Use a Karatsuba-based reduction for large moduli.

[u/mdw/catacomb] / mpx-kmul.c

/* -*-c-*-
 *
 * $Id: mpx-kmul.c,v 1.1 1999/12/10 23:23:51 mdw Exp $
 *
 * Karatsuba's multiplication algorithm
 *
 * (c) 1999 Straylight/Edgeware
 */

/*----- Licensing notice --------------------------------------------------* 
 *
 * This file is part of Catacomb.
 *
 * Catacomb is free software; you can redistribute it and/or modify
 * it under the terms of the GNU Library General Public License as
 * published by the Free Software Foundation; either version 2 of the
 * License, or (at your option) any later version.
 * 
 * Catacomb 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 Library General Public License for more details.
 * 
 * You should have received a copy of the GNU Library General Public
 * License along with Catacomb; if not, write to the Free
 * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
 * MA 02111-1307, USA.
 */

/*----- Revision history --------------------------------------------------* 
 *
 * $Log: mpx-kmul.c,v $
 * Revision 1.1  1999/12/10 23:23:51  mdw
 * Karatsuba-Ofman multiplication algorithm.
 *
 */

/*----- Header files ------------------------------------------------------*/

#include <stdio.h>

#include "mpx.h"

/*----- Tweakables --------------------------------------------------------*/

/* --- @KARATSUBA_CUTOFF@ --- *
 *
 * If either of the arguments to @mpx_kmul@ contains this number of words or
 * fewer, the job is dumped out to @mpx_umul@ instead.  Reduce the size when
 * testing, to ensure better coverage.
 */

#ifdef TEST_RIG
#  undef KARATSUBA_CUTOFF
#  define KARATSUBA_CUTOFF 2
#endif

/*----- Addition macros ---------------------------------------------------*/

#define UADD(dv, av, avl) do {						\
  mpw *_dv = (dv);							\
  const mpw *_av = (av), *_avl = (avl);					\
  mpw _c = 0;								\
									\
  while (_av < _avl) {							\
    mpw _a, _b;								\
    mpd _x;								\
    _a = *_av++;							\
    _b = *_dv;								\
    _x = (mpd)_a + (mpd)_b + _c;					\
    *_dv++ = MPW(_x);							\
    _c = _x >> MPW_BITS;						\
  }									\
  while (_c) {								\
    mpd _x = (mpd)*_dv + (mpd)_c;					\
    *_dv++ = MPW(_x);							\
    _c = _x >> MPW_BITS;						\
  }									\
} while (0)

#define UADD2(dv, dvl, av, avl, bv, bvl) do {				\
  mpw *_dv = (dv), *_dvl = (dvl);					\
  const mpw *_av = (av), *_avl = (avl);					\
  const mpw *_bv = (bv), *_bvl = (bvl);					\
  mpw _c = 0;								\
									\
  while (_av < _avl || _bv < _bvl) {					\
    mpw _a, _b;								\
    mpd _x;								\
    _a = (_av < _avl) ? *_av++ : 0;					\
    _b = (_bv < _bvl) ? *_bv++ : 0;					\
    _x = (mpd)_a + (mpd)_b + _c;					\
    *_dv++ = MPW(_x);							\
    _c = _x >> MPW_BITS;						\
  }									\
  *_dv++ = _c;								\
  while (_dv < _dvl)							\
    *_dv++ = 0;								\
} while (0)

#define USUB(dv, av, avl) do {						\
  mpw *_dv = (dv);							\
  const mpw *_av = (av), *_avl = (avl);					\
  mpw _c = 0;								\
									\
  while (_av < _avl) {							\
    mpw _a, _b;								\
    mpd _x;								\
    _a = *_av++;							\
    _b = *_dv;								\
    _x = (mpd)_b - (mpd)_a - _c;					\
    *_dv++ = MPW(_x);							\
    if (_x >> MPW_BITS)							\
      _c = 1;								\
    else								\
      _c = 0;								\
  }									\
  while (_c) {								\
    mpd _x = (mpd)*_dv - (mpd)_c;					\
    *_dv++ = MPW(_x);							\
    if (_x >> MPW_BITS)							\
      _c = 1;								\
    else								\
      _c = 0;								\
  }									\
} while (0)

/*----- Main code ---------------------------------------------------------*/

/* --- @mpx_kmul@ --- *
 *
 * Arguments:	@mpw *dv, *dvl@ = pointer to destination buffer
 *		@const mpw *av, *avl@ = pointer to first argument
 *		@const mpw *bv, *bvl@ = pointer to second argument
 *		@mpw *sv, *svl@ = pointer to scratch workspace
 *
 * Returns:	---
 *
 * Use:		Multiplies two multiprecision integers using Karatsuba's
 *		algorithm.  This is rather faster than traditional long
 *		multiplication (e.g., @mpx_umul@) on large numbers, although
 *		more expensive on small ones.
 *
 *		The destination must be twice as large as the larger
 *		argument.  The scratch space must be twice as large as the
 *		larger argument, plus the magic number @KARATSUBA_SLOP@.
 *		(Actually, a number of words proportional to the depth of
 *		recursion, but since recusion is strongly bounded by memory,
 *		I can replace it with a constant as long as it's `big
 *		enough'.)
 */

void mpx_kmul(mpw *dv, mpw *dvl,
	      const mpw *av, const mpw *avl,
	      const mpw *bv, const mpw *bvl,
	      mpw *sv, mpw *svl)
{
  const mpw *avm, *bvm;
  size_t m;

  /* --- Dispose of easy cases to @mpx_umul@ --- *
   *
   * Karatsuba is only a win on large numbers, because of all the
   * recursiveness and bookkeeping.  The recursive calls make a quick check
   * to see whether to bottom out to @mpx_umul@ which should help quite a
   * lot, but sometimes the only way to know is to make sure...
   */

  MPX_SHRINK(av, avl);
  MPX_SHRINK(bv, bvl);

  if (avl - av <= KARATSUBA_CUTOFF || bvl - bv <= KARATSUBA_CUTOFF) {
    mpx_umul(dv, dvl, av, avl, bv, bvl);
    return;
  }

  /* --- How the algorithm works --- *
   *
   * Let %$A = xb + y$% and %$B = ub + v$%.  Then, simply by expanding, %$AB
   * = x u b^2 + b(x v + y u) + y v$%.  That's not helped any, because I've
   * got four multiplications, each four times easier than the one I started
   * with.  However, note that I can rewrite the coefficient of %$b$% as
   * %$xv + yu = (x + y)(u + v) - xu - yv$%.  The terms %$xu$% and %$yv$%
   * I've already calculated, and that leaves only one more multiplication to
   * do.  So now I have three multiplications, each four times easier, and
   * that's a win.
   */

  /* --- First things --- *
   *
   * Sort out where to break the factors in half.  I'll choose the midpoint
   * of the largest one, since this minimizes the amount of work I have to do
   * most effectively.
   */

  if (avl - av > bvl - bv) {
    m = (avl - av + 1) >> 1;
    avm = av + m;
    if (bvl - bv > m)
      bvm = bv + m;
    else
      bvm = bvl;
  } else {
    m = (bvl - bv + 1) >> 1;
    bvm = bv + m;
    if (avl - av > m)
      avm = av + m;
    else
      avm = avl;
  }

  /* --- Sort out the middle term --- *
   *
   * I'm going to keep track of the carry by hand rather than pass it down to
   * the next level, because it means multiplication by one or zero, which I
   * can do easily myself.
   */

  {
    unsigned f = 0;
    enum {
      carry_a = 1,
      carry_b = 2
    };

    mpw *bsv = sv + m, *ssv = bsv + m;
    mpw *rdv = dv + m, *rdvl = rdv + 2 * m;

    UADD2(sv, bsv + 1, av, avm, avm, avl);
    if (*bsv)
      f |= carry_a;
    UADD2(bsv, ssv + 1, bv, bvm, bvm, bvl);
    if (*ssv)
      f |= carry_b;
    MPX_ZERO(dv, rdv);
    if (m > KARATSUBA_CUTOFF)
      mpx_kmul(rdv, rdvl, sv, bsv, bsv, ssv, ssv, svl);
    else
      mpx_umul(rdv, rdvl, sv, bsv, bsv, ssv);
    MPX_ZERO(rdvl, dvl);
    rdv += m; rdvl += m;
    if (f & carry_b)
      UADD(rdv, sv, bsv);
    if (f & carry_a)
      UADD(rdv, bsv, ssv);
    if (!(~f & (carry_a | carry_b)))
      MPX_UADDN(rdv + m, rdvl, 1);
  }

  /* --- Sort out the other two terms --- */

  {
    mpw *ssv = sv + 2 * m;
    mpw *tdv = dv + m;
    mpw *rdv = tdv + m;

    if (m > KARATSUBA_CUTOFF)
      mpx_kmul(sv, ssv, avm, avl, bvm, bvl, ssv, svl);
    else
      mpx_umul(sv, ssv, avm, avl, bvm, bvl);
    UADD(rdv, sv, ssv);
    USUB(tdv, sv, ssv);
    
    if (m > KARATSUBA_CUTOFF)
      mpx_kmul(sv, ssv, av, avm, bv, bvm, ssv, svl);
    else
      mpx_umul(sv, ssv, av, avm, bv, bvm);
    USUB(tdv, sv, ssv);
    UADD(dv, sv, ssv);
  }
}

/*----- Test rig ----------------------------------------------------------*/

#ifdef TEST_RIG

#include <mLib/alloc.h>
#include <mLib/testrig.h>

#include "mpscan.h"

#define ALLOC(v, vl, sz) do {                                           \
  size_t _sz = (sz);                                                    \
  mpw *_vv = xmalloc(MPWS(_sz));                                        \
  mpw *_vvl = _vv + _sz;                                                \
  (v) = _vv;                                                            \
  (vl) = _vvl;                                                          \
} while (0)

#define LOAD(v, vl, d) do {                                             \
  const dstr *_d = (d);                                                 \
  mpw *_v, *_vl;                                                        \
  ALLOC(_v, _vl, MPW_RQ(_d->len));                                      \
  mpx_loadb(_v, _vl, _d->buf, _d->len);                                 \
  (v) = _v;                                                             \
  (vl) = _vl;                                                           \
} while (0)

#define MAX(x, y) ((x) > (y) ? (x) : (y))

static void dumpmp(const char *msg, const mpw *v, const mpw *vl)
{
  fputs(msg, stderr);
  MPX_SHRINK(v, vl);
  while (v < vl)
    fprintf(stderr, " %08lx", (unsigned long)*--vl);
  fputc('\n', stderr);
}

static int umul(dstr *v)
{
  mpw *a, *al;
  mpw *b, *bl;
  mpw *c, *cl;
  mpw *d, *dl;
  mpw *s, *sl;
  size_t m;
  int ok = 1;

  LOAD(a, al, &v[0]);
  LOAD(b, bl, &v[1]);
  LOAD(c, cl, &v[2]);
  m = MAX(al - a, bl - b) + 1;
  ALLOC(d, dl, 2 * m);
  ALLOC(s, sl, 2 * m + 32);

  mpx_kmul(d, dl, a, al, b, bl, s, sl);
  if (MPX_UCMP(d, dl, !=, c, cl)) {
    fprintf(stderr, "\n*** umul failed\n");
    dumpmp("       a", a, al);
    dumpmp("       b", b, bl);
    dumpmp("expected", c, cl);
    dumpmp("  result", d, dl);
    ok = 0;
  }

  free(a); free(b); free(c); free(d); free(s);
  return (ok);
}

static test_chunk defs[] = {
  { "umul", umul, { &type_hex, &type_hex, &type_hex, 0 } },
  { 0, 0, { 0 } }
};

int main(int argc, char *argv[])
{
  test_run(argc, argv, defs, SRCDIR"/tests/mpx");
  return (0);
}

#endif

/*----- That's all, folks -------------------------------------------------*/