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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
===============================================================================*/
/****************************************************************************
bernoulli_zmod.c: Finds Bernoulli numbers B_{2k}
Based on the implementation in SAGE written by David Harvey
Uses zmod_polys for calculation.
Copyright (C) 2007, David Howden
*****************************************************************************/
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <gmp.h>
#include "flint.h"
#include "long_extras.h"
#include "zmod_poly.h"
#define TRUE 1
#define FALSE 0
/*
Debugging function
*/
// void print_var(char *name, unsigned long value)
// {
// printf("%s = %d\n", name, value);
// }
/*
Computes the bernoulli numbers B_0, B_2, ..., B_{p-3}
for prime p
Requires that res be allocated for (p-1)/2 unsigned longs
which will hold the result.
If returns 0, then the factoring of p has failed, otherwise
will always return 1.
*/
int bernoulli_mod_p(unsigned long *res, unsigned long p)
{
FLINT_ASSERT(p > 2);
FLINT_ASSERT(z_isprime(p) == 1);
unsigned long g, g_inv, g_sqr, g_sqr_inv;
double p_inv = z_precompute_inverse(p);
g = z_primitive_root_precomp(p, p_inv);
if(!g)
{
return FALSE;
}
g_inv = z_invert(g, p);
g_sqr = z_mulmod_precomp(g, g, p, p_inv);
g_sqr_inv = z_mulmod_precomp(g_inv, g_inv, p, p_inv);
unsigned long poly_size = (p-1)/2;
int is_odd = poly_size % 2;
unsigned long g_power, g_power_inv;
g_power = g_inv;
g_power_inv = 1;
// constant is (g-1)/2 mod p
unsigned long constant;
if(g % 2)
{
constant = (g-1)/2;
}
else
{
constant = (g+p-1)/2;
}
// fudge holds g^{i^2}, fudge_inv holds g^{-i^2}
unsigned long fudge, fudge_inv;
fudge = fudge_inv = 1;
// compute the polynomials F(X) and G(X)
zmod_poly_t F, G;
zmod_poly_init2(F, p, poly_size);
zmod_poly_init2(G, p, poly_size);
unsigned long i, temp, h;
for(i = 0; i < poly_size; i++)
{
// compute h(g^i)/g^i (h(x) is as in latex notes)
temp = g * g_power;
h = z_mulmod_precomp(p + constant - (temp / p), g_power_inv, p, p_inv);
g_power = z_mod_precomp(temp, p, p_inv);
g_power_inv = z_mulmod_precomp(g_power_inv, g_inv, p, p_inv);
// store coefficient g^{i^2} h(g^i)/g^i
zmod_poly_set_coeff_ui(G, i, z_mulmod_precomp(h, fudge, p, p_inv));
zmod_poly_set_coeff_ui(F, i, fudge_inv);
// update fudge and fudge_inv
fudge = z_mulmod_precomp(z_mulmod_precomp(fudge, g_power, p, p_inv), z_mulmod_precomp(g_power, g, p, p_inv), p, p_inv);
fudge_inv = z_mulmod_precomp(z_mulmod_precomp(fudge_inv, g_power_inv, p, p_inv), z_mulmod_precomp(g_power_inv, g, p, p_inv), p, p_inv);
}
zmod_poly_set_coeff_ui(F, 0, 0);
// step 2: multiply the polynomials...
zmod_poly_t product;
zmod_poly_init(product, p);
zmod_poly_mul_KS(product, G, F, 0);
// step 3: assemble the result...
unsigned long g_sqr_power, value;
g_sqr_power = g_sqr;
fudge = g;
res[0] = 1L;
unsigned long value_coeff_ui;
for(i = 1; i < poly_size; i++)
{
value = zmod_poly_get_coeff_ui(product, i + poly_size);
if(is_odd)
{
value = z_mod_precomp(zmod_poly_get_coeff_ui(G, i) + zmod_poly_get_coeff_ui(product, i) + p - value, p, p_inv);
}
else
{
value = z_mod_precomp(zmod_poly_get_coeff_ui(G, i) + zmod_poly_get_coeff_ui(product, i) + value, p, p_inv);
}
value = z_mulmod_precomp(z_mulmod_precomp(z_mulmod_precomp(4L, i, p, p_inv), fudge, p, p_inv), value, p, p_inv);
value = z_mulmod_precomp(value, z_invert(p+1L-g_sqr_power, p), p, p_inv);
res[i] = value;
g_sqr_power = z_mulmod_precomp(g_sqr_power, g, p, p_inv);
fudge = z_mulmod_precomp(fudge, g_sqr_power, p, p_inv);
g_sqr_power = z_mulmod_precomp(g_sqr_power, g, p, p_inv);
}
zmod_poly_clear(product);
zmod_poly_clear(F);
zmod_poly_clear(G);
return TRUE;
}
/*
Verifies that the ouput of bernoulli_mod_p above is correct.
Takes the result from bernoulli_mod_p (res - an array of (p-1)/2
unsigned longs), and the prime p.
Returns 0 if res is incorrect, 1 if res is correct.
*/
int verify_bernoulli_mod_p(unsigned long *res, unsigned long p)
{
unsigned long N, i, product, sum, value, element;
double p_inv;
N = (p-1)/2;
product = 1L;
sum = 0L;
p_inv = z_precompute_inverse(p);
for(i = 0; i < N; i++)
{
element = res[i];
// if((signed long)element < 0)
// {
// printf("NEGATIVE NUMBER!!!!!\n");
// }
// if(element > p)
// {
// printf("OVERFLOW!!!!!\n");
// }
value = z_mulmod_precomp(z_mulmod_precomp(product, 2*i+1L, p, p_inv), element, p, p_inv);
sum = z_mod_precomp(sum + value, p, p_inv);
product = z_mulmod_precomp(product, 4L, p, p_inv);
}
if(z_mod_precomp(sum + 2L, p, p_inv))
{
// i = 0;
// printf("Error occurred, output:\n");
// while (i < N)
// {
// printf("%d\n", res[i]);
// i++;
// }
return FALSE;
}
return TRUE;
}
/*
Test function for bernoulli_mod_p
Calculates bernoulli_mod_p for the prime p and verifies the result.
Returs 0 if incorrect, and 1 if correct.
*/
int test_bernoulli_mod_p(unsigned long p)
{
unsigned long *res = (unsigned long*) flint_stack_alloc((p-1)/2);
if(!bernoulli_mod_p(res, p))
{
printf("Could not factor p = %d\n", p);
flint_stack_release();
return FALSE;
}
int result = verify_bernoulli_mod_p(res, p);
flint_stack_release();
return result;
}
int main (int argc, char const *argv[])
{
if (argc == 2)
{
unsigned long n = atoi(argv[1]);
n = z_nextprime(n);
printf("Computing bernoulli_mod_p(%ld)... ", n);
if (!test_bernoulli_mod_p(n))
{
printf("Failed\n");
}
else
{
printf("Done\n");
}
return 0;
}
unsigned long p = 2;
unsigned long tests = 2000;
unsigned long fail = 0;
for(unsigned long i = 0; i < tests; i++)
{
p = z_nextprime(p);
if(!test_bernoulli_mod_p(p))
{
printf("Fails on p = %d\n", p);
fail++;
}
else
{
printf("Works on p = %d\n", p);
}
}
printf("\nResults: %d OK, %d FAILED.\n", tests - fail, fail);
return 0;
}
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