| 1 | /* -*-c-*- |
| 2 | * |
| 3 | * Compute square roots modulo a prime |
| 4 | * |
| 5 | * (c) 2000 Straylight/Edgeware |
| 6 | */ |
| 7 | |
| 8 | /*----- Licensing notice --------------------------------------------------* |
| 9 | * |
| 10 | * This file is part of Catacomb. |
| 11 | * |
| 12 | * Catacomb is free software; you can redistribute it and/or modify |
| 13 | * it under the terms of the GNU Library General Public License as |
| 14 | * published by the Free Software Foundation; either version 2 of the |
| 15 | * License, or (at your option) any later version. |
| 16 | * |
| 17 | * Catacomb is distributed in the hope that it will be useful, |
| 18 | * but WITHOUT ANY WARRANTY; without even the implied warranty of |
| 19 | * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the |
| 20 | * GNU Library General Public License for more details. |
| 21 | * |
| 22 | * You should have received a copy of the GNU Library General Public |
| 23 | * License along with Catacomb; if not, write to the Free |
| 24 | * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, |
| 25 | * MA 02111-1307, USA. |
| 26 | */ |
| 27 | |
| 28 | /*----- Header files ------------------------------------------------------*/ |
| 29 | |
| 30 | #include "fibrand.h" |
| 31 | #include "grand.h" |
| 32 | #include "mp.h" |
| 33 | #include "mpmont.h" |
| 34 | #include "mprand.h" |
| 35 | |
| 36 | /*----- Main code ---------------------------------------------------------*/ |
| 37 | |
| 38 | /* --- @mp_modsqrt@ --- * |
| 39 | * |
| 40 | * Arguments: @mp *d@ = destination integer |
| 41 | * @mp *a@ = source integer |
| 42 | * @mp *p@ = modulus (must be prime) |
| 43 | * |
| 44 | * Returns: If %$a$% is a quadratic residue, a square root of %$a$%; else |
| 45 | * a null pointer. |
| 46 | * |
| 47 | * Use: Returns an integer %$x$% such that %$x^2 \equiv a \pmod{p}$%, |
| 48 | * if one exists; else a null pointer. This function will not |
| 49 | * work if %$p$% is composite: you must factor the modulus, take |
| 50 | * a square root mod each factor, and recombine the results |
| 51 | * using the Chinese Remainder Theorem. |
| 52 | * |
| 53 | * We guarantee that the square root returned is the smallest |
| 54 | * one (i.e., the `positive' square root). |
| 55 | */ |
| 56 | |
| 57 | mp *mp_modsqrt(mp *d, mp *a, mp *p) |
| 58 | { |
| 59 | mpmont mm; |
| 60 | size_t i, s; |
| 61 | mp *b, *c; |
| 62 | mp *ainv; |
| 63 | mp *r, *A, *aa; |
| 64 | mp *t; |
| 65 | grand *gr; |
| 66 | |
| 67 | /* --- Cope if %$a \not\in Q_p$% --- */ |
| 68 | |
| 69 | if (mp_jacobi(a, p) != 1) { |
| 70 | mp_drop(d); |
| 71 | return (0); |
| 72 | } |
| 73 | |
| 74 | /* --- Choose some quadratic non-residue --- */ |
| 75 | |
| 76 | gr = fibrand_create(0); |
| 77 | b = MP_NEW; |
| 78 | do b = mprand_range(b, p, gr, 0); while (mp_jacobi(b, p) != -1); |
| 79 | gr->ops->destroy(gr); |
| 80 | |
| 81 | /* --- Some initial setup --- */ |
| 82 | |
| 83 | mpmont_create(&mm, p); |
| 84 | ainv = mp_modinv(MP_NEW, a, p); /* %$a^{-1} \bmod p$% */ |
| 85 | ainv = mpmont_mul(&mm, ainv, ainv, mm.r2); |
| 86 | t = mp_sub(MP_NEW, p, MP_ONE); |
| 87 | t = mp_odd(t, t, &s); /* %$2^s t = p - 1$% */ |
| 88 | b = mpmont_mul(&mm, b, b, mm.r2); |
| 89 | c = mpmont_expr(&mm, b, b, t); /* %$b^t \bmod p$% */ |
| 90 | t = mp_add(t, t, MP_ONE); |
| 91 | t = mp_lsr(t, t, 1); /* %$(t + 1)/2$% */ |
| 92 | a = mpmont_mul(&mm, MP_NEW, a, mm.r2); |
| 93 | r = mpmont_expr(&mm, a, a, t); /* %$a^{(t+1)/2} \bmod p$% */ |
| 94 | |
| 95 | /* --- Now for the main loop --- * |
| 96 | * |
| 97 | * Let %$g = c^{-2}$%; we know that %$g$% is a generator of the order- |
| 98 | * %$2^{s-1}$% subgroup mod %$p$%. We also know that %$A = a^t = r^2/a$% |
| 99 | * is an element of this group. If we can determine %$m$% such that |
| 100 | * %$g^m = A$% then %$a^{(t+1)/2}/g^{m/2} = r c^m$% is the square root we |
| 101 | * seek. |
| 102 | * |
| 103 | * Write %$m = m_0 + 2 m'$%. Then %$A^{2^{s-1}} = g^{m_0 2^{s-1}}$%, which |
| 104 | * is %$1$% if %$m_0 = 0$% or %$-1$% if %$m_0 = 1$% (modulo %$p$%). Then |
| 105 | * %$A/g^{m_0} = (g^2)^{m'}$% and we can proceed inductively. The end |
| 106 | * result will me %$A/g^m$%. |
| 107 | * |
| 108 | * Note that this loop keeps track of (what will be) %$r c^m$%, since this |
| 109 | * is the result we want, and computes $A/g^m = r^2/a$% on demand. |
| 110 | */ |
| 111 | |
| 112 | A = mp_sqr(t, r); A = mpmont_reduce(&mm, A, A); |
| 113 | A = mpmont_mul(&mm, A, A, ainv); /* %$x^t/g^m$% */ |
| 114 | |
| 115 | while (s-- > 1) { |
| 116 | aa = MP_COPY(A); |
| 117 | for (i = 1; i < s; i++) |
| 118 | { aa = mp_sqr(aa, aa); aa = mpmont_reduce(&mm, aa, aa); } |
| 119 | if (!MP_EQ(aa, mm.r)) { |
| 120 | r = mpmont_mul(&mm, r, r, c); |
| 121 | A = mp_sqr(A, r); A = mpmont_reduce(&mm, A, A); |
| 122 | A = mpmont_mul(&mm, A, A, ainv); /* %$x^t/g^m$% */ |
| 123 | } |
| 124 | c = mp_sqr(c, c); c = mpmont_reduce(&mm, c, c); |
| 125 | MP_DROP(aa); |
| 126 | } |
| 127 | |
| 128 | /* --- We want the smaller square root --- */ |
| 129 | |
| 130 | d = mpmont_reduce(&mm, d, r); |
| 131 | r = mp_sub(r, p, d); |
| 132 | if (MP_CMP(r, <, d)) { mp *tt = r; r = d; d = tt; } |
| 133 | |
| 134 | /* --- Clear away all the temporaries --- */ |
| 135 | |
| 136 | mp_drop(ainv); |
| 137 | mp_drop(r); mp_drop(c); |
| 138 | mp_drop(A); |
| 139 | mpmont_destroy(&mm); |
| 140 | |
| 141 | /* --- Done --- */ |
| 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; |
| 162 | else if (MP_EQ(r, rr)) |
| 163 | ok = 1; |
| 164 | |
| 165 | if (!ok) { |
| 166 | fputs("\n*** fail\n", stderr); |
| 167 | fputs("a = ", stderr); mp_writefile(a, stderr, 10); fputc('\n', stderr); |
| 168 | fputs("p = ", stderr); mp_writefile(p, stderr, 10); fputc('\n', stderr); |
| 169 | if (r) { |
| 170 | fputs("r = ", stderr); |
| 171 | mp_writefile(r, stderr, 10); |
| 172 | fputc('\n', stderr); |
| 173 | } else |
| 174 | fputs("r = <undef>\n", stderr); |
| 175 | fputs("rr = ", stderr); mp_writefile(rr, stderr, 10); fputc('\n', stderr); |
| 176 | ok = 0; |
| 177 | } |
| 178 | |
| 179 | mp_drop(a); |
| 180 | mp_drop(p); |
| 181 | mp_drop(r); |
| 182 | mp_drop(rr); |
| 183 | assert(mparena_count(MPARENA_GLOBAL) == 0); |
| 184 | return (ok); |
| 185 | } |
| 186 | |
| 187 | static test_chunk tests[] = { |
| 188 | { "modsqrt", verify, { &type_mp, &type_mp, &type_mp, 0 } }, |
| 189 | { 0, 0, { 0 } } |
| 190 | }; |
| 191 | |
| 192 | int main(int argc, char *argv[]) |
| 193 | { |
| 194 | sub_init(); |
| 195 | test_run(argc, argv, tests, SRCDIR "/t/mp"); |
| 196 | return (0); |
| 197 | } |
| 198 | |
| 199 | #endif |
| 200 | |
| 201 | /*----- That's all, folks -------------------------------------------------*/ |