cpython/Modules/_decimal/libmpdec/crt.c

/*
 * Copyright (c) 2008-2020 Stefan Krah. All rights reserved.
 *
 * Redistribution and use in source and binary forms, with or without
 * modification, are permitted provided that the following conditions
 * are met:
 *
 * 1. Redistributions of source code must retain the above copyright
 *    notice, this list of conditions and the following disclaimer.
 *
 * 2. Redistributions in binary form must reproduce the above copyright
 *    notice, this list of conditions and the following disclaimer in the
 *    documentation and/or other materials provided with the distribution.
 *
 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS "AS IS" AND
 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
 * ARE DISCLAIMED.  IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
 * SUCH DAMAGE.
 */


#include "mpdecimal.h"

#include <assert.h>

#include "constants.h"
#include "crt.h"
#include "numbertheory.h"
#include "typearith.h"
#include "umodarith.h"


/* Bignum: Chinese Remainder Theorem, extends the maximum transform length. */


/* Multiply P1P2 by v, store result in w. */
static inline void
_crt_mulP1P2_3(mpd_uint_t w[3], mpd_uint_t v)
{ … }

/* Add 3 words from v to w. The result is known to fit in w. */
static inline void
_crt_add3(mpd_uint_t w[3], mpd_uint_t v[3])
{ … }

/* Divide 3 words in u by v, store result in w, return remainder. */
static inline mpd_uint_t
_crt_div3(mpd_uint_t *w, const mpd_uint_t *u, mpd_uint_t v)
{ … }


/*
 * Chinese Remainder Theorem:
 * Algorithm from Joerg Arndt, "Matters Computational",
 * Chapter 37.4.1 [http://www.jjj.de/fxt/]
 *
 * See also Knuth, TAOCP, Volume 2, 4.3.2, exercise 7.
 */

/*
 * CRT with carry: x1, x2, x3 contain numbers modulo p1, p2, p3. For each
 * triple of members of the arrays, find the unique z modulo p1*p2*p3, with
 * zmax = p1*p2*p3 - 1.
 *
 * In each iteration of the loop, split z into result[i] = z % MPD_RADIX
 * and carry = z / MPD_RADIX. Let N be the size of carry[] and cmax the
 * maximum carry.
 *
 * Limits for the 32-bit build:
 *
 *   N    = 2**96
 *   cmax = 7711435591312380274
 *
 * Limits for the 64 bit build:
 *
 *   N    = 2**192
 *   cmax = 627710135393475385904124401220046371710
 *
 * The following statements hold for both versions:
 *
 *   1) cmax + zmax < N, so the addition does not overflow.
 *
 *   2) (cmax + zmax) / MPD_RADIX == cmax.
 *
 *   3) If c <= cmax, then c_next = (c + zmax) / MPD_RADIX <= cmax.
 */
void
crt3(mpd_uint_t *x1, mpd_uint_t *x2, mpd_uint_t *x3, mpd_size_t rsize)
{ … }