243 lines
6.2 KiB
C
243 lines
6.2 KiB
C
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/*=============================================================================
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This file is part of FLINT.
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FLINT is free software; you can redistribute it and/or modify
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it under the terms of the GNU General Public License as published by
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the Free Software Foundation; either version 2 of the License, or
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(at your option) any later version.
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FLINT is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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GNU General Public License for more details.
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You should have received a copy of the GNU General Public License
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along with FLINT; if not, write to the Free Software
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Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
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=============================================================================*/
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/******************************************************************************
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Copyright (C) 2011 Jan Tuitman
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Copyright (C) 2011, 2012 Sebastian Pancratz
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******************************************************************************/
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#include "padic.h"
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/*
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Returns whether \code{op} has a square root modulo $p^N$ and if
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so sets \code{rop} to such an element.
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Assumes that \code{op} is a unit modulo $p^N$. Assumes $p$ is an
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odd prime.
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In the current implementation, allows aliasing.
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*/
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static int _padic_sqrt_p(fmpz_t rop, const fmpz_t op, const fmpz_t p, slong N)
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{
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int ans;
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if (N == 1)
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{
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ans = fmpz_sqrtmod(rop, op, p);
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return ans;
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}
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else
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{
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slong *e, i, n;
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fmpz *W, *pow, *u;
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e = _padic_lifts_exps(&n, N);
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W = _fmpz_vec_init(2 + 2 * n);
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pow = W + 2;
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u = W + (2 + n);
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_padic_lifts_pows(pow, e, n, p);
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/* Compute reduced units */
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{
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fmpz_mod(u, op, pow);
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}
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for (i = 1; i < n; i++)
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{
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fmpz_mod(u + i, u + (i - 1), pow + i);
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}
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/*
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Run Newton iteration for the inverse square root,
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using the update formula
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z := z - z (u z^2 - 1) / 2
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for all but the last step. The last step is
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replaced with
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b := u z mod p^{N'}
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z := b + z (u - b^2) / 2 mod p^{N}.
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*/
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i = n - 1;
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{
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ans = fmpz_sqrtmod(rop, u + i, p);
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if (!ans)
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goto exit;
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fmpz_invmod(rop, rop, p);
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}
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for (i--; i >= 1; i--)
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{
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fmpz_mul(W, rop, rop);
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fmpz_mul(W + 1, u + i, W);
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fmpz_sub_ui(W + 1, W + 1, 1);
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if (fmpz_is_odd(W + 1))
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fmpz_add(W + 1, W + 1, pow + i);
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fmpz_fdiv_q_2exp(W + 1, W + 1, 1);
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fmpz_mul(W, W + 1, rop);
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fmpz_sub(rop, rop, W);
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fmpz_mod(rop, rop, pow + i);
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}
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{
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fmpz_mul(W, u + 1, rop);
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fmpz_mul(W + 1, W, W);
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fmpz_sub(W + 1, u + 0, W + 1);
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if (fmpz_is_odd(W + 1))
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fmpz_add(W + 1, W + 1, pow + 0);
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fmpz_fdiv_q_2exp(W + 1, W + 1, 1);
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fmpz_mul(rop, rop, W + 1);
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fmpz_add(rop, W, rop);
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fmpz_mod(rop, rop, pow + 0);
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}
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exit:
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flint_free(e);
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_fmpz_vec_clear(W, 2 + 2 * n);
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return ans;
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}
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}
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/*
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Returns whether \code{op} has a square root modulo $2^N$ and if
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so sets \code{rop} to such an element.
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Assumes that \code{op} is a unit modulo $2^N$.
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In the current implementation, allows aliasing.
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*/
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static int _padic_sqrt_2(fmpz_t rop, const fmpz_t op, slong N)
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{
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if (fmpz_fdiv_ui(op, 8) != 1)
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return 0;
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if (N <= 3)
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{
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fmpz_one(rop);
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}
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else
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{
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slong *e, i, n;
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fmpz *W, *u;
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i = FLINT_CLOG2(N);
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/* Compute sequence of exponents */
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e = flint_malloc((i + 2) * sizeof(slong));
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for (e[i = 0] = N; e[i] > 3; i++)
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e[i + 1] = (e[i] + 3) / 2;
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n = i + 1;
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W = _fmpz_vec_init(2 + n);
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u = W + 2;
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/* Compute reduced units */
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{
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fmpz_fdiv_r_2exp(u, op, e[0]);
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}
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for (i = 1; i < n; i++)
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{
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fmpz_fdiv_r_2exp(u + i, u + (i - 1), e[i]);
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}
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/* Run Newton iteration */
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fmpz_one(rop);
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for (i = n - 2; i >= 1; i--) /* z := z - z (a z^2 - 1) / 2 */
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{
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fmpz_mul(W, rop, rop);
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fmpz_mul(W + 1, u + i, W);
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fmpz_sub_ui(W + 1, W + 1, 1);
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fmpz_fdiv_q_2exp(W + 1, W + 1, 1);
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fmpz_mul(W, W + 1, rop);
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fmpz_sub(rop, rop, W);
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fmpz_fdiv_r_2exp(rop, rop, e[i]);
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}
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{
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fmpz_mul(W, u + 1, rop);
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fmpz_mul(W + 1, W, W);
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fmpz_sub(W + 1, u + 0, W + 1);
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fmpz_fdiv_q_2exp(W + 1, W + 1, 1);
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fmpz_mul(rop, rop, W + 1);
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fmpz_add(rop, W, rop);
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}
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fmpz_fdiv_r_2exp(rop, rop, e[0]);
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flint_free(e);
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_fmpz_vec_clear(W, 2 + n);
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}
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return 1;
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}
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int _padic_sqrt(fmpz_t rop, const fmpz_t op, const fmpz_t p, slong N)
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{
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if (fmpz_equal_ui(p, 2))
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{
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return _padic_sqrt_2(rop, op, N);
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}
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else
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{
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return _padic_sqrt_p(rop, op, p, N);
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}
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}
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int padic_sqrt(padic_t rop, const padic_t op, const padic_ctx_t ctx)
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{
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if (padic_is_zero(op))
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{
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padic_zero(rop);
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return 1;
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}
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if (padic_val(op) & WORD(1))
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{
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return 0;
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}
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padic_val(rop) = padic_val(op) / 2;
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/*
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In this case, if there is a square root it will be
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zero modulo $p^N$. We only have to establish whether
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or not the element \code{op} is a square.
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*/
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if (padic_val(rop) >= padic_prec(rop))
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{
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int ans;
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if (fmpz_equal_ui(ctx->p, 2))
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{
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ans = (fmpz_fdiv_ui(padic_unit(op), 8) == 1);
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}
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else
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{
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ans = fmpz_sqrtmod(padic_unit(rop), padic_unit(op), ctx->p);
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}
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padic_zero(rop);
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return ans;
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}
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return _padic_sqrt(padic_unit(rop),
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padic_unit(op), ctx->p, padic_prec(rop) - padic_val(rop));
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}
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