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/*=============================================================================
This file is part of FLINT.
FLINT is free software; you can redistribute it and/or modify
it under the terms of the GNU General Public License as published by
the Free Software Foundation; either version 2 of the License, or
(at your option) any later version.
FLINT 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 General Public License for more details.
You should have received a copy of the GNU General Public License
along with FLINT; if not, write to the Free Software
Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
=============================================================================*/
/******************************************************************************
Copyright (C) 2006, 2011 William Hart
******************************************************************************/
#include <stdio.h>
#include <gmp.h>
#include <math.h>
#include "flint.h"
#include "ulong_extras.h"
#include "longlong.h"
#include "qsieve.h"
/* Array of possible Knuth-Schroeppel multipliers */
static const mp_limb_t multipliers[] = {1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15,
17, 19, 21, 22, 23, 26, 29, 30, 31,
33, 34, 35, 37, 38, 41, 42, 43, 47};
/* Number of possible Knuth-Schroeppel multipliers */
#define KS_MULTIPLIERS (sizeof(multipliers)/sizeof(mp_limb_t))
/*
Try to compute a multiplier k such that there are a lot of small primes
which are quadratic residues modulo kn. If a small weight of n is found
during this process it is returned.
*/
mp_limb_t qsieve_ll_knuth_schroeppel(qs_t qs_inf)
{
float weights[KS_MULTIPLIERS]; /* array of Knuth-Schroeppel weights */
float best_weight = -10.0f; /* best weight so far */
ulong i;
ulong num_primes, max;
float logpdivp;
mp_limb_t nmod8, mod8, p, nmod, pinv, mult;
int kron, jac;
if ((qs_inf->lo & 1) == 0) /* check 2 is not a factor */
return 2;
/* initialise weights for each multiplier k depending on kn mod 8 */
nmod8 = qs_inf->lo % 8; /* n modulo 8 */
for (i = 0; i < KS_MULTIPLIERS; i++)
{
mod8 = ((nmod8*multipliers[i]) % 8); /* kn modulo 8 */
weights[i] = 0.34657359; /* ln2/2 */
if (mod8 == 1) weights[i] *= 4.0;
if (mod8 == 5) weights[i] *= 2.0;
weights[i] -= (log((float) multipliers[i]) / 2.0);
}
/*
maximum number of primes to try
may not exceed number of factor base primes (recall k and 2 are factor base primes)
*/
max = FLINT_MIN(qs_inf->ks_primes, qs_inf->num_primes - 2);
#if QS_DEBUG
flint_printf("Checking %wd Knuth-Schroeppel primes\n", max);
#endif
p = 3;
for (num_primes = 0; num_primes < max; num_primes++)
{
pinv = n_preinvert_limb(p); /* compute precomputed inverse */
logpdivp = log((float) p) / (float) p; /* log p / p */
nmod = n_ll_mod_preinv(qs_inf->hi, qs_inf->lo, p, pinv);
if (nmod == 0) return p; /* we found a small factor */
kron = 1; /* n mod p is even, not handled by n_jacobi */
while ((nmod % 2) == 0)
{
if ((p % 8) == 3 || (p % 8) == 5) kron *= -1;
nmod /= 2;
}
kron *= n_jacobi(nmod, p);
for (i = 0; i < KS_MULTIPLIERS; i++)
{
mult = multipliers[i];
if (mult >= p)
mult = n_mod2_preinv(mult, p, pinv); /* k mod p */
if (mult == 0) weights[i] += logpdivp; /* kn == 0 mod p */
else
{
jac = 1;
while ((mult % 2) == 0) /* k mod p is even, not handled by n_jacobi */
{
if ((p % 8) == 3 || (p % 8) == 5) jac *= -1;
mult /= 2;
}
if (kron*jac*n_jacobi(mult, p) == 1) /* kn is a square mod p */
weights[i] += 2.0*logpdivp;
}
}
p = n_nextprime(p, 0);
}
/* search for the multiplier with the best weight and set qs_inf->k */
for (i = 0; i < KS_MULTIPLIERS; i++)
{
if (weights[i] > best_weight)
{
best_weight = weights[i];
qs_inf->k = multipliers[i];
}
}
#if QS_DEBUG
flint_printf("Using multiplier %wd\n", qs_inf->k);
#endif
return 0; /* we didn't find any small factors */
}
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