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

/* -*-c-*-
 *
 * $Id: mpx-kmul.c,v 1.2 1999/12/11 10:58:02 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 1.2  1999/12/11 10:58:02  mdw
 * Remove tweakable comments.
 *
 * Revision 1.1  1999/12/10 23:23:51  mdw
 * Karatsuba-Ofman multiplication algorithm.
 *
 */

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

#include <stdio.h>

#include "mpx.h"

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

#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@.
 */

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 -------------------------------------------------*/
Commit	Line	Data
a86e33af	1	/* --c--
a86e33af	2	*
1b756626	3	* $Id: mpx-kmul.c,v 1.2 1999/12/11 10:58:02 mdw Exp $
a86e33af	4	*
	5	* Karatsuba's multiplication algorithm
	6	*
	7	* (c) 1999 Straylight/Edgeware
	8	*/
	9
	10	/----- Licensing notice --------------------------------------------------
	11	*
	12	* This file is part of Catacomb.
	13	*
	14	* Catacomb is free software; you can redistribute it and/or modify
	15	* it under the terms of the GNU Library General Public License as
	16	* published by the Free Software Foundation; either version 2 of the
	17	* License, or (at your option) any later version.
	18	*
	19	* Catacomb is distributed in the hope that it will be useful,
	20	* but WITHOUT ANY WARRANTY; without even the implied warranty of
	21	* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
	22	* GNU Library General Public License for more details.
	23	*
	24	* You should have received a copy of the GNU Library General Public
	25	* License along with Catacomb; if not, write to the Free
	26	* Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
	27	* MA 02111-1307, USA.
	28	*/
	29
	30	/----- Revision history --------------------------------------------------
	31	*
	32	* $Log: mpx-kmul.c,v $
1b756626	33	* Revision 1.2 1999/12/11 10:58:02 mdw
	34	* Remove tweakable comments.
	35	*
a86e33af	36	* Revision 1.1 1999/12/10 23:23:51 mdw
	37	* Karatsuba-Ofman multiplication algorithm.
	38	*
	39	*/
	40
	41	/----- Header files ------------------------------------------------------/
	42
	43	#include <stdio.h>
	44
	45	#include "mpx.h"
	46
	47	/----- Tweakables --------------------------------------------------------/
	48
a86e33af	49	#ifdef TEST_RIG
	50	# undef KARATSUBA_CUTOFF
	51	# define KARATSUBA_CUTOFF 2
	52	#endif
	53
	54	/----- Addition macros ---------------------------------------------------/
	55
	56	#define UADD(dv, av, avl) do { \
	57	mpw *_dv = (dv); \
	58	const mpw _av = (av), _avl = (avl); \
	59	mpw _c = 0; \
	60	\
	61	while (_av < _avl) { \
	62	mpw _a, _b; \
	63	mpd _x; \
	64	_a = *_av++; \
	65	_b = *_dv; \
	66	_x = (mpd)_a + (mpd)_b + _c; \
	67	*_dv++ = MPW(_x); \
	68	_c = _x >> MPW_BITS; \
	69	} \
	70	while (_c) { \
	71	mpd _x = (mpd)*_dv + (mpd)_c; \
	72	*_dv++ = MPW(_x); \
	73	_c = _x >> MPW_BITS; \
	74	} \
	75	} while (0)
	76
	77	#define UADD2(dv, dvl, av, avl, bv, bvl) do { \
	78	mpw _dv = (dv), _dvl = (dvl); \
	79	const mpw _av = (av), _avl = (avl); \
	80	const mpw _bv = (bv), _bvl = (bvl); \
	81	mpw _c = 0; \
	82	\
	83	while (_av < _avl \|\| _bv < _bvl) { \
	84	mpw _a, _b; \
	85	mpd _x; \
	86	_a = (_av < _avl) ? *_av++ : 0; \
	87	_b = (_bv < _bvl) ? *_bv++ : 0; \
	88	_x = (mpd)_a + (mpd)_b + _c; \
	89	*_dv++ = MPW(_x); \
	90	_c = _x >> MPW_BITS; \
	91	} \
	92	*_dv++ = _c; \
	93	while (_dv < _dvl) \
	94	*_dv++ = 0; \
	95	} while (0)
	96
	97	#define USUB(dv, av, avl) do { \
	98	mpw *_dv = (dv); \
	99	const mpw _av = (av), _avl = (avl); \
	100	mpw _c = 0; \
	101	\
	102	while (_av < _avl) { \
	103	mpw _a, _b; \
	104	mpd _x; \
	105	_a = *_av++; \
	106	_b = *_dv; \
	107	_x = (mpd)_b - (mpd)_a - _c; \
	108	*_dv++ = MPW(_x); \
	109	if (_x >> MPW_BITS) \
	110	_c = 1; \
	111	else \
	112	_c = 0; \
113	} \
114	while (_c) { \
115	mpd _x = (mpd)*_dv - (mpd)_c; \
116	*_dv++ = MPW(_x); \
117	if (_x >> MPW_BITS) \
118	_c = 1; \
119	else \
120	_c = 0; \
121	} \
122	} while (0)
123
124	/----- Main code ---------------------------------------------------------/
125
126	/* --- @mpx_kmul@ --- *
127	*
128	* Arguments: @mpw dv, dvl@ = pointer to destination buffer
129	* @const mpw av, avl@ = pointer to first argument
130	* @const mpw bv, bvl@ = pointer to second argument
131	* @mpw sv, svl@ = pointer to scratch workspace
132	*
133	* Returns: ---
134	*
135	* Use: Multiplies two multiprecision integers using Karatsuba's
136	* algorithm. This is rather faster than traditional long
137	* multiplication (e.g., @mpx_umul@) on large numbers, although
138	* more expensive on small ones.
139	*
140	* The destination must be twice as large as the larger
141	* argument. The scratch space must be twice as large as the
142	* larger argument, plus the magic number @KARATSUBA_SLOP@.
a86e33af	143	*/
	144
	145	void mpx_kmul(mpw dv, mpw dvl,
	146	const mpw av, const mpw avl,
	147	const mpw bv, const mpw bvl,
	148	mpw sv, mpw svl)
	149	{
	150	const mpw avm, bvm;
	151	size_t m;
	152
	153	/* --- Dispose of easy cases to @mpx_umul@ --- *
	154	*
	155	* Karatsuba is only a win on large numbers, because of all the
	156	* recursiveness and bookkeeping. The recursive calls make a quick check
	157	* to see whether to bottom out to @mpx_umul@ which should help quite a
	158	* lot, but sometimes the only way to know is to make sure...
	159	*/
	160
	161	MPX_SHRINK(av, avl);
	162	MPX_SHRINK(bv, bvl);
	163
	164	if (avl - av <= KARATSUBA_CUTOFF \|\| bvl - bv <= KARATSUBA_CUTOFF) {
	165	mpx_umul(dv, dvl, av, avl, bv, bvl);
	166	return;
	167	}
	168
	169	/* --- How the algorithm works --- *
	170	*
	171	* Let %$A = xb + y$% and %$B = ub + v$%. Then, simply by expanding, %$AB
	172	* = x u b^2 + b(x v + y u) + y v$%. That's not helped any, because I've
	173	* got four multiplications, each four times easier than the one I started
	174	* with. However, note that I can rewrite the coefficient of %$b$% as
	175	* %$xv + yu = (x + y)(u + v) - xu - yv$%. The terms %$xu$% and %$yv$%
	176	* I've already calculated, and that leaves only one more multiplication to
	177	* do. So now I have three multiplications, each four times easier, and
	178	* that's a win.
	179	*/
	180
	181	/* --- First things --- *
	182	*
	183	* Sort out where to break the factors in half. I'll choose the midpoint
	184	* of the largest one, since this minimizes the amount of work I have to do
	185	* most effectively.
	186	*/
	187
	188	if (avl - av > bvl - bv) {
	189	m = (avl - av + 1) >> 1;
	190	avm = av + m;
	191	if (bvl - bv > m)
	192	bvm = bv + m;
	193	else
	194	bvm = bvl;
	195	} else {
	196	m = (bvl - bv + 1) >> 1;
	197	bvm = bv + m;
	198	if (avl - av > m)
	199	avm = av + m;
	200	else
	201	avm = avl;
	202	}
	203
	204	/* --- Sort out the middle term --- *
	205	*
	206	* I'm going to keep track of the carry by hand rather than pass it down to
207	* the next level, because it means multiplication by one or zero, which I
208	* can do easily myself.
209	*/
210
211	{
212	unsigned f = 0;
213	enum {
214	carry_a = 1,
215	carry_b = 2
216	};
217
218	mpw bsv = sv + m, ssv = bsv + m;
219	mpw rdv = dv + m, rdvl = rdv + 2 * m;
220
221	UADD2(sv, bsv + 1, av, avm, avm, avl);
222	if (*bsv)
223	f \|= carry_a;
224	UADD2(bsv, ssv + 1, bv, bvm, bvm, bvl);
225	if (*ssv)
226	f \|= carry_b;
227	MPX_ZERO(dv, rdv);
228	if (m > KARATSUBA_CUTOFF)
229	mpx_kmul(rdv, rdvl, sv, bsv, bsv, ssv, ssv, svl);
230	else
231	mpx_umul(rdv, rdvl, sv, bsv, bsv, ssv);
232	MPX_ZERO(rdvl, dvl);
233	rdv += m; rdvl += m;
234	if (f & carry_b)
235	UADD(rdv, sv, bsv);
236	if (f & carry_a)
237	UADD(rdv, bsv, ssv);
238	if (!(~f & (carry_a \| carry_b)))
239	MPX_UADDN(rdv + m, rdvl, 1);
240	}
241
242	/* --- Sort out the other two terms --- */
243
244	{
245	mpw ssv = sv + 2 m;
246	mpw *tdv = dv + m;
247	mpw *rdv = tdv + m;
248
249	if (m > KARATSUBA_CUTOFF)
250	mpx_kmul(sv, ssv, avm, avl, bvm, bvl, ssv, svl);
251	else
252	mpx_umul(sv, ssv, avm, avl, bvm, bvl);
253	UADD(rdv, sv, ssv);
254	USUB(tdv, sv, ssv);
255
256	if (m > KARATSUBA_CUTOFF)
257	mpx_kmul(sv, ssv, av, avm, bv, bvm, ssv, svl);
258	else
259	mpx_umul(sv, ssv, av, avm, bv, bvm);
260	USUB(tdv, sv, ssv);
261	UADD(dv, sv, ssv);
262	}
263	}
264
265	/----- Test rig ----------------------------------------------------------/
266
267	#ifdef TEST_RIG
268
269	#include <mLib/alloc.h>
270	#include <mLib/testrig.h>
271
272	#include "mpscan.h"
273
274	#define ALLOC(v, vl, sz) do { \
275	size_t _sz = (sz); \
276	mpw *_vv = xmalloc(MPWS(_sz)); \
277	mpw *_vvl = _vv + _sz; \
278	(v) = _vv; \
279	(vl) = _vvl; \
280	} while (0)
281
282	#define LOAD(v, vl, d) do { \
283	const dstr *_d = (d); \
284	mpw _v, _vl; \
285	ALLOC(_v, _vl, MPW_RQ(_d->len)); \
286	mpx_loadb(_v, _vl, _d->buf, _d->len); \
287	(v) = _v; \
288	(vl) = _vl; \
289	} while (0)
290
291	#define MAX(x, y) ((x) > (y) ? (x) : (y))
292
293	static void dumpmp(const char msg, const mpw v, const mpw *vl)
294	{
295	fputs(msg, stderr);
296	MPX_SHRINK(v, vl);
297	while (v < vl)
298	fprintf(stderr, " %08lx", (unsigned long)*--vl);
299	fputc('\n', stderr);
300	}
301
302	static int umul(dstr *v)
303	{
304	mpw a, al;
305	mpw b, bl;
306	mpw c, cl;
307	mpw d, dl;
308	mpw s, sl;
309	size_t m;
310	int ok = 1;
311
312	LOAD(a, al, &v[0]);
313	LOAD(b, bl, &v[1]);
314	LOAD(c, cl, &v[2]);
315	m = MAX(al - a, bl - b) + 1;
316	ALLOC(d, dl, 2 * m);
317	ALLOC(s, sl, 2 * m + 32);
318
319	mpx_kmul(d, dl, a, al, b, bl, s, sl);
320	if (MPX_UCMP(d, dl, !=, c, cl)) {
321	fprintf(stderr, "\n*** umul failed\n");
322	dumpmp(" a", a, al);
323	dumpmp(" b", b, bl);
324	dumpmp("expected", c, cl);
325	dumpmp(" result", d, dl);
326	ok = 0;
327	}
328
329	free(a); free(b); free(c); free(d); free(s);
330	return (ok);
331	}
332
333	static test_chunk defs[] = {
334	{ "umul", umul, { &type_hex, &type_hex, &type_hex, 0 } },
335	{ 0, 0, { 0 } }
336	};
337
338	int main(int argc, char *argv[])
339	{
340	test_run(argc, argv, defs, SRCDIR"/tests/mpx");
341	return (0);
342	}
343
344	#endif
345
346	/----- That's all, folks -------------------------------------------------/