From: mdw Date: Sat, 11 Dec 1999 10:57:43 +0000 (+0000) Subject: Karatsuba squaring algorithm. X-Git-Url: https://git.distorted.org.uk/u/mdw/catacomb/commitdiff_plain/5bf74deaebe06922cc0e03bd9118f207b31f211e Karatsuba squaring algorithm. --- diff --git a/mp-arith.c b/mp-arith.c index 6f00656..cd6b0bd 100644 --- a/mp-arith.c +++ b/mp-arith.c @@ -1,6 +1,6 @@ /* -*-c-*- * - * $Id: mp-arith.c,v 1.2 1999/12/10 23:18:39 mdw Exp $ + * $Id: mp-arith.c,v 1.3 1999/12/11 10:57:43 mdw Exp $ * * Basic arithmetic on multiprecision integers * @@ -30,6 +30,9 @@ /*----- Revision history --------------------------------------------------* * * $Log: mp-arith.c,v $ + * Revision 1.3 1999/12/11 10:57:43 mdw + * Karatsuba squaring algorithm. + * * Revision 1.2 1999/12/10 23:18:39 mdw * Change interface for suggested destinations. * @@ -242,7 +245,7 @@ mp *mp_sqr(mp *d, mp *a) mpw *s; m = 2 * (m + 1) + 32; s = MP_ALLOC(m); - mpx_kmul(d->v, d->vl, a->v, a->vl, a->v, a->vl, s, s + m); + mpx_ksqr(d->v, d->vl, a->v, a->vl, s, s + m); MP_FREE(s); } else mpx_usqr(d->v, d->vl, a->v, a->vl); diff --git a/mpx-ksqr.c b/mpx-ksqr.c new file mode 100644 index 0000000..45c49c9 --- /dev/null +++ b/mpx-ksqr.c @@ -0,0 +1,259 @@ +/* -*-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 + +#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 +#include + +#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 -------------------------------------------------*/ diff --git a/mpx.h b/mpx.h index 23c0b8d..509da02 100644 --- a/mpx.h +++ b/mpx.h @@ -1,6 +1,6 @@ /* -*-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 * @@ -30,6 +30,9 @@ /*----- 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. * @@ -487,10 +490,8 @@ extern void mpx_umuln(mpw */*dv*/, mpw */*dvl*/, 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; \ @@ -515,6 +516,49 @@ extern void mpx_umlan(mpw */*dv*/, mpw */*dvl*/, 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 @@ -530,38 +574,37 @@ extern void mpx_usqr(mpw */*dv*/, mpw */*dvl*/, * 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 -------------------------------------------------*/