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/*
Copyright (C) 2009 William Hart
Copyright (C) 2011 Sebastian Pancratz
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 "ulong_extras.h"
ulong n_sqrtmod(ulong a, ulong p)
{
slong i, r, m, iter;
ulong p1, k, b, g, bpow, gpow, res;
ulong pinv;
if (a <= 1)
{
return a;
}
/* just do a brute force search */
if (p < 600)
{
ulong t, t2;
if (p > 50 && n_jacobi_unsigned(a, p) == -1)
return 0;
t = t2 = 0;
while (t < (p - 1) / 2)
{
t2 = n_addmod(t2, 2*t + 1, p);
t++;
if (t2 == a)
return t;
}
return 0;
}
if (n_is_square(p)) /* modulus is a square */
return 0;
if ((p & 1) == 0) /* modulus is even */
return 0;
pinv = n_preinvert_limb(p);
if (n_jacobi_unsigned(a, p) == -1)
return 0;
if ((p & UWORD(3)) == 3)
{
return n_powmod2_ui_preinv(a, (p + 1)/4, p, pinv); /* p == 2^B - 1 isn't prime */
}
if ((p & UWORD(7)) == 5)
{
b = n_powmod2_ui_preinv(a, (p + 3)/8, p, pinv); /* p == 2^B - 3 isn't prime */
g = n_mulmod2_preinv(b, b, p, pinv);
if (g == a)
return b;
g = n_powmod2_ui_preinv(2, (p - 1)/4, p, pinv);
return n_mulmod2_preinv(g, b, p, pinv);
}
r = 0;
p1 = p - 1;
do {
p1 >>= UWORD(1);
r++;
} while ((p1 & UWORD(1)) == 0);
b = n_powmod2_ui_preinv(a, p1, p, pinv);
for (k = 3; ; k+=2) /* 2 is a quadratic residue mod p = 8k + 1 */
{
if (n_jacobi_unsigned(k, p) == -1) break;
}
g = n_powmod2_ui_preinv(k, p1, p, pinv);
res = n_powmod2_ui_preinv(a, (p1 + 1) / 2, p, pinv);
iter = r - 1; /* maximum number of iterations possible if p is prime */
while (b != 1)
{
bpow = b;
m = 0;
do
{
bpow = n_mulmod2_preinv(bpow, bpow, p, pinv);
m++;
} while (m < r && bpow != 1);
gpow = g;
for (i = 1; i < r - m; i++)
{
gpow = n_mulmod2_preinv(gpow, gpow, p, pinv);
}
res = n_mulmod2_preinv(res, gpow, p, pinv);
g = n_mulmod2_preinv(gpow, gpow, p, pinv);
b = n_mulmod2_preinv(b, g, p, pinv);
r = m;
if (iter-- == 0)
return 0; /* too many iterations, p is not prime, prevents hang */
}
return res;
}