diff options
Diffstat (limited to 'libtommath/bn_mp_sqrtmod_prime.c')
| -rw-r--r-- | libtommath/bn_mp_sqrtmod_prime.c | 124 |
1 files changed, 124 insertions, 0 deletions
diff --git a/libtommath/bn_mp_sqrtmod_prime.c b/libtommath/bn_mp_sqrtmod_prime.c new file mode 100644 index 0000000..968729e --- /dev/null +++ b/libtommath/bn_mp_sqrtmod_prime.c @@ -0,0 +1,124 @@ +#include <tommath_private.h> +#ifdef BN_MP_SQRTMOD_PRIME_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 is free for all purposes without any express + * guarantee it works. + */ + +/* 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(mp_int *n, 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, 0) == MP_EQ) { + mp_zero(ret); + return MP_OKAY; + } + if (mp_cmp_d(prime, 2) == 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, 4, &i)) != MP_OKAY) goto cleanup; + if (i == 3) { + if ((res = mp_add_d(prime, 1, &t1)) != MP_OKAY) goto cleanup; + if ((res = mp_div_2(&t1, &t1)) != MP_OKAY) goto cleanup; + if ((res = mp_div_2(&t1, &t1)) != MP_OKAY) goto cleanup; + if ((res = mp_exptmod(n, &t1, prime, ret)) != MP_OKAY) goto cleanup; + res = MP_OKAY; + goto cleanup; + } + + /* NOW: Tonelli-Shanks algorithm */ + + /* factor out powers of 2 from prime-1, defining Q and S as: prime-1 = Q*2^S */ + if ((res = mp_copy(prime, &Q)) != MP_OKAY) goto cleanup; + if ((res = mp_sub_d(&Q, 1, &Q)) != MP_OKAY) goto cleanup; + /* Q = prime - 1 */ + mp_zero(&S); + /* S = 0 */ + while (mp_iseven(&Q) != MP_NO) { + if ((res = mp_div_2(&Q, &Q)) != MP_OKAY) goto cleanup; + /* Q = Q / 2 */ + if ((res = mp_add_d(&S, 1, &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, 2)) != 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, 1, &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, 1, &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; + /* R = n ^ ((Q + 1) / 2) mod prime */ + if ((res = mp_exptmod(n, &Q, prime, &T)) != MP_OKAY) goto cleanup; + /* T = n ^ Q mod prime */ + if ((res = mp_copy(&S, &M)) != MP_OKAY) goto cleanup; + /* M = S */ + if ((res = mp_set_int(&two, 2)) != MP_OKAY) goto cleanup; + + res = MP_VAL; + while (1) { + if ((res = mp_copy(&T, &t1)) != MP_OKAY) goto cleanup; + i = 0; + while (1) { + if (mp_cmp_d(&t1, 1) == MP_EQ) break; + if ((res = mp_exptmod(&t1, &two, prime, &t1)) != MP_OKAY) goto cleanup; + i++; + } + if (i == 0) { + if ((res = mp_copy(&R, ret)) != MP_OKAY) goto cleanup; + res = MP_OKAY; + goto cleanup; + } + if ((res = mp_sub_d(&M, i, &t1)) != MP_OKAY) goto cleanup; + if ((res = mp_sub_d(&t1, 1, &t1)) != MP_OKAY) goto cleanup; + if ((res = mp_exptmod(&two, &t1, prime, &t1)) != MP_OKAY) goto cleanup; + /* t1 = 2 ^ (M - i - 1) */ + if ((res = mp_exptmod(&C, &t1, prime, &t1)) != MP_OKAY) goto cleanup; + /* t1 = C ^ (2 ^ (M - i - 1)) mod prime */ + if ((res = mp_sqrmod(&t1, prime, &C)) != MP_OKAY) goto cleanup; + /* C = (t1 * t1) mod prime */ + if ((res = mp_mulmod(&R, &t1, prime, &R)) != MP_OKAY) goto cleanup; + /* R = (R * t1) mod prime */ + if ((res = mp_mulmod(&T, &C, prime, &T)) != MP_OKAY) goto cleanup; + /* T = (T * C) mod prime */ + mp_set(&M, i); + /* M = i */ + } + +cleanup: + mp_clear_multi(&t1, &C, &Q, &S, &Z, &M, &T, &R, &two, NULL); + return res; +} + +#endif |