3 * $Id: mpx-kmul.c,v 1.4 2000/06/17 11:42:11 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 1.4 2000/06/17 11:42:11 mdw
34 * Moved the Karatsuba macros into a separate file for better sharing.
35 * Fixed some comments.
37 * Revision 1.3 1999/12/13 15:35:01 mdw
38 * Simplify and improve.
40 * Revision 1.2 1999/12/11 10:58:02 mdw
41 * Remove tweakable comments.
43 * Revision 1.1 1999/12/10 23:23:51 mdw
44 * Karatsuba-Ofman multiplication algorithm.
48 /*----- Header files ------------------------------------------------------*/
56 /*----- Tweakables --------------------------------------------------------*/
59 # undef KARATSUBA_CUTOFF
60 # define KARATSUBA_CUTOFF 2
63 /*----- Main code ---------------------------------------------------------*/
65 /* --- @mpx_kmul@ --- *
67 * Arguments: @mpw *dv, *dvl@ = pointer to destination buffer
68 * @const mpw *av, *avl@ = pointer to first argument
69 * @const mpw *bv, *bvl@ = pointer to second argument
70 * @mpw *sv, *svl@ = pointer to scratch workspace
74 * Use: Multiplies two multiprecision integers using Karatsuba's
75 * algorithm. This is rather faster than traditional long
76 * multiplication (e.g., @mpx_umul@) on large numbers, although
77 * more expensive on small ones.
79 * The destination must be twice as large as the larger
80 * argument. The scratch space must be twice as large as the
81 * larger argument, plus the magic number @KARATSUBA_SLOP@.
84 void mpx_kmul(mpw
*dv
, mpw
*dvl
,
85 const mpw
*av
, const mpw
*avl
,
86 const mpw
*bv
, const mpw
*bvl
,
92 /* --- Dispose of easy cases to @mpx_umul@ --- *
94 * Karatsuba is only a win on large numbers, because of all the
95 * recursiveness and bookkeeping. The recursive calls make a quick check
96 * to see whether to bottom out to @mpx_umul@ which should help quite a
97 * lot, but sometimes the only way to know is to make sure...
103 if (avl
- av
<= KARATSUBA_CUTOFF
|| bvl
- bv
<= KARATSUBA_CUTOFF
) {
104 mpx_umul(dv
, dvl
, av
, avl
, bv
, bvl
);
108 /* --- How the algorithm works --- *
110 * Let %$A = xb + y$% and %$B = ub + v$%. Then, simply by expanding,
111 * %$AB = x u b^2 + b(x v + y u) + y v$%. That's not helped any, because
112 * I've got four multiplications, each four times easier than the one I
113 * started with. However, note that I can rewrite the coefficient of %$b$%
114 * as %$xv + yu = (x + y)(u + v) - xu - yv$%. The terms %$xu$% and %$yv$%
115 * I've already calculated, and that leaves only one more multiplication to
116 * do. So now I have three multiplications, each four times easier, and
120 /* --- First things --- *
122 * Sort out where to break the factors in half. I'll choose the midpoint
123 * of the largest one, since this minimizes the amount of work I have to do
127 if (avl
- av
> bvl
- bv
) {
128 m
= (avl
- av
+ 1) >> 1;
135 m
= (bvl
- bv
+ 1) >> 1;
143 assert(((void)"Destination too small for Karatsuba multiply",
145 assert(((void)"Not enough workspace for Karatsuba multiply",
148 /* --- Sort out the middle term --- */
151 mpw
*bsv
= sv
+ m
+ 1, *ssv
= bsv
+ m
+ 1;
152 mpw
*rdv
= dv
+ m
, *rdvl
= rdv
+ 2 * (m
+ 2);
154 UADD2(sv
, bsv
, av
, avm
, avm
, avl
);
155 UADD2(bsv
, ssv
, bv
, bvm
, bvm
, bvl
);
156 if (m
> KARATSUBA_CUTOFF
)
157 mpx_kmul(rdv
, rdvl
, sv
, bsv
, bsv
, ssv
, ssv
, svl
);
159 mpx_umul(rdv
, rdvl
, sv
, bsv
, bsv
, ssv
);
162 /* --- Sort out the other two terms --- */
165 mpw
*svm
= sv
+ m
, *svn
= svm
+ m
, *ssv
= svn
+ 4;
169 if (avl
== avm
|| bvl
== bvm
)
170 MPX_ZERO(rdv
+ m
+ 1, dvl
);
172 if (m
> KARATSUBA_CUTOFF
)
173 mpx_kmul(sv
, ssv
, avm
, avl
, bvm
, bvl
, ssv
, svl
);
175 mpx_umul(sv
, ssv
, avm
, avl
, bvm
, bvl
);
176 MPX_COPY(rdv
+ m
+ 1, dvl
, svm
+ 1, svn
);
177 UADD(rdv
, sv
, svm
+ 1);
181 if (m
> KARATSUBA_CUTOFF
)
182 mpx_kmul(sv
, ssv
, av
, avm
, bv
, bvm
, ssv
, svl
);
184 mpx_umul(sv
, ssv
, av
, avm
, bv
, bvm
);
185 MPX_COPY(dv
, tdv
, sv
, svm
);
191 /*----- Test rig ----------------------------------------------------------*/
195 #include <mLib/alloc.h>
196 #include <mLib/testrig.h>
200 #define ALLOC(v, vl, sz) do { \
202 mpw *_vv = xmalloc(MPWS(_sz)); \
203 mpw *_vvl = _vv + _sz; \
208 #define LOAD(v, vl, d) do { \
209 const dstr *_d = (d); \
211 ALLOC(_v, _vl, MPW_RQ(_d->len)); \
212 mpx_loadb(_v, _vl, _d->buf, _d->len); \
217 #define MAX(x, y) ((x) > (y) ? (x) : (y))
219 static void dumpmp(const char *msg
, const mpw
*v
, const mpw
*vl
)
224 fprintf(stderr
, " %08lx", (unsigned long)*--vl
);
228 static int umul(dstr
*v
)
241 m
= MAX(al
- a
, bl
- b
) + 1;
243 ALLOC(s
, sl
, 2 * m
+ 32);
245 mpx_kmul(d
, dl
, a
, al
, b
, bl
, s
, sl
);
246 if (MPX_UCMP(d
, dl
, !=, c
, cl
)) {
247 fprintf(stderr
, "\n*** umul failed\n");
250 dumpmp("expected", c
, cl
);
251 dumpmp(" result", d
, dl
);
255 free(a
); free(b
); free(c
); free(d
); free(s
);
259 static test_chunk defs
[] = {
260 { "umul", umul
, { &type_hex
, &type_hex
, &type_hex
, 0 } },
264 int main(int argc
, char *argv
[])
266 test_run(argc
, argv
, defs
, SRCDIR
"/tests/mpx");
272 /*----- That's all, folks -------------------------------------------------*/