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1 | /* -*-c-*- |
2 | * |
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3 | * $Id: mp-modsqrt.c,v 1.3 2001/02/03 12:00:29 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 | |
30 | /*----- Revision history --------------------------------------------------* |
31 | * |
32 | * $Log: mp-modsqrt.c,v $ |
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33 | * Revision 1.3 2001/02/03 12:00:29 mdw |
34 | * Now @mp_drop@ checks its argument is non-NULL before attempting to free |
35 | * it. Note that the macro version @MP_DROP@ doesn't do this. |
36 | * |
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37 | * Revision 1.2 2000/10/08 12:02:21 mdw |
38 | * Use @MP_EQ@ instead of @MP_CMP@. |
39 | * |
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40 | * Revision 1.1 2000/06/22 19:01:31 mdw |
41 | * Compute square roots in a prime field. |
42 | * |
43 | */ |
44 | |
45 | /*----- Header files ------------------------------------------------------*/ |
46 | |
47 | #include "fibrand.h" |
48 | #include "grand.h" |
49 | #include "mp.h" |
50 | #include "mpmont.h" |
51 | #include "mprand.h" |
52 | |
53 | /*----- Main code ---------------------------------------------------------*/ |
54 | |
55 | /* --- @mp_modsqrt@ --- * |
56 | * |
57 | * Arguments: @mp *d@ = destination integer |
58 | * @mp *a@ = source integer |
59 | * @mp *p@ = modulus (must be prime) |
60 | * |
61 | * Returns: If %$a$% is a quadratic residue, a square root of %$a$%; else |
62 | * a null pointer. |
63 | * |
64 | * Use: Returns an integer %$x$% such that %$x^2 \equiv a \pmod{p}$%, |
65 | * if one exists; else a null pointer. This function will not |
66 | * work if %$p$% is composite: you must factor the modulus, take |
67 | * a square root mod each factor, and recombine the results |
68 | * using the Chinese Remainder Theorem. |
69 | */ |
70 | |
71 | mp *mp_modsqrt(mp *d, mp *a, mp *p) |
72 | { |
73 | mpmont mm; |
74 | mp *t; |
75 | size_t s; |
76 | mp *b; |
77 | mp *ainv; |
78 | mp *c, *r; |
79 | size_t i, j; |
80 | mp *dd, *mone; |
81 | |
82 | /* --- Cope if %$a \not\in Q_p$% --- */ |
83 | |
84 | if (mp_jacobi(a, p) != 1) { |
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85 | mp_drop(d); |
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86 | return (0); |
87 | } |
88 | |
89 | /* --- Choose some quadratic non-residue --- */ |
90 | |
91 | { |
92 | grand *g = fibrand_create(0); |
93 | |
94 | b = MP_NEW; |
95 | do |
96 | b = mprand_range(b, p, g, 0); |
97 | while (mp_jacobi(b, p) != -1); |
98 | g->ops->destroy(g); |
99 | } |
100 | |
101 | /* --- Find the inverse of %$a$% --- */ |
102 | |
103 | ainv = MP_NEW; |
104 | mp_gcd(0, &ainv, 0, a, p); |
105 | |
106 | /* --- Split %$p - 1$% into a power of two and an odd number --- */ |
107 | |
108 | t = mp_sub(MP_NEW, p, MP_ONE); |
109 | t = mp_odd(t, t, &s); |
110 | |
111 | /* --- Now to really get going --- */ |
112 | |
113 | mpmont_create(&mm, p); |
114 | c = mpmont_expr(&mm, b, b, t); |
115 | t = mp_add(t, t, MP_ONE); |
116 | t = mp_lsr(t, t, 1); |
117 | r = mpmont_expr(&mm, t, a, t); |
118 | ainv = mpmont_mul(&mm, ainv, ainv, mm.r2); |
119 | |
120 | mone = mp_sub(MP_NEW, p, mm.r); |
121 | |
122 | dd = MP_NEW; |
123 | |
124 | for (i = 1; i < s; i++) { |
125 | |
126 | /* --- Compute %$d_0 = r^2a^{-1}$% --- */ |
127 | |
128 | dd = mp_sqr(dd, r); |
129 | dd = mpmont_reduce(&mm, dd, dd); |
130 | dd = mpmont_mul(&mm, dd, dd, ainv); |
131 | |
132 | /* --- Now %$d = d_0^{s - i - 1}$% --- */ |
133 | |
134 | for (j = i; j < s - 1; j++) { |
135 | dd = mp_sqr(dd, dd); |
136 | dd = mpmont_reduce(&mm, dd, dd); |
137 | } |
138 | |
139 | /* --- Fiddle at the end --- */ |
140 | |
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141 | if (MP_EQ(dd, mone)) |
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142 | r = mpmont_mul(&mm, r, r, c); |
143 | c = mp_sqr(c, c); |
144 | c = mpmont_reduce(&mm, c, c); |
145 | } |
146 | |
147 | /* --- Done, so tidy up --- */ |
148 | |
149 | d = mpmont_reduce(&mm, d, r); |
150 | mp_drop(ainv); |
151 | mp_drop(r); mp_drop(c); |
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152 | mp_drop(dd); |
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153 | mp_drop(mone); |
154 | mpmont_destroy(&mm); |
155 | |
156 | return (d); |
157 | } |
158 | |
159 | /*----- Test rig ----------------------------------------------------------*/ |
160 | |
161 | #ifdef TEST_RIG |
162 | |
163 | #include <mLib/testrig.h> |
164 | |
165 | static int verify(dstr *v) |
166 | { |
167 | mp *a = *(mp **)v[0].buf; |
168 | mp *p = *(mp **)v[1].buf; |
169 | mp *rr = *(mp **)v[2].buf; |
170 | mp *r = mp_modsqrt(MP_NEW, a, p); |
171 | int ok = 0; |
172 | |
173 | if (!r) |
174 | ok = 0; |
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175 | else if (MP_EQ(r, rr)) |
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176 | ok = 1; |
177 | else { |
178 | r = mp_sub(r, p, r); |
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179 | if (MP_EQ(r, rr)) |
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180 | ok = 1; |
181 | } |
182 | |
183 | if (!ok) { |
184 | fputs("\n*** fail\n", stderr); |
185 | fputs("a = ", stderr); mp_writefile(a, stderr, 10); fputc('\n', stderr); |
186 | fputs("p = ", stderr); mp_writefile(p, stderr, 10); fputc('\n', stderr); |
187 | if (r) { |
188 | fputs("r = ", stderr); |
189 | mp_writefile(r, stderr, 10); |
190 | fputc('\n', stderr); |
191 | } else |
192 | fputs("r = <undef>\n", stderr); |
193 | fputs("rr = ", stderr); mp_writefile(rr, stderr, 10); fputc('\n', stderr); |
194 | ok = 0; |
195 | } |
196 | |
197 | mp_drop(a); |
198 | mp_drop(p); |
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199 | mp_drop(r); |
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200 | mp_drop(rr); |
201 | assert(mparena_count(MPARENA_GLOBAL) == 0); |
202 | return (ok); |
203 | } |
204 | |
205 | static test_chunk tests[] = { |
206 | { "modsqrt", verify, { &type_mp, &type_mp, &type_mp, 0 } }, |
207 | { 0, 0, { 0 } } |
208 | }; |
209 | |
210 | int main(int argc, char *argv[]) |
211 | { |
212 | sub_init(); |
213 | test_run(argc, argv, tests, SRCDIR "/tests/mp"); |
214 | return (0); |
215 | } |
216 | |
217 | #endif |
218 | |
219 | /*----- That's all, folks -------------------------------------------------*/ |