/* LibTomMath, multiple-precision integer library -- Tom St Denis
*
* LibTomMath is library that provides for multiple-precision
* integer arithmetic as well as number theoretic functionality.
*
* The library is designed directly after the MPI library by
* Michael Fromberger but has been written from scratch with
* additional optimizations in place.
*
* The library is free for all purposes without any express
* guarantee it works.
*
* Tom St Denis, tomstdenis@iahu.ca, http://math.libtomcrypt.org
*/
#include <tommath.h>
/* c = |a| * |b| using Karatsuba Multiplication using three half size multiplications
*
* Let B represent the radix [e.g. 2**DIGIT_BIT] and let n represent half of the number of digits in the min(a,b)
*
* a = a1 * B^n + a0
* b = b1 * B^n + b0
*
* Then, a * b => a1b1 * B^2n + ((a1 - b1)(a0 - b0) + a0b0 + a1b1) * B + a0b0
*
* Note that a1b1 and a0b0 are used twice and only need to be computed once. So in total
* three half size (half # of digit) multiplications are performed, a0b0, a1b1 and (a1-b1)(a0-b0)
*
* Note that a multiplication of half the digits requires 1/4th the number of single precision
* multiplications so in total after one call 25% of the single precision multiplications are saved.
* Note also that the call to mp_mul can end up back in this function if the a0, a1, b0, or b1 are above
* the threshold. This is known as divide-and-conquer and leads to the famous O(N^lg(3)) or O(N^1.584) work which
* is asymptopically lower than the standard O(N^2) that the baseline/comba methods use. Generally though the
* overhead of this method doesn't pay off until a certain size (N ~ 80) is reached.
*/
int
mp_karatsuba_mul (mp_int * a, mp_int * b, mp_int * c)
{
mp_int x0, x1, y0, y1, t1, t2, x0y0, x1y1;
int B, err;
err = MP_MEM;
/* min # of digits */
B = MIN (a->used, b->used);
/* now divide in two */
B = B / 2;
/* init copy all the temps */
if (mp_init_size (&x0, B) != MP_OKAY)
goto ERR;
if (mp_init_size (&x1, a->used - B) != MP_OKAY)
goto X0;
if (mp_init_size (&y0, B) != MP_OKAY)
goto X1;
if (mp_init_size (&y1, b->used - B) != MP_OKAY)
goto Y0;
/* init temps */
if (mp_init_size (&t1, B * 2) != MP_OKAY)
goto Y1;
if (mp_init_size (&t2, B * 2) != MP_OKAY)
goto T1;
if (mp_init_size (&x0y0, B * 2) != MP_OKAY)
goto T2;
if (mp_init_size (&x1y1, B * 2) != MP_OKAY)
goto X0Y0;
/* now shift the digits */
x0.sign = x1.sign = a->sign;
y0.sign = y1.sign = b->sign;
x0.used = y0.used = B;
x1.used = a->used - B;
y1.used = b->used - B;
{
register int x;
register mp_digit *tmpa, *tmpb, *tmpx, *tmpy;
/* we copy the digits directly instead of using higher level functions
* since we also need to shift the digits
*/
tmpa = a->dp;
tmpb = b->dp;
tmpx = x0.dp;
tmpy = y0.dp;
for (x = 0; x < B; x++) {
*tmpx++ = *tmpa++;
*tmpy++ = *tmpb++;
}
tmpx = x1.dp;
for (x = B; x < a->used; x++) {
*tmpx++ = *tmpa++;
}
tmpy = y1.dp;
for (x = B; x < b->used; x++) {
*tmpy++ = *tmpb++;
}
}
/* only need to clamp the lower words since by definition the upper words x1/y1 must
* have a known number of digits
*/
mp_clamp (&x0);
mp_clamp (&y0);
/* now calc the products x0y0 and x1y1 */
if (mp_mul (&x0, &y0, &x0y0) != MP_OKAY)
goto X1Y1; /* x0y0 = x0*y0 */
if (mp_mul (&x1, &y1, &x1y1) != MP_OKAY)
goto X1Y1; /* x1y1 = x1*y1 */
/* now calc x1-x0 and y1-y0 */
if (mp_sub (&x1, &x0, &t1) != MP_OKAY)
goto X1Y1; /* t1 = x1 - x0 */
if (mp_sub (&y1, &y0, &t2) != MP_OKAY)
goto X1Y1; /* t2 = y1 - y0 */
if (mp_mul (&t1, &t2, &t1) != MP_OKAY)
goto X1Y1; /* t1 = (x1 - x0) * (y1 - y0) */
/* add x0y0 */
if (mp_add (&x0y0, &x1y1, &t2) != MP_OKAY)
goto X1Y1; /* t2 = x0y0 + x1y1 */
if (mp_sub (&t2, &t1, &t1) != MP_OKAY)
goto X1Y1; /* t1 = x0y0 + x1y1 - (x1-x0)*(y1-y0) */
/* shift by B */
if (mp_lshd (&t1, B) != MP_OKAY)
goto X1Y1; /* t1 = (x0y0 + x1y1 - (x1-x0)*(y1-y0))<<B */
if (mp_lshd (&x1y1, B * 2) != MP_OKAY)
goto X1Y1; /* x1y1 = x1y1 << 2*B */
if (mp_add (&x0y0, &t1, &t1) != MP_OKAY)
goto X1Y1; /* t1 = x0y0 + t1 */
if (mp_add (&t1, &x1y1, c) != MP_OKAY)
goto X1Y1; /* t1 = x0y0 + t1 + x1y1 */
err = MP_OKAY;
X1Y1:mp_clear (&x1y1);
X0Y0:mp_clear (&x0y0);
T2:mp_clear (&t2);
T1:mp_clear (&t1);
Y1:mp_clear (&y1);
Y0:mp_clear (&y0);
X1:mp_clear (&x1);
X0:mp_clear (&x0);
ERR:
return err;
}
