9f11b970 |
1 | /* -*-c-*- |
2 | * |
3 | * $Id: mp-modsqrt.c,v 1.1 2000/06/22 19:01:31 mdw Exp $ |
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 $ |
33 | * Revision 1.1 2000/06/22 19:01:31 mdw |
34 | * Compute square roots in a prime field. |
35 | * |
36 | */ |
37 | |
38 | /*----- Header files ------------------------------------------------------*/ |
39 | |
40 | #include "fibrand.h" |
41 | #include "grand.h" |
42 | #include "mp.h" |
43 | #include "mpmont.h" |
44 | #include "mprand.h" |
45 | |
46 | /*----- Main code ---------------------------------------------------------*/ |
47 | |
48 | /* --- @mp_modsqrt@ --- * |
49 | * |
50 | * Arguments: @mp *d@ = destination integer |
51 | * @mp *a@ = source integer |
52 | * @mp *p@ = modulus (must be prime) |
53 | * |
54 | * Returns: If %$a$% is a quadratic residue, a square root of %$a$%; else |
55 | * a null pointer. |
56 | * |
57 | * Use: Returns an integer %$x$% such that %$x^2 \equiv a \pmod{p}$%, |
58 | * if one exists; else a null pointer. This function will not |
59 | * work if %$p$% is composite: you must factor the modulus, take |
60 | * a square root mod each factor, and recombine the results |
61 | * using the Chinese Remainder Theorem. |
62 | */ |
63 | |
64 | mp *mp_modsqrt(mp *d, mp *a, mp *p) |
65 | { |
66 | mpmont mm; |
67 | mp *t; |
68 | size_t s; |
69 | mp *b; |
70 | mp *ainv; |
71 | mp *c, *r; |
72 | size_t i, j; |
73 | mp *dd, *mone; |
74 | |
75 | /* --- Cope if %$a \not\in Q_p$% --- */ |
76 | |
77 | if (mp_jacobi(a, p) != 1) { |
78 | if (d) |
79 | mp_drop(d); |
80 | return (0); |
81 | } |
82 | |
83 | /* --- Choose some quadratic non-residue --- */ |
84 | |
85 | { |
86 | grand *g = fibrand_create(0); |
87 | |
88 | b = MP_NEW; |
89 | do |
90 | b = mprand_range(b, p, g, 0); |
91 | while (mp_jacobi(b, p) != -1); |
92 | g->ops->destroy(g); |
93 | } |
94 | |
95 | /* --- Find the inverse of %$a$% --- */ |
96 | |
97 | ainv = MP_NEW; |
98 | mp_gcd(0, &ainv, 0, a, p); |
99 | |
100 | /* --- Split %$p - 1$% into a power of two and an odd number --- */ |
101 | |
102 | t = mp_sub(MP_NEW, p, MP_ONE); |
103 | t = mp_odd(t, t, &s); |
104 | |
105 | /* --- Now to really get going --- */ |
106 | |
107 | mpmont_create(&mm, p); |
108 | c = mpmont_expr(&mm, b, b, t); |
109 | t = mp_add(t, t, MP_ONE); |
110 | t = mp_lsr(t, t, 1); |
111 | r = mpmont_expr(&mm, t, a, t); |
112 | ainv = mpmont_mul(&mm, ainv, ainv, mm.r2); |
113 | |
114 | mone = mp_sub(MP_NEW, p, mm.r); |
115 | |
116 | dd = MP_NEW; |
117 | |
118 | for (i = 1; i < s; i++) { |
119 | |
120 | /* --- Compute %$d_0 = r^2a^{-1}$% --- */ |
121 | |
122 | dd = mp_sqr(dd, r); |
123 | dd = mpmont_reduce(&mm, dd, dd); |
124 | dd = mpmont_mul(&mm, dd, dd, ainv); |
125 | |
126 | /* --- Now %$d = d_0^{s - i - 1}$% --- */ |
127 | |
128 | for (j = i; j < s - 1; j++) { |
129 | dd = mp_sqr(dd, dd); |
130 | dd = mpmont_reduce(&mm, dd, dd); |
131 | } |
132 | |
133 | /* --- Fiddle at the end --- */ |
134 | |
135 | if (MP_CMP(dd, ==, mone)) |
136 | r = mpmont_mul(&mm, r, r, c); |
137 | c = mp_sqr(c, c); |
138 | c = mpmont_reduce(&mm, c, c); |
139 | } |
140 | |
141 | /* --- Done, so tidy up --- */ |
142 | |
143 | d = mpmont_reduce(&mm, d, r); |
144 | mp_drop(ainv); |
145 | mp_drop(r); mp_drop(c); |
146 | if (dd) |
147 | mp_drop(dd); |
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; |
170 | else if (MP_CMP(r, ==, rr)) |
171 | ok = 1; |
172 | else { |
173 | r = mp_sub(r, p, r); |
174 | if (MP_CMP(r, ==, rr)) |
175 | ok = 1; |
176 | } |
177 | |
178 | if (!ok) { |
179 | fputs("\n*** fail\n", stderr); |
180 | fputs("a = ", stderr); mp_writefile(a, stderr, 10); fputc('\n', stderr); |
181 | fputs("p = ", stderr); mp_writefile(p, stderr, 10); fputc('\n', stderr); |
182 | if (r) { |
183 | fputs("r = ", stderr); |
184 | mp_writefile(r, stderr, 10); |
185 | fputc('\n', stderr); |
186 | } else |
187 | fputs("r = <undef>\n", stderr); |
188 | fputs("rr = ", stderr); mp_writefile(rr, stderr, 10); fputc('\n', stderr); |
189 | ok = 0; |
190 | } |
191 | |
192 | mp_drop(a); |
193 | mp_drop(p); |
194 | if (r) |
195 | mp_drop(r); |
196 | mp_drop(rr); |
197 | assert(mparena_count(MPARENA_GLOBAL) == 0); |
198 | return (ok); |
199 | } |
200 | |
201 | static test_chunk tests[] = { |
202 | { "modsqrt", verify, { &type_mp, &type_mp, &type_mp, 0 } }, |
203 | { 0, 0, { 0 } } |
204 | }; |
205 | |
206 | int main(int argc, char *argv[]) |
207 | { |
208 | sub_init(); |
209 | test_run(argc, argv, tests, SRCDIR "/tests/mp"); |
210 | return (0); |
211 | } |
212 | |
213 | #endif |
214 | |
215 | /*----- That's all, folks -------------------------------------------------*/ |