numbertheory.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 <stdlib.h>

#include "bits.h"
#include "numbertheory.h"
#include "umodarith.h"


/* Bignum: Initialize the Number Theoretic Transform. */


/*
 * Return the nth root of unity in F(p). This corresponds to e**((2*pi*i)/n)
 * in the Fourier transform. We have w**n == 1 (mod p).
 *    n := transform length.
 *    sign := -1 for forward transform, 1 for backward transform.
 *    modnum := one of {P1, P2, P3}.
 */
mpd_uint_t
_mpd_getkernel(mpd_uint_t n, int sign, int modnum)
{
    mpd_uint_t umod, p, r, xi;
#ifdef PPRO
    double dmod;
    uint32_t dinvmod[3];
#endif

    SETMODULUS(modnum);
    r = mpd_roots[modnum]; /* primitive root of F(p) */
    p = umod;
    xi = (p-1) / n;

    if (sign == -1)
        return POWMOD(r, (p-1-xi));
    else
        return POWMOD(r, xi);
}

/*
 * Initialize and return transform parameters.
 *    n := transform length.
 *    sign := -1 for forward transform, 1 for backward transform.
 *    modnum := one of {P1, P2, P3}.
 */
struct fnt_params *
_mpd_init_fnt_params(mpd_size_t n, int sign, int modnum)
{
    struct fnt_params *tparams;
    mpd_uint_t umod;
#ifdef PPRO
    double dmod;
    uint32_t dinvmod[3];
#endif
    mpd_uint_t kernel, w;
    mpd_uint_t i;
    mpd_size_t nhalf;

    assert(ispower2(n));
    assert(sign == -1 || sign == 1);
    assert(P1 <= modnum && modnum <= P3);

    nhalf = n/2;
    tparams = mpd_sh_alloc(sizeof *tparams, nhalf, sizeof (mpd_uint_t));
    if (tparams == NULL) {
        return NULL;
    }

    SETMODULUS(modnum);
    kernel = _mpd_getkernel(n, sign, modnum);

    tparams->modnum = modnum;
    tparams->modulus = umod;
    tparams->kernel = kernel;

    /* wtable[] := w**0, w**1, ..., w**(nhalf-1) */
    w = 1;
    for (i = 0; i < nhalf; i++) {
        tparams->wtable[i] = w;
        w = MULMOD(w, kernel);
    }

    return tparams;
}

/* Initialize wtable of size three. */
void
_mpd_init_w3table(mpd_uint_t w3table[3], int sign, int modnum)
{
    mpd_uint_t umod;
#ifdef PPRO
    double dmod;
    uint32_t dinvmod[3];
#endif
    mpd_uint_t kernel;

    SETMODULUS(modnum);
    kernel = _mpd_getkernel(3, sign, modnum);

    w3table[0] = 1;
    w3table[1] = kernel;
    w3table[2] = POWMOD(kernel, 2);
}