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