/* 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>
/* find the n'th root of an integer
*
* Result found such that (c)^b <= a and (c+1)^b > a
*
* This algorithm uses Newton's approximation x[i+1] = x[i] - f(x[i])/f'(x[i])
* which will find the root in log(N) time where each step involves a fair bit. This
* is not meant to find huge roots [square and cube at most].
*/
int
mp_n_root (mp_int * a, mp_digit b, mp_int * c)
{
mp_int t1, t2, t3;
int res, neg;
/* input must be positive if b is even */
if ((b & 1) == 0 && a->sign == MP_NEG) {
return MP_VAL;
}
if ((res = mp_init (&t1)) != MP_OKAY) {
return res;
}
if ((res = mp_init (&t2)) != MP_OKAY) {
goto __T1;
}
if ((res = mp_init (&t3)) != MP_OKAY) {
goto __T2;
}
/* if a is negative fudge the sign but keep track */
neg = a->sign;
a->sign = MP_ZPOS;
/* t2 = 2 */
mp_set (&t2, 2);
do {
/* t1 = t2 */
if ((res = mp_copy (&t2, &t1)) != MP_OKAY) {
goto __T3;
}
/* t2 = t1 - ((t1^b - a) / (b * t1^(b-1))) */
if ((res = mp_expt_d (&t1, b - 1, &t3)) != MP_OKAY) { /* t3 = t1^(b-1) */
goto __T3;
}
/* numerator */
if ((res = mp_mul (&t3, &t1, &t2)) != MP_OKAY) { /* t2 = t1^b */
goto __T3;
}
if ((res = mp_sub (&t2, a, &t2)) != MP_OKAY) { /* t2 = t1^b - a */
goto __T3;
}
if ((res = mp_mul_d (&t3, b, &t3)) != MP_OKAY) { /* t3 = t1^(b-1) * b */
goto __T3;
}
if ((res = mp_div (&t2, &t3, &t3, NULL)) != MP_OKAY) { /* t3 = (t1^b - a)/(b * t1^(b-1)) */
goto __T3;
}
if ((res = mp_sub (&t1, &t3, &t2)) != MP_OKAY) {
goto __T3;
}
}
while (mp_cmp (&t1, &t2) != MP_EQ);
/* result can be off by a few so check */
for (;;) {
if ((res = mp_expt_d (&t1, b, &t2)) != MP_OKAY) {
goto __T3;
}
if (mp_cmp (&t2, a) == MP_GT) {
if ((res = mp_sub_d (&t1, 1, &t1)) != MP_OKAY) {
goto __T3;
}
} else {
break;
}
}
/* reset the sign of a first */
a->sign = neg;
/* set the result */
mp_exch (&t1, c);
/* set the sign of the result */
c->sign = neg;
res = MP_OKAY;
__T3:mp_clear (&t3);
__T2:mp_clear (&t2);
__T1:mp_clear (&t1);
return res;
}
