;
-025 \}
-026 return mp_copy (b, a);
-027 \}
-\end{alltt}
-\end{small}
-
-This will initialize \textbf{a} and make it a verbatim copy of the contents of \textbf{b}. Note that
-\textbf{a} will have its own memory allocated which means that \textbf{b} may be cleared after the call
-and \textbf{a} will be left intact.
-
-\subsection{Multiple Integer Initializations And Clearings}
-Occasionally a function will require a series of mp\_int data types to be made available. The mp\_init\_multi algorithm
-is provided to simplify such cases. The purpose of this algorithm is to initialize a variable length array of mp\_int
-structures at once. As a result algorithms that require multiple integers only has to use
-one algorithm to initialize all the mp\_int variables.
-
-\begin{figure}[here]
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_init\_multi}. \\
-\textbf{Input}. Variable length array of mp\_int variables of length $k$. \\
-\textbf{Output}. The array is initialized such that each each mp\_int is ready to use. \\
-\hline \\
-1. for $n$ from 0 to $k - 1$ do \\
-\hspace{+3mm}1.1. Initialize the $n$'th mp\_int (\textit{mp\_init}) \\
-\hspace{+3mm}1.2. If initialization failed then do \\
-\hspace{+6mm}1.2.1. for $j$ from $0$ to $n$ do \\
-\hspace{+9mm}1.2.1.1. Free the $j$'th mp\_int (\textit{mp\_clear}) \\
-\hspace{+6mm}1.2.2. Return(\textit{MP\_MEM}) \\
-2. Return(\textit{MP\_OKAY}) \\
-\hline
-\end{tabular}
-\end{center}
-\caption{Algorithm mp\_init\_multi}
-\end{figure}
-
-\textbf{Algorithm mp\_init\_multi.}
-The algorithm will initialize the array of mp\_int variables one at a time. As soon as an runtime error is detected (\textit{step 1.2}) all of
-the previously initialized variables are cleared. The goal is an ``all or nothing'' initialization which allows for quick recovery from runtime
-errors.
-
-Similarly to clear a variable length array of mp\_int structures the mp\_clear\_multi algorithm will be used.
-
-Consider the following snippet which demonstrates how to use both routines.
-\begin{small}
-\begin{verbatim}
-#include <tommath.h>
-#include <stdio.h>
-#include <stdlib.h>
-int main(void)
-{
- mp_int num1, num2, num3;
- int err;
-
- if ((err = mp_init_multi(&num1, &num2, &num3, NULL)) !- MP_OKAY) {
- printf("Error: %d\n", err);
- return EXIT_FAILURE;
- }
-
- /* at this point num1/num2/num3 are ready */
-
- /* free them */
- mp_clear_multi(&num1, &num2, &num3, NULL);
-
- return EXIT_SUCCESS;
-}
-\end{verbatim}
-\end{small}
-
-Note how both lists are terminated with the \textbf{NULL} variable. This indicates to the algorithms to stop fetching parameters off
-of the stack. If it is not present the functions will most likely cause a segmentation fault.
-
-\index{bn\_mp\_multi.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_multi.c
-\vspace{-3mm}
-\begin{alltt}
-016 #include <stdarg.h>
-017
-018 int mp_init_multi(mp_int *mp, ...)
-019 \{
-020 mp_err res = MP_OKAY; /* Assume ok until proven otherwise */
-021 int n = 0; /* Number of ok inits */
-022 mp_int* cur_arg = mp;
-023 va_list args;
-024
-025 va_start(args, mp); /* init args to next argument from caller */
-026 while (cur_arg != NULL) \{
-027 if (mp_init(cur_arg) != MP_OKAY) \{
-028 /* Oops - error! Back-track and mp_clear what we already
-029 succeeded in init-ing, then return error.
-030 */
-031 va_list clean_args;
-032
-033 /* end the current list */
-034 va_end(args);
-035
-036 /* now start cleaning up */
-037 cur_arg = mp;
-038 va_start(clean_args, mp);
-039 while (n--) \{
-040 mp_clear(cur_arg);
-041 cur_arg = va_arg(clean_args, mp_int*);
-042 \}
-043 va_end(clean_args);
-044 res = MP_MEM;
-045 break;
-046 \}
-047 n++;
-048 cur_arg = va_arg(args, mp_int*);
-049 \}
-050 va_end(args);
-051 return res; /* Assumed ok, if error flagged above. */
-052 \}
-053
-054 void mp_clear_multi(mp_int *mp, ...)
-055 \{
-056 mp_int* next_mp = mp;
-057 va_list args;
-058 va_start(args, mp);
-059 while (next_mp != NULL) \{
-060 mp_clear(next_mp);
-061 next_mp = va_arg(args, mp_int*);
-062 \}
-063 va_end(args);
-064 \}
-\end{alltt}
-\end{small}
-
-Both routines are implemented in the same source file since they are typically used in conjunction with each other.
-
-\section{Maintenance}
-A small useful collection of mp\_int maintenance functions will also prove useful.
-
-\subsection{Augmenting Integer Precision}
-When storing a value in an mp\_int sufficient digits must be available to accomodate the entire value without
-loss of precision. Quite often the size of the array given by the \textbf{alloc} member is large enough to simply
-increase the \textbf{used} digit count. However, when the size of the array is too small it must be re-sized
-appropriately to accomodate the result. The mp\_grow algorithm will provide this functionality.
-
-\begin{figure}[here]
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_grow}. \\
-\textbf{Input}. An mp\_int $a$ and an integer $b$. \\
-\textbf{Output}. $a$ is expanded to accomodate $b$ digits. \\
-\hline \\
-1. if $a.alloc \ge b$ then return(\textit{MP\_OKAY}) \\
-2. $u \leftarrow b\mbox{ (mod }MP\_PREC\mbox{)}$ \\
-3. $v \leftarrow b + 2 \cdot MP\_PREC - u$ \\
-4. Re-Allocate the array of digits $a$ to size $v$ \\
-5. If the allocation failed then return(\textit{MP\_MEM}). \\
-6. for n from a.alloc to $v - 1$ do \\
-\hspace{+3mm}6.1 $a_n \leftarrow 0$ \\
-7. $a.alloc \leftarrow v$ \\
-8. Return(\textit{MP\_OKAY}) \\
-\hline
-\end{tabular}
-\end{center}
-\caption{Algorithm mp\_grow}
-\end{figure}
-
-\textbf{Algorithm mp\_grow.}
-Step one will prevent a re-allocation from being performed if it was not required. This is useful to prevent mp\_ints
-from growing excessively in code that erroneously calls mp\_grow. Similar to mp\_init\_size the requested digit count
-is padded to provide more digits than requested.
-
-In step four it is assumed that the reallocation leaves the lower $a.alloc$ digits intact. This is much akin to how the
-\textit{realloc} function from the standard C library works. Since the newly allocated digits are assumed to contain
-undefined values they are also initially zeroed.
-
-\index{bn\_mp\_grow.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_grow.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* grow as required */
-018 int
-019 mp_grow (mp_int * a, int size)
-020 \{
-021 int i;
-022
-023 /* if the alloc size is smaller alloc more ram */
-024 if (a->alloc < size) \{
-025 /* ensure there are always at least MP_PREC digits extra on top */
-026 size += (MP_PREC * 2) - (size & (MP_PREC - 1));
-027
-028 a->dp = OPT_CAST realloc (a->dp, sizeof (mp_digit) * size);
-029 if (a->dp == NULL) \{
-030 return MP_MEM;
-031 \}
-032
-033 /* zero excess digits */
-034 i = a->alloc;
-035 a->alloc = size;
-036 for (; i < a->alloc; i++) \{
-037 a->dp[i] = 0;
-038 \}
-039 \}
-040 return MP_OKAY;
-041 \}
-\end{alltt}
-\end{small}
-
-The first step is to see if we actually need to perform a re-allocation at all. This is tested for on line
-24. Similar to mp\_init\_size the same code on line 26 was used to resize the
-digits requested. A simple for loop from line 34 to line 38 will zero all digits that were above the
-old \textbf{alloc} limit to make sure the integer is in a known state.
-
-\subsection{Clamping Excess Digits}
-When a function anticipates a result will be $n$ digits it is simpler to assume this is true within the body of
-the function. For example, a multiplication of a $i$ digit number by a $j$ digit produces a result of at most
-$i + j$ digits. It is entirely possible that the result is $i + j - 1$ though, with no final carry into the last
-position. However, suppose the destination had to be first expanded (\textit{via mp\_grow}) to accomodate $i + j - 1$
-digits than further expanded to accomodate the final carry. That would be a considerable waste of time since heap
-operations are relatively slow.
-
-The ideal solution is to always assume the result is $i + j$ and fix up the \textbf{used} count after the function
-terminates. This way a single heap operation (\textit{at most}) is required. However, if the result was not checked
-there would be an excess high order zero digit.
-
-For example, suppose the product of two integers was $x_n = (0x_{n-1}x_{n-2}...x_0)_{\beta}$. The leading zero digit
-will not contribute to the precision of the result. In fact, through subsequent operations more leading zero digits would
-accumulate to the point the size of the integer would be prohibitive. As a result even though the precision is very
-low the representation is excessively large.
-
-The mp\_clamp algorithm is designed to solve this very problem. It will trim leading zeros by decrementing the
-\textbf{used} count until a non-zero leading digit is found. Also in this system, zero is considered to be a positive
-number which means that if the \textbf{used} count is decremented to zero the sign must be set to \textbf{MP\_ZPOS}.
-
-\begin{figure}[here]
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_clamp}. \\
-\textbf{Input}. An mp\_int $a$ \\
-\textbf{Output}. Any excess leading zero digits of $a$ are removed \\
-\hline \\
-1. while $a.used > 0$ and $a_{a.used - 1} = 0$ do \\
-\hspace{+3mm}1.1 $a.used \leftarrow a.used - 1$ \\
-2. if $a.used = 0$ then do \\
-\hspace{+3mm}2.1 $a.sign \leftarrow MP\_ZPOS$ \\
-\hline \\
-\end{tabular}
-\end{center}
-\caption{Algorithm mp\_clamp}
-\end{figure}
-
-\textbf{Algorithm mp\_clamp.}
-As can be expected this algorithm is very simple. The loop on step one is expected to iterate only once or twice at
-the most. For example, this will happen in cases where there is not a carry to fill the last position. Step two fixes the sign for
-when all of the digits are zero to ensure that the mp\_int is valid at all times.
-
-\index{bn\_mp\_clamp.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_clamp.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* trim unused digits
-018 *
-019 * This is used to ensure that leading zero digits are
-020 * trimed and the leading "used" digit will be non-zero
-021 * Typically very fast. Also fixes the sign if there
-022 * are no more leading digits
-023 */
-024 void
-025 mp_clamp (mp_int * a)
-026 \{
-027 while (a->used > 0 && a->dp[a->used - 1] == 0) \{
-028 --(a->used);
-029 \}
-030 if (a->used == 0) \{
-031 a->sign = MP_ZPOS;
-032 \}
-033 \}
-\end{alltt}
-\end{small}
-
-Note on line 27 how to test for the \textbf{used} count is made on the left of the \&\& operator. In the C programming
-language the terms to \&\& are evaluated left to right with a boolean short-circuit if any condition fails. This is
-important since if the \textbf{used} is zero the test on the right would fetch below the array. That is obviously
-undesirable. The parenthesis on line 28 is used to make sure the \textbf{used} count is decremented and not
-the pointer ``a''.
-
-\section*{Exercises}
-\begin{tabular}{cl}
-$\left [ 1 \right ]$ & Discuss the relevance of the \textbf{used} member of the mp\_int structure. \\
- & \\
-$\left [ 1 \right ]$ & Discuss the consequences of not using padding when performing allocations. \\
- & \\
-$\left [ 2 \right ]$ & Estimate an ideal value for \textbf{MP\_PREC} when performing 1024-bit RSA \\
- & encryption when $\beta = 2^{28}$. \\
- & \\
-$\left [ 1 \right ]$ & Discuss the relevance of the algorithm mp\_clamp. What does it prevent? \\
- & \\
-$\left [ 1 \right ]$ & Give an example of when the algorithm mp\_init\_copy might be useful. \\
- & \\
-\end{tabular}
-
-
-\chapter{Basic Operations}
-\section{Copying an Integer}
-After the various house-keeping routines are in place, simple algorithms can be designed to take advantage of them. Being able
-to make a verbatim copy of an integer is a very useful function to have. To copy an integer the mp\_copy algorithm will be used.
-
-\newpage\begin{figure}[here]
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_copy}. \\
-\textbf{Input}. An mp\_int $a$ and $b$. \\
-\textbf{Output}. Store a copy of $a$ in $b$. \\
-\hline \\
-1. Check if $a$ and $b$ point to the same location in memory. \\
-2. If true then return(\textit{MP\_OKAY}). \\
-3. If $b.alloc < a.used$ then grow $b$ to $a.used$ digits. (\textit{mp\_grow}) \\
-4. If failed to grow then return(\textit{MP\_MEM}). \\
-5. for $n$ from 0 to $a.used - 1$ do \\
-\hspace{3mm}5.1 $b_{n} \leftarrow a_{n}$ \\
-6. if $a.used < b.used - 1$ then \\
-\hspace{3mm}6.1. for $n$ from $a.used$ to $b.used - 1$ do \\
-\hspace{6mm}6.1.1 $b_{n} \leftarrow 0$ \\
-7. $b.used \leftarrow a.used$ \\
-8. $b.sign \leftarrow a.sign$ \\
-9. return(\textit{MP\_OKAY}) \\
-\hline
-\end{tabular}
-\end{center}
-\caption{Algorithm mp\_copy}
-\end{figure}
-
-\textbf{Algorithm mp\_copy.}
-Step 1 and 2 make sure that the two mp\_ints are unique. This allows the user to call the copy function with
-potentially the same input and not waste time. Step 3 and 4 ensure that the destination is large enough to
-hold a copy of the input $a$. Note that the \textbf{used} member of $b$ may be smaller than the \textbf{used}
-member of $a$ but a memory re-allocation is only required if the \textbf{alloc} member of $b$ is smaller. This
-prevents trivial memory reallocations.
-
-Step 5 copies the digits from $a$ to $b$ while step 6 ensures that if initially $\vert b \vert > \vert a \vert$,
-the more significant digits of $b$ will be zeroed. Finally steps 7 and 8 copies the \textbf{used} and \textbf{sign} members over
-which completes the copy operation.
-
-\index{bn\_mp\_copy.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_copy.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* copy, b = a */
-018 int
-019 mp_copy (mp_int * a, mp_int * b)
-020 \{
-021 int res, n;
-022
-023 /* if dst == src do nothing */
-024 if (a == b) \{
-025 return MP_OKAY;
-026 \}
-027
-028 /* grow dest */
-029 if ((res = mp_grow (b, a->used)) != MP_OKAY) \{
-030 return res;
-031 \}
-032
-033 /* zero b and copy the parameters over */
-034 \{
-035 register mp_digit *tmpa, *tmpb;
-036
-037 /* pointer aliases */
-038 tmpa = a->dp;
-039 tmpb = b->dp;
-040
-041 /* copy all the digits */
-042 for (n = 0; n < a->used; n++) \{
-043 *tmpb++ = *tmpa++;
-044 \}
-045
-046 /* clear high digits */
-047 for (; n < b->used; n++) \{
-048 *tmpb++ = 0;
-049 \}
-050 \}
-051 b->used = a->used;
-052 b->sign = a->sign;
-053 return MP_OKAY;
-054 \}
-\end{alltt}
-\end{small}
-
-Source lines 23-31 do the initial house keeping. That is to see if the input is unique and if so to
-make sure there is enough room. If not enough space is available it returns the error and leaves the destination variable
-intact.
-
-The inner loop of the copy operation is contained between lines 34 and 50. Many LibTomMath routines are designed with this source code style
-in mind, making aliases to shorten lengthy pointers (\textit{see line 38 and 39}) for rapid use. Also the
-use of nested braces creates a simple way to denote various portions of code that reside on various work levels. Here, the copy loop is at the
-$O(n)$ level.
-
-\section{Zeroing an Integer}
-Reseting an mp\_int to the default state is a common step in many algorithms. The mp\_zero algorithm will be the algorithm used to
-perform this task.
-
-\begin{figure}[here]
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_zero}. \\
-\textbf{Input}. An mp\_int $a$ \\
-\textbf{Output}. Zero the contents of $a$ \\
-\hline \\
-1. $a.used \leftarrow 0$ \\
-2. $a.sign \leftarrow$ MP\_ZPOS \\
-3. for $n$ from 0 to $a.alloc - 1$ do \\
-\hspace{3mm}3.1 $a_n \leftarrow 0$ \\
-\hline
-\end{tabular}
-\end{center}
-\caption{Algorithm mp\_zero}
-\end{figure}
-
-\textbf{Algorithm mp\_zero.}
-This algorithm simply resets a mp\_int to the default state.
-
-\index{bn\_mp\_zero.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_zero.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* set to zero */
-018 void
-019 mp_zero (mp_int * a)
-020 \{
-021 a->sign = MP_ZPOS;
-022 a->used = 0;
-023 memset (a->dp, 0, sizeof (mp_digit) * a->alloc);
-024 \}
-\end{alltt}
-\end{small}
-
-After the function is completed, all of the digits are zeroed, the \textbf{used} count is zeroed and the
-\textbf{sign} variable is set to \textbf{MP\_ZPOS}.
-
-\section{Sign Manipulation}
-\subsection{Absolute Value}
-With the mp\_int representation of an integer, calculating the absolute value is trivial. The mp\_abs algorithm will compute
-the absolute value of an mp\_int.
-
-\begin{figure}[here]
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_abs}. \\
-\textbf{Input}. An mp\_int $a$ \\
-\textbf{Output}. Computes $b = \vert a \vert$ \\
-\hline \\
-1. Copy $a$ to $b$. (\textit{mp\_copy}) \\
-2. If the copy failed return(\textit{MP\_MEM}). \\
-3. $b.sign \leftarrow MP\_ZPOS$ \\
-4. Return(\textit{MP\_OKAY}) \\
-\hline
-\end{tabular}
-\end{center}
-\caption{Algorithm mp\_abs}
-\end{figure}
-
-\textbf{Algorithm mp\_abs.}
-This algorithm computes the absolute of an mp\_int input. As can be expected the algorithm is very trivial.
-
-\index{bn\_mp\_abs.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_abs.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* b = |a|
-018 *
-019 * Simple function copies the input and fixes the sign to positive
-020 */
-021 int
-022 mp_abs (mp_int * a, mp_int * b)
-023 \{
-024 int res;
-025 if ((res = mp_copy (a, b)) != MP_OKAY) \{
-026 return res;
-027 \}
-028 b->sign = MP_ZPOS;
-029 return MP_OKAY;
-030 \}
-\end{alltt}
-\end{small}
-
-\subsection{Integer Negation}
-With the mp\_int representation of an integer, calculating the negation is also trivial. The mp\_neg algorithm will compute
-the negative of an mp\_int input.
-
-\newpage\begin{figure}[here]
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_neg}. \\
-\textbf{Input}. An mp\_int $a$ \\
-\textbf{Output}. Computes $b = -a$ \\
-\hline \\
-1. Copy $a$ to $b$. (\textit{mp\_copy}) \\
-2. If the copy failed return(\textit{MP\_MEM}). \\
-3. If $a.sign = MP\_ZPOS$ then do \\
-\hspace{3mm}3.1 $b.sign = MP\_NEG$. \\
-4. else do \\
-\hspace{3mm}4.1 $b.sign = MP\_ZPOS$. \\
-5. Return(\textit{MP\_OKAY}) \\
-\hline
-\end{tabular}
-\end{center}
-\caption{Algorithm mp\_neg}
-\end{figure}
-
-\textbf{Algorithm mp\_neg.}
-This algorithm computes the negation of an input.
-
-\index{bn\_mp\_neg.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_neg.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* b = -a */
-018 int
-019 mp_neg (mp_int * a, mp_int * b)
-020 \{
-021 int res;
-022 if ((res = mp_copy (a, b)) != MP_OKAY) \{
-023 return res;
-024 \}
-025 b->sign = (a->sign == MP_ZPOS) ? MP_NEG : MP_ZPOS;
-026 return MP_OKAY;
-027 \}
-\end{alltt}
-\end{small}
-
-\section{Small Constants}
-\subsection{Setting Small Constants}
-Often a mp\_int must be set to a relatively small value such as $1$ or $2$. For these cases the mp\_set algorithm is useful.
-
-\newpage\begin{figure}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_set}. \\
-\textbf{Input}. An mp\_int $a$ and a digit $b$ \\
-\textbf{Output}. Make $a$ equivalent to $b$ \\
-\hline \\
-1. Zero $a$ (\textit{mp\_zero}). \\
-2. $a_0 \leftarrow b \mbox{ (mod }\beta\mbox{)}$ \\
-3. $a.used \leftarrow \left \lbrace \begin{array}{ll}
- 1 & \mbox{if }a_0 > 0 \\
- 0 & \mbox{if }a_0 = 0
- \end{array} \right .$ \\
-\hline
-\end{tabular}
-\end{center}
-\caption{Algorithm mp\_set}
-\end{figure}
-
-\textbf{Algorithm mp\_set.}
-This algorithm sets a mp\_int to a small single digit value. Step number 1 ensures that the integer is reset to the default state. The
-single digit is set (\textit{modulo $\beta$}) and the \textbf{used} count is adjusted accordingly.
-
-\index{bn\_mp\_set.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_set.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* set to a digit */
-018 void
-019 mp_set (mp_int * a, mp_digit b)
-020 \{
-021 mp_zero (a);
-022 a->dp[0] = b & MP_MASK;
-023 a->used = (a->dp[0] != 0) ? 1 : 0;
-024 \}
-\end{alltt}
-\end{small}
-
-Line 21 calls mp\_zero() to clear the mp\_int and reset the sign. Line 22 copies the digit
-into the least significant location. Note the usage of a new constant \textbf{MP\_MASK}. This constant is used to quickly
-reduce an integer modulo $\beta$. Since $\beta$ is of the form $2^k$ for any suitable $k$ it suffices to perform a binary AND with
-$MP\_MASK = 2^k - 1$ to perform the reduction. Finally line 23 will set the \textbf{used} member with respect to the
-digit actually set. This function will always make the integer positive.
-
-One important limitation of this function is that it will only set one digit. The size of a digit is not fixed, meaning source that uses
-this function should take that into account. Meaning that only trivially small constants can be set using this function.
-
-\subsection{Setting Large Constants}
-To overcome the limitations of the mp\_set algorithm the mp\_set\_int algorithm is provided. It accepts a ``long''
-data type as input and will always treat it as a 32-bit integer.
-
-\begin{figure}[here]
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_set\_int}. \\
-\textbf{Input}. An mp\_int $a$ and a ``long'' integer $b$ \\
-\textbf{Output}. Make $a$ equivalent to $b$ \\
-\hline \\
-1. Zero $a$ (\textit{mp\_zero}) \\
-2. for $n$ from 0 to 7 do \\
-\hspace{3mm}2.1 $a \leftarrow a \cdot 16$ (\textit{mp\_mul2d}) \\
-\hspace{3mm}2.2 $u \leftarrow \lfloor b / 2^{4(7 - n)} \rfloor \mbox{ (mod }16\mbox{)}$\\
-\hspace{3mm}2.3 $a_0 \leftarrow a_0 + u$ \\
-\hspace{3mm}2.4 $a.used \leftarrow a.used + 1$ \\
-3. Clamp excess used digits (\textit{mp\_clamp}) \\
-\hline
-\end{tabular}
-\end{center}
-\caption{Algorithm mp\_set\_int}
-\end{figure}
-
-\textbf{Algorithm mp\_set\_int.}
-The algorithm performs eight iterations of a simple loop where in each iteration four bits from the source are added to the
-mp\_int. Step 2.1 will multiply the current result by sixteen making room for four more bits in the less significant positions. In step 2.2 the
-next four bits from the source are extracted and are added to the mp\_int. The \textbf{used} digit count is
-incremented to reflect the addition. The \textbf{used} digit counter is incremented since if any of the leading digits were zero the mp\_int would have
-zero digits used and the newly added four bits would be ignored.
-
-Excess zero digits are trimmed in steps 2.1 and 3 by using higher level algorithms mp\_mul2d and mp\_clamp.
-
-\index{bn\_mp\_set\_int.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_set\_int.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* set a 32-bit const */
-018 int
-019 mp_set_int (mp_int * a, unsigned int b)
-020 \{
-021 int x, res;
-022
-023 mp_zero (a);
-024 /* set four bits at a time */
-025 for (x = 0; x < 8; x++) \{
-026 /* shift the number up four bits */
-027 if ((res = mp_mul_2d (a, 4, a)) != MP_OKAY) \{
-028 return res;
-029 \}
-030
-031 /* OR in the top four bits of the source */
-032 a->dp[0] |= (b >> 28) & 15;
-033
-034 /* shift the source up to the next four bits */
-035 b <<= 4;
-036
-037 /* ensure that digits are not clamped off */
-038 a->used += 1;
-039 \}
-040 mp_clamp (a);
-041 return MP_OKAY;
-042 \}
-\end{alltt}
-\end{small}
-
-This function sets four bits of the number at a time to handle all practical \textbf{DIGIT\_BIT} sizes. The weird
-addition on line 38 ensures that the newly added in bits are added to the number of digits. While it may not
-seem obvious as to why the digit counter does not grow exceedingly large it is because of the shift on line 27
-as well as the call to mp\_clamp() on line 40. Both functions will clamp excess leading digits which keeps
-the number of used digits low.
-
-\section{Comparisons}
-\subsection{Unsigned Comparisions}
-Comparing a multiple precision integer is performed with the exact same algorithm used to compare two decimal numbers. For example,
-to compare $1,234$ to $1,264$ the digits are extracted by their positions. That is we compare $1 \cdot 10^3 + 2 \cdot 10^2 + 3 \cdot 10^1 + 4 \cdot 10^0$
-to $1 \cdot 10^3 + 2 \cdot 10^2 + 6 \cdot 10^1 + 4 \cdot 10^0$ by comparing single digits at a time starting with the highest magnitude
-positions. If any leading digit of one integer is greater than a digit in the same position of another integer then obviously it must be greater.
-
-The first comparision routine that will be developed is the unsigned magnitude compare which will perform a comparison based on the digits of two
-mp\_int variables alone. It will ignore the sign of the two inputs. Such a function is useful when an absolute comparison is required or if the
-signs are known to agree in advance.
-
-To facilitate working with the results of the comparison functions three constants are required.
-
-\begin{figure}[here]
-\begin{center}
-\begin{tabular}{|r|l|}
-\hline \textbf{Constant} & \textbf{Meaning} \\
-\hline \textbf{MP\_GT} & Greater Than \\
-\hline \textbf{MP\_EQ} & Equal To \\
-\hline \textbf{MP\_LT} & Less Than \\
-\hline
-\end{tabular}
-\end{center}
-\caption{Comparison Return Codes}
-\end{figure}
-
-\begin{figure}[here]
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_cmp\_mag}. \\
-\textbf{Input}. Two mp\_ints $a$ and $b$. \\
-\textbf{Output}. Unsigned comparison results ($a$ to the left of $b$). \\
-\hline \\
-1. If $a.used > b.used$ then return(\textit{MP\_GT}) \\
-2. If $a.used < b.used$ then return(\textit{MP\_LT}) \\
-3. for n from $a.used - 1$ to 0 do \\
-\hspace{+3mm}3.1 if $a_n > b_n$ then return(\textit{MP\_GT}) \\
-\hspace{+3mm}3.2 if $a_n < b_n$ then return(\textit{MP\_LT}) \\
-4. Return(\textit{MP\_EQ}) \\
-\hline
-\end{tabular}
-\end{center}
-\caption{Algorithm mp\_cmp\_mag}
-\end{figure}
-
-\textbf{Algorithm mp\_cmp\_mag.}
-By saying ``$a$ to the left of $b$'' it is meant that the comparison is with respect to $a$, that is if $a$ is greater than $b$ it will return
-\textbf{MP\_GT} and similar with respect to when $a = b$ and $a < b$. The first two steps compare the number of digits used in both $a$ and $b$.
-Obviously if the digit counts differ there would be an imaginary zero digit in the smaller number where the leading digit of the larger number is.
-If both have the same number of digits than the actual digits themselves must be compared starting at the leading digit.
-
-By step three both inputs must have the same number of digits so its safe to start from either $a.used - 1$ or $b.used - 1$ and count down to
-the zero'th digit. If after all of the digits have been compared, no difference is found, the algorithm returns \textbf{MP\_EQ}.
-
-\index{bn\_mp\_cmp\_mag.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_cmp\_mag.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* compare maginitude of two ints (unsigned) */
-018 int
-019 mp_cmp_mag (mp_int * a, mp_int * b)
-020 \{
-021 int n;
-022
-023 /* compare based on # of non-zero digits */
-024 if (a->used > b->used) \{
-025 return MP_GT;
-026 \}
-027
-028 if (a->used < b->used) \{
-029 return MP_LT;
-030 \}
-031
-032 /* compare based on digits */
-033 for (n = a->used - 1; n >= 0; n--) \{
-034 if (a->dp[n] > b->dp[n]) \{
-035 return MP_GT;
-036 \}
-037
-038 if (a->dp[n] < b->dp[n]) \{
-039 return MP_LT;
-040 \}
-041 \}
-042 return MP_EQ;
-043 \}
-\end{alltt}
-\end{small}
-
-The two if statements on lines 24 and 28 compare the number of digits in the two inputs. These two are performed before all of the digits
-are compared since it is a very cheap test to perform and can potentially save considerable time. The implementation given is also not valid
-without those two statements. $b.alloc$ may be smaller than $a.used$, meaning that undefined values will be read from $b$ past the end of the
-array of digits.
-
-\subsection{Signed Comparisons}
-Comparing with sign considerations is also fairly critical in several routines (\textit{division for example}). Based on an unsigned magnitude
-comparison a trivial signed comparison algorithm can be written.
-
-\begin{figure}[here]
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_cmp}. \\
-\textbf{Input}. Two mp\_ints $a$ and $b$ \\
-\textbf{Output}. Signed Comparison Results ($a$ to the left of $b$) \\
-\hline \\
-1. if $a.sign = MP\_NEG$ and $b.sign = MP\_ZPOS$ then return(\textit{MP\_LT}) \\
-2. if $a.sign = MP\_ZPOS$ and $b.sign = MP\_NEG$ then return(\textit{MP\_GT}) \\
-3. if $a.sign = MP\_NEG$ then \\
-\hspace{+3mm}3.1 Return the unsigned comparison of $b$ and $a$ (\textit{mp\_cmp\_mag}) \\
-4 Otherwise \\
-\hspace{+3mm}4.1 Return the unsigned comparison of $a$ and $b$ \\
-\hline
-\end{tabular}
-\end{center}
-\caption{Algorithm mp\_cmp}
-\end{figure}
-
-\textbf{Algorithm mp\_cmp.}
-The first two steps compare the signs of the two inputs. If the signs do not agree then it can return right away with the appropriate
-comparison code. When the signs are equal the digits of the inputs must be compared to determine the correct result. In step
-three the unsigned comparision flips the order of the arguments since they are both negative. For instance, if $-a > -b$ then
-$\vert a \vert < \vert b \vert$. Step number four will compare the two when they are both positive.
-
-\index{bn\_mp\_cmp.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_cmp.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* compare two ints (signed)*/
-018 int
-019 mp_cmp (mp_int * a, mp_int * b)
-020 \{
-021 /* compare based on sign */
-022 if (a->sign == MP_NEG && b->sign == MP_ZPOS) \{
-023 return MP_LT;
-024 \}
-025
-026 if (a->sign == MP_ZPOS && b->sign == MP_NEG) \{
-027 return MP_GT;
-028 \}
-029
-030 /* compare digits */
-031 if (a->sign == MP_NEG) \{
-032 /* if negative compare opposite direction */
-033 return mp_cmp_mag(b, a);
-034 \} else \{
-035 return mp_cmp_mag(a, b);
-036 \}
-037 \}
-\end{alltt}
-\end{small}
-
-The two if statements on lines 22 and 26 perform the initial sign comparison. If the signs are not the equal then which ever
-has the positive sign is larger. At line 31, the inputs are compared based on magnitudes. If the signs were both negative then
-the unsigned comparison is performed in the opposite direction (\textit{line 33}). Otherwise, the signs are assumed to
-be both positive and a forward direction unsigned comparison is performed.
-
-\section*{Exercises}
-\begin{tabular}{cl}
-$\left [ 2 \right ]$ & Modify algorithm mp\_set\_int to accept as input a variable length array of bits. \\
- & \\
-$\left [ 3 \right ]$ & Give the probability that algorithm mp\_cmp\_mag will have to compare $k$ digits \\
- & of two random digits (of equal magnitude) before a difference is found. \\
- & \\
-$\left [ 1 \right ]$ & Suggest a simple method to speed up the implementation of mp\_cmp\_mag based \\
- & on the observations made in the previous problem. \\
- &
-\end{tabular}
-
-\chapter{Basic Arithmetic}
-\section{Building Blocks}
-At this point algorithms for initialization, clearing, zeroing, copying, comparing and setting small constants have been
-established. The next logical set of algorithms to develop are addition, subtraction and digit shifting algorithms. These
-algorithms make use of the lower level algorithms and are the cruicial building block for the multiplication algorithms. It is very important
-that these algorithms are highly optimized. On their own they are simple $O(n)$ algorithms but they can be called from higher level algorithms
-which easily places them at $O(n^2)$ or even $O(n^3)$ work levels.
-
-All nine algorithms within this chapter make use of the logical bit shift operations denoted by $<<$ and $>>$ for left and right
-logical shifts respectively. A logical shift is analogous to sliding the decimal point of radix-10 representations. For example, the real
-number $0.9345$ is equivalent to $93.45\%$ which is found by sliding the the decimal two places to the right (\textit{multiplying by $10^2$}).
-Mathematically a logical shift is equivalent to a division or multiplication by a power of two.
-For example, $a << k = a \cdot 2^k$ while $a >> k = \lfloor a/2^k \rfloor$.
-
-One significant difference between a logical shift and the way decimals are shifted is that digits below the zero'th position are removed
-from the number. For example, consider $1101_2 >> 1$ using decimal notation this would produce $110.1_2$. However, with a logical shift the
-result is $110_2$.
-
-\section{Addition and Subtraction}
-In normal fixed precision arithmetic negative numbers are easily represented by subtraction from the modulus. For example, with 32-bit integers
-$a - b\mbox{ (mod }2^{32}\mbox{)}$ is the same as $a + (2^{32} - b) \mbox{ (mod }2^{32}\mbox{)}$ since $2^{32} \equiv 0 \mbox{ (mod }2^{32}\mbox{)}$.
-As a result subtraction can be performed with a trivial series of logical operations and an addition.
-
-However, in multiple precision arithmetic negative numbers are not represented in the same way. Instead a sign flag is used to keep track of the
-sign of the integer. As a result signed addition and subtraction are actually implemented as conditional usage of lower level addition or
-subtraction algorithms with the sign fixed up appropriately.
-
-The lower level algorithms will add or subtract integers without regard to the sign flag. That is they will add or subtract the magnitude of
-the integers respectively.
-
-\subsection{Low Level Addition}
-An unsigned addition of multiple precision integers is performed with the same long-hand algorithm used to add decimal numbers. That is to add the
-trailing digits first and propagate the resulting carry upwards. Since this is a lower level algorithm the name will have a ``s\_'' prefix.
-Historically that convention stems from the MPI library where ``s\_'' stood for static functions that were hidden from the developer entirely.
-
-\newpage
-\begin{figure}[!here]
-\begin{center}
-\begin{small}
-\begin{tabular}{l}
-\hline Algorithm \textbf{s\_mp\_add}. \\
-\textbf{Input}. Two mp\_ints $a$ and $b$ \\
-\textbf{Output}. The unsigned addition $c = \vert a \vert + \vert b \vert$. \\
-\hline \\
-1. if $a.used > b.used$ then \\
-\hspace{+3mm}1.1 $min \leftarrow b.used$ \\
-\hspace{+3mm}1.2 $max \leftarrow a.used$ \\
-\hspace{+3mm}1.3 $x \leftarrow a$ \\
-2. else \\
-\hspace{+3mm}2.1 $min \leftarrow a.used$ \\
-\hspace{+3mm}2.2 $max \leftarrow b.used$ \\
-\hspace{+3mm}2.3 $x \leftarrow b$ \\
-3. If $c.alloc < max + 1$ then grow $c$ to hold at least $max + 1$ digits (\textit{mp\_grow}) \\
-4. If failed to grow $c$ return(\textit{MP\_MEM}) \\
-5. $oldused \leftarrow c.used$ \\
-6. $c.used \leftarrow max + 1$ \\
-7. $u \leftarrow 0$ \\
-8. for $n$ from $0$ to $min - 1$ do \\
-\hspace{+3mm}8.1 $c_n \leftarrow a_n + b_n + u$ \\
-\hspace{+3mm}8.2 $u \leftarrow c_n >> lg(\beta)$ \\
-\hspace{+3mm}8.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\
-9. if $min \ne max$ then do \\
-\hspace{+3mm}9.1 for $n$ from $min$ to $max - 1$ do \\
-\hspace{+6mm}9.1.1 $c_n \leftarrow x_n + u$ \\
-\hspace{+6mm}9.1.2 $u \leftarrow c_n >> lg(\beta)$ \\
-\hspace{+6mm}9.1.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\
-10. $c_{max} \leftarrow u$ \\
-11. if $olduse > max$ then \\
-\hspace{+3mm}11.1 for $n$ from $max + 1$ to $olduse - 1$ do \\
-\hspace{+6mm}11.1.1 $c_n \leftarrow 0$ \\
-12. Clamp excess digits in $c$. (\textit{mp\_clamp}) \\
-13. Return(\textit{MP\_OKAY}) \\
-\hline
-\end{tabular}
-\end{small}
-\end{center}
-\caption{Algorithm s\_mp\_add}
-\end{figure}
-
-\textbf{Algorithm s\_mp\_add.}
-This algorithm is loosely based on algorithm 14.7 of HAC \cite[pp. 594]{HAC} but has been extended to allow the inputs to have different magnitudes.
-Coincidentally the description of algorithm A in Knuth \cite[pp. 266]{TAOCPV2} shares the same deficiency as the algorithm from \cite{HAC}. Even the
-MIX pseudo machine code presented by Knuth \cite[pp. 266-267]{TAOCPV2} is incapable of handling inputs which are of different magnitudes.
-
-Steps 1 and 2 will sort the two inputs based on their \textbf{used} digit count. This allows the inputs to have varying magnitudes which not
-only makes it more efficient than the trivial algorithm presented in the references but more flexible. The variable $min$ is given the lowest
-digit count while $max$ is given the highest digit count. If both inputs have the same \textbf{used} digit count both $min$ and $max$ are
-set to the same value. The variable $x$ is an \textit{alias} for the largest input and not meant to be a copy of it. After the inputs are sorted,
-steps 3 and 4 will ensure that the destination $c$ can accommodate the result. The old \textbf{used} count from $c$ is copied to
-$oldused$ so that excess digits can be cleared later, and the new \textbf{used} count is set to $max+1$, so that a carry from the most significant
-word can be handled.
-
-At step 7 the carry variable $u$ is set to zero and the first part of the addition loop can begin. The first step of the loop (\textit{8.1}) adds
-digits from the two inputs together along with the carry variable $u$. The following step extracts the carry bit by shifting the result of the
-preceding step right by $lg(\beta)$ positions. The shift to extract the carry is similar to how carry extraction works with decimal addition.
-
-Consider adding $77$ to $65$, the first addition of the first column is $7 + 5$ which produces the result $12$. The trailing digit of the result
-is $2 \equiv 12 \mbox{ (mod }10\mbox{)}$ and the carry is found by dividing (\textit{and ignoring the remainder}) $12$ by the radix or in this case $10$. The
-division and multiplication of $10$ is simply a logical right or left shift, respectively, of the digits. In otherwords the carry can be extracted
-by shifting one digit to the right.
-
-Note that $lg()$ is simply the base two logarithm such that $lg(2^k) = k$. This implies that $lg(\beta)$ is the number of bits in a radix-$\beta$
-digit. Therefore, a logical shift right of the summand by $lg(\beta)$ will extract the carry. The final step of the loop reduces the digit
-modulo the radix $\beta$ to ensure it is in range.
-
-After step 8 the smallest input (\textit{or both if they are the same magnitude}) has been exhausted. Step 9 decides whether
-the inputs were of equal magnitude. If not than another loop similar to that in step 8, must be executed. The loop at step
-number 9.1 differs from the previous loop since it only adds the mp\_int $x$ along with the carry.
-
-Step 10 finishes the addition phase by copying the final carry to the highest location in the result $c_{max}$. Step 11 ensures that
-leading digits that were originally present in $c$ are cleared. Finally excess leading digits are clamped and the algorithm returns success.
-
-\index{bn\_s\_mp\_add.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_add.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* low level addition, based on HAC pp.594, Algorithm 14.7 */
-018 int
-019 s_mp_add (mp_int * a, mp_int * b, mp_int * c)
-020 \{
-021 mp_int *x;
-022 int olduse, res, min, max;
-023
-024 /* find sizes, we let |a| <= |b| which means we have to sort
-025 * them. "x" will point to the input with the most digits
-026 */
-027 if (a->used > b->used) \{
-028 min = b->used;
-029 max = a->used;
-030 x = a;
-031 \} else \{
-032 min = a->used;
-033 max = b->used;
-034 x = b;
-035 \}
-036
-037 /* init result */
-038 if (c->alloc < max + 1) \{
-039 if ((res = mp_grow (c, max + 1)) != MP_OKAY) \{
-040 return res;
-041 \}
-042 \}
-043
-044 /* get old used digit count and set new one */
-045 olduse = c->used;
-046 c->used = max + 1;
-047
-048 \{
-049 register mp_digit u, *tmpa, *tmpb, *tmpc;
-050 register int i;
-051
-052 /* alias for digit pointers */
-053
-054 /* first input */
-055 tmpa = a->dp;
-056
-057 /* second input */
-058 tmpb = b->dp;
-059
-060 /* destination */
-061 tmpc = c->dp;
-062
-063 /* zero the carry */
-064 u = 0;
-065 for (i = 0; i < min; i++) \{
-066 /* Compute the sum at one digit, T[i] = A[i] + B[i] + U */
-067 *tmpc = *tmpa++ + *tmpb++ + u;
-068
-069 /* U = carry bit of T[i] */
-070 u = *tmpc >> ((mp_digit)DIGIT_BIT);
-071
-072 /* take away carry bit from T[i] */
-073 *tmpc++ &= MP_MASK;
-074 \}
-075
-076 /* now copy higher words if any, that is in A+B
-077 * if A or B has more digits add those in
-078 */
-079 if (min != max) \{
-080 for (; i < max; i++) \{
-081 /* T[i] = X[i] + U */
-082 *tmpc = x->dp[i] + u;
-083
-084 /* U = carry bit of T[i] */
-085 u = *tmpc >> ((mp_digit)DIGIT_BIT);
-086
-087 /* take away carry bit from T[i] */
-088 *tmpc++ &= MP_MASK;
-089 \}
-090 \}
-091
-092 /* add carry */
-093 *tmpc++ = u;
-094
-095 /* clear digits above oldused */
-096 for (i = c->used; i < olduse; i++) \{
-097 *tmpc++ = 0;
-098 \}
-099 \}
-100
-101 mp_clamp (c);
-102 return MP_OKAY;
-103 \}
-\end{alltt}
-\end{small}
-
-Lines 27 to 35 perform the initial sorting of the inputs and determine the $min$ and $max$ variables. Note that $x$ is a pointer to a
-mp\_int assigned to the largest input, in effect it is a local alias. Lines 37 to 42 ensure that the destination is grown to
-accomodate the result of the addition.
-
-Similar to the implementation of mp\_copy this function uses the braced code and local aliases coding style. The three aliases that are on
-lines 55, 58 and 61 represent the two inputs and destination variables respectively. These aliases are used to ensure the
-compiler does not have to dereference $a$, $b$ or $c$ (respectively) to access the digits of the respective mp\_int.
-
-The initial carry $u$ is cleared on line 64, note that $u$ is of type mp\_digit which ensures type compatibility within the
-implementation. The initial addition loop begins on line 65 and ends on line 74. Similarly the conditional addition loop
-begins on line 80 and ends on line 90. The addition is finished with the final carry being stored in $tmpc$ on line 93.
-Note the ``++'' operator on the same line. After line 93 $tmpc$ will point to the $c.used$'th digit of the mp\_int $c$. This is useful
-for the next loop on lines 96 to 99 which set any old upper digits to zero.
-
-\subsection{Low Level Subtraction}
-The low level unsigned subtraction algorithm is very similar to the low level unsigned addition algorithm. The principle difference is that the
-unsigned subtraction algorithm requires the result to be positive. That is when computing $a - b$ the condition $\vert a \vert \ge \vert b\vert$ must
-be met for this algorithm to function properly. Keep in mind this low level algorithm is not meant to be used in higher level algorithms directly.
-This algorithm as will be shown can be used to create functional signed addition and subtraction algorithms.
-
-
-For this algorithm a new variable is required to make the description simpler. Recall from section 1.3.1 that a mp\_digit must be able to represent
-the range $0 \le x < 2\beta$ for the algorithms to work correctly. However, it is allowable that a mp\_digit represent a larger range of values. For
-this algorithm we will assume that the variable $\gamma$ represents the number of bits available in a
-mp\_digit (\textit{this implies $2^{\gamma} > \beta$}).
-
-For example, the default for LibTomMath is to use a ``unsigned long'' for the mp\_digit ``type'' while $\beta = 2^{28}$. In ISO C an ``unsigned long''
-data type must be able to represent $0 \le x < 2^{32}$ meaning that in this case $\gamma = 32$.
-
-\newpage\begin{figure}[!here]
-\begin{center}
-\begin{small}
-\begin{tabular}{l}
-\hline Algorithm \textbf{s\_mp\_sub}. \\
-\textbf{Input}. Two mp\_ints $a$ and $b$ ($\vert a \vert \ge \vert b \vert$) \\
-\textbf{Output}. The unsigned subtraction $c = \vert a \vert - \vert b \vert$. \\
-\hline \\
-1. $min \leftarrow b.used$ \\
-2. $max \leftarrow a.used$ \\
-3. If $c.alloc < max$ then grow $c$ to hold at least $max$ digits. (\textit{mp\_grow}) \\
-4. If the reallocation failed return(\textit{MP\_MEM}). \\
-5. $oldused \leftarrow c.used$ \\
-6. $c.used \leftarrow max$ \\
-7. $u \leftarrow 0$ \\
-8. for $n$ from $0$ to $min - 1$ do \\
-\hspace{3mm}8.1 $c_n \leftarrow a_n - b_n - u$ \\
-\hspace{3mm}8.2 $u \leftarrow c_n >> (\gamma - 1)$ \\
-\hspace{3mm}8.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\
-9. if $min < max$ then do \\
-\hspace{3mm}9.1 for $n$ from $min$ to $max - 1$ do \\
-\hspace{6mm}9.1.1 $c_n \leftarrow a_n - u$ \\
-\hspace{6mm}9.1.2 $u \leftarrow c_n >> (\gamma - 1)$ \\
-\hspace{6mm}9.1.3 $c_n \leftarrow c_n \mbox{ (mod }\beta\mbox{)}$ \\
-10. if $oldused > max$ then do \\
-\hspace{3mm}10.1 for $n$ from $max$ to $oldused - 1$ do \\
-\hspace{6mm}10.1.1 $c_n \leftarrow 0$ \\
-11. Clamp excess digits of $c$. (\textit{mp\_clamp}). \\
-12. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{small}
-\end{center}
-\caption{Algorithm s\_mp\_sub}
-\end{figure}
-
-\textbf{Algorithm s\_mp\_sub.}
-This algorithm performs the unsigned subtraction of two mp\_int variables under the restriction that the result must be positive. That is when
-passing variables $a$ and $b$ the condition that $\vert a \vert \ge \vert b \vert$ must be met for the algorithm to function correctly. This
-algorithm is loosely based on algorithm 14.9 \cite[pp. 595]{HAC} and is similar to algorithm S in \cite[pp. 267]{TAOCPV2} as well. As was the case
-of the algorithm s\_mp\_add both other references lack discussion concerning various practical details such as when the inputs differ in magnitude.
-
-The initial sorting of the inputs is trivial in this algorithm since $a$ is guaranteed to have at least the same magnitude of $b$. Steps 1 and 2
-set the $min$ and $max$ variables. Unlike the addition routine there is guaranteed to be no carry which means that the final result can be at
-most $max$ digits in length as opposed to $max + 1$. Similar to the addition algorithm the \textbf{used} count of $c$ is copied locally and
-set to the maximal count for the operation.
-
-The subtraction loop that begins on step 8 is essentially the same as the addition loop of algorithm s\_mp\_add except single precision
-subtraction is used instead. Note the use of the $\gamma$ variable to extract the carry (\textit{also known as the borrow}) within the subtraction
-loops. Under the assumption that two's complement single precision arithmetic is used this will successfully extract the desired carry.
-
-For example, consider subtracting $0101_2$ from $0100_2$ where $\gamma = 4$ and $\beta = 2$. The least significant bit will force a carry upwards to
-the third bit which will be set to zero after the borrow. After the very first bit has been subtracted $4 - 1 \equiv 0011_2$ will remain, When the
-third bit of $0101_2$ is subtracted from the result it will cause another carry. In this case though the carry will be forced to propagate all the
-way to the most significant bit.
-
-Recall that $\beta < 2^{\gamma}$. This means that if a carry does occur just before the $lg(\beta)$'th bit it will propagate all the way to the most
-significant bit. Thus, the high order bits of the mp\_digit that are not part of the actual digit will either be all zero, or all one. All that
-is needed is a single zero or one bit for the carry. Therefore a single logical shift right by $\gamma - 1$ positions is sufficient to extract the
-carry. This method of carry extraction may seem awkward but the reason for it becomes apparent when the implementation is discussed.
-
-If $b$ has a smaller magnitude than $a$ then step 9 will force the carry and copy operation to propagate through the larger input $a$ into $c$. Step
-10 will ensure that any leading digits of $c$ above the $max$'th position are zeroed.
-
-\index{bn\_s\_mp\_sub.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_sub.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* low level subtraction (assumes |a| > |b|), HAC pp.595 Algorithm 14.9 */
-018 int
-019 s_mp_sub (mp_int * a, mp_int * b, mp_int * c)
-020 \{
-021 int olduse, res, min, max;
-022
-023 /* find sizes */
-024 min = b->used;
-025 max = a->used;
-026
-027 /* init result */
-028 if (c->alloc < max) \{
-029 if ((res = mp_grow (c, max)) != MP_OKAY) \{
-030 return res;
-031 \}
-032 \}
-033 olduse = c->used;
-034 c->used = max;
-035
-036 \{
-037 register mp_digit u, *tmpa, *tmpb, *tmpc;
-038 register int i;
-039
-040 /* alias for digit pointers */
-041 tmpa = a->dp;
-042 tmpb = b->dp;
-043 tmpc = c->dp;
-044
-045 /* set carry to zero */
-046 u = 0;
-047 for (i = 0; i < min; i++) \{
-048 /* T[i] = A[i] - B[i] - U */
-049 *tmpc = *tmpa++ - *tmpb++ - u;
-050
-051 /* U = carry bit of T[i]
-052 * Note this saves performing an AND operation since
-053 * if a carry does occur it will propagate all the way to the
-054 * MSB. As a result a single shift is enough to get the carry
-055 */
-056 u = *tmpc >> ((mp_digit)(CHAR_BIT * sizeof (mp_digit) - 1));
-057
-058 /* Clear carry from T[i] */
-059 *tmpc++ &= MP_MASK;
-060 \}
-061
-062 /* now copy higher words if any, e.g. if A has more digits than B */
-063 for (; i < max; i++) \{
-064 /* T[i] = A[i] - U */
-065 *tmpc = *tmpa++ - u;
-066
-067 /* U = carry bit of T[i] */
-068 u = *tmpc >> ((mp_digit)(CHAR_BIT * sizeof (mp_digit) - 1));
-069
-070 /* Clear carry from T[i] */
-071 *tmpc++ &= MP_MASK;
-072 \}
-073
-074 /* clear digits above used (since we may not have grown result above) */
-
-075 for (i = c->used; i < olduse; i++) \{
-076 *tmpc++ = 0;
-077 \}
-078 \}
-079
-080 mp_clamp (c);
-081 return MP_OKAY;
-082 \}
-083
-\end{alltt}
-\end{small}
-
-Line 24 and 25 perform the initial hardcoded sorting of the inputs. In reality the $min$ and $max$ variables are only aliases and are only
-used to make the source code easier to read. Again the pointer alias optimization is used within this algorithm. Lines 41, 42 and 43 initialize the aliases for
-$a$, $b$ and $c$ respectively.
-
-The first subtraction loop occurs on lines 46 through 60. The theory behind the subtraction loop is exactly the same as that for
-the addition loop. As remarked earlier there is an implementation reason for using the ``awkward'' method of extracting the carry
-(\textit{see line 56}). The traditional method for extracting the carry would be to shift by $lg(\beta)$ positions and logically AND
-the least significant bit. The AND operation is required because all of the bits above the $\lg(\beta)$'th bit will be set to one after a carry
-occurs from subtraction. This carry extraction requires two relatively cheap operations to extract the carry. The other method is to simply
-shift the most significant bit to the least significant bit thus extracting the carry with a single cheap operation. This optimization only works on
-twos compliment machines which is a safe assumption to make.
-
-If $a$ has a larger magnitude than $b$ an additional loop (\textit{see lines 63 through 72}) is required to propagate the carry through
-$a$ and copy the result to $c$.
-
-\subsection{High Level Addition}
-Now that both lower level addition and subtraction algorithms have been established an effective high level signed addition algorithm can be
-established. This high level addition algorithm will be what other algorithms and developers will use to perform addition of mp\_int data
-types.
-
-Recall from section 5.2 that an mp\_int represents an integer with an unsigned mantissa (\textit{the array of digits}) and a \textbf{sign}
-flag. A high level addition is actually performed as a series of eight separate cases which can be optimized down to three unique cases.
-
-\begin{figure}[!here]
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_add}. \\
-\textbf{Input}. Two mp\_ints $a$ and $b$ \\
-\textbf{Output}. The signed addition $c = a + b$. \\
-\hline \\
-1. if $a.sign = b.sign$ then do \\
-\hspace{3mm}1.1 $c.sign \leftarrow a.sign$ \\
-\hspace{3mm}1.2 $c \leftarrow \vert a \vert + \vert b \vert$ (\textit{s\_mp\_add})\\
-2. else do \\
-\hspace{3mm}2.1 if $\vert a \vert < \vert b \vert$ then do (\textit{mp\_cmp\_mag}) \\
-\hspace{6mm}2.1.1 $c.sign \leftarrow b.sign$ \\
-\hspace{6mm}2.1.2 $c \leftarrow \vert b \vert - \vert a \vert$ (\textit{s\_mp\_sub}) \\
-\hspace{3mm}2.2 else do \\
-\hspace{6mm}2.2.1 $c.sign \leftarrow a.sign$ \\
-\hspace{6mm}2.2.2 $c \leftarrow \vert a \vert - \vert b \vert$ \\
-3. If any of the lower level operations failed return(\textit{MP\_MEM}) \\
-4. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\caption{Algorithm mp\_add}
-\end{figure}
-
-\textbf{Algorithm mp\_add.}
-This algorithm performs the signed addition of two mp\_int variables. There is no reference algorithm to draw upon from either \cite{TAOCPV2} or
-\cite{HAC} since they both only provide unsigned operations. The algorithm is fairly straightforward but restricted since subtraction can only
-produce positive results.
-
-\begin{figure}[here]
-\begin{small}
-\begin{center}
-\begin{tabular}{|c|c|c|c|c|}
-\hline \textbf{Sign of $a$} & \textbf{Sign of $b$} & \textbf{$\vert a \vert > \vert b \vert $} & \textbf{Unsigned Operation} & \textbf{Result Sign Flag} \\
-\hline $+$ & $+$ & Yes & $c = a + b$ & $a.sign$ \\
-\hline $+$ & $+$ & No & $c = a + b$ & $a.sign$ \\
-\hline $-$ & $-$ & Yes & $c = a + b$ & $a.sign$ \\
-\hline $-$ & $-$ & No & $c = a + b$ & $a.sign$ \\
-\hline &&&&\\
-
-\hline $+$ & $-$ & No & $c = b - a$ & $b.sign$ \\
-\hline $-$ & $+$ & No & $c = b - a$ & $b.sign$ \\
-
-\hline &&&&\\
-
-\hline $+$ & $-$ & Yes & $c = a - b$ & $a.sign$ \\
-\hline $-$ & $+$ & Yes & $c = a - b$ & $a.sign$ \\
-
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Addition Guide Chart}
-\label{fig:AddChart}
-\end{figure}
-
-Figure~\ref{fig:AddChart} lists all of the eight possible input combinations and is sorted to show that only three specific cases need to be handled. The
-return code of the unsigned operations at step 1.2, 2.1.2 and 2.2.2 are forwarded to step 3 to check for errors. This simplifies the description
-of the algorithm considerably and best follows how the implementation actually was achieved.
-
-Also note how the \textbf{sign} is set before the unsigned addition or subtraction is performed. Recall from the descriptions of algorithms
-s\_mp\_add and s\_mp\_sub that the mp\_clamp function is used at the end to trim excess digits. The mp\_clamp algorithm will set the \textbf{sign}
-to \textbf{MP\_ZPOS} when the \textbf{used} digit count reaches zero.
-
-For example, consider performing $-a + a$ with algorithm mp\_add. By the description of the algorithm the sign is set to \textbf{MP\_NEG} which would
-produce a result of $-0$. However, since the sign is set first then the unsigned addition is performed the subsequent usage of algorithm mp\_clamp
-within algorithm s\_mp\_add will force $-0$ to become $0$.
-
-\index{bn\_mp\_add.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_add.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* high level addition (handles signs) */
-018 int
-019 mp_add (mp_int * a, mp_int * b, mp_int * c)
-020 \{
-021 int sa, sb, res;
-022
-023 /* get sign of both inputs */
-024 sa = a->sign;
-025 sb = b->sign;
-026
-027 /* handle two cases, not four */
-028 if (sa == sb) \{
-029 /* both positive or both negative */
-030 /* add their magnitudes, copy the sign */
-031 c->sign = sa;
-032 res = s_mp_add (a, b, c);
-033 \} else \{
-034 /* one positive, the other negative */
-035 /* subtract the one with the greater magnitude from */
-036 /* the one of the lesser magnitude. The result gets */
-037 /* the sign of the one with the greater magnitude. */
-038 if (mp_cmp_mag (a, b) == MP_LT) \{
-039 c->sign = sb;
-040 res = s_mp_sub (b, a, c);
-041 \} else \{
-042 c->sign = sa;
-043 res = s_mp_sub (a, b, c);
-044 \}
-045 \}
-046 return res;
-047 \}
-048
-\end{alltt}
-\end{small}
-
-The source code follows the algorithm fairly closely. The most notable new source code addition is the usage of the $res$ integer variable which
-is used to pass result of the unsigned operations forward. Unlike in the algorithm, the variable $res$ is merely returned as is without
-explicitly checking it and returning the constant \textbf{MP\_OKAY}. The observation is this algorithm will succeed or fail only if the lower
-level functions do so. Returning their return code is sufficient.
-
-\subsection{High Level Subtraction}
-The high level signed subtraction algorithm is essentially the same as the high level signed addition algorithm.
-
-\newpage\begin{figure}[!here]
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_sub}. \\
-\textbf{Input}. Two mp\_ints $a$ and $b$ \\
-\textbf{Output}. The signed subtraction $c = a - b$. \\
-\hline \\
-1. if $a.sign \ne b.sign$ then do \\
-\hspace{3mm}1.1 $c.sign \leftarrow a.sign$ \\
-\hspace{3mm}1.2 $c \leftarrow \vert a \vert + \vert b \vert$ (\textit{s\_mp\_add}) \\
-2. else do \\
-\hspace{3mm}2.1 if $\vert a \vert \ge \vert b \vert$ then do (\textit{mp\_cmp\_mag}) \\
-\hspace{6mm}2.1.1 $c.sign \leftarrow a.sign$ \\
-\hspace{6mm}2.1.2 $c \leftarrow \vert a \vert - \vert b \vert$ (\textit{s\_mp\_sub}) \\
-\hspace{3mm}2.2 else do \\
-\hspace{6mm}2.2.1 $c.sign \leftarrow \left \lbrace \begin{array}{ll}
- MP\_ZPOS & \mbox{if }a.sign = MP\_NEG \\
- MP\_NEG & \mbox{otherwise} \\
- \end{array} \right .$ \\
-\hspace{6mm}2.2.2 $c \leftarrow \vert b \vert - \vert a \vert$ \\
-3. If any of the lower level operations failed return(\textit{MP\_MEM}). \\
-4. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\caption{Algorithm mp\_sub}
-\end{figure}
-
-\textbf{Algorithm mp\_sub.}
-This algorithm performs the signed subtraction of two inputs. Similar to algorithm mp\_add there is no reference in either \cite{TAOCPV2} or
-\cite{HAC}. Also this algorithm is restricted by algorithm s\_mp\_sub. The following chart lists the eight possible inputs and
-the operations required.
-
-\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{|c|c|c|c|c|}
-\hline \textbf{Sign of $a$} & \textbf{Sign of $b$} & \textbf{$\vert a \vert \ge \vert b \vert $} & \textbf{Unsigned Operation} & \textbf{Result Sign Flag} \\
-\hline $+$ & $-$ & Yes & $c = a + b$ & $a.sign$ \\
-\hline $+$ & $-$ & No & $c = a + b$ & $a.sign$ \\
-\hline $-$ & $+$ & Yes & $c = a + b$ & $a.sign$ \\
-\hline $-$ & $+$ & No & $c = a + b$ & $a.sign$ \\
-\hline &&&& \\
-\hline $+$ & $+$ & Yes & $c = a - b$ & $a.sign$ \\
-\hline $-$ & $-$ & Yes & $c = a - b$ & $a.sign$ \\
-\hline &&&& \\
-\hline $+$ & $+$ & No & $c = b - a$ & $\mbox{opposite of }a.sign$ \\
-\hline $-$ & $-$ & No & $c = b - a$ & $\mbox{opposite of }a.sign$ \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Subtraction Guide Chart}
-\end{figure}
-
-Similar to the case of algorithm mp\_add the \textbf{sign} is set first before the unsigned addition or subtraction. That is to prevent the
-algorithm from producing $-a - -a = -0$ as a result.
-
-\index{bn\_mp\_sub.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_sub.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* high level subtraction (handles signs) */
-018 int
-019 mp_sub (mp_int * a, mp_int * b, mp_int * c)
-020 \{
-021 int sa, sb, res;
-022
-023 sa = a->sign;
-024 sb = b->sign;
-025
-026 if (sa != sb) \{
-027 /* subtract a negative from a positive, OR */
-028 /* subtract a positive from a negative. */
-029 /* In either case, ADD their magnitudes, */
-030 /* and use the sign of the first number. */
-031 c->sign = sa;
-032 res = s_mp_add (a, b, c);
-033 \} else \{
-034 /* subtract a positive from a positive, OR */
-035 /* subtract a negative from a negative. */
-036 /* First, take the difference between their */
-037 /* magnitudes, then... */
-038 if (mp_cmp_mag (a, b) != MP_LT) \{
-039 /* Copy the sign from the first */
-040 c->sign = sa;
-041 /* The first has a larger or equal magnitude */
-042 res = s_mp_sub (a, b, c);
-043 \} else \{
-044 /* The result has the *opposite* sign from */
-045 /* the first number. */
-046 c->sign = (sa == MP_ZPOS) ? MP_NEG : MP_ZPOS;
-047 /* The second has a larger magnitude */
-048 res = s_mp_sub (b, a, c);
-049 \}
-050 \}
-051 return res;
-052 \}
-053
-\end{alltt}
-\end{small}
-
-Much like the implementation of algorithm mp\_add the variable $res$ is used to catch the return code of the unsigned addition or subtraction operations
-and forward it to the end of the function. On line 38 the ``not equal to'' \textbf{MP\_LT} expression is used to emulate a
-``greater than or equal to'' comparison.
-
-\section{Bit and Digit Shifting}
-It is quite common to think of a multiple precision integer as a polynomial in $x$, that is $y = f(\beta)$ where $f(x) = \sum_{i=0}^{n-1} a_i x^i$.
-This notation arises within discussion of Montgomery and Diminished Radix Reduction as well as Karatsuba multiplication and squaring.
-
-In order to facilitate operations on polynomials in $x$ as above a series of simple ``digit'' algorithms have to be established. That is to shift
-the digits left or right as well to shift individual bits of the digits left and right. It is important to note that not all ``shift'' operations
-are on radix-$\beta$ digits.
-
-\subsection{Multiplication by Two}
-
-In a binary system where the radix is a power of two multiplication by two not only arises often in other algorithms it is a fairly efficient
-operation to perform. A single precision logical shift left is sufficient to multiply a single digit by two.
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_mul\_2}. \\
-\textbf{Input}. One mp\_int $a$ \\
-\textbf{Output}. $b = 2a$. \\
-\hline \\
-1. If $b.alloc < a.used + 1$ then grow $b$ to hold $a.used + 1$ digits. (\textit{mp\_grow}) \\
-2. If the reallocation failed return(\textit{MP\_MEM}). \\
-3. $oldused \leftarrow b.used$ \\
-4. $b.used \leftarrow a.used$ \\
-5. $r \leftarrow 0$ \\
-6. for $n$ from 0 to $a.used - 1$ do \\
-\hspace{3mm}6.1 $rr \leftarrow a_n >> (lg(\beta) - 1)$ \\
-\hspace{3mm}6.2 $b_n \leftarrow (a_n << 1) + r \mbox{ (mod }\beta\mbox{)}$ \\
-\hspace{3mm}6.3 $r \leftarrow rr$ \\
-7. If $r \ne 0$ then do \\
-\hspace{3mm}7.1 $b_{n + 1} \leftarrow r$ \\
-\hspace{3mm}7.2 $b.used \leftarrow b.used + 1$ \\
-8. If $b.used < oldused - 1$ then do \\
-\hspace{3mm}8.1 for $n$ from $b.used$ to $oldused - 1$ do \\
-\hspace{6mm}8.1.1 $b_n \leftarrow 0$ \\
-9. $b.sign \leftarrow a.sign$ \\
-10. Return(\textit{MP\_OKAY}).\\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_mul\_2}
-\end{figure}
-
-\textbf{Algorithm mp\_mul\_2.}
-This algorithm will quickly multiply a mp\_int by two provided $\beta$ is a power of two. Neither \cite{TAOCPV2} nor \cite{HAC} describe such
-an algorithm despite the fact it arises often in other algorithms. The algorithm is setup much like the lower level algorithm s\_mp\_add since
-it is for all intents and purposes equivalent to the operation $b = \vert a \vert + \vert a \vert$.
-
-Step 1 and 2 grow the input as required to accomodate the maximum number of \textbf{used} digits in the result. The initial \textbf{used} count
-is set to $a.used$ at step 4. Only if there is a final carry will the \textbf{used} count require adjustment.
-
-Step 6 is an optimization implementation of the addition loop for this specific case. That is since the two values being added together
-are the same there is no need to perform two reads from the digits of $a$. Step 6.1 performs a single precision shift on the current digit $a_n$ to
-obtain what will be the carry for the next iteration. Step 6.2 calculates the $n$'th digit of the result as single precision shift of $a_n$ plus
-the previous carry. Recall from section 5.1 that $a_n << 1$ is equivalent to $a_n \cdot 2$. An iteration of the addition loop is finished with
-forwarding the carry to the next iteration.
-
-Step 7 takes care of any final carry by setting the $a.used$'th digit of the result to the carry and augmenting the \textbf{used} count of $b$.
-Step 8 clears any leading digits of $b$ in case it originally had a larger magnitude than $a$.
-
-\index{bn\_mp\_mul\_2.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_mul\_2.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* b = a*2 */
-018 int
-019 mp_mul_2 (mp_int * a, mp_int * b)
-020 \{
-021 int x, res, oldused;
-022
-023 /* grow to accomodate result */
-024 if (b->alloc < a->used + 1) \{
-025 if ((res = mp_grow (b, a->used + 1)) != MP_OKAY) \{
-026 return res;
-027 \}
-028 \}
-029
-030 oldused = b->used;
-031 b->used = a->used;
-032
-033 \{
-034 register mp_digit r, rr, *tmpa, *tmpb;
-035
-036 /* alias for source */
-037 tmpa = a->dp;
-038
-039 /* alias for dest */
-040 tmpb = b->dp;
-041
-042 /* carry */
-043 r = 0;
-044 for (x = 0; x < a->used; x++) \{
-045
-046 /* get what will be the *next* carry bit from the
-047 * MSB of the current digit
-048 */
-049 rr = *tmpa >> ((mp_digit)(DIGIT_BIT - 1));
-050
-051 /* now shift up this digit, add in the carry [from the previous] */
-052 *tmpb++ = ((*tmpa++ << ((mp_digit)1)) | r) & MP_MASK;
-053
-054 /* copy the carry that would be from the source
-055 * digit into the next iteration
-056 */
-057 r = rr;
-058 \}
-059
-060 /* new leading digit? */
-061 if (r != 0) \{
-062 /* add a MSB which is always 1 at this point */
-063 *tmpb = 1;
-064 ++b->used;
-065 \}
-066
-067 /* now zero any excess digits on the destination
-068 * that we didn't write to
-069 */
-070 tmpb = b->dp + b->used;
-071 for (x = b->used; x < oldused; x++) \{
-072 *tmpb++ = 0;
-073 \}
-074 \}
-075 b->sign = a->sign;
-076 return MP_OKAY;
-077 \}
-\end{alltt}
-\end{small}
-
-This implementation is essentially an optimized implementation of s\_mp\_add for the case of doubling an input. The only noteworthy difference
-is the use of the logical shift operator on line 52 to perform a single precision doubling.
-
-\subsection{Division by Two}
-A division by two can just as easily be accomplished with a logical shift right as multiplication by two can be with a logical shift left.
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_div\_2}. \\
-\textbf{Input}. One mp\_int $a$ \\
-\textbf{Output}. $b = a/2$. \\
-\hline \\
-1. If $b.alloc < a.used$ then grow $b$ to hold $a.used$ digits. (\textit{mp\_grow}) \\
-2. If the reallocation failed return(\textit{MP\_MEM}). \\
-3. $oldused \leftarrow b.used$ \\
-4. $b.used \leftarrow a.used$ \\
-5. $r \leftarrow 0$ \\
-6. for $n$ from $b.used - 1$ to $0$ do \\
-\hspace{3mm}6.1 $rr \leftarrow a_n \mbox{ (mod }2\mbox{)}$\\
-\hspace{3mm}6.2 $b_n \leftarrow (a_n >> 1) + (r << (lg(\beta) - 1)) \mbox{ (mod }\beta\mbox{)}$ \\
-\hspace{3mm}6.3 $r \leftarrow rr$ \\
-7. If $b.used < oldused - 1$ then do \\
-\hspace{3mm}7.1 for $n$ from $b.used$ to $oldused - 1$ do \\
-\hspace{6mm}7.1.1 $b_n \leftarrow 0$ \\
-8. $b.sign \leftarrow a.sign$ \\
-9. Clamp excess digits of $b$. (\textit{mp\_clamp}) \\
-10. Return(\textit{MP\_OKAY}).\\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_div\_2}
-\end{figure}
-
-\textbf{Algorithm mp\_div\_2.}
-This algorithm will divide an mp\_int by two using logical shifts to the right. Like mp\_mul\_2 it uses a modified low level addition
-core as the basis of the algorithm. Unlike mp\_mul\_2 the shift operations work from the leading digit to the trailing digit. The algorithm
-could be written to work from the trailing digit to the leading digit however, it would have to stop one short of $a.used - 1$ digits to prevent
-reading past the end of the array of digits.
-
-Essentially the loop at step 6 is similar to that of mp\_mul\_2 except the logical shifts go in the opposite direction and the carry is at the
-least significant bit not the most significant bit.
-
-\index{bn\_mp\_div\_2.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_div\_2.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* b = a/2 */
-018 int
-019 mp_div_2 (mp_int * a, mp_int * b)
-020 \{
-021 int x, res, oldused;
-022
-023 /* copy */
-024 if (b->alloc < a->used) \{
-025 if ((res = mp_grow (b, a->used)) != MP_OKAY) \{
-026 return res;
-027 \}
-028 \}
-029
-030 oldused = b->used;
-031 b->used = a->used;
-032 \{
-033 register mp_digit r, rr, *tmpa, *tmpb;
-034
-035 /* source alias */
-036 tmpa = a->dp + b->used - 1;
-037
-038 /* dest alias */
-039 tmpb = b->dp + b->used - 1;
-040
-041 /* carry */
-042 r = 0;
-043 for (x = b->used - 1; x >= 0; x--) \{
-044 /* get the carry for the next iteration */
-045 rr = *tmpa & 1;
-046
-047 /* shift the current digit, add in carry and store */
-048 *tmpb-- = (*tmpa-- >> 1) | (r << (DIGIT_BIT - 1));
-049
-050 /* forward carry to next iteration */
-051 r = rr;
-052 \}
-053
-054 /* zero excess digits */
-055 tmpb = b->dp + b->used;
-056 for (x = b->used; x < oldused; x++) \{
-057 *tmpb++ = 0;
-058 \}
-059 \}
-060 b->sign = a->sign;
-061 mp_clamp (b);
-062 return MP_OKAY;
-063 \}
-\end{alltt}
-\end{small}
-
-\section{Polynomial Basis Operations}
-Recall from section 5.3 that any integer can be represented as a polynomial in $x$ as $y = f(\beta)$. Such a representation is also known as
-the polynomial basis \cite[pp. 48]{ROSE}. Given such a notation a multiplication or division by $x$ amounts to shifting whole digits a single
-place. The need for such operations arises in several other higher level algorithms such as Barrett and Montgomery reduction, integer
-division and Karatsuba multiplication.
-
-Converting from an array of digits to polynomial basis is very simple. Consider the integer $y \equiv (a_2, a_1, a_0)_{\beta}$ and recall that
-$y = \sum_{i=0}^{2} a_i \beta^i$. Simply replace $\beta$ with $x$ and the expression is in polynomial basis. For example, $f(x) = 8x + 9$ is the
-polynomial basis representation for $89$ using radix ten. That is, $f(10) = 8(10) + 9 = 89$.
-
-\subsection{Multiplication by $x$}
-
-Given a polynomial in $x$ such as $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_0$ multiplying by $x$ amounts to shifting the coefficients up one
-degree. In this case $f(x) \cdot x = a_n x^{n+1} + a_{n-1} x^n + ... + a_0 x$. From a scalar basis point of view multiplying by $x$ is equivalent to
-multiplying by the integer $\beta$.
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_lshd}. \\
-\textbf{Input}. One mp\_int $a$ and an integer $b$ \\
-\textbf{Output}. $a \leftarrow a \cdot \beta^b$ (equivalent to multiplication by $x^b$). \\
-\hline \\
-1. If $b \le 0$ then return(\textit{MP\_OKAY}). \\
-2. If $a.alloc < a.used + b$ then grow $a$ to at least $a.used + b$ digits. (\textit{mp\_grow}). \\
-3. If the reallocation failed return(\textit{MP\_MEM}). \\
-4. $a.used \leftarrow a.used + b$ \\
-5. $i \leftarrow a.used - 1$ \\
-6. $j \leftarrow a.used - 1 - b$ \\
-7. for $n$ from $a.used - 1$ to $b$ do \\
-\hspace{3mm}7.1 $a_{i} \leftarrow a_{j}$ \\
-\hspace{3mm}7.2 $i \leftarrow i - 1$ \\
-\hspace{3mm}7.3 $j \leftarrow j - 1$ \\
-8. for $n$ from 0 to $b - 1$ do \\
-\hspace{3mm}8.1 $a_n \leftarrow 0$ \\
-9. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_lshd}
-\end{figure}
-
-\textbf{Algorithm mp\_lshd.}
-This algorithm multiplies an mp\_int by the $b$'th power of $x$. This is equivalent to multiplying by $\beta^b$. The algorithm differs
-from the other algorithms presented so far as it performs the operation in place instead storing the result in a separate location. The
-motivation behind this change is due to the way this function is typically used. Algorithms such as mp\_add store the result in an optionally
-different third mp\_int because the original inputs are often still required. Algorithm mp\_lshd (\textit{and similarly algorithm mp\_rshd}) is
-typically used on values where the original value is no longer required. The algorithm will return success immediately if
-$b \le 0$ since the rest of algorithm is only valid when $b > 0$.
-
-First the destination $a$ is grown as required to accomodate the result. The counters $i$ and $j$ are used to form a \textit{sliding window} over
-the digits of $a$ of length $b$. The head of the sliding window is at $i$ (\textit{the leading digit}) and the tail at $j$ (\textit{the trailing digit}).
-The loop on step 7 copies the digit from the tail to the head. In each iteration the window is moved down one digit. The last loop on
-step 8 sets the lower $b$ digits to zero.
-
-\newpage
-\begin{center}
-\begin{figure}[here]
-\includegraphics{pics/sliding_window}
-\caption{Sliding Window Movement}
-\end{figure}
-\end{center}
-
-\index{bn\_mp\_lshd.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_lshd.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* shift left a certain amount of digits */
-018 int
-019 mp_lshd (mp_int * a, int b)
-020 \{
-021 int x, res;
-022
-023 /* if its less than zero return */
-024 if (b <= 0) \{
-025 return MP_OKAY;
-026 \}
-027
-028 /* grow to fit the new digits */
-029 if (a->alloc < a->used + b) \{
-030 if ((res = mp_grow (a, a->used + b)) != MP_OKAY) \{
-031 return res;
-032 \}
-033 \}
-034
-035 \{
-036 register mp_digit *top, *bottom;
-037
-038 /* increment the used by the shift amount then copy upwards */
-039 a->used += b;
-040
-041 /* top */
-042 top = a->dp + a->used - 1;
-043
-044 /* base */
-045 bottom = a->dp + a->used - 1 - b;
-046
-047 /* much like mp_rshd this is implemented using a sliding window
-048 * except the window goes the otherway around. Copying from
-049 * the bottom to the top. see bn_mp_rshd.c for more info.
-050 */
-051 for (x = a->used - 1; x >= b; x--) \{
-052 *top-- = *bottom--;
-053 \}
-054
-055 /* zero the lower digits */
-056 top = a->dp;
-057 for (x = 0; x < b; x++) \{
-058 *top++ = 0;
-059 \}
-060 \}
-061 return MP_OKAY;
-062 \}
-\end{alltt}
-\end{small}
-
-The if statement on line 24 ensures that the $b$ variable is greater than zero. The \textbf{used} count is incremented by $b$ before
-the copy loop begins. This elminates the need for an additional variable in the for loop. The variable $top$ on line 42 is an alias
-for the leading digit while $bottom$ on line 45 is an alias for the trailing edge. The aliases form a window of exactly $b$ digits
-over the input.
-
-\subsection{Division by $x$}
-
-Division by powers of $x$ is easily achieved by shifting the digits right and removing any that will end up to the right of the zero'th digit.
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_rshd}. \\
-\textbf{Input}. One mp\_int $a$ and an integer $b$ \\
-\textbf{Output}. $a \leftarrow a / \beta^b$ (Divide by $x^b$). \\
-\hline \\
-1. If $b \le 0$ then return. \\
-2. If $a.used \le b$ then do \\
-\hspace{3mm}2.1 Zero $a$. (\textit{mp\_zero}). \\
-\hspace{3mm}2.2 Return. \\
-3. $i \leftarrow 0$ \\
-4. $j \leftarrow b$ \\
-5. for $n$ from 0 to $a.used - b - 1$ do \\
-\hspace{3mm}5.1 $a_i \leftarrow a_j$ \\
-\hspace{3mm}5.2 $i \leftarrow i + 1$ \\
-\hspace{3mm}5.3 $j \leftarrow j + 1$ \\
-6. for $n$ from $a.used - b$ to $a.used - 1$ do \\
-\hspace{3mm}6.1 $a_n \leftarrow 0$ \\
-7. $a.used \leftarrow a.used - b$ \\
-8. Return. \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_rshd}
-\end{figure}
-
-\textbf{Algorithm mp\_rshd.}
-This algorithm divides the input in place by the $b$'th power of $x$. It is analogous to dividing by a $\beta^b$ but much quicker since
-it does not require single precision division. This algorithm does not actually return an error code as it cannot fail.
-
-If the input $b$ is less than one the algorithm quickly returns without performing any work. If the \textbf{used} count is less than or equal
-to the shift count $b$ then it will simply zero the input and return.
-
-After the trivial cases of inputs have been handled the sliding window is setup. Much like the case of algorithm mp\_lshd a sliding window that
-is $b$ digits wide is used to copy the digits. Unlike mp\_lshd the window slides in the opposite direction from the trailing to the leading digit.
-Also the digits are copied from the leading to the trailing edge.
-
-Once the window copy is complete the upper digits must be zeroed and the \textbf{used} count decremented.
-
-\index{bn\_mp\_rshd.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_rshd.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* shift right a certain amount of digits */
-018 void
-019 mp_rshd (mp_int * a, int b)
-020 \{
-021 int x;
-022
-023 /* if b <= 0 then ignore it */
-024 if (b <= 0) \{
-025 return;
-026 \}
-027
-028 /* if b > used then simply zero it and return */
-029 if (a->used <= b) \{
-030 mp_zero (a);
-031 return;
-032 \}
-033
-034 \{
-035 register mp_digit *bottom, *top;
-036
-037 /* shift the digits down */
-038
-039 /* bottom */
-040 bottom = a->dp;
-041
-042 /* top [offset into digits] */
-043 top = a->dp + b;
-044
-045 /* this is implemented as a sliding window where
-046 * the window is b-digits long and digits from
-047 * the top of the window are copied to the bottom
-048 *
-049 * e.g.
-050
-051 b-2 | b-1 | b0 | b1 | b2 | ... | bb | ---->
-052 /\symbol{92} | ---->
-053 \symbol{92}-------------------/ ---->
-054 */
-055 for (x = 0; x < (a->used - b); x++) \{
-056 *bottom++ = *top++;
-057 \}
-058
-059 /* zero the top digits */
-060 for (; x < a->used; x++) \{
-061 *bottom++ = 0;
-062 \}
-063 \}
-064
-065 /* remove excess digits */
-066 a->used -= b;
-067 \}
-\end{alltt}
-\end{small}
-
-The only noteworthy element of this routine is the lack of a return type.
-
--- Will update later to give it a return type...Tom
-
-\section{Powers of Two}
-
-Now that algorithms for moving single bits as well as whole digits exist algorithms for moving the ``in between'' distances are required. For
-example, to quickly multiply by $2^k$ for any $k$ without using a full multiplier algorithm would prove useful. Instead of performing single
-shifts $k$ times to achieve a multiplication by $2^{\pm k}$ a mixture of whole digit shifting and partial digit shifting is employed.
-
-\subsection{Multiplication by Power of Two}
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_mul\_2d}. \\
-\textbf{Input}. One mp\_int $a$ and an integer $b$ \\
-\textbf{Output}. $c \leftarrow a \cdot 2^b$. \\
-\hline \\
-1. $c \leftarrow a$. (\textit{mp\_copy}) \\
-2. If $c.alloc < c.used + \lfloor b / lg(\beta) \rfloor + 2$ then grow $c$ accordingly. \\
-3. If the reallocation failed return(\textit{MP\_MEM}). \\
-4. If $b \ge lg(\beta)$ then \\
-\hspace{3mm}4.1 $c \leftarrow c \cdot \beta^{\lfloor b / lg(\beta) \rfloor}$ (\textit{mp\_lshd}). \\
-\hspace{3mm}4.2 If step 4.1 failed return(\textit{MP\_MEM}). \\
-5. $d \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\
-6. If $d \ne 0$ then do \\
-\hspace{3mm}6.1 $mask \leftarrow 2^d$ \\
-\hspace{3mm}6.2 $r \leftarrow 0$ \\
-\hspace{3mm}6.3 for $n$ from $0$ to $c.used - 1$ do \\
-\hspace{6mm}6.3.1 $rr \leftarrow c_n >> (lg(\beta) - d) \mbox{ (mod }mask\mbox{)}$ \\
-\hspace{6mm}6.3.2 $c_n \leftarrow (c_n << d) + r \mbox{ (mod }\beta\mbox{)}$ \\
-\hspace{6mm}6.3.3 $r \leftarrow rr$ \\
-\hspace{3mm}6.4 If $r > 0$ then do \\
-\hspace{6mm}6.4.1 $c_{c.used} \leftarrow r$ \\
-\hspace{6mm}6.4.2 $c.used \leftarrow c.used + 1$ \\
-7. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_mul\_2d}
-\end{figure}
-
-\textbf{Algorithm mp\_mul\_2d.}
-This algorithm multiplies $a$ by $2^b$ and stores the result in $c$. The algorithm uses algorithm mp\_lshd and a derivative of algorithm mp\_mul\_2 to
-quickly compute the product.
-
-First the algorithm will multiply $a$ by $x^{\lfloor b / lg(\beta) \rfloor}$ which will ensure that the remainder multiplicand is less than
-$\beta$. For example, if $b = 37$ and $\beta = 2^{28}$ then this step will multiply by $x$ leaving a multiplication by $2^{37 - 28} = 2^{9}$
-left.
-
-After the digits have been shifted appropriately at most $lg(\beta) - 1$ shifts are left to perform. Step 5 calculates the number of remaining shifts
-required. If it is non-zero a modified shift loop is used to calculate the remaining product.
-Essentially the loop is a generic version of algorith mp\_mul2 designed to handle any shift count in the range $1 \le x < lg(\beta)$. The $mask$
-variable is used to extract the upper $d$ bits to form the carry for the next iteration.
-
-This algorithm is loosely measured as a $O(2n)$ algorithm which means that if the input is $n$-digits that it takes $2n$ ``time'' to
-complete. It is possible to optimize this algorithm down to a $O(n)$ algorithm at a cost of making the algorithm slightly harder to follow.
-
-\index{bn\_mp\_mul\_2d.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_mul\_2d.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* NOTE: This routine requires updating. For instance the c->used = c->all
- oc bit
-018 is wrong. We should just shift c->used digits then set the carry as c->d
- p[c->used] = carry
-019
-020 To be fixed for LTM 0.18
-021 */
-022
-023 /* shift left by a certain bit count */
-024 int
-025 mp_mul_2d (mp_int * a, int b, mp_int * c)
-026 \{
-027 mp_digit d;
-028 int res;
-029
-030 /* copy */
-031 if (a != c) \{
-032 if ((res = mp_copy (a, c)) != MP_OKAY) \{
-033 return res;
-034 \}
-035 \}
-036
-037 if (c->alloc < (int)(c->used + b/DIGIT_BIT + 2)) \{
-038 if ((res = mp_grow (c, c->used + b / DIGIT_BIT + 2)) != MP_OKAY) \{
-039 return res;
-040 \}
-041 \}
-042
-043 /* shift by as many digits in the bit count */
-044 if (b >= (int)DIGIT_BIT) \{
-045 if ((res = mp_lshd (c, b / DIGIT_BIT)) != MP_OKAY) \{
-046 return res;
-047 \}
-048 \}
-049 c->used = c->alloc;
-050
-051 /* shift any bit count < DIGIT_BIT */
-052 d = (mp_digit) (b % DIGIT_BIT);
-053 if (d != 0) \{
-054 register mp_digit *tmpc, mask, r, rr;
-055 register int x;
-056
-057 /* bitmask for carries */
-058 mask = (((mp_digit)1) << d) - 1;
-059
-060 /* alias */
-061 tmpc = c->dp;
-062
-063 /* carry */
-064 r = 0;
-065 for (x = 0; x < c->used; x++) \{
-066 /* get the higher bits of the current word */
-067 rr = (*tmpc >> (DIGIT_BIT - d)) & mask;
-068
-069 /* shift the current word and OR in the carry */
-070 *tmpc = ((*tmpc << d) | r) & MP_MASK;
-071 ++tmpc;
-072
-073 /* set the carry to the carry bits of the current word */
-074 r = rr;
-075 \}
-076 \}
-077 mp_clamp (c);
-078 return MP_OKAY;
-079 \}
-\end{alltt}
-\end{small}
-
-Notes to be revised when code is updated. -- Tom
-
-\subsection{Division by Power of Two}
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_div\_2d}. \\
-\textbf{Input}. One mp\_int $a$ and an integer $b$ \\
-\textbf{Output}. $c \leftarrow \lfloor a / 2^b \rfloor, d \leftarrow a \mbox{ (mod }2^b\mbox{)}$. \\
-\hline \\
-1. If $b \le 0$ then do \\
-\hspace{3mm}1.1 $c \leftarrow a$ (\textit{mp\_copy}) \\
-\hspace{3mm}1.2 $d \leftarrow 0$ (\textit{mp\_zero}) \\
-\hspace{3mm}1.3 Return(\textit{MP\_OKAY}). \\
-2. $c \leftarrow a$ \\
-3. $d \leftarrow a \mbox{ (mod }2^b\mbox{)}$ (\textit{mp\_mod\_2d}) \\
-4. If $b \ge lg(\beta)$ then do \\
-\hspace{3mm}4.1 $c \leftarrow \lfloor c/\beta^{\lfloor b/lg(\beta) \rfloor} \rfloor$ (\textit{mp\_rshd}). \\
-5. $k \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\
-6. If $k \ne 0$ then do \\
-\hspace{3mm}6.1 $mask \leftarrow 2^k$ \\
-\hspace{3mm}6.2 $r \leftarrow 0$ \\
-\hspace{3mm}6.3 for $n$ from $c.used - 1$ to $0$ do \\
-\hspace{6mm}6.3.1 $rr \leftarrow c_n \mbox{ (mod }mask\mbox{)}$ \\
-\hspace{6mm}6.3.2 $c_n \leftarrow (c_n >> k) + (r << (lg(\beta) - k))$ \\
-\hspace{6mm}6.3.3 $r \leftarrow rr$ \\
-7. Clamp excess digits of $c$. (\textit{mp\_clamp}) \\
-8. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_div\_2d}
-\end{figure}
-
-\textbf{Algorithm mp\_div\_2d.}
-This algorithm will divide an input $a$ by $2^b$ and produce the quotient and remainder. The algorithm is designed much like algorithm
-mp\_mul\_2d by first using whole digit shifts then single precision shifts. This algorithm will also produce the remainder of the division
-by using algorithm mp\_mod\_2d.
-
-\index{bn\_mp\_div\_2d.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_div\_2d.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* shift right by a certain bit count (store quotient in c, optional remaind
- er in d) */
-018 int
-019 mp_div_2d (mp_int * a, int b, mp_int * c, mp_int * d)
-020 \{
-021 mp_digit D, r, rr;
-022 int x, res;
-023 mp_int t;
-024
-025
-026 /* if the shift count is <= 0 then we do no work */
-027 if (b <= 0) \{
-028 res = mp_copy (a, c);
-029 if (d != NULL) \{
-030 mp_zero (d);
-031 \}
-032 return res;
-033 \}
-034
-035 if ((res = mp_init (&t)) != MP_OKAY) \{
-036 return res;
-037 \}
-038
-039 /* get the remainder */
-040 if (d != NULL) \{
-041 if ((res = mp_mod_2d (a, b, &t)) != MP_OKAY) \{
-042 mp_clear (&t);
-043 return res;
-044 \}
-045 \}
-046
-047 /* copy */
-048 if ((res = mp_copy (a, c)) != MP_OKAY) \{
-049 mp_clear (&t);
-050 return res;
-051 \}
-052
-053 /* shift by as many digits in the bit count */
-054 if (b >= (int)DIGIT_BIT) \{
-055 mp_rshd (c, b / DIGIT_BIT);
-056 \}
-057
-058 /* shift any bit count < DIGIT_BIT */
-059 D = (mp_digit) (b % DIGIT_BIT);
-060 if (D != 0) \{
-061 register mp_digit *tmpc, mask;
-062
-063 /* mask */
-064 mask = (((mp_digit)1) << D) - 1;
-065
-066 /* alias */
-067 tmpc = c->dp + (c->used - 1);
-068
-069 /* carry */
-070 r = 0;
-071 for (x = c->used - 1; x >= 0; x--) \{
-072 /* get the lower bits of this word in a temp */
-073 rr = *tmpc & mask;
-074
-075 /* shift the current word and mix in the carry bits from the previous
- word */
-076 *tmpc = (*tmpc >> D) | (r << (DIGIT_BIT - D));
-077 --tmpc;
-078
-079 /* set the carry to the carry bits of the current word found above */
-080 r = rr;
-081 \}
-082 \}
-083 mp_clamp (c);
-084 if (d != NULL) \{
-085 mp_exch (&t, d);
-086 \}
-087 mp_clear (&t);
-088 return MP_OKAY;
-089 \}
-\end{alltt}
-\end{small}
-
-The implementation of algorithm mp\_div\_2d is slightly different than the algorithm specifies. The remainder $d$ may be optionally
-ignored by passing \textbf{NULL} as the pointer to the mp\_int variable. The temporary mp\_int variable $t$ is used to hold the
-result of the remainder operation until the end. This allows $d$ and $a$ to represent the same mp\_int without modifying $a$ before
-the quotient is obtained.
-
-The remainder of the source code is essentially the same as the source code for mp\_mul\_2d. (-- Fix this paragraph up later, Tom).
-
-\subsection{Remainder of Division by Power of Two}
-
-The last algorithm in the series of polynomial basis power of two algorithms is calculating the remainder of division by $2^b$. This
-algorithm benefits from the fact that in twos complement arithmetic $a \mbox{ (mod }2^b\mbox{)}$ is the same as $a$ AND $2^b - 1$.
-
-\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_mod\_2d}. \\
-\textbf{Input}. One mp\_int $a$ and an integer $b$ \\
-\textbf{Output}. $c \leftarrow a \mbox{ (mod }2^b\mbox{)}$. \\
-\hline \\
-1. If $b \le 0$ then do \\
-\hspace{3mm}1.1 $c \leftarrow 0$ (\textit{mp\_zero}) \\
-\hspace{3mm}1.2 Return(\textit{MP\_OKAY}). \\
-2. If $b > a.used \cdot lg(\beta)$ then do \\
-\hspace{3mm}2.1 $c \leftarrow a$ (\textit{mp\_copy}) \\
-\hspace{3mm}2.2 Return the result of step 2.1. \\
-3. $c \leftarrow a$ \\
-4. If step 3 failed return(\textit{MP\_MEM}). \\
-5. for $n$ from $\lceil b / lg(\beta) \rceil$ to $c.used$ do \\
-\hspace{3mm}5.1 $c_n \leftarrow 0$ \\
-6. $k \leftarrow b \mbox{ (mod }lg(\beta)\mbox{)}$ \\
-7. $c_{\lfloor b / lg(\beta) \rfloor} \leftarrow c_{\lfloor b / lg(\beta) \rfloor} \mbox{ (mod }2^{k}\mbox{)}$. \\
-8. Clamp excess digits of $c$. (\textit{mp\_clamp}) \\
-9. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_mod\_2d}
-\end{figure}
-
-\textbf{Algorithm mp\_mod\_2d.}
-This algorithm will quickly calculate the value of $a \mbox{ (mod }2^b\mbox{)}$. First if $b$ is less than or equal to zero the
-result is set to zero. If $b$ is greater than the number of bits in $a$ then it simply copies $a$ to $c$ and returns. Otherwise, $a$
-is copied to $b$, leading digits are removed and the remaining leading digit is trimed to the exact bit count.
-
-\index{bn\_mp\_mod\_2d.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_mod\_2d.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* calc a value mod 2\b */
-018 int
-019 mp_mod_2d (mp_int * a, int b, mp_int * c)
-020 \{
-021 int x, res;
-022
-023
-024 /* if b is <= 0 then zero the int */
-025 if (b <= 0) \{
-026 mp_zero (c);
-027 return MP_OKAY;
-028 \}
-029
-030 /* if the modulus is larger than the value than return */
-031 if (b > (int) (a->used * DIGIT_BIT)) \{
-032 res = mp_copy (a, c);
-033 return res;
-034 \}
-035
-036 /* copy */
-037 if ((res = mp_copy (a, c)) != MP_OKAY) \{
-038 return res;
-039 \}
-040
-041 /* zero digits above the last digit of the modulus */
-042 for (x = (b / DIGIT_BIT) + ((b % DIGIT_BIT) == 0 ? 0 : 1); x < c->used; x+
- +) \{
-043 c->dp[x] = 0;
-044 \}
-045 /* clear the digit that is not completely outside/inside the modulus */
-046 c->dp[b / DIGIT_BIT] &=
-047 (mp_digit) ((((mp_digit) 1) << (((mp_digit) b) % DIGIT_BIT)) - ((mp_digi
- t) 1));
-048 mp_clamp (c);
-049 return MP_OKAY;
-050 \}
-\end{alltt}
-\end{small}
-
--- Add comments later, Tom.
-
-\section*{Exercises}
-\begin{tabular}{cl}
-$\left [ 3 \right ] $ & Devise an algorithm that performs $a \cdot 2^b$ for generic values of $b$ \\
- & in $O(n)$ time. \\
- &\\
-$\left [ 3 \right ] $ & Devise an efficient algorithm to multiply by small low hamming \\
- & weight values such as $3$, $5$ and $9$. Extend it to handle all values \\
- & upto $64$ with a hamming weight less than three. \\
- &\\
-$\left [ 2 \right ] $ & Modify the preceding algorithm to handle values of the form \\
- & $2^k - 1$ as well. \\
- &\\
-$\left [ 3 \right ] $ & Using only algorithms mp\_mul\_2, mp\_div\_2 and mp\_add create an \\
- & algorithm to multiply two integers in roughly $O(2n^2)$ time for \\
- & any $n$-bit input. Note that the time of addition is ignored in the \\
- & calculation. \\
- & \\
-$\left [ 5 \right ] $ & Improve the previous algorithm to have a working time of at most \\
- & $O \left (2^{(k-1)}n + \left ({2n^2 \over k} \right ) \right )$ for an appropriate choice of $k$. Again ignore \\
- & the cost of addition. \\
- & \\
-$\left [ 2 \right ] $ & Devise a chart to find optimal values of $k$ for the previous problem \\
- & for $n = 64 \ldots 1024$ in steps of $64$. \\
- & \\
-$\left [ 2 \right ] $ & Using only algorithms mp\_abs and mp\_sub devise another method for \\
- & calculating the result of a signed comparison. \\
- &
-\end{tabular}
-
-\chapter{Multiplication and Squaring}
-\section{The Multipliers}
-For most number theoretic problems including certain public key cryptographic algorithms, the ``multipliers'' form the most important subset of
-algorithms of any multiple precision integer package. The set of multiplier algorithms include integer multiplication, squaring and modular reduction
-where in each of the algorithms single precision multiplication is the dominant operation performed. This chapter will discuss integer multiplication
-and squaring, leaving modular reductions for the subsequent chapter.
-
-The importance of the multiplier algorithms is for the most part driven by the fact that certain popular public key algorithms are based on modular
-exponentiation, that is computing $d \equiv a^b \mbox{ (mod }c\mbox{)}$ for some arbitrary choice of $a$, $b$, $c$ and $d$. During a modular
-exponentiation the majority\footnote{Roughly speaking a modular exponentiation will spend about 40\% of the time performing modular reductions,
-35\% of the time performing squaring and 25\% of the time performing multiplications.} of the processor time is spent performing single precision
-multiplications.
-
-For centuries general purpose multiplication has required a lengthly $O(n^2)$ process, whereby each digit of one multiplicand has to be multiplied
-against every digit of the other multiplicand. Traditional long-hand multiplication is based on this process; while the techniques can differ the
-overall algorithm used is essentially the same. Only ``recently'' have faster algorithms been studied. First Karatsuba multiplication was discovered in
-1962. This algorithm can multiply two numbers with considerably fewer single precision multiplications when compared to the long-hand approach.
-This technique led to the discovery of polynomial basis algorithms (\textit{good reference?}) and subquently Fourier Transform based solutions.
-
-\section{Multiplication}
-\subsection{The Baseline Multiplication}
-\index{baseline multiplication}
-Computing the product of two integers in software can be achieved using a trivial adaptation of the standard $O(n^2)$ long-hand multiplication
-algorithm that school children are taught. The algorithm is considered an $O(n^2)$ algoritn since for two $n$-digit inputs $n^2$ single precision
-multiplications are required. More specifically for a $m$ and $n$ digit input $m \cdot n$ single precision multiplications are required. To
-simplify most discussions, it will be assumed that the inputs have comparable number of digits.
-
-The ``baseline multiplication'' algorithm is designed to act as the ``catch-all'' algorithm, only to be used when the faster algorithms cannot be
-used. This algorithm does not use any particularly interesting optimizations and should ideally be avoided if possible. One important
-facet of this algorithm, is that it has been modified to only produce a certain amount of output digits as resolution. The importance of this
-modification will become evident during the discussion of Barrett modular reduction. Recall that for a $n$ and $m$ digit input the product
-will be at most $n + m$ digits. Therefore, this algorithm can be reduced to a full multiplier by having it produce $n + m$ digits of the product.
-
-Recall from sub-section 5.2.2 the definition of $\gamma$ as the number of bits in the type \textbf{mp\_digit}. We shall now extend the variable set to
-include $\alpha$ which shall represent the number of bits in the type \textbf{mp\_word}. This implies that $2^{\alpha} > 2 \cdot \beta^2$. The
-constant $\delta = 2^{\alpha - 2lg(\beta)}$ will represent the maximal weight of any column in a product (\textit{see sub-section 6.2.2 for more information}).
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{s\_mp\_mul\_digs}. \\
-\textbf{Input}. mp\_int $a$, mp\_int $b$ and an integer $digs$ \\
-\textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert \mbox{ (mod }\beta^{digs}\mbox{)}$. \\
-\hline \\
-1. If min$(a.used, b.used) < \delta$ then do \\
-\hspace{3mm}1.1 Calculate $c = \vert a \vert \cdot \vert b \vert$ by the Comba method (\textit{see algorithm~\ref{fig:COMBAMULT}}). \\
-\hspace{3mm}1.2 Return the result of step 1.1 \\
-\\
-Allocate and initialize a temporary mp\_int. \\
-2. Init $t$ to be of size $digs$ \\
-3. If step 2 failed return(\textit{MP\_MEM}). \\
-4. $t.used \leftarrow digs$ \\
-\\
-Compute the product. \\
-5. for $ix$ from $0$ to $a.used - 1$ do \\
-\hspace{3mm}5.1 $u \leftarrow 0$ \\
-\hspace{3mm}5.2 $pb \leftarrow \mbox{min}(b.used, digs - ix)$ \\
-\hspace{3mm}5.3 If $pb < 1$ then goto step 6. \\
-\hspace{3mm}5.4 for $iy$ from $0$ to $pb - 1$ do \\
-\hspace{6mm}5.4.1 $\hat r \leftarrow t_{iy + ix} + a_{ix} \cdot b_{iy} + u$ \\
-\hspace{6mm}5.4.2 $t_{iy + ix} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\
-\hspace{6mm}5.4.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\
-\hspace{3mm}5.5 if $ix + pb < digs$ then do \\
-\hspace{6mm}5.5.1 $t_{ix + pb} \leftarrow u$ \\
-6. Clamp excess digits of $t$. \\
-7. Swap $c$ with $t$ \\
-8. Clear $t$ \\
-9. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm s\_mp\_mul\_digs}
-\end{figure}
-
-\textbf{Algorithm s\_mp\_mul\_digs.}
-This algorithm computes the unsigned product of two inputs $a$ and $b$, limited to an output precision of $digs$ digits. While it may seem
-a bit awkward to modify the function from its simple $O(n^2)$ description, the usefulness of partial multipliers will arise in a subsequent
-algorithm. The algorithm is loosely based on algorithm 14.12 from \cite[pp. 595]{HAC} and is similar to Algorithm M of Knuth \cite[pp. 268]{TAOCPV2}.
-Algorithm s\_mp\_mul\_digs differs from these cited references since it can produce a variable output precision regardless of the precision of the
-inputs.
-
-The first thing this algorithm checks for is whether a Comba multiplier can be used instead. If the minimum digit count of either
-input is less than $\delta$, then the Comba method may be used instead. After the Comba method is ruled out, the baseline algorithm begins. A
-temporary mp\_int variable $t$ is used to hold the intermediate result of the product. This allows the algorithm to be used to
-compute products when either $a = c$ or $b = c$ without overwriting the inputs.
-
-All of step 5 is the infamous $O(n^2)$ multiplication loop slightly modified to only produce upto $digs$ digits of output. The $pb$ variable
-is given the count of digits to read from $b$ inside the nested loop. If $pb \le 1$ then no more output digits can be produced and the algorithm
-will exit the loop. The best way to think of the loops are as a series of $pb \times 1$ multiplications. That is, in each pass of the
-innermost loop $a_{ix}$ is multiplied against $b$ and the result is added (\textit{with an appropriate shift}) to $t$.
-
-For example, consider multiplying $576$ by $241$. That is equivalent to computing $10^0(1)(576) + 10^1(4)(576) + 10^2(2)(576)$ which is best
-visualized in the following table.
-
-\begin{figure}[here]
-\begin{center}
-\begin{tabular}{|c|c|c|c|c|c|l|}
-\hline && & 5 & 7 & 6 & \\
-\hline $\times$&& & 2 & 4 & 1 & \\
-\hline &&&&&&\\
- && & 5 & 7 & 6 & $10^0(1)(576)$ \\
- &2 & 3 & 6 & 1 & 6 & $10^1(4)(576) + 10^0(1)(576)$ \\
- 1 & 3 & 8 & 8 & 1 & 6 & $10^2(2)(576) + 10^1(4)(576) + 10^0(1)(576)$ \\
-\hline
-\end{tabular}
-\end{center}
-\caption{Long-Hand Multiplication Diagram}
-\end{figure}
-
-Each row of the product is added to the result after being shifted to the left (\textit{multiplied by a power of the radix}) by the appropriate
-count. That is in pass $ix$ of the inner loop the product is added starting at the $ix$'th digit of the reult.
-
-Step 5.4.1 introduces the hat symbol (\textit{e.g. $\hat r$}) which represents a double precision variable. The multiplication on that step
-is assumed to be a double wide output single precision multiplication. That is, two single precision variables are multiplied to produce a
-double precision result. The step is somewhat optimized from a long-hand multiplication algorithm because the carry from the addition in step
-5.4.1 is propagated through the nested loop. If the carry was not propagated immediately it would overflow the single precision digit
-$t_{ix+iy}$ and the result would be lost.
-
-At step 5.5 the nested loop is finished and any carry that was left over should be forwarded. The carry does not have to be added to the $ix+pb$'th
-digit since that digit is assumed to be zero at this point. However, if $ix + pb \ge digs$ the carry is not set as it would make the result
-exceed the precision requested.
-
-\index{bn\_s\_mp\_mul\_digs.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_mul\_digs.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* multiplies |a| * |b| and only computes upto digs digits of result
-018 * HAC pp. 595, Algorithm 14.12 Modified so you can control how
-019 * many digits of output are created.
-020 */
-021 int
-022 s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs)
-023 \{
-024 mp_int t;
-025 int res, pa, pb, ix, iy;
-026 mp_digit u;
-027 mp_word r;
-028 mp_digit tmpx, *tmpt, *tmpy;
-029
-030 /* can we use the fast multiplier? */
-031 if (((digs) < MP_WARRAY) &&
-032 MIN (a->used, b->used) <
-033 (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) \{
-034 return fast_s_mp_mul_digs (a, b, c, digs);
-035 \}
-036
-037 if ((res = mp_init_size (&t, digs)) != MP_OKAY) \{
-038 return res;
-039 \}
-040 t.used = digs;
-041
-042 /* compute the digits of the product directly */
-043 pa = a->used;
-044 for (ix = 0; ix < pa; ix++) \{
-045 /* set the carry to zero */
-046 u = 0;
-047
-048 /* limit ourselves to making digs digits of output */
-049 pb = MIN (b->used, digs - ix);
-050
-051 /* setup some aliases */
-052 /* copy of the digit from a used within the nested loop */
-053 tmpx = a->dp[ix];
-054
-055 /* an alias for the destination shifted ix places */
-056 tmpt = t.dp + ix;
-057
-058 /* an alias for the digits of b */
-059 tmpy = b->dp;
-060
-061 /* compute the columns of the output and propagate the carry */
-062 for (iy = 0; iy < pb; iy++) \{
-063 /* compute the column as a mp_word */
-064 r = ((mp_word) *tmpt) +
-065 ((mp_word) tmpx) * ((mp_word) * tmpy++) +
-066 ((mp_word) u);
-067
-068 /* the new column is the lower part of the result */
-069 *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK));
-070
-071 /* get the carry word from the result */
-072 u = (mp_digit) (r >> ((mp_word) DIGIT_BIT));
-073 \}
-074 /* set carry if it is placed below digs */
-075 if (ix + iy < digs) \{
-076 *tmpt = u;
-077 \}
-078 \}
-079
-080 mp_clamp (&t);
-081 mp_exch (&t, c);
-082
-083 mp_clear (&t);
-084 return MP_OKAY;
-085 \}
-\end{alltt}
-\end{small}
-
-Lines 31 to 35 determine if the Comba method can be used first. The conditions for using the Comba routine are that min$(a.used, b.used) < \delta$ and
-the number of digits of output is less than \textbf{MP\_WARRAY}. This new constant is used to control
-the stack usage in the Comba routines. By default it is set to $\delta$ but can be reduced when memory is at a premium.
-
-Of particular importance is the calculation of the $ix+iy$'th column on lines 64, 65 and 66. Note how all of the
-variables are cast to the type \textbf{mp\_word}, which is also the type of variable $\hat r$. That is to ensure that double precision operations
-are used instead of single precision. The multiplication on line 65 makes use of a specific GCC optimizer behaviour. On the outset it looks like
-the compiler will have to use a double precision multiplication to produce the result required. Such an operation would be horribly slow on most
-processors and drag this to a crawl. However, GCC is smart enough to realize that double wide output single precision multipliers can be used. For
-example, the instruction ``MUL'' on the x86 processor can multiply two 32-bit values and produce a 64-bit result.
-
-\subsection{Faster Multiplication by the ``Comba'' Method}
-
-One of the huge drawbacks of the ``baseline'' algorithms is that at the $O(n^2)$ level the carry must be computed and propagated upwards. This
-makes the nested loop very sequential and hard to unroll and implement in parallel. The ``Comba'' \cite{COMBA} method is named after little known
-(\textit{in cryptographic venues}) Paul G. Comba who described a method of implementing fast multipliers that do not require nested
-carry fixup operations. As an interesting aside it seems that Paul Barrett describes a similar technique in
-his 1986 paper \cite{BARRETT} written five years before.
-
-At the heart of the Comba technique is once again the long-hand algorithm. Except in this case a slight twist is placed on how
-the columns of the result are produced. In the standard long-hand algorithm rows of products are produced then added together to form the
-final result. In the baseline algorithm the columns are added together after each iteration to get the result instantaneously.
-
-In the Comba algorithm the columns of the result are produced entirely independently of each other. That is at the $O(n^2)$ level a
-simple multiplication and addition step is performed. The carries of the columns are propagated after the nested loop to reduce the amount
-of work requiored. Succintly the first step of the algorithm is to compute the product vector $\vec x$ as follows.
-
-\begin{equation}
-\vec x_n = \sum_{i+j = n} a_ib_j, \forall n \in \lbrace 0, 1, 2, \ldots, i + j \rbrace
-\end{equation}
-
-Where $\vec x_n$ is the $n'th$ column of the output vector. Consider the following example which computes the vector $\vec x$ for the multiplication
-of $576$ and $241$.
-
-\newpage\begin{figure}[here]
-\begin{small}
-\begin{center}
-\begin{tabular}{|c|c|c|c|c|c|}
- \hline & & 5 & 7 & 6 & First Input\\
- \hline $\times$ & & 2 & 4 & 1 & Second Input\\
-\hline & & $1 \cdot 5 = 5$ & $1 \cdot 7 = 7$ & $1 \cdot 6 = 6$ & First pass \\
- & $4 \cdot 5 = 20$ & $4 \cdot 7+5=33$ & $4 \cdot 6+7=31$ & 6 & Second pass \\
- $2 \cdot 5 = 10$ & $2 \cdot 7 + 20 = 34$ & $2 \cdot 6+33=45$ & 31 & 6 & Third pass \\
-\hline 10 & 34 & 45 & 31 & 6 & Final Result \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Comba Multiplication Diagram}
-\end{figure}
-
-At this point the vector $x = \left < 10, 34, 45, 31, 6 \right >$ is the result of the first step of the Comba multipler.
-Now the columns must be fixed by propagating the carry upwards. The resultant vector will have one extra dimension over the input vector which is
-congruent to adding a leading zero digit.
-
-\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{Comba Fixup}. \\
-\textbf{Input}. Vector $\vec x$ of dimension $k$ \\
-\textbf{Output}. Vector $\vec x$ such that the carries have been propagated. \\
-\hline \\
-1. for $n$ from $0$ to $k - 1$ do \\
-\hspace{3mm}1.1 $\vec x_{n+1} \leftarrow \vec x_{n+1} + \lfloor \vec x_{n}/\beta \rfloor$ \\
-\hspace{3mm}1.2 $\vec x_{n} \leftarrow \vec x_{n} \mbox{ (mod }\beta\mbox{)}$ \\
-2. Return($\vec x$). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm Comba Fixup}
-\end{figure}
-
-With that algorithm and $k = 5$ and $\beta = 10$ the following vector is produced $\vec x= \left < 1, 3, 8, 8, 1, 6 \right >$. In this case
-$241 \cdot 576$ is in fact $138816$ and the procedure succeeded. If the algorithm is correct and as will be demonstrated shortly more
-efficient than the baseline algorithm why not simply always use this algorithm?
-
-\subsubsection{Column Weight.}
-At the nested $O(n^2)$ level the Comba method adds the product of two single precision variables to each column of the output
-independently. A serious obstacle is if the carry is lost, due to lack of precision before the algorithm has a chance to fix
-the carries. For example, in the multiplication of two three-digit numbers the third column of output will be the sum of
-three single precision multiplications. If the precision of the accumulator for the output digits is less then $3 \cdot (\beta - 1)^2$ then
-an overflow can occur and the carry information will be lost. For any $m$ and $n$ digit inputs the maximum weight of any column is
-min$(m, n)$ which is fairly obvious.
-
-The maximum number of terms in any column of a product is known as the ``column weight'' and strictly governs when the algorithm can be used. Recall
-from earlier that a double precision type has $\alpha$ bits of resolution and a single precision digit has $lg(\beta)$ bits of precision. Given these
-two quantities we must not violate the following
-
-\begin{equation}
-k \cdot \left (\beta - 1 \right )^2 < 2^{\alpha}
-\end{equation}
-
-Which reduces to
-
-\begin{equation}
-k \cdot \left ( \beta^2 - 2\beta + 1 \right ) < 2^{\alpha}
-\end{equation}
-
-Let $\rho = lg(\beta)$ represent the number of bits in a single precision digit. By further re-arrangement of the equation the final solution is
-found.
-
-\begin{equation}
-k < {{2^{\alpha}} \over {\left (2^{2\rho} - 2^{\rho + 1} + 1 \right )}}
-\end{equation}
-
-The defaults for LibTomMath are $\beta = 2^{28}$ and $\alpha = 2^{64}$ which means that $k$ is bounded by $k < 257$. In this configuration
-the smaller input may not have more than $256$ digits if the Comba method is to be used. This is quite satisfactory for most applications since
-$256$ digits would allow for numbers in the range of $0 \le x < 2^{7168}$ which, is much larger than most public key cryptographic algorithms require.
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{fast\_s\_mp\_mul\_digs}. \\
-\textbf{Input}. mp\_int $a$, mp\_int $b$ and an integer $digs$ \\
-\textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert \mbox{ (mod }\beta^{digs}\mbox{)}$. \\
-\hline \\
-Place an array of \textbf{MP\_WARRAY} double precision digits named $\hat W$ on the stack. \\
-1. If $c.alloc < digs$ then grow $c$ to $digs$ digits. (\textit{mp\_grow}) \\
-2. If step 1 failed return(\textit{MP\_MEM}).\\
-\\
-Zero the temporary array $\hat W$. \\
-3. for $n$ from $0$ to $digs - 1$ do \\
-\hspace{3mm}3.1 $\hat W_n \leftarrow 0$ \\
-\\
-Compute the columns. \\
-4. for $ix$ from $0$ to $a.used - 1$ do \\
-\hspace{3mm}4.1 $pb \leftarrow \mbox{min}(b.used, digs - ix)$ \\
-\hspace{3mm}4.2 If $pb < 1$ then goto step 5. \\
-\hspace{3mm}4.3 for $iy$ from $0$ to $pb - 1$ do \\
-\hspace{6mm}4.3.1 $\hat W_{ix+iy} \leftarrow \hat W_{ix+iy} + a_{ix}b_{iy}$ \\
-\\
-Propagate the carries upwards. \\
-5. $oldused \leftarrow c.used$ \\
-6. $c.used \leftarrow digs$ \\
-7. If $digs > 1$ then do \\
-\hspace{3mm}7.1. for $ix$ from $1$ to $digs - 1$ do \\
-\hspace{6mm}7.1.1 $\hat W_{ix} \leftarrow \hat W_{ix} + \lfloor \hat W_{ix-1} / \beta \rfloor$ \\
-\hspace{6mm}7.1.2 $c_{ix - 1} \leftarrow \hat W_{ix - 1} \mbox{ (mod }\beta\mbox{)}$ \\
-8. else do \\
-\hspace{3mm}8.1 $ix \leftarrow 0$ \\
-9. $c_{ix} \leftarrow \hat W_{ix} \mbox{ (mod }\beta\mbox{)}$ \\
-\\
-Zero excess digits. \\
-10. If $digs < oldused$ then do \\
-\hspace{3mm}10.1 for $n$ from $digs$ to $oldused - 1$ do \\
-\hspace{6mm}10.1.1 $c_n \leftarrow 0$ \\
-11. Clamp excessive digits of $c$. (\textit{mp\_clamp}) \\
-12. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm fast\_s\_mp\_mul\_digs}
-\label{fig:COMBAMULT}
-\end{figure}
-
-\textbf{Algorithm fast\_s\_mp\_mul\_digs.}
-This algorithm performs the unsigned multiplication of $a$ and $b$ using the Comba method limited to $digs$ digits of precision. The algorithm
-essentially peforms the same calculation as algorithm s\_mp\_mul\_digs, just much faster.
-
-The array $\hat W$ is meant to be on the stack when the algorithm is used. The size of the array does not change which is ideal. Note also that
-unlike algorithm s\_mp\_mul\_digs no temporary mp\_int is required since the result is calculated directly in $\hat W$.
-
-The $O(n^2)$ loop on step four is where the Comba method's advantages begin to show through in comparison to the baseline algorithm. The lack of
-a carry variable or propagation in this loop allows the loop to be performed with only single precision multiplication and additions. Now that each
-iteration of the inner loop can be performed independent of the others the inner loop can be performed with a high level of parallelism.
-
-To measure the benefits of the Comba method over the baseline method consider the number of operations that are required. If the
-cost in terms of time of a multiply and addition is $p$ and the cost of a carry propagation is $q$ then a baseline multiplication would require
-$O \left ((p + q)n^2 \right )$ time to multiply two $n$-digit numbers. The Comba method requires only $O(pn^2 + qn)$ time, however in practice,
-the speed increase is actually much more. With $O(n)$ space the algorithm can be reduced to $O(pn + qn)$ time by implementing the $n$ multiply
-and addition operations in the nested loop in parallel.
-
-\index{bn\_fast\_s\_mp\_mul\_digs.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_fast\_s\_mp\_mul\_digs.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* Fast (comba) multiplier
-018 *
-019 * This is the fast column-array [comba] multiplier. It is
-020 * designed to compute the columns of the product first
-021 * then handle the carries afterwards. This has the effect
-022 * of making the nested loops that compute the columns very
-023 * simple and schedulable on super-scalar processors.
-024 *
-025 * This has been modified to produce a variable number of
-026 * digits of output so if say only a half-product is required
-027 * you don't have to compute the upper half (a feature
-028 * required for fast Barrett reduction).
-029 *
-030 * Based on Algorithm 14.12 on pp.595 of HAC.
-031 *
-032 */
-033 int
-034 fast_s_mp_mul_digs (mp_int * a, mp_int * b, mp_int * c, int digs)
-035 \{
-036 int olduse, res, pa, ix;
-037 mp_word W[MP_WARRAY];
-038
-039 /* grow the destination as required */
-040 if (c->alloc < digs) \{
-041 if ((res = mp_grow (c, digs)) != MP_OKAY) \{
-042 return res;
-043 \}
-044 \}
-045
-046 /* clear temp buf (the columns) */
-047 memset (W, 0, sizeof (mp_word) * digs);
-048
-049 /* calculate the columns */
-050 pa = a->used;
-051 for (ix = 0; ix < pa; ix++) \{
-052 /* this multiplier has been modified to allow you to
-053 * control how many digits of output are produced.
-054 * So at most we want to make upto "digs" digits of output.
-055 *
-056 * this adds products to distinct columns (at ix+iy) of W
-057 * note that each step through the loop is not dependent on
-058 * the previous which means the compiler can easily unroll
-059 * the loop without scheduling problems
-060 */
-061 \{
-062 register mp_digit tmpx, *tmpy;
-063 register mp_word *_W;
-064 register int iy, pb;
-065
-066 /* alias for the the word on the left e.g. A[ix] * A[iy] */
-067 tmpx = a->dp[ix];
-068
-069 /* alias for the right side */
-070 tmpy = b->dp;
-071
-072 /* alias for the columns, each step through the loop adds a new
-073 term to each column
-074 */
-075 _W = W + ix;
-076
-077 /* the number of digits is limited by their placement. E.g.
-078 we avoid multiplying digits that will end up above the # of
-079 digits of precision requested
-080 */
-081 pb = MIN (b->used, digs - ix);
-082
-083 for (iy = 0; iy < pb; iy++) \{
-084 *_W++ += ((mp_word) tmpx) * ((mp_word) * tmpy++);
-085 \}
-086 \}
-087
-088 \}
-089
-090 /* setup dest */
-091 olduse = c->used;
-092 c->used = digs;
-093
-094 \{
-095 register mp_digit *tmpc;
-096
-097 /* At this point W[] contains the sums of each column. To get the
-098 * correct result we must take the extra bits from each column and
-099 * carry them down
-100 *
-101 * Note that while this adds extra code to the multiplier it
-102 * saves time since the carry propagation is removed from the
-103 * above nested loop.This has the effect of reducing the work
-104 * from N*(N+N*c)==N**2 + c*N**2 to N**2 + N*c where c is the
-105 * cost of the shifting. On very small numbers this is slower
-106 * but on most cryptographic size numbers it is faster.
-107 */
-108 tmpc = c->dp;
-109 for (ix = 1; ix < digs; ix++) \{
-110 W[ix] += (W[ix - 1] >> ((mp_word) DIGIT_BIT));
-111 *tmpc++ = (mp_digit) (W[ix - 1] & ((mp_word) MP_MASK));
-112 \}
-113 *tmpc++ = (mp_digit) (W[digs - 1] & ((mp_word) MP_MASK));
-114
-115 /* clear unused */
-116 for (; ix < olduse; ix++) \{
-117 *tmpc++ = 0;
-118 \}
-119 \}
-120
-121 mp_clamp (c);
-122 return MP_OKAY;
-123 \}
-\end{alltt}
-\end{small}
-
-The memset on line 47 clears the initial $\hat W$ array to zero in a single step. Like the slower baseline multiplication
-implementation a series of aliases (\textit{lines 67, 70 and 75}) are used to simplify the inner $O(n^2)$ loop.
-In this case a new alias $\_\hat W$ has been added which refers to the double precision columns offset by $ix$ in each pass.
-
-The inner loop on lines 83, 84 and 85 is where the algorithm will spend the majority of the time, which is why it has been
-stripped to the bones of any extra baggage\footnote{Hence the pointer aliases.}. On x86 processors the multiplication and additions amount to at the
-very least five instructions (\textit{two loads, two additions, one multiply}) while on the ARMv4 processors they amount to only three
-(\textit{one load, one store, one multiply-add}). For both of the x86 and ARMv4 processors the GCC compiler performs a good job at unrolling the loop
-and scheduling the instructions so there are very few dependency stalls.
-
-In theory the difference between the baseline and comba algorithms is a mere $O(qn)$ time difference. However, in the $O(n^2)$ nested loop of the
-baseline method there are dependency stalls as the algorithm must wait for the multiplier to finish before propagating the carry to the next
-digit. As a result fewer of the often multiple execution units\footnote{The AMD Athlon has three execution units and the Intel P4 has four.} can
-be simultaneously used.
-
-\subsection{Polynomial Basis Multiplication}
-To break the $O(n^2)$ barrier in multiplication requires a completely different look at integer multiplication. In the following algorithms
-the use of polynomial basis representation for two integers $a$ and $b$ as $f(x) = \sum_{i=0}^{n} a_i x^i$ and
-$g(x) = \sum_{i=0}^{n} b_i x^i$ respectively, is required. In this system both $f(x)$ and $g(x)$ have $n + 1$ terms and are of the $n$'th degree.
-
-The product $a \cdot b \equiv f(x)g(x)$ is the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$. The coefficients $w_i$ will
-directly yield the desired product when $\beta$ is substituted for $x$. The direct solution to solve for the $2n + 1$ coefficients
-requires $O(n^2)$ time and would in practice be slower than the Comba technique.
-
-However, numerical analysis theory indicates that only $2n + 1$ distinct points in $W(x)$ are required to determine the values of the $2n + 1$ unknown
-coefficients. This means by finding $\zeta_y = W(y)$ for $2n + 1$ small values of $y$ the coefficients of $W(x)$ can be found with
-Gaussian elimination. This technique is also occasionally refered to as the \textit{interpolation technique} (\textit{references please...}) since in
-effect an interpolation based on $2n + 1$ points will yield a polynomial equivalent to $W(x)$.
-
-The coefficients of the polynomial $W(x)$ are unknown which makes finding $W(y)$ for any value of $y$ impossible. However, since
-$W(x) = f(x)g(x)$ the equivalent $\zeta_y = f(y) g(y)$ can be used in its place. The benefit of this technique stems from the
-fact that $f(y)$ and $g(y)$ are much smaller than either $a$ or $b$ respectively. As a result finding the $2n + 1$ relations required
-by multiplying $f(y)g(y)$ involves multiplying integers that are much smaller than either of the inputs.
-
-When picking points to gather relations there are always three obvious points to choose, $y = 0, 1$ and $ \infty$. The $\zeta_0$ term
-is simply the product $W(0) = w_0 = a_0 \cdot b_0$. The $\zeta_1$ term is the product
-$W(1) = \left (\sum_{i = 0}^{n} a_i \right ) \left (\sum_{i = 0}^{n} b_i \right )$. The third point $\zeta_{\infty}$ is less obvious but rather
-simple to explain. The $2n + 1$'th coefficient of $W(x)$ is numerically equivalent to the most significant column in an integer multiplication.
-The point at $\infty$ is used symbolically to represent the most significant column, that is $W(\infty) = w_{2n} = a_nb_n$. Note that the
-points at $y = 0$ and $\infty$ yield the coefficients $w_0$ and $w_{2n}$ directly.
-
-If more points are required they should be of small values and powers of two such as $2^q$ and the related \textit{mirror points}
-$\left (2^q \right )^{2n} \cdot \zeta_{2^{-q}}$ for small values of $q$. The term ``mirror point'' stems from the fact that
-$\left (2^q \right )^{2n} \cdot \zeta_{2^{-q}}$ can be calculated in the exact opposite fashion as $\zeta_{2^q}$. For
-example, when $n = 2$ and $q = 1$ then following two equations are equivalent to the point $\zeta_{2}$ and its mirror.
-
-\begin{eqnarray}
-\zeta_{2} = f(2)g(2) = (4a_2 + 2a_1 + a_0)(4b_2 + 2b_1 + b_0) \nonumber \\
-16 \cdot \zeta_{1 \over 2} = 4f({1\over 2}) \cdot 4g({1 \over 2}) = (a_2 + 2a_1 + 4a_0)(b_2 + 2b_1 + 4b_0)
-\end{eqnarray}
-
-Using such points will allow the values of $f(y)$ and $g(y)$ to be independently calculated using only left shifts. For example, when $n = 2$ the
-polynomial $f(2^q)$ is equal to $2^q((2^qa_2) + a_1) + a_0$. This technique of polynomial representation is known as Horner's method.
-
-As a general rule of the algorithm when the inputs are split into $n$ parts each there are $2n - 1$ multiplications. Each multiplication is of
-multiplicands that have $n$ times fewer digits than the inputs. The asymptotic running time of this algorithm is
-$O \left ( k^{lg_n(2n - 1)} \right )$ for $k$ digit inputs (\textit{assuming they have the same number of digits}). Figure~\ref{fig:exponent}
-summarizes the exponents for various values of $n$.
-
-\begin{figure}
-\begin{center}
-\begin{tabular}{|c|c|c|}
-\hline \textbf{Split into $n$ Parts} & \textbf{Exponent} & \textbf{Notes}\\
-\hline $2$ & $1.584962501$ & This is Karatsuba Multiplication. \\
-\hline $3$ & $1.464973520$ & This is Toom-Cook Multiplication. \\
-\hline $4$ & $1.403677461$ &\\
-\hline $5$ & $1.365212389$ &\\
-\hline $10$ & $1.278753601$ &\\
-\hline $100$ & $1.149426538$ &\\
-\hline $1000$ & $1.100270931$ &\\
-\hline $10000$ & $1.075252070$ &\\
-\hline
-\end{tabular}
-\end{center}
-\caption{Asymptotic Running Time of Polynomial Basis Multiplication}
-\label{fig:exponent}
-\end{figure}
-
-At first it may seem like a good idea to choose $n = 1000$ since the exponent is approximately $1.1$. However, the overhead
-of solving for the 2001 terms of $W(x)$ will certainly consume any savings the algorithm could offer for all but exceedingly large
-numbers.
-
-\subsubsection{Cutoff Point}
-The polynomial basis multiplication algorithms all require fewer single precision multiplications than a straight Comba approach. However,
-the algorithms incur an overhead (\textit{at the $O(n)$ work level}) since they require a system of equations to be solved. This makes the
-polynomial basis approach more costly to use with small inputs.
-
-Let $m$ represent the number of digits in the multiplicands (\textit{assume both multiplicands have the same number of digits}). There exists a
-point $y$ such that when $m < y$ the polynomial basis algorithms are more costly than Comba, when $m = y$ they are roughly the same cost and
-when $m > y$ the Comba methods are slower than the polynomial basis algorithms.
-
-The exact location of $y$ depends on several key architectural elements of the computer platform in question.
-
-\begin{enumerate}
-\item The ratio of clock cycles for single precision multiplication versus other simpler operations such as addition, shifting, etc. For example
-on the AMD Athlon the ratio is roughly $17 : 1$ while on the Intel P4 it is $29 : 1$. The higher the ratio in favour of multiplication the lower
-the cutoff point $y$ will be.
-
-\item The complexity of the linear system of equations (\textit{for the coefficients of $W(x)$}) is. Generally speaking as the number of splits
-grows the complexity grows substantially. Ideally solving the system will only involve addition, subtraction and shifting of integers. This
-directly reflects on the ratio previous mentioned.
-
-\item To a lesser extent memory bandwidth and function call overheads. Provided the values are in the processor cache this is less of an
-influence over the cutoff point.
-
-\end{enumerate}
-
-A clean cutoff point separation occurs when a point $y$ is found such that all of the cutoff point conditions are met. For example, if the point
-is too low then there will be values of $m$ such that $m > y$ and the Comba method is still faster. Finding the cutoff points is fairly simple when
-a high resolution timer is available.
-
-\subsection{Karatsuba Multiplication}
-Karatsuba \cite{KARA} multiplication when originally proposed in 1962 was among the first set of algorithms to break the $O(n^2)$ barrier for
-general purpose multiplication. Given two polynomial basis representations $f(x) = ax + b$ and $g(x) = cx + d$, Karatsuba proved with
-light algebra \cite{KARAP} that the following polynomial is equivalent to multiplication of the two integers the polynomials represent.
-
-\begin{equation}
-f(x) \cdot g(x) = acx^2 + ((a - b)(c - d) + ac + bd)x + bd
-\end{equation}
-
-Using the observation that $ac$ and $bd$ could be re-used only three half sized multiplications would be required to produce the product. Applying
-this algorithm recursively, the work factor becomes $O(n^{lg(3)})$ which is substantially better than the work factor $O(n^2)$ of the Comba technique. It turns
-out what Karatsuba did not know or at least did not publish was that this is simply polynomial basis multiplication with the points
-$\zeta_0$, $\zeta_{\infty}$ and $-\zeta_{-1}$. Consider the resultant system of equations.
-
-\begin{center}
-\begin{tabular}{rcrcrcrc}
-$\zeta_{0}$ & $=$ & & & & & $w_0$ \\
-$-\zeta_{-1}$ & $=$ & $-w_2$ & $+$ & $w_1$ & $-$ & $w_0$ \\
-$\zeta_{\infty}$ & $=$ & $w_2$ & & & & \\
-\end{tabular}
-\end{center}
-
-By adding the first and last equation to the equation in the middle the term $w_1$ can be isolated and all three coefficients solved for. The simplicity
-of this system of equations has made Karatsuba fairly popular. In fact the cutoff point is often fairly low\footnote{With LibTomMath 0.18 it is 70 and 109 digits for the Intel P4 and AMD Athlon respectively.}
-making it an ideal algorithm to speed up certain public key cryptosystems such as RSA and Diffie-Hellman. It is worth noting that the point
-$\zeta_1$ could be substituted for $-\zeta_{-1}$. In this case the first and third row are subtracted instead of added to the second row.
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_karatsuba\_mul}. \\
-\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\
-\textbf{Output}. $c \leftarrow \vert a \vert \cdot \vert b \vert$ \\
-\hline \\
-1. Init the following mp\_int variables: $x0$, $x1$, $y0$, $y1$, $t1$, $x0y0$, $x1y1$.\\
-2. If step 2 failed then return(\textit{MP\_MEM}). \\
-\\
-Split the input. e.g. $a = x1 \cdot \beta^B + x0$ \\
-3. $B \leftarrow \mbox{min}(a.used, b.used)/2$ \\
-4. $x0 \leftarrow a \mbox{ (mod }\beta^B\mbox{)}$ (\textit{mp\_mod\_2d}) \\
-5. $y0 \leftarrow b \mbox{ (mod }\beta^B\mbox{)}$ \\
-6. $x1 \leftarrow \lfloor a / \beta^B \rfloor$ (\textit{mp\_rshd}) \\
-7. $y1 \leftarrow \lfloor b / \beta^B \rfloor$ \\
-\\
-Calculate the three products. \\
-8. $x0y0 \leftarrow x0 \cdot y0$ (\textit{mp\_mul}) \\
-9. $x1y1 \leftarrow x1 \cdot y1$ \\
-10. $t1 \leftarrow x1 - x0$ (\textit{mp\_sub}) \\
-11. $x0 \leftarrow y1 - y0$ \\
-12. $t1 \leftarrow t1 \cdot x0$ \\
-\\
-Calculate the middle term. \\
-13. $x0 \leftarrow x0y0 + x1y1$ \\
-14. $t1 \leftarrow x0 - t1$ \\
-\\
-Calculate the final product. \\
-15. $t1 \leftarrow t1 \cdot \beta^B$ (\textit{mp\_lshd}) \\
-16. $x1y1 \leftarrow x1y1 \cdot \beta^{2B}$ \\
-17. $t1 \leftarrow x0y0 + t1$ \\
-18. $c \leftarrow t1 + x1y1$ \\
-19. Clear all of the temporary variables. \\
-20. Return(\textit{MP\_OKAY}).\\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_karatsuba\_mul}
-\end{figure}
-
-\textbf{Algorithm mp\_karatsuba\_mul.}
-This algorithm computes the unsigned product of two inputs using the Karatsuba multiplication algorithm. It is loosely based on the description
-from Knuth \cite[pp. 294-295]{TAOCPV2}.
-
-\index{radix point}
-In order to split the two inputs into their respective halves, a suitable \textit{radix point} must be chosen. The radix point chosen must
-be used for both of the inputs meaning that it must be smaller than the smallest input. Step 3 chooses the radix point $B$ as half of the
-smallest input \textbf{used} count. After the radix point is chosen the inputs are split into lower and upper halves. Step 4 and 5
-compute the lower halves. Step 6 and 7 computer the upper halves.
-
-After the halves have been computed the three intermediate half-size products must be computed. Step 8 and 9 compute the trivial products
-$x0 \cdot y0$ and $x1 \cdot y1$. The mp\_int $x0$ is used as a temporary variable after $x1 - x0$ has been computed. By using $x0$ instead
-of an additional temporary variable, the algorithm can avoid an addition memory allocation operation.
-
-The remaining steps 13 through 18 compute the Karatsuba polynomial through a variety of digit shifting and addition operations.
-
-\index{bn\_mp\_karatsuba\_mul.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_karatsuba\_mul.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* c = |a| * |b| using Karatsuba Multiplication using
-018 * three half size multiplications
-019 *
-020 * Let B represent the radix [e.g. 2**DIGIT_BIT] and
-021 * let n represent half of the number of digits in
-022 * the min(a,b)
-023 *
-024 * a = a1 * B**n + a0
-025 * b = b1 * B**n + b0
-026 *
-027 * Then, a * b =>
-028 a1b1 * B**2n + ((a1 - a0)(b1 - b0) + a0b0 + a1b1) * B + a0b0
-029 *
-030 * Note that a1b1 and a0b0 are used twice and only need to be
-031 * computed once. So in total three half size (half # of
-032 * digit) multiplications are performed, a0b0, a1b1 and
-033 * (a1-b1)(a0-b0)
-034 *
-035 * Note that a multiplication of half the digits requires
-036 * 1/4th the number of single precision multiplications so in
-037 * total after one call 25% of the single precision multiplications
-038 * are saved. Note also that the call to mp_mul can end up back
-039 * in this function if the a0, a1, b0, or b1 are above the threshold.
-040 * This is known as divide-and-conquer and leads to the famous
-041 * O(N**lg(3)) or O(N**1.584) work which is asymptopically lower than
-042 * the standard O(N**2) that the baseline/comba methods use.
-043 * Generally though the overhead of this method doesn't pay off
-044 * until a certain size (N ~ 80) is reached.
-045 */
-046 int
-047 mp_karatsuba_mul (mp_int * a, mp_int * b, mp_int * c)
-048 \{
-049 mp_int x0, x1, y0, y1, t1, x0y0, x1y1;
-050 int B, err;
-051
-052 /* default the return code to an error */
-053 err = MP_MEM;
-054
-055 /* min # of digits */
-056 B = MIN (a->used, b->used);
-057
-058 /* now divide in two */
-059 B = B / 2;
-060
-061 /* init copy all the temps */
-062 if (mp_init_size (&x0, B) != MP_OKAY)
-063 goto ERR;
-064 if (mp_init_size (&x1, a->used - B) != MP_OKAY)
-065 goto X0;
-066 if (mp_init_size (&y0, B) != MP_OKAY)
-067 goto X1;
-068 if (mp_init_size (&y1, b->used - B) != MP_OKAY)
-069 goto Y0;
-070
-071 /* init temps */
-072 if (mp_init_size (&t1, B * 2) != MP_OKAY)
-073 goto Y1;
-074 if (mp_init_size (&x0y0, B * 2) != MP_OKAY)
-075 goto T1;
-076 if (mp_init_size (&x1y1, B * 2) != MP_OKAY)
-077 goto X0Y0;
-078
-079 /* now shift the digits */
-080 x0.sign = x1.sign = a->sign;
-081 y0.sign = y1.sign = b->sign;
-082
-083 x0.used = y0.used = B;
-084 x1.used = a->used - B;
-085 y1.used = b->used - B;
-086
-087 \{
-088 register int x;
-089 register mp_digit *tmpa, *tmpb, *tmpx, *tmpy;
-090
-091 /* we copy the digits directly instead of using higher level functions
-092 * since we also need to shift the digits
-093 */
-094 tmpa = a->dp;
-095 tmpb = b->dp;
-096
-097 tmpx = x0.dp;
-098 tmpy = y0.dp;
-099 for (x = 0; x < B; x++) \{
-100 *tmpx++ = *tmpa++;
-101 *tmpy++ = *tmpb++;
-102 \}
-103
-104 tmpx = x1.dp;
-105 for (x = B; x < a->used; x++) \{
-106 *tmpx++ = *tmpa++;
-107 \}
-108
-109 tmpy = y1.dp;
-110 for (x = B; x < b->used; x++) \{
-111 *tmpy++ = *tmpb++;
-112 \}
-113 \}
-114
-115 /* only need to clamp the lower words since by definition the
-116 * upper words x1/y1 must have a known number of digits
-117 */
-118 mp_clamp (&x0);
-119 mp_clamp (&y0);
-120
-121 /* now calc the products x0y0 and x1y1 */
-122 /* after this x0 is no longer required, free temp [x0==t2]! */
-123 if (mp_mul (&x0, &y0, &x0y0) != MP_OKAY)
-124 goto X1Y1; /* x0y0 = x0*y0 */
-125 if (mp_mul (&x1, &y1, &x1y1) != MP_OKAY)
-126 goto X1Y1; /* x1y1 = x1*y1 */
-127
-128 /* now calc x1-x0 and y1-y0 */
-129 if (mp_sub (&x1, &x0, &t1) != MP_OKAY)
-130 goto X1Y1; /* t1 = x1 - x0 */
-131 if (mp_sub (&y1, &y0, &x0) != MP_OKAY)
-132 goto X1Y1; /* t2 = y1 - y0 */
-133 if (mp_mul (&t1, &x0, &t1) != MP_OKAY)
-134 goto X1Y1; /* t1 = (x1 - x0) * (y1 - y0) */
-135
-136 /* add x0y0 */
-137 if (mp_add (&x0y0, &x1y1, &x0) != MP_OKAY)
-138 goto X1Y1; /* t2 = x0y0 + x1y1 */
-139 if (mp_sub (&x0, &t1, &t1) != MP_OKAY)
-140 goto X1Y1; /* t1 = x0y0 + x1y1 - (x1-x0)*(y1-y0) */
-141
-142 /* shift by B */
-143 if (mp_lshd (&t1, B) != MP_OKAY)
-144 goto X1Y1; /* t1 = (x0y0 + x1y1 - (x1-x0)*(y1-y0))<<B */
-145 if (mp_lshd (&x1y1, B * 2) != MP_OKAY)
-146 goto X1Y1; /* x1y1 = x1y1 << 2*B */
-147
-148 if (mp_add (&x0y0, &t1, &t1) != MP_OKAY)
-149 goto X1Y1; /* t1 = x0y0 + t1 */
-150 if (mp_add (&t1, &x1y1, c) != MP_OKAY)
-151 goto X1Y1; /* t1 = x0y0 + t1 + x1y1 */
-152
-153 /* Algorithm succeeded set the return code to MP_OKAY */
-154 err = MP_OKAY;
-155
-156 X1Y1:mp_clear (&x1y1);
-157 X0Y0:mp_clear (&x0y0);
-158 T1:mp_clear (&t1);
-159 Y1:mp_clear (&y1);
-160 Y0:mp_clear (&y0);
-161 X1:mp_clear (&x1);
-162 X0:mp_clear (&x0);
-163 ERR:
-164 return err;
-165 \}
-\end{alltt}
-\end{small}
-
-The new coding element in this routine, not seen in previous routines, is the usage of goto statements. The conventional
-wisdom is that goto statements should be avoided. This is generally true, however when every single function call can fail, it makes sense
-to handle error recovery with a single piece of code. Lines 62 to 76 handle initializing all of the temporary variables
-required. Note how each of the if statements goes to a different label in case of failure. This allows the routine to correctly free only
-the temporaries that have been successfully allocated so far.
-
-The temporary variables are all initialized using the mp\_init\_size routine since they are expected to be large. This saves the
-additional reallocation that would have been necessary. Also $x0$, $x1$, $y0$ and $y1$ have to be able to hold at least their respective
-number of digits for the next section of code.
-
-The first algebraic portion of the algorithm is to split the two inputs into their halves. However, instead of using mp\_mod\_2d and mp\_rshd
-to extract the halves, the respective code has been placed inline within the body of the function. To initialize the halves, the \textbf{used} and
-\textbf{sign} members are copied first. The first for loop on line 99 copies the lower halves. Since they are both the same magnitude it
-is simpler to calculate both lower halves in a single loop. The for loop on lines 105 and 110 calculate the upper halves $x1$ and
-$y1$ respectively.
-
-By inlining the calculation of the halves, the Karatsuba multiplier has a slightly lower overhead and can be used for smaller magnitude inputs.
-
-When line 154 is reached, the algorithm has completed succesfully. The ``error status'' variable $err$ is set to \textbf{MP\_OKAY} so that
-the same code that handles errors can be used to clear the temporary variables and return.
-
-\subsection{Toom-Cook $3$-Way Multiplication}
-Toom-Cook $3$-Way \cite{TOOM} multiplication is essentially the polynomial basis algorithm for $n = 3$ except that the points are
-chosen such that $\zeta$ is easy to compute and the resulting system of equations easy to reduce. Here, the points $\zeta_{0}$,
-$16 \cdot \zeta_{1 \over 2}$, $\zeta_1$, $\zeta_2$ and $\zeta_{\infty}$ make up the five required points to solve for the coefficients
-of the $W(x)$.
-
-With the five relations that Toom-Cook specifies, the following system of equations is formed.
-
-\begin{center}
-\begin{tabular}{rcrcrcrcrcr}
-$\zeta_0$ & $=$ & $0w_4$ & $+$ & $0w_3$ & $+$ & $0w_2$ & $+$ & $0w_1$ & $+$ & $1w_0$ \\
-$16 \cdot \zeta_{1 \over 2}$ & $=$ & $1w_4$ & $+$ & $2w_3$ & $+$ & $4w_2$ & $+$ & $8w_1$ & $+$ & $16w_0$ \\
-$\zeta_1$ & $=$ & $1w_4$ & $+$ & $1w_3$ & $+$ & $1w_2$ & $+$ & $1w_1$ & $+$ & $1w_0$ \\
-$\zeta_2$ & $=$ & $16w_4$ & $+$ & $8w_3$ & $+$ & $4w_2$ & $+$ & $2w_1$ & $+$ & $1w_0$ \\
-$\zeta_{\infty}$ & $=$ & $1w_4$ & $+$ & $0w_3$ & $+$ & $0w_2$ & $+$ & $0w_1$ & $+$ & $0w_0$ \\
-\end{tabular}
-\end{center}
-
-A trivial solution to this matrix requires $12$ subtractions, two multiplications by a small power of two, two divisions by a small power
-of two, two divisions by three and one multiplication by three. All of these $19$ sub-operations require less than quadratic time, meaning that
-the algorithm can be faster than a baseline multiplication. However, the greater complexity of this algorithm places the cutoff point
-(\textbf{TOOM\_MUL\_CUTOFF}) where Toom-Cook becomes more efficient much higher than the Karatsuba cutoff point.
-
-\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_toom\_mul}. \\
-\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\
-\textbf{Output}. $c \leftarrow a \cdot b $ \\
-\hline \\
-Split $a$ and $b$ into three pieces. E.g. $a = a_2 \beta^{2k} + a_1 \beta^{k} + a_0$ \\
-1. $k \leftarrow \lfloor \mbox{min}(a.used, b.used) / 3 \rfloor$ \\
-2. $a_0 \leftarrow a \mbox{ (mod }\beta^{k}\mbox{)}$ \\
-3. $a_1 \leftarrow \lfloor a / \beta^k \rfloor$, $a_1 \leftarrow a_1 \mbox{ (mod }\beta^{k}\mbox{)}$ \\
-4. $a_2 \leftarrow \lfloor a / \beta^{2k} \rfloor$, $a_2 \leftarrow a_2 \mbox{ (mod }\beta^{k}\mbox{)}$ \\
-5. $b_0 \leftarrow a \mbox{ (mod }\beta^{k}\mbox{)}$ \\
-6. $b_1 \leftarrow \lfloor a / \beta^k \rfloor$, $b_1 \leftarrow b_1 \mbox{ (mod }\beta^{k}\mbox{)}$ \\
-7. $b_2 \leftarrow \lfloor a / \beta^{2k} \rfloor$, $b_2 \leftarrow b_2 \mbox{ (mod }\beta^{k}\mbox{)}$ \\
-\\
-Find the five equations for $w_0, w_1, ..., w_4$. \\
-8. $w_0 \leftarrow a_0 \cdot b_0$ \\
-9. $w_4 \leftarrow a_2 \cdot b_2$ \\
-10. $tmp_1 \leftarrow 2 \cdot a_0$, $tmp_1 \leftarrow a_1 + tmp_1$, $tmp_1 \leftarrow 2 \cdot tmp_1$, $tmp_1 \leftarrow tmp_1 + a_2$ \\
-11. $tmp_2 \leftarrow 2 \cdot b_0$, $tmp_2 \leftarrow b_1 + tmp_2$, $tmp_2 \leftarrow 2 \cdot tmp_2$, $tmp_2 \leftarrow tmp_2 + b_2$ \\
-12. $w_1 \leftarrow tmp_1 \cdot tmp_2$ \\
-13. $tmp_1 \leftarrow 2 \cdot a_2$, $tmp_1 \leftarrow a_1 + tmp_1$, $tmp_1 \leftarrow 2 \cdot tmp_1$, $tmp_1 \leftarrow tmp_1 + a_0$ \\
-14. $tmp_2 \leftarrow 2 \cdot b_2$, $tmp_2 \leftarrow b_1 + tmp_2$, $tmp_2 \leftarrow 2 \cdot tmp_2$, $tmp_2 \leftarrow tmp_2 + b_0$ \\
-15. $w_3 \leftarrow tmp_1 \cdot tmp_2$ \\
-16. $tmp_1 \leftarrow a_0 + a_1$, $tmp_1 \leftarrow tmp_1 + a_2$, $tmp_2 \leftarrow b_0 + b_1$, $tmp_2 \leftarrow tmp_2 + b_2$ \\
-17. $w_2 \leftarrow tmp_1 \cdot tmp_2$ \\
-\\
-Continued on the next page.\\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_toom\_mul}
-\end{figure}
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_toom\_mul} (continued). \\
-\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\
-\textbf{Output}. $c \leftarrow a \cdot b $ \\
-\hline \\
-Now solve the system of equations. \\
-18. $w_1 \leftarrow w_4 - w_1$, $w_3 \leftarrow w_3 - w_0$ \\
-19. $w_1 \leftarrow \lfloor w_1 / 2 \rfloor$, $w_3 \leftarrow \lfloor w_3 / 2 \rfloor$ \\
-20. $w_2 \leftarrow w_2 - w_0$, $w_2 \leftarrow w_2 - w_4$ \\
-21. $w_1 \leftarrow w_1 - w_2$, $w_3 \leftarrow w_3 - w_2$ \\
-22. $tmp_1 \leftarrow 8 \cdot w_0$, $w_1 \leftarrow w_1 - tmp_1$, $tmp_1 \leftarrow 8 \cdot w_4$, $w_3 \leftarrow w_3 - tmp_1$ \\
-23. $w_2 \leftarrow 3 \cdot w_2$, $w_2 \leftarrow w_2 - w_1$, $w_2 \leftarrow w_2 - w_3$ \\
-24. $w_1 \leftarrow w_1 - w_2$, $w_3 \leftarrow w_3 - w_2$ \\
-25. $w_1 \leftarrow \lfloor w_1 / 3 \rfloor, w_3 \leftarrow \lfloor w_3 / 3 \rfloor$ \\
-\\
-Now substitute $\beta^k$ for $x$ by shifting $w_0, w_1, ..., w_4$. \\
-26. for $n$ from $1$ to $4$ do \\
-\hspace{3mm}26.1 $w_n \leftarrow w_n \cdot \beta^{nk}$ \\
-27. $c \leftarrow w_0 + w_1$, $c \leftarrow c + w_2$, $c \leftarrow c + w_3$, $c \leftarrow c + w_4$ \\
-28. Return(\textit{MP\_OKAY}) \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_toom\_mul (continued)}
-\end{figure}
-
-\textbf{Algorithm mp\_toom\_mul.}
-This algorithm computes the product of two mp\_int variables $a$ and $b$ using the Toom-Cook approach. Compared to the Karatsuba multiplication, this
-algorithm has a lower asymptotic running time of approximately $O(n^{1.464})$ but at an obvious cost in overhead. In this
-description, several statements have been compounded to save space. The intention is that the statements are executed from left to right across
-any given step.
-
-The two inputs $a$ and $b$ are first split into three $k$-digit integers $a_0, a_1, a_2$ and $b_0, b_1, b_2$ respectively. From these smaller
-integers the coefficients of the polynomial basis representations $f(x)$ and $g(x)$ are known and can be used to find the relations required.
-
-The first two relations $w_0$ and $w_4$ are the points $\zeta_{0}$ and $\zeta_{\infty}$ respectively. The relation $w_1, w_2$ and $w_3$ correspond
-to the points $16 \cdot \zeta_{1 \over 2}, \zeta_{2}$ and $\zeta_{1}$ respectively. These are found using logical shifts to independently find
-$f(y)$ and $g(y)$ which significantly speeds up the algorithm.
-
-After the five relations $w_0, w_1, \ldots, w_4$ have been computed, the system they represent must be solved in order for the unknown coefficients
-$w_1, w_2$ and $w_3$ to be isolated. The steps 18 through 25 perform the system reduction required as previously described. Each step of
-the reduction represents the comparable matrix operation that would be performed had this been performed by pencil. For example, step 18 indicates
-that row $1$ must be subtracted from row $4$ and simultaneously row $0$ subtracted from row $3$.
-
-Once the coeffients have been isolated, the polynomial $W(x) = \sum_{i=0}^{2n} w_i x^i$ is known. By substituting $\beta^{k}$ for $x$, the integer
-result $a \cdot b$ is produced.
-
-\index{bn\_mp\_toom\_mul.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_toom\_mul.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* multiplication using the Toom-Cook 3-way algorithm */
-018 int
-019 mp_toom_mul(mp_int *a, mp_int *b, mp_int *c)
-020 \{
-021 mp_int w0, w1, w2, w3, w4, tmp1, tmp2, a0, a1, a2, b0, b1, b2;
-022 int res, B;
-023
-024 /* init temps */
-025 if ((res = mp_init_multi(&w0, &w1, &w2, &w3, &w4,
-026 &a0, &a1, &a2, &b0, &b1,
-027 &b2, &tmp1, &tmp2, NULL)) != MP_OKAY) \{
-028 return res;
-029 \}
-030
-031 /* B */
-032 B = MIN(a->used, b->used) / 3;
-033
-034 /* a = a2 * B**2 + a1 * B + a0 */
-035 if ((res = mp_mod_2d(a, DIGIT_BIT * B, &a0)) != MP_OKAY) \{
-036 goto ERR;
-037 \}
-038
-039 if ((res = mp_copy(a, &a1)) != MP_OKAY) \{
-040 goto ERR;
-041 \}
-042 mp_rshd(&a1, B);
-043 mp_mod_2d(&a1, DIGIT_BIT * B, &a1);
-044
-045 if ((res = mp_copy(a, &a2)) != MP_OKAY) \{
-046 goto ERR;
-047 \}
-048 mp_rshd(&a2, B*2);
-049
-050 /* b = b2 * B**2 + b1 * B + b0 */
-051 if ((res = mp_mod_2d(b, DIGIT_BIT * B, &b0)) != MP_OKAY) \{
-052 goto ERR;
-053 \}
-054
-055 if ((res = mp_copy(b, &b1)) != MP_OKAY) \{
-056 goto ERR;
-057 \}
-058 mp_rshd(&b1, B);
-059 mp_mod_2d(&b1, DIGIT_BIT * B, &b1);
-060
-061 if ((res = mp_copy(b, &b2)) != MP_OKAY) \{
-062 goto ERR;
-063 \}
-064 mp_rshd(&b2, B*2);
-065
-066 /* w0 = a0*b0 */
-067 if ((res = mp_mul(&a0, &b0, &w0)) != MP_OKAY) \{
-068 goto ERR;
-069 \}
-070
-071 /* w4 = a2 * b2 */
-072 if ((res = mp_mul(&a2, &b2, &w4)) != MP_OKAY) \{
-073 goto ERR;
-074 \}
-075
-076 /* w1 = (a2 + 2(a1 + 2a0))(b2 + 2(b1 + 2b0)) */
-077 if ((res = mp_mul_2(&a0, &tmp1)) != MP_OKAY) \{
-078 goto ERR;
-079 \}
-080 if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) \{
-081 goto ERR;
-082 \}
-083 if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) \{
-084 goto ERR;
-085 \}
-086 if ((res = mp_add(&tmp1, &a2, &tmp1)) != MP_OKAY) \{
-087 goto ERR;
-088 \}
-089
-090 if ((res = mp_mul_2(&b0, &tmp2)) != MP_OKAY) \{
-091 goto ERR;
-092 \}
-093 if ((res = mp_add(&tmp2, &b1, &tmp2)) != MP_OKAY) \{
-094 goto ERR;
-095 \}
-096 if ((res = mp_mul_2(&tmp2, &tmp2)) != MP_OKAY) \{
-097 goto ERR;
-098 \}
-099 if ((res = mp_add(&tmp2, &b2, &tmp2)) != MP_OKAY) \{
-100 goto ERR;
-101 \}
-102
-103 if ((res = mp_mul(&tmp1, &tmp2, &w1)) != MP_OKAY) \{
-104 goto ERR;
-105 \}
-106
-107 /* w3 = (a0 + 2(a1 + 2a2))(b0 + 2(b1 + 2b2)) */
-108 if ((res = mp_mul_2(&a2, &tmp1)) != MP_OKAY) \{
-109 goto ERR;
-110 \}
-111 if ((res = mp_add(&tmp1, &a1, &tmp1)) != MP_OKAY) \{
-112 goto ERR;
-113 \}
-114 if ((res = mp_mul_2(&tmp1, &tmp1)) != MP_OKAY) \{
-115 goto ERR;
-116 \}
-117 if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) \{
-118 goto ERR;
-119 \}
-120
-121 if ((res = mp_mul_2(&b2, &tmp2)) != MP_OKAY) \{
-122 goto ERR;
-123 \}
-124 if ((res = mp_add(&tmp2, &b1, &tmp2)) != MP_OKAY) \{
-125 goto ERR;
-126 \}
-127 if ((res = mp_mul_2(&tmp2, &tmp2)) != MP_OKAY) \{
-128 goto ERR;
-129 \}
-130 if ((res = mp_add(&tmp2, &b0, &tmp2)) != MP_OKAY) \{
-131 goto ERR;
-132 \}
-133
-134 if ((res = mp_mul(&tmp1, &tmp2, &w3)) != MP_OKAY) \{
-135 goto ERR;
-136 \}
-137
-138
-139 /* w2 = (a2 + a1 + a0)(b2 + b1 + b0) */
-140 if ((res = mp_add(&a2, &a1, &tmp1)) != MP_OKAY) \{
-141 goto ERR;
-142 \}
-143 if ((res = mp_add(&tmp1, &a0, &tmp1)) != MP_OKAY) \{
-144 goto ERR;
-145 \}
-146 if ((res = mp_add(&b2, &b1, &tmp2)) != MP_OKAY) \{
-147 goto ERR;
-148 \}
-149 if ((res = mp_add(&tmp2, &b0, &tmp2)) != MP_OKAY) \{
-150 goto ERR;
-151 \}
-152 if ((res = mp_mul(&tmp1, &tmp2, &w2)) != MP_OKAY) \{
-153 goto ERR;
-154 \}
-155
-156 /* now solve the matrix
-157
-158 0 0 0 0 1
-159 1 2 4 8 16
-160 1 1 1 1 1
-161 16 8 4 2 1
-162 1 0 0 0 0
-163
-164 using 12 subtractions, 4 shifts,
-165 2 small divisions and 1 small multiplication
-166 */
-167
-168 /* r1 - r4 */
-169 if ((res = mp_sub(&w1, &w4, &w1)) != MP_OKAY) \{
-170 goto ERR;
-171 \}
-172 /* r3 - r0 */
-173 if ((res = mp_sub(&w3, &w0, &w3)) != MP_OKAY) \{
-174 goto ERR;
-175 \}
-176 /* r1/2 */
-177 if ((res = mp_div_2(&w1, &w1)) != MP_OKAY) \{
-178 goto ERR;
-179 \}
-180 /* r3/2 */
-181 if ((res = mp_div_2(&w3, &w3)) != MP_OKAY) \{
-182 goto ERR;
-183 \}
-184 /* r2 - r0 - r4 */
-185 if ((res = mp_sub(&w2, &w0, &w2)) != MP_OKAY) \{
-186 goto ERR;
-187 \}
-188 if ((res = mp_sub(&w2, &w4, &w2)) != MP_OKAY) \{
-189 goto ERR;
-190 \}
-191 /* r1 - r2 */
-192 if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) \{
-193 goto ERR;
-194 \}
-195 /* r3 - r2 */
-196 if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) \{
-197 goto ERR;
-198 \}
-199 /* r1 - 8r0 */
-200 if ((res = mp_mul_2d(&w0, 3, &tmp1)) != MP_OKAY) \{
-201 goto ERR;
-202 \}
-203 if ((res = mp_sub(&w1, &tmp1, &w1)) != MP_OKAY) \{
-204 goto ERR;
-205 \}
-206 /* r3 - 8r4 */
-207 if ((res = mp_mul_2d(&w4, 3, &tmp1)) != MP_OKAY) \{
-208 goto ERR;
-209 \}
-210 if ((res = mp_sub(&w3, &tmp1, &w3)) != MP_OKAY) \{
-211 goto ERR;
-212 \}
-213 /* 3r2 - r1 - r3 */
-214 if ((res = mp_mul_d(&w2, 3, &w2)) != MP_OKAY) \{
-215 goto ERR;
-216 \}
-217 if ((res = mp_sub(&w2, &w1, &w2)) != MP_OKAY) \{
-218 goto ERR;
-219 \}
-220 if ((res = mp_sub(&w2, &w3, &w2)) != MP_OKAY) \{
-221 goto ERR;
-222 \}
-223 /* r1 - r2 */
-224 if ((res = mp_sub(&w1, &w2, &w1)) != MP_OKAY) \{
-225 goto ERR;
-226 \}
-227 /* r3 - r2 */
-228 if ((res = mp_sub(&w3, &w2, &w3)) != MP_OKAY) \{
-229 goto ERR;
-230 \}
-231 /* r1/3 */
-232 if ((res = mp_div_3(&w1, &w1, NULL)) != MP_OKAY) \{
-233 goto ERR;
-234 \}
-235 /* r3/3 */
-236 if ((res = mp_div_3(&w3, &w3, NULL)) != MP_OKAY) \{
-237 goto ERR;
-238 \}
-239
-240 /* at this point shift W[n] by B*n */
-241 if ((res = mp_lshd(&w1, 1*B)) != MP_OKAY) \{
-242 goto ERR;
-243 \}
-244 if ((res = mp_lshd(&w2, 2*B)) != MP_OKAY) \{
-245 goto ERR;
-246 \}
-247 if ((res = mp_lshd(&w3, 3*B)) != MP_OKAY) \{
-248 goto ERR;
-249 \}
-250 if ((res = mp_lshd(&w4, 4*B)) != MP_OKAY) \{
-251 goto ERR;
-252 \}
-253
-254 if ((res = mp_add(&w0, &w1, c)) != MP_OKAY) \{
-255 goto ERR;
-256 \}
-257 if ((res = mp_add(&w2, &w3, &tmp1)) != MP_OKAY) \{
-258 goto ERR;
-259 \}
-260 if ((res = mp_add(&w4, &tmp1, &tmp1)) != MP_OKAY) \{
-261 goto ERR;
-262 \}
-263 if ((res = mp_add(&tmp1, c, c)) != MP_OKAY) \{
-264 goto ERR;
-265 \}
-266
-267 ERR:
-268 mp_clear_multi(&w0, &w1, &w2, &w3, &w4,
-269 &a0, &a1, &a2, &b0, &b1,
-270 &b2, &tmp1, &tmp2, NULL);
-271 return res;
-272 \}
-273
-\end{alltt}
-\end{small}
-
--- Comments to be added during editing phase.
-
-\subsection{Signed Multiplication}
-Now that algorithms to handle multiplications of every useful dimensions have been developed, a rather simple finishing touch is required. So far all
-of the multiplication algorithms have been unsigned multiplications which leaves only a signed multiplication algorithm to be established.
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_mul}. \\
-\textbf{Input}. mp\_int $a$ and mp\_int $b$ \\
-\textbf{Output}. $c \leftarrow a \cdot b$ \\
-\hline \\
-1. If $a.sign = b.sign$ then \\
-\hspace{3mm}1.1 $sign = MP\_ZPOS$ \\
-2. else \\
-\hspace{3mm}2.1 $sign = MP\_ZNEG$ \\
-3. If min$(a.used, b.used) \ge TOOM\_MUL\_CUTOFF$ then \\
-\hspace{3mm}3.1 $c \leftarrow a \cdot b$ using algorithm mp\_toom\_mul \\
-4. else if min$(a.used, b.used) \ge KARATSUBA\_MUL\_CUTOFF$ then \\
-\hspace{3mm}4.1 $c \leftarrow a \cdot b$ using algorithm mp\_karatsuba\_mul \\
-5. else \\
-\hspace{3mm}5.1 $digs \leftarrow a.used + b.used + 1$ \\
-\hspace{3mm}5.2 If $digs < MP\_ARRAY$ and min$(a.used, b.used) \le \delta$ then \\
-\hspace{6mm}5.2.1 $c \leftarrow a \cdot b \mbox{ (mod }\beta^{digs}\mbox{)}$ using algorithm fast\_s\_mp\_mul\_digs. \\
-\hspace{3mm}5.3 else \\
-\hspace{6mm}5.3.1 $c \leftarrow a \cdot b \mbox{ (mod }\beta^{digs}\mbox{)}$ using algorithm s\_mp\_mul\_digs. \\
-6. $c.sign \leftarrow sign$ \\
-7. Return the result of the unsigned multiplication performed. \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_mul}
-\end{figure}
-
-\textbf{Algorithm mp\_mul.}
-This algorithm performs the signed multiplication of two inputs. It will make use of any of the three unsigned multiplication algorithms
-available when the input is of appropriate size. The \textbf{sign} of the result is not set until the end of the algorithm since algorithm
-s\_mp\_mul\_digs will clear it.
-
-\index{bn\_mp\_mul.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_mul.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* high level multiplication (handles sign) */
-018 int
-019 mp_mul (mp_int * a, mp_int * b, mp_int * c)
-020 \{
-021 int res, neg;
-022 neg = (a->sign == b->sign) ? MP_ZPOS : MP_NEG;
-023
-024 if (MIN (a->used, b->used) >= TOOM_MUL_CUTOFF) \{
-025 res = mp_toom_mul(a, b, c);
-026 \} else if (MIN (a->used, b->used) >= KARATSUBA_MUL_CUTOFF) \{
-027 res = mp_karatsuba_mul (a, b, c);
-028 \} else \{
-029
-030 /* can we use the fast multiplier?
-031 *
-032 * The fast multiplier can be used if the output will
-033 * have less than MP_WARRAY digits and the number of
-034 * digits won't affect carry propagation
-035 */
-036 int digs = a->used + b->used + 1;
-037
-038 if ((digs < MP_WARRAY) &&
-039 MIN(a->used, b->used) <=
-040 (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) \{
-041 res = fast_s_mp_mul_digs (a, b, c, digs);
-042 \} else \{
-043 res = s_mp_mul (a, b, c);
-044 \}
-045
-046 \}
-047 c->sign = neg;
-048 return res;
-049 \}
-\end{alltt}
-\end{small}
-
-The implementation is rather simplistic and is not particularly noteworthy. Line 22 computes the sign of the result using the ``?''
-operator from the C programming language. Line 40 computes $\delta$ using the fact that $1 << k$ is equal to $2^k$.
-
-\section{Squaring}
-
-Squaring is a special case of multiplication where both multiplicands are equal. At first it may seem like there is no significant optimization
-available but in fact there is. Consider the multiplication of $576$ against $241$. In total there will be nine single precision multiplications
-performed which are $1\cdot 6$, $1 \cdot 7$, $1 \cdot 5$, $4 \cdot 6$, $4 \cdot 7$, $4 \cdot 5$, $2 \cdot 6$, $2 \cdot 7$ and $2 \cdot 5$. Now consider
-the multiplication of $123$ against $123$. The nine products are $3 \cdot 3$, $3 \cdot 2$, $3 \cdot 1$, $2 \cdot 3$, $2 \cdot 2$, $2 \cdot 1$,
-$1 \cdot 3$, $1 \cdot 2$ and $1 \cdot 1$. On closer inspection some of the products are equivalent. For example, $3 \cdot 2 = 2 \cdot 3$
-and $3 \cdot 1 = 1 \cdot 3$.
-
-For any $n$-digit input, there are ${{\left (n^2 + n \right)}\over 2}$ possible unique single precision multiplications required compared to the $n^2$
-required for multiplication. The following diagram gives an example of the operations required.
-
-\begin{figure}[here]
-\begin{center}
-\begin{tabular}{ccccc|c}
-&&1&2&3&\\
-$\times$ &&1&2&3&\\
-\hline && $3 \cdot 1$ & $3 \cdot 2$ & $3 \cdot 3$ & Row 0\\
- & $2 \cdot 1$ & $2 \cdot 2$ & $2 \cdot 3$ && Row 1 \\
- $1 \cdot 1$ & $1 \cdot 2$ & $1 \cdot 3$ &&& Row 2 \\
-\end{tabular}
-\end{center}
-\caption{Squaring Optimization Diagram}
-\end{figure}
-
-Starting from zero and numbering the columns from right to left a very simple pattern becomes obvious. For the purposes of this discussion let $x$
-represent the number being squared. The first observation is that in row $k$ the $2k$'th column of the product has a $\left (x_k \right)^2$ term in it.
-
-The second observation is that every column $j$ in row $k$ where $j \ne 2k$ is part of a double product. Every non-square term of a column will
-appear twice hence the name ``double product''. Every odd column is made up entirely of double products. In fact every column is made up of double
-products and at most one square (\textit{see the exercise section}).
-
-The third and final observation is that for row $k$ the first unique non-square term, that is, one that hasn't already appeared in an earlier row,
-occurs at column $2k + 1$. For example, on row $1$ of the previous squaring, column one is part of the double product with column one from row zero.
-Column two of row one is a square and column three is the first unique column.
-
-\subsection{The Baseline Squaring Algorithm}
-The baseline squaring algorithm is meant to be a catch-all squaring algorithm. It will handle any of the input sizes that the faster routines
-will not handle.
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{s\_mp\_sqr}. \\
-\textbf{Input}. mp\_int $a$ \\
-\textbf{Output}. $b \leftarrow a^2$ \\
-\hline \\
-1. Init a temporary mp\_int of at least $2 \cdot a.used +1$ digits. (\textit{mp\_init\_size}) \\
-2. If step 1 failed return(\textit{MP\_MEM}) \\
-3. $t.used \leftarrow 2 \cdot a.used + 1$ \\
-4. For $ix$ from 0 to $a.used - 1$ do \\
-\hspace{3mm}Calculate the square. \\
-\hspace{3mm}4.1 $\hat r \leftarrow t_{2ix} + \left (a_{ix} \right )^2$ \\
-\hspace{3mm}4.2 $t_{2ix} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\
-\hspace{3mm}Calculate the double products after the square. \\
-\hspace{3mm}4.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\
-\hspace{3mm}4.4 For $iy$ from $ix + 1$ to $a.used - 1$ do \\
-\hspace{6mm}4.4.1 $\hat r \leftarrow 2 \cdot a_{ix}a_{iy} + t_{ix + iy} + u$ \\
-\hspace{6mm}4.4.2 $t_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\
-\hspace{6mm}4.4.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\
-\hspace{3mm}Set the last carry. \\
-\hspace{3mm}4.5 While $u > 0$ do \\
-\hspace{6mm}4.5.1 $iy \leftarrow iy + 1$ \\
-\hspace{6mm}4.5.2 $\hat r \leftarrow t_{ix + iy} + u$ \\
-\hspace{6mm}4.5.3 $t_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\
-\hspace{6mm}4.5.4 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\
-5. Clamp excess digits of $t$. (\textit{mp\_clamp}) \\
-6. Exchange $b$ and $t$. \\
-7. Clear $t$ (\textit{mp\_clear}) \\
-8. Return(\textit{MP\_OKAY}) \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm s\_mp\_sqr}
-\end{figure}
-
-\textbf{Algorithm s\_mp\_sqr.}
-This algorithm computes the square of an input using the three observations on squaring. It is based fairly faithfully on algorithm 14.16 of HAC
-\cite[pp.596-597]{HAC}. Similar to algorithm s\_mp\_mul\_digs, a temporary mp\_int is allocated to hold the result of the squaring. This allows the
-destination mp\_int to be the same as the source mp\_int.
-
-The outer loop of this algorithm begins on step 4. It is best to think of the outer loop as walking down the rows of the partial results, while
-the inner loop computes the columns of the partial result. Step 4.1 and 4.2 compute the square term for each row, and step 4.3 and 4.4 propagate
-the carry and compute the double products.
-
-The requirement that a mp\_word be able to represent the range $0 \le x < 2 \beta^2$ arises from this
-very algorithm. The product $a_{ix}a_{iy}$ will lie in the range $0 \le x \le \beta^2 - 2\beta + 1$ which is obviously less than $\beta^2$ meaning that
-when it is multiplied by two, it can be properly represented by a mp\_word.
-
-Similar to algorithm s\_mp\_mul\_digs, after every pass of the inner loop, the destination is correctly set to the sum of all of the partial
-results calculated so far. This involves expensive carry propagation which will be eliminated in the next algorithm.
-
-\index{bn\_s\_mp\_sqr.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_sqr.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* low level squaring, b = a*a, HAC pp.596-597, Algorithm 14.16 */
-018 int
-019 s_mp_sqr (mp_int * a, mp_int * b)
-020 \{
-021 mp_int t;
-022 int res, ix, iy, pa;
-023 mp_word r;
-024 mp_digit u, tmpx, *tmpt;
-025
-026 pa = a->used;
-027 if ((res = mp_init_size (&t, 2*pa + 1)) != MP_OKAY) \{
-028 return res;
-029 \}
-030 t.used = 2*pa + 1;
-031
-032 for (ix = 0; ix < pa; ix++) \{
-033 /* first calculate the digit at 2*ix */
-034 /* calculate double precision result */
-035 r = ((mp_word) t.dp[2*ix]) +
-036 ((mp_word) a->dp[ix]) * ((mp_word) a->dp[ix]);
-037
-038 /* store lower part in result */
-039 t.dp[2*ix] = (mp_digit) (r & ((mp_word) MP_MASK));
-040
-041 /* get the carry */
-042 u = (r >> ((mp_word) DIGIT_BIT));
-043
-044 /* left hand side of A[ix] * A[iy] */
-045 tmpx = a->dp[ix];
-046
-047 /* alias for where to store the results */
-048 tmpt = t.dp + (2*ix + 1);
-049
-050 for (iy = ix + 1; iy < pa; iy++) \{
-051 /* first calculate the product */
-052 r = ((mp_word) tmpx) * ((mp_word) a->dp[iy]);
-053
-054 /* now calculate the double precision result, note we use
-055 * addition instead of *2 since it's easier to optimize
-056 */
-057 r = ((mp_word) * tmpt) + r + r + ((mp_word) u);
-058
-059 /* store lower part */
-060 *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK));
-061
-062 /* get carry */
-063 u = (r >> ((mp_word) DIGIT_BIT));
-064 \}
-065 /* propagate upwards */
-066 while (u != ((mp_digit) 0)) \{
-067 r = ((mp_word) * tmpt) + ((mp_word) u);
-068 *tmpt++ = (mp_digit) (r & ((mp_word) MP_MASK));
-069 u = (r >> ((mp_word) DIGIT_BIT));
-070 \}
-071 \}
-072
-073 mp_clamp (&t);
-074 mp_exch (&t, b);
-075 mp_clear (&t);
-076 return MP_OKAY;
-077 \}
-\end{alltt}
-\end{small}
-
-Inside the outer loop (\textit{see line 32}) the square term is calculated on line 35. Line 42 extracts the carry from the square
-term. Aliases for $a_{ix}$ and $t_{ix+iy}$ are initialized on lines 45 and 48 respectively. The doubling is performed using two
-additions (\textit{see line 57}) since it is usually faster than shifting,if not at least as fast.
-
-\subsection{Faster Squaring by the ``Comba'' Method}
-A major drawback to the baseline method is the requirement for single precision shifting inside the $O(n^2)$ nested loop. Squaring has an additional
-drawback that it must double the product inside the inner loop as well. As for multiplication, the Comba technique can be used to eliminate these
-performance hazards.
-
-The first obvious solution is to make an array of mp\_words which will hold all of the columns. This will indeed eliminate all of the carry
-propagation operations from the inner loop. However, the inner product must still be doubled $O(n^2)$ times. The solution stems from the simple fact
-that $2a + 2b + 2c = 2(a + b + c)$. That is the sum of all of the double products is equal to double the sum of all the products. For example,
-$ab + ba + ac + ca = 2ab + 2ac = 2(ab + ac)$.
-
-However, we cannot simply double all of the columns, since the squares appear only once per row. The most practical solution is to have two mp\_word
-arrays. One array will hold the squares and the other array will hold the double products. With both arrays the doubling and carry propagation can be
-moved to a $O(n)$ work level outside the $O(n^2)$ level.
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{fast\_s\_mp\_sqr}. \\
-\textbf{Input}. mp\_int $a$ \\
-\textbf{Output}. $b \leftarrow a^2$ \\
-\hline \\
-Place two arrays of \textbf{MP\_WARRAY} mp\_words named $\hat W$ and $\hat {X}$ on the stack. \\
-1. If $b.alloc < 2a.used + 1$ then grow $b$ to $2a.used + 1$ digits. (\textit{mp\_grow}). \\
-2. If step 1 failed return(\textit{MP\_MEM}). \\
-3. for $ix$ from $0$ to $2a.used + 1$ do \\
-\hspace{3mm}3.1 $\hat W_{ix} \leftarrow 0$ \\
-\hspace{3mm}3.2 $\hat {X}_{ix} \leftarrow 0$ \\
-4. for $ix$ from $0$ to $a.used - 1$ do \\
-\hspace{3mm}Compute the square.\\
-\hspace{3mm}4.1 $\hat {X}_{ix+ix} \leftarrow \left ( a_ix \right )^2$ \\
-\\
-\hspace{3mm}Compute the double products.\\
-\hspace{3mm}4.2 for $iy$ from $ix + 1$ to $a.used - 1$ do \\
-\hspace{6mm}4.2.1 $\hat W_{ix+iy} \leftarrow \hat W_{ix+iy} + a_{ix}a_{iy}$ \\
-5. $oldused \leftarrow b.used$ \\
-6. $b.used \leftarrow 2a.used + 1$ \\
-\\
-Double the products and propagate the carries simultaneously. \\
-7. $\hat W_0 \leftarrow 2 \hat W_0 + \hat {X}_0$ \\
-8. for $ix$ from $1$ to $2a.used$ do \\
-\hspace{3mm}8.1 $\hat W_{ix} \leftarrow 2 \hat W_{ix} + \hat {X}_{ix}$ \\
-\hspace{3mm}8.2 $\hat W_{ix} \leftarrow \hat W_{ix} + \lfloor \hat W_{ix - 1} / \beta \rfloor$ \\
-\hspace{3mm}8.3 $b_{ix-1} \leftarrow W_{ix-1} \mbox{ (mod }\beta\mbox{)}$ \\
-9. $b_{2a.used} \leftarrow \hat W_{2a.used} \mbox{ (mod }\beta\mbox{)}$ \\
-10. if $2a.used + 1 < oldused$ then do \\
-\hspace{3mm}10.1 for $ix$ from $2a.used + 1$ to $oldused$ do \\
-\hspace{6mm}10.1.1 $b_{ix} \leftarrow 0$ \\
-11. Clamp excess digits from $b$. (\textit{mp\_clamp}) \\
-12. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm fast\_s\_mp\_sqr}
-\end{figure}
-
-\textbf{Algorithm fast\_s\_mp\_sqr.}
-This algorithm computes the square of an input using the Comba technique. It is designed to be a replacement for algorithm s\_mp\_sqr when
-the number of input digits is less than \textbf{MP\_WARRAY} and less than $\delta \over 2$.
-
-This routine requires two arrays of mp\_words to be placed on the stack. The first array $\hat W$ will hold the double products and the second
-array $\hat X$ will hold the squares. Though only at most $MP\_WARRAY \over 2$ words of $\hat X$ are used, it has proven faster on most
-processors to simply make it a full size array.
-
-The loop on step 3 will zero the two arrays to prepare them for the squaring step. Step 4.1 computes the squares of the product. Note how
-it simply assigns the value into the $\hat X$ array. The nested loop on step 4.2 computes the doubles of the products. This loop
-computes the sum of the products for each column. They are not doubled until later.
-
-After the squaring loop, the products stored in $\hat W$ musted be doubled and the carries propagated forwards. It makes sense to do both
-operations at the same time. The expression $\hat W_{ix} \leftarrow 2 \hat W_{ix} + \hat {X}_{ix}$ computes the sum of the double product and the
-squares in place.
-
-\index{bn\_fast\_s\_mp\_sqr.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_fast\_s\_mp\_sqr.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* fast squaring
-018 *
-019 * This is the comba method where the columns of the product
-020 * are computed first then the carries are computed. This
-021 * has the effect of making a very simple inner loop that
-022 * is executed the most
-023 *
-024 * W2 represents the outer products and W the inner.
-025 *
-026 * A further optimizations is made because the inner
-027 * products are of the form "A * B * 2". The *2 part does
-028 * not need to be computed until the end which is good
-029 * because 64-bit shifts are slow!
-030 *
-031 * Based on Algorithm 14.16 on pp.597 of HAC.
-032 *
-033 */
-034 int
-035 fast_s_mp_sqr (mp_int * a, mp_int * b)
-036 \{
-037 int olduse, newused, res, ix, pa;
-038 mp_word W2[MP_WARRAY], W[MP_WARRAY];
-039
-040 /* calculate size of product and allocate as required */
-041 pa = a->used;
-042 newused = pa + pa + 1;
-043 if (b->alloc < newused) \{
-044 if ((res = mp_grow (b, newused)) != MP_OKAY) \{
-045 return res;
-046 \}
-047 \}
-048
-049 /* zero temp buffer (columns)
-050 * Note that there are two buffers. Since squaring requires
-051 * a outter and inner product and the inner product requires
-052 * computing a product and doubling it (a relatively expensive
-053 * op to perform n**2 times if you don't have to) the inner and
-054 * outer products are computed in different buffers. This way
-055 * the inner product can be doubled using n doublings instead of
-056 * n**2
-057 */
-058 memset (W, 0, newused * sizeof (mp_word));
-059 memset (W2, 0, newused * sizeof (mp_word));
-060
-061 /* This computes the inner product. To simplify the inner N**2 loop
-062 * the multiplication by two is done afterwards in the N loop.
-063 */
-064 for (ix = 0; ix < pa; ix++) \{
-065 /* compute the outer product
-066 *
-067 * Note that every outer product is computed
-068 * for a particular column only once which means that
-069 * there is no need todo a double precision addition
-070 */
-071 W2[ix + ix] = ((mp_word) a->dp[ix]) * ((mp_word) a->dp[ix]);
-072
-073 \{
-074 register mp_digit tmpx, *tmpy;
-075 register mp_word *_W;
-076 register int iy;
-077
-078 /* copy of left side */
-079 tmpx = a->dp[ix];
-080
-081 /* alias for right side */
-082 tmpy = a->dp + (ix + 1);
-083
-084 /* the column to store the result in */
-085 _W = W + (ix + ix + 1);
-086
-087 /* inner products */
-088 for (iy = ix + 1; iy < pa; iy++) \{
-089 *_W++ += ((mp_word) tmpx) * ((mp_word) * tmpy++);
-090 \}
-091 \}
-092 \}
-093
-094 /* setup dest */
-095 olduse = b->used;
-096 b->used = newused;
-097
-098 /* now compute digits */
-099 \{
-100 register mp_digit *tmpb;
-101
-102 /* double first value, since the inner products are
-103 * half of what they should be
-104 */
-105 W[0] += W[0] + W2[0];
-106
-107 tmpb = b->dp;
-108 for (ix = 1; ix < newused; ix++) \{
-109 /* double/add next digit */
-110 W[ix] += W[ix] + W2[ix];
-111
-112 W[ix] = W[ix] + (W[ix - 1] >> ((mp_word) DIGIT_BIT));
-113 *tmpb++ = (mp_digit) (W[ix - 1] & ((mp_word) MP_MASK));
-114 \}
-115 /* set the last value. Note even if the carry is zero
-116 * this is required since the next step will not zero
-117 * it if b originally had a value at b->dp[2*a.used]
-118 */
-119 *tmpb++ = (mp_digit) (W[(newused) - 1] & ((mp_word) MP_MASK));
-120
-121 /* clear high digits */
-122 for (; ix < olduse; ix++) \{
-123 *tmpb++ = 0;
-124 \}
-125 \}
-126
-127 mp_clamp (b);
-128 return MP_OKAY;
-129 \}
-\end{alltt}
-\end{small}
-
--- Write something deep and insightful later, Tom.
-
-\subsection{Polynomial Basis Squaring}
-The same algorithm that performs optimal polynomial basis multiplication can be used to perform polynomial basis squaring. The minor exception
-is that $\zeta_y = f(y)g(y)$ is actually equivalent to $\zeta_y = f(y)^2$ since $f(y) = g(y)$. Instead of performing $2n + 1$
-multiplications to find the $\zeta$ relations, squaring operations are performed instead.
-
-\subsection{Karatsuba Squaring}
-Let $f(x) = ax + b$ represent the polynomial basis representation of a number to square.
-Let $h(x) = \left ( f(x) \right )^2$ represent the square of the polynomial. The Karatsuba equation can be modified to square a
-number with the following equation.
-
-\begin{equation}
-h(x) = a^2x^2 + \left (a^2 + b^2 - (a - b)^2 \right )x + b^2
-\end{equation}
-
-Upon closer inspection this equation only requires the calculation of three half-sized squares: $a^2$, $b^2$ and $(a - b)^2$. As in
-Karatsuba multiplication, this algorithm can be applied recursively on the input and will achieve an asymptotic running time of
-$O \left ( n^{lg(3)} \right )$.
-
-You might ask yourself, if the asymptotic time of Karatsuba squaring and multiplication is the same, why not simply use the multiplication algorithm
-instead? The answer to this arises from the cutoff point for squaring. As in multiplication there exists a cutoff point, at which the
-time required for a Comba based squaring and a Karatsuba based squaring meet. Due to the overhead inherent in the Karatsuba method, the cutoff
-point is fairly high. For example, on an AMD Athlon XP processor with $\beta = 2^{28}$, the cutoff point is around 127 digits.
-
-Consider squaring a 200 digit number with this technique. It will be split into two 100 digit halves which are subsequently squared.
-The 100 digit halves will not be squared using Karatsuba, but instead using the faster Comba based squaring algorithm. If Karatsuba multiplication
-were used instead, the 100 digit numbers would be squared with a slower Comba based multiplication.
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_karatsuba\_sqr}. \\
-\textbf{Input}. mp\_int $a$ \\
-\textbf{Output}. $b \leftarrow a^2$ \\
-\hline \\
-1. Initialize the following temporary mp\_ints: $x0$, $x1$, $t1$, $t2$, $x0x0$ and $x1x1$. \\
-2. If any of the initializations on step 1 failed return(\textit{MP\_MEM}). \\
-\\
-Split the input. e.g. $a = x1\beta^B + x0$ \\
-3. $B \leftarrow \lfloor a.used / 2 \rfloor$ \\
-4. $x0 \leftarrow a \mbox{ (mod }\beta^B\mbox{)}$ (\textit{mp\_mod\_2d}) \\
-5. $x1 \leftarrow \lfloor a / \beta^B \rfloor$ (\textit{mp\_lshd}) \\
-\\
-Calculate the three squares. \\
-6. $x0x0 \leftarrow x0^2$ (\textit{mp\_sqr}) \\
-7. $x1x1 \leftarrow x1^2$ \\
-8. $t1 \leftarrow x1 - x0$ (\textit{mp\_sub}) \\
-9. $t1 \leftarrow t1^2$ \\
-\\
-Compute the middle term. \\
-10. $t2 \leftarrow x0x0 + x1x1$ (\textit{s\_mp\_add}) \\
-11. $t1 \leftarrow t2 - t1$ \\
-\\
-Compute final product. \\
-12. $t1 \leftarrow t1\beta^B$ (\textit{mp\_lshd}) \\
-13. $x1x1 \leftarrow x1x1\beta^{2B}$ \\
-14. $t1 \leftarrow t1 + x0x0$ \\
-15. $b \leftarrow t1 + x1x1$ \\
-16. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_karatsuba\_sqr}
-\end{figure}
-
-\textbf{Algorithm mp\_karatsuba\_sqr.}
-This algorithm computes the square of an input $a$ using the Karatsuba technique. This algorithm is very similar to the Karatsuba based
-multiplication algorithm with the exception that the three half-size multiplications have been replaced with three half-size squarings.
-
-The radix point for squaring is simply placed exactly in the middle of the digits when the input has an odd number of digits, otherwise it is
-placed just below the middle. Step 3, 4 and 5 compute the two halves required using $B$
-as the radix point. The first two squares in steps 6 and 7 are rather straightforward while the last square is of a more compact form.
-
-By expanding $\left (x1 - x0 \right )^2$, the $x1^2$ and $x0^2$ terms in the middle disappear, that is $x1^2 + x0^2 - (x1 - x0)^2 = 2 \cdot x0 \cdot x1$.
-Now if $5n$ single precision additions and a squaring of $n$-digits is faster than multiplying two $n$-digit numbers and doubling then
-this method is faster. Assuming no further recursions occur, the difference can be estimated with the following inequality.
-
-Let $p$ represent the cost of a single precision addition and $q$ the cost of a single precision multiplication both in terms of time\footnote{Or
-machine clock cycles.}.
-
-\begin{equation}
-5pn +{{q(n^2 + n)} \over 2} \le pn + qn^2
-\end{equation}
-
-For example, on an AMD Athlon XP processor $p = {1 \over 3}$ and $q = 6$. This implies that the following inequality should hold.
-\begin{center}
-\begin{tabular}{rcl}
-${5n \over 3} + 3n^2 + 3n$ & $<$ & ${n \over 3} + 6n^2$ \\
-${5 \over 3} + 3n + 3$ & $<$ & ${1 \over 3} + 6n$ \\
-${13 \over 9}$ & $<$ & $n$ \\
-\end{tabular}
-\end{center}
-
-This results in a cutoff point around $n = 2$. As a consequence it is actually faster to compute the middle term the ``long way'' on processors
-where multiplication is substantially slower\footnote{On the Athlon there is a 1:17 ratio between clock cycles for addition and multiplication. On
-the Intel P4 processor this ratio is 1:29 making this method even more beneficial. The only common exception is the ARMv4 processor which has a
-ratio of 1:7. } than simpler operations such as addition.
-
-\index{bn\_mp\_karatsuba\_sqr.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_karatsuba\_sqr.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* Karatsuba squaring, computes b = a*a using three
-018 * half size squarings
-019 *
-020 * See comments of mp_karatsuba_mul for details. It
-021 * is essentially the same algorithm but merely
-022 * tuned to perform recursive squarings.
-023 */
-024 int
-025 mp_karatsuba_sqr (mp_int * a, mp_int * b)
-026 \{
-027 mp_int x0, x1, t1, t2, x0x0, x1x1;
-028 int B, err;
-029
-030 err = MP_MEM;
-031
-032 /* min # of digits */
-033 B = a->used;
-034
-035 /* now divide in two */
-036 B = B / 2;
-037
-038 /* init copy all the temps */
-039 if (mp_init_size (&x0, B) != MP_OKAY)
-040 goto ERR;
-041 if (mp_init_size (&x1, a->used - B) != MP_OKAY)
-042 goto X0;
-043
-044 /* init temps */
-045 if (mp_init_size (&t1, a->used * 2) != MP_OKAY)
-046 goto X1;
-047 if (mp_init_size (&t2, a->used * 2) != MP_OKAY)
-048 goto T1;
-049 if (mp_init_size (&x0x0, B * 2) != MP_OKAY)
-050 goto T2;
-051 if (mp_init_size (&x1x1, (a->used - B) * 2) != MP_OKAY)
-052 goto X0X0;
-053
-054 \{
-055 register int x;
-056 register mp_digit *dst, *src;
-057
-058 src = a->dp;
-059
-060 /* now shift the digits */
-061 dst = x0.dp;
-062 for (x = 0; x < B; x++) \{
-063 *dst++ = *src++;
-064 \}
-065
-066 dst = x1.dp;
-067 for (x = B; x < a->used; x++) \{
-068 *dst++ = *src++;
-069 \}
-070 \}
-071
-072 x0.used = B;
-073 x1.used = a->used - B;
-074
-075 mp_clamp (&x0);
-076
-077 /* now calc the products x0*x0 and x1*x1 */
-078 if (mp_sqr (&x0, &x0x0) != MP_OKAY)
-079 goto X1X1; /* x0x0 = x0*x0 */
-080 if (mp_sqr (&x1, &x1x1) != MP_OKAY)
-081 goto X1X1; /* x1x1 = x1*x1 */
-082
-083 /* now calc (x1-x0)**2 */
-084 if (mp_sub (&x1, &x0, &t1) != MP_OKAY)
-085 goto X1X1; /* t1 = x1 - x0 */
-086 if (mp_sqr (&t1, &t1) != MP_OKAY)
-087 goto X1X1; /* t1 = (x1 - x0) * (x1 - x0) */
-088
-089 /* add x0y0 */
-090 if (s_mp_add (&x0x0, &x1x1, &t2) != MP_OKAY)
-091 goto X1X1; /* t2 = x0x0 + x1x1 */
-092 if (mp_sub (&t2, &t1, &t1) != MP_OKAY)
-093 goto X1X1; /* t1 = x0x0 + x1x1 - (x1-x0)*(x1-x0) */
-094
-095 /* shift by B */
-096 if (mp_lshd (&t1, B) != MP_OKAY)
-097 goto X1X1; /* t1 = (x0x0 + x1x1 - (x1-x0)*(x1-x0))<<B */
-098 if (mp_lshd (&x1x1, B * 2) != MP_OKAY)
-099 goto X1X1; /* x1x1 = x1x1 << 2*B */
-100
-101 if (mp_add (&x0x0, &t1, &t1) != MP_OKAY)
-102 goto X1X1; /* t1 = x0x0 + t1 */
-103 if (mp_add (&t1, &x1x1, b) != MP_OKAY)
-104 goto X1X1; /* t1 = x0x0 + t1 + x1x1 */
-105
-106 err = MP_OKAY;
-107
-108 X1X1:mp_clear (&x1x1);
-109 X0X0:mp_clear (&x0x0);
-110 T2:mp_clear (&t2);
-111 T1:mp_clear (&t1);
-112 X1:mp_clear (&x1);
-113 X0:mp_clear (&x0);
-114 ERR:
-115 return err;
-116 \}
-\end{alltt}
-\end{small}
-
-This implementation is largely based on the implementation of algorithm mp\_karatsuba\_mul. It uses the same inline style to copy and
-shift the input into the two halves. The loop from line 54 to line 70 has been modified since only one input exists. The \textbf{used}
-count of both $x0$ and $x1$ is fixed up and $x0$ is clamped before the calculations begin. At this point $x1$ and $x0$ are valid equivalents
-to the respective halves as if mp\_rshd and mp\_mod\_2d had been used.
-
-By inlining the copy and shift operations the cutoff point for Karatsuba multiplication can be lowered. On the Athlon the cutoff point
-is exactly at the point where Comba squaring can no longer be used (\textit{128 digits}). On slower processors such as the Intel P4
-it is actually below the Comba limit (\textit{at 110 digits}).
-
-This routine uses the same error trap coding style as mp\_karatsuba\_sqr. As the temporary variables are initialized errors are redirected to
-the error trap higher up. If the algorithm completes without error the error code is set to \textbf{MP\_OKAY} and mp\_clears are executed normally.
-
-\textit{Last paragraph sucks. re-write! -- Tom}
-
-\subsection{Toom-Cook Squaring}
-The Toom-Cook squaring algorithm mp\_toom\_sqr is heavily based on the algorithm mp\_toom\_mul with the exception that squarings are used
-instead of multiplication to find the five relations.. The reader is encouraged to read the description of the latter algorithm and try to
-derive their own Toom-Cook squaring algorithm.
-
-\subsection{High Level Squaring}
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_sqr}. \\
-\textbf{Input}. mp\_int $a$ \\
-\textbf{Output}. $b \leftarrow a^2$ \\
-\hline \\
-1. If $a.used \ge TOOM\_SQR\_CUTOFF$ then \\
-\hspace{3mm}1.1 $b \leftarrow a^2$ using algorithm mp\_toom\_sqr \\
-2. else if $a.used \ge KARATSUBA\_SQR\_CUTOFF$ then \\
-\hspace{3mm}2.1 $b \leftarrow a^2$ using algorithm mp\_karatsuba\_sqr \\
-3. else \\
-\hspace{3mm}3.1 $digs \leftarrow a.used + b.used + 1$ \\
-\hspace{3mm}3.2 If $digs < MP\_ARRAY$ and $a.used \le \delta$ then \\
-\hspace{6mm}3.2.1 $b \leftarrow a^2$ using algorithm fast\_s\_mp\_sqr. \\
-\hspace{3mm}3.3 else \\
-\hspace{6mm}3.3.1 $b \leftarrow a^2$ using algorithm s\_mp\_sqr. \\
-4. $b.sign \leftarrow MP\_ZPOS$ \\
-5. Return the result of the unsigned squaring performed. \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_sqr}
-\end{figure}
-
-\textbf{Algorithm mp\_sqr.}
-This algorithm computes the square of the input using one of four different algorithms. If the input is very large and has at least
-\textbf{TOOM\_SQR\_CUTOFF} or \textbf{KARATSUBA\_SQR\_CUTOFF} digits then either the Toom-Cook or the Karatsuba Squaring algorithm is used. If
-neither of the polynomial basis algorithms should be used then either the Comba or baseline algorithm is used.
-
-\index{bn\_mp\_sqr.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_sqr.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* computes b = a*a */
-018 int
-019 mp_sqr (mp_int * a, mp_int * b)
-020 \{
-021 int res;
-022 if (a->used >= TOOM_SQR_CUTOFF) \{
-023 res = mp_toom_sqr(a, b);
-024 \} else if (a->used >= KARATSUBA_SQR_CUTOFF) \{
-025 res = mp_karatsuba_sqr (a, b);
-026 \} else \{
-027
-028 /* can we use the fast multiplier? */
-029 if ((a->used * 2 + 1) < MP_WARRAY &&
-030 a->used <
-031 (1 << (sizeof(mp_word) * CHAR_BIT - 2*DIGIT_BIT - 1))) \{
-032 res = fast_s_mp_sqr (a, b);
-033 \} else \{
-034 res = s_mp_sqr (a, b);
-035 \}
-036 \}
-037 b->sign = MP_ZPOS;
-038 return res;
-039 \}
-\end{alltt}
-\end{small}
-
-\section*{Exercises}
-\begin{tabular}{cl}
-$\left [ 3 \right ] $ & Devise an efficient algorithm for selection of the radix point to handle inputs \\
- & that have different number of digits in Karatsuba multiplication. \\
- & \\
-$\left [ 3 \right ] $ & In section 6.3 the fact that every column of a squaring is made up \\
- & of double products and at most one square is stated. Prove this statement. \\
- & \\
-$\left [ 2 \right ] $ & In the Comba squaring algorithm half of the $\hat X$ variables are not used. \\
- & Revise algorithm fast\_s\_mp\_sqr to shrink the $\hat X$ array. \\
- & \\
-$\left [ 3 \right ] $ & Prove the equation for Karatsuba squaring. \\
- & \\
-$\left [ 1 \right ] $ & Prove that Karatsuba squaring requires $O \left (n^{lg(3)} \right )$ time. \\
- & \\
-$\left [ 2 \right ] $ & Determine the minimal ratio between addition and multiplication clock cycles \\
- & required for equation $6.7$ to be true. \\
- & \\
-\end{tabular}
-
-\chapter{Modular Reduction}
-\section{Basics of Modular Reduction}
-\index{modular residue}
-Modular reduction is an operation that arises quite often within public key cryptography algorithms and various number theoretic algorithms,
-such as factoring. Modular reduction algorithms are the third class of algorithms of the ``multipliers'' set. A number $a$ is said to be reduced
-modulo another number $b$ by finding the remainder of the division $a/b$.
-
-Modular reduction is equivalent to solving for $r$ in the following equation. $a = bq + r$ where $q = \lfloor a/b \rfloor$. The result
-$r$ is said to be ``congruent to $a$ modulo $b$'' which is also written as $r \equiv a \mbox{ (mod }b\mbox{)}$. In other vernacular $r$ is known as the
-``modular residue'' which leads to ``quadratic residue''\footnote{That's fancy talk for $b \equiv a^2 \mbox{ (mod }p\mbox{)}$.} and
-other forms of residues.
-
-\index{modulus}
-Modular reductions are normally used to form finite groups such as fields and rings. For example, in the RSA public key algorithm \cite{RSAPAPER}
-two private primes $p$ and $q$ are chosen which when multiplied $n = pq$ forms a composite modulus. When operations such as multiplication and
-squaring are performed on units of the ring $\Z_n$ a finite multiplicative sub-group is formed.
-
-Modular reductions have a variety of other useful properties. For example, a number $x$ is a square if and only if it is a quadratic
-residue modulo a prime. With a finite set of primes $B = \left < p_0, p_1, \ldots, p_n \right >$ a quick test for whether $x$ is square or not can
-be performed\footnote{Provided none of the primes from $B$ divide $x$.}. Consider the figure~\ref{fig:QR} with the candiate $x = 955621$ a simple
-set of modular reductions modulo $3, 5, \ldots, 11$ may detect whether $x$ is a square or not. In this case $955621 \equiv 7 \mbox{ (mod }11\mbox{)}$
-and since $7$ is not a quadratic residue modulo $11$ the number $955621$ is not a square.
-
-\begin{figure}
-\begin{center}
-\begin{tabular}{|c|l|}
-\hline \textbf{Prime} & \textbf{Quadratic Residues} \\
-\hline $3$ & $1$ \\
-\hline $5$ & $1, 4$ \\
-\hline $7$ & $1, 2, 4$ \\
-\hline $11$ & $1, 3, 4, 5, 9$ \\
-\hline
-\end{tabular}
-\end{center}
-\caption{Quadratic Residues for primes less than $13$}
-\label{fig:QR}
-\end{figure}
-
-The most common usage for performance driven modular reductions is in modular exponentiation algorithms. That is to compute
-$d = a^b \mbox{ (mod }c\mbox{)}$ as fast as possible. As will be discussed in the subsequent chapter there exists fast algorithms for computing
-modular exponentiations without having to perform (\textit{in this example}) $b - 1$ multiplications. These algorithms will produce partial
-results in the range $0 \le x < c^2$ which can be taken advantage of to create several efficient algorithms.
-
-\section{The Barrett Reduction}
-The Barrett reduction algorithm \cite{BARRETT} was inspired by fast division algorithms which multiply by the reciprocal to emulate
-division. Barretts observation was that the residue $c$ of $a$ modulo $b$ is equal to
-
-\begin{equation}
-c = a - b \cdot \lfloor a/b \rfloor
-\end{equation}
-
-Since algorithms such as modular exponentiation would be using the same modulus extensively, typical DSP intuition would indicate the next step
-would be to replace $a/b$ by a multiplication by the reciprocal. However, DSP intuition on its own will not work as these numbers are considerably
-larger than the precision of common DSP floating point data types. It would take another common optimization to optimize the algorithm.
-
-\subsection{Fixed Point Arithmetic}
-The trick used to optimize the above equation is based on a technique of emulating floating point data types with fixed precision integers. Fixed
-point arithmetic would vastly popularlize the ``3d-shooter'' genre of games in the mid 1990s when floating point units were fairly slow. The idea behind
-fixed point arithmetic is to take a normal $k$-bit integer data type and break it into $p$-bit integer and a $q$-bit fraction part
-(\textit{where $p+q = k$}).
-
-In this system a $k$-bit integer $n$ would actually represent $n/2^q$. For example, with $q = 4$ the integer $n = 37$ would actually represent the
-value $2.3125$. To multiply two fixed point numbers the integers are multiplied using traditional arithmetic and subsequently normalized. For example,
-with $q = 4$ to multiply the integers $9$ and $5$ they must be converted to fixed point first by multiplying by $2^q$. Let $a = 9(2^q)$
-represent the fixed point representation of $9$ and $b = 5(2^q)$ represent the fixed point representation of $5$. The product $ab$ is equal to
-$45(2^{2q})$ which when normalized produces $45(2^q)$.
-
-Using fixed point arithmetic division can be easily achieved by multiplying by the reciprocal. If $2^q$ is equivalent to one than $2^q/b$ is
-equivalent to $1/b$ using real arithmetic. Using this fact dividing an integer $a$ by another integer $b$ can be achieved with the following
-expression.
-
-\begin{equation}
-\lfloor (a \cdot (\lfloor 2^q / b \rfloor))/2^q \rfloor
-\end{equation}
-
-The precision of the division is proportional to the value of $q$. If the divisor $b$ is used frequently as is the case with
-modular exponentiation pre-computing $2^q/b$ will allow a division to be performed with a multiplication and a right shift. Both operations
-are considerably faster than division on most processors.
-
-Consider dividing $19$ by $5$. The correct result is $\lfloor 19/5 \rfloor = 3$. With $q = 3$ the reciprocal is $\lfloor 2^q/5 \rfloor = 1$ which
-leads to a product of $19$ which when divided by $2^q$ produces $2$. However, with $q = 4$ the reciprocal is $\lfloor 2^q/5 \rfloor = 3$ and
-the result of the emulated division is $\lfloor 3 \cdot 19 / 2^q \rfloor = 3$ which is correct.
-
-Plugging this form of divison into the original equation the following modular residue equation arises.
-
-\begin{equation}
-c = a - b \cdot \lfloor (a \cdot (\lfloor 2^q / b \rfloor))/2^q \rfloor
-\end{equation}
-
-Using the notation from \cite{BARRETT} the value of $\lfloor 2^q / b \rfloor$ will be represented by the $\mu$ symbol. Using the $\mu$
-variable also helps re-inforce the idea that it is meant to be computed once and re-used.
-
-\begin{equation}
-c = a - b \cdot \lfloor (a \cdot \mu)/2^q \rfloor
-\end{equation}
-
-Provided that $2^q > b^2$ this algorithm will produce a quotient that is either exactly correct or off by a value of one. Let $n$ represent
-the number of digits in $b$. This algorithm requires approximately $2n^2$ single precision multiplications to produce the quotient and
-another $n^2$ single precision multiplications to find the residue. In total $3n^2$ single precision multiplications are required to
-reduce the number.
-
-For example, if $b = 1179677$ and $q = 41$ ($2^q > b^2$), then the reciprocal $\mu$ is equal to $\lfloor 2^q / b \rfloor = 1864089$. Consider reducing
-$a = 180388626447$ modulo $b$ using the above reduction equation. The quotient using the new formula is $\lfloor (a \cdot \mu) / 2^q \rfloor = 152913$.
-By subtracting $152913b$ from $a$ the correct residue $a \equiv 677346 \mbox{ (mod }b\mbox{)}$ is found.
-
-\subsection{Choosing a Radix Point}
-Using the fixed point representation a modular reduction can be performed with $3n^2$ single precision multiplications. If that were the best
-that could be achieved a full division might as well be used in its place. The key to optimizing the reduction is to reduce the precision of
-the initial multiplication that finds the quotient.
-
-Let $a$ represent the number of which the residue is sought. Let $b$ represent the modulus used to find the residue. Let $m$ represent
-the number of digits in $b$. For the purposes of this discussion we will assume that the number of digits in $a$ is $2m$. Dividing $a$ by
-$b$ is the same as dividing a $2m$ digit integer by a $m$ digit integer. Digits below the $m - 1$'th digit of $a$ will contribute at most a value
-of $1$ to the quotient because $\beta^k < b$ for any $0 \le k \le m - 1$.
-
-Since those digits do not contribute much to the quotient the observation is that they might as well be zero. However, if the digits
-``might as well be zero'' they might as well not be there in the first place. Let $q_0 = \lfloor a/\beta^{m-1} \rfloor$ represent the input
-with the zeroes trimmed. Now the modular reduction is trimmed to the almost equivalent equation
-
-\begin{equation}
-c = a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor
-\end{equation}
-
-Note that the original divisor $2^q$ has been replaced with $\beta^{m+1}$. Also note that the exponent on the divisor when added to the amount $q_0$
-was shifted by equals $2m$. If the optimization had not been performed the divisor would have the exponent $2m$ so in the end the exponents
-do ``add up''. Using the above equation the quotient $\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ can be off from the true quotient by at most
-two implying that $0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$. By first subtracting $b$ times the quotient and then
-conditionally subtracting $b$ once or twice the residue is found.
-
-The quotient is now found using $(m + 1)(m) = m^2 + m$ single precision multiplications and the residue with an additional $m^2$ single
-precision multiplications. In total $2m^2 + m$ single precision multiplications are required which is considerably faster than the original
-attempt.
-
-For example, let $\beta = 10$ represent the radix of the digits. Let $b = 9999$ represent the modulus which implies $m = 4$. Let $a = 99929878$
-represent the value of which the residue is desired. In this case $q = 8$ since $10^7 < 9999^2$ meaning that $\mu = \lfloor \beta^{q}/b \rfloor = 10001$.
-With the new observation the multiplicand for the quotient is equal to $q_0 = \lfloor a / \beta^{m - 1} \rfloor = 99929$. The quotient is then
-$\lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor = 9993$. Subtracting $9993b$ from $a$ and the correct residue $a \equiv 9871 \mbox{ (mod }b\mbox{)}$
-is found.
-
-\subsection{Trimming the Quotient}
-So far the reduction algorithm has been optimized from $3m^2$ single precision multiplications down to $2m^2 + m$ single precision multiplications. As
-it stands now the algorithm is already fairly fast compared to a full integer division algorithm. However, there is still room for
-optimization.
-
-After the first multiplication inside the quotient ($q_0 \cdot \mu$) the value is shifted right by $m + 1$ places effectively nullifying the lower
-half of the product. It would be nice to be able to remove those digits from the product to effectively cut down the number of single precision
-multiplications. If the number of digits in the modulus $m$ is far less than $\beta$ a full product is not required for the algorithm to work properly.
-In fact the lower $m - 2$ digits will not affect the upper half of the product at all and do not need to be computed.
-
-The value of $\mu$ is a $m$-digit number and $q_0$ is a $m + 1$ digit number. Using a full multiplier $(m + 1)(m) = m^2 + m$ single precision
-multiplications would be required. Using a multiplier that will only produce digits at and above the $m - 1$'th digit reduces the number
-of single precision multiplications to ${m^2 + m} \over 2$ single precision multiplications.
-
-\subsection{Trimming the Residue}
-After the quotient has been calculated it is used to reduce the input. As previously noted the algorithm is not exact and it can be off by a small
-multiple of the modulus, that is $0 \le a - b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor < 3b$. If $b$ is $m$ digits than the
-result of reduction equation is a value of at most $m + 1$ digits (\textit{provided $3 < \beta$}) implying that the upper $m - 1$ digits are
-implicitly zero.
-
-The next optimization arises from this very fact. Instead of computing $b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ using a full
-$O(m^2)$ multiplication algorithm only the lower $m+1$ digits of the product have to be computed. Similarly the value of $a$ can
-be reduced modulo $\beta^{m+1}$ before the multiple of $b$ is subtracted which simplifes the subtraction as well. A multiplication that produces
-only the lower $m+1$ digits requires ${m^2 + 3m - 2} \over 2$ single precision multiplications.
-
-With both optimizations in place the algorithm is the algorithm Barrett proposed. It requires $m^2 + 2m - 1$ single precision multiplications which
-is considerably faster than the straightforward $3m^2$ method.
-
-\subsection{The Barrett Algorithm}
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_reduce}. \\
-\textbf{Input}. mp\_int $a$, mp\_int $b$ and $\mu = \lfloor \beta^{2m}/b \rfloor$ $(0 \le a < b^2, b > 1)$ \\
-\textbf{Output}. $c \leftarrow a \mbox{ (mod }b\mbox{)}$ \\
-\hline \\
-Let $m$ represent the number of digits in $b$. \\
-1. Make a copy of $a$ and store it in $q$. (\textit{mp\_init\_copy}) \\
-2. $q \leftarrow \lfloor q / \beta^{m - 1} \rfloor$ (\textit{mp\_rshd}) \\
-\\
-Produce the quotient. \\
-3. $q \leftarrow q \cdot \mu$ (\textit{note: only produce digits at or above $m-1$}) \\
-4. $q \leftarrow \lfloor q / \beta^{m + 1} \rfloor$ \\
-\\
-Subtract the multiple of modulus from the input. \\
-5. $c \leftarrow a \mbox{ (mod }\beta^{m+1}\mbox{)}$ (\textit{mp\_mod\_2d}) \\
-6. $q \leftarrow q \cdot b \mbox{ (mod }\beta^{m+1}\mbox{)}$ (\textit{s\_mp\_mul\_digs}) \\
-7. $c \leftarrow c - q$ (\textit{mp\_sub}) \\
-\\
-Add $\beta^{m+1}$ if a carry occured. \\
-8. If $c < 0$ then (\textit{mp\_cmp\_d}) \\
-\hspace{3mm}8.1 $q \leftarrow 1$ (\textit{mp\_set}) \\
-\hspace{3mm}8.2 $q \leftarrow q \cdot \beta^{m+1}$ (\textit{mp\_lshd}) \\
-\hspace{3mm}8.3 $c \leftarrow c + q$ \\
-\\
-Now subtract the modulus if the residue is too large (e.g. quotient too small). \\
-9. While $c \ge b$ do (\textit{mp\_cmp}) \\
-\hspace{3mm}9.1 $c \leftarrow c - b$ \\
-10. Clear $q$. \\
-11. Return(\textit{MP\_OKAY}) \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_reduce}
-\end{figure}
-
-\textbf{Algorithm mp\_reduce.}
-This algorithm will reduce the input $a$ modulo $b$ in place using the Barrett algorithm. It is loosely based on algorithm 14.42 of HAC
-\cite[pp. 602]{HAC} which is based on the paper from Paul Barrett \cite{BARRETT}. The algorithm has several restrictions and assumptions which must be adhered to
-for the algorithm to work.
-
-First the modulus $b$ is assumed to be positive and greater than one. If the modulus were less than or equal to one than subtracting
-a multiple of it would either accomplish nothing or actually enlarge the input. The input $a$ must be in the range $0 \le a < b^2$ in order
-for the quotient to have enough precision. Technically the algorithm will still work if $a \ge b^2$ but it will take much longer to finish. The
-value of $\mu$ is passed as an argument to this algorithm and is assumed to be setup before the algorithm is used.
-
-Recall that the multiplication for the quotient on step 3 must only produce digits at or above the $m-1$'th position. An algorithm called
-$s\_mp\_mul\_high\_digs$ which has not been presented is used to accomplish this task. This optimal algorithm can only be used if the number
-of digits in $b$ is very much smaller than $\beta$.
-
-After the multiple of the modulus has been subtracted from $a$ the residue must be fixed up in case its negative. While it is known that
-$a \ge b \cdot \lfloor (q_0 \cdot \mu) / \beta^{m+1} \rfloor$ only the lower $m+1$ digits are being used to compute the residue. In this case
-the invariant $\beta^{m+1}$ must be added to the residue to make it positive again.
-
-The while loop at step 9 will subtract $b$ until the residue is less than $b$. If the algorithm is performed correctly this step is only
-performed upto two times. However, if $a \ge b^2$ than it will iterate substantially more times than it should.
-
-\index{bn\_mp\_reduce.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* reduces x mod m, assumes 0 < x < m**2, mu is
-018 * precomputed via mp_reduce_setup.
-019 * From HAC pp.604 Algorithm 14.42
-020 */
-021 int
-022 mp_reduce (mp_int * x, mp_int * m, mp_int * mu)
-023 \{
-024 mp_int q;
-025 int res, um = m->used;
-026
-027 /* q = x */
-028 if ((res = mp_init_copy (&q, x)) != MP_OKAY) \{
-029 return res;
-030 \}
-031
-032 /* q1 = x / b**(k-1) */
-033 mp_rshd (&q, um - 1);
-034
-035 /* according to HAC this is optimization is ok */
-036 if (((unsigned long) m->used) > (((mp_digit)1) << (DIGIT_BIT - 1))) \{
-037 if ((res = mp_mul (&q, mu, &q)) != MP_OKAY) \{
-038 goto CLEANUP;
-039 \}
-040 \} else \{
-041 if ((res = s_mp_mul_high_digs (&q, mu, &q, um - 1)) != MP_OKAY) \{
-042 goto CLEANUP;
-043 \}
-044 \}
-045
-046 /* q3 = q2 / b**(k+1) */
-047 mp_rshd (&q, um + 1);
-048
-049 /* x = x mod b**(k+1), quick (no division) */
-050 if ((res = mp_mod_2d (x, DIGIT_BIT * (um + 1), x)) != MP_OKAY) \{
-051 goto CLEANUP;
-052 \}
-053
-054 /* q = q * m mod b**(k+1), quick (no division) */
-055 if ((res = s_mp_mul_digs (&q, m, &q, um + 1)) != MP_OKAY) \{
-056 goto CLEANUP;
-057 \}
-058
-059 /* x = x - q */
-060 if ((res = mp_sub (x, &q, x)) != MP_OKAY) \{
-061 goto CLEANUP;
-062 \}
-063
-064 /* If x < 0, add b**(k+1) to it */
-065 if (mp_cmp_d (x, 0) == MP_LT) \{
-066 mp_set (&q, 1);
-067 if ((res = mp_lshd (&q, um + 1)) != MP_OKAY)
-068 goto CLEANUP;
-069 if ((res = mp_add (x, &q, x)) != MP_OKAY)
-070 goto CLEANUP;
-071 \}
-072
-073 /* Back off if it's too big */
-074 while (mp_cmp (x, m) != MP_LT) \{
-075 if ((res = s_mp_sub (x, m, x)) != MP_OKAY) \{
-076 break;
-077 \}
-078 \}
-079
-080 CLEANUP:
-081 mp_clear (&q);
-082
-083 return res;
-084 \}
-\end{alltt}
-\end{small}
-
-The first multiplication that determines the quotient can be performed by only producing the digits from $m - 1$ and up. This essentially halves
-the number of single precision multiplications required. However, the optimization is only safe if $\beta$ is much larger than the number of digits
-in the modulus. In the source code this is evaluated on lines 36 to 44 where algorithm s\_mp\_mul\_high\_digs is used when it is
-safe to do so.
-
-\subsection{The Barrett Setup Algorithm}
-In order to use algorithm mp\_reduce the value of $\mu$ must be calculated in advance. Ideally this value should be computed once and stored for
-future use so that the Barrett algorithm can be used without delay.
-
-\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_reduce\_setup}. \\
-\textbf{Input}. mp\_int $a$ ($a > 1$) \\
-\textbf{Output}. $\mu \leftarrow \lfloor \beta^{2m}/a \rfloor$ \\
-\hline \\
-1. $\mu \leftarrow 2^{2 \cdot lg(\beta) \cdot m}$ (\textit{mp\_2expt}) \\
-2. $\mu \leftarrow \lfloor \mu / b \rfloor$ (\textit{mp\_div}) \\
-3. Return(\textit{MP\_OKAY}) \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_reduce\_setup}
-\end{figure}
-
-\textbf{Algorithm mp\_reduce\_setup.}
-This algorithm computes the reciprocal $\mu$ required for Barrett reduction. First $\beta^{2m}$ is calculated as $2^{2 \cdot lg(\beta) \cdot m}$ which
-is equivalent and much faster. The final value is computed by taking the integer quotient of $\lfloor \mu / b \rfloor$.
-
-\index{bn\_mp\_reduce\_setup.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_setup.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* pre-calculate the value required for Barrett reduction
-018 * For a given modulus "b" it calulates the value required in "a"
-019 */
-020 int
-021 mp_reduce_setup (mp_int * a, mp_int * b)
-022 \{
-023 int res;
-024
-025 if ((res = mp_2expt (a, b->used * 2 * DIGIT_BIT)) != MP_OKAY) \{
-026 return res;
-027 \}
-028 return mp_div (a, b, a, NULL);
-029 \}
-\end{alltt}
-\end{small}
-
-This simple routine calculates the reciprocal $\mu$ required by Barrett reduction. Note the extended usage of algorithm mp\_div where the variable
-which would received the remainder is passed as NULL. As will be discussed in section 9.1 the division routine allows both the quotient and the
-remainder to be passed as NULL meaning to ignore the value.
-
-\section{The Montgomery Reduction}
-Montgomery reduction\footnote{Thanks to Niels Ferguson for his insightful explanation of the algorithm.} \cite{MONT} is by far the most interesting
-form of reduction in common use. It computes a modular residue which is not actually equal to the residue of the input yet instead equal to a
-residue times a constant. However, as perplexing as this may sound the algorithm is relatively simple and very efficient.
-
-Throughout this entire section the variable $n$ will represent the modulus used to form the residue. As will be discussed shortly the value of
-$n$ must be odd. The variable $x$ will represent the quantity of which the residue is sought. Similar to the Barrett algorithm the input
-is restricted to $0 \le x < n^2$. To begin the description some simple number theory facts must be established.
-
-\textbf{Fact 1.} Adding $n$ to $x$ does not change the residue since in effect it adds one to the quotient $\lfloor x / n \rfloor$. Another way
-to explain this is that $n$ (\textit{or multiples of $n$}) is congruent to zero modulo $n$. Adding zero will not change the value of the residue.
-
-\textbf{Fact 2.} If $x$ is even then performing a division by two in $\Z$ is congruent to $x \cdot 2^{-1} \mbox{ (mod }n\mbox{)}$. Actually
-this is an application of the fact that if $x$ is evenly divisible by any $k \in \Z$ then division in $\Z$ will be congruent to
-multiplication by $k^{-1}$ modulo $n$.
-
-From these two simple facts the following simple algorithm can be derived.
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{Montgomery Reduction}. \\
-\textbf{Input}. Integer $x$, $n$ and $k$ \\
-\textbf{Output}. $2^{-k}x \mbox{ (mod }n\mbox{)}$ \\
-\hline \\
-1. for $t$ from $1$ to $k$ do \\
-\hspace{3mm}1.1 If $x$ is odd then \\
-\hspace{6mm}1.1.1 $x \leftarrow x + n$ \\
-\hspace{3mm}1.2 $x \leftarrow x/2$ \\
-2. Return $x$. \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm Montgomery Reduction}
-\end{figure}
-
-The algorithm reduces the input one bit at a time using the two congruencies stated previously. Inside the loop $n$, which is odd, is
-added to $x$ if $x$ is odd. This forces $x$ to be even which allows the division by two in $\Z$ to be congruent to a modular division by two. Since
-$x$ is assumed to be initially much larger than $n$ the addition of $n$ will contribute an insignificant magnitude to $x$. Let $r$ represent the
-final result of the Montgomery algorithm. If $k > lg(n)$ and $0 \le x < n^2$ then the final result is limited to
-$0 \le r < \lfloor x/2^k \rfloor + n$. As a result at most a single subtraction is required to get the residue desired.
-
-\begin{figure}[here]
-\begin{small}
-\begin{center}
-\begin{tabular}{|c|l|}
-\hline \textbf{Step number ($t$)} & \textbf{Result ($x$)} \\
-\hline $1$ & $x + n = 5812$, $x/2 = 2906$ \\
-\hline $2$ & $x/2 = 1453$ \\
-\hline $3$ & $x + n = 1710$, $x/2 = 855$ \\
-\hline $4$ & $x + n = 1112$, $x/2 = 556$ \\
-\hline $5$ & $x/2 = 278$ \\
-\hline $6$ & $x/2 = 139$ \\
-\hline $7$ & $x + n = 396$, $x/2 = 198$ \\
-\hline $8$ & $x/2 = 99$ \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Example of Montgomery Reduction (I)}
-\label{fig:MONT1}
-\end{figure}
-
-Consider the example in figure~\ref{fig:MONT1} which reduces $x = 5555$ modulo $n = 257$ when $k = 8$. The final result $r = 99$ which is actually
-$2^{-8} \cdot 5555 \mbox{ (mod }257\mbox{)}$ can reveal the residue $x \equiv 158$ by multiplying by $2^8$ modulo $n$.
-
-Let $k = \lfloor lg(n) \rfloor + 1$ represent the number of bits in $n$. The current algorithm requires $2k^2$ single precision shifts
-and $k^2$ single precision additions. At this rate the algorithm is most certainly slower than Barrett reduction and not terribly useful.
-Fortunately there exists an alternative representation of the algorithm.
-
-\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{Montgomery Reduction} (modified I). \\
-\textbf{Input}. Integer $x$, $n$ and $k$ \\
-\textbf{Output}. $2^{-k}x \mbox{ (mod }n\mbox{)}$ \\
-\hline \\
-1. for $t$ from $0$ to $k - 1$ do \\
-\hspace{3mm}1.1 If the $t$'th bit of $x$ is one then \\
-\hspace{6mm}1.1.1 $x \leftarrow x + 2^tn$ \\
-2. Return $x/2^k$. \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm Montgomery Reduction (modified I)}
-\end{figure}
-
-This algorithm is equivalent since $2^tn$ is a multiple of $n$ and the lower $k$ bits of $x$ are zero by step 2. The number of single
-precision shifts has now been reduced from $2k^2$ to $k^2 + k$ which is only a small improvement.
-
-\begin{figure}[here]
-\begin{small}
-\begin{center}
-\begin{tabular}{|c|l|}
-\hline \textbf{Step number ($t$)} & \textbf{Result ($x$)} \\
-\hline $1$ & $x + 2^{0}n = 5812$ \\
-\hline $2$ & $5812$ \\
-\hline $3$ & $x + 2^{2}n = 6840$ \\
-\hline $4$ & $x + 2^{3}n = 8896$ \\
-\hline $5$ & $8896$ \\
-\hline $6$ & $8896$ \\
-\hline $7$ & $x + 2^{6}n = 25344$ \\
-\hline $8$ & $25344$ \\
-\hline -- & $x/2^k = 99$ \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Example of Montgomery Reduction (II)}
-\label{fig:MONT2}
-\end{figure}
-
-Figure~\ref{fig:MONT2} demonstrates the modified algorithm reducing $x = 4093$ modulo $n = 257$ with $k = 8$.
-With this algorithm a single shift right at the end is the only right shift required to reduce the input instead of $k$ right shifts inside the
-loop. Note that for the iterations $t = 2, 5, 6$ and $8$ where the result $x$ is not changed. In those iterations the $t$'th bit of $x$ is
-zero and the appropriate multiple of $n$ does not need to be added to force the $t$'th bit of the result to zero.
-
-\subsection{Digit Based Montgomery Reduction}
-Instead of computing the reduction on a bit-by-bit basis it is actually much faster to compute it on digit-by-digit basis. Consider the
-previous algorithm re-written to compute the Montgomery reduction in this new fashion.
-
-\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{Montgomery Reduction} (modified II). \\
-\textbf{Input}. Integer $x$, $n$ and $k$ \\
-\textbf{Output}. $\beta^{-k}x \mbox{ (mod }n\mbox{)}$ \\
-\hline \\
-1. for $t$ from $0$ to $k - 1$ do \\
-\hspace{3mm}1.1 $x \leftarrow x + \mu n \beta^t$ \\
-2. Return $x/\beta^k$. \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm Montgomery Reduction (modified II)}
-\end{figure}
-
-The value $\mu n \beta^t$ is a multiple of the modulus $n$ meaning that it will not change the residue. If the first digit of
-the value $\mu n \beta^t$ equals the negative (modulo $\beta$) of the $t$'th digit of $x$ then the addition will result in a zero digit. This
-problem breaks down to solving the following congruency.
-
-\begin{center}
-\begin{tabular}{rcl}
-$x_t + \mu n_0$ & $\equiv$ & $0 \mbox{ (mod }\beta\mbox{)}$ \\
-$\mu n_0$ & $\equiv$ & $-x_t \mbox{ (mod }\beta\mbox{)}$ \\
-$\mu$ & $\equiv$ & $-x_t/n_0 \mbox{ (mod }\beta\mbox{)}$ \\
-\end{tabular}
-\end{center}
-
-In each iteration of the loop on step 1 a new value of $\mu$ must be calculated. The value of $-1/n_0 \mbox{ (mod }\beta\mbox{)}$ is used
-extensively in this algorithm and should be precomputed. Let $\rho$ represent the negative of the modular inverse of $n_0$ modulo $\beta$.
-
-For example, let $\beta = 10$ represent the radix. Let $n = 17$ represent the modulus which implies $k = 2$ and $\rho \equiv 7$. Let $x = 33$
-represent the value to reduce.
-
-\newpage\begin{figure}
-\begin{center}
-\begin{tabular}{|c|c|c|}
-\hline \textbf{Step ($t$)} & \textbf{Value of $x$} & \textbf{Value of $\mu$} \\
-\hline -- & $33$ & --\\
-\hline $0$ & $33 + \mu n = 50$ & $1$ \\
-\hline $1$ & $50 + \mu n \beta = 900$ & $5$ \\
-\hline
-\end{tabular}
-\end{center}
-\caption{Example of Montgomery Reduction}
-\end{figure}
-
-The final result $900$ is then divided by $\beta^k$ to produce the final result $9$. The first observation is that $9 \nequiv x \mbox{ (mod }n\mbox{)}$
-which implies the result is not the modular residue of $x$ modulo $n$. However, recall that the residue is actually multiplied by $\beta^{-k}$ in
-the algorithm. To get the true residue the value must be multiplied by $\beta^k$. In this case $\beta^k \equiv 15 \mbox{ (mod }n\mbox{)}$ and
-the correct residue is $9 \cdot 15 \equiv 16 \mbox{ (mod }n\mbox{)}$.
-
-\subsection{Baseline Montgomery Reduction}
-The baseline Montgomery reduction algorithm will produce the residue for any size input. It is designed to be a catch-all algororithm for
-Montgomery reductions.
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_montgomery\_reduce}. \\
-\textbf{Input}. mp\_int $x$, mp\_int $n$ and a digit $\rho \equiv -1/n_0 \mbox{ (mod }n\mbox{)}$. \\
-\hspace{11.5mm}($0 \le x < n^2, n > 1, (n, \beta) = 1, \beta^k > n$) \\
-\textbf{Output}. $\beta^{-k}x \mbox{ (mod }n\mbox{)}$ \\
-\hline \\
-1. $digs \leftarrow 2n.used + 1$ \\
-2. If $digs < MP\_ARRAY$ and $m.used < \delta$ then \\
-\hspace{3mm}2.1 Use algorithm fast\_mp\_montgomery\_reduce instead. \\
-\\
-Setup $x$ for the reduction. \\
-3. If $x.alloc < digs$ then grow $x$ to $digs$ digits. \\
-4. $x.used \leftarrow digs$ \\
-\\
-Eliminate the lower $k$ digits. \\
-5. For $ix$ from $0$ to $k - 1$ do \\
-\hspace{3mm}5.1 $\mu \leftarrow x_{ix} \cdot \rho \mbox{ (mod }\beta\mbox{)}$ \\
-\hspace{3mm}5.2 $u \leftarrow 0$ \\
-\hspace{3mm}5.3 For $iy$ from $0$ to $k - 1$ do \\
-\hspace{6mm}5.3.1 $\hat r \leftarrow \mu n_{iy} + x_{ix + iy} + u$ \\
-\hspace{6mm}5.3.2 $x_{ix + iy} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\
-\hspace{6mm}5.3.3 $u \leftarrow \lfloor \hat r / \beta \rfloor$ \\
-\hspace{3mm}5.4 While $u > 0$ do \\
-\hspace{6mm}5.4.1 $iy \leftarrow iy + 1$ \\
-\hspace{6mm}5.4.2 $x_{ix + iy} \leftarrow x_{ix + iy} + u$ \\
-\hspace{6mm}5.4.3 $u \leftarrow \lfloor x_{ix+iy} / \beta \rfloor$ \\
-\hspace{6mm}5.4.4 $x_{ix + iy} \leftarrow x_{ix+iy} \mbox{ (mod }\beta\mbox{)}$ \\
-\\
-Divide by $\beta^k$ and fix up as required. \\
-6. $x \leftarrow \lfloor x / \beta^k \rfloor$ \\
-7. If $x \ge n$ then \\
-\hspace{3mm}7.1 $x \leftarrow x - n$ \\
-8. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_montgomery\_reduce}
-\end{figure}
-
-\textbf{Algorithm mp\_montgomery\_reduce.}
-This algorithm reduces the input $x$ modulo $n$ in place using the Montgomery reduction algorithm. The algorithm is loosely based
-on algorithm 14.32 of \cite[pp.601]{HAC} except it merges the multiplication of $\mu n \beta^t$ with the addition in the inner loop. The
-restrictions on this algorithm are fairly easy to adapt to. First $0 \le x < n^2$ bounds the input to numbers in the same range as
-for the Barrett algorithm. Additionally $n > 1$ will ensure a modular inverse $\rho$ exists. $\rho$ must be calculated in
-advance of this algorithm. Finally the variable $k$ is fixed and a pseudonym for $n.used$.
-
-Step 2 decides whether a faster Montgomery algorithm can be used. It is based on the Comba technique meaning that there are limits on
-the size of the input. This algorithm is discussed in sub-section 7.3.3.
-
-Step 5 is the main reduction loop of the algorithm. The value of $\mu$ is calculated once per iteration in the outer loop. The inner loop
-calculates $x + \mu n \beta^{ix}$ by multiplying $\mu n$ and adding the result to $x$ shifted by $ix$ digits. Both the addition and
-multiplication are performed in the same loop to save time and memory. Step 5.4 will handle any additional carries that escape the inner loop.
-
-Using a quick inspection this algorithm requires $n$ single precision multiplications for the outer loop and $n^2$ single precision multiplications
-in the inner loop. In total $n^2 + n$ single precision multiplications which compares favourably to Barrett at $n^2 + 2n - 1$ single precision
-multiplications.
-
-\index{bn\_mp\_montgomery\_reduce.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_montgomery\_reduce.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* computes xR**-1 == x (mod N) via Montgomery Reduction */
-018 int
-019 mp_montgomery_reduce (mp_int * x, mp_int * n, mp_digit rho)
-020 \{
-021 int ix, res, digs;
-022 mp_digit mu;
-023
-024 /* can the fast reduction [comba] method be used?
-025 *
-026 * Note that unlike in mp_mul you're safely allowed *less*
-027 * than the available columns [255 per default] since carries
-028 * are fixed up in the inner loop.
-029 */
-030 digs = n->used * 2 + 1;
-031 if ((digs < MP_WARRAY) &&
-032 n->used <
-033 (1 << ((CHAR_BIT * sizeof (mp_word)) - (2 * DIGIT_BIT)))) \{
-034 return fast_mp_montgomery_reduce (x, n, rho);
-035 \}
-036
-037 /* grow the input as required */
-038 if (x->alloc < digs) \{
-039 if ((res = mp_grow (x, digs)) != MP_OKAY) \{
-040 return res;
-041 \}
-042 \}
-043 x->used = digs;
-044
-045 for (ix = 0; ix < n->used; ix++) \{
-046 /* mu = ai * m' mod b */
-047 mu = (x->dp[ix] * rho) & MP_MASK;
-048
-049 /* a = a + mu * m * b**i */
-050 \{
-051 register int iy;
-052 register mp_digit *tmpn, *tmpx, u;
-053 register mp_word r;
-054
-055 /* aliases */
-056 tmpn = n->dp;
-057 tmpx = x->dp + ix;
-058
-059 /* set the carry to zero */
-060 u = 0;
-061
-062 /* Multiply and add in place */
-063 for (iy = 0; iy < n->used; iy++) \{
-064 r = ((mp_word) mu) * ((mp_word) * tmpn++) +
-065 ((mp_word) u) + ((mp_word) * tmpx);
-066 u = (r >> ((mp_word) DIGIT_BIT));
-067 *tmpx++ = (r & ((mp_word) MP_MASK));
-068 \}
-069 /* propagate carries */
-070 while (u) \{
-071 *tmpx += u;
-072 u = *tmpx >> DIGIT_BIT;
-073 *tmpx++ &= MP_MASK;
-074 \}
-075 \}
-076 \}
-077
-078 /* x = x/b**n.used */
-079 mp_clamp(x);
-080 mp_rshd (x, n->used);
-081
-082 /* if A >= m then A = A - m */
-083 if (mp_cmp_mag (x, n) != MP_LT) \{
-084 return s_mp_sub (x, n, x);
-085 \}
-086
-087 return MP_OKAY;
-088 \}
-\end{alltt}
-\end{small}
-
-This is the baseline implementation of the Montgomery reduction algorithm. Lines 30 to 35 determine if the Comba based
-routine can be used instead. Line 47 computes the value of $\mu$ for that particular iteration of the outer loop.
-
-The multiplication $\mu n \beta^{ix}$ is performed in one step in the inner loop. The alias $tmpx$ refers to the $ix$'th digit of $x$ and
-the alias $tmpn$ refers to the modulus $n$.
-
-\subsection{Faster ``Comba'' Montgomery Reduction}
-
-The Montgomery reduction requires fewer single precision multiplications than a Barrett reduction, however it is much slower due to the serial
-nature of the inner loop. The Barrett reduction algorithm requires two slightly modified multipliers which can be implemented with the Comba
-technique. The Montgomery reduction algorithm cannot directly use the Comba technique to any significant advantage since the inner loop calculates
-a $k \times 1$ product $k$ times.
-
-The biggest obstacle is that at the $ix$'th iteration of the outer loop the value of $x_{ix}$ is required to calculate $\mu$. This means the
-carries from $0$ to $ix - 1$ must have been propagated upwards to form a valid $ix$'th digit. The solution as it turns out is very simple.
-Perform a Comba like multiplier and inside the outer loop just after the inner loop fix up the $ix + 1$'th digit by forwarding the carry.
-
-With this change in place the Montgomery reduction algorithm can be performed with a Comba style multiplication loop which substantially increases
-the speed of the algorithm.
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{fast\_mp\_montgomery\_reduce}. \\
-\textbf{Input}. mp\_int $x$, mp\_int $n$ and a digit $\rho \equiv -1/n_0 \mbox{ (mod }n\mbox{)}$. \\
-\hspace{11.5mm}($0 \le x < n^2, n > 1, (n, \beta) = 1, \beta^k > n$) \\
-\textbf{Output}. $\beta^{-k}x \mbox{ (mod }n\mbox{)}$ \\
-\hline \\
-Place an array of \textbf{MP\_WARRAY} mp\_word variables called $\hat W$ on the stack. \\
-1. if $x.alloc < n.used + 1$ then grow $x$ to $n.used + 1$ digits. \\
-Copy the digits of $x$ into the array $\hat W$ \\
-2. For $ix$ from $0$ to $x.used - 1$ do \\
-\hspace{3mm}2.1 $\hat W_{ix} \leftarrow x_{ix}$ \\
-3. For $ix$ from $x.used$ to $2n.used - 1$ do \\
-\hspace{3mm}3.1 $\hat W_{ix} \leftarrow 0$ \\
-Elimiate the lower $k$ digits. \\
-4. for $ix$ from $0$ to $n.used - 1$ do \\
-\hspace{3mm}4.1 $\mu \leftarrow \hat W_{ix} \cdot \rho \mbox{ (mod }\beta\mbox{)}$ \\
-\hspace{3mm}4.2 For $iy$ from $0$ to $n.used - 1$ do \\
-\hspace{6mm}4.2.1 $\hat W_{iy + ix} \leftarrow \hat W_{iy + ix} + \mu \cdot n_{iy}$ \\
-\hspace{3mm}4.3 $\hat W_{ix + 1} \leftarrow \hat W_{ix + 1} + \lfloor \hat W_{ix} / \beta \rfloor$ \\
-Propagate carries upwards. \\
-5. for $ix$ from $n.used$ to $2n.used + 1$ do \\
-\hspace{3mm}5.1 $\hat W_{ix + 1} \leftarrow \hat W_{ix + 1} + \lfloor \hat W_{ix} / \beta \rfloor$ \\
-Shift right and reduce modulo $\beta$ simultaneously. \\
-6. for $ix$ from $0$ to $n.used + 1$ do \\
-\hspace{3mm}6.1 $x_{ix} \leftarrow \hat W_{ix + n.used} \mbox{ (mod }\beta\mbox{)}$ \\
-Zero excess digits and fixup $x$. \\
-7. if $x.used > n.used + 1$ then do \\
-\hspace{3mm}7.1 for $ix$ from $n.used + 1$ to $x.used - 1$ do \\
-\hspace{6mm}7.1.1 $x_{ix} \leftarrow 0$ \\
-8. $x.used \leftarrow n.used + 1$ \\
-9. Clamp excessive digits of $x$. \\
-10. If $x \ge n$ then \\
-\hspace{3mm}10.1 $x \leftarrow x - n$ \\
-11. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm fast\_mp\_montgomery\_reduce}
-\end{figure}
-
-\textbf{Algorithm fast\_mp\_montgomery\_reduce.}
-This algorithm will compute the Montgomery reduction of $x$ modulo $n$ using the Comba technique. It is on most computer platforms significantly
-faster than algorithm mp\_montgomery\_reduce and algorithm mp\_reduce (\textit{Barrett reduction}). The algorithm has the same restrictions
-on the input as the baseline reduction algorithm. An additional two restrictions are imposed on this algorithm. The number of digits $k$ in the
-the modulus $n$ must not violate $MP\_WARRAY > 2k +1$ and $n < \delta$. When $\beta = 2^{28}$ this algorithm can be used to reduce modulo
-a modulus of at most $3,556$ bits in length.
-
-As in the other Comba reduction algorithms there is a $\hat W$ array which stores the columns of the product. It is initially filled with the
-contents of $x$ with the excess digits zeroed. The reduction loop is very similar the to the baseline loop at heart. The multiplication on step
-4.1 can be single precision only since $ab \mbox{ (mod }\beta\mbox{)} \equiv (a \mbox{ mod }\beta)(b \mbox{ mod }\beta)$. Some multipliers such
-as those on the ARM processors take a variable length time to complete depending on the number of bytes of result it must produce. By performing
-a single precision multiplication instead half the amount of time is spent.
-
-Also note that digit $\hat W_{ix}$ must have the carry from the $ix - 1$'th digit propagated upwards in order for this to work. That is what step
-4.3 will do. In effect over the $n.used$ iterations of the outer loop the $n.used$'th lower columns all have the their carries propagated forwards. Note
-how the upper bits of those same words are not reduced modulo $\beta$. This is because those values will be discarded shortly and there is no
-point.
-
-Step 5 will propgate the remainder of the carries upwards. On step 6 the columns are reduced modulo $\beta$ and shifted simultaneously as they are
-stored in the destination $x$.
-
-\index{bn\_fast\_mp\_montgomery\_reduce.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_fast\_mp\_montgomery\_reduce.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* computes xR**-1 == x (mod N) via Montgomery Reduction
-018 *
-019 * This is an optimized implementation of mp_montgomery_reduce
-020 * which uses the comba method to quickly calculate the columns of the
-021 * reduction.
-022 *
-023 * Based on Algorithm 14.32 on pp.601 of HAC.
-024 */
-025 int
-026 fast_mp_montgomery_reduce (mp_int * x, mp_int * n, mp_digit rho)
-027 \{
-028 int ix, res, olduse;
-029 mp_word W[MP_WARRAY];
-030
-031 /* get old used count */
-032 olduse = x->used;
-033
-034 /* grow a as required */
-035 if (x->alloc < n->used + 1) \{
-036 if ((res = mp_grow (x, n->used + 1)) != MP_OKAY) \{
-037 return res;
-038 \}
-039 \}
-040
-041 \{
-042 register mp_word *_W;
-043 register mp_digit *tmpx;
-044
-045 _W = W;
-046 tmpx = x->dp;
-047
-048 /* copy the digits of a into W[0..a->used-1] */
-049 for (ix = 0; ix < x->used; ix++) \{
-050 *_W++ = *tmpx++;
-051 \}
-052
-053 /* zero the high words of W[a->used..m->used*2] */
-054 for (; ix < n->used * 2 + 1; ix++) \{
-055 *_W++ = 0;
-056 \}
-057 \}
-058
-059 for (ix = 0; ix < n->used; ix++) \{
-060 /* mu = ai * m' mod b
-061 *
-062 * We avoid a double precision multiplication (which isn't required)
-063 * by casting the value down to a mp_digit. Note this requires
-064 * that W[ix-1] have the carry cleared (see after the inner loop)
-065 */
-066 register mp_digit mu;
-067 mu = (((mp_digit) (W[ix] & MP_MASK)) * rho) & MP_MASK;
-068
-069 /* a = a + mu * m * b**i
-070 *
-071 * This is computed in place and on the fly. The multiplication
-072 * by b**i is handled by offseting which columns the results
-073 * are added to.
-074 *
-075 * Note the comba method normally doesn't handle carries in the
-076 * inner loop In this case we fix the carry from the previous
-077 * column since the Montgomery reduction requires digits of the
-078 * result (so far) [see above] to work. This is
-079 * handled by fixing up one carry after the inner loop. The
-080 * carry fixups are done in order so after these loops the
-081 * first m->used words of W[] have the carries fixed
-082 */
-083 \{
-084 register int iy;
-085 register mp_digit *tmpn;
-086 register mp_word *_W;
-087
-088 /* alias for the digits of the modulus */
-089 tmpn = n->dp;
-090
-091 /* Alias for the columns set by an offset of ix */
-092 _W = W + ix;
-093
-094 /* inner loop */
-095 for (iy = 0; iy < n->used; iy++) \{
-096 *_W++ += ((mp_word) mu) * ((mp_word) * tmpn++);
-097 \}
-098 \}
-099
-100 /* now fix carry for next digit, W[ix+1] */
-101 W[ix + 1] += W[ix] >> ((mp_word) DIGIT_BIT);
-102 \}
-103
-104
-105 \{
-106 register mp_digit *tmpx;
-107 register mp_word *_W, *_W1;
-108
-109 /* nox fix rest of carries */
-110 _W1 = W + ix;
-111 _W = W + ++ix;
-112
-113 for (; ix <= n->used * 2 + 1; ix++) \{
-114 *_W++ += *_W1++ >> ((mp_word) DIGIT_BIT);
-115 \}
-116
-117 /* copy out, A = A/b**n
-118 *
-119 * The result is A/b**n but instead of converting from an
-120 * array of mp_word to mp_digit than calling mp_rshd
-121 * we just copy them in the right order
-122 */
-123 tmpx = x->dp;
-124 _W = W + n->used;
-125
-126 for (ix = 0; ix < n->used + 1; ix++) \{
-127 *tmpx++ = *_W++ & ((mp_word) MP_MASK);
-128 \}
-129
-130 /* zero oldused digits, if the input a was larger than
-131 * m->used+1 we'll have to clear the digits */
-132 for (; ix < olduse; ix++) \{
-133 *tmpx++ = 0;
-134 \}
-135 \}
-136
-137 /* set the max used and clamp */
-138 x->used = n->used + 1;
-139 mp_clamp (x);
-140
-141 /* if A >= m then A = A - m */
-142 if (mp_cmp_mag (x, n) != MP_LT) \{
-143 return s_mp_sub (x, n, x);
-144 \}
-145 return MP_OKAY;
-146 \}
-\end{alltt}
-\end{small}
-
-The $\hat W$ array is first filled with digits of $x$ on line 49 then the rest of the digits are zeroed on line 54. Both loops share
-the same alias variables to make the code easier to read.
-
-The value of $\mu$ is calculated in an interesting fashion. First the value $\hat W_{ix}$ is reduced modulo $\beta$ and cast to a mp\_digit. This
-forces the compiler to use a single precision multiplication and prevents any concerns about loss of precision. Line 101 fixes the carry
-for the next iteration of the loop by propagating the carry from $\hat W_{ix}$ to $\hat W_{ix+1}$.
-
-The for loop on line 113 propagates the rest of the carries upwards through the columns. The for loop on line 126 reduces the columns
-modulo $\beta$ and shifts them $k$ places at the same time. The alias $\_ \hat W$ actually refers to the array $\hat W$ starting at the $n.used$'th
-digit, that is $\_ \hat W_{t} = \hat W_{n.used + t}$.
-
-\subsection{Montgomery Setup}
-To calculate the variable $\rho$ a relatively simple algorithm will be required.
-
-\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_montgomery\_setup}. \\
-\textbf{Input}. mp\_int $n$ ($n > 1$ and $(n, 2) = 1$) \\
-\textbf{Output}. $\rho \equiv -1/n_0 \mbox{ (mod }\beta\mbox{)}$ \\
-\hline \\
-1. $b \leftarrow n_0$ \\
-2. If $b$ is even return(\textit{MP\_VAL}) \\
-3. $x \leftarrow ((b + 2) \mbox{ AND } 4) << 1) + b$ \\
-4. for $k$ from 0 to $3$ do \\
-\hspace{3mm}4.1 $x \leftarrow x \cdot (2 - bx)$ \\
-5. $\rho \leftarrow \beta - x \mbox{ (mod }\beta\mbox{)}$ \\
-6. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_montgomery\_setup}
-\end{figure}
-
-\textbf{Algorithm mp\_montgomery\_setup.}
-This algorithm will calculate the value of $\rho$ required within the Montgomery reduction algorithms. It uses a very interesting trick
-to calculate $1/n_0$ when $\beta$ is a power of two.
-
-\index{bn\_mp\_montgomery\_setup.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_montgomery\_setup.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* setups the montgomery reduction stuff */
-018 int
-019 mp_montgomery_setup (mp_int * n, mp_digit * rho)
-020 \{
-021 mp_digit x, b;
-022
-023 /* fast inversion mod 2**k
-024 *
-025 * Based on the fact that
-026 *
-027 * XA = 1 (mod 2**n) => (X(2-XA)) A = 1 (mod 2**2n)
-028 * => 2*X*A - X*X*A*A = 1
-029 * => 2*(1) - (1) = 1
-030 */
-031 b = n->dp[0];
-032
-033 if ((b & 1) == 0) \{
-034 return MP_VAL;
-035 \}
-036
-037 x = (((b + 2) & 4) << 1) + b; /* here x*a==1 mod 2**4 */
-038 x *= 2 - b * x; /* here x*a==1 mod 2**8 */
-039 #if !defined(MP_8BIT)
-040 x *= 2 - b * x; /* here x*a==1 mod 2**16 */
-041 #endif
-042 #if defined(MP_64BIT) || !(defined(MP_8BIT) || defined(MP_16BIT))
-043 x *= 2 - b * x; /* here x*a==1 mod 2**32 */
-044 #endif
-045 #ifdef MP_64BIT
-046 x *= 2 - b * x; /* here x*a==1 mod 2**64 */
-047 #endif
-048
-049 /* rho = -1/m mod b */
-050 *rho = (((mp_digit) 1 << ((mp_digit) DIGIT_BIT)) - x) & MP_MASK;
-051
-052 return MP_OKAY;
-053 \}
-\end{alltt}
-\end{small}
-
-This source code computes the value of $\rho$ required to perform Montgomery reduction. It has been modified to avoid performing excess
-multiplications when $\beta$ is not the default 28-bits.
-
-\section{The Diminished Radix Algorithm}
-The Diminished Radix method of modular reduction \cite{DRMET} is a fairly clever technique which can be more efficient than either the Barrett
-or Montgomery methods for certain forms of moduli. The technique is based on the following simple congruence.
-
-\begin{equation}
-(x \mbox{ mod } n) + k \lfloor x / n \rfloor \equiv x \mbox{ (mod }(n - k)\mbox{)}
-\end{equation}
-
-This observation was used in the MMB \cite{MMB} block cipher to create a diffusion primitive. It used the fact that if $n = 2^{31}$ and $k=1$ that
-then a x86 multiplier could produce the 62-bit product and use the ``shrd'' instruction to perform a double-precision right shift. The proof
-of the above equation is very simple. First write $x$ in the product form.
-
-\begin{equation}
-x = qn + r
-\end{equation}
-
-Now reduce both sides modulo $(n - k)$.
-
-\begin{equation}
-x \equiv qk + r \mbox{ (mod }(n-k)\mbox{)}
-\end{equation}
-
-The variable $n$ reduces modulo $n - k$ to $k$. By putting $q = \lfloor x/n \rfloor$ and $r = x \mbox{ mod } n$
-into the equation the original congruence is reproduced, thus concluding the proof. The following algorithm is based on this observation.
-
-\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{Diminished Radix Reduction}. \\
-\textbf{Input}. Integer $x$, $n$, $k$ \\
-\textbf{Output}. $x \mbox{ mod } (n - k)$ \\
-\hline \\
-1. $q \leftarrow \lfloor x / n \rfloor$ \\
-2. $q \leftarrow k \cdot q$ \\
-3. $x \leftarrow x \mbox{ (mod }n\mbox{)}$ \\
-4. $x \leftarrow x + q$ \\
-5. If $x \ge (n - k)$ then \\
-\hspace{3mm}5.1 $x \leftarrow x - (n - k)$ \\
-\hspace{3mm}5.2 Goto step 1. \\
-6. Return $x$ \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm Diminished Radix Reduction}
-\label{fig:DR}
-\end{figure}
-
-This algorithm will reduce $x$ modulo $n - k$ and return the residue. If $0 \le x < (n - k)^2$ then the algorithm will loop almost always
-once or twice and occasionally three times. For simplicity sake the value of $x$ is bounded by the following simple polynomial.
-
-\begin{equation}
-0 \le x < n^2 + k^2 - 2nk
-\end{equation}
-
-The true bound is $0 \le x < (n - k - 1)^2$ but this has quite a few more terms. The value of $q$ after step 1 is bounded by the following.
-
-\begin{equation}
-q < n - 2k - k^2/n
-\end{equation}
-
-Since $k^2$ is going to be considerably smaller than $n$ that term will always be zero. The value of $x$ after step 3 is bounded trivially as
-$0 \le x < n$. By step four the sum $x + q$ is bounded by
-
-\begin{equation}
-0 \le q + x < (k + 1)n - 2k^2 - 1
-\end{equation}
-
-With a second pass $q$ will be loosely bounded by $0 \le q < k^2$ after step 2 while $x$ will still be loosely bounded by $0 \le x < n$ after step 3. After the second pass it is highly unlike that the
-sum in step 4 will exceed $n - k$. In practice fewer than three passes of the algorithm are required to reduce virtually every input in the
-range $0 \le x < (n - k - 1)^2$.
-
-\begin{figure}
-\begin{small}
-\begin{center}
-\begin{tabular}{|l|}
-\hline
-$x = 123456789, n = 256, k = 3$ \\
-\hline $q \leftarrow \lfloor x/n \rfloor = 482253$ \\
-$q \leftarrow q*k = 1446759$ \\
-$x \leftarrow x \mbox{ mod } n = 21$ \\
-$x \leftarrow x + q = 1446780$ \\
-$x \leftarrow x - (n - k) = 1446527$ \\
-\hline
-$q \leftarrow \lfloor x/n \rfloor = 5650$ \\
-$q \leftarrow q*k = 16950$ \\
-$x \leftarrow x \mbox{ mod } n = 127$ \\
-$x \leftarrow x + q = 17077$ \\
-$x \leftarrow x - (n - k) = 16824$ \\
-\hline
-$q \leftarrow \lfloor x/n \rfloor = 65$ \\
-$q \leftarrow q*k = 195$ \\
-$x \leftarrow x \mbox{ mod } n = 184$ \\
-$x \leftarrow x + q = 379$ \\
-$x \leftarrow x - (n - k) = 126$ \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Example Diminished Radix Reduction}
-\label{fig:EXDR}
-\end{figure}
-
-Figure~\ref{fig:EXDR} demonstrates the reduction of $x = 123456789$ modulo $n - k = 253$ when $n = 256$ and $k = 3$. Note that even while $x$
-is considerably larger than $(n - k - 1)^2 = 63504$ the algorithm still converges on the modular residue exceedingly fast. In this case only
-three passes were required to find the residue $x \equiv 126$.
-
-
-\subsection{Choice of Moduli}
-On the surface this algorithm looks like a very expensive algorithm. It requires a couple of subtractions followed by multiplication and other
-modular reductions. The usefulness of this algorithm becomes exceedingly clear when an appropriate moduli is chosen.
-
-Division in general is a very expensive operation to perform. The one exception is when the division is by a power of the radix of representation used.
-Division by ten for example is simple for pencil and paper mathematics since it amounts to shifting the decimal place to the right. Similarly division
-by two (\textit{or powers of two}) is very simple for binary computers to perform. It would therefore seem logical to choose $n$ of the form $2^p$
-which would imply that $\lfloor x / n \rfloor$ is a simple shift of $x$ right $p$ bits.
-
-However, there is one operation related to division of power of twos that is even faster than this. If $n = \beta^p$ then the division may be
-performed by moving whole digits to the right $p$ places. In practice division by $\beta^p$ is much faster than division by $2^p$ for any $p$.
-Also with the choice of $n = \beta^p$ reducing $x$ modulo $n$ requires zeroing the digits above the $p-1$'th digit of $x$.
-
-Throughout the next section the term ``restricted modulus'' will refer to a modulus of the form $\beta^p - k$ where as the term ``unrestricted
-modulus'' will refer to a modulus of the form $2^p - k$. The word ``restricted'' in this case refers to the fact that it is based on the
-$2^p$ logic except $p$ must be a multiple of $lg(\beta)$.
-
-\subsection{Choice of $k$}
-Now that division and reduction (\textit{step 1 and 3 of figure~\ref{fig:DR}}) have been optimized to simple digit operations the multiplication by $k$
-in step 2 is the most expensive operation. Fortunately the choice of $k$ is not terribly limited. For all intents and purposes it might
-as well be a single digit. The smaller the value of $k$ is the faster the algorithm will be.
-
-\subsection{Restricted Diminished Radix Reduction}
-The restricted Diminished Radix algorithm can quickly reduce an input modulo a modulus of the form $n = \beta^p - k$. This algorithm can reduce
-an input $x$ within the range $0 \le x < n^2$ using only a couple passes of the algorithm demonstrated in figure~\ref{fig:DR}. The implementation
-of this algorithm has been optimized to avoid additional overhead associated with a division by $\beta^p$, the multiplication by $k$ or the addition
-of $x$ and $q$. The resulting algorithm is very efficient and can lead to substantial improvements over Barrett and Montgomery reduction when modular
-exponentiations are performed.
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_dr\_reduce}. \\
-\textbf{Input}. mp\_int $x$, $n$ and a mp\_digit $k = \beta - n_0$ \\
-\hspace{11.5mm}($0 \le x < n^2$, $n > 1$, $0 < k < \beta$) \\
-\textbf{Output}. $x \mbox{ mod } n$ \\
-\hline \\
-1. $m \leftarrow n.used$ \\
-2. If $x.alloc < 2m$ then grow $x$ to $2m$ digits. \\
-3. $\mu \leftarrow 0$ \\
-4. for $i$ from $0$ to $m - 1$ do \\
-\hspace{3mm}4.1 $\hat r \leftarrow k \cdot x_{m+i} + x_{i} + \mu$ \\
-\hspace{3mm}4.2 $x_{i} \leftarrow \hat r \mbox{ (mod }\beta\mbox{)}$ \\
-\hspace{3mm}4.3 $\mu \leftarrow \lfloor \hat r / \beta \rfloor$ \\
-5. $x_{m} \leftarrow \mu$ \\
-6. for $i$ from $m + 1$ to $x.used - 1$ do \\
-\hspace{3mm}6.1 $x_{i} \leftarrow 0$ \\
-7. Clamp excess digits of $x$. \\
-8. If $x \ge n$ then \\
-\hspace{3mm}8.1 $x \leftarrow x - n$ \\
-\hspace{3mm}8.2 Goto step 3. \\
-9. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_dr\_reduce}
-\end{figure}
-
-\textbf{Algorithm mp\_dr\_reduce.}
-This algorithm will perform the Dimished Radix reduction of $x$ modulo $n$. It has similar restrictions to that of the Barrett reduction
-with the addition that $n$ must be of the form $n = \beta^m - k$ where $0 < k <\beta$.
-
-This algorithm essentially implements the pseudo-code in figure~\ref{fig:DR} except with a slight optimization. The division by $\beta^m$, multiplication by $k$
-and addition of $x \mbox{ mod }\beta^m$ are all performed simultaneously inside the loop on step 4. The division by $\beta^m$ is emulated by accessing
-the term at the $m+i$'th position which is subsequently multiplied by $k$ and added to the term at the $i$'th position. After the loop the $m$'th
-digit is set to the carry and the upper digits are zeroed. Steps 5 and 6 emulate the reduction modulo $\beta^m$ that should have happend to
-$x$ before the addition of the multiple of the upper half.
-
-At step 8 if $x$ is still larger than $n$ another pass of the algorithm is required. First $n$ is subtracted from $x$ and then the algorithm resumes
-at step 3.
-
-\index{bn\_mp\_dr\_reduce.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_dr\_reduce.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* reduce "x" in place modulo "n" using the Diminished Radix algorithm.
-018 *
-019 * Based on algorithm from the paper
-020 *
-021 * "Generating Efficient Primes for Discrete Log Cryptosystems"
-022 * Chae Hoon Lim, Pil Loong Lee,
-023 * POSTECH Information Research Laboratories
-024 *
-025 * The modulus must be of a special format [see manual]
-026 *
-027 * Has been modified to use algorithm 7.10 from the LTM book instead
-028 */
-029 int
-030 mp_dr_reduce (mp_int * x, mp_int * n, mp_digit k)
-031 \{
-032 int err, i, m;
-033 mp_word r;
-034 mp_digit mu, *tmpx1, *tmpx2;
-035
-036 /* m = digits in modulus */
-037 m = n->used;
-038
-039 /* ensure that "x" has at least 2m digits */
-040 if (x->alloc < m + m) \{
-041 if ((err = mp_grow (x, m + m)) != MP_OKAY) \{
-042 return err;
-043 \}
-044 \}
-045
-046 /* top of loop, this is where the code resumes if
-047 * another reduction pass is required.
-048 */
-049 top:
-050 /* aliases for digits */
-051 /* alias for lower half of x */
-052 tmpx1 = x->dp;
-053
-054 /* alias for upper half of x, or x/B**m */
-055 tmpx2 = x->dp + m;
-056
-057 /* set carry to zero */
-058 mu = 0;
-059
-060 /* compute (x mod B**m) + mp * [x/B**m] inline and inplace */
-061 for (i = 0; i < m; i++) \{
-062 r = ((mp_word)*tmpx2++) * ((mp_word)k) + *tmpx1 + mu;
-063 *tmpx1++ = r & MP_MASK;
-064 mu = r >> ((mp_word)DIGIT_BIT);
-065 \}
-066
-067 /* set final carry */
-068 *tmpx1++ = mu;
-069
-070 /* zero words above m */
-071 for (i = m + 1; i < x->used; i++) \{
-072 *tmpx1++ = 0;
-073 \}
-074
-075 /* clamp, sub and return */
-076 mp_clamp (x);
-077
-078 /* if x >= n then subtract and reduce again
-079 * Each successive "recursion" makes the input smaller and smaller.
-080 */
-081 if (mp_cmp_mag (x, n) != MP_LT) \{
-082 s_mp_sub(x, n, x);
-083 goto top;
-084 \}
-085 return MP_OKAY;
-086 \}
-\end{alltt}
-\end{small}
-
-The first step is to grow $x$ as required to $2m$ digits since the reduction is performed in place on $x$. The label on line 49 is where
-the algorithm will resume if further reduction passes are required. In theory it could be placed at the top of the function however, the size of
-the modulus and question of whether $x$ is large enough are invariant after the first pass meaning that it would be a waste of time.
-
-The aliases $tmpx1$ and $tmpx2$ refer to the digits of $x$ where the latter is offset by $m$ digits. By reading digits from $x$ offset by $m$ digits
-a division by $\beta^m$ can be simulated virtually for free. The loop on line 61 performs the bulk of the work (\textit{corresponds to step 4 of algorithm 7.11})
-in this algorithm.
-
-By line 68 the pointer $tmpx1$ points to the $m$'th digit of $x$ which is where the final carry will be placed. Similarly by line 71 the
-same pointer will point to the $m+1$'th digit where the zeroes will be placed.
-
-Since the algorithm is only valid if both $x$ and $n$ are greater than zero an unsigned comparison suffices to determine if another pass is required.
-With the same logic at line 82 the value of $x$ is known to be greater than or equal to $n$ meaning that an unsigned subtraction can be used
-as well. Since the destination of the subtraction is the larger of the inputs the call to algorithm s\_mp\_sub cannot fail and the return code
-does not need to be checked.
-
-\subsubsection{Setup}
-To setup the restricted Diminished Radix algorithm the value $k = \beta - n_0$ is required. This algorithm is not really complicated but provided for
-completeness.
-
-\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_dr\_setup}. \\
-\textbf{Input}. mp\_int $n$ \\
-\textbf{Output}. $k = \beta - n_0$ \\
-\hline \\
-1. $k \leftarrow \beta - n_0$ \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_dr\_setup}
-\end{figure}
-
-\index{bn\_mp\_dr\_setup.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_dr\_setup.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* determines the setup value */
-018 void mp_dr_setup(mp_int *a, mp_digit *d)
-019 \{
-020 /* the casts are required if DIGIT_BIT is one less than
-021 * the number of bits in a mp_digit [e.g. DIGIT_BIT==31]
-022 */
-023 *d = (mp_digit)((((mp_word)1) << ((mp_word)DIGIT_BIT)) -
-024 ((mp_word)a->dp[0]));
-025 \}
-026
-\end{alltt}
-\end{small}
-
-\subsubsection{Modulus Detection}
-Another algorithm which will be useful is the ability to detect a restricted Diminished Radix modulus. An integer is said to be
-of restricted Diminished Radix form if all of the digits are equal to $\beta - 1$ except the trailing digit which may be any value.
-
-\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_dr\_is\_modulus}. \\
-\textbf{Input}. mp\_int $n$ \\
-\textbf{Output}. $1$ if $n$ is in D.R form, $0$ otherwise \\
-\hline
-1. If $n.used < 2$ then return($0$). \\
-2. for $ix$ from $1$ to $n.used - 1$ do \\
-\hspace{3mm}2.1 If $n_{ix} \ne \beta - 1$ return($0$). \\
-3. Return($1$). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_dr\_is\_modulus}
-\end{figure}
-
-\textbf{Algorithm mp\_dr\_is\_modulus.}
-This algorithm determines if a value is in Diminished Radix form. Step 1 rejects obvious cases where fewer than two digits are
-in the mp\_int. Step 2 tests all but the first digit to see if they are equal to $\beta - 1$. If the algorithm manages to get to
-step 3 then $n$ must of Diminished Radix form.
-
-\index{bn\_mp\_dr\_is\_modulus.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_dr\_is\_modulus.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* determines if a number is a valid DR modulus */
-018 int mp_dr_is_modulus(mp_int *a)
-019 \{
-020 int ix;
-021
-022 /* must be at least two digits */
-023 if (a->used < 2) \{
-024 return 0;
-025 \}
-026
-027 for (ix = 1; ix < a->used; ix++) \{
-028 if (a->dp[ix] != MP_MASK) \{
-029 return 0;
-030 \}
-031 \}
-032 return 1;
-033 \}
-034
-\end{alltt}
-\end{small}
-
-\subsection{Unrestricted Diminished Radix Reduction}
-The unrestricted Diminished Radix algorithm allows modular reductions to be performed when the modulus is of the form $2^p - k$. This algorithm
-is a straightforward adaptation of algorithm~\ref{fig:DR}.
-
-In general the restricted Diminished Radix reduction algorithm is much faster since it has considerably lower overhead. However, this new
-algorithm is much faster than either Montgomery or Barrett reduction when the moduli are of the appropriate form.
-
-\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_reduce\_2k}. \\
-\textbf{Input}. mp\_int $a$ and $n$. mp\_digit $k$ \\
-\hspace{11.5mm}($a \ge 0$, $n > 1$, $0 < k < \beta$, $n + k$ is a power of two) \\
-\textbf{Output}. $a \mbox{ (mod }n\mbox{)}$ \\
-\hline
-1. $p \leftarrow \lceil lg(n) \rceil$ (\textit{mp\_count\_bits}) \\
-2. While $a \ge n$ do \\
-\hspace{3mm}2.1 $q \leftarrow \lfloor a / 2^p \rfloor$ (\textit{mp\_div\_2d}) \\
-\hspace{3mm}2.2 $a \leftarrow a \mbox{ (mod }2^p\mbox{)}$ (\textit{mp\_mod\_2d}) \\
-\hspace{3mm}2.3 $q \leftarrow q \cdot k$ (\textit{mp\_mul\_d}) \\
-\hspace{3mm}2.4 $a \leftarrow a - q$ (\textit{s\_mp\_sub}) \\
-\hspace{3mm}2.5 If $a \ge n$ then do \\
-\hspace{6mm}2.5.1 $a \leftarrow a - n$ \\
-3. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_reduce\_2k}
-\end{figure}
-
-\textbf{Algorithm mp\_reduce\_2k.}
-This algorithm quickly reduces an input $a$ modulo an unrestricted Diminished Radix modulus $n$. Division by $2^p$ is emulated with a right
-shift which makes the algorithm fairly inexpensive to use.
-
-\index{bn\_mp\_reduce\_2k.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_2k.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* reduces a modulo n where n is of the form 2**p - k */
-018 int
-019 mp_reduce_2k(mp_int *a, mp_int *n, mp_digit k)
-020 \{
-021 mp_int q;
-022 int p, res;
-023
-024 if ((res = mp_init(&q)) != MP_OKAY) \{
-025 return res;
-026 \}
-027
-028 p = mp_count_bits(n);
-029 top:
-030 /* q = a/2**p, a = a mod 2**p */
-031 if ((res = mp_div_2d(a, p, &q, a)) != MP_OKAY) \{
-032 goto ERR;
-033 \}
-034
-035 if (k != 1) \{
-036 /* q = q * k */
-037 if ((res = mp_mul_d(&q, k, &q)) != MP_OKAY) \{
-038 goto ERR;
-039 \}
-040 \}
-041
-042 /* a = a + q */
-043 if ((res = s_mp_add(a, &q, a)) != MP_OKAY) \{
-044 goto ERR;
-045 \}
-046
-047 if (mp_cmp_mag(a, n) != MP_LT) \{
-048 s_mp_sub(a, n, a);
-049 goto top;
-050 \}
-051
-052 ERR:
-053 mp_clear(&q);
-054 return res;
-055 \}
-056
-\end{alltt}
-\end{small}
-
-The algorithm mp\_count\_bits calculates the number of bits in an mp\_int which is used to find the initial value of $p$. The call to mp\_div\_2d
-on line 31 calculates both the quotient $q$ and the remainder $a$ required. By doing both in a single function call the code size
-is kept fairly small. The multiplication by $k$ is only performed if $k > 1$. This allows reductions modulo $2^p - 1$ to be performed without
-any multiplications.
-
-The unsigned s\_mp\_add, mp\_cmp\_mag and s\_mp\_sub are used in place of their full sign counterparts since the inputs are only valid if they are
-positive. By using the unsigned versions the overhead is kept to a minimum.
-
-\subsubsection{Unrestricted Setup}
-To setup this reduction algorithm the value of $k = 2^p - n$ is required.
-
-\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_reduce\_2k\_setup}. \\
-\textbf{Input}. mp\_int $n$ \\
-\textbf{Output}. $k = 2^p - n$ \\
-\hline
-1. $p \leftarrow \lceil lg(n) \rceil$ (\textit{mp\_count\_bits}) \\
-2. $x \leftarrow 2^p$ (\textit{mp\_2expt}) \\
-3. $x \leftarrow x - n$ (\textit{mp\_sub}) \\
-4. $k \leftarrow x_0$ \\
-5. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_reduce\_2k\_setup}
-\end{figure}
-
-\textbf{Algorithm mp\_reduce\_2k\_setup.}
-This algorithm computes the value of $k$ required for the algorithm mp\_reduce\_2k. By making a temporary variable $x$ equal to $2^p$ a subtraction
-is sufficient to solve for $k$. Alternatively if $n$ has more than one digit the value of $k$ is simply $\beta - n_0$.
-
-\index{bn\_mp\_reduce\_2k\_setup.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_2k\_setup.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* determines the setup value */
-018 int
-019 mp_reduce_2k_setup(mp_int *a, mp_digit *d)
-020 \{
-021 int res, p;
-022 mp_int tmp;
-023
-024 if ((res = mp_init(&tmp)) != MP_OKAY) \{
-025 return res;
-026 \}
-027
-028 p = mp_count_bits(a);
-029 if ((res = mp_2expt(&tmp, p)) != MP_OKAY) \{
-030 mp_clear(&tmp);
-031 return res;
-032 \}
-033
-034 if ((res = s_mp_sub(&tmp, a, &tmp)) != MP_OKAY) \{
-035 mp_clear(&tmp);
-036 return res;
-037 \}
-038
-039 *d = tmp.dp[0];
-040 mp_clear(&tmp);
-041 return MP_OKAY;
-042 \}
-\end{alltt}
-\end{small}
-
-\subsubsection{Unrestricted Detection}
-An integer $n$ is a valid unrestricted Diminished Radix modulus if either of the following are true.
-
-\begin{enumerate}
-\item The number has only one digit.
-\item The number has more than one digit and every bit from the $\beta$'th to the most significant is one.
-\end{enumerate}
-
-If either condition is true than there is a power of two namely $2^p$ such that $0 < 2^p - n < \beta$. If the input is only
-one digit than it will always be of the correct form. Otherwise all of the bits above the first digit must be one. This arises from the fact
-that there will be value of $k$ that when added to the modulus causes a carry in the first digit which propagates all the way to the most
-significant bit. The resulting sum will be a power of two.
-
-\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_reduce\_is\_2k}. \\
-\textbf{Input}. mp\_int $n$ \\
-\textbf{Output}. $1$ if of proper form, $0$ otherwise \\
-\hline
-1. If $n.used = 0$ then return($0$). \\
-2. If $n.used = 1$ then return($1$). \\
-3. $p \leftarrow \rceil lg(n) \lceil$ (\textit{mp\_count\_bits}) \\
-4. for $x$ from $lg(\beta)$ to $p$ do \\
-\hspace{3mm}4.1 If the ($x \mbox{ mod }lg(\beta)$)'th bit of the $\lfloor x / lg(\beta) \rfloor$ of $n$ is zero then return($0$). \\
-5. Return($1$). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_reduce\_is\_2k}
-\end{figure}
-
-\textbf{Algorithm mp\_reduce\_is\_2k.}
-This algorithm quickly determines if a modulus is of the form required for algorithm mp\_reduce\_2k to function properly.
-
-\index{bn\_mp\_reduce\_is\_2k.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_reduce\_is\_2k.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* determines if mp_reduce_2k can be used */
-018 int
-019 mp_reduce_is_2k(mp_int *a)
-020 \{
-021 int ix, iy;
-022
-023 if (a->used == 0) \{
-024 return 0;
-025 \} else if (a->used == 1) \{
-026 return 1;
-027 \} else if (a->used > 1) \{
-028 iy = mp_count_bits(a);
-029 for (ix = DIGIT_BIT; ix < iy; ix++) \{
-030 if ((a->dp[ix/DIGIT_BIT] &
-031 ((mp_digit)1 << (mp_digit)(ix % DIGIT_BIT))) == 0) \{
-032 return 0;
-033 \}
-034 \}
-035 \}
-036 return 1;
-037 \}
-038
-\end{alltt}
-\end{small}
-
-
-
-\section{Algorithm Comparison}
-So far three very different algorithms for modular reduction have been discussed. Each of the algorithms have their own strengths and weaknesses
-that makes having such a selection very useful. The following table sumarizes the three algorithms along with comparisons of work factors. Since
-all three algorithms have the restriction that $0 \le x < n^2$ and $n > 1$ those limitations are not included in the table.
-
-\begin{center}
-\begin{small}
-\begin{tabular}{|c|c|c|c|c|c|}
-\hline \textbf{Method} & \textbf{Work Required} & \textbf{Limitations} & \textbf{$m = 8$} & \textbf{$m = 32$} & \textbf{$m = 64$} \\
-\hline Barrett & $m^2 + 2m - 1$ & None & $79$ & $1087$ & $4223$ \\
-\hline Montgomery & $m^2 + m$ & $n$ must be odd & $72$ & $1056$ & $4160$ \\
-\hline D.R. & $2m$ & $n = \beta^m - k$ & $16$ & $64$ & $128$ \\
-\hline
-\end{tabular}
-\end{small}
-\end{center}
-
-In theory Montgomery and Barrett reductions would require roughly the same amount of time to complete. However, in practice since Montgomery
-reduction can be written as a single function with the Comba technique it is much faster. Barrett reduction suffers from the overhead of
-calling the half precision multipliers, addition and division by $\beta$ algorithms.
-
-For almost every cryptographic algorithm Montgomery reduction is the algorithm of choice. The one set of algorithms where Diminished Radix reduction truly
-shines are based on the discrete logarithm problem such as Diffie-Hellman \cite{DH} and ElGamal \cite{ELGAMAL}. In these algorithms
-primes of the form $\beta^m - k$ can be found and shared amongst users. These primes will allow the Diminished Radix algorithm to be used in
-modular exponentiation to greatly speed up the operation.
-
-
-
-\section*{Exercises}
-\begin{tabular}{cl}
-$\left [ 3 \right ]$ & Prove that the ``trick'' in algorithm mp\_montgomery\_setup actually \\
- & calculates the correct value of $\rho$. \\
- & \\
-$\left [ 2 \right ]$ & Devise an algorithm to reduce modulo $n + k$ for small $k$ quickly. \\
- & \\
-$\left [ 4 \right ]$ & Prove that the pseudo-code algorithm ``Diminished Radix Reduction'' \\
- & (\textit{figure~\ref{fig:DR}}) terminates. Also prove the probability that it will \\
- & terminate within $1 \le k \le 10$ iterations. \\
- & \\
-\end{tabular}
-
-
-\chapter{Exponentiation}
-Exponentiation is the operation of raising one variable to the power of another, for example, $a^b$. A variant of exponentiation, computed
-in a finite field or ring, is called modular exponentiation. This latter style of operation is typically used in public key
-cryptosystems such as RSA and Diffie-Hellman. The ability to quickly compute modular exponentiations is of great benefit to any
-such cryptosystem and many methods have been sought to speed it up.
-
-\section{Exponentiation Basics}
-A trivial algorithm would simply multiply $a$ against itself $b - 1$ times to compute the exponentiation desired. However, as $b$ grows in size
-the number of multiplications becomes prohibitive. Imagine what would happen if $b$ $\approx$ $2^{1024}$ as is the case when computing an RSA signature
-with a $1024$-bit key. Such a calculation could never be completed as it would take simply far too long.
-
-Fortunately there is a very simple algorithm based on the laws of exponents. Recall that $lg_a(a^b) = b$ and that $lg_a(a^ba^c) = b + c$ which
-are two trivial relationships between the base and the exponent. Let $b_i$ represent the $i$'th bit of $b$ starting from the least
-significant bit. If $b$ is a $k$-bit integer than the following equation is true.
-
-\begin{equation}
-a^b = \prod_{i=0}^{k-1} a^{2^i \cdot b_i}
-\end{equation}
-
-By taking the base $a$ logarithm of both sides of the equation the following equation is the result.
-
-\begin{equation}
-b = \sum_{i=0}^{k-1}2^i \cdot b_i
-\end{equation}
-
-The term $a^{2^i}$ can be found from the $i - 1$'th term by squaring the term since $\left ( a^{2^i} \right )^2$ is equal to
-$a^{2^{i+1}}$. This observation forms the basis of essentially all fast exponentiation algorithms. It requires $k$ squarings and on average
-$k \over 2$ multiplications to compute the result. This is indeed quite an improvement over simply multiplying by $a$ a total of $b-1$ times.
-
-While this current method is a considerable speed up there are further improvements to be made. For example, the $a^{2^i}$ term does not need to
-be computed in an auxilary variable. Consider the following equivalent algorithm.
-
-\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{Left to Right Exponentiation}. \\
-\textbf{Input}. Integer $a$, $b$ and $k$ \\
-\textbf{Output}. $c = a^b$ \\
-\hline \\
-1. $c \leftarrow 1$ \\
-2. for $i$ from $k - 1$ to $0$ do \\
-\hspace{3mm}2.1 $c \leftarrow c^2$ \\
-\hspace{3mm}2.2 $c \leftarrow c \cdot a^{b_i}$ \\
-3. Return $c$. \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Left to Right Exponentiation}
-\label{fig:LTOR}
-\end{figure}
-
-This algorithm starts from the most significant bit and works towards the least significant bit. When the $i$'th bit of $b$ is set $a$ is
-multiplied against the current product. In each iteration the product is squared which doubles the exponent of the individual terms of the
-product.
-
-For example, let $b = 101100_2 \equiv 44_{10}$. The following chart demonstrates the actions of the algorithm.
-
-\newpage\begin{figure}
-\begin{center}
-\begin{tabular}{|c|c|}
-\hline \textbf{Value of $i$} & \textbf{Value of $c$} \\
-\hline - & $1$ \\
-\hline $5$ & $a$ \\
-\hline $4$ & $a^2$ \\
-\hline $3$ & $a^4 \cdot a$ \\
-\hline $2$ & $a^8 \cdot a^2 \cdot a$ \\
-\hline $1$ & $a^{16} \cdot a^4 \cdot a^2$ \\
-\hline $0$ & $a^{32} \cdot a^8 \cdot a^4$ \\
-\hline
-\end{tabular}
-\end{center}
-\caption{Example of Left to Right Exponentiation}
-\end{figure}
-
-When the product $a^{32} \cdot a^8 \cdot a^4$ is simplified it is equal $a^{44}$ which is the desired exponentiation. This particular algorithm is
-called ``Left to Right'' because it reads the exponent in that order. All of the exponentiation algorithms that will be presented are of this nature.
-
-\subsection{Single Digit Exponentiation}
-The first algorithm in the series of exponentiation algorithms will be an unbounded algorithm where the exponent is a single digit. It is intended
-to be used when a small power of an input is required (\textit{e.g. $a^5$}). It is faster than simply multiplying $b - 1$ times for all values of
-$b$ that are greater than three.
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_expt\_d}. \\
-\textbf{Input}. mp\_int $a$ and mp\_digit $b$ \\
-\textbf{Output}. $c = a^b$ \\
-\hline \\
-1. $g \leftarrow a$ (\textit{mp\_init\_copy}) \\
-2. $c \leftarrow 1$ (\textit{mp\_set}) \\
-3. for $x$ from 1 to $lg(\beta)$ do \\
-\hspace{3mm}3.1 $c \leftarrow c^2$ (\textit{mp\_sqr}) \\
-\hspace{3mm}3.2 If $b$ AND $2^{lg(\beta) - 1} \ne 0$ then \\
-\hspace{6mm}3.2.1 $c \leftarrow c \cdot g$ (\textit{mp\_mul}) \\
-\hspace{3mm}3.3 $b \leftarrow b << 1$ \\
-4. Clear $g$. \\
-5. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_expt\_d}
-\end{figure}
-
-\textbf{Algorithm mp\_expt\_d.}
-This algorithm computes the value of $a$ raised to the power of a single digit $b$. It uses the left to right exponentiation algorithm to
-quickly compute the exponentiation. It is loosely based on algorithm 14.79 of HAC \cite[pp. 615]{HAC} with the difference that the
-exponent is a fixed width.
-
-A copy of $a$ is made first to allow destination variable $c$ be the same as the source variable $a$. The result is set to the initial value of
-$1$ in the subsequent step.
-
-Inside the loop the exponent is read from the most significant bit first down to the least significant bit. First $c$ is invariably squared
-on step 3.1. In the following step if the most significant bit of $b$ is one the copy of $a$ is multiplied against $c$. The value
-of $b$ is shifted left one bit to make the next bit down from the most signficant bit the new most significant bit. In effect each
-iteration of the loop moves the bits of the exponent $b$ upwards to the most significant location.
-
-\index{bn\_mp\_expt\_d.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_expt\_d.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* calculate c = a**b using a square-multiply algorithm */
-018 int
-019 mp_expt_d (mp_int * a, mp_digit b, mp_int * c)
-020 \{
-021 int res, x;
-022 mp_int g;
-023
-024 if ((res = mp_init_copy (&g, a)) != MP_OKAY) \{
-025 return res;
-026 \}
-027
-028 /* set initial result */
-029 mp_set (c, 1);
-030
-031 for (x = 0; x < (int) DIGIT_BIT; x++) \{
-032 /* square */
-033 if ((res = mp_sqr (c, c)) != MP_OKAY) \{
-034 mp_clear (&g);
-035 return res;
-036 \}
-037
-038 /* if the bit is set multiply */
-039 if ((b & (mp_digit) (((mp_digit)1) << (DIGIT_BIT - 1))) != 0) \{
-040 if ((res = mp_mul (c, &g, c)) != MP_OKAY) \{
-041 mp_clear (&g);
-042 return res;
-043 \}
-044 \}
-045
-046 /* shift to next bit */
-047 b <<= 1;
-048 \}
-049
-050 mp_clear (&g);
-051 return MP_OKAY;
-052 \}
-\end{alltt}
-\end{small}
-
--- Some note later.
-
-\section{$k$-ary Exponentiation}
-When calculating an exponentiation the most time consuming bottleneck is the multiplications which are in general a small factor
-slower than squaring. Recall from the previous algorithm that $b_{i}$ refers to the $i$'th bit of the exponent $b$. Suppose instead it referred to
-the $i$'th $k$-bit digit of the exponent of $b$. For $k = 1$ the definitions are synonymous and for $k > 1$ algorithm~\ref{fig:KARY}
-computes the same exponentiation. A group of $k$ bits from the exponent is called a \textit{window}. That is it is a small window on only a
-portion of the entire exponent. Consider the following modification to the basic left to right exponentiation algorithm.
-
-\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{$k$-ary Exponentiation}. \\
-\textbf{Input}. Integer $a$, $b$, $k$ and $t$ \\
-\textbf{Output}. $c = a^b$ \\
-\hline \\
-1. $c \leftarrow 1$ \\
-2. for $i$ from $t - 1$ to $0$ do \\
-\hspace{3mm}2.1 $c \leftarrow c^{2^k} $ \\
-\hspace{3mm}2.2 Extract the $i$'th $k$-bit word from $b$ and store it in $g$. \\
-\hspace{3mm}2.3 $c \leftarrow c \cdot a^g$ \\
-3. Return $c$. \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{$k$-ary Exponentiation}
-\label{fig:KARY}
-\end{figure}
-
-The squaring on step 2.1 can be calculated by squaring the value $c$ successively $k$ times. If the values of $a^g$ for $0 < g < 2^k$ have been
-precomputed this algorithm requires only $t$ multiplications and $tk$ squarings. The table can be generated with $2^{k - 1} - 1$ squarings and
-$2^{k - 1} + 1$ multiplications. This algorithm assumes that the number of bits in the exponent is evenly divisible by $k$.
-However, when it is not the remaining $0 < x \le k - 1$ bits can be handled with algorithm~\ref{fig:LTOR}.
-
-Suppose $k = 4$ and $t = 100$. This modified algorithm will require $109$ multiplications and $408$ squarings to compute the exponentiation. The
-original algorithm would on average have required $200$ multiplications and $400$ squrings to compute the same value. The total number of squarings
-has increased slightly but the number of multiplications has nearly halved.
-
-\subsection{Optimal Values of $k$}
-An optimal value of $k$ will minimize $2^{k} + \lceil n / k \rceil + n - 1$ for a fixed number of bits in the exponent $n$. The simplest
-approach is to brute force search amongst the values $k = 2, 3, \ldots, 8$ for the lowest result. Table~\ref{fig:OPTK} lists optimal values of $k$
-for various exponent sizes and compares the number of multiplication and squarings required against algorithm~\ref{fig:LTOR}.
-
-\begin{figure}[here]
-\begin{center}
-\begin{small}
-\begin{tabular}{|c|c|c|c|c|c|}
-\hline \textbf{Exponent (bits)} & \textbf{Optimal $k$} & \textbf{Work at $k$} & \textbf{Work with ~\ref{fig:LTOR}} \\
-\hline $16$ & $2$ & $27$ & $24$ \\
-\hline $32$ & $3$ & $49$ & $48$ \\
-\hline $64$ & $3$ & $92$ & $96$ \\
-\hline $128$ & $4$ & $175$ & $192$ \\
-\hline $256$ & $4$ & $335$ & $384$ \\
-\hline $512$ & $5$ & $645$ & $768$ \\
-\hline $1024$ & $6$ & $1257$ & $1536$ \\
-\hline $2048$ & $6$ & $2452$ & $3072$ \\
-\hline $4096$ & $7$ & $4808$ & $6144$ \\
-\hline
-\end{tabular}
-\end{small}
-\end{center}
-\caption{Optimal Values of $k$ for $k$-ary Exponentiation}
-\label{fig:OPTK}
-\end{figure}
-
-\subsection{Sliding-Window Exponentiation}
-A simple modification to the previous algorithm is only generate the upper half of the table in the range $2^{k-1} \le g < 2^k$. Essentially
-this is a table for all values of $g$ where the most significant bit of $g$ is a one. However, in order for this to be allowed in the
-algorithm values of $g$ in the range $0 \le g < 2^{k-1}$ must be avoided.
-
-Table~\ref{fig:OPTK2} lists optimal values of $k$ for various exponent sizes and compares the work required against algorithm~\ref{fig:KARY}.
-
-\begin{figure}[here]
-\begin{center}
-\begin{small}
-\begin{tabular}{|c|c|c|c|c|c|}
-\hline \textbf{Exponent (bits)} & \textbf{Optimal $k$} & \textbf{Work at $k$} & \textbf{Work with ~\ref{fig:KARY}} \\
-\hline $16$ & $3$ & $24$ & $27$ \\
-\hline $32$ & $3$ & $45$ & $49$ \\
-\hline $64$ & $4$ & $87$ & $92$ \\
-\hline $128$ & $4$ & $167$ & $175$ \\
-\hline $256$ & $5$ & $322$ & $335$ \\
-\hline $512$ & $6$ & $628$ & $645$ \\
-\hline $1024$ & $6$ & $1225$ & $1257$ \\
-\hline $2048$ & $7$ & $2403$ & $2452$ \\
-\hline $4096$ & $8$ & $4735$ & $4808$ \\
-\hline
-\end{tabular}
-\end{small}
-\end{center}
-\caption{Optimal Values of $k$ for Sliding Window Exponentiation}
-\label{fig:OPTK2}
-\end{figure}
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{Sliding Window $k$-ary Exponentiation}. \\
-\textbf{Input}. Integer $a$, $b$, $k$ and $t$ \\
-\textbf{Output}. $c = a^b$ \\
-\hline \\
-1. $c \leftarrow 1$ \\
-2. for $i$ from $t - 1$ to $0$ do \\
-\hspace{3mm}2.1 If the $i$'th bit of $b$ is a zero then \\
-\hspace{6mm}2.1.1 $c \leftarrow c^2$ \\
-\hspace{3mm}2.2 else do \\
-\hspace{6mm}2.2.1 $c \leftarrow c^{2^k}$ \\
-\hspace{6mm}2.2.2 Extract the $k$ bits from $(b_{i}b_{i-1}\ldots b_{i-(k-1)})$ and store it in $g$. \\
-\hspace{6mm}2.2.3 $c \leftarrow c \cdot a^g$ \\
-\hspace{6mm}2.2.4 $i \leftarrow i - k$ \\
-3. Return $c$. \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Sliding Window $k$-ary Exponentiation}
-\end{figure}
-
-Similar to the previous algorithm this algorithm must have a special handler when fewer than $k$ bits are left in the exponent. While this
-algorithm requires the same number of squarings it can potentially have fewer multiplications. The pre-computed table $a^g$ is also half
-the size as the previous table.
-
-Consider the exponent $b = 111101011001000_2 \equiv 31432_{10}$ with $k = 3$ using both algorithms. The first algorithm will divide the exponent up as
-the following five $3$-bit words $b \equiv \left ( 111, 101, 011, 001, 000 \right )_{2}$. The second algorithm will break the
-exponent as $b \equiv \left ( 111, 101, 0, 110, 0, 100, 0 \right )_{2}$. The single digit $0$ in the second representation are where
-a single squaring took place instead of a squaring and multiplication. In total the first method requires $10$ multiplications and $18$
-squarings. The second method requires $8$ multiplications and $18$ squarings.
-
-In general the sliding window method is never slower than the generic $k$-ary method and often it is slightly faster.
-
-\section{Modular Exponentiation}
-
-Modular exponentiation is essentially computing the power of a base within a finite field or ring. For example, computing
-$d \equiv a^b \mbox{ (mod }c\mbox{)}$ is a modular exponentiation. Instead of first computing $a^b$ and then reducing it
-modulo $c$ the intermediate result is reduced modulo $c$ after every squaring or multiplication operation.
-
-This guarantees that any intermediate result is bounded by $0 \le d \le c^2 - 2c + 1$ and can be reduced modulo $c$ quickly using
-one of the algorithms presented in chapter seven.
-
-Before the actual modular exponentiation algorithm can be written a wrapper algorithm must be written first. This algorithm
-will allow the exponent $b$ to be negative which is computed as $c \equiv \left (1 / a \right )^{\vert b \vert} \mbox{(mod }d\mbox{)}$. The
-value of $(1/a) \mbox{ mod }c$ is computed using the modular inverse (\textit{see section 10.4}). If no inverse exists the algorithm
-terminates with an error.
-
-\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_exptmod}. \\
-\textbf{Input}. mp\_int $a$, $b$ and $c$ \\
-\textbf{Output}. $y \equiv g^x \mbox{ (mod }p\mbox{)}$ \\
-\hline \\
-1. If $c.sign = MP\_NEG$ return(\textit{MP\_VAL}). \\
-2. If $b.sign = MP\_NEG$ then \\
-\hspace{3mm}2.1 $g' \leftarrow g^{-1} \mbox{ (mod }c\mbox{)}$ \\
-\hspace{3mm}2.2 $x' \leftarrow \vert x \vert$ \\
-\hspace{3mm}2.3 Compute $d \equiv g'^{x'} \mbox{ (mod }c\mbox{)}$ via recursion. \\
-3. if $p$ is odd \textbf{OR} $p$ is a D.R. modulus then \\
-\hspace{3mm}3.1 Compute $y \equiv g^{x} \mbox{ (mod }p\mbox{)}$ via algorithm mp\_exptmod\_fast. \\
-4. else \\
-\hspace{3mm}4.1 Compute $y \equiv g^{x} \mbox{ (mod }p\mbox{)}$ via algorithm s\_mp\_exptmod. \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_exptmod}
-\end{figure}
-
-\textbf{Algorithm mp\_exptmod.}
-The first algorithm which actually performs modular exponentiation is algorithm s\_mp\_exptmod. It is a sliding window $k$-ary algorithm
-which uses Barrett reduction to reduce the product modulo $p$. The second algorithm mp\_exptmod\_fast performs the same operation
-except it uses either Montgomery or Diminished Radix reduction. The two latter reduction algorithms are clumped in the same exponentiation
-algorithm since their arguments are essentially the same (\textit{two mp\_ints and one mp\_digit}).
-
-\index{bn\_mp\_exptmod.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_exptmod.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017
-018 /* this is a shell function that calls either the normal or Montgomery
-019 * exptmod functions. Originally the call to the montgomery code was
-020 * embedded in the normal function but that wasted alot of stack space
-021 * for nothing (since 99% of the time the Montgomery code would be called)
-022 */
-023 int
-024 mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y)
-025 \{
-026 int dr;
-027
-028 /* modulus P must be positive */
-029 if (P->sign == MP_NEG) \{
-030 return MP_VAL;
-031 \}
-032
-033 /* if exponent X is negative we have to recurse */
-034 if (X->sign == MP_NEG) \{
-035 mp_int tmpG, tmpX;
-036 int err;
-037
-038 /* first compute 1/G mod P */
-039 if ((err = mp_init(&tmpG)) != MP_OKAY) \{
-040 return err;
-041 \}
-042 if ((err = mp_invmod(G, P, &tmpG)) != MP_OKAY) \{
-043 mp_clear(&tmpG);
-044 return err;
-045 \}
-046
-047 /* now get |X| */
-048 if ((err = mp_init(&tmpX)) != MP_OKAY) \{
-049 mp_clear(&tmpG);
-050 return err;
-051 \}
-052 if ((err = mp_abs(X, &tmpX)) != MP_OKAY) \{
-053 mp_clear_multi(&tmpG, &tmpX, NULL);
-054 return err;
-055 \}
-056
-057 /* and now compute (1/G)**|X| instead of G**X [X < 0] */
-058 err = mp_exptmod(&tmpG, &tmpX, P, Y);
-059 mp_clear_multi(&tmpG, &tmpX, NULL);
-060 return err;
-061 \}
-062
-063 dr = mp_dr_is_modulus(P);
-064 if (dr == 0) \{
-065 dr = mp_reduce_is_2k(P) << 1;
-066 \}
-067
-068 /* if the modulus is odd or dr != 0 use the fast method */
-069 if (mp_isodd (P) == 1 || dr != 0) \{
-070 return mp_exptmod_fast (G, X, P, Y, dr);
-071 \} else \{
-072 return s_mp_exptmod (G, X, P, Y);
-073 \}
-074 \}
-075
-\end{alltt}
-\end{small}
-
-In order to keep the algorithms in a known state the first step on line 29 is to reject any negative modulus as input. If the exponent is
-negative the algorithm tries to perform a modular exponentiation with the modular inverse of the base $G$. The temporary variable $tmpG$ is assigned
-the modular inverse of $G$ and $tmpX$ is assigned the absolute value of $X$. The algorithm will recuse with these new values with a positive
-exponent.
-
-If the exponent is positive the algorithm resumes the exponentiation. Line 63 determines if the modulus is of the restricted Diminished Radix
-form. If it is not line 65 attempts to determine if it is of a unrestricted Diminished Radix form. The integer $dr$ will take on one
-of three values.
-
-\begin{enumerate}
-\item $dr = 0$ means that the modulus is not of either restricted or unrestricted Diminished Radix form.
-\item $dr = 1$ means that the modulus is of restricted Diminished Radix form.
-\item $dr = 2$ means that the modulus is of unrestricted Diminished Radix form.
-\end{enumerate}
-
-Line 69 determines if the fast modular exponentiation algorithm can be used. It is allowed if $dr \ne 0$ or if the modulus is odd. Otherwise,
-the slower s\_mp\_exptmod algorithm is used which uses Barrett reduction.
-
-\subsection{Barrett Modular Exponentiation}
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{s\_mp\_exptmod}. \\
-\textbf{Input}. mp\_int $a$, $b$ and $c$ \\
-\textbf{Output}. $y \equiv g^x \mbox{ (mod }p\mbox{)}$ \\
-\hline \\
-1. $k \leftarrow lg(x)$ \\
-2. $winsize \leftarrow \left \lbrace \begin{array}{ll}
- 2 & \mbox{if }k \le 7 \\
- 3 & \mbox{if }7 < k \le 36 \\
- 4 & \mbox{if }36 < k \le 140 \\
- 5 & \mbox{if }140 < k \le 450 \\
- 6 & \mbox{if }450 < k \le 1303 \\
- 7 & \mbox{if }1303 < k \le 3529 \\
- 8 & \mbox{if }3529 < k \\
- \end{array} \right .$ \\
-3. Initialize $2^{winsize}$ mp\_ints in an array named $M$ and one mp\_int named $\mu$ \\
-4. Calculate the $\mu$ required for Barrett Reduction (\textit{mp\_reduce\_setup}). \\
-5. $M_1 \leftarrow g \mbox{ (mod }p\mbox{)}$ \\
-\\
-Setup the table of small powers of $g$. First find $g^{2^{winsize}}$ and then all multiples of it. \\
-6. $k \leftarrow 2^{winsize - 1}$ \\
-7. $M_{k} \leftarrow M_1$ \\
-8. for $ix$ from 0 to $winsize - 2$ do \\
-\hspace{3mm}8.1 $M_k \leftarrow \left ( M_k \right )^2$ (\textit{mp\_sqr}) \\
-\hspace{3mm}8.2 $M_k \leftarrow M_k \mbox{ (mod }p\mbox{)}$ (\textit{mp\_reduce}) \\
-9. for $ix$ from $2^{winsize - 1} + 1$ to $2^{winsize} - 1$ do \\
-\hspace{3mm}9.1 $M_{ix} \leftarrow M_{ix - 1} \cdot M_{1}$ (\textit{mp\_mul}) \\
-\hspace{3mm}9.2 $M_{ix} \leftarrow M_{ix} \mbox{ (mod }p\mbox{)}$ (\textit{mp\_reduce}) \\
-10. $res \leftarrow 1$ \\
-\\
-Start Sliding Window. \\
-11. $mode \leftarrow 0, bitcnt \leftarrow 1, buf \leftarrow 0, digidx \leftarrow x.used - 1, bitcpy \leftarrow 0, bitbuf \leftarrow 0$ \\
-12. Loop \\
-\hspace{3mm}12.1 $bitcnt \leftarrow bitcnt - 1$ \\
-\hspace{3mm}12.2 If $bitcnt = 0$ then do \\
-\hspace{6mm}12.2.1 If $digidx = -1$ goto step 13. \\
-\hspace{6mm}12.2.2 $buf \leftarrow x_{digidx}$ \\
-\hspace{6mm}12.2.3 $digidx \leftarrow digidx - 1$ \\
-\hspace{6mm}12.2.4 $bitcnt \leftarrow lg(\beta)$ \\
-Continued on next page. \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm s\_mp\_exptmod}
-\end{figure}
-
-\newpage\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{s\_mp\_exptmod} (\textit{continued}). \\
-\textbf{Input}. mp\_int $a$, $b$ and $c$ \\
-\textbf{Output}. $y \equiv g^x \mbox{ (mod }p\mbox{)}$ \\
-\hline \\
-\hspace{3mm}12.3 $y \leftarrow (buf >> (lg(\beta) - 1))$ AND $1$ \\
-\hspace{3mm}12.4 $buf \leftarrow buf << 1$ \\
-\hspace{3mm}12.5 if $mode = 0$ and $y = 0$ then goto step 12. \\
-\hspace{3mm}12.6 if $mode = 1$ and $y = 0$ then do \\
-\hspace{6mm}12.6.1 $res \leftarrow res^2$ \\
-\hspace{6mm}12.6.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\
-\hspace{6mm}12.6.3 Goto step 12. \\
-\hspace{3mm}12.7 $bitcpy \leftarrow bitcpy + 1$ \\
-\hspace{3mm}12.8 $bitbuf \leftarrow bitbuf + (y << (winsize - bitcpy))$ \\
-\hspace{3mm}12.9 $mode \leftarrow 2$ \\
-\hspace{3mm}12.10 If $bitcpy = winsize$ then do \\
-\hspace{6mm}Window is full so perform the squarings and single multiplication. \\
-\hspace{6mm}12.10.1 for $ix$ from $0$ to $winsize -1$ do \\
-\hspace{9mm}12.10.1.1 $res \leftarrow res^2$ \\
-\hspace{9mm}12.10.1.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\
-\hspace{6mm}12.10.2 $res \leftarrow res \cdot M_{bitbuf}$ \\
-\hspace{6mm}12.10.3 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\
-\hspace{6mm}Reset the window. \\
-\hspace{6mm}12.10.4 $bitcpy \leftarrow 0, bitbuf \leftarrow 0, mode \leftarrow 1$ \\
-\\
-No more windows left. Check for residual bits of exponent. \\
-13. If $mode = 2$ and $bitcpy > 0$ then do \\
-\hspace{3mm}13.1 for $ix$ form $0$ to $bitcpy - 1$ do \\
-\hspace{6mm}13.1.1 $res \leftarrow res^2$ \\
-\hspace{6mm}13.1.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\
-\hspace{6mm}13.1.3 $bitbuf \leftarrow bitbuf << 1$ \\
-\hspace{6mm}13.1.4 If $bitbuf$ AND $2^{winsize} \ne 0$ then do \\
-\hspace{9mm}13.1.4.1 $res \leftarrow res \cdot M_{1}$ \\
-\hspace{9mm}13.1.4.2 $res \leftarrow res \mbox{ (mod }p\mbox{)}$ \\
-14. $y \leftarrow res$ \\
-15. Clear $res$, $mu$ and the $M$ array. \\
-16. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm s\_mp\_exptmod (continued)}
-\end{figure}
-
-\textbf{Algorithm s\_mp\_exptmod.}
-This algorithm computes the $x$'th power of $g$ modulo $p$ and stores the result in $y$. It takes advantage of the Barrett reduction
-algorithm to keep the product small throughout the algorithm.
-
-The first two steps determine the optimal window size based on the number of bits in the exponent. The larger the exponent the
-larger the window size becomes. After a window size $winsize$ has been chosen an array of $2^{winsize}$ mp\_int variables is allocated. This
-table will hold the values of $g^x \mbox{ (mod }p\mbox{)}$ for $2^{winsize - 1} \le x < 2^{winsize}$.
-
-After the table is allocated the first power of $g$ is found. Since $g \ge p$ is allowed it must be first reduced modulo $p$ to make
-the rest of the algorithm more efficient. The first element of the table at $2^{winsize - 1}$ is found by squaring $M_1$ successively $winsize - 2$
-times. The rest of the table elements are found by multiplying the previous element by $M_1$ modulo $p$.
-
-Now that the table is available the sliding window may begin. The following list describes the functions of all the variables in the window.
-\begin{enumerate}
-\item The variable $mode$ dictates how the bits of the exponent are interpreted.
-\begin{enumerate}
- \item When $mode = 0$ the bits are ignored since no non-zero bit of the exponent has been seen yet. For example, if the exponent were simply
- $1$ then there would be $lg(\beta) - 1$ zero bits before the first non-zero bit. In this case bits are ignored until a non-zero bit is found.
- \item When $mode = 1$ a non-zero bit has been seen before and a new $winsize$-bit window has not been formed yet. In this mode leading $0$ bits
- are read and a single squaring is performed. If a non-zero bit is read a new window is created.
- \item When $mode = 2$ the algorithm is in the middle of forming a window and new bits are appended to the window from the most significant bit
- downwards.
-\end{enumerate}
-\item The variable $bitcnt$ indicates how many bits are left in the current digit of the exponent left to be read. When it reaches zero a new digit
- is fetched from the exponent.
-\item The variable $buf$ holds the currently read digit of the exponent.
-\item The variable $digidx$ is an index into the exponents digits. It starts at the leading digit $x.used - 1$ and moves towards the trailing digit.
-\item The variable $bitcpy$ indicates how many bits are in the currently formed window. When it reaches $winsize$ the window is flushed and
- the appropriate operations performed.
-\item The variable $bitbuf$ holds the current bits of the window being formed.
-\end{enumerate}
-
-All of step 12 is the window processing loop. It will iterate while there are digits available form the exponent to read. The first step
-inside this loop is to extract a new digit if no more bits are available in the current digit. If there are no bits left a new digit is
-read and if there are no digits left than the loop terminates.
-
-After a digit is made available step 12.3 will extract the most significant bit of the current digit and move all other bits in the digit
-upwards. In effect the digit is read from most significant bit to least significant bit and since the digits are read from leading to
-trailing edges the entire exponent is read from most significant bit to least significant bit.
-
-At step 12.5 if the $mode$ and currently extracted bit $y$ are both zero the bit is ignored and the next bit is read. This prevents the
-algorithm from having to perform trivial squaring and reduction operations before the first non-zero bit is read. Step 12.6 and 12.7-10 handle
-the two cases of $mode = 1$ and $mode = 2$ respectively.
-
-\begin{center}
-\begin{figure}[here]
-\includegraphics{pics/expt_state}
-\caption{Sliding Window State Diagram}
-\end{figure}
-\end{center}
-
-By step 13 there are no more digits left in the exponent. However, there may be partial bits in the window left. If $mode = 2$ then
-a Left-to-Right algorithm is used to process the remaining few bits.
-
-\index{bn\_s\_mp\_exptmod.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_s\_mp\_exptmod.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 int
-018 s_mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y)
-019 \{
-020 mp_int M[256], res, mu;
-021 mp_digit buf;
-022 int err, bitbuf, bitcpy, bitcnt, mode, digidx, x, y, winsize;
-023
-024 /* find window size */
-025 x = mp_count_bits (X);
-026 if (x <= 7) \{
-027 winsize = 2;
-028 \} else if (x <= 36) \{
-029 winsize = 3;
-030 \} else if (x <= 140) \{
-031 winsize = 4;
-032 \} else if (x <= 450) \{
-033 winsize = 5;
-034 \} else if (x <= 1303) \{
-035 winsize = 6;
-036 \} else if (x <= 3529) \{
-037 winsize = 7;
-038 \} else \{
-039 winsize = 8;
-040 \}
-041
-042 #ifdef MP_LOW_MEM
-043 if (winsize > 5) \{
-044 winsize = 5;
-045 \}
-046 #endif
-047
-048 /* init M array */
-049 for (x = 0; x < (1 << winsize); x++) \{
-050 if ((err = mp_init_size (&M[x], 1)) != MP_OKAY) \{
-051 for (y = 0; y < x; y++) \{
-052 mp_clear (&M[y]);
-053 \}
-054 return err;
-055 \}
-056 \}
-057
-058 /* create mu, used for Barrett reduction */
-059 if ((err = mp_init (&mu)) != MP_OKAY) \{
-060 goto __M;
-061 \}
-062 if ((err = mp_reduce_setup (&mu, P)) != MP_OKAY) \{
-063 goto __MU;
-064 \}
-065
-066 /* create M table
-067 *
-068 * The M table contains powers of the base,
-069 * e.g. M[x] = G**x mod P
-070 *
-071 * The first half of the table is not
-072 * computed though accept for M[0] and M[1]
-073 */
-074 if ((err = mp_mod (G, P, &M[1])) != MP_OKAY) \{
-075 goto __MU;
-076 \}
-077
-078 /* compute the value at M[1<<(winsize-1)] by squaring
-079 * M[1] (winsize-1) times
-080 */
-081 if ((err = mp_copy (&M[1], &M[1 << (winsize - 1)])) != MP_OKAY) \{
-082 goto __MU;
-083 \}
-084
-085 for (x = 0; x < (winsize - 1); x++) \{
-086 if ((err = mp_sqr (&M[1 << (winsize - 1)],
-087 &M[1 << (winsize - 1)])) != MP_OKAY) \{
-088 goto __MU;
-089 \}
-090 if ((err = mp_reduce (&M[1 << (winsize - 1)], P, &mu)) != MP_OKAY) \{
-091 goto __MU;
-092 \}
-093 \}
-094
-095 /* create upper table */
-096 for (x = (1 << (winsize - 1)) + 1; x < (1 << winsize); x++) \{
-097 if ((err = mp_mul (&M[x - 1], &M[1], &M[x])) != MP_OKAY) \{
-098 goto __MU;
-099 \}
-100 if ((err = mp_reduce (&M[x], P, &mu)) != MP_OKAY) \{
-101 goto __MU;
-102 \}
-103 \}
-104
-105 /* setup result */
-106 if ((err = mp_init (&res)) != MP_OKAY) \{
-107 goto __MU;
-108 \}
-109 mp_set (&res, 1);
-110
-111 /* set initial mode and bit cnt */
-112 mode = 0;
-113 bitcnt = 1;
-114 buf = 0;
-115 digidx = X->used - 1;
-116 bitcpy = 0;
-117 bitbuf = 0;
-118
-119 for (;;) \{
-120 /* grab next digit as required */
-121 if (--bitcnt == 0) \{
-122 if (digidx == -1) \{
-123 break;
-124 \}
-125 buf = X->dp[digidx--];
-126 bitcnt = (int) DIGIT_BIT;
-127 \}
-128
-129 /* grab the next msb from the exponent */
-130 y = (buf >> (mp_digit)(DIGIT_BIT - 1)) & 1;
-131 buf <<= (mp_digit)1;
-132
-133 /* if the bit is zero and mode == 0 then we ignore it
-134 * These represent the leading zero bits before the first 1 bit
-135 * in the exponent. Technically this opt is not required but it
-136 * does lower the # of trivial squaring/reductions used
-137 */
-138 if (mode == 0 && y == 0)
-139 continue;
-140
-141 /* if the bit is zero and mode == 1 then we square */
-142 if (mode == 1 && y == 0) \{
-143 if ((err = mp_sqr (&res, &res)) != MP_OKAY) \{
-144 goto __RES;
-145 \}
-146 if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) \{
-147 goto __RES;
-148 \}
-149 continue;
-150 \}
-151
-152 /* else we add it to the window */
-153 bitbuf |= (y << (winsize - ++bitcpy));
-154 mode = 2;
-155
-156 if (bitcpy == winsize) \{
-157 /* ok window is filled so square as required and multiply */
-158 /* square first */
-159 for (x = 0; x < winsize; x++) \{
-160 if ((err = mp_sqr (&res, &res)) != MP_OKAY) \{
-161 goto __RES;
-162 \}
-163 if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) \{
-164 goto __RES;
-165 \}
-166 \}
-167
-168 /* then multiply */
-169 if ((err = mp_mul (&res, &M[bitbuf], &res)) != MP_OKAY) \{
-170 goto __MU;
-171 \}
-172 if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) \{
-173 goto __MU;
-174 \}
-175
-176 /* empty window and reset */
-177 bitcpy = 0;
-178 bitbuf = 0;
-179 mode = 1;
-180 \}
-181 \}
-182
-183 /* if bits remain then square/multiply */
-184 if (mode == 2 && bitcpy > 0) \{
-185 /* square then multiply if the bit is set */
-186 for (x = 0; x < bitcpy; x++) \{
-187 if ((err = mp_sqr (&res, &res)) != MP_OKAY) \{
-188 goto __RES;
-189 \}
-190 if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) \{
-191 goto __RES;
-192 \}
-193
-194 bitbuf <<= 1;
-195 if ((bitbuf & (1 << winsize)) != 0) \{
-196 /* then multiply */
-197 if ((err = mp_mul (&res, &M[1], &res)) != MP_OKAY) \{
-198 goto __RES;
-199 \}
-200 if ((err = mp_reduce (&res, P, &mu)) != MP_OKAY) \{
-201 goto __RES;
-202 \}
-203 \}
-204 \}
-205 \}
-206
-207 mp_exch (&res, Y);
-208 err = MP_OKAY;
-209 __RES:mp_clear (&res);
-210 __MU:mp_clear (&mu);
-211 __M:
-212 for (x = 0; x < (1 << winsize); x++) \{
-213 mp_clear (&M[x]);
-214 \}
-215 return err;
-216 \}
-\end{alltt}
-\end{small}
-
-Lines 26 through 40 determine the optimal window size based on the length of the exponent in bits. The window divisions are sorted
-from smallest to greatest so that in each \textbf{if} statement only one condition must be tested. For example, by the \textbf{if} statement
-on line 32 the value of $x$ is already known to be greater than $140$.
-
-The conditional piece of code beginning on line @42,define@ allows the window size to be restricted to five bits. This logic is used to ensure
-the table of precomputed powers of $G$ remains relatively small.
-
-The for loop on line 49 initializes the $M$ array while lines 59 and 62 compute the value of $\mu$ required for
-Barrett reduction.
-
--- More later.
-
-\section{Quick Power of Two}
-Calculating $b = 2^a$ can be performed much quicker than with any of the previous algorithms. Recall that a logical shift left $m << k$ is
-equivalent to $m \cdot 2^k$. By this logic when $m = 1$ a quick power of two can be achieved.
-
-\begin{figure}[!here]
-\begin{small}
-\begin{center}
-\begin{tabular}{l}
-\hline Algorithm \textbf{mp\_2expt}. \\
-\textbf{Input}. integer $b$ \\
-\textbf{Output}. $a \leftarrow 2^b$ \\
-\hline \\
-1. $a \leftarrow 0$ \\
-2. If $a.alloc < \lfloor b / lg(\beta) \rfloor + 1$ then grow $a$ appropriately. \\
-3. $a.used \leftarrow \lfloor b / lg(\beta) \rfloor + 1$ \\
-4. $a_{\lfloor b / lg(\beta) \rfloor} \leftarrow 1 << (b \mbox{ mod } lg(\beta))$ \\
-5. Return(\textit{MP\_OKAY}). \\
-\hline
-\end{tabular}
-\end{center}
-\end{small}
-\caption{Algorithm mp\_2expt}
-\end{figure}
-
-\textbf{Algorithm mp\_2expt.}
-
-\index{bn\_mp\_2expt.c}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: bn\_mp\_2expt.c
-\vspace{-3mm}
-\begin{alltt}
-016
-017 /* computes a = 2**b
-018 *
-019 * Simple algorithm which zeroes the int, grows it then just sets one bit
-020 * as required.
-021 */
-022 int
-023 mp_2expt (mp_int * a, int b)
-024 \{
-025 int res;
-026
-027 mp_zero (a);
-028 if ((res = mp_grow (a, b / DIGIT_BIT + 1)) != MP_OKAY) \{
-029 return res;
-030 \}
-031 a->used = b / DIGIT_BIT + 1;
-032 a->dp[b / DIGIT_BIT] = 1 << (b % DIGIT_BIT);
-033
-034 return MP_OKAY;
-035 \}
-\end{alltt}
-\end{small}
-
-\chapter{Higher Level Algorithms}
-
-This chapter discusses the various higher level algorithms that are required to complete a well rounded multiple precision integer package. These
-routines are less performance oriented than the algorithms of chapters five, six and seven but are no less important.
-
-The first section describes a method of integer division with remainder that is universally well known. It provides the signed division logic
-for the package. The subsequent section discusses a set of algorithms which allow a single digit to be the 2nd operand for a variety of operations.
-These algorithms serve mostly to simplify other algorithms where small constants are required. The last two sections discuss how to manipulate
-various representations of integers. For example, converting from an mp\_int to a string of character.
-
-\section{Integer Division with Remainder}
-
-Integer division aside from modular exponentiation is most intensive algorithm to compute.
-
-
-\section{Single Digit Helpers}
-\subsection{Single Digit Addition}
-\subsection{Single Digit Subtraction}
-\subsection{Single Digit Multiplication}
-\subsection{Single Digit Division}
-\subsection{Single Digit Modulo}
-\subsection{Single Digit Root Extraction}
-\section{Random Number Generation}
-\section{Formatted Output}
-\subsection{Getting The Output Size}
-\subsection{Generating Radix-n Output}
-\subsection{Reading Radix-n Input}
-\section{Unformatted Output}
-\subsection{Getting The Output Size}
-\subsection{Generating Output}
-\subsection{Reading Input}
-
-\chapter{Number Theoretic Algorithms}
-\section{Greatest Common Divisor}
-\section{Least Common Multiple}
-\section{Jacobi Symbol Computation}
-\section{Modular Inverse}
-\subsection{General Case}
-\subsection{Odd Moduli}
-\section{Primality Tests}
-\subsection{Trial Division}
-\subsection{The Fermat Test}
-\subsection{The Miller-Rabin Test}
-\subsection{Primality Test in a Bottle}
-\subsection{The Next Prime}
-\section{Root Extraction}
-
-\backmatter
-\appendix
-\begin{thebibliography}{ABCDEF}
-\bibitem[1]{TAOCPV2}
-Donald Knuth, \textit{The Art of Computer Programming}, Third Edition, Volume Two, Seminumerical Algorithms, Addison-Wesley, 1998
-
-\bibitem[2]{HAC}
-A. Menezes, P. van Oorschot, S. Vanstone, \textit{Handbook of Applied Cryptography}, CRC Press, 1996
-
-\bibitem[3]{ROSE}
-Michael Rosing, \textit{Implementing Elliptic Curve Cryptography}, Manning Publications, 1999
-
-\bibitem[4]{COMBA}
-Paul G. Comba, \textit{Exponentiation Cryptosystems on the IBM PC}. IBM Systems Journal 29(4): 526-538 (1990)
-
-\bibitem[5]{KARA}
-A. Karatsuba, Doklay Akad. Nauk SSSR 145 (1962), pp.293-294
-
-\bibitem[6]{KARAP}
-Andre Weimerskirch and Christof Paar, \textit{Generalizations of the Karatsuba Algorithm for Polynomial Multiplication}, Submitted to Design, Codes and Cryptography, March 2002
-
-\bibitem[7]{BARRETT}
-Paul Barrett, \textit{Implementing the Rivest Shamir and Adleman Public Key Encryption Algorithm on a Standard Digital Signal Processor}, Advances in Cryptology, Crypto '86, Springer-Verlag.
-
-\bibitem[8]{MONT}
-P.L.Montgomery. \textit{Modular multiplication without trial division}. Mathematics of Computation, 44(170):519-521, April 1985.
-
-\bibitem[9]{DRMET}
-Chae Hoon Lim and Pil Joong Lee, \textit{Generating Efficient Primes for Discrete Log Cryptosystems}, POSTECH Information Research Laboratories
-
-\bibitem[10]{MMB}
-J. Daemen and R. Govaerts and J. Vandewalle, \textit{Block ciphers based on Modular Arithmetic}, State and {P}rogress in the {R}esearch of {C}ryptography, 1993, pp. 80-89
-
-\end{thebibliography}
-
-\input{tommath.ind}
-
-\chapter{Appendix}
-\subsection*{Appendix A -- Source Listing of tommath.h}
-
-The following is the source listing of the header file ``tommath.h'' for the LibTomMath project. It contains many of
-the definitions used throughout the code such as \textbf{mp\_int}, \textbf{MP\_PREC} and so on. The header is
-presented here for completeness.
-
-\index{tommath.h}
-\vspace{+3mm}\begin{small}
-\hspace{-5.1mm}{\bf File}: tommath.h
-\vspace{-3mm}
-\begin{alltt}
-001 /* LibTomMath, multiple-precision integer library -- Tom St Denis
-002 *
-003 * LibTomMath is library that provides for multiple-precision
-004 * integer arithmetic as well as number theoretic functionality.
-005 *
-006 * The library is designed directly after the MPI library by
-007 * Michael Fromberger but has been written from scratch with
-008 * additional optimizations in place.
-009 *
-010 * The library is free for all purposes without any express
-011 * guarantee it works.
-012 *
-013 * Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org
-014 */
-015 #ifndef BN_H_
-016 #define BN_H_
-017
-018 #include <stdio.h>
-019 #include <string.h>
-020 #include <stdlib.h>
-021 #include <ctype.h>
-022 #include <limits.h>
-023
-024 #undef MIN
-025 #define MIN(x,y) ((x)<(y)?(x):(y))
-026 #undef MAX
-027 #define MAX(x,y) ((x)>(y)?(x):(y))
-028
-029 #ifdef __cplusplus
-030 extern "C" \{
-031
-032 /* C++ compilers don't like assigning void * to mp_digit * */
-033 #define OPT_CAST (mp_digit *)
-034
-035 #else
-036
-037 /* C on the other hand doesn't care */
-038 #define OPT_CAST
-039
-040 #endif
-041
-042 /* some default configurations.
-043 *
-044 * A "mp_digit" must be able to hold DIGIT_BIT + 1 bits
-045 * A "mp_word" must be able to hold 2*DIGIT_BIT + 1 bits
-046 *
-047 * At the very least a mp_digit must be able to hold 7 bits
-048 * [any size beyond that is ok provided it doesn't overflow the data type]
-049 */
-050 #ifdef MP_8BIT
-051 typedef unsigned char mp_digit;
-052 typedef unsigned short mp_word;
-053 #elif defined(MP_16BIT)
-054 typedef unsigned short mp_digit;
-055 typedef unsigned long mp_word;
-056 #elif defined(MP_64BIT)
-057 /* for GCC only on supported platforms */
-058 #ifndef CRYPT
-059 typedef unsigned long long ulong64;
-060 typedef signed long long long64;
-061 #endif
-062
-063 typedef ulong64 mp_digit;
-064 typedef unsigned long mp_word __attribute__ ((mode(TI)));
-065
-066 #define DIGIT_BIT 60
-067 #else
-068 /* this is the default case, 28-bit digits */
-069
-070 /* this is to make porting into LibTomCrypt easier :-) */
-071 #ifndef CRYPT
-072 #if defined(_MSC_VER) || defined(__BORLANDC__)
-073 typedef unsigned __int64 ulong64;
-074 typedef signed __int64 long64;
-075 #else
-076 typedef unsigned long long ulong64;
-077 typedef signed long long long64;
-078 #endif
-079 #endif
-080
-081 typedef unsigned long mp_digit;
-082 typedef ulong64 mp_word;
-083
-084 #ifdef MP_31BIT
-085 #define DIGIT_BIT 31
-086 #else
-087 #define DIGIT_BIT 28
-088 #endif
-089 #endif
-090
-091 /* otherwise the bits per digit is calculated automatically from the size of
- a mp_digit */
-092 #ifndef DIGIT_BIT
-093 #define DIGIT_BIT ((CHAR_BIT * sizeof(mp_digit) - 1)) /* bits per di
- git */
-094 #endif
-095
-096
-097 #define MP_DIGIT_BIT DIGIT_BIT
-098 #define MP_MASK ((((mp_digit)1)<<((mp_digit)DIGIT_BIT))-((mp_digit)
- 1))
-099 #define MP_DIGIT_MAX MP_MASK
-100
-101 /* equalities */
-102 #define MP_LT -1 /* less than */
-103 #define MP_EQ 0 /* equal to */
-104 #define MP_GT 1 /* greater than */
-105
-106 #define MP_ZPOS 0 /* positive integer */
-107 #define MP_NEG 1 /* negative */
-108
-109 #define MP_OKAY 0 /* ok result */
-110 #define MP_MEM -2 /* out of mem */
-111 #define MP_VAL -3 /* invalid input */
-112 #define MP_RANGE MP_VAL
-113
-114 typedef int mp_err;
-115
-116 /* you'll have to tune these... */
-117 extern int KARATSUBA_MUL_CUTOFF,
-118 KARATSUBA_SQR_CUTOFF,
-119 TOOM_MUL_CUTOFF,
-120 TOOM_SQR_CUTOFF;
-121
-122 /* various build options */
-123 #define MP_PREC 64 /* default digits of precision (must
- be power of two) */
-124
-125 /* define this to use lower memory usage routines (exptmods mostly) */
-126 /* #define MP_LOW_MEM */
-127
-128 /* size of comba arrays, should be at least 2 * 2**(BITS_PER_WORD - BITS_PER
- _DIGIT*2) */
-129 #define MP_WARRAY (1 << (sizeof(mp_word) * CHAR_BIT - 2 * DIGI
- T_BIT + 1))
-130
-131 typedef struct \{
-132 int used, alloc, sign;
-133 mp_digit *dp;
-134 \} mp_int;
-135
-136 #define USED(m) ((m)->used)
-137 #define DIGIT(m,k) ((m)->dp[k])
-138 #define SIGN(m) ((m)->sign)
-139
-140 /* ---> init and deinit bignum functions <--- */
-141
-142 /* init a bignum */
-143 int mp_init(mp_int *a);
-144
-145 /* free a bignum */
-146 void mp_clear(mp_int *a);
-147
-148 /* init a null terminated series of arguments */
-149 int mp_init_multi(mp_int *mp, ...);
-150
-151 /* clear a null terminated series of arguments */
-152 void mp_clear_multi(mp_int *mp, ...);
-153
-154 /* exchange two ints */
-155 void mp_exch(mp_int *a, mp_int *b);
-156
-157 /* shrink ram required for a bignum */
-158 int mp_shrink(mp_int *a);
-159
-160 /* grow an int to a given size */
-161 int mp_grow(mp_int *a, int size);
-162
-163 /* init to a given number of digits */
-164 int mp_init_size(mp_int *a, int size);
-165
-166 /* ---> Basic Manipulations <--- */
-167
-168 #define mp_iszero(a) (((a)->used == 0) ? 1 : 0)
-169 #define mp_iseven(a) (((a)->used == 0 || (((a)->dp[0] & 1) == 0)) ? 1 : 0)
-170 #define mp_isodd(a) (((a)->used > 0 && (((a)->dp[0] & 1) == 1)) ? 1 : 0)
-171
-172 /* set to zero */
-173 void mp_zero(mp_int *a);
-174
-175 /* set to a digit */
-176 void mp_set(mp_int *a, mp_digit b);
-177
-178 /* set a 32-bit const */
-179 int mp_set_int(mp_int *a, unsigned int b);
-180
-181 /* copy, b = a */
-182 int mp_copy(mp_int *a, mp_int *b);
-183
-184 /* inits and copies, a = b */
-185 int mp_init_copy(mp_int *a, mp_int *b);
-186
-187 /* trim unused digits */
-188 void mp_clamp(mp_int *a);
-189
-190 /* ---> digit manipulation <--- */
-191
-192 /* right shift by "b" digits */
-193 void mp_rshd(mp_int *a, int b);
-194
-195 /* left shift by "b" digits */
-196 int mp_lshd(mp_int *a, int b);
-197
-198 /* c = a / 2**b */
-199 int mp_div_2d(mp_int *a, int b, mp_int *c, mp_int *d);
-200
-201 /* b = a/2 */
-202 int mp_div_2(mp_int *a, mp_int *b);
-203
-204 /* c = a * 2**b */
-205 int mp_mul_2d(mp_int *a, int b, mp_int *c);
-206
-207 /* b = a*2 */
-208 int mp_mul_2(mp_int *a, mp_int *b);
-209
-210 /* c = a mod 2**d */
-211 int mp_mod_2d(mp_int *a, int b, mp_int *c);
-212
-213 /* computes a = 2**b */
-214 int mp_2expt(mp_int *a, int b);
-215
-216 /* makes a pseudo-random int of a given size */
-217 int mp_rand(mp_int *a, int digits);
-218
-219 /* ---> binary operations <--- */
-220 /* c = a XOR b */
-221 int mp_xor(mp_int *a, mp_int *b, mp_int *c);
-222
-223 /* c = a OR b */
-224 int mp_or(mp_int *a, mp_int *b, mp_int *c);
-225
-226 /* c = a AND b */
-227 int mp_and(mp_int *a, mp_int *b, mp_int *c);
-228
-229 /* ---> Basic arithmetic <--- */
-230
-231 /* b = -a */
-232 int mp_neg(mp_int *a, mp_int *b);
-233
-234 /* b = |a| */
-235 int mp_abs(mp_int *a, mp_int *b);
-236
-237 /* compare a to b */
-238 int mp_cmp(mp_int *a, mp_int *b);
-239
-240 /* compare |a| to |b| */
-241 int mp_cmp_mag(mp_int *a, mp_int *b);
-242
-243 /* c = a + b */
-244 int mp_add(mp_int *a, mp_int *b, mp_int *c);
-245
-246 /* c = a - b */
-247 int mp_sub(mp_int *a, mp_int *b, mp_int *c);
-248
-249 /* c = a * b */
-250 int mp_mul(mp_int *a, mp_int *b, mp_int *c);
-251
-252 /* b = a*a */
-253 int mp_sqr(mp_int *a, mp_int *b);
-254
-255 /* a/b => cb + d == a */
-256 int mp_div(mp_int *a, mp_int *b, mp_int *c, mp_int *d);
-257
-258 /* c = a mod b, 0 <= c < b */
-259 int mp_mod(mp_int *a, mp_int *b, mp_int *c);
-260
-261 /* ---> single digit functions <--- */
-262
-263 /* compare against a single digit */
-264 int mp_cmp_d(mp_int *a, mp_digit b);
-265
-266 /* c = a + b */
-267 int mp_add_d(mp_int *a, mp_digit b, mp_int *c);
-268
-269 /* c = a - b */
-270 int mp_sub_d(mp_int *a, mp_digit b, mp_int *c);
-271
-272 /* c = a * b */
-273 int mp_mul_d(mp_int *a, mp_digit b, mp_int *c);
-274
-275 /* a/b => cb + d == a */
-276 int mp_div_d(mp_int *a, mp_digit b, mp_int *c, mp_digit *d);
-277
-278 /* a/3 => 3c + d == a */
-279 int mp_div_3(mp_int *a, mp_int *c, mp_digit *d);
-280
-281 /* c = a**b */
-282 int mp_expt_d(mp_int *a, mp_digit b, mp_int *c);
-283
-284 /* c = a mod b, 0 <= c < b */
-285 int mp_mod_d(mp_int *a, mp_digit b, mp_digit *c);
-286
-287 /* ---> number theory <--- */
-288
-289 /* d = a + b (mod c) */
-290 int mp_addmod(mp_int *a, mp_int *b, mp_int *c, mp_int *d);
-291
-292 /* d = a - b (mod c) */
-293 int mp_submod(mp_int *a, mp_int *b, mp_int *c, mp_int *d);
-294
-295 /* d = a * b (mod c) */
-296 int mp_mulmod(mp_int *a, mp_int *b, mp_int *c, mp_int *d);
-297
-298 /* c = a * a (mod b) */
-299 int mp_sqrmod(mp_int *a, mp_int *b, mp_int *c);
-300
-301 /* c = 1/a (mod b) */
-302 int mp_invmod(mp_int *a, mp_int *b, mp_int *c);
-303
-304 /* c = (a, b) */
-305 int mp_gcd(mp_int *a, mp_int *b, mp_int *c);
-306
-307 /* c = [a, b] or (a*b)/(a, b) */
-308 int mp_lcm(mp_int *a, mp_int *b, mp_int *c);
-309
-310 /* finds one of the b'th root of a, such that |c|**b <= |a|
-311 *
-312 * returns error if a < 0 and b is even
-313 */
-314 int mp_n_root(mp_int *a, mp_digit b, mp_int *c);
-315
-316 /* shortcut for square root */
-317 #define mp_sqrt(a, b) mp_n_root(a, 2, b)
-318
-319 /* computes the jacobi c = (a | n) (or Legendre if b is prime) */
-320 int mp_jacobi(mp_int *a, mp_int *n, int *c);
-321
-322 /* used to setup the Barrett reduction for a given modulus b */
-323 int mp_reduce_setup(mp_int *a, mp_int *b);
-324
-325 /* Barrett Reduction, computes a (mod b) with a precomputed value c
-326 *
-327 * Assumes that 0 < a <= b*b, note if 0 > a > -(b*b) then you can merely
-328 * compute the reduction as -1 * mp_reduce(mp_abs(a)) [pseudo code].
-329 */
-330 int mp_reduce(mp_int *a, mp_int *b, mp_int *c);
-331
-332 /* setups the montgomery reduction */
-333 int mp_montgomery_setup(mp_int *a, mp_digit *mp);
-334
-335 /* computes a = B**n mod b without division or multiplication useful for
-336 * normalizing numbers in a Montgomery system.
-337 */
-338 int mp_montgomery_calc_normalization(mp_int *a, mp_int *b);
-339
-340 /* computes x/R == x (mod N) via Montgomery Reduction */
-341 int mp_montgomery_reduce(mp_int *a, mp_int *m, mp_digit mp);
-342
-343 /* returns 1 if a is a valid DR modulus */
-344 int mp_dr_is_modulus(mp_int *a);
-345
-346 /* sets the value of "d" required for mp_dr_reduce */
-347 void mp_dr_setup(mp_int *a, mp_digit *d);
-348
-349 /* reduces a modulo b using the Diminished Radix method */
-350 int mp_dr_reduce(mp_int *a, mp_int *b, mp_digit mp);
-351
-352 /* returns true if a can be reduced with mp_reduce_2k */
-353 int mp_reduce_is_2k(mp_int *a);
-354
-355 /* determines k value for 2k reduction */
-356 int mp_reduce_2k_setup(mp_int *a, mp_digit *d);
-357
-358 /* reduces a modulo b where b is of the form 2**p - k [0 <= a] */
-359 int mp_reduce_2k(mp_int *a, mp_int *n, mp_digit k);
-360
-361 /* d = a**b (mod c) */
-362 int mp_exptmod(mp_int *a, mp_int *b, mp_int *c, mp_int *d);
-363
-364 /* ---> Primes <--- */
-365
-366 /* number of primes */
-367 #ifdef MP_8BIT
-368 #define PRIME_SIZE 31
-369 #else
-370 #define PRIME_SIZE 256
-371 #endif
-372
-373 /* table of first PRIME_SIZE primes */
-374 extern const mp_digit __prime_tab[];
-375
-376 /* result=1 if a is divisible by one of the first PRIME_SIZE primes */
-377 int mp_prime_is_divisible(mp_int *a, int *result);
-378
-379 /* performs one Fermat test of "a" using base "b".
-380 * Sets result to 0 if composite or 1 if probable prime
-381 */
-382 int mp_prime_fermat(mp_int *a, mp_int *b, int *result);
-383
-384 /* performs one Miller-Rabin test of "a" using base "b".
-385 * Sets result to 0 if composite or 1 if probable prime
-386 */
-387 int mp_prime_miller_rabin(mp_int *a, mp_int *b, int *result);
-388
-389 /* performs t rounds of Miller-Rabin on "a" using the first
-390 * t prime bases. Also performs an initial sieve of trial
-391 * division. Determines if "a" is prime with probability
-392 * of error no more than (1/4)**t.
-393 *
-394 * Sets result to 1 if probably prime, 0 otherwise
-395 */
-396 int mp_prime_is_prime(mp_int *a, int t, int *result);
-397
-398 /* finds the next prime after the number "a" using "t" trials
-399 * of Miller-Rabin.
-400 */
-401 int mp_prime_next_prime(mp_int *a, int t);
-402
-403
-404 /* ---> radix conversion <--- */
-405 int mp_count_bits(mp_int *a);
-406
-407 int mp_unsigned_bin_size(mp_int *a);
-408 int mp_read_unsigned_bin(mp_int *a, unsigned char *b, int c);
-409 int mp_to_unsigned_bin(mp_int *a, unsigned char *b);
-410
-411 int mp_signed_bin_size(mp_int *a);
-412 int mp_read_signed_bin(mp_int *a, unsigned char *b, int c);
-413 int mp_to_signed_bin(mp_int *a, unsigned char *b);
-414
-415 int mp_read_radix(mp_int *a, char *str, int radix);
-416 int mp_toradix(mp_int *a, char *str, int radix);
-417 int mp_radix_size(mp_int *a, int radix);
-418
-419 int mp_fread(mp_int *a, int radix, FILE *stream);
-420 int mp_fwrite(mp_int *a, int radix, FILE *stream);
-421
-422 #define mp_read_raw(mp, str, len) mp_read_signed_bin((mp), (str), (len))
-423 #define mp_raw_size(mp) mp_signed_bin_size(mp)
-424 #define mp_toraw(mp, str) mp_to_signed_bin((mp), (str))
-425 #define mp_read_mag(mp, str, len) mp_read_unsigned_bin((mp), (str), (len))
-426 #define mp_mag_size(mp) mp_unsigned_bin_size(mp)
-427 #define mp_tomag(mp, str) mp_to_unsigned_bin((mp), (str))
-428
-429 #define mp_tobinary(M, S) mp_toradix((M), (S), 2)
-430 #define mp_tooctal(M, S) mp_toradix((M), (S), 8)
-431 #define mp_todecimal(M, S) mp_toradix((M), (S), 10)
-432 #define mp_tohex(M, S) mp_toradix((M), (S), 16)
-433
-434 /* lowlevel functions, do not call! */
-435 int s_mp_add(mp_int *a, mp_int *b, mp_int *c);
-436 int s_mp_sub(mp_int *a, mp_int *b, mp_int *c);
-437 #define s_mp_mul(a, b, c) s_mp_mul_digs(a, b, c, (a)->used + (b)->used + 1)
-438 int fast_s_mp_mul_digs(mp_int *a, mp_int *b, mp_int *c, int digs);
-439 int s_mp_mul_digs(mp_int *a, mp_int *b, mp_int *c, int digs);
-440 int fast_s_mp_mul_high_digs(mp_int *a, mp_int *b, mp_int *c, int digs);
-441 int s_mp_mul_high_digs(mp_int *a, mp_int *b, mp_int *c, int digs);
-442 int fast_s_mp_sqr(mp_int *a, mp_int *b);
-443 int s_mp_sqr(mp_int *a, mp_int *b);
-444 int mp_karatsuba_mul(mp_int *a, mp_int *b, mp_int *c);
-445 int mp_toom_mul(mp_int *a, mp_int *b, mp_int *c);
-446 int mp_karatsuba_sqr(mp_int *a, mp_int *b);
-447 int mp_toom_sqr(mp_int *a, mp_int *b);
-448 int fast_mp_invmod(mp_int *a, mp_int *b, mp_int *c);
-449 int fast_mp_montgomery_reduce(mp_int *a, mp_int *m, mp_digit mp);
-450 int mp_exptmod_fast(mp_int *G, mp_int *X, mp_int *P, mp_int *Y, int mode);
-451 int s_mp_exptmod (mp_int * G, mp_int * X, mp_int * P, mp_int * Y);
-452 void bn_reverse(unsigned char *s, int len);
-453
-454 #ifdef __cplusplus
-455 \}
-456 #endif
-457
-458 #endif
-459
-\end{alltt}
-\end{small}
-
-\end{document}
\ No newline at end of file