3 * $Id: mpx-kmul.c,v 1.5 2000/07/29 17:04:02 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 1.5 2000/07/29 17:04:02 mdw
34 * Remove useless header `mpscan.h'.
36 * Revision 1.4 2000/06/17 11:42:11 mdw
37 * Moved the Karatsuba macros into a separate file for better sharing.
38 * Fixed some comments.
40 * Revision 1.3 1999/12/13 15:35:01 mdw
41 * Simplify and improve.
43 * Revision 1.2 1999/12/11 10:58:02 mdw
44 * Remove tweakable comments.
46 * Revision 1.1 1999/12/10 23:23:51 mdw
47 * Karatsuba-Ofman multiplication algorithm.
51 /*----- Header files ------------------------------------------------------*/
59 /*----- Tweakables --------------------------------------------------------*/
62 # undef KARATSUBA_CUTOFF
63 # define KARATSUBA_CUTOFF 2
66 /*----- Main code ---------------------------------------------------------*/
68 /* --- @mpx_kmul@ --- *
70 * Arguments: @mpw *dv, *dvl@ = pointer to destination buffer
71 * @const mpw *av, *avl@ = pointer to first argument
72 * @const mpw *bv, *bvl@ = pointer to second argument
73 * @mpw *sv, *svl@ = pointer to scratch workspace
77 * Use: Multiplies two multiprecision integers using Karatsuba's
78 * algorithm. This is rather faster than traditional long
79 * multiplication (e.g., @mpx_umul@) on large numbers, although
80 * more expensive on small ones.
82 * The destination must be twice as large as the larger
83 * argument. The scratch space must be twice as large as the
84 * larger argument, plus the magic number @KARATSUBA_SLOP@.
87 void mpx_kmul(mpw
*dv
, mpw
*dvl
,
88 const mpw
*av
, const mpw
*avl
,
89 const mpw
*bv
, const mpw
*bvl
,
95 /* --- Dispose of easy cases to @mpx_umul@ --- *
97 * Karatsuba is only a win on large numbers, because of all the
98 * recursiveness and bookkeeping. The recursive calls make a quick check
99 * to see whether to bottom out to @mpx_umul@ which should help quite a
100 * lot, but sometimes the only way to know is to make sure...
106 if (avl
- av
<= KARATSUBA_CUTOFF
|| bvl
- bv
<= KARATSUBA_CUTOFF
) {
107 mpx_umul(dv
, dvl
, av
, avl
, bv
, bvl
);
111 /* --- How the algorithm works --- *
113 * Let %$A = xb + y$% and %$B = ub + v$%. Then, simply by expanding,
114 * %$AB = x u b^2 + b(x v + y u) + y v$%. That's not helped any, because
115 * I've got four multiplications, each four times easier than the one I
116 * started with. However, note that I can rewrite the coefficient of %$b$%
117 * as %$xv + yu = (x + y)(u + v) - xu - yv$%. The terms %$xu$% and %$yv$%
118 * I've already calculated, and that leaves only one more multiplication to
119 * do. So now I have three multiplications, each four times easier, and
123 /* --- First things --- *
125 * Sort out where to break the factors in half. I'll choose the midpoint
126 * of the largest one, since this minimizes the amount of work I have to do
130 if (avl
- av
> bvl
- bv
) {
131 m
= (avl
- av
+ 1) >> 1;
138 m
= (bvl
- bv
+ 1) >> 1;
146 assert(((void)"Destination too small for Karatsuba multiply",
148 assert(((void)"Not enough workspace for Karatsuba multiply",
151 /* --- Sort out the middle term --- */
154 mpw
*bsv
= sv
+ m
+ 1, *ssv
= bsv
+ m
+ 1;
155 mpw
*rdv
= dv
+ m
, *rdvl
= rdv
+ 2 * (m
+ 2);
157 UADD2(sv
, bsv
, av
, avm
, avm
, avl
);
158 UADD2(bsv
, ssv
, bv
, bvm
, bvm
, bvl
);
159 if (m
> KARATSUBA_CUTOFF
)
160 mpx_kmul(rdv
, rdvl
, sv
, bsv
, bsv
, ssv
, ssv
, svl
);
162 mpx_umul(rdv
, rdvl
, sv
, bsv
, bsv
, ssv
);
165 /* --- Sort out the other two terms --- */
168 mpw
*svm
= sv
+ m
, *svn
= svm
+ m
, *ssv
= svn
+ 4;
172 if (avl
== avm
|| bvl
== bvm
)
173 MPX_ZERO(rdv
+ m
+ 1, dvl
);
175 if (m
> KARATSUBA_CUTOFF
)
176 mpx_kmul(sv
, ssv
, avm
, avl
, bvm
, bvl
, ssv
, svl
);
178 mpx_umul(sv
, ssv
, avm
, avl
, bvm
, bvl
);
179 MPX_COPY(rdv
+ m
+ 1, dvl
, svm
+ 1, svn
);
180 UADD(rdv
, sv
, svm
+ 1);
184 if (m
> KARATSUBA_CUTOFF
)
185 mpx_kmul(sv
, ssv
, av
, avm
, bv
, bvm
, ssv
, svl
);
187 mpx_umul(sv
, ssv
, av
, avm
, bv
, bvm
);
188 MPX_COPY(dv
, tdv
, sv
, svm
);
194 /*----- Test rig ----------------------------------------------------------*/
198 #include <mLib/alloc.h>
199 #include <mLib/testrig.h>
201 #define ALLOC(v, vl, sz) do { \
203 mpw *_vv = xmalloc(MPWS(_sz)); \
204 mpw *_vvl = _vv + _sz; \
209 #define LOAD(v, vl, d) do { \
210 const dstr *_d = (d); \
212 ALLOC(_v, _vl, MPW_RQ(_d->len)); \
213 mpx_loadb(_v, _vl, _d->buf, _d->len); \
218 #define MAX(x, y) ((x) > (y) ? (x) : (y))
220 static void dumpmp(const char *msg
, const mpw
*v
, const mpw
*vl
)
225 fprintf(stderr
, " %08lx", (unsigned long)*--vl
);
229 static int umul(dstr
*v
)
242 m
= MAX(al
- a
, bl
- b
) + 1;
244 ALLOC(s
, sl
, 2 * m
+ 32);
246 mpx_kmul(d
, dl
, a
, al
, b
, bl
, s
, sl
);
247 if (MPX_UCMP(d
, dl
, !=, c
, cl
)) {
248 fprintf(stderr
, "\n*** umul failed\n");
251 dumpmp("expected", c
, cl
);
252 dumpmp(" result", d
, dl
);
256 free(a
); free(b
); free(c
); free(d
); free(s
);
260 static test_chunk defs
[] = {
261 { "umul", umul
, { &type_hex
, &type_hex
, &type_hex
, 0 } },
265 int main(int argc
, char *argv
[])
267 test_run(argc
, argv
, defs
, SRCDIR
"/tests/mpx");
273 /*----- That's all, folks -------------------------------------------------*/