/* -*-c-*-
*
- * $Id: mpx-ksqr.c,v 1.2 1999/12/13 15:35:01 mdw Exp $
+ * $Id: mpx-ksqr.c,v 1.8 2004/04/08 01:36:15 mdw Exp $
*
* Karatsuba-based squaring algorithm
*
* MA 02111-1307, USA.
*/
-/*----- Revision history --------------------------------------------------*
- *
- * $Log: mpx-ksqr.c,v $
- * Revision 1.2 1999/12/13 15:35:01 mdw
- * Simplify and improve.
- *
- * Revision 1.1 1999/12/11 10:57:43 mdw
- * Karatsuba squaring algorithm.
- *
- */
-
/*----- Header files ------------------------------------------------------*/
#include <assert.h>
#include <stdio.h>
#include "mpx.h"
+#include "karatsuba.h"
/*----- Tweakables --------------------------------------------------------*/
#ifdef TEST_RIG
-# undef KARATSUBA_CUTOFF
-# define KARATSUBA_CUTOFF 2
+# undef MPK_THRESH
+# define MPK_THRESH 4
#endif
-/*----- Addition macros ---------------------------------------------------*/
-
-#define ULSL1(dv, av, avl) do { \
- mpw *_dv = (dv); \
- const mpw *_av = (av), *_avl = (avl); \
- mpw _c = 0; \
- \
- while (_av < _avl) { \
- mpw _x = *_av++; \
- *_dv++ = MPW(_c | (_x << 1)); \
- _c = MPW(_x >> (MPW_BITS - 1)); \
- } \
- *_dv++ = _c; \
-} while (0)
-
-#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)
-
/*----- Main code ---------------------------------------------------------*/
/* --- @mpx_ksqr@ --- *
* large numbers, although more expensive on small ones, and
* rather simpler than full-blown Karatsuba multiplication.
*
- * The destination must be twice as large as the argument. The
- * scratch space must be twice as large as the argument, plus
- * the magic number @KARATSUBA_SLOP@.
+ * The destination must be three times as large as the larger
+ * argument. The scratch space must be five times as large as
+ * the larger argument.
*/
void mpx_ksqr(mpw *dv, mpw *dvl,
MPX_SHRINK(av, avl);
- if (avl - av <= KARATSUBA_CUTOFF) {
+ if (avl - av <= MPK_THRESH) {
mpx_usqr(dv, dvl, av, avl);
return;
}
/* --- How the algorithm works --- *
*
- * Unlike Karatsuba's identity for multiplication which isn't particularly
- * obvious, the identity for multiplication is known to all schoolchildren.
- * Let %$A = xb + y$%. Then %$A^2 = x^2 b^x + 2 x y b + y^2$%. So now I
- * have three multiplications, each four times easier, and that's a win.
+ * The identity for squaring is known to all schoolchildren.
+ * Let %$A = xb + y$%. Then %$A^2 = x^2 b^2 + 2 x y b + y^2$%. Now,
+ * %$(x + y)^2 - x^2 - y^2 = 2 x y$%, which means I only need to do three
+ * squarings.
*/
/* --- First things --- *
m = (avl - av + 1) >> 1;
avm = av + m;
- assert(((void)"Destination too small for Karatsuba square",
- dvl - dv >= 4 * m));
- assert(((void)"Not enough workspace for Karatsuba square",
- svl - sv >= 4 * m));
-
/* --- Sort out everything --- */
{
mpw *tdv = dv + m;
mpw *rdv = tdv + m;
- /* --- The cross term in the middle needs a multiply --- *
- *
- * This isn't actually true, since %$x y = ((x + y)^2 - (x - y)^2)/4%.
- * But that's two squarings, versus one multiplication.
- */
-
- if (m > KARATSUBA_CUTOFF)
- mpx_kmul(sv, ssv, av, avm, avm, avl, ssv, svl);
+ assert(rdv + m + 4 < dvl);
+ assert(ssv < svl);
+ UADD2(sv, svm, av, avm, avm, avl);
+ if (m > MPK_THRESH)
+ mpx_ksqr(tdv, rdv + m + 4, sv, svm + 1, ssv, svl);
else
- mpx_umul(sv, ssv, av, avm, avm, avl);
- ULSL1(tdv, sv, svn);
+ mpx_usqr(tdv, rdv + m + 4, sv, svm + 1);
- if (m > KARATSUBA_CUTOFF)
+ if (m > MPK_THRESH)
mpx_ksqr(sv, ssv, avm, avl, ssv, svl);
else
mpx_usqr(sv, ssv, avm, avl);
MPX_COPY(rdv + m + 1, dvl, svm + 1, svn);
UADD(rdv, sv, svm + 1);
+ USUB(tdv, sv, svn);
- if (m > KARATSUBA_CUTOFF)
+ if (m > MPK_THRESH)
mpx_ksqr(sv, ssv, av, avm, ssv, svl);
else
mpx_usqr(sv, ssv, av, avm);
MPX_COPY(dv, tdv, sv, svm);
UADD(tdv, svm, svn);
+ USUB(tdv, sv, svn);
}
}
#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)); \
LOAD(a, al, &v[0]);
LOAD(c, cl, &v[1]);
m = al - a + 1;
- ALLOC(d, dl, 2 * m);
- ALLOC(s, sl, 2 * m + 32);
+ ALLOC(d, dl, 3 * m);
+ ALLOC(s, sl, 5 * m);
mpx_ksqr(d, dl, a, al, s, sl);
- if (MPX_UCMP(d, dl, !=, c, cl)) {
+ if (!mpx_ueq(d, dl, c, cl)) {
fprintf(stderr, "\n*** usqr failed\n");
dumpmp(" a", a, al);
dumpmp("expected", c, cl);