--- /dev/null
+/* -*-c-*-
+ *
+ * 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 "/t/mp");
+ return (0);
+}
+
+#endif
+
+/*----- That's all, folks -------------------------------------------------*/
projects / u / mdw / catacomb / blobdiff