--- /dev/null
+/* -*-c-*-
+ *
+ * $Id: mpx-ksqr.c,v 1.1 1999/12/11 10:57:43 mdw Exp $
+ *
+ * Karatsuba-based squaring algorithm
+ *
+ * (c) 1999 Straylight/Edgeware
+ */
+
+/*----- Licensing notice --------------------------------------------------*
+ *
+ * This file is part of Catacomb.
+ *
+ * Catacomb is free software; you can redistribute it and/or modify
+ * it under the terms of the GNU Library General Public License as
+ * published by the Free Software Foundation; either version 2 of the
+ * License, or (at your option) any later version.
+ *
+ * Catacomb is distributed in the hope that it will be useful,
+ * but WITHOUT ANY WARRANTY; without even the implied warranty of
+ * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+ * GNU Library General Public License for more details.
+ *
+ * You should have received a copy of the GNU Library General Public
+ * License along with Catacomb; if not, write to the Free
+ * Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
+ * MA 02111-1307, USA.
+ */
+
+/*----- Revision history --------------------------------------------------*
+ *
+ * $Log: mpx-ksqr.c,v $
+ * Revision 1.1 1999/12/11 10:57:43 mdw
+ * Karatsuba squaring algorithm.
+ *
+ */
+
+/*----- Header files ------------------------------------------------------*/
+
+#include <stdio.h>
+
+#include "mpx.h"
+
+/*----- Tweakables --------------------------------------------------------*/
+
+#ifdef TEST_RIG
+# undef KARATSUBA_CUTOFF
+# define KARATSUBA_CUTOFF 2
+#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@ --- *
+ *
+ * Arguments: @mpw *dv, *dvl@ = pointer to destination buffer
+ * @const mpw *av, *avl@ = pointer to first argument
+ * @mpw *sv, *svl@ = pointer to scratch workspace
+ *
+ * Returns: ---
+ *
+ * Use: Squares a multiprecision integers using something similar to
+ * Karatsuba's multiplication algorithm. This is rather faster
+ * than traditional long multiplication (e.g., @mpx_umul@) on
+ * 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@.
+ */
+
+void mpx_ksqr(mpw *dv, mpw *dvl,
+ const mpw *av, const mpw *avl,
+ mpw *sv, mpw *svl)
+{
+ const mpw *avm;
+ size_t m;
+
+ /* --- Dispose of easy cases to @mpx_usqr@ --- *
+ *
+ * Karatsuba is only a win on large numbers, because of all the
+ * recursiveness and bookkeeping. The recursive calls make a quick check
+ * to see whether to bottom out to @mpx_usqr@ which should help quite a
+ * lot, but sometimes the only way to know is to make sure...
+ */
+
+ MPX_SHRINK(av, avl);
+
+ if (avl - av <= KARATSUBA_CUTOFF) {
+ 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.
+ */
+
+ /* --- First things --- *
+ *
+ * Sort out where to break the factor in half.
+ */
+
+ m = (avl - av + 1) >> 1;
+ avm = av + m;
+
+ /* --- Sort out everything --- */
+
+ {
+ mpw *ssv = sv + 2 * m;
+ 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);
+ else
+ mpx_umul(sv, ssv, av, avm, avm, avl);
+ ULSL1(tdv, sv, ssv);
+ MPX_ZERO(dv, tdv);
+ MPX_ZERO(rdv + m + 1, dvl);
+
+ if (m > KARATSUBA_CUTOFF)
+ mpx_ksqr(sv, ssv, avm, avl, ssv, svl);
+ else
+ mpx_usqr(sv, ssv, avm, avl);
+ UADD(rdv, sv, ssv);
+
+ if (m > KARATSUBA_CUTOFF)
+ mpx_ksqr(sv, ssv, av, avm, ssv, svl);
+ else
+ mpx_usqr(sv, ssv, av, avm);
+ UADD(dv, sv, ssv);
+ }
+}
+
+/*----- Test rig ----------------------------------------------------------*/
+
+#ifdef TEST_RIG
+
+#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)); \
+ mpw *_vvl = _vv + _sz; \
+ (v) = _vv; \
+ (vl) = _vvl; \
+} while (0)
+
+#define LOAD(v, vl, d) do { \
+ const dstr *_d = (d); \
+ mpw *_v, *_vl; \
+ ALLOC(_v, _vl, MPW_RQ(_d->len)); \
+ mpx_loadb(_v, _vl, _d->buf, _d->len); \
+ (v) = _v; \
+ (vl) = _vl; \
+} while (0)
+
+#define MAX(x, y) ((x) > (y) ? (x) : (y))
+
+static void dumpmp(const char *msg, const mpw *v, const mpw *vl)
+{
+ fputs(msg, stderr);
+ MPX_SHRINK(v, vl);
+ while (v < vl)
+ fprintf(stderr, " %08lx", (unsigned long)*--vl);
+ fputc('\n', stderr);
+}
+
+static int usqr(dstr *v)
+{
+ mpw *a, *al;
+ mpw *c, *cl;
+ mpw *d, *dl;
+ mpw *s, *sl;
+ size_t m;
+ int ok = 1;
+
+ 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);
+
+ mpx_ksqr(d, dl, a, al, s, sl);
+ if (MPX_UCMP(d, dl, !=, c, cl)) {
+ fprintf(stderr, "\n*** usqr failed\n");
+ dumpmp(" a", a, al);
+ dumpmp("expected", c, cl);
+ dumpmp(" result", d, dl);
+ ok = 0;
+ }
+
+ free(a); free(c); free(d); free(s);
+ return (ok);
+}
+
+static test_chunk defs[] = {
+ { "usqr", usqr, { &type_hex, &type_hex, 0 } },
+ { 0, 0, { 0 } }
+};
+
+int main(int argc, char *argv[])
+{
+ test_run(argc, argv, defs, SRCDIR"/tests/mpx");
+ return (0);
+}
+
+#endif
+
+/*----- That's all, folks -------------------------------------------------*/
/* -*-c-*-
*
- * $Id: mpx.h,v 1.7 1999/12/11 01:51:28 mdw Exp $
+ * $Id: mpx.h,v 1.8 1999/12/11 10:57:43 mdw Exp $
*
* Low level multiprecision arithmetic
*
/*----- Revision history --------------------------------------------------*
*
* $Log: mpx.h,v $
+ * Revision 1.8 1999/12/11 10:57:43 mdw
+ * Karatsuba squaring algorithm.
+ *
* Revision 1.7 1999/12/11 01:51:28 mdw
* Change Karatsuba parameters slightly.
*
mpw _cc = 0; \
mpd _m = (m); \
\
- while (_av < _avl) { \
+ while (_dv < _dvl && _av < _avl) { \
mpd _x; \
- if (_dv >= _dvl) \
- break; \
_x = (mpd)*_dv + (mpd)_m * (mpd)*_av++ + _cc; \
*_dv++ = MPW(_x); \
_cc = _x >> MPW_BITS; \
extern void mpx_usqr(mpw */*dv*/, mpw */*dvl*/,
const mpw */*av*/, const mpw */*avl*/);
+/* --- @mpx_udiv@ --- *
+ *
+ * Arguments: @mpw *qv, *qvl@ = quotient vector base and limit
+ * @mpw *rv, *rvl@ = dividend/remainder vector base and limit
+ * @const mpw *dv, *dvl@ = divisor vector base and limit
+ * @mpw *sv, *svl@ = scratch workspace
+ *
+ * Returns: ---
+ *
+ * Use: Performs unsigned integer division. If the result overflows
+ * the quotient vector, high-order bits are discarded. (Clearly
+ * the remainder vector can't overflow.) The various vectors
+ * may not overlap in any way. Yes, I know it's a bit odd
+ * requiring the dividend to be in the result position but it
+ * does make some sense really. The remainder must have
+ * headroom for at least two extra words. The scratch space
+ * must be at least one word larger than the divisor.
+ */
+
+extern void mpx_udiv(mpw */*qv*/, mpw */*qvl*/, mpw */*rv*/, mpw */*rvl*/,
+ const mpw */*dv*/, const mpw */*dvl*/,
+ mpw */*sv*/, mpw */*svl*/);
+
+/*----- Karatsuba multiplication algorithms -------------------------------*/
+
+/* --- @KARATSUBA_CUTOFF@ --- *
+ *
+ * This is the limiting length for using Karatsuba algorithms. It's best to
+ * use the simpler classical multiplication method on numbers smaller than
+ * this.
+ */
+
+#define KARATSUBA_CUTOFF 16
+
+/* --- @KARATSUBA_SLOP@ --- *
+ *
+ * The extra number of words required as scratch space by the Karatsuba
+ * routines. This is a (generous) guess, since the actual amount of space
+ * required is proportional to the recursion depth.
+ */
+
+#define KARATSUBA_SLOP 32
+
/* --- @mpx_kmul@ --- *
*
* Arguments: @mpw *dv, *dvl@ = pointer to destination buffer
* more expensive on small ones.
*
* The destination and scratch buffers must be twice as large as
- * the larger argument.
+ * the larger argument. The scratch space must be twice as
+ * large as the larger argument, plus the magic number
+ * @KARATSUBA_SLOP@.
*/
-#define KARATSUBA_CUTOFF 20
-#define KARATSUBA_SLOP 32
-
extern void mpx_kmul(mpw */*dv*/, mpw */*dvl*/,
const mpw */*av*/, const mpw */*avl*/,
const mpw */*bv*/, const mpw */*bvl*/,
mpw */*sv*/, mpw */*svl*/);
-/* --- @mpx_udiv@ --- *
+/* --- @mpx_ksqr@ --- *
*
- * Arguments: @mpw *qv, *qvl@ = quotient vector base and limit
- * @mpw *rv, *rvl@ = dividend/remainder vector base and limit
- * @const mpw *dv, *dvl@ = divisor vector base and limit
- * @mpw *sv, *svl@ = scratch workspace
+ * Arguments: @mpw *dv, *dvl@ = pointer to destination buffer
+ * @const mpw *av, *avl@ = pointer to first argument
+ * @mpw *sv, *svl@ = pointer to scratch workspace
*
* Returns: ---
*
- * Use: Performs unsigned integer division. If the result overflows
- * the quotient vector, high-order bits are discarded. (Clearly
- * the remainder vector can't overflow.) The various vectors
- * may not overlap in any way. Yes, I know it's a bit odd
- * requiring the dividend to be in the result position but it
- * does make some sense really. The remainder must have
- * headroom for at least two extra words. The scratch space
- * must be at least one word larger than the divisor.
+ * Use: Squares a multiprecision integers using something similar to
+ * Karatsuba's multiplication algorithm. This is rather faster
+ * than traditional long multiplication (e.g., @mpx_umul@) on
+ * 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@.
*/
-extern void mpx_udiv(mpw */*qv*/, mpw */*qvl*/, mpw */*rv*/, mpw */*rvl*/,
- const mpw */*dv*/, const mpw */*dvl*/,
+extern void mpx_ksqr(mpw */*dv*/, mpw */*dvl*/,
+ const mpw */*av*/, const mpw */*avl*/,
mpw */*sv*/, mpw */*svl*/);
/*----- That's all, folks -------------------------------------------------*/
projects / u / mdw / catacomb / commitdiff