Commit | Line | Data |
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9f11b970 | 1 | /* -*-c-*- |
2 | * | |
9f11b970 | 3 | * Compute square roots modulo a prime |
4 | * | |
5 | * (c) 2000 Straylight/Edgeware | |
6 | */ | |
7 | ||
45c0fd36 | 8 | /*----- Licensing notice --------------------------------------------------* |
9f11b970 | 9 | * |
10 | * This file is part of Catacomb. | |
11 | * | |
12 | * Catacomb is free software; you can redistribute it and/or modify | |
13 | * it under the terms of the GNU Library General Public License as | |
14 | * published by the Free Software Foundation; either version 2 of the | |
15 | * License, or (at your option) any later version. | |
45c0fd36 | 16 | * |
9f11b970 | 17 | * Catacomb is distributed in the hope that it will be useful, |
18 | * but WITHOUT ANY WARRANTY; without even the implied warranty of | |
19 | * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the | |
20 | * GNU Library General Public License for more details. | |
45c0fd36 | 21 | * |
9f11b970 | 22 | * You should have received a copy of the GNU Library General Public |
23 | * License along with Catacomb; if not, write to the Free | |
24 | * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, | |
25 | * MA 02111-1307, USA. | |
26 | */ | |
27 | ||
9f11b970 | 28 | /*----- Header files ------------------------------------------------------*/ |
29 | ||
30 | #include "fibrand.h" | |
31 | #include "grand.h" | |
32 | #include "mp.h" | |
33 | #include "mpmont.h" | |
34 | #include "mprand.h" | |
35 | ||
36 | /*----- Main code ---------------------------------------------------------*/ | |
37 | ||
38 | /* --- @mp_modsqrt@ --- * | |
39 | * | |
40 | * Arguments: @mp *d@ = destination integer | |
41 | * @mp *a@ = source integer | |
42 | * @mp *p@ = modulus (must be prime) | |
43 | * | |
44 | * Returns: If %$a$% is a quadratic residue, a square root of %$a$%; else | |
45 | * a null pointer. | |
46 | * | |
47 | * Use: Returns an integer %$x$% such that %$x^2 \equiv a \pmod{p}$%, | |
48 | * if one exists; else a null pointer. This function will not | |
49 | * work if %$p$% is composite: you must factor the modulus, take | |
50 | * a square root mod each factor, and recombine the results | |
51 | * using the Chinese Remainder Theorem. | |
222c8a43 MW |
52 | * |
53 | * We guarantee that the square root returned is the smallest | |
54 | * one (i.e., the `positive' square root). | |
9f11b970 | 55 | */ |
56 | ||
57 | mp *mp_modsqrt(mp *d, mp *a, mp *p) | |
58 | { | |
59 | mpmont mm; | |
60 | mp *t; | |
61 | size_t s; | |
62 | mp *b; | |
63 | mp *ainv; | |
64 | mp *c, *r; | |
65 | size_t i, j; | |
66 | mp *dd, *mone; | |
67 | ||
68 | /* --- Cope if %$a \not\in Q_p$% --- */ | |
69 | ||
70 | if (mp_jacobi(a, p) != 1) { | |
f1140c41 | 71 | mp_drop(d); |
9f11b970 | 72 | return (0); |
73 | } | |
74 | ||
75 | /* --- Choose some quadratic non-residue --- */ | |
76 | ||
77 | { | |
78 | grand *g = fibrand_create(0); | |
79 | ||
80 | b = MP_NEW; | |
81 | do | |
82 | b = mprand_range(b, p, g, 0); | |
83 | while (mp_jacobi(b, p) != -1); | |
84 | g->ops->destroy(g); | |
85 | } | |
86 | ||
87 | /* --- Find the inverse of %$a$% --- */ | |
88 | ||
b817bfc6 | 89 | ainv = mp_modinv(MP_NEW, a, p); |
45c0fd36 | 90 | |
9f11b970 | 91 | /* --- Split %$p - 1$% into a power of two and an odd number --- */ |
92 | ||
93 | t = mp_sub(MP_NEW, p, MP_ONE); | |
94 | t = mp_odd(t, t, &s); | |
95 | ||
96 | /* --- Now to really get going --- */ | |
97 | ||
98 | mpmont_create(&mm, p); | |
b0b682aa | 99 | b = mpmont_mul(&mm, b, b, mm.r2); |
9f11b970 | 100 | c = mpmont_expr(&mm, b, b, t); |
101 | t = mp_add(t, t, MP_ONE); | |
102 | t = mp_lsr(t, t, 1); | |
b0b682aa | 103 | dd = mpmont_mul(&mm, MP_NEW, a, mm.r2); |
104 | r = mpmont_expr(&mm, t, dd, t); | |
105 | mp_drop(dd); | |
9f11b970 | 106 | ainv = mpmont_mul(&mm, ainv, ainv, mm.r2); |
107 | ||
108 | mone = mp_sub(MP_NEW, p, mm.r); | |
109 | ||
110 | dd = MP_NEW; | |
111 | ||
112 | for (i = 1; i < s; i++) { | |
113 | ||
114 | /* --- Compute %$d_0 = r^2a^{-1}$% --- */ | |
115 | ||
116 | dd = mp_sqr(dd, r); | |
117 | dd = mpmont_reduce(&mm, dd, dd); | |
118 | dd = mpmont_mul(&mm, dd, dd, ainv); | |
119 | ||
78ec50fa | 120 | /* --- Now %$d = d_0^{2^{s - i - 1}}$% --- */ |
9f11b970 | 121 | |
122 | for (j = i; j < s - 1; j++) { | |
123 | dd = mp_sqr(dd, dd); | |
124 | dd = mpmont_reduce(&mm, dd, dd); | |
125 | } | |
126 | ||
127 | /* --- Fiddle at the end --- */ | |
128 | ||
4b536f42 | 129 | if (MP_EQ(dd, mone)) |
9f11b970 | 130 | r = mpmont_mul(&mm, r, r, c); |
131 | c = mp_sqr(c, c); | |
132 | c = mpmont_reduce(&mm, c, c); | |
133 | } | |
134 | ||
222c8a43 MW |
135 | /* --- Done, so tidy up --- * |
136 | * | |
137 | * Canonify the answer. | |
138 | */ | |
9f11b970 | 139 | |
140 | d = mpmont_reduce(&mm, d, r); | |
222c8a43 MW |
141 | r = mp_sub(r, p, d); |
142 | if (MP_CMP(r, <, d)) { mp *tt = r; r = d; d = tt; } | |
9f11b970 | 143 | mp_drop(ainv); |
144 | mp_drop(r); mp_drop(c); | |
f1140c41 | 145 | mp_drop(dd); |
9f11b970 | 146 | mp_drop(mone); |
147 | mpmont_destroy(&mm); | |
148 | ||
149 | return (d); | |
150 | } | |
151 | ||
152 | /*----- Test rig ----------------------------------------------------------*/ | |
153 | ||
154 | #ifdef TEST_RIG | |
155 | ||
156 | #include <mLib/testrig.h> | |
157 | ||
158 | static int verify(dstr *v) | |
159 | { | |
160 | mp *a = *(mp **)v[0].buf; | |
161 | mp *p = *(mp **)v[1].buf; | |
162 | mp *rr = *(mp **)v[2].buf; | |
163 | mp *r = mp_modsqrt(MP_NEW, a, p); | |
164 | int ok = 0; | |
165 | ||
166 | if (!r) | |
167 | ok = 0; | |
4b536f42 | 168 | else if (MP_EQ(r, rr)) |
9f11b970 | 169 | ok = 1; |
9f11b970 | 170 | |
171 | if (!ok) { | |
172 | fputs("\n*** fail\n", stderr); | |
173 | fputs("a = ", stderr); mp_writefile(a, stderr, 10); fputc('\n', stderr); | |
174 | fputs("p = ", stderr); mp_writefile(p, stderr, 10); fputc('\n', stderr); | |
175 | if (r) { | |
45c0fd36 | 176 | fputs("r = ", stderr); |
9f11b970 | 177 | mp_writefile(r, stderr, 10); |
178 | fputc('\n', stderr); | |
179 | } else | |
45c0fd36 | 180 | fputs("r = <undef>\n", stderr); |
9f11b970 | 181 | fputs("rr = ", stderr); mp_writefile(rr, stderr, 10); fputc('\n', stderr); |
182 | ok = 0; | |
183 | } | |
184 | ||
185 | mp_drop(a); | |
186 | mp_drop(p); | |
f1140c41 | 187 | mp_drop(r); |
9f11b970 | 188 | mp_drop(rr); |
189 | assert(mparena_count(MPARENA_GLOBAL) == 0); | |
190 | return (ok); | |
191 | } | |
192 | ||
193 | static test_chunk tests[] = { | |
194 | { "modsqrt", verify, { &type_mp, &type_mp, &type_mp, 0 } }, | |
195 | { 0, 0, { 0 } } | |
196 | }; | |
197 | ||
198 | int main(int argc, char *argv[]) | |
199 | { | |
200 | sub_init(); | |
0f00dc4c | 201 | test_run(argc, argv, tests, SRCDIR "/t/mp"); |
9f11b970 | 202 | return (0); |
203 | } | |
204 | ||
205 | #endif | |
206 | ||
207 | /*----- That's all, folks -------------------------------------------------*/ |