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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
=============================================================================*/
/******************************************************************************
Copyright (C) 2011 Jan Tuitman
Copyright (C) 2011, 2012 Sebastian Pancratz
******************************************************************************/
#include "padic.h"
/*
Returns whether \code{op} has a square root modulo $p^N$ and if
so sets \code{rop} to such an element.
Assumes that \code{op} is a unit modulo $p^N$. Assumes $p$ is an
odd prime.
In the current implementation, allows aliasing.
*/
static int _padic_sqrt_p(fmpz_t rop, const fmpz_t op, const fmpz_t p, slong N)
{
int ans;
if (N == 1)
{
ans = fmpz_sqrtmod(rop, op, p);
return ans;
}
else
{
slong *e, i, n;
fmpz *W, *pow, *u;
e = _padic_lifts_exps(&n, N);
W = _fmpz_vec_init(2 + 2 * n);
pow = W + 2;
u = W + (2 + n);
_padic_lifts_pows(pow, e, n, p);
/* Compute reduced units */
{
fmpz_mod(u, op, pow);
}
for (i = 1; i < n; i++)
{
fmpz_mod(u + i, u + (i - 1), pow + i);
}
/*
Run Newton iteration for the inverse square root,
using the update formula
z := z - z (u z^2 - 1) / 2
for all but the last step. The last step is
replaced with
b := u z mod p^{N'}
z := b + z (u - b^2) / 2 mod p^{N}.
*/
i = n - 1;
{
ans = fmpz_sqrtmod(rop, u + i, p);
if (!ans)
goto exit;
fmpz_invmod(rop, rop, p);
}
for (i--; i >= 1; i--)
{
fmpz_mul(W, rop, rop);
fmpz_mul(W + 1, u + i, W);
fmpz_sub_ui(W + 1, W + 1, 1);
if (fmpz_is_odd(W + 1))
fmpz_add(W + 1, W + 1, pow + i);
fmpz_fdiv_q_2exp(W + 1, W + 1, 1);
fmpz_mul(W, W + 1, rop);
fmpz_sub(rop, rop, W);
fmpz_mod(rop, rop, pow + i);
}
{
fmpz_mul(W, u + 1, rop);
fmpz_mul(W + 1, W, W);
fmpz_sub(W + 1, u + 0, W + 1);
if (fmpz_is_odd(W + 1))
fmpz_add(W + 1, W + 1, pow + 0);
fmpz_fdiv_q_2exp(W + 1, W + 1, 1);
fmpz_mul(rop, rop, W + 1);
fmpz_add(rop, W, rop);
fmpz_mod(rop, rop, pow + 0);
}
exit:
flint_free(e);
_fmpz_vec_clear(W, 2 + 2 * n);
return ans;
}
}
/*
Returns whether \code{op} has a square root modulo $2^N$ and if
so sets \code{rop} to such an element.
Assumes that \code{op} is a unit modulo $2^N$.
In the current implementation, allows aliasing.
*/
static int _padic_sqrt_2(fmpz_t rop, const fmpz_t op, slong N)
{
if (fmpz_fdiv_ui(op, 8) != 1)
return 0;
if (N <= 3)
{
fmpz_one(rop);
}
else
{
slong *e, i, n;
fmpz *W, *u;
i = FLINT_CLOG2(N);
/* Compute sequence of exponents */
e = flint_malloc((i + 2) * sizeof(slong));
for (e[i = 0] = N; e[i] > 3; i++)
e[i + 1] = (e[i] + 3) / 2;
n = i + 1;
W = _fmpz_vec_init(2 + n);
u = W + 2;
/* Compute reduced units */
{
fmpz_fdiv_r_2exp(u, op, e[0]);
}
for (i = 1; i < n; i++)
{
fmpz_fdiv_r_2exp(u + i, u + (i - 1), e[i]);
}
/* Run Newton iteration */
fmpz_one(rop);
for (i = n - 2; i >= 1; i--) /* z := z - z (a z^2 - 1) / 2 */
{
fmpz_mul(W, rop, rop);
fmpz_mul(W + 1, u + i, W);
fmpz_sub_ui(W + 1, W + 1, 1);
fmpz_fdiv_q_2exp(W + 1, W + 1, 1);
fmpz_mul(W, W + 1, rop);
fmpz_sub(rop, rop, W);
fmpz_fdiv_r_2exp(rop, rop, e[i]);
}
{
fmpz_mul(W, u + 1, rop);
fmpz_mul(W + 1, W, W);
fmpz_sub(W + 1, u + 0, W + 1);
fmpz_fdiv_q_2exp(W + 1, W + 1, 1);
fmpz_mul(rop, rop, W + 1);
fmpz_add(rop, W, rop);
}
fmpz_fdiv_r_2exp(rop, rop, e[0]);
flint_free(e);
_fmpz_vec_clear(W, 2 + n);
}
return 1;
}
int _padic_sqrt(fmpz_t rop, const fmpz_t op, const fmpz_t p, slong N)
{
if (fmpz_equal_ui(p, 2))
{
return _padic_sqrt_2(rop, op, N);
}
else
{
return _padic_sqrt_p(rop, op, p, N);
}
}
int padic_sqrt(padic_t rop, const padic_t op, const padic_ctx_t ctx)
{
if (padic_is_zero(op))
{
padic_zero(rop);
return 1;
}
if (padic_val(op) & WORD(1))
{
return 0;
}
padic_val(rop) = padic_val(op) / 2;
/*
In this case, if there is a square root it will be
zero modulo $p^N$. We only have to establish whether
or not the element \code{op} is a square.
*/
if (padic_val(rop) >= padic_prec(rop))
{
int ans;
if (fmpz_equal_ui(ctx->p, 2))
{
ans = (fmpz_fdiv_ui(padic_unit(op), 8) == 1);
}
else
{
ans = fmpz_sqrtmod(padic_unit(rop), padic_unit(op), ctx->p);
}
padic_zero(rop);
return ans;
}
return _padic_sqrt(padic_unit(rop),
padic_unit(op), ctx->p, padic_prec(rop) - padic_val(rop));
}
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