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comparison libtommath/bn_mp_sqrtmod_prime.c @ 1436:60fc6476e044

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Update to libtommath v1.0
author Matt Johnston <matt@ucc.asn.au>
date Sat, 24 Jun 2017 22:37:14 +0800
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children f52919ffd3b1
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1435:f849a5ca2efc 1436:60fc6476e044
1 #include <tommath_private.h>
2 #ifdef BN_MP_SQRTMOD_PRIME_C
3 /* LibTomMath, multiple-precision integer library -- Tom St Denis
4 *
5 * LibTomMath is a library that provides multiple-precision
6 * integer arithmetic as well as number theoretic functionality.
7 *
8 * The library is free for all purposes without any express
9 * guarantee it works.
10 */
11
12 /* Tonelli-Shanks algorithm
13 * https://en.wikipedia.org/wiki/Tonelli%E2%80%93Shanks_algorithm
14 * https://gmplib.org/list-archives/gmp-discuss/2013-April/005300.html
15 *
16 */
17
18 int mp_sqrtmod_prime(mp_int *n, mp_int *prime, mp_int *ret)
19 {
20 int res, legendre;
21 mp_int t1, C, Q, S, Z, M, T, R, two;
22 mp_digit i;
23
24 /* first handle the simple cases */
25 if (mp_cmp_d(n, 0) == MP_EQ) {
26 mp_zero(ret);
27 return MP_OKAY;
28 }
29 if (mp_cmp_d(prime, 2) == MP_EQ) return MP_VAL; /* prime must be odd */
30 if ((res = mp_jacobi(n, prime, &legendre)) != MP_OKAY) return res;
31 if (legendre == -1) return MP_VAL; /* quadratic non-residue mod prime */
32
33 if ((res = mp_init_multi(&t1, &C, &Q, &S, &Z, &M, &T, &R, &two, NULL)) != MP_OKAY) {
34 return res;
35 }
36
37 /* SPECIAL CASE: if prime mod 4 == 3
38 * compute directly: res = n^(prime+1)/4 mod prime
39 * Handbook of Applied Cryptography algorithm 3.36
40 */
41 if ((res = mp_mod_d(prime, 4, &i)) != MP_OKAY) goto cleanup;
42 if (i == 3) {
43 if ((res = mp_add_d(prime, 1, &t1)) != MP_OKAY) goto cleanup;
44 if ((res = mp_div_2(&t1, &t1)) != MP_OKAY) goto cleanup;
45 if ((res = mp_div_2(&t1, &t1)) != MP_OKAY) goto cleanup;
46 if ((res = mp_exptmod(n, &t1, prime, ret)) != MP_OKAY) goto cleanup;
47 res = MP_OKAY;
48 goto cleanup;
49 }
50
51 /* NOW: Tonelli-Shanks algorithm */
52
53 /* factor out powers of 2 from prime-1, defining Q and S as: prime-1 = Q*2^S */
54 if ((res = mp_copy(prime, &Q)) != MP_OKAY) goto cleanup;
55 if ((res = mp_sub_d(&Q, 1, &Q)) != MP_OKAY) goto cleanup;
56 /* Q = prime - 1 */
57 mp_zero(&S);
58 /* S = 0 */
59 while (mp_iseven(&Q) != MP_NO) {
60 if ((res = mp_div_2(&Q, &Q)) != MP_OKAY) goto cleanup;
61 /* Q = Q / 2 */
62 if ((res = mp_add_d(&S, 1, &S)) != MP_OKAY) goto cleanup;
63 /* S = S + 1 */
64 }
65
66 /* find a Z such that the Legendre symbol (Z|prime) == -1 */
67 if ((res = mp_set_int(&Z, 2)) != MP_OKAY) goto cleanup;
68 /* Z = 2 */
69 while(1) {
70 if ((res = mp_jacobi(&Z, prime, &legendre)) != MP_OKAY) goto cleanup;
71 if (legendre == -1) break;
72 if ((res = mp_add_d(&Z, 1, &Z)) != MP_OKAY) goto cleanup;
73 /* Z = Z + 1 */
74 }
75
76 if ((res = mp_exptmod(&Z, &Q, prime, &C)) != MP_OKAY) goto cleanup;
77 /* C = Z ^ Q mod prime */
78 if ((res = mp_add_d(&Q, 1, &t1)) != MP_OKAY) goto cleanup;
79 if ((res = mp_div_2(&t1, &t1)) != MP_OKAY) goto cleanup;
80 /* t1 = (Q + 1) / 2 */
81 if ((res = mp_exptmod(n, &t1, prime, &R)) != MP_OKAY) goto cleanup;
82 /* R = n ^ ((Q + 1) / 2) mod prime */
83 if ((res = mp_exptmod(n, &Q, prime, &T)) != MP_OKAY) goto cleanup;
84 /* T = n ^ Q mod prime */
85 if ((res = mp_copy(&S, &M)) != MP_OKAY) goto cleanup;
86 /* M = S */
87 if ((res = mp_set_int(&two, 2)) != MP_OKAY) goto cleanup;
88
89 res = MP_VAL;
90 while (1) {
91 if ((res = mp_copy(&T, &t1)) != MP_OKAY) goto cleanup;
92 i = 0;
93 while (1) {
94 if (mp_cmp_d(&t1, 1) == MP_EQ) break;
95 if ((res = mp_exptmod(&t1, &two, prime, &t1)) != MP_OKAY) goto cleanup;
96 i++;
97 }
98 if (i == 0) {
99 if ((res = mp_copy(&R, ret)) != MP_OKAY) goto cleanup;
100 res = MP_OKAY;
101 goto cleanup;
102 }
103 if ((res = mp_sub_d(&M, i, &t1)) != MP_OKAY) goto cleanup;
104 if ((res = mp_sub_d(&t1, 1, &t1)) != MP_OKAY) goto cleanup;
105 if ((res = mp_exptmod(&two, &t1, prime, &t1)) != MP_OKAY) goto cleanup;
106 /* t1 = 2 ^ (M - i - 1) */
107 if ((res = mp_exptmod(&C, &t1, prime, &t1)) != MP_OKAY) goto cleanup;
108 /* t1 = C ^ (2 ^ (M - i - 1)) mod prime */
109 if ((res = mp_sqrmod(&t1, prime, &C)) != MP_OKAY) goto cleanup;
110 /* C = (t1 * t1) mod prime */
111 if ((res = mp_mulmod(&R, &t1, prime, &R)) != MP_OKAY) goto cleanup;
112 /* R = (R * t1) mod prime */
113 if ((res = mp_mulmod(&T, &C, prime, &T)) != MP_OKAY) goto cleanup;
114 /* T = (T * C) mod prime */
115 mp_set(&M, i);
116 /* M = i */
117 }
118
119 cleanup:
120 mp_clear_multi(&t1, &C, &Q, &S, &Z, &M, &T, &R, &two, NULL);
121 return res;
122 }
123
124 #endif