119 lines
4.4 KiB
C
119 lines
4.4 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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mp_int t1, C, Q, S, Z, M, T, R, two;
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mp_digit i;
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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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if (mp_cmp_d(prime, 2uL) == MP_EQ) return MP_VAL; /* prime must be odd */
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if ((err = mp_kronecker(n, prime, &legendre)) != MP_OKAY) return err;
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if (legendre == -1) return MP_VAL; /* quadratic non-residue mod prime */
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if ((err = mp_init_multi(&t1, &C, &Q, &S, &Z, &M, &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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if ((err = mp_mod_d(prime, 4uL, &i)) != MP_OKAY) goto cleanup;
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if (i == 3u) {
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if ((err = mp_add_d(prime, 1uL, &t1)) != MP_OKAY) goto cleanup;
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if ((err = mp_div_2(&t1, &t1)) != MP_OKAY) goto cleanup;
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if ((err = mp_div_2(&t1, &t1)) != MP_OKAY) goto cleanup;
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if ((err = mp_exptmod(n, &t1, prime, ret)) != MP_OKAY) goto cleanup;
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err = MP_OKAY;
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goto cleanup;
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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 cleanup;
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if ((err = mp_sub_d(&Q, 1uL, &Q)) != MP_OKAY) goto cleanup;
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/* Q = prime - 1 */
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mp_zero(&S);
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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 cleanup;
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/* Q = Q / 2 */
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if ((err = mp_add_d(&S, 1uL, &S)) != MP_OKAY) goto cleanup;
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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 cleanup;
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if (legendre == -1) break;
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if ((err = mp_add_d(&Z, 1uL, &Z)) != MP_OKAY) goto cleanup;
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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 cleanup;
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/* C = Z ^ Q mod prime */
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if ((err = mp_add_d(&Q, 1uL, &t1)) != MP_OKAY) goto cleanup;
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if ((err = mp_div_2(&t1, &t1)) != MP_OKAY) goto cleanup;
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/* t1 = (Q + 1) / 2 */
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if ((err = mp_exptmod(n, &t1, prime, &R)) != MP_OKAY) goto cleanup;
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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 cleanup;
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/* T = n ^ Q mod prime */
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if ((err = mp_copy(&S, &M)) != MP_OKAY) goto cleanup;
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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 cleanup;
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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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if ((err = mp_exptmod(&t1, &two, prime, &t1)) != MP_OKAY) goto cleanup;
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i++;
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}
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if (i == 0u) {
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if ((err = mp_copy(&R, ret)) != MP_OKAY) goto cleanup;
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err = MP_OKAY;
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goto cleanup;
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}
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if ((err = mp_sub_d(&M, i, &t1)) != MP_OKAY) goto cleanup;
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if ((err = mp_sub_d(&t1, 1uL, &t1)) != MP_OKAY) goto cleanup;
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if ((err = mp_exptmod(&two, &t1, prime, &t1)) != MP_OKAY) goto cleanup;
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/* t1 = 2 ^ (M - i - 1) */
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if ((err = mp_exptmod(&C, &t1, prime, &t1)) != MP_OKAY) goto cleanup;
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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 cleanup;
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/* C = (t1 * t1) mod prime */
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if ((err = mp_mulmod(&R, &t1, prime, &R)) != MP_OKAY) goto cleanup;
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/* R = (R * t1) mod prime */
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if ((err = mp_mulmod(&T, &C, prime, &T)) != MP_OKAY) goto cleanup;
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/* T = (T * C) mod prime */
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mp_set(&M, i);
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/* M = i */
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}
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cleanup:
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mp_clear_multi(&t1, &C, &Q, &S, &Z, &M, &T, &R, &two, NULL);
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return err;
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}
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#endif
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