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src/ltm/bn_mp_sqrtmod_prime.c  view on Meta::CPAN

 */

/* Tonelli-Shanks algorithm
 * https://en.wikipedia.org/wiki/Tonelli%E2%80%93Shanks_algorithm
 * https://gmplib.org/list-archives/gmp-discuss/2013-April/005300.html
 *
 */

int mp_sqrtmod_prime(const mp_int *n, const mp_int *prime, mp_int *ret)
{
   int res, legendre;
   mp_int t1, C, Q, S, Z, M, T, R, two;
   mp_digit i;

   /* first handle the simple cases */
   if (mp_cmp_d(n, 0uL) == MP_EQ) {
      mp_zero(ret);
      return MP_OKAY;
   }
   if (mp_cmp_d(prime, 2uL) == MP_EQ)                            return MP_VAL; /* prime must be odd */
   if ((res = mp_jacobi(n, prime, &legendre)) != MP_OKAY)        return res;
   if (legendre == -1)                                           return MP_VAL; /* quadratic non-residue mod prime */

   if ((res = mp_init_multi(&t1, &C, &Q, &S, &Z, &M, &T, &R, &two, NULL)) != MP_OKAY) {
      return res;
   }

   /* SPECIAL CASE: if prime mod 4 == 3
    * compute directly: res = n^(prime+1)/4 mod prime
    * Handbook of Applied Cryptography algorithm 3.36
    */
   if ((res = mp_mod_d(prime, 4uL, &i)) != MP_OKAY)               goto cleanup;

src/ltm/bn_mp_sqrtmod_prime.c  view on Meta::CPAN

      if ((res = mp_div_2(&Q, &Q)) != MP_OKAY)                    goto cleanup;
      /* Q = Q / 2 */
      if ((res = mp_add_d(&S, 1uL, &S)) != MP_OKAY)               goto cleanup;
      /* S = S + 1 */
   }

   /* find a Z such that the Legendre symbol (Z|prime) == -1 */
   if ((res = mp_set_int(&Z, 2uL)) != MP_OKAY)                    goto cleanup;
   /* Z = 2 */
   while (1) {
      if ((res = mp_jacobi(&Z, prime, &legendre)) != MP_OKAY)     goto cleanup;
      if (legendre == -1) break;
      if ((res = mp_add_d(&Z, 1uL, &Z)) != MP_OKAY)               goto cleanup;
      /* Z = Z + 1 */
   }

   if ((res = mp_exptmod(&Z, &Q, prime, &C)) != MP_OKAY)         goto cleanup;
   /* C = Z ^ Q mod prime */
   if ((res = mp_add_d(&Q, 1uL, &t1)) != MP_OKAY)                goto cleanup;
   if ((res = mp_div_2(&t1, &t1)) != MP_OKAY)                    goto cleanup;
   /* t1 = (Q + 1) / 2 */
   if ((res = mp_exptmod(n, &t1, prime, &R)) != MP_OKAY)         goto cleanup;