/* -*-c-*-
*
- * $Id: mp-jacobi.c,v 1.2 1999/12/10 23:19:02 mdw Exp $
+ * $Id$
*
* Compute Jacobi symbol
*
* (c) 1999 Straylight/Edgeware
*/
-/*----- Licensing notice --------------------------------------------------*
+/*----- Licensing notice --------------------------------------------------*
*
* This file is part of Catacomb.
*
* 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.
*/
-/*----- Revision history --------------------------------------------------*
- *
- * $Log: mp-jacobi.c,v $
- * Revision 1.2 1999/12/10 23:19:02 mdw
- * Improve error-checking.
- *
- * Revision 1.1 1999/11/22 20:50:37 mdw
- * Add support for computing Jacobi symbols.
- *
- */
-
/*----- Header files ------------------------------------------------------*/
#include "mp.h"
/* --- @mp_jacobi@ --- *
*
- * Arguments: @mp *a@ = an integer less than @n@
- * @mp *n@ = an odd integer
+ * Arguments: @mp *a@ = an integer
+ * @mp *n@ = another integer
*
* Returns: @-1@, @0@ or @1@ -- the Jacobi symbol %$J(a, n)$%.
*
- * Use: Computes the Jacobi symbol. 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.
+ * 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);
+ }
- /* --- Take copies of the arguments --- */
-
- a = MP_COPY(a);
- n = MP_COPY(n);
-
- /* --- Main recursive mess, flattened out into something nice --- */
-
- for (;;) {
-
- /* --- Some simple special cases --- */
-
- MP_SHRINK(a);
+ /* --- Deal with powers of two --- *
+ *
+ * This implicitly takes a copy of %$n$%. Copy %$a$% at the same time to
+ * make cleanup easier.
+ */
- if (MP_LEN(a) == 0) {
+ 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;
+ }
- /* --- Find the power-of-two factor in @a@ --- */
+ /* --- Deal with negative %$n$% --- */
- {
- mpscan sc;
- mpw nn;
- unsigned e;
+ if (MP_NEGP(n)) {
+ n = mp_neg(n, n);
+ if (MP_NEGP(a))
+ s = -s;
+ }
- /* --- Scan for a set bit --- */
+ /* --- Check for unit %$n$% --- */
- MP_SCAN(&sc, a);
- e = 0;
- while (MP_STEP(&sc) && !MP_BIT(&sc))
- e++;
+ if (MP_EQ(n, MP_ONE))
+ goto done;
- /* --- Do the shift --- */
+ /* --- Reduce %$a$% modulo %$n$% --- */
- if (e)
- a = mp_lsr(a, a, e);
+ if (MP_NEGP(a) || MP_CMP(a, >=, n))
+ mp_div(0, &a, a, n);
- /* --- Maybe adjust the sign of @s@ --- */
+ /* --- Main recursive mess, flattened out into something nice --- */
- nn = n->v[0] & 7;
- if ((e & 1) && (nn == 3 || nn == 5))
- s = -s;
+ for (;;) {
+ mpw nn;
+ size_t e;
- if (MP_LEN(a) == 1 && a->v[0] == 1)
- goto done;
+ /* --- Some simple special cases --- */
- if ((nn & 3) == 3 && (a->v[0] & 3) == 3)
- s = -s;
+ 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);
projects / u / mdw / catacomb / blobdiff