3 * $Id: mpx-kmul.c,v 1.3 1999/12/13 15:35:01 mdw Exp $
5 * Karatsuba's multiplication algorithm
7 * (c) 1999 Straylight/Edgeware
10 /*----- Licensing notice --------------------------------------------------*
12 * This file is part of Catacomb.
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.
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.
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,
30 /*----- Revision history --------------------------------------------------*
32 * $Log: mpx-kmul.c,v $
33 * Revision 1.3 1999/12/13 15:35:01 mdw
34 * Simplify and improve.
36 * Revision 1.2 1999/12/11 10:58:02 mdw
37 * Remove tweakable comments.
39 * Revision 1.1 1999/12/10 23:23:51 mdw
40 * Karatsuba-Ofman multiplication algorithm.
44 /*----- Header files ------------------------------------------------------*/
51 /*----- Tweakables --------------------------------------------------------*/
54 # undef KARATSUBA_CUTOFF
55 # define KARATSUBA_CUTOFF 2
58 /*----- Addition macros ---------------------------------------------------*/
60 #define UADD(dv, av, avl) do { \
62 const mpw *_av = (av), *_avl = (avl); \
65 while (_av < _avl) { \
70 _x = (mpd)_a + (mpd)_b + _c; \
72 _c = _x >> MPW_BITS; \
75 mpd _x = (mpd)*_dv + (mpd)_c; \
77 _c = _x >> MPW_BITS; \
81 #define UADD2(dv, dvl, av, avl, bv, bvl) do { \
82 mpw *_dv = (dv), *_dvl = (dvl); \
83 const mpw *_av = (av), *_avl = (avl); \
84 const mpw *_bv = (bv), *_bvl = (bvl); \
87 while (_av < _avl || _bv < _bvl) { \
90 _a = (_av < _avl) ? *_av++ : 0; \
91 _b = (_bv < _bvl) ? *_bv++ : 0; \
92 _x = (mpd)_a + (mpd)_b + _c; \
94 _c = _x >> MPW_BITS; \
101 #define USUB(dv, av, avl) do { \
103 const mpw *_av = (av), *_avl = (avl); \
106 while (_av < _avl) { \
111 _x = (mpd)_b - (mpd)_a - _c; \
113 if (_x >> MPW_BITS) \
119 mpd _x = (mpd)*_dv - (mpd)_c; \
121 if (_x >> MPW_BITS) \
128 /*----- Main code ---------------------------------------------------------*/
130 /* --- @mpx_kmul@ --- *
132 * Arguments: @mpw *dv, *dvl@ = pointer to destination buffer
133 * @const mpw *av, *avl@ = pointer to first argument
134 * @const mpw *bv, *bvl@ = pointer to second argument
135 * @mpw *sv, *svl@ = pointer to scratch workspace
139 * Use: Multiplies two multiprecision integers using Karatsuba's
140 * algorithm. This is rather faster than traditional long
141 * multiplication (e.g., @mpx_umul@) on large numbers, although
142 * more expensive on small ones.
144 * The destination must be twice as large as the larger
145 * argument. The scratch space must be twice as large as the
146 * larger argument, plus the magic number @KARATSUBA_SLOP@.
149 void mpx_kmul(mpw
*dv
, mpw
*dvl
,
150 const mpw
*av
, const mpw
*avl
,
151 const mpw
*bv
, const mpw
*bvl
,
154 const mpw
*avm
, *bvm
;
157 /* --- Dispose of easy cases to @mpx_umul@ --- *
159 * Karatsuba is only a win on large numbers, because of all the
160 * recursiveness and bookkeeping. The recursive calls make a quick check
161 * to see whether to bottom out to @mpx_umul@ which should help quite a
162 * lot, but sometimes the only way to know is to make sure...
168 if (avl
- av
<= KARATSUBA_CUTOFF
|| bvl
- bv
<= KARATSUBA_CUTOFF
) {
169 mpx_umul(dv
, dvl
, av
, avl
, bv
, bvl
);
173 /* --- How the algorithm works --- *
175 * Let %$A = xb + y$% and %$B = ub + v$%. Then, simply by expanding, %$AB
176 * = x u b^2 + b(x v + y u) + y v$%. That's not helped any, because I've
177 * got four multiplications, each four times easier than the one I started
178 * with. However, note that I can rewrite the coefficient of %$b$% as
179 * %$xv + yu = (x + y)(u + v) - xu - yv$%. The terms %$xu$% and %$yv$%
180 * I've already calculated, and that leaves only one more multiplication to
181 * do. So now I have three multiplications, each four times easier, and
185 /* --- First things --- *
187 * Sort out where to break the factors in half. I'll choose the midpoint
188 * of the largest one, since this minimizes the amount of work I have to do
192 if (avl
- av
> bvl
- bv
) {
193 m
= (avl
- av
+ 1) >> 1;
200 m
= (bvl
- bv
+ 1) >> 1;
208 assert(((void)"Destination too small for Karatsuba multiply",
210 assert(((void)"Not enough workspace for Karatsuba multiply",
213 /* --- Sort out the middle term --- */
216 mpw
*bsv
= sv
+ m
+ 1, *ssv
= bsv
+ m
+ 1;
217 mpw
*rdv
= dv
+ m
, *rdvl
= rdv
+ 2 * (m
+ 2);
219 UADD2(sv
, bsv
, av
, avm
, avm
, avl
);
220 UADD2(bsv
, ssv
, bv
, bvm
, bvm
, bvl
);
221 if (m
> KARATSUBA_CUTOFF
)
222 mpx_kmul(rdv
, rdvl
, sv
, bsv
, bsv
, ssv
, ssv
, svl
);
224 mpx_umul(rdv
, rdvl
, sv
, bsv
, bsv
, ssv
);
227 /* --- Sort out the other two terms --- */
230 mpw
*svm
= sv
+ m
, *svn
= svm
+ m
, *ssv
= svn
+ 4;
234 if (avl
== avm
|| bvl
== bvm
)
235 MPX_ZERO(rdv
+ m
+ 1, dvl
);
237 if (m
> KARATSUBA_CUTOFF
)
238 mpx_kmul(sv
, ssv
, avm
, avl
, bvm
, bvl
, ssv
, svl
);
240 mpx_umul(sv
, ssv
, avm
, avl
, bvm
, bvl
);
241 MPX_COPY(rdv
+ m
+ 1, dvl
, svm
+ 1, svn
);
242 UADD(rdv
, sv
, svm
+ 1);
246 if (m
> KARATSUBA_CUTOFF
)
247 mpx_kmul(sv
, ssv
, av
, avm
, bv
, bvm
, ssv
, svl
);
249 mpx_umul(sv
, ssv
, av
, avm
, bv
, bvm
);
250 MPX_COPY(dv
, tdv
, sv
, svm
);
256 /*----- Test rig ----------------------------------------------------------*/
260 #include <mLib/alloc.h>
261 #include <mLib/testrig.h>
265 #define ALLOC(v, vl, sz) do { \
267 mpw *_vv = xmalloc(MPWS(_sz)); \
268 mpw *_vvl = _vv + _sz; \
273 #define LOAD(v, vl, d) do { \
274 const dstr *_d = (d); \
276 ALLOC(_v, _vl, MPW_RQ(_d->len)); \
277 mpx_loadb(_v, _vl, _d->buf, _d->len); \
282 #define MAX(x, y) ((x) > (y) ? (x) : (y))
284 static void dumpmp(const char *msg
, const mpw
*v
, const mpw
*vl
)
289 fprintf(stderr
, " %08lx", (unsigned long)*--vl
);
293 static int umul(dstr
*v
)
306 m
= MAX(al
- a
, bl
- b
) + 1;
308 ALLOC(s
, sl
, 2 * m
+ 32);
310 mpx_kmul(d
, dl
, a
, al
, b
, bl
, s
, sl
);
311 if (MPX_UCMP(d
, dl
, !=, c
, cl
)) {
312 fprintf(stderr
, "\n*** umul failed\n");
315 dumpmp("expected", c
, cl
);
316 dumpmp(" result", d
, dl
);
320 free(a
); free(b
); free(c
); free(d
); free(s
);
324 static test_chunk defs
[] = {
325 { "umul", umul
, { &type_hex
, &type_hex
, &type_hex
, 0 } },
329 int main(int argc
, char *argv
[])
331 test_run(argc
, argv
, defs
, SRCDIR
"/tests/mpx");
337 /*----- That's all, folks -------------------------------------------------*/