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 "umodarith.h"
#include "typearith.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)
{
    mpd_uint_t hi1, hi2, lo;

    _mpd_mul_words(&hi1, &lo, LH_P1P2, v);
    w[0] = lo;

    _mpd_mul_words(&hi2, &lo, UH_P1P2, v);
    lo = hi1 + lo;
    if (lo < hi1) hi2++;

    w[1] = lo;
    w[2] = hi2;
}

/* 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])
{
    mpd_uint_t carry;
    mpd_uint_t s;

    s = w[0] + v[0];
    carry = (s < w[0]);
    w[0] = s;

    s = w[1] + (v[1] + carry);
    carry = (s < w[1]);
    w[1] = s;

    w[2] = w[2] + (v[2] + carry);
}

/* 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)
{
    mpd_uint_t r1 = u[2];
    mpd_uint_t r2;

    if (r1 < v) {
        w[2] = 0;
    }
    else {
        _mpd_div_word(&w[2], &r1, u[2], v); /* GCOV_NOT_REACHED */
    }

    _mpd_div_words(&w[1], &r2, r1, u[1], v);
    _mpd_div_words(&w[0], &r1, r2, u[0], v);

    return r1;
}


/*
 * 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)
{
    mpd_uint_t p1 = mpd_moduli[P1];
    mpd_uint_t umod;
#ifdef PPRO
    double dmod;
    uint32_t dinvmod[3];
#endif
    mpd_uint_t a1, a2, a3;
    mpd_uint_t s;
    mpd_uint_t z[3], t[3];
    mpd_uint_t carry[3] = {0,0,0};
    mpd_uint_t hi, lo;
    mpd_size_t i;

    for (i = 0; i < rsize; i++) {

        a1 = x1[i];
        a2 = x2[i];
        a3 = x3[i];

        SETMODULUS(P2);
        s = ext_submod(a2, a1, umod);
        s = MULMOD(s, INV_P1_MOD_P2);

        _mpd_mul_words(&hi, &lo, s, p1);
        lo = lo + a1;
        if (lo < a1) hi++;

        SETMODULUS(P3);
        s = dw_submod(a3, hi, lo, umod);
        s = MULMOD(s, INV_P1P2_MOD_P3);

        z[0] = lo;
        z[1] = hi;
        z[2] = 0;

        _crt_mulP1P2_3(t, s);
        _crt_add3(z, t);
        _crt_add3(carry, z);

        x1[i] = _crt_div3(carry, carry, MPD_RADIX);
    }

    assert(carry[0] == 0 && carry[1] == 0 && carry[2] == 0);
}