/* Exercise mpz_*_kronecker_*() and mpz_jacobi() functions. */
/*
Copyright (C) 1999, 2000 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., 59 Temple Place - Suite 330, Boston,
MA 02111-1307, USA.
*/
/* With no arguments the various Kronecker/Jacobi symbol routines are
checked against some test data and a lot of derived data.
To check the test data against PARI-GP, run
t-jac -p | gp -q
It takes a while because the output from "t-jac -p" is big.
Enhancements:
More big test cases than those given by check_squares_zi would be good. */
#include <stdio.h>
#include "gmp.h"
#include "gmp-impl.h"
int option_pari = 0;
unsigned long
mpz_mod4 (mpz_srcptr z)
{
mpz_t m;
unsigned long ret;
mpz_init (m);
mpz_fdiv_r_2exp (m, z, 2);
ret = mpz_get_ui (m);
mpz_clear (m);
return ret;
}
int
mpz_fits_ulimb_p (mpz_srcptr z)
{
return (SIZ(z) == 1 || SIZ(z) == 0);
}
mp_limb_t
mpz_get_ulimb (mpz_srcptr z)
{
if (SIZ(z) == 0)
return 0;
else
return PTR(z)[0];
}
void
try_base (mp_limb_t a, mp_limb_t b, int answer)
{
int got;
if ((b & 1) == 0 || b == 1 || a > b)
return;
got = mpn_jacobi_base (a, b, 0);
if (got != answer)
{
printf ("mpn_jacobi_base (%lu, %lu) is %d should be %d\n",
a, b, got, answer);
exit (1);
}
}
void
try_zi_ui (mpz_srcptr a, unsigned long b, int answer)
{
int got;
got = mpz_kronecker_ui (a, b);
if (got != answer)
{
printf ("mpz_kronecker_ui (");
mpz_out_str (stdout, 10, a);
printf (", %lu) is %d should be %d\n", b, got, answer);
exit (1);
}
}
void
try_zi_si (mpz_srcptr a, long b, int answer)
{
int got;
got = mpz_kronecker_si (a, b);
if (got != answer)
{
printf ("mpz_kronecker_si (");
mpz_out_str (stdout, 10, a);
printf (", %ld) is %d should be %d\n", b, got, answer);
exit (1);
}
}
void
try_ui_zi (unsigned long a, mpz_srcptr b, int answer)
{
int got;
got = mpz_ui_kronecker (a, b);
if (got != answer)
{
printf ("mpz_ui_kronecker (%lu, ", a);
mpz_out_str (stdout, 10, b);
printf (") is %d should be %d\n", got, answer);
exit (1);
}
}
void
try_si_zi (int a, mpz_srcptr b, int answer)
{
int got;
got = mpz_si_kronecker (a, b);
if (got != answer)
{
printf ("mpz_si_kronecker (%d, ", a);
mpz_out_str (stdout, 10, b);
printf (") is %d should be %d\n", got, answer);
exit (1);
}
}
void
try_zi_zi (mpz_srcptr a, mpz_srcptr b, int answer)
{
int got;
/* gmp 2.0.2 mpz_jacobi() doesn't handle negative or even b */
if (mpz_sgn (b) <= 0 || mpz_even_p (b))
return;
got = mpz_jacobi (a, b);
if (got != answer)
{
printf ("mpz_jacobi (");
mpz_out_str (stdout, 10, a);
printf (", ");
mpz_out_str (stdout, 10, b);
printf (") is %d should be %d\n", got, answer);
exit (1);
}
}
void
try_pari (mpz_srcptr a, mpz_srcptr b, int answer)
{
printf ("try(");
mpz_out_str (stdout, 10, a);
printf (",");
mpz_out_str (stdout, 10, b);
printf (",%d)\n", answer);
}
void
try_each (mpz_srcptr a, mpz_srcptr b, int answer)
{
if (option_pari)
{
try_pari (a, b, answer);
return;
}
if (mpz_fits_ulimb_p (a) && mpz_fits_ulimb_p (b))
try_base (mpz_get_ulimb (a), mpz_get_ulimb (b), answer);
if (mpz_fits_ulong_p (b))
try_zi_ui (a, mpz_get_ui (b), answer);
if (mpz_fits_slong_p (b))
try_zi_si (a, mpz_get_si (b), answer);
if (mpz_fits_ulong_p (a))
try_ui_zi (mpz_get_ui (a), b, answer);
if (mpz_fits_sint_p (a))
try_si_zi (mpz_get_si (a), b, answer);
try_zi_zi (a, b, answer);
}
/* Try (a/b) and (a/-b). */
void
try_pn (mpz_srcptr a, mpz_srcptr b_orig, int answer)
{
mpz_t b;
mpz_init_set (b, b_orig);
try_each (a, b, answer);
mpz_neg (b, b);
if (mpz_sgn (a) < 0)
answer = -answer;
try_each (a, b, answer);
mpz_clear (b);
}
/* Try (a+k*p/b) for various k, using the fact (a/b) is periodic in a with
period p. For b>0, p=b if b!=2mod4 or p=4*b if b==2mod4. */
void
try_periodic_num (mpz_srcptr a_orig, mpz_srcptr b, int answer)
{
mpz_t a, a_period;
int i;
if (mpz_sgn (b) <= 0)
return;
mpz_init_set (a, a_orig);
mpz_init_set (a_period, b);
if (mpz_mod4 (b) == 2)
mpz_mul_ui (a_period, a_period, 4);
/* don't bother with these tests if they're only going to produce
even/even */
if (mpz_even_p (a) && mpz_even_p (b) && mpz_even_p (a_period))
goto done;
for (i = 0; i < 10; i++)
{
mpz_add (a, a, a_period);
try_pn (a, b, answer);
}
mpz_set (a, a_orig);
for (i = 0; i < 10; i++)
{
mpz_sub (a, a, a_period);
try_pn (a, b, answer);
}
done:
mpz_clear (a);
mpz_clear (a_period);
}
/* Try (a/b+k*p) for various k, using the fact (a/b) is periodic in b of
period p.
period p
a==0,1mod4 a
a==2mod4 4*a
a==3mod4 and b odd 4*a
a==3mod4 and b even 8*a
In Henri Cohen's book the period is given as 4*a for all a==2,3mod4, but
a counterexample would seem to be (3/2)=-1 which with (3/14)=+1 doesn't
have period 4*a (but rather 8*a with (3/26)=-1). Maybe the plain 4*a is
to be read as applying to a plain Jacobi symbol with b odd, rather than
the Kronecker extension to b even. */
void
try_periodic_den (mpz_srcptr a, mpz_srcptr b_orig, int answer)
{
mpz_t b, b_period;
int i;
if (mpz_sgn (a) == 0 || mpz_sgn (b_orig) == 0)
return;
mpz_init_set (b, b_orig);
mpz_init_set (b_period, a);
if (mpz_mod4 (a) == 3 && mpz_even_p (b))
mpz_mul_ui (b_period, b_period, 8);
else if (mpz_mod4 (a) >= 2)
mpz_mul_ui (b_period, b_period, 4);
/* don't bother with these tests if they're only going to produce
even/even */
if (mpz_even_p (a) && mpz_even_p (b) && mpz_even_p (b_period))
goto done;
for (i = 0; i < 10; i++)
{
mpz_add (b, b, b_period);
try_pn (a, b, answer);
}
mpz_set (b, b_orig);
for (i = 0; i < 10; i++)
{
mpz_sub (b, b, b_period);
try_pn (a, b, answer);
}
done:
mpz_clear (b);
mpz_clear (b_period);
}
/* Try (a/b*2^k) for various k. If it happens mpz_ui_kronecker() gets (a/2)
wrong it will show up as wrong answers demanded. */
void
try_2den (mpz_srcptr a, mpz_srcptr b_orig, int answer)
{
mpz_t b;
int i;
int answer_two;
/* don't bother when b==0 */
if (mpz_sgn (b_orig) == 0)
return;
mpz_init_set (b, b_orig);
answer_two = mpz_kronecker_ui (a, 2);
for (i = 0; i < 3 * BITS_PER_MP_LIMB; i++)
{
answer *= answer_two;
mpz_mul_2exp (b, b, 1);
try_pn (a, b, answer);
}
mpz_clear (b);
}
/* Try (a*2^k/b) for various k. If it happens mpz_ui_kronecker() gets (2/b)
wrong it will show up as wrong answers demanded. */
void
try_2num (mpz_srcptr a_orig, mpz_srcptr b, int answer)
{
mpz_t a;
int i;
int answer_twos;
/* don't bother when a==0 */
if (mpz_sgn (a_orig) == 0)
return;
mpz_init_set (a, a_orig);
answer_twos = mpz_ui_kronecker (2, b);
for (i = 0; i < 3 * BITS_PER_MP_LIMB; i++)
{
answer *= answer_twos;
mpz_mul_2exp (a, a, 1);
try_pn (a, b, answer);
}
mpz_clear (a);
}
/* The try_2num() and try_2den() routines don't in turn call
try_periodic_num() and try_periodic_den() because it hugely increases the
number of tests performed, without obviously increasing coverage.
Useful extra derived cases can be added here. */
void
try_all (mpz_t a, mpz_t b, int answer)
{
try_pn (a, b, answer);
try_periodic_num (a, b, answer);
try_periodic_den (a, b, answer);
try_2num (a, b, answer);
try_2den (a, b, answer);
}
void
check_data (void)
{
static const struct {
const char *a;
const char *b;
int answer;
} data[] = {
/* Note that the various derived checks in try_all() reduce the cases
that need to be given here. */
/* some zeros */
{ "0", "0", 0 },
{ "0", "2", 0 },
{ "0", "6", 0 },
{ "5", "0", 0 },
{ "24", "60", 0 },
/* (a/1) = 1, any a
In particular note (0/1)=1 so that (a/b)=(a mod b/b). */
{ "0", "1", 1 },
{ "1", "1", 1 },
{ "2", "1", 1 },
{ "3", "1", 1 },
{ "4", "1", 1 },
{ "5", "1", 1 },
/* (0/b) = 0, b != 1 */
{ "0", "3", 0 },
{ "0", "5", 0 },
{ "0", "7", 0 },
{ "0", "9", 0 },
{ "0", "11", 0 },
{ "0", "13", 0 },
{ "0", "15", 0 },
/* (1/b) = 1 */
{ "1", "1", 1 },
{ "1", "3", 1 },
{ "1", "5", 1 },
{ "1", "7", 1 },
{ "1", "9", 1 },
{ "1", "11", 1 },
/* (-1/b) = (-1)^((b-1)/2) which is -1 for b==3 mod 4 */
{ "-1", "1", 1 },
{ "-1", "3", -1 },
{ "-1", "5", 1 },
{ "-1", "7", -1 },
{ "-1", "9", 1 },
{ "-1", "11", -1 },
{ "-1", "13", 1 },
{ "-1", "15", -1 },
{ "-1", "17", 1 },
{ "-1", "19", -1 },
/* (2/b) = (-1)^((b^2-1)/8) which is -1 for b==3,5 mod 8.
try_2num() will exercise multiple powers of 2 in the numerator. */
{ "2", "1", 1 },
{ "2", "3", -1 },
{ "2", "5", -1 },
{ "2", "7", 1 },
{ "2", "9", 1 },
{ "2", "11", -1 },
{ "2", "13", -1 },
{ "2", "15", 1 },
{ "2", "17", 1 },
/* (-2/b) = (-1)^((b^2-1)/8)*(-1)^((b-1)/2) which is -1 for b==5,7mod8.
try_2num() will exercise multiple powers of 2 in the numerator, which
will test that the shift in mpz_si_kronecker() uses unsigned not
signed. */
{ "-2", "1", 1 },
{ "-2", "3", 1 },
{ "-2", "5", -1 },
{ "-2", "7", -1 },
{ "-2", "9", 1 },
{ "-2", "11", 1 },
{ "-2", "13", -1 },
{ "-2", "15", -1 },
{ "-2", "17", 1 },
/* (a/2)=(2/a).
try_2den() will exercise multiple powers of 2 in the denominator. */
{ "3", "2", -1 },
{ "5", "2", -1 },
{ "7", "2", 1 },
{ "9", "2", 1 },
{ "11", "2", -1 },
/* Harriet Griffin, "Elementary Theory of Numbers", page 155, various
examples. */
{ "2", "135", 1 },
{ "135", "19", -1 },
{ "2", "19", -1 },
{ "19", "135", 1 },
{ "173", "135", 1 },
{ "38", "135", 1 },
{ "135", "173", 1 },
{ "173", "5", -1 },
{ "3", "5", -1 },
{ "5", "173", -1 },
{ "173", "3", -1 },
{ "2", "3", -1 },
{ "3", "173", -1 },
{ "253", "21", 1 },
{ "1", "21", 1 },
{ "21", "253", 1 },
{ "21", "11", -1 },
{ "-1", "11", -1 },
/* Griffin page 147 */
{ "-1", "17", 1 },
{ "2", "17", 1 },
{ "-2", "17", 1 },
{ "-1", "89", 1 },
{ "2", "89", 1 },
/* Griffin page 148 */
{ "89", "11", 1 },
{ "1", "11", 1 },
{ "89", "3", -1 },
{ "2", "3", -1 },
{ "3", "89", -1 },
{ "11", "89", 1 },
{ "33", "89", -1 },
/* H. Davenport, "The Higher Arithmetic", page 65, the quadratic
residues and non-residues mod 19. */
{ "1", "19", 1 },
{ "4", "19", 1 },
{ "5", "19", 1 },
{ "6", "19", 1 },
{ "7", "19", 1 },
{ "9", "19", 1 },
{ "11", "19", 1 },
{ "16", "19", 1 },
{ "17", "19", 1 },
{ "2", "19", -1 },
{ "3", "19", -1 },
{ "8", "19", -1 },
{ "10", "19", -1 },
{ "12", "19", -1 },
{ "13", "19", -1 },
{ "14", "19", -1 },
{ "15", "19", -1 },
{ "18", "19", -1 },
/* Residues and non-residues mod 13 */
{ "0", "13", 0 },
{ "1", "13", 1 },
{ "2", "13", -1 },
{ "3", "13", 1 },
{ "4", "13", 1 },
{ "5", "13", -1 },
{ "6", "13", -1 },
{ "7", "13", -1 },
{ "8", "13", -1 },
{ "9", "13", 1 },
{ "10", "13", 1 },
{ "11", "13", -1 },
{ "12", "13", 1 },
/* various */
{ "5", "7", -1 },
{ "15", "17", 1 },
{ "67", "89", 1 },
};
int i, answer;
mpz_t a, b;
mpz_init (a);
mpz_init (b);
for (i = 0; i < numberof (data); i++)
{
mpz_set_str (a, data[i].a, 0);
mpz_set_str (b, data[i].b, 0);
answer = data[i].answer;
try_all (a, b, data[i].answer);
}
mpz_clear (a);
mpz_clear (b);
}
/* (a^2/b)=1 if gcd(a,b)=1, or (a^2/b)=0 if gcd(a,b)!=1. */
void
check_squares_zi (void)
{
mpz_t a, b, g;
int i, answer;
mpz_init (a);
mpz_init (b);
mpz_init (g);
for (i = 0; i < 200; i++)
{
mpz_random2 (a, rand() % 50);
mpz_random2 (b, rand() % 50);
mpz_mul (a, a, b);
mpz_gcd (g, a, b);
if (mpz_cmp_ui (g, 1) == 0)
answer = 1;
else
answer = 0;
try_all (a, b, answer);
}
mpz_clear (a);
mpz_clear (b);
mpz_clear (g);
}
int
main (int argc, char *argv[])
{
if (argc >= 2 && strcmp (argv[1], "-p") == 0)
{
option_pari = 1;
printf ("\
try(a,b,answer) =\n\
{\n\
if (kronecker(a,b) != answer,\n\
print(\"wrong at \", a, \",\", b,\n\
\" expected \", answer,\n\
\" pari says \", kronecker(a,b)))\n\
}\n");
}
check_data ();
check_squares_zi ();
exit (0);
}
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