| 1 | /* -*-c-*- |
| 2 | * |
| 3 | * Compute Jacobi symbol |
| 4 | * |
| 5 | * (c) 1999 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 "mp.h" |
| 31 | |
| 32 | /*----- Main code ---------------------------------------------------------*/ |
| 33 | |
| 34 | /* --- @mp_jacobi@ --- * |
| 35 | * |
| 36 | * Arguments: @mp *a@ = an integer |
| 37 | * @mp *n@ = another integer |
| 38 | * |
| 39 | * Returns: @-1@, @0@ or @1@ -- the Jacobi symbol %$J(a, n)$%. |
| 40 | * |
| 41 | * Use: Computes the Kronecker symbol %$\jacobi{a}{n}$%. If @n@ is |
| 42 | * prime, this is the Legendre symbol and is equal to 1 if and |
| 43 | * only if @a@ is a quadratic residue mod @n@. The result is |
| 44 | * zero if and only if @a@ and @n@ have a common factor greater |
| 45 | * than one. |
| 46 | * |
| 47 | * If @n@ is composite, then this computes the Kronecker symbol |
| 48 | * |
| 49 | * %$\jacobi{a}{n}=\jacobi{a}{u}\prod_i\jacobi{a}{p_i}^{e_i}$% |
| 50 | * |
| 51 | * where %$n = u p_0^{e_0} \ldots p_{n-1}^{e_{n-1}}$% is the |
| 52 | * prime factorization of %$n$%. The missing bits are: |
| 53 | * |
| 54 | * * %$\jacobi{a}{1} = 1$%; |
| 55 | * * %$\jacobi{a}{-1} = 1$% if @a@ is negative, or 1 if |
| 56 | * positive; |
| 57 | * * %$\jacobi{a}{0} = 0$%; |
| 58 | * * %$\jacobi{a}{2}$ is 0 if @a@ is even, 1 if @a@ is |
| 59 | * congruent to 1 or 7 (mod 8), or %$-1$% otherwise. |
| 60 | * |
| 61 | * If %$n$% is positive and odd, then this is the Jacobi |
| 62 | * symbol. (The Kronecker symbol is a consistant domain |
| 63 | * extension; the Jacobi symbol was implemented first, and the |
| 64 | * name stuck.) |
| 65 | */ |
| 66 | |
| 67 | int mp_jacobi(mp *a, mp *n) |
| 68 | { |
| 69 | int s = 1; |
| 70 | size_t p2; |
| 71 | |
| 72 | /* --- Handle zero specially --- * |
| 73 | * |
| 74 | * I can't find any specific statement for what to do when %$n = 0$%; PARI |
| 75 | * opts to set %$\jacobi{\pm1}{0} = \pm 1$% and %$\jacobi{a}{0} = 0$% for |
| 76 | * other %$a$%. |
| 77 | */ |
| 78 | |
| 79 | if (MP_ZEROP(n)) { |
| 80 | if (MP_EQ(a, MP_ONE)) return (+1); |
| 81 | else if (MP_EQ(a, MP_MONE)) return (-1); |
| 82 | else return (0); |
| 83 | } |
| 84 | |
| 85 | /* --- Deal with powers of two --- * |
| 86 | * |
| 87 | * This implicitly takes a copy of %$n$%. Copy %$a$% at the same time to |
| 88 | * make cleanup easier. |
| 89 | */ |
| 90 | |
| 91 | MP_COPY(a); |
| 92 | n = mp_odd(MP_NEW, n, &p2); |
| 93 | if (p2) { |
| 94 | if (MP_EVENP(a)) { |
| 95 | s = 0; |
| 96 | goto done; |
| 97 | } else if ((p2 & 1) && ((a->v[0] & 7) == 3 || (a->v[0] & 7) == 5)) |
| 98 | s = -s; |
| 99 | } |
| 100 | |
| 101 | /* --- Deal with negative %$n$% --- */ |
| 102 | |
| 103 | if (MP_NEGP(n)) { |
| 104 | n = mp_neg(n, n); |
| 105 | if (MP_NEGP(a)) |
| 106 | s = -s; |
| 107 | } |
| 108 | |
| 109 | /* --- Check for unit %$n$% --- */ |
| 110 | |
| 111 | if (MP_EQ(n, MP_ONE)) |
| 112 | goto done; |
| 113 | |
| 114 | /* --- Reduce %$a$% modulo %$n$% --- */ |
| 115 | |
| 116 | if (MP_NEGP(a) || MP_CMP(a, >=, n)) |
| 117 | mp_div(0, &a, a, n); |
| 118 | |
| 119 | /* --- Main recursive mess, flattened out into something nice --- */ |
| 120 | |
| 121 | for (;;) { |
| 122 | mpw nn; |
| 123 | size_t e; |
| 124 | |
| 125 | /* --- Some simple special cases --- */ |
| 126 | |
| 127 | MP_SHRINK(a); |
| 128 | if (MP_ZEROP(a)) { |
| 129 | s = 0; |
| 130 | goto done; |
| 131 | } |
| 132 | |
| 133 | /* --- Main case with powers of two --- */ |
| 134 | |
| 135 | a = mp_odd(a, a, &e); |
| 136 | nn = n->v[0] & 7; |
| 137 | if ((e & 1) && (nn == 3 || nn == 5)) |
| 138 | s = -s; |
| 139 | if (MP_LEN(a) == 1 && a->v[0] == 1) |
| 140 | goto done; |
| 141 | if ((nn & 3) == 3 && (a->v[0] & 3) == 3) |
| 142 | s = -s; |
| 143 | |
| 144 | /* --- Reduce and swap --- */ |
| 145 | |
| 146 | mp_div(0, &n, n, a); |
| 147 | { mp *t = n; n = a; a = t; } |
| 148 | } |
| 149 | |
| 150 | /* --- Wrap everything up --- */ |
| 151 | |
| 152 | done: |
| 153 | MP_DROP(a); |
| 154 | MP_DROP(n); |
| 155 | return (s); |
| 156 | } |
| 157 | |
| 158 | /*----- Test rig ----------------------------------------------------------*/ |
| 159 | |
| 160 | #ifdef TEST_RIG |
| 161 | |
| 162 | #include <mLib/testrig.h> |
| 163 | |
| 164 | static int verify(dstr *v) |
| 165 | { |
| 166 | mp *a = *(mp **)v[0].buf; |
| 167 | mp *n = *(mp **)v[1].buf; |
| 168 | int s = *(int *)v[2].buf; |
| 169 | int j = mp_jacobi(a, n); |
| 170 | int ok = 1; |
| 171 | |
| 172 | if (s != j) { |
| 173 | fputs("\n*** fail", stderr); |
| 174 | fputs("a = ", stderr); mp_writefile(a, stderr, 10); fputc('\n', stderr); |
| 175 | fputs("n = ", stderr); mp_writefile(n, stderr, 10); fputc('\n', stderr); |
| 176 | fprintf(stderr, "s = %i\n", s); |
| 177 | fprintf(stderr, "j = %i\n", j); |
| 178 | ok = 0; |
| 179 | } |
| 180 | |
| 181 | mp_drop(a); |
| 182 | mp_drop(n); |
| 183 | assert(mparena_count(MPARENA_GLOBAL) == 0); |
| 184 | return (ok); |
| 185 | } |
| 186 | |
| 187 | static test_chunk tests[] = { |
| 188 | { "jacobi", verify, { &type_mp, &type_mp, &type_int, 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 -------------------------------------------------*/ |