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mpir/gen-psqr.c

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/* Generate perfect square testing data.
Copyright 2002, 2003, 2004 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 Lesser General Public License as published by
the Free Software Foundation; either version 2.1 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 Lesser General Public
License for more details.
You should have received a copy of the GNU Lesser General Public License
along with the GNU MP Library; see the file COPYING.LIB. If not, write to
the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston,
MA 02110-1301, USA. */
#include <stdio.h>
#include <stdlib.h>
#include "dumbmp.c"
/* The aim of this program is to choose either mpn_mod_34lsub1 or mpn_mod_1
(plus a PERFSQR_PP modulus), and generate tables indicating quadratic
residues and non-residues modulo small factors of that modulus.
For the usual 32 or 64 bit cases mpn_mod_34lsub1 gets used. That
function exists specifically because 2^24-1 and 2^48-1 have nice sets of
prime factors. For other limb sizes it's considered, but if it doesn't
have good factors then mpn_mod_1 will be used instead.
When mpn_mod_1 is used, the modulus PERFSQR_PP is created from a
selection of small primes, chosen to fill PERFSQR_MOD_BITS of a limb,
with that bit count chosen so (2*GMP_LIMB_BITS)*2^PERFSQR_MOD_BITS <=
GMP_LIMB_MAX, allowing PERFSQR_MOD_IDX in mpn/generic/perfsqr.c to do its
calculation within a single limb.
In either case primes can be combined to make divisors. The table data
then effectively indicates remainders which are quadratic residues mod
all the primes. This sort of combining reduces the number of steps
needed after mpn_mod_34lsub1 or mpn_mod_1, saving code size and time.
Nothing is gained or lost in terms of detections, the same total fraction
of non-residues will be identified.
Nothing particularly sophisticated is attempted for combining factors to
make divisors. This is probably a kind of knapsack problem so it'd be
too hard to attempt anything completely general. For the usual 32 and 64
bit limbs we get a good enough result just pairing the biggest and
smallest which fit together, repeatedly.
Another aim is to get powerful combinations, ie. divisors which identify
biggest fraction of non-residues, and have those run first. Again for
the usual 32 and 64 bits it seems good enough just to pair for big
divisors then sort according to the resulting fraction of non-residues
identified.
Also in this program, a table sq_res_0x100 of residues modulo 256 is
generated. This simply fills bits into limbs of the appropriate
build-time GMP_LIMB_BITS each.
*/
/* Normally we aren't using const in gen*.c programs, so as not to have to
bother figuring out if it works, but using it with f_cmp_divisor and
f_cmp_fraction avoids warnings from the qsort calls. */
/* Same tests as gmp.h. */
#if defined (__STDC__) \
|| defined (__cplusplus) \
|| defined (_AIX) \
|| defined (__DECC) \
|| (defined (__mips) && defined (_SYSTYPE_SVR4)) \
|| defined (_MSC_VER) \
|| defined (_WIN32)
#define HAVE_CONST 1
#endif
#if ! HAVE_CONST
#define const
#endif
mpz_t *sq_res_0x100; /* table of limbs */
int nsq_res_0x100; /* elements in sq_res_0x100 array */
int sq_res_0x100_num; /* squares in sq_res_0x100 */
double sq_res_0x100_fraction; /* sq_res_0x100_num / 256 */
int mod34_bits; /* 3*GMP_NUMB_BITS/4 */
int mod_bits; /* bits from PERFSQR_MOD_34 or MOD_PP */
int max_divisor; /* all divisors <= max_divisor */
int max_divisor_bits; /* ceil(log2(max_divisor)) */
double total_fraction; /* of squares */
mpz_t pp; /* product of primes, or 0 if mod_34lsub1 used */
mpz_t pp_norm; /* pp shifted so NUMB high bit set */
mpz_t pp_inverted; /* invert_limb style inverse */
mpz_t mod_mask; /* 2^mod_bits-1 */
char mod34_excuse[128]; /* why mod_34lsub1 not used (if it's not) */
/* raw list of divisors of 2^mod34_bits-1 or pp, just to show in a comment */
struct rawfactor_t {
int divisor;
int multiplicity;
};
struct rawfactor_t *rawfactor;
int nrawfactor;
/* factors of 2^mod34_bits-1 or pp and associated data, after combining etc */
struct factor_t {
int divisor;
mpz_t inverse; /* 1/divisor mod 2^mod_bits */
mpz_t mask; /* indicating squares mod divisor */
double fraction; /* squares/total */
};
struct factor_t *factor;
int nfactor; /* entries in use in factor array */
int factor_alloc; /* entries allocated to factor array */
int
f_cmp_divisor (const void *parg, const void *qarg)
{
const struct factor_t *p, *q;
p = parg;
q = qarg;
if (p->divisor > q->divisor)
return 1;
else if (p->divisor < q->divisor)
return -1;
else
return 0;
}
int
f_cmp_fraction (const void *parg, const void *qarg)
{
const struct factor_t *p, *q;
p = parg;
q = qarg;
if (p->fraction > q->fraction)
return 1;
else if (p->fraction < q->fraction)
return -1;
else
return 0;
}
/* Remove array[idx] by copying the remainder down, and adjust narray
accordingly. */
#define COLLAPSE_ELEMENT(array, idx, narray) \
do { \
mem_copyi ((char *) &(array)[idx], \
(char *) &(array)[idx+1], \
((narray)-((idx)+1)) * sizeof (array[0])); \
(narray)--; \
} while (0)
/* return n*2^p mod m */
int
mul_2exp_mod (int n, int p, int m)
{
int i;
for (i = 0; i < p; i++)
n = (2 * n) % m;
return n;
}
/* return -n mod m */
int
neg_mod (int n, int m)
{
ASSERT (n >= 0 && n < m);
return (n == 0 ? 0 : m-n);
}
/* Set "mask" to a value such that "mask & (1<<idx)" is non-zero if
"-(idx<<mod_bits)" can be a square modulo m. */
void
square_mask (mpz_t mask, int m)
{
int p, i, r, idx;
p = mul_2exp_mod (1, mod_bits, m);
p = neg_mod (p, m);
mpz_set_ui (mask, 0L);
for (i = 0; i < m; i++)
{
r = (i * i) % m;
idx = (r * p) % m;
mpz_setbit (mask, (unsigned long) idx);
}
}
void
generate_sq_res_0x100 (int limb_bits)
{
int i, res;
nsq_res_0x100 = (0x100 + limb_bits - 1) / limb_bits;
sq_res_0x100 = (mpz_t *) xmalloc (nsq_res_0x100 * sizeof (*sq_res_0x100));
for (i = 0; i < nsq_res_0x100; i++)
mpz_init_set_ui (sq_res_0x100[i], 0L);
for (i = 0; i < 0x100; i++)
{
res = (i * i) % 0x100;
mpz_setbit (sq_res_0x100[res / limb_bits],
(unsigned long) (res % limb_bits));
}
sq_res_0x100_num = 0;
for (i = 0; i < nsq_res_0x100; i++)
sq_res_0x100_num += mpz_popcount (sq_res_0x100[i]);
sq_res_0x100_fraction = (double) sq_res_0x100_num / 256.0;
}
void
generate_mod (int limb_bits, int nail_bits)
{
int numb_bits = limb_bits - nail_bits;
int i, divisor;
mpz_init_set_ui (pp, 0L);
mpz_init_set_ui (pp_norm, 0L);
mpz_init_set_ui (pp_inverted, 0L);
/* no more than limb_bits many factors in a one limb modulus (and of
course in reality nothing like that many) */
factor_alloc = limb_bits;
factor = (struct factor_t *) xmalloc (factor_alloc * sizeof (*factor));
rawfactor = (struct rawfactor_t *)
xmalloc (factor_alloc * sizeof (*rawfactor));
if (numb_bits % 4 != 0)
{
strcpy (mod34_excuse, "GMP_NUMB_BITS % 4 != 0");
goto use_pp;
}
max_divisor = 2*limb_bits;
max_divisor_bits = log2_ceil (max_divisor);
if (numb_bits / 4 < max_divisor_bits)
{
/* Wind back to one limb worth of max_divisor, if that will let us use
mpn_mod_34lsub1. */
max_divisor = limb_bits;
max_divisor_bits = log2_ceil (max_divisor);
if (numb_bits / 4 < max_divisor_bits)
{
strcpy (mod34_excuse, "GMP_NUMB_BITS / 4 too small");
goto use_pp;
}
}
{
/* Can use mpn_mod_34lsub1, find small factors of 2^mod34_bits-1. */
mpz_t m, q, r;
int multiplicity;
mod34_bits = (numb_bits / 4) * 3;
/* mpn_mod_34lsub1 returns a full limb value, PERFSQR_MOD_34 folds it at
the mod34_bits mark, adding the two halves for a remainder of at most
mod34_bits+1 many bits */
mod_bits = mod34_bits + 1;
mpz_init_set_ui (m, 1L);
mpz_mul_2exp (m, m, mod34_bits);
mpz_sub_ui (m, m, 1L);
mpz_init (q);
mpz_init (r);
for (i = 3; i <= max_divisor; i++)
{
if (! isprime (i))
continue;
mpz_tdiv_qr_ui (q, r, m, (unsigned long) i);
if (mpz_sgn (r) != 0)
continue;
/* if a repeated prime is found it's used as an i^n in one factor */
divisor = 1;
multiplicity = 0;
do
{
if (divisor > max_divisor / i)
break;
multiplicity++;
mpz_set (m, q);
mpz_tdiv_qr_ui (q, r, m, (unsigned long) i);
}
while (mpz_sgn (r) == 0);
ASSERT (nrawfactor < factor_alloc);
rawfactor[nrawfactor].divisor = i;
rawfactor[nrawfactor].multiplicity = multiplicity;
nrawfactor++;
}
mpz_clear (m);
mpz_clear (q);
mpz_clear (r);
}
if (nrawfactor <= 2)
{
mpz_t new_pp;
sprintf (mod34_excuse, "only %d small factor%s",
nrawfactor, nrawfactor == 1 ? "" : "s");
use_pp:
/* reset to two limbs of max_divisor, in case the mpn_mod_34lsub1 code
tried with just one */
max_divisor = 2*limb_bits;
max_divisor_bits = log2_ceil (max_divisor);
mpz_init (new_pp);
nrawfactor = 0;
mod_bits = MIN (numb_bits, limb_bits - max_divisor_bits);
/* one copy of each small prime */
mpz_set_ui (pp, 1L);
for (i = 3; i <= max_divisor; i++)
{
if (! isprime (i))
continue;
mpz_mul_ui (new_pp, pp, (unsigned long) i);
if (mpz_sizeinbase (new_pp, 2) > mod_bits)
break;
mpz_set (pp, new_pp);
ASSERT (nrawfactor < factor_alloc);
rawfactor[nrawfactor].divisor = i;
rawfactor[nrawfactor].multiplicity = 1;
nrawfactor++;
}
/* Plus an extra copy of one or more of the primes selected, if that
still fits in max_divisor and the total in mod_bits. Usually only
3 or 5 will be candidates */
for (i = nrawfactor-1; i >= 0; i--)
{
if (rawfactor[i].divisor > max_divisor / rawfactor[i].divisor)
continue;
mpz_mul_ui (new_pp, pp, (unsigned long) rawfactor[i].divisor);
if (mpz_sizeinbase (new_pp, 2) > mod_bits)
continue;
mpz_set (pp, new_pp);
rawfactor[i].multiplicity++;
}
mod_bits = mpz_sizeinbase (pp, 2);
mpz_set (pp_norm, pp);
while (mpz_sizeinbase (pp_norm, 2) < numb_bits)
mpz_add (pp_norm, pp_norm, pp_norm);
mpz_preinv_invert (pp_inverted, pp_norm, numb_bits);
mpz_clear (new_pp);
}
/* start the factor array */
for (i = 0; i < nrawfactor; i++)
{
int j;
ASSERT (nfactor < factor_alloc);
factor[nfactor].divisor = 1;
for (j = 0; j < rawfactor[i].multiplicity; j++)
factor[nfactor].divisor *= rawfactor[i].divisor;
nfactor++;
}
combine:
/* Combine entries in the factor array. Combine the smallest entry with
the biggest one that will fit with it (ie. under max_divisor), then
repeat that with the new smallest entry. */
qsort (factor, nfactor, sizeof (factor[0]), f_cmp_divisor);
for (i = nfactor-1; i >= 1; i--)
{
if (factor[i].divisor <= max_divisor / factor[0].divisor)
{
factor[0].divisor *= factor[i].divisor;
COLLAPSE_ELEMENT (factor, i, nfactor);
goto combine;
}
}
total_fraction = 1.0;
for (i = 0; i < nfactor; i++)
{
mpz_init (factor[i].inverse);
mpz_invert_ui_2exp (factor[i].inverse,
(unsigned long) factor[i].divisor,
(unsigned long) mod_bits);
mpz_init (factor[i].mask);
square_mask (factor[i].mask, factor[i].divisor);
/* fraction of possible squares */
factor[i].fraction = (double) mpz_popcount (factor[i].mask)
/ factor[i].divisor;
/* total fraction of possible squares */
total_fraction *= factor[i].fraction;
}
/* best tests first (ie. smallest fraction) */
qsort (factor, nfactor, sizeof (factor[0]), f_cmp_fraction);
}
void
print (int limb_bits, int nail_bits)
{
int i;
mpz_t mhi, mlo;
printf ("/* This file generated by gen-psqr.c - DO NOT EDIT. */\n");
printf ("\n");
printf ("#if GMP_LIMB_BITS != %d || GMP_NAIL_BITS != %d\n",
limb_bits, nail_bits);
printf ("Error, error, this data is for %d bit limb and %d bit nail\n",
limb_bits, nail_bits);
printf ("#endif\n");
printf ("\n");
printf ("/* Non-zero bit indicates a quadratic residue mod 0x100.\n");
printf (" This test identifies %.2f%% as non-squares (%d/256). */\n",
(1.0 - sq_res_0x100_fraction) * 100.0,
0x100 - sq_res_0x100_num);
printf ("static const mp_limb_t\n");
printf ("sq_res_0x100[%d] = {\n", nsq_res_0x100);
for (i = 0; i < nsq_res_0x100; i++)
{
printf (" CNST_LIMB(0x");
mpz_out_str (stdout, 16, sq_res_0x100[i]);
printf ("),\n");
}
printf ("};\n");
printf ("\n");
if (mpz_sgn (pp) != 0)
{
printf ("/* mpn_mod_34lsub1 not used due to %s */\n", mod34_excuse);
printf ("/* PERFSQR_PP = ");
}
else
printf ("/* 2^%d-1 = ", mod34_bits);
for (i = 0; i < nrawfactor; i++)
{
if (i != 0)
printf (" * ");
printf ("%d", rawfactor[i].divisor);
if (rawfactor[i].multiplicity != 1)
printf ("^%d", rawfactor[i].multiplicity);
}
printf (" %s*/\n", mpz_sgn (pp) == 0 ? "... " : "");
printf ("#define PERFSQR_MOD_BITS %d\n", mod_bits);
if (mpz_sgn (pp) != 0)
{
printf ("#define PERFSQR_PP CNST_LIMB(0x");
mpz_out_str (stdout, 16, pp);
printf (")\n");
printf ("#define PERFSQR_PP_NORM CNST_LIMB(0x");
mpz_out_str (stdout, 16, pp_norm);
printf (")\n");
printf ("#define PERFSQR_PP_INVERTED CNST_LIMB(0x");
mpz_out_str (stdout, 16, pp_inverted);
printf (")\n");
}
printf ("\n");
mpz_init (mhi);
mpz_init (mlo);
printf ("/* This test identifies %.2f%% as non-squares. */\n",
(1.0 - total_fraction) * 100.0);
printf ("#define PERFSQR_MOD_TEST(up, usize) \\\n");
printf (" do { \\\n");
printf (" mp_limb_t r; \\\n");
if (mpz_sgn (pp) != 0)
printf (" PERFSQR_MOD_PP (r, up, usize); \\\n");
else
printf (" PERFSQR_MOD_34 (r, up, usize); \\\n");
for (i = 0; i < nfactor; i++)
{
printf (" \\\n");
printf (" /* %5.2f%% */ \\\n",
(1.0 - factor[i].fraction) * 100.0);
printf (" PERFSQR_MOD_%d (r, CNST_LIMB(%2d), CNST_LIMB(0x",
factor[i].divisor <= limb_bits ? 1 : 2,
factor[i].divisor);
mpz_out_str (stdout, 16, factor[i].inverse);
printf ("), \\\n");
printf (" CNST_LIMB(0x");
if ( factor[i].divisor <= limb_bits)
{
mpz_out_str (stdout, 16, factor[i].mask);
}
else
{
mpz_tdiv_r_2exp (mlo, factor[i].mask, (unsigned long) limb_bits);
mpz_tdiv_q_2exp (mhi, factor[i].mask, (unsigned long) limb_bits);
mpz_out_str (stdout, 16, mhi);
printf ("), CNST_LIMB(0x");
mpz_out_str (stdout, 16, mlo);
}
printf (")); \\\n");
}
printf (" } while (0)\n");
printf ("\n");
printf ("/* Grand total sq_res_0x100 and PERFSQR_MOD_TEST, %.2f%% non-squares. */\n",
(1.0 - (total_fraction * 44.0/256.0)) * 100.0);
printf ("\n");
printf ("/* helper for tests/mpz/t-perfsqr.c */\n");
printf ("#define PERFSQR_DIVISORS { 256,");
for (i = 0; i < nfactor; i++)
printf (" %d,", factor[i].divisor);
printf (" }\n");
mpz_clear (mhi);
mpz_clear (mlo);
}
int
main (int argc, char *argv[])
{
int limb_bits, nail_bits;
if (argc != 3)
{
fprintf (stderr, "Usage: gen-psqr <limbbits> <nailbits>\n");
exit (1);
}
limb_bits = atoi (argv[1]);
nail_bits = atoi (argv[2]);
if (limb_bits <= 0
|| nail_bits < 0
|| nail_bits >= limb_bits)
{
fprintf (stderr, "Invalid limb/nail bits: %d %d\n",
limb_bits, nail_bits);
exit (1);
}
generate_sq_res_0x100 (limb_bits);
generate_mod (limb_bits, nail_bits);
print (limb_bits, nail_bits);
return 0;
}
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