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1 | /* -*-c-*- |
2 | * |
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3 | * $Id: mp-modsqrt.c,v 1.4 2001/06/16 12:56:38 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 1.4 2001/06/16 12:56:38 mdw |
34 | * Fixes for interface change to @mpmont_expr@ and @mpmont_mexpr@. |
35 | * |
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36 | * Revision 1.3 2001/02/03 12:00:29 mdw |
37 | * Now @mp_drop@ checks its argument is non-NULL before attempting to free |
38 | * it. Note that the macro version @MP_DROP@ doesn't do this. |
39 | * |
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40 | * Revision 1.2 2000/10/08 12:02:21 mdw |
41 | * Use @MP_EQ@ instead of @MP_CMP@. |
42 | * |
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43 | * Revision 1.1 2000/06/22 19:01:31 mdw |
44 | * Compute square roots in a prime field. |
45 | * |
46 | */ |
47 | |
48 | /*----- Header files ------------------------------------------------------*/ |
49 | |
50 | #include "fibrand.h" |
51 | #include "grand.h" |
52 | #include "mp.h" |
53 | #include "mpmont.h" |
54 | #include "mprand.h" |
55 | |
56 | /*----- Main code ---------------------------------------------------------*/ |
57 | |
58 | /* --- @mp_modsqrt@ --- * |
59 | * |
60 | * Arguments: @mp *d@ = destination integer |
61 | * @mp *a@ = source integer |
62 | * @mp *p@ = modulus (must be prime) |
63 | * |
64 | * Returns: If %$a$% is a quadratic residue, a square root of %$a$%; else |
65 | * a null pointer. |
66 | * |
67 | * Use: Returns an integer %$x$% such that %$x^2 \equiv a \pmod{p}$%, |
68 | * if one exists; else a null pointer. This function will not |
69 | * work if %$p$% is composite: you must factor the modulus, take |
70 | * a square root mod each factor, and recombine the results |
71 | * using the Chinese Remainder Theorem. |
72 | */ |
73 | |
74 | mp *mp_modsqrt(mp *d, mp *a, mp *p) |
75 | { |
76 | mpmont mm; |
77 | mp *t; |
78 | size_t s; |
79 | mp *b; |
80 | mp *ainv; |
81 | mp *c, *r; |
82 | size_t i, j; |
83 | mp *dd, *mone; |
84 | |
85 | /* --- Cope if %$a \not\in Q_p$% --- */ |
86 | |
87 | if (mp_jacobi(a, p) != 1) { |
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88 | mp_drop(d); |
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89 | return (0); |
90 | } |
91 | |
92 | /* --- Choose some quadratic non-residue --- */ |
93 | |
94 | { |
95 | grand *g = fibrand_create(0); |
96 | |
97 | b = MP_NEW; |
98 | do |
99 | b = mprand_range(b, p, g, 0); |
100 | while (mp_jacobi(b, p) != -1); |
101 | g->ops->destroy(g); |
102 | } |
103 | |
104 | /* --- Find the inverse of %$a$% --- */ |
105 | |
106 | ainv = MP_NEW; |
107 | mp_gcd(0, &ainv, 0, a, p); |
108 | |
109 | /* --- Split %$p - 1$% into a power of two and an odd number --- */ |
110 | |
111 | t = mp_sub(MP_NEW, p, MP_ONE); |
112 | t = mp_odd(t, t, &s); |
113 | |
114 | /* --- Now to really get going --- */ |
115 | |
116 | mpmont_create(&mm, p); |
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117 | b = mpmont_mul(&mm, b, b, mm.r2); |
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118 | c = mpmont_expr(&mm, b, b, t); |
119 | t = mp_add(t, t, MP_ONE); |
120 | t = mp_lsr(t, t, 1); |
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121 | dd = mpmont_mul(&mm, MP_NEW, a, mm.r2); |
122 | r = mpmont_expr(&mm, t, dd, t); |
123 | mp_drop(dd); |
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124 | ainv = mpmont_mul(&mm, ainv, ainv, mm.r2); |
125 | |
126 | mone = mp_sub(MP_NEW, p, mm.r); |
127 | |
128 | dd = MP_NEW; |
129 | |
130 | for (i = 1; i < s; i++) { |
131 | |
132 | /* --- Compute %$d_0 = r^2a^{-1}$% --- */ |
133 | |
134 | dd = mp_sqr(dd, r); |
135 | dd = mpmont_reduce(&mm, dd, dd); |
136 | dd = mpmont_mul(&mm, dd, dd, ainv); |
137 | |
138 | /* --- Now %$d = d_0^{s - i - 1}$% --- */ |
139 | |
140 | for (j = i; j < s - 1; j++) { |
141 | dd = mp_sqr(dd, dd); |
142 | dd = mpmont_reduce(&mm, dd, dd); |
143 | } |
144 | |
145 | /* --- Fiddle at the end --- */ |
146 | |
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147 | if (MP_EQ(dd, mone)) |
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148 | r = mpmont_mul(&mm, r, r, c); |
149 | c = mp_sqr(c, c); |
150 | c = mpmont_reduce(&mm, c, c); |
151 | } |
152 | |
153 | /* --- Done, so tidy up --- */ |
154 | |
155 | d = mpmont_reduce(&mm, d, r); |
156 | mp_drop(ainv); |
157 | mp_drop(r); mp_drop(c); |
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158 | mp_drop(dd); |
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159 | mp_drop(mone); |
160 | mpmont_destroy(&mm); |
161 | |
162 | return (d); |
163 | } |
164 | |
165 | /*----- Test rig ----------------------------------------------------------*/ |
166 | |
167 | #ifdef TEST_RIG |
168 | |
169 | #include <mLib/testrig.h> |
170 | |
171 | static int verify(dstr *v) |
172 | { |
173 | mp *a = *(mp **)v[0].buf; |
174 | mp *p = *(mp **)v[1].buf; |
175 | mp *rr = *(mp **)v[2].buf; |
176 | mp *r = mp_modsqrt(MP_NEW, a, p); |
177 | int ok = 0; |
178 | |
179 | if (!r) |
180 | ok = 0; |
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181 | else if (MP_EQ(r, rr)) |
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182 | ok = 1; |
183 | else { |
184 | r = mp_sub(r, p, r); |
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185 | if (MP_EQ(r, rr)) |
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186 | ok = 1; |
187 | } |
188 | |
189 | if (!ok) { |
190 | fputs("\n*** fail\n", stderr); |
191 | fputs("a = ", stderr); mp_writefile(a, stderr, 10); fputc('\n', stderr); |
192 | fputs("p = ", stderr); mp_writefile(p, stderr, 10); fputc('\n', stderr); |
193 | if (r) { |
194 | fputs("r = ", stderr); |
195 | mp_writefile(r, stderr, 10); |
196 | fputc('\n', stderr); |
197 | } else |
198 | fputs("r = <undef>\n", stderr); |
199 | fputs("rr = ", stderr); mp_writefile(rr, stderr, 10); fputc('\n', stderr); |
200 | ok = 0; |
201 | } |
202 | |
203 | mp_drop(a); |
204 | mp_drop(p); |
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205 | mp_drop(r); |
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206 | mp_drop(rr); |
207 | assert(mparena_count(MPARENA_GLOBAL) == 0); |
208 | return (ok); |
209 | } |
210 | |
211 | static test_chunk tests[] = { |
212 | { "modsqrt", verify, { &type_mp, &type_mp, &type_mp, 0 } }, |
213 | { 0, 0, { 0 } } |
214 | }; |
215 | |
216 | int main(int argc, char *argv[]) |
217 | { |
218 | sub_init(); |
219 | test_run(argc, argv, tests, SRCDIR "/tests/mp"); |
220 | return (0); |
221 | } |
222 | |
223 | #endif |
224 | |
225 | /*----- That's all, folks -------------------------------------------------*/ |