X-Git-Url: https://git.distorted.org.uk/u/mdw/catacomb/blobdiff_plain/ba6e6b64033b1f9de49feccb5c9cd438354481f7..0f00dc4c8eb47e67bc0f148c2dd109f73a451e0a:/math/mp-jacobi.c

diff --git a/math/mp-jacobi.c b/math/mp-jacobi.c
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+/* -*-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 -------------------------------------------------*/