3 * Karatsuba's multiplication algorithm
5 * (c) 1999 Straylight/Edgeware
8 /*----- Licensing notice --------------------------------------------------*
10 * This file is part of Catacomb.
12 * Catacomb is free software; you can redistribute it and/or modify
13 * it under the terms of the GNU Library General Public License as
14 * published by the Free Software Foundation; either version 2 of the
15 * License, or (at your option) any later version.
17 * Catacomb is distributed in the hope that it will be useful,
18 * but WITHOUT ANY WARRANTY; without even the implied warranty of
19 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
20 * GNU Library General Public License for more details.
22 * You should have received a copy of the GNU Library General Public
23 * License along with Catacomb; if not, write to the Free
24 * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
28 /*----- Header files ------------------------------------------------------*/
34 #include "karatsuba.h"
36 /*----- Tweakables --------------------------------------------------------*/
40 # define MPK_THRESH 4 /* Smallest possible correct value */
43 /*----- Main code ---------------------------------------------------------*/
45 /* --- @mpx_kmul@ --- *
47 * Arguments: @mpw *dv, *dvl@ = pointer to destination buffer
48 * @const mpw *av, *avl@ = pointer to first argument
49 * @const mpw *bv, *bvl@ = pointer to second argument
50 * @mpw *sv, *svl@ = pointer to scratch workspace
54 * Use: Multiplies two multiprecision integers using Karatsuba's
55 * algorithm. This is rather faster than traditional long
56 * multiplication (e.g., @mpx_umul@) on large numbers, although
57 * more expensive on small ones.
59 * The destination must be three times as large as the larger
60 * argument. The scratch space must be five times as large as
61 * the larger argument.
64 void mpx_kmul(mpw
*dv
, mpw
*dvl
,
65 const mpw
*av
, const mpw
*avl
,
66 const mpw
*bv
, const mpw
*bvl
,
72 /* --- Dispose of easy cases to @mpx_umul@ --- *
74 * Karatsuba is only a win on large numbers, because of all the
75 * recursiveness and bookkeeping. The recursive calls make a quick check
76 * to see whether to bottom out to @mpx_umul@ which should help quite a
77 * lot, but sometimes the only way to know is to make sure...
83 if (avl
- av
<= MPK_THRESH
|| bvl
- bv
<= MPK_THRESH
) {
84 mpx_umul(dv
, dvl
, av
, avl
, bv
, bvl
);
88 /* --- How the algorithm works --- *
90 * Let %$A = xb + y$% and %$B = ub + v$%. Then, simply by expanding,
91 * %$AB = x u b^2 + b(x v + y u) + y v$%. That's not helped any, because
92 * I've got four multiplications, each four times easier than the one I
93 * started with. However, note that I can rewrite the coefficient of %$b$%
94 * as %$xv + yu = (x + y)(u + v) - xu - yv$%. The terms %$xu$% and %$yv$%
95 * I've already calculated, and that leaves only one more multiplication to
96 * do. So now I have three multiplications, each four times easier, and
100 /* --- First things --- *
102 * Sort out where to break the factors in half. I'll choose the midpoint
103 * of the larger one, since this minimizes the amount of work I have to do
107 if (avl
- av
> bvl
- bv
) {
108 m
= (avl
- av
+ 1) >> 1;
115 m
= (bvl
- bv
+ 1) >> 1;
123 /* --- Sort out the middle term --- */
126 mpw
*bsv
= sv
+ m
+ 1, *ssv
= bsv
+ m
+ 1;
127 mpw
*rdv
= dv
+ m
, *rdvl
= rdv
+ 2 * (m
+ 2);
131 UADD2(sv
, bsv
, av
, avm
, avm
, avl
);
132 UADD2(bsv
, ssv
, bv
, bvm
, bvm
, bvl
);
134 mpx_kmul(rdv
, rdvl
, sv
, bsv
, bsv
, ssv
, ssv
, svl
);
136 mpx_umul(rdv
, rdvl
, sv
, bsv
, bsv
, ssv
);
139 /* --- Sort out the other two terms --- */
142 mpw
*svm
= sv
+ m
, *svn
= svm
+ m
, *ssv
= svn
+ 4;
146 if (avl
== avm
|| bvl
== bvm
)
147 MPX_ZERO(rdv
+ m
+ 1, dvl
);
150 mpx_kmul(sv
, ssv
, avm
, avl
, bvm
, bvl
, ssv
, svl
);
152 mpx_umul(sv
, ssv
, avm
, avl
, bvm
, bvl
);
153 MPX_COPY(rdv
+ m
+ 1, dvl
, svm
+ 1, svn
);
154 UADD(rdv
, sv
, svm
+ 1);
159 mpx_kmul(sv
, ssv
, av
, avm
, bv
, bvm
, ssv
, svl
);
161 mpx_umul(sv
, ssv
, av
, avm
, bv
, bvm
);
162 MPX_COPY(dv
, tdv
, sv
, svm
);
168 /*----- Test rig ----------------------------------------------------------*/
172 #include <mLib/alloc.h>
173 #include <mLib/testrig.h>
175 #define ALLOC(v, vl, sz) do { \
177 mpw *_vv = xmalloc(MPWS(_sz)); \
178 mpw *_vvl = _vv + _sz; \
183 #define LOAD(v, vl, d) do { \
184 const dstr *_d = (d); \
186 ALLOC(_v, _vl, MPW_RQ(_d->len)); \
187 mpx_loadb(_v, _vl, _d->buf, _d->len); \
192 #define MAX(x, y) ((x) > (y) ? (x) : (y))
194 static void dumpmp(const char *msg
, const mpw
*v
, const mpw
*vl
)
199 fprintf(stderr
, " %08lx", (unsigned long)*--vl
);
203 static int umul(dstr
*v
)
216 m
= MAX(al
- a
, bl
- b
) + 1;
220 mpx_kmul(d
, dl
, a
, al
, b
, bl
, s
, sl
);
221 if (!mpx_ueq(d
, dl
, c
, cl
)) {
222 fprintf(stderr
, "\n*** umul failed\n");
225 dumpmp("expected", c
, cl
);
226 dumpmp(" result", d
, dl
);
230 xfree(a
); xfree(b
); xfree(c
); xfree(d
); xfree(s
);
234 static test_chunk defs
[] = {
235 { "umul", umul
, { &type_hex
, &type_hex
, &type_hex
, 0 } },
239 int main(int argc
, char *argv
[])
241 test_run(argc
, argv
, defs
, SRCDIR
"/t/mpx");
247 /*----- That's all, folks -------------------------------------------------*/