c3-lang/libtommath/mp_sqrtmod_prime.c

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• Hash : fb305e09
  Author :
  Date : 2020-08-05T15:18:59
  

  Additional input checks and a test for b \cong 0 (mod a) in test_mp_sqrtmod_prime
  to go along with it.
  

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• mp_sqrtmod_prime.c

• #include "tommath_private.h"
  #ifdef MP_SQRTMOD_PRIME_C
  /* LibTomMath, multiple-precision integer library -- Tom St Denis */
  /* SPDX-License-Identifier: Unlicense */
  
  /* Tonelli-Shanks algorithm
   * https://en.wikipedia.org/wiki/Tonelli%E2%80%93Shanks_algorithm
   * https://gmplib.org/list-archives/gmp-discuss/2013-April/005300.html
   *
   */
  
  mp_err mp_sqrtmod_prime(const mp_int *n, const mp_int *prime, mp_int *ret)
  {
     mp_err err;
     int legendre;
     /* The type is "int" because of the types in the mp_int struct.
        Don't forget to change them here when you change them there! */
     int S, M, i;
     mp_int t1, C, Q, Z, T, R, two;
  
     /* first handle the simple cases */
     if (mp_cmp_d(n, 0uL) == MP_EQ) {
        mp_zero(ret);
        return MP_OKAY;
     }
     /* "prime" must be odd and > 2 */
     if (mp_iseven(prime) || (mp_cmp_d(prime, 3uL) == MP_LT))      return MP_VAL;
     if ((err = mp_kronecker(n, prime, &legendre)) != MP_OKAY)     return err;
     /* n \not\cong 0 (mod p) and n \cong r^2 (mod p) for some r \in N^+ */
     if (legendre != 1)                                            return MP_VAL;
  
     if ((err = mp_init_multi(&t1, &C, &Q, &Z, &T, &R, &two, NULL)) != MP_OKAY) {
        return err;
     }
  
     /* SPECIAL CASE: if prime mod 4 == 3
      * compute directly: err = n^(prime+1)/4 mod prime
      * Handbook of Applied Cryptography algorithm 3.36
      */
     /* x%4 == x&3 for x in N and x>0 */
     if ((prime->dp[0] & 3u) == 3u) {
        if ((err = mp_add_d(prime, 1uL, &t1)) != MP_OKAY)           goto LBL_END;
        if ((err = mp_div_2(&t1, &t1)) != MP_OKAY)                  goto LBL_END;
        if ((err = mp_div_2(&t1, &t1)) != MP_OKAY)                  goto LBL_END;
        if ((err = mp_exptmod(n, &t1, prime, ret)) != MP_OKAY)      goto LBL_END;
        err = MP_OKAY;
        goto LBL_END;
     }
  
     /* NOW: Tonelli-Shanks algorithm */
  
     /* factor out powers of 2 from prime-1, defining Q and S as: prime-1 = Q*2^S */
     if ((err = mp_copy(prime, &Q)) != MP_OKAY)                    goto LBL_END;
     if ((err = mp_sub_d(&Q, 1uL, &Q)) != MP_OKAY)                 goto LBL_END;
     /* Q = prime - 1 */
     S = 0;
     /* S = 0 */
     while (mp_iseven(&Q)) {
        if ((err = mp_div_2(&Q, &Q)) != MP_OKAY)                    goto LBL_END;
        /* Q = Q / 2 */
        S++;
        /* S = S + 1 */
     }
  
     /* find a Z such that the Legendre symbol (Z|prime) == -1 */
     mp_set(&Z, 2uL);
     /* Z = 2 */
     for (;;) {
        if ((err = mp_kronecker(&Z, prime, &legendre)) != MP_OKAY)     goto LBL_END;
        /* If "prime" (p) is an odd prime Jacobi(k|p) = 0 for k \cong 0 (mod p) */
        /* but there is at least one non-quadratic residue before k>=p if p is an odd prime. */
        if (legendre == 0) {
           err = MP_VAL;
           goto LBL_END;
        }
        if (legendre == -1) break;
        if ((err = mp_add_d(&Z, 1uL, &Z)) != MP_OKAY)               goto LBL_END;
        /* Z = Z + 1 */
     }
  
     if ((err = mp_exptmod(&Z, &Q, prime, &C)) != MP_OKAY)         goto LBL_END;
     /* C = Z ^ Q mod prime */
     if ((err = mp_add_d(&Q, 1uL, &t1)) != MP_OKAY)                goto LBL_END;
     if ((err = mp_div_2(&t1, &t1)) != MP_OKAY)                    goto LBL_END;
     /* t1 = (Q + 1) / 2 */
     if ((err = mp_exptmod(n, &t1, prime, &R)) != MP_OKAY)         goto LBL_END;
     /* R = n ^ ((Q + 1) / 2) mod prime */
     if ((err = mp_exptmod(n, &Q, prime, &T)) != MP_OKAY)          goto LBL_END;
     /* T = n ^ Q mod prime */
     M = S;
     /* M = S */
     mp_set(&two, 2uL);
  
     for (;;) {
        if ((err = mp_copy(&T, &t1)) != MP_OKAY)                    goto LBL_END;
        i = 0;
        for (;;) {
           if (mp_cmp_d(&t1, 1uL) == MP_EQ) break;
           /*  No exponent in the range 0 < i < M found
              (M is at least 1 in the first round because "prime" > 2) */
           if (M == i) {
              err = MP_VAL;
              goto LBL_END;
           }
           if ((err = mp_exptmod(&t1, &two, prime, &t1)) != MP_OKAY) goto LBL_END;
           i++;
        }
        if (i == 0) {
           if ((err = mp_copy(&R, ret)) != MP_OKAY)                  goto LBL_END;
           err = MP_OKAY;
           goto LBL_END;
        }
        mp_set_i32(&t1, M - i - 1);
        if ((err = mp_exptmod(&two, &t1, prime, &t1)) != MP_OKAY)   goto LBL_END;
        /* t1 = 2 ^ (M - i - 1) */
        if ((err = mp_exptmod(&C, &t1, prime, &t1)) != MP_OKAY)     goto LBL_END;
        /* t1 = C ^ (2 ^ (M - i - 1)) mod prime */
        if ((err = mp_sqrmod(&t1, prime, &C)) != MP_OKAY)           goto LBL_END;
        /* C = (t1 * t1) mod prime */
        if ((err = mp_mulmod(&R, &t1, prime, &R)) != MP_OKAY)       goto LBL_END;
        /* R = (R * t1) mod prime */
        if ((err = mp_mulmod(&T, &C, prime, &T)) != MP_OKAY)        goto LBL_END;
        /* T = (T * C) mod prime */
        M = i;
        /* M = i */
     }
  
  LBL_END:
     mp_clear_multi(&t1, &C, &Q, &Z, &T, &R, &two, NULL);
     return err;
  }
  
  #endif
  

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