+++ /dev/null
-/* -*-c-*-
- *
- * $Id$
- *
- * Compute Jacobi symbol
- *
- * (c) 1999 Straylight/Edgeware
- */
-
-/*----- Licensing notice --------------------------------------------------*
- *
- * This file is part of Catacomb.
- *
- * Catacomb is free software; you can redistribute it and/or modify
- * it under the terms of the GNU Library General Public License as
- * published by the Free Software Foundation; either version 2 of the
- * License, or (at your option) any later version.
- *
- * Catacomb is distributed in the hope that it will be useful,
- * but WITHOUT ANY WARRANTY; without even the implied warranty of
- * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
- * GNU Library General Public License for more details.
- *
- * You should have received a copy of the GNU Library General Public
- * License along with Catacomb; if not, write to the Free
- * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
- * MA 02111-1307, USA.
- */
-
-/*----- Header files ------------------------------------------------------*/
-
-#include "mp.h"
-
-/*----- Main code ---------------------------------------------------------*/
-
-/* --- @mp_jacobi@ --- *
- *
- * Arguments: @mp *a@ = an integer
- * @mp *n@ = another integer
- *
- * Returns: @-1@, @0@ or @1@ -- the Jacobi symbol %$J(a, n)$%.
- *
- * Use: Computes the Kronecker symbol %$\jacobi{a}{n}$%. If @n@ is
- * prime, this is the Legendre symbol and is equal to 1 if and
- * only if @a@ is a quadratic residue mod @n@. The result is
- * zero if and only if @a@ and @n@ have a common factor greater
- * than one.
- *
- * If @n@ is composite, then this computes the Kronecker symbol
- *
- * %$\jacobi{a}{n}=\jacobi{a}{u}\prod_i\jacobi{a}{p_i}^{e_i}$%
- *
- * where %$n = u p_0^{e_0} \ldots p_{n-1}^{e_{n-1}}$% is the
- * prime factorization of %$n$%. The missing bits are:
- *
- * * %$\jacobi{a}{1} = 1$%;
- * * %$\jacobi{a}{-1} = 1$% if @a@ is negative, or 1 if
- * positive;
- * * %$\jacobi{a}{0} = 0$%;
- * * %$\jacobi{a}{2}$ is 0 if @a@ is even, 1 if @a@ is
- * congruent to 1 or 7 (mod 8), or %$-1$% otherwise.
- *
- * If %$n$% is positive and odd, then this is the Jacobi
- * symbol. (The Kronecker symbol is a consistant domain
- * extension; the Jacobi symbol was implemented first, and the
- * name stuck.)
- */
-
-int mp_jacobi(mp *a, mp *n)
-{
- int s = 1;
- size_t p2;
-
- /* --- Handle zero specially --- *
- *
- * I can't find any specific statement for what to do when %$n = 0$%; PARI
- * opts to set %$\jacobi{\pm1}{0} = \pm 1$% and %$\jacobi{a}{0} = 0$% for
- * other %$a$%.
- */
-
- if (MP_ZEROP(n)) {
- if (MP_EQ(a, MP_ONE)) return (+1);
- else if (MP_EQ(a, MP_MONE)) return (-1);
- else return (0);
- }
-
- /* --- Deal with powers of two --- *
- *
- * This implicitly takes a copy of %$n$%. Copy %$a$% at the same time to
- * make cleanup easier.
- */
-
- MP_COPY(a);
- n = mp_odd(MP_NEW, n, &p2);
- if (p2) {
- if (MP_EVENP(a)) {
- s = 0;
- goto done;
- } else if ((p2 & 1) && ((a->v[0] & 7) == 3 || (a->v[0] & 7) == 5))
- s = -s;
- }
-
- /* --- Deal with negative %$n$% --- */
-
- if (MP_NEGP(n)) {
- n = mp_neg(n, n);
- if (MP_NEGP(a))
- s = -s;
- }
-
- /* --- Check for unit %$n$% --- */
-
- if (MP_EQ(n, MP_ONE))
- goto done;
-
- /* --- Reduce %$a$% modulo %$n$% --- */
-
- if (MP_NEGP(a) || MP_CMP(a, >=, n))
- mp_div(0, &a, a, n);
-
- /* --- Main recursive mess, flattened out into something nice --- */
-
- for (;;) {
- mpw nn;
- size_t e;
-
- /* --- Some simple special cases --- */
-
- MP_SHRINK(a);
- if (MP_ZEROP(a)) {
- s = 0;
- goto done;
- }
-
- /* --- Main case with powers of two --- */
-
- a = mp_odd(a, a, &e);
- nn = n->v[0] & 7;
- if ((e & 1) && (nn == 3 || nn == 5))
- s = -s;
- if (MP_LEN(a) == 1 && a->v[0] == 1)
- goto done;
- if ((nn & 3) == 3 && (a->v[0] & 3) == 3)
- s = -s;
-
- /* --- Reduce and swap --- */
-
- mp_div(0, &n, n, a);
- { mp *t = n; n = a; a = t; }
- }
-
- /* --- Wrap everything up --- */
-
-done:
- MP_DROP(a);
- MP_DROP(n);
- return (s);
-}
-
-/*----- Test rig ----------------------------------------------------------*/
-
-#ifdef TEST_RIG
-
-#include <mLib/testrig.h>
-
-static int verify(dstr *v)
-{
- mp *a = *(mp **)v[0].buf;
- mp *n = *(mp **)v[1].buf;
- int s = *(int *)v[2].buf;
- int j = mp_jacobi(a, n);
- int ok = 1;
-
- if (s != j) {
- fputs("\n*** fail", stderr);
- fputs("a = ", stderr); mp_writefile(a, stderr, 10); fputc('\n', stderr);
- fputs("n = ", stderr); mp_writefile(n, stderr, 10); fputc('\n', stderr);
- fprintf(stderr, "s = %i\n", s);
- fprintf(stderr, "j = %i\n", j);
- ok = 0;
- }
-
- mp_drop(a);
- mp_drop(n);
- assert(mparena_count(MPARENA_GLOBAL) == 0);
- return (ok);
-}
-
-static test_chunk tests[] = {
- { "jacobi", verify, { &type_mp, &type_mp, &type_int, 0 } },
- { 0, 0, { 0 } }
-};
-
-int main(int argc, char *argv[])
-{
- sub_init();
- test_run(argc, argv, tests, SRCDIR "/tests/mp");
- return (0);
-}
-
-#endif
-
-/*----- That's all, folks -------------------------------------------------*/
projects / u / mdw / catacomb / blobdiff