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/*
Copyright (C) 2014 William Hart
This file is part of FLINT.
FLINT is free software: you can redistribute it and/or modify it under
the terms of the GNU Lesser General Public License (LGPL) as published
by the Free Software Foundation; either version 3 of the License, or
(at your option) any later version. See <https://www.gnu.org/licenses/>.
*/
#include "fmpz.h"
#include "fmpz_vec.h"
int fmpz_divisor_in_residue_class_lenstra(fmpz_t fac, const fmpz_t n, const fmpz_t r, const fmpz_t s)
{
fmpz_t r1, r2, a0, a1, b0, b1, c0, c1, q, d, d1, s1, s2, ns2;
slong i;
int res = 0;
fmpz_init(r1);
fmpz_init(r2);
fmpz_init(a0);
fmpz_init(a1);
fmpz_init(b0);
fmpz_init(b1);
fmpz_init(c0);
fmpz_init(c1);
fmpz_init(q);
fmpz_init(d);
fmpz_init(d1);
fmpz_init(s1);
fmpz_init(s2);
fmpz_init(ns2);
/* ns2 = n/s^2 */
fmpz_mul(ns2, s, s);
fmpz_tdiv_q(ns2, n, ns2);
/* initialise */
fmpz_invmod(r1, r, s); /* r1 = r^-1 mod s */
fmpz_mul(r2, r1, n);
fmpz_mod(r2, r2, s); /* r2 = r1*n mod s */
fmpz_set(a0, s); /* a0 = s */
fmpz_mul(a1, r1, r2); /* a1 = r1*r2 mod s */
fmpz_mod(a1, a1, s);
if (fmpz_is_zero(a1))
fmpz_add(a1, a1, s); /* 0 < a1 <= s */
fmpz_zero(b0);
fmpz_one(b1);
fmpz_zero(c0);
fmpz_mul(c1, r, r2); /* c1 = (n - r*r2)/s * r1 mod s */
fmpz_sub(c1, n, c1);
fmpz_divexact(c1, c1, s);
fmpz_mul(c1, c1, r1);
fmpz_mod(c1, c1, s);
/* deal with a0, b0, c0 */
if (!fmpz_is_one(r) && !fmpz_equal(n, r) && fmpz_divisible(n, r))
{
fmpz_set(fac, r);
res = 1;
goto cleanup;
}
for (i = 1; ; i++)
{
if ((i & 1) == 0)
{
fmpz_mod(s1, c1, s);
fmpz_neg(s2, s);
} else
{
fmpz_mul(s2, a1, b1); /* s2 = a1*b1 */
fmpz_add(s1, s2, ns2); /* s1 = a1*b1 + n/s^2 */
fmpz_mod(q, s1, s); /* s1 largest integer < a1*b1 + n/s^2 congruent to c1 mod s */
if (fmpz_cmp(q, c1) < 0)
fmpz_sub(s1, s1, s);
fmpz_sub(s1, s1, q);
fmpz_add(s1, s1, c1);
fmpz_add(s2, s2, s2); /* s2 = 2*a1*b1 - 1 */
fmpz_sub_ui(s2, s2, 1);
}
while (fmpz_cmp(s1, s2) > 0) /* for each value s1 in range */
{
fmpz_mul(d, s, s1); /* d = (s1*s + a1*r + b1*r2)^2 - 4*a1*b1*n */
fmpz_addmul(d, a1, r);
fmpz_addmul(d, b1, r2);
fmpz_set(d1, d); /* d1 = s1*s + a1*r + b1*r2 */
fmpz_mul(d, d, d);
fmpz_mul(q, a1, b1);
fmpz_mul(q, q, n);
fmpz_submul_ui(d, q, 4);
if (fmpz_is_square(d)) /* divisor exists, roots are (d1 +/- sqrt(d))/2 */
{
fmpz_sqrt(d, d);
fmpz_add(d, d, d1);
fmpz_tdiv_q_2exp(d, d, 1);
if (fmpz_is_zero(a1))
{
fmpz_tdiv_q(fac, s1, b1); /* y*b1 = s1 */
fmpz_mul(fac, fac, s); /* check if ys + r2 is factor */
fmpz_add(fac, fac, r2);
fmpz_abs(fac, fac);
if (!fmpz_is_zero(fac) && !fmpz_is_one(fac) && !fmpz_equal(fac, n) && fmpz_divisible(n, fac))
{
res = 1;
break;
}
} else if (fmpz_is_zero(b1))
{
fmpz_tdiv_q(fac, s1, a1); /* x*a1 = s1 */
fmpz_mul(fac, fac, s); /* check if xs + r is factor */
fmpz_add(fac, fac, r);
fmpz_abs(fac, fac);
if (!fmpz_is_zero(fac) && !fmpz_is_one(fac) && !fmpz_equal(fac, n) && fmpz_divisible(n, fac))
{
res = 1;
break;
}
} else
{
/* either d/a1 or d/b1 is a divisor of n */
fmpz_tdiv_q(fac, d, a1);
fmpz_abs(fac, fac);
if (!fmpz_is_zero(fac) && !fmpz_is_one(fac) && !fmpz_equal(fac, n) && fmpz_divisible(n, fac))
{
res = 1;
break;
}
fmpz_tdiv_q(fac, d, b1);
fmpz_abs(fac, fac);
if (!fmpz_is_zero(fac) && !fmpz_is_one(fac) && !fmpz_equal(fac, n) && fmpz_divisible(n, fac))
{
res = 1;
break;
}
}
}
fmpz_sub(s1, s1, s);
}
if (fmpz_is_zero(a1) || res == 1) /* Euclidean chain has terminated */
break;
/*
Euclidean chain:
a1, a0 = a0 - q*a1, a1,
where 0 <= a1 < a0 for i even
and 0 < a1 <= a0 for i odd
*/
fmpz_tdiv_qr(q, a0, a0, a1);
if ((i & 1) == 0 && fmpz_is_zero(a0))
{
fmpz_sub_ui(q, q, 1);
fmpz_add(a0, a0, a1);
}
fmpz_swap(a0, a1);
/* b1, b0 = b0 - q*b1, b1 */
fmpz_submul(b0, q, b1);
fmpz_swap(b0, b1);
/* c1, c0 = c0 - q*c1, c1 mod s */
fmpz_submul(c0, q, c1);
fmpz_mod(c0, c0, s);
fmpz_swap(c0, c1);
}
cleanup:
fmpz_clear(r1);
fmpz_clear(r2);
fmpz_clear(a0);
fmpz_clear(a1);
fmpz_clear(b0);
fmpz_clear(b1);
fmpz_clear(c0);
fmpz_clear(c1);
fmpz_clear(q);
fmpz_clear(d);
fmpz_clear(d1);
fmpz_clear(s1);
fmpz_clear(s2);
fmpz_clear(ns2);
return res;
}