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9f11b970 | 1 | /* -*-c-*- |
2 | * | |
b817bfc6 | 3 | * $Id: mp-modsqrt.c,v 1.5 2004/04/08 01:36:15 mdw Exp $ |
9f11b970 | 4 | * |
5 | * Compute square roots modulo a prime | |
6 | * | |
7 | * (c) 2000 Straylight/Edgeware | |
8 | */ | |
9 | ||
45c0fd36 | 10 | /*----- Licensing notice --------------------------------------------------* |
9f11b970 | 11 | * |
12 | * This file is part of Catacomb. | |
13 | * | |
14 | * Catacomb is free software; you can redistribute it and/or modify | |
15 | * it under the terms of the GNU Library General Public License as | |
16 | * published by the Free Software Foundation; either version 2 of the | |
17 | * License, or (at your option) any later version. | |
45c0fd36 | 18 | * |
9f11b970 | 19 | * Catacomb is distributed in the hope that it will be useful, |
20 | * but WITHOUT ANY WARRANTY; without even the implied warranty of | |
21 | * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the | |
22 | * GNU Library General Public License for more details. | |
45c0fd36 | 23 | * |
9f11b970 | 24 | * You should have received a copy of the GNU Library General Public |
25 | * License along with Catacomb; if not, write to the Free | |
26 | * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, | |
27 | * MA 02111-1307, USA. | |
28 | */ | |
29 | ||
9f11b970 | 30 | /*----- Header files ------------------------------------------------------*/ |
31 | ||
32 | #include "fibrand.h" | |
33 | #include "grand.h" | |
34 | #include "mp.h" | |
35 | #include "mpmont.h" | |
36 | #include "mprand.h" | |
37 | ||
38 | /*----- Main code ---------------------------------------------------------*/ | |
39 | ||
40 | /* --- @mp_modsqrt@ --- * | |
41 | * | |
42 | * Arguments: @mp *d@ = destination integer | |
43 | * @mp *a@ = source integer | |
44 | * @mp *p@ = modulus (must be prime) | |
45 | * | |
46 | * Returns: If %$a$% is a quadratic residue, a square root of %$a$%; else | |
47 | * a null pointer. | |
48 | * | |
49 | * Use: Returns an integer %$x$% such that %$x^2 \equiv a \pmod{p}$%, | |
50 | * if one exists; else a null pointer. This function will not | |
51 | * work if %$p$% is composite: you must factor the modulus, take | |
52 | * a square root mod each factor, and recombine the results | |
53 | * using the Chinese Remainder Theorem. | |
222c8a43 MW |
54 | * |
55 | * We guarantee that the square root returned is the smallest | |
56 | * one (i.e., the `positive' square root). | |
9f11b970 | 57 | */ |
58 | ||
59 | mp *mp_modsqrt(mp *d, mp *a, mp *p) | |
60 | { | |
61 | mpmont mm; | |
62 | mp *t; | |
63 | size_t s; | |
64 | mp *b; | |
65 | mp *ainv; | |
66 | mp *c, *r; | |
67 | size_t i, j; | |
68 | mp *dd, *mone; | |
69 | ||
70 | /* --- Cope if %$a \not\in Q_p$% --- */ | |
71 | ||
72 | if (mp_jacobi(a, p) != 1) { | |
f1140c41 | 73 | mp_drop(d); |
9f11b970 | 74 | return (0); |
75 | } | |
76 | ||
77 | /* --- Choose some quadratic non-residue --- */ | |
78 | ||
79 | { | |
80 | grand *g = fibrand_create(0); | |
81 | ||
82 | b = MP_NEW; | |
83 | do | |
84 | b = mprand_range(b, p, g, 0); | |
85 | while (mp_jacobi(b, p) != -1); | |
86 | g->ops->destroy(g); | |
87 | } | |
88 | ||
89 | /* --- Find the inverse of %$a$% --- */ | |
90 | ||
b817bfc6 | 91 | ainv = mp_modinv(MP_NEW, a, p); |
45c0fd36 | 92 | |
9f11b970 | 93 | /* --- Split %$p - 1$% into a power of two and an odd number --- */ |
94 | ||
95 | t = mp_sub(MP_NEW, p, MP_ONE); | |
96 | t = mp_odd(t, t, &s); | |
97 | ||
98 | /* --- Now to really get going --- */ | |
99 | ||
100 | mpmont_create(&mm, p); | |
b0b682aa | 101 | b = mpmont_mul(&mm, b, b, mm.r2); |
9f11b970 | 102 | c = mpmont_expr(&mm, b, b, t); |
103 | t = mp_add(t, t, MP_ONE); | |
104 | t = mp_lsr(t, t, 1); | |
b0b682aa | 105 | dd = mpmont_mul(&mm, MP_NEW, a, mm.r2); |
106 | r = mpmont_expr(&mm, t, dd, t); | |
107 | mp_drop(dd); | |
9f11b970 | 108 | ainv = mpmont_mul(&mm, ainv, ainv, mm.r2); |
109 | ||
110 | mone = mp_sub(MP_NEW, p, mm.r); | |
111 | ||
112 | dd = MP_NEW; | |
113 | ||
114 | for (i = 1; i < s; i++) { | |
115 | ||
116 | /* --- Compute %$d_0 = r^2a^{-1}$% --- */ | |
117 | ||
118 | dd = mp_sqr(dd, r); | |
119 | dd = mpmont_reduce(&mm, dd, dd); | |
120 | dd = mpmont_mul(&mm, dd, dd, ainv); | |
121 | ||
78ec50fa | 122 | /* --- Now %$d = d_0^{2^{s - i - 1}}$% --- */ |
9f11b970 | 123 | |
124 | for (j = i; j < s - 1; j++) { | |
125 | dd = mp_sqr(dd, dd); | |
126 | dd = mpmont_reduce(&mm, dd, dd); | |
127 | } | |
128 | ||
129 | /* --- Fiddle at the end --- */ | |
130 | ||
4b536f42 | 131 | if (MP_EQ(dd, mone)) |
9f11b970 | 132 | r = mpmont_mul(&mm, r, r, c); |
133 | c = mp_sqr(c, c); | |
134 | c = mpmont_reduce(&mm, c, c); | |
135 | } | |
136 | ||
222c8a43 MW |
137 | /* --- Done, so tidy up --- * |
138 | * | |
139 | * Canonify the answer. | |
140 | */ | |
9f11b970 | 141 | |
142 | d = mpmont_reduce(&mm, d, r); | |
222c8a43 MW |
143 | r = mp_sub(r, p, d); |
144 | if (MP_CMP(r, <, d)) { mp *tt = r; r = d; d = tt; } | |
9f11b970 | 145 | mp_drop(ainv); |
146 | mp_drop(r); mp_drop(c); | |
f1140c41 | 147 | mp_drop(dd); |
9f11b970 | 148 | mp_drop(mone); |
149 | mpmont_destroy(&mm); | |
150 | ||
151 | return (d); | |
152 | } | |
153 | ||
154 | /*----- Test rig ----------------------------------------------------------*/ | |
155 | ||
156 | #ifdef TEST_RIG | |
157 | ||
158 | #include <mLib/testrig.h> | |
159 | ||
160 | static int verify(dstr *v) | |
161 | { | |
162 | mp *a = *(mp **)v[0].buf; | |
163 | mp *p = *(mp **)v[1].buf; | |
164 | mp *rr = *(mp **)v[2].buf; | |
165 | mp *r = mp_modsqrt(MP_NEW, a, p); | |
166 | int ok = 0; | |
167 | ||
168 | if (!r) | |
169 | ok = 0; | |
4b536f42 | 170 | else if (MP_EQ(r, rr)) |
9f11b970 | 171 | ok = 1; |
9f11b970 | 172 | |
173 | if (!ok) { | |
174 | fputs("\n*** fail\n", stderr); | |
175 | fputs("a = ", stderr); mp_writefile(a, stderr, 10); fputc('\n', stderr); | |
176 | fputs("p = ", stderr); mp_writefile(p, stderr, 10); fputc('\n', stderr); | |
177 | if (r) { | |
45c0fd36 | 178 | fputs("r = ", stderr); |
9f11b970 | 179 | mp_writefile(r, stderr, 10); |
180 | fputc('\n', stderr); | |
181 | } else | |
45c0fd36 | 182 | fputs("r = <undef>\n", stderr); |
9f11b970 | 183 | fputs("rr = ", stderr); mp_writefile(rr, stderr, 10); fputc('\n', stderr); |
184 | ok = 0; | |
185 | } | |
186 | ||
187 | mp_drop(a); | |
188 | mp_drop(p); | |
f1140c41 | 189 | mp_drop(r); |
9f11b970 | 190 | mp_drop(rr); |
191 | assert(mparena_count(MPARENA_GLOBAL) == 0); | |
192 | return (ok); | |
193 | } | |
194 | ||
195 | static test_chunk tests[] = { | |
196 | { "modsqrt", verify, { &type_mp, &type_mp, &type_mp, 0 } }, | |
197 | { 0, 0, { 0 } } | |
198 | }; | |
199 | ||
200 | int main(int argc, char *argv[]) | |
201 | { | |
202 | sub_init(); | |
203 | test_run(argc, argv, tests, SRCDIR "/tests/mp"); | |
204 | return (0); | |
205 | } | |
206 | ||
207 | #endif | |
208 | ||
209 | /*----- That's all, folks -------------------------------------------------*/ |