| 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 -------------------------------------------------*/ |