tommath/bn_mp_n_root_ex.c

#include "tommath_private.h"
#ifdef BN_MP_N_ROOT_EX_C
/* LibTomMath, multiple-precision integer library -- Tom St Denis
 *
 * LibTomMath is a library that provides multiple-precision
 * integer arithmetic as well as number theoretic functionality.
 *
 * The library was designed directly after the MPI library by
 * Michael Fromberger but has been written from scratch with
 * additional optimizations in place.
 *
 * SPDX-License-Identifier: Unlicense
 */

/* 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, etc].
 */
int mp_n_root_ex(const mp_int *a, mp_digit b, mp_int *c, int fast)
{
   mp_int  t1, t2, t3, a_;
   int     res;

   /* input must be positive if b is even */
   if (((b & 1u) == 0u) && (a->sign == MP_NEG)) {
      return MP_VAL;
   }

   if ((res = mp_init(&t1)) != MP_OKAY) {
      return res;
   }

   if ((res = mp_init(&t2)) != MP_OKAY) {
      goto LBL_T1;
   }

   if ((res = mp_init(&t3)) != MP_OKAY) {
      goto LBL_T2;
   }

   /* if a is negative fudge the sign but keep track */
   a_ = *a;
   a_.sign = MP_ZPOS;

   /* t2 = 2 */
   mp_set(&t2, 2uL);

   do {
      /* t1 = t2 */
      if ((res = mp_copy(&t2, &t1)) != MP_OKAY) {
         goto LBL_T3;
      }

      /* t2 = t1 - ((t1**b - a) / (b * t1**(b-1))) */

      /* t3 = t1**(b-1) */
      if ((res = mp_expt_d_ex(&t1, b - 1u, &t3, fast)) != MP_OKAY) {
         goto LBL_T3;
      }

      /* numerator */
      /* t2 = t1**b */
      if ((res = mp_mul(&t3, &t1, &t2)) != MP_OKAY) {
         goto LBL_T3;
      }

      /* t2 = t1**b - a */
      if ((res = mp_sub(&t2, &a_, &t2)) != MP_OKAY) {
         goto LBL_T3;
      }

      /* denominator */
      /* t3 = t1**(b-1) * b  */
      if ((res = mp_mul_d(&t3, b, &t3)) != MP_OKAY) {
         goto LBL_T3;
      }

      /* t3 = (t1**b - a)/(b * t1**(b-1)) */
      if ((res = mp_div(&t2, &t3, &t3, NULL)) != MP_OKAY) {
         goto LBL_T3;
      }

      if ((res = mp_sub(&t1, &t3, &t2)) != MP_OKAY) {
         goto LBL_T3;
      }
   }  while (mp_cmp(&t1, &t2) != MP_EQ);

   /* result can be off by a few so check */
   for (;;) {
      if ((res = mp_expt_d_ex(&t1, b, &t2, fast)) != MP_OKAY) {
         goto LBL_T3;
      }

      if (mp_cmp(&t2, &a_) == MP_GT) {
         if ((res = mp_sub_d(&t1, 1uL, &t1)) != MP_OKAY) {
            goto LBL_T3;
         }
      } else {
         break;
      }
   }

   /* set the result */
   mp_exch(&t1, c);

   /* set the sign of the result */
   c->sign = a->sign;

   res = MP_OKAY;

LBL_T3:
   mp_clear(&t3);
LBL_T2:
   mp_clear(&t2);
LBL_T1:
   mp_clear(&t1);
   return res;
}
#endif

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