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/* mpz_legendre (op1, op2).
Contributed by Bennet Yee (bsy) at Carnegie-Mellon University
Copyright (C) 1992, 1996 Free Software Foundation, Inc.
This file is part of the GNU MP Library.
The GNU MP Library 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.
The GNU MP Library 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 the GNU MP Library; see the file COPYING.LIB. If not, write to
the Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
MA 02111-1307, USA. */
#include "gmp.h"
#if defined (DEBUG)
#include <stdio.h>
#endif
/* Precondition: both p and q are positive */
int
#if __STDC__
mpz_legendre (mpz_srcptr pi, mpz_srcptr qi)
#else
mpz_legendre (pi, qi)
mpz_srcptr pi, qi;
#endif
{
mpz_t p, q, qdiv2;
#ifdef Q_MINUS_1
mpz_t q_minus_1;
#endif
mpz_ptr mtmp;
register mpz_ptr pptr, qptr;
register int retval = 1;
register unsigned long int s;
pptr = p;
mpz_init_set (pptr, pi);
qptr = q;
mpz_init_set (qptr, qi);
#ifdef Q_MINUS_1
mpz_init (q_minus_1);
#endif
mpz_init (qdiv2);
tail_recurse2:
#ifdef DEBUG
printf ("tail_recurse2: p=");
mpz_out_str (stdout, 10, pptr);
printf ("\nq=");
mpz_out_str (stdout, 10, qptr);
putchar ('\n');
#endif
s = mpz_scan1 (qptr, 0);
if (s) mpz_tdiv_q_2exp (qptr, qptr, s); /* J(a,2) = 1 */
#ifdef DEBUG
printf ("2 factor decomposition: p=");
mpz_out_str (stdout, 10, pptr);
printf ("\nq=");
mpz_out_str (stdout, 10, qptr);
putchar ('\n');
#endif
/* postcondition q odd */
if (!mpz_cmp_ui (qptr, 1L)) /* J(a,1) = 1 */
goto done;
mpz_mod (pptr, pptr, qptr); /* J(a,q) = J(b,q) when a == b mod q */
#ifdef DEBUG
printf ("mod out by q: p=");
mpz_out_str (stdout, 10, pptr);
printf ("\nq=");
mpz_out_str (stdout, 10, qptr);
putchar ('\n');
#endif
/* quick calculation to get approximate size first */
/* precondition: p < q */
if ((mpz_sizeinbase (pptr, 2) + 1 >= mpz_sizeinbase (qptr,2))
&& (mpz_tdiv_q_2exp (qdiv2, qptr, 1L), mpz_cmp (pptr, qdiv2) > 0))
{
/* p > q/2 */
mpz_sub (pptr, qptr, pptr);
/* J(-1,q) = (-1)^((q-1)/2), q odd */
if (mpz_get_ui (qptr) & 2)
retval = -retval;
}
/* p < q/2 */
#ifdef Q_MINUS_1
mpz_sub_ui (q_minus_q, qptr, 1L);
#endif
tail_recurse: /* we use tail_recurse only if q has not changed */
#ifdef DEBUG
printf ("tail_recurse1: p=");
mpz_out_str (stdout, 10, pptr);
printf ("\nq=");
mpz_out_str (stdout, 10, qptr);
putchar ('\n');
#endif
/*
* J(0,q) = 0
* this occurs only if gcd(p,q) != 1 which is never true for
* Legendre function.
*/
if (!mpz_cmp_ui (pptr, 0L))
{
retval = 0;
goto done;
}
if (!mpz_cmp_ui (pptr, 1L))
{
/* J(1,q) = 1 */
/* retval *= 1; */
goto done;
}
#ifdef Q_MINUS_1
if (!mpz_cmp (pptr, q_minus_1))
{
/* J(-1,q) = (-1)^((q-1)/2) */
if (mpz_get_ui (qptr) & 2)
retval = -retval;
/* else retval *= 1; */
goto done;
}
#endif
/*
* we do not handle J(xy,q) except for x==2
* since we do not want to factor
*/
if ((s = mpz_scan1 (pptr, 0)) != 0)
{
/*
* J(2,q) = (-1)^((q^2-1)/8)
*
* Note that q odd guarantees that q^2-1 is divisible by 8:
* Let a: q=2a+1. q^2 = 4a^2+4a+1, (q^2-1)/8 = a(a+1)/2, qed
*
* Now, note that this means that the low two bits of _a_
* (or the low bits of q shifted over by 1 determines
* the factor).
*/
mpz_tdiv_q_2exp (pptr, pptr, s);
/* even powers of 2 gives J(2,q)^{2n} = 1 */
if (s & 1)
{
s = mpz_get_ui (qptr) >> 1;
s = s * (s + 1);
if (s & 2)
retval = -retval;
}
goto tail_recurse;
}
/*
* we know p is odd since we have cast out 2s
* precondition that q is odd guarantees both odd.
*
* quadratic reciprocity
* J(p,q) = (-1)^((p-1)(q-1)/4) * J(q,p)
*/
if ((s = mpz_scan1 (pptr, 1)) <= 2 && (s + mpz_scan1 (qptr, 1)) <= 2)
retval = -retval;
mtmp = pptr; pptr = qptr; qptr = mtmp;
goto tail_recurse2;
done:
mpz_clear (p);
mpz_clear (q);
mpz_clear (qdiv2);
#ifdef Q_MINUS_1
mpz_clear (q_minus_1);
#endif
return retval;
}
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