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1 | /* -*-c-*- |
2 | * |
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3 | * $Id: mp-modsqrt.c,v 1.5 2004/04/08 01:36:15 mdw Exp $ |
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4 | * |
5 | * Compute square roots modulo a prime |
6 | * |
7 | * (c) 2000 Straylight/Edgeware |
8 | */ |
9 | |
10 | /*----- Licensing notice --------------------------------------------------* |
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. |
18 | * |
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. |
23 | * |
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 | |
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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. |
54 | */ |
55 | |
56 | mp *mp_modsqrt(mp *d, mp *a, mp *p) |
57 | { |
58 | mpmont mm; |
59 | mp *t; |
60 | size_t s; |
61 | mp *b; |
62 | mp *ainv; |
63 | mp *c, *r; |
64 | size_t i, j; |
65 | mp *dd, *mone; |
66 | |
67 | /* --- Cope if %$a \not\in Q_p$% --- */ |
68 | |
69 | if (mp_jacobi(a, p) != 1) { |
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70 | mp_drop(d); |
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71 | return (0); |
72 | } |
73 | |
74 | /* --- Choose some quadratic non-residue --- */ |
75 | |
76 | { |
77 | grand *g = fibrand_create(0); |
78 | |
79 | b = MP_NEW; |
80 | do |
81 | b = mprand_range(b, p, g, 0); |
82 | while (mp_jacobi(b, p) != -1); |
83 | g->ops->destroy(g); |
84 | } |
85 | |
86 | /* --- Find the inverse of %$a$% --- */ |
87 | |
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88 | ainv = mp_modinv(MP_NEW, a, p); |
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89 | |
90 | /* --- Split %$p - 1$% into a power of two and an odd number --- */ |
91 | |
92 | t = mp_sub(MP_NEW, p, MP_ONE); |
93 | t = mp_odd(t, t, &s); |
94 | |
95 | /* --- Now to really get going --- */ |
96 | |
97 | mpmont_create(&mm, p); |
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98 | b = mpmont_mul(&mm, b, b, mm.r2); |
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99 | c = mpmont_expr(&mm, b, b, t); |
100 | t = mp_add(t, t, MP_ONE); |
101 | t = mp_lsr(t, t, 1); |
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102 | dd = mpmont_mul(&mm, MP_NEW, a, mm.r2); |
103 | r = mpmont_expr(&mm, t, dd, t); |
104 | mp_drop(dd); |
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105 | ainv = mpmont_mul(&mm, ainv, ainv, mm.r2); |
106 | |
107 | mone = mp_sub(MP_NEW, p, mm.r); |
108 | |
109 | dd = MP_NEW; |
110 | |
111 | for (i = 1; i < s; i++) { |
112 | |
113 | /* --- Compute %$d_0 = r^2a^{-1}$% --- */ |
114 | |
115 | dd = mp_sqr(dd, r); |
116 | dd = mpmont_reduce(&mm, dd, dd); |
117 | dd = mpmont_mul(&mm, dd, dd, ainv); |
118 | |
119 | /* --- Now %$d = d_0^{s - i - 1}$% --- */ |
120 | |
121 | for (j = i; j < s - 1; j++) { |
122 | dd = mp_sqr(dd, dd); |
123 | dd = mpmont_reduce(&mm, dd, dd); |
124 | } |
125 | |
126 | /* --- Fiddle at the end --- */ |
127 | |
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128 | if (MP_EQ(dd, mone)) |
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129 | r = mpmont_mul(&mm, r, r, c); |
130 | c = mp_sqr(c, c); |
131 | c = mpmont_reduce(&mm, c, c); |
132 | } |
133 | |
134 | /* --- Done, so tidy up --- */ |
135 | |
136 | d = mpmont_reduce(&mm, d, r); |
137 | mp_drop(ainv); |
138 | mp_drop(r); mp_drop(c); |
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139 | mp_drop(dd); |
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140 | mp_drop(mone); |
141 | mpmont_destroy(&mm); |
142 | |
143 | return (d); |
144 | } |
145 | |
146 | /*----- Test rig ----------------------------------------------------------*/ |
147 | |
148 | #ifdef TEST_RIG |
149 | |
150 | #include <mLib/testrig.h> |
151 | |
152 | static int verify(dstr *v) |
153 | { |
154 | mp *a = *(mp **)v[0].buf; |
155 | mp *p = *(mp **)v[1].buf; |
156 | mp *rr = *(mp **)v[2].buf; |
157 | mp *r = mp_modsqrt(MP_NEW, a, p); |
158 | int ok = 0; |
159 | |
160 | if (!r) |
161 | ok = 0; |
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162 | else if (MP_EQ(r, rr)) |
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163 | ok = 1; |
164 | else { |
165 | r = mp_sub(r, p, r); |
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166 | if (MP_EQ(r, rr)) |
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167 | ok = 1; |
168 | } |
169 | |
170 | if (!ok) { |
171 | fputs("\n*** fail\n", stderr); |
172 | fputs("a = ", stderr); mp_writefile(a, stderr, 10); fputc('\n', stderr); |
173 | fputs("p = ", stderr); mp_writefile(p, stderr, 10); fputc('\n', stderr); |
174 | if (r) { |
175 | fputs("r = ", stderr); |
176 | mp_writefile(r, stderr, 10); |
177 | fputc('\n', stderr); |
178 | } else |
179 | fputs("r = <undef>\n", stderr); |
180 | fputs("rr = ", stderr); mp_writefile(rr, stderr, 10); fputc('\n', stderr); |
181 | ok = 0; |
182 | } |
183 | |
184 | mp_drop(a); |
185 | mp_drop(p); |
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186 | mp_drop(r); |
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187 | mp_drop(rr); |
188 | assert(mparena_count(MPARENA_GLOBAL) == 0); |
189 | return (ok); |
190 | } |
191 | |
192 | static test_chunk tests[] = { |
193 | { "modsqrt", verify, { &type_mp, &type_mp, &type_mp, 0 } }, |
194 | { 0, 0, { 0 } } |
195 | }; |
196 | |
197 | int main(int argc, char *argv[]) |
198 | { |
199 | sub_init(); |
200 | test_run(argc, argv, tests, SRCDIR "/tests/mp"); |
201 | return (0); |
202 | } |
203 | |
204 | #endif |
205 | |
206 | /*----- That's all, folks -------------------------------------------------*/ |