/* Exercise mpz_*_kronecker_*() and mpz_jacobi() functions.
Copyright 1999, 2000, 2001, 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 3 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. If not, see http://www.gnu.org/licenses/. */
/* 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 <stdlib.h>
#include <string.h>
#include "gmp.h"
#include "gmp-impl.h"
#include "tests.h"
#ifdef _LONG_LONG_LIMB
#define LL(l,ll) ll
#else
#define LL(l,ll) l
#endif
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 (LL("mpn_jacobi_base (%lu, %lu) is %d should be %d\n",
"mpn_jacobi_base (%llu, %llu) is %d should be %d\n"),
a, b, got, answer);
abort ();
}
}
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);
abort ();
}
}
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);
abort ();
}
}
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);
abort ();
}
}
void
try_si_zi (long a, mpz_srcptr b, int answer)
{
int got;
got = mpz_si_kronecker (a, b);
if (got != answer)
{
printf ("mpz_si_kronecker (%ld, ", a);
mpz_out_str (stdout, 10, b);
printf (") is %d should be %d\n", got, answer);
abort ();
}
}
/* Don't bother checking mpz_jacobi, since it only differs for b even, and
we don't have an actual expected answer for it. tests/devel/try.c does
some checks though. */
void
try_zi_zi (mpz_srcptr a, mpz_srcptr b, int answer)
{
int got;
got = mpz_kronecker (a, b);
if (got != answer)
{
printf ("mpz_kronecker (");
mpz_out_str (stdout, 10, a);
printf (", ");
mpz_out_str (stdout, 10, b);
printf (") is %d should be %d\n", got, answer);
abort ();
}
}
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 < 6; i++)
{
mpz_add (a, a, a_period);
try_pn (a, b, answer);
}
mpz_set (a, a_orig);
for (i = 0; i < 6; 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, 8L);
else if (mpz_mod4 (a) >= 2)
mpz_mul_ui (b_period, b_period, 4L);
/* 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 < 6; i++)
{
mpz_add (b, b, b_period);
try_pn (a, b, answer);
}
mpz_set (b, b_orig);
for (i = 0; i < 6; i++)
{
mpz_sub (b, b, b_period);
try_pn (a, b, answer);
}
done:
mpz_clear (b);
mpz_clear (b_period);
}
static const unsigned long ktable[] = {
0, 1, 2, 3, 4, 5, 6, 7,
GMP_NUMB_BITS-1, GMP_NUMB_BITS, GMP_NUMB_BITS+1,
2*GMP_NUMB_BITS-1, 2*GMP_NUMB_BITS, 2*GMP_NUMB_BITS+1,
3*GMP_NUMB_BITS-1, 3*GMP_NUMB_BITS, 3*GMP_NUMB_BITS+1
};
/* Try (a/b*2^k) for various k. */
void
try_2den (mpz_srcptr a, mpz_srcptr b_orig, int answer)
{
mpz_t b;
int kindex;
int answer_a2, answer_k;
unsigned long k;
/* don't bother when b==0 */
if (mpz_sgn (b_orig) == 0)
return;
mpz_init_set (b, b_orig);
/* (a/2) is 0 if a even, 1 if a==1 or 7 mod 8, -1 if a==3 or 5 mod 8 */
answer_a2 = (mpz_even_p (a) ? 0
: (((SIZ(a) >= 0 ? PTR(a)[0] : -PTR(a)[0]) + 2) & 7) < 4 ? 1
: -1);
for (kindex = 0; kindex < numberof (ktable); kindex++)
{
k = ktable[kindex];
/* answer_k = answer*(answer_a2^k) */
answer_k = (answer_a2 == 0 && k != 0 ? 0
: (k & 1) == 1 && answer_a2 == -1 ? -answer
: answer);
mpz_mul_2exp (b, b_orig, k);
try_pn (a, b, answer_k);
}
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 kindex;
int answer_2b, answer_k;
unsigned long k;
/* don't bother when a==0 */
if (mpz_sgn (a_orig) == 0)
return;
mpz_init (a);
/* (2/b) is 0 if b even, 1 if b==1 or 7 mod 8, -1 if b==3 or 5 mod 8 */
answer_2b = (mpz_even_p (b) ? 0
: (((SIZ(b) >= 0 ? PTR(b)[0] : -PTR(b)[0]) + 2) & 7) < 4 ? 1
: -1);
for (kindex = 0; kindex < numberof (ktable); kindex++)
{
k = ktable[kindex];
/* answer_k = answer*(answer_2b^k) */
answer_k = (answer_2b == 0 && k != 0 ? 0
: (k & 1) == 1 && answer_2b == -1 ? -answer
: answer);
mpz_mul_2exp (a, a_orig, k);
try_pn (a, b, answer_k);
}
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 },
/* special values inducing a==b==1 at the end of jac_or_kron() */
{ "0x10000000000000000000000000000000000000000000000001",
"0x10000000000000000000000000000000000000000000000003", 1 },
};
int i;
mpz_t a, b;
mpz_init (a);
mpz_init (b);
for (i = 0; i < numberof (data); i++)
{
mpz_set_str_or_abort (a, data[i].a, 0);
mpz_set_str_or_abort (b, data[i].b, 0);
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.
This includes when a=0 or b=0. */
void
check_squares_zi (void)
{
gmp_randstate_ptr rands = RANDS;
mpz_t a, b, g;
int i, answer;
mp_size_t size_range, an, bn;
mpz_t bs;
mpz_init (bs);
mpz_init (a);
mpz_init (b);
mpz_init (g);
for (i = 0; i < 50; i++)
{
mpz_urandomb (bs, rands, 32);
size_range = mpz_get_ui (bs) % 10 + 2;
mpz_urandomb (bs, rands, size_range);
an = mpz_get_ui (bs);
mpz_rrandomb (a, rands, an);
mpz_urandomb (bs, rands, size_range);
bn = mpz_get_ui (bs);
mpz_rrandomb (b, rands, bn);
mpz_gcd (g, a, b);
if (mpz_cmp_ui (g, 1L) == 0)
answer = 1;
else
answer = 0;
mpz_mul (a, a, a);
try_all (a, b, answer);
}
mpz_clear (bs);
mpz_clear (a);
mpz_clear (b);
mpz_clear (g);
}
/* Check the handling of asize==0, make sure it isn't affected by the low
limb. */
void
check_a_zero (void)
{
mpz_t a, b;
mpz_init_set_ui (a, 0);
mpz_init (b);
mpz_set_ui (b, 1L);
PTR(a)[0] = 0;
try_all (a, b, 1); /* (0/1)=1 */
PTR(a)[0] = 1;
try_all (a, b, 1); /* (0/1)=1 */
mpz_set_si (b, -1L);
PTR(a)[0] = 0;
try_all (a, b, 1); /* (0/-1)=1 */
PTR(a)[0] = 1;
try_all (a, b, 1); /* (0/-1)=1 */
mpz_set_ui (b, 0);
PTR(a)[0] = 0;
try_all (a, b, 0); /* (0/0)=0 */
PTR(a)[0] = 1;
try_all (a, b, 0); /* (0/0)=0 */
mpz_set_ui (b, 2);
PTR(a)[0] = 0;
try_all (a, b, 0); /* (0/2)=0 */
PTR(a)[0] = 1;
try_all (a, b, 0); /* (0/2)=0 */
mpz_clear (a);
mpz_clear (b);
}
int
main (int argc, char *argv[])
{
tests_start ();
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 ();
check_a_zero ();
tests_end ();
exit (0);
}
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