--- /dev/null
+/* -*-c-*-
+ *
+ * $Id: gfreduce.c,v 1.2 2004/03/21 22:52:06 mdw Exp $
+ *
+ * Efficient reduction modulo sparse binary polynomials
+ *
+ * (c) 2004 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: gfreduce.c,v $
+ * Revision 1.2 2004/03/21 22:52:06 mdw
+ * Merge and close elliptic curve branch.
+ *
+ * Revision 1.1.2.1 2004/03/21 22:39:46 mdw
+ * Elliptic curves on binary fields work.
+ *
+ */
+
+/*----- Header files ------------------------------------------------------*/
+
+#include <mLib/alloc.h>
+#include <mLib/darray.h>
+#include <mLib/macros.h>
+
+#include "gf.h"
+#include "gfreduce.h"
+#include "gfreduce-exp.h"
+#include "fibrand.h"
+#include "mprand.h"
+
+/*----- Data structures ---------------------------------------------------*/
+
+DA_DECL(instr_v, gfreduce_instr);
+
+/*----- Main code ---------------------------------------------------------*/
+
+/* --- What's going on here? --- *
+ *
+ * Let's face it, @gfx_div@ sucks. It works (I hope), but it's not in any
+ * sense fast. Here, we do efficient reduction modulo sparse polynomials.
+ *
+ * Suppose we have a polynomial @X@ we're trying to reduce mod @P@. If we
+ * take the topmost nonzero word of @X@, call it @w@, then we can eliminate
+ * it by subtracting off @w P x^{k}@ for an appropriate value of @k@. The
+ * trick is in observing that if @P@ is sparse we can do this multiplication
+ * and subtraction efficiently, just by XORing appropriate shifts of @w@ into
+ * @X@.
+ *
+ * The first tricky bit is in working out when to stop. I'll use eight-bit
+ * words to demonstrate what I'm talking about.
+ *
+ * xxxxxx xxxxxxxx xxxxxxxx xxxxxxxx xxxxxxxx xxxxxxxx
+ * 001ppppp pppppppp pppppppp pppppppp
+ * |<rp>|
+ * |<------------ bp ------------->|
+ * |<------------ nw --------------->|
+ *
+ * The trick of taking whole words off of @X@ stops working when there are
+ * only @nw@ words left. Then we have to mask off the bottom bits of @w@
+ * before continuing.
+ */
+
+/* --- @gfreduce_create@ --- *
+ *
+ * Arguments: @gfreduce *r@ = structure to fill in
+ * @mp *x@ = a (hopefully sparse) polynomial
+ *
+ * Returns: ---
+ *
+ * Use: Initializes a context structure for reduction.
+ */
+
+void gfreduce_create(gfreduce *r, mp *p)
+{
+ instr_v iv = DA_INIT;
+ unsigned long d, dw;
+ mpscan sc;
+ unsigned long i;
+ gfreduce_instr *ip;
+ unsigned f = 0;
+ size_t w, ww, wi, wl, ll;
+
+ /* --- Sort out the easy stuff --- */
+
+ d = mp_bits(p); assert(d); d--;
+ r->lim = d/MPW_BITS;
+ dw = d%MPW_BITS;
+ if (!dw)
+ r->mask = 0;
+ else {
+ r->mask = MPW(((mpw)-1) << dw);
+ r->lim++;
+ }
+ r->p = mp_copy(p);
+
+ /* --- Stash a new instruction --- */
+
+#define INSTR(op_, arg_) do { \
+ DA_ENSURE(&iv, 1); \
+ DA(&iv)[DA_LEN(&iv)].op = (op_); \
+ DA(&iv)[DA_LEN(&iv)].arg = (arg_); \
+ DA_EXTEND(&iv, 1); \
+} while (0)
+
+#define f_lsr 1u
+
+ w = (d + MPW_BITS - 1)/MPW_BITS;
+ INSTR(GFRI_LOAD, w);
+ wi = DA_LEN(&iv);
+ f = 0;
+ ll = 0;
+ for (i = 0, mp_scan(&sc, p); mp_step(&sc) && i < d; i++) {
+ if (!mp_bit(&sc))
+ continue;
+ ww = (d - i + MPW_BITS - 1)/MPW_BITS;
+ if (ww != w) {
+ wl = DA_LEN(&iv);
+ INSTR(GFRI_STORE, w);
+ if (!ll)
+ ll = DA_LEN(&iv);
+ if (!(f & f_lsr))
+ INSTR(GFRI_LOAD, ww);
+ else {
+ INSTR(GFRI_LOAD, w - 1);
+ for (; wi < wl; wi++) {
+ ip = &DA(&iv)[wi];
+ assert(ip->op == GFRI_LSL);
+ if (ip->arg)
+ INSTR(GFRI_LSR, MPW_BITS - ip->arg);
+ }
+ if (w - 1 != ww) {
+ INSTR(GFRI_STORE, w - 1);
+ INSTR(GFRI_LOAD, ww);
+ }
+ f &= ~f_lsr;
+ }
+ w = ww;
+ wi = DA_LEN(&iv);
+ }
+ INSTR(GFRI_LSL, (i - d)%MPW_BITS);
+ if ((i - d)%MPW_BITS)
+ f |= f_lsr;
+ }
+ wl = DA_LEN(&iv);
+ INSTR(GFRI_STORE, w);
+ if (!ll)
+ ll = DA_LEN(&iv);
+ if (f & f_lsr) {
+ INSTR(GFRI_LOAD, w - 1);
+ for (; wi < wl; wi++) {
+ ip = &DA(&iv)[wi];
+ assert(ip->op == GFRI_LSL);
+ if (ip->arg)
+ INSTR(GFRI_LSR, MPW_BITS - ip->arg);
+ }
+ INSTR(GFRI_STORE, w - 1);
+ }
+
+#undef INSTR
+
+ r->in = DA_LEN(&iv);
+ r->iv = xmalloc(r->in * sizeof(gfreduce_instr));
+ r->liv = r->iv + ll;
+ memcpy(r->iv, DA(&iv), r->in * sizeof(gfreduce_instr));
+ DA_DESTROY(&iv);
+}
+
+/* --- @gfreduce_destroy@ --- *
+ *
+ * Arguments: @gfreduce *r@ = structure to free
+ *
+ * Returns: ---
+ *
+ * Use: Reclaims the resources from a reduction context.
+ */
+
+void gfreduce_destroy(gfreduce *r)
+{
+ mp_drop(r->p);
+ xfree(r->iv);
+}
+
+/* --- @gfreduce_dump@ --- *
+ *
+ * Arguments: @gfreduce *r@ = structure to dump
+ * @FILE *fp@ = file to dump on
+ *
+ * Returns: ---
+ *
+ * Use: Dumps a reduction context.
+ */
+
+void gfreduce_dump(gfreduce *r, FILE *fp)
+{
+ size_t i;
+
+ fprintf(fp, "poly = "); mp_writefile(r->p, fp, 16);
+ fprintf(fp, "\n lim = %lu; mask = %lx\n",
+ (unsigned long)r->lim, (unsigned long)r->mask);
+ for (i = 0; i < r->in; i++) {
+ static const char *opname[] = { "load", "lsl", "lsr", "store" };
+ assert(r->iv[i].op < N(opname));
+ fprintf(fp, " %s %lu\n",
+ opname[r->iv[i].op],
+ (unsigned long)r->iv[i].arg);
+ }
+}
+
+/* --- @gfreduce_do@ --- *
+ *
+ * Arguments: @gfreduce *r@ = reduction context
+ * @mp *d@ = destination
+ * @mp *x@ = source
+ *
+ * Returns: Destination, @x@ reduced modulo the reduction poly.
+ */
+
+static void run(const gfreduce_instr *i, const gfreduce_instr *il,
+ mpw *v, mpw z)
+{
+ mpw w = 0;
+
+ for (; i < il; i++) {
+ switch (i->op) {
+ case GFRI_LOAD: w = *(v - i->arg); break;
+ case GFRI_LSL: w ^= z << i->arg; break;
+ case GFRI_LSR: w ^= z >> i->arg; break;
+ case GFRI_STORE: *(v - i->arg) = MPW(w); break;
+ default: abort();
+ }
+ }
+}
+
+mp *gfreduce_do(gfreduce *r, mp *d, mp *x)
+{
+ mpw *v, *vl;
+ const gfreduce_instr *il;
+ mpw z;
+
+ /* --- Try to reuse the source's space --- */
+
+ MP_COPY(x);
+ if (d) MP_DROP(d);
+ MP_DEST(x, MP_LEN(x), x->f);
+
+ /* --- Do the reduction --- */
+
+ il = r->iv + r->in;
+ if (MP_LEN(x) >= r->lim) {
+ v = x->v + r->lim;
+ vl = x->vl;
+ while (vl-- > v) {
+ while (*vl) {
+ z = *vl;
+ *vl = 0;
+ run(r->iv, il, vl, z);
+ }
+ }
+ if (r->mask) {
+ while (*vl & r->mask) {
+ z = *vl & r->mask;
+ *vl &= ~r->mask;
+ run(r->iv, il, vl, z);
+ }
+ }
+ }
+
+ /* --- Done --- */
+
+ MP_SHRINK(x);
+ return (x);
+}
+
+/* --- @gfreduce_sqrt@ --- *
+ *
+ * Arguments: @gfreduce *r@ = pointer to reduction context
+ * @mp *d@ = destination
+ * @mp *x@ = some polynomial
+ *
+ * Returns: The square root of @x@ modulo @r->p@, or null.
+ */
+
+mp *gfreduce_sqrt(gfreduce *r, mp *d, mp *x)
+{
+ mp *y = MP_COPY(x);
+ mp *z, *spare = MP_NEW;
+ unsigned long m = mp_bits(r->p) - 1;
+ unsigned long i;
+
+ for (i = 0; i < m - 1; i++) {
+ mp *t = gf_sqr(spare, y);
+ spare = y;
+ y = gfreduce_do(r, t, t);
+ }
+ z = gf_sqr(spare, y);
+ z = gfreduce_do(r, z, z);
+ if (!MP_EQ(x, z)) {
+ mp_drop(y);
+ y = 0;
+ }
+ mp_drop(z);
+ mp_drop(d);
+ return (y);
+}
+
+/* --- @gfreduce_trace@ --- *
+ *
+ * Arguments: @gfreduce *r@ = pointer to reduction context
+ * @mp *x@ = some polynomial
+ *
+ * Returns: The trace of @x@. (%$\Tr(x)=x + x^2 + \cdots + x^{2^{m-1}}$%
+ * if %$x \in \gf{2^m}$%).
+ */
+
+int gfreduce_trace(gfreduce *r, mp *x)
+{
+ mp *y = MP_COPY(x);
+ mp *spare = MP_NEW;
+ unsigned long m = mp_bits(r->p) - 1;
+ unsigned long i;
+ int rc;
+
+ for (i = 0; i < m - 1; i++) {
+ mp *t = gf_sqr(spare, y);
+ spare = y;
+ y = gfreduce_do(r, t, t);
+ y = gf_add(y, y, x);
+ }
+ rc = !MP_ISZERO(y);
+ mp_drop(spare);
+ mp_drop(y);
+ return (rc);
+}
+
+/* --- @gfreduce_halftrace@ --- *
+ *
+ * Arguments: @gfreduce *r@ = pointer to reduction context
+ * @mp *d@ = destination
+ * @mp *x@ = some polynomial
+ *
+ * Returns: The half-trace of @x@.
+ * (%$\HfTr(x)= x + x^{2^2} + \cdots + x^{2^{m-1}}$%
+ * if %$x \in \gf{2^m}$% with %$m$% odd).
+ */
+
+mp *gfreduce_halftrace(gfreduce *r, mp *d, mp *x)
+{
+ mp *y = MP_COPY(x);
+ mp *spare = MP_NEW;
+ unsigned long m = mp_bits(r->p) - 1;
+ unsigned long i;
+
+ mp_drop(d);
+ for (i = 0; i < m - 1; i += 2) {
+ mp *t = gf_sqr(spare, y);
+ spare = y;
+ y = gfreduce_do(r, t, t);
+ t = gf_sqr(spare, y);
+ spare = y;
+ y = gfreduce_do(r, t, t);
+ y = gf_add(y, y, x);
+ }
+ mp_drop(spare);
+ return (y);
+}
+
+/* --- @gfreduce_quadsolve@ --- *
+ *
+ * Arguments: @gfreduce *r@ = pointer to reduction context
+ * @mp *d@ = destination
+ * @mp *x@ = some polynomial
+ *
+ * Returns: A polynomial @y@ such that %$y^2 + y = x$%, or null.
+ */
+
+mp *gfreduce_quadsolve(gfreduce *r, mp *d, mp *x)
+{
+ unsigned long m = mp_bits(r->p) - 1;
+ mp *t;
+
+ MP_COPY(x);
+ if (m & 1)
+ d = gfreduce_halftrace(r, d, x);
+ else {
+ mp *z, *w, *rho = MP_NEW;
+ mp *spare = MP_NEW;
+ grand *fr = fibrand_create(0);
+ unsigned long i;
+
+ for (;;) {
+ rho = mprand(rho, m, fr, 0);
+ z = MP_ZERO;
+ w = MP_COPY(rho);
+ for (i = 0; i < m - 1; i++) {
+ t = gf_sqr(spare, z); spare = z; z = gfreduce_do(r, t, t);
+ t = gf_sqr(spare, w); spare = w; w = gfreduce_do(r, t, t);
+ t = gf_mul(spare, w, x); t = gfreduce_do(r, t, t); spare = t;
+ z = gf_add(z, z, t);
+ w = gf_add(w, w, rho);
+ }
+ if (!MP_ISZERO(w))
+ break;
+ MP_DROP(z);
+ MP_DROP(w);
+ }
+ if (d) MP_DROP(d);
+ MP_DROP(w);
+ MP_DROP(spare);
+ MP_DROP(rho);
+ fr->ops->destroy(fr);
+ d = z;
+ }
+
+ t = gf_sqr(MP_NEW, d); t = gfreduce_do(r, t, t); t = gf_add(t, t, d);
+ if (!MP_EQ(t, x)) {
+ MP_DROP(d);
+ d = 0;
+ }
+ MP_DROP(t);
+ MP_DROP(x);
+ d->v[0] &= ~(mpw)1;
+ return (d);
+}
+
+/* --- @gfreduce_exp@ --- *
+ *
+ * Arguments: @gfreduce *gr@ = pointer to reduction context
+ * @mp *d@ = fake destination
+ * @mp *a@ = base
+ * @mp *e@ = exponent
+ *
+ * Returns: Result, %$a^e \bmod m$%.
+ */
+
+mp *gfreduce_exp(gfreduce *gr, mp *d, mp *a, mp *e)
+{
+ mp *x = MP_ONE;
+ mp *spare = (e->f & MP_BURN) ? MP_NEWSEC : MP_NEW;
+
+ MP_SHRINK(e);
+ if (!MP_LEN(e))
+ ;
+ else if (MP_LEN(e) < EXP_THRESH)
+ EXP_SIMPLE(x, a, e);
+ else
+ EXP_WINDOW(x, a, e);
+ mp_drop(d);
+ mp_drop(spare);
+ return (x);
+}
+
+/*----- Test rig ----------------------------------------------------------*/
+
+#ifdef TEST_RIG
+
+#define MP(x) mp_readstring(MP_NEW, #x, 0, 0)
+
+static int vreduce(dstr *v)
+{
+ mp *d = *(mp **)v[0].buf;
+ mp *n = *(mp **)v[1].buf;
+ mp *r = *(mp **)v[2].buf;
+ mp *c;
+ int ok = 1;
+ gfreduce rr;
+
+ gfreduce_create(&rr, d);
+ c = gfreduce_do(&rr, MP_NEW, n);
+ if (!MP_EQ(c, r)) {
+ fprintf(stderr, "\n*** reduction failed\n*** ");
+ gfreduce_dump(&rr, stderr);
+ fprintf(stderr, "\n*** n = "); mp_writefile(n, stderr, 16);
+ fprintf(stderr, "\n*** r = "); mp_writefile(r, stderr, 16);
+ fprintf(stderr, "\n*** c = "); mp_writefile(c, stderr, 16);
+ fprintf(stderr, "\n");
+ ok = 0;
+ }
+ gfreduce_destroy(&rr);
+ mp_drop(n); mp_drop(d); mp_drop(r); mp_drop(c);
+ assert(mparena_count(MPARENA_GLOBAL) == 0);
+ return (ok);
+}
+
+static int vmodexp(dstr *v)
+{
+ mp *p = *(mp **)v[0].buf;
+ mp *g = *(mp **)v[1].buf;
+ mp *x = *(mp **)v[2].buf;
+ mp *r = *(mp **)v[3].buf;
+ mp *c;
+ int ok = 1;
+ gfreduce rr;
+
+ gfreduce_create(&rr, p);
+ c = gfreduce_exp(&rr, MP_NEW, g, x);
+ if (!MP_EQ(c, r)) {
+ fprintf(stderr, "\n*** modexp failed\n*** ");
+ fprintf(stderr, "\n*** p = "); mp_writefile(p, stderr, 16);
+ fprintf(stderr, "\n*** g = "); mp_writefile(g, stderr, 16);
+ fprintf(stderr, "\n*** x = "); mp_writefile(x, stderr, 16);
+ fprintf(stderr, "\n*** c = "); mp_writefile(c, stderr, 16);
+ fprintf(stderr, "\n*** r = "); mp_writefile(r, stderr, 16);
+ fprintf(stderr, "\n");
+ ok = 0;
+ }
+ gfreduce_destroy(&rr);
+ mp_drop(p); mp_drop(g); mp_drop(r); mp_drop(x); mp_drop(c);
+ assert(mparena_count(MPARENA_GLOBAL) == 0);
+ return (ok);
+}
+
+static int vsqrt(dstr *v)
+{
+ mp *p = *(mp **)v[0].buf;
+ mp *x = *(mp **)v[1].buf;
+ mp *r = *(mp **)v[2].buf;
+ mp *c;
+ int ok = 1;
+ gfreduce rr;
+
+ gfreduce_create(&rr, p);
+ c = gfreduce_sqrt(&rr, MP_NEW, x);
+ if (!MP_EQ(c, r)) {
+ fprintf(stderr, "\n*** sqrt failed\n*** ");
+ fprintf(stderr, "\n*** p = "); mp_writefile(p, stderr, 16);
+ fprintf(stderr, "\n*** x = "); mp_writefile(x, stderr, 16);
+ fprintf(stderr, "\n*** c = "); mp_writefile(c, stderr, 16);
+ fprintf(stderr, "\n*** r = "); mp_writefile(r, stderr, 16);
+ fprintf(stderr, "\n");
+ ok = 0;
+ }
+ gfreduce_destroy(&rr);
+ mp_drop(p); mp_drop(r); mp_drop(x); mp_drop(c);
+ assert(mparena_count(MPARENA_GLOBAL) == 0);
+ return (ok);
+}
+
+static int vtr(dstr *v)
+{
+ mp *p = *(mp **)v[0].buf;
+ mp *x = *(mp **)v[1].buf;
+ int r = *(int *)v[2].buf, c;
+ int ok = 1;
+ gfreduce rr;
+
+ gfreduce_create(&rr, p);
+ c = gfreduce_trace(&rr, x);
+ if (c != r) {
+ fprintf(stderr, "\n*** trace failed\n*** ");
+ fprintf(stderr, "\n*** p = "); mp_writefile(p, stderr, 16);
+ fprintf(stderr, "\n*** x = "); mp_writefile(x, stderr, 16);
+ fprintf(stderr, "\n*** c = %d", c);
+ fprintf(stderr, "\n*** r = %d", r);
+ fprintf(stderr, "\n");
+ ok = 0;
+ }
+ gfreduce_destroy(&rr);
+ mp_drop(p); mp_drop(x);
+ assert(mparena_count(MPARENA_GLOBAL) == 0);
+ return (ok);
+}
+
+static int vhftr(dstr *v)
+{
+ mp *p = *(mp **)v[0].buf;
+ mp *x = *(mp **)v[1].buf;
+ mp *r = *(mp **)v[2].buf;
+ mp *c;
+ int ok = 1;
+ gfreduce rr;
+
+ gfreduce_create(&rr, p);
+ c = gfreduce_halftrace(&rr, MP_NEW, x);
+ if (!MP_EQ(c, r)) {
+ fprintf(stderr, "\n*** halftrace failed\n*** ");
+ fprintf(stderr, "\n*** p = "); mp_writefile(p, stderr, 16);
+ fprintf(stderr, "\n*** x = "); mp_writefile(x, stderr, 16);
+ fprintf(stderr, "\n*** c = "); mp_writefile(c, stderr, 16);
+ fprintf(stderr, "\n*** r = "); mp_writefile(r, stderr, 16);
+ fprintf(stderr, "\n");
+ ok = 0;
+ }
+ gfreduce_destroy(&rr);
+ mp_drop(p); mp_drop(r); mp_drop(x); mp_drop(c);
+ assert(mparena_count(MPARENA_GLOBAL) == 0);
+ return (ok);
+}
+
+static int vquad(dstr *v)
+{
+ mp *p = *(mp **)v[0].buf;
+ mp *x = *(mp **)v[1].buf;
+ mp *r = *(mp **)v[2].buf;
+ mp *c;
+ int ok = 1;
+ gfreduce rr;
+
+ gfreduce_create(&rr, p);
+ c = gfreduce_quadsolve(&rr, MP_NEW, x);
+ if (!MP_EQ(c, r)) {
+ fprintf(stderr, "\n*** quadsolve failed\n*** ");
+ fprintf(stderr, "\n*** p = "); mp_writefile(p, stderr, 16);
+ fprintf(stderr, "\n*** x = "); mp_writefile(x, stderr, 16);
+ fprintf(stderr, "\n*** c = "); mp_writefile(c, stderr, 16);
+ fprintf(stderr, "\n*** r = "); mp_writefile(r, stderr, 16);
+ fprintf(stderr, "\n");
+ ok = 0;
+ }
+ gfreduce_destroy(&rr);
+ mp_drop(p); mp_drop(r); mp_drop(x); mp_drop(c);
+ assert(mparena_count(MPARENA_GLOBAL) == 0);
+ return (ok);
+}
+
+static test_chunk defs[] = {
+ { "reduce", vreduce, { &type_mp, &type_mp, &type_mp, 0 } },
+ { "modexp", vmodexp, { &type_mp, &type_mp, &type_mp, &type_mp, 0 } },
+ { "sqrt", vsqrt, { &type_mp, &type_mp, &type_mp, 0 } },
+ { "trace", vtr, { &type_mp, &type_mp, &type_int, 0 } },
+ { "halftrace", vhftr, { &type_mp, &type_mp, &type_mp, 0 } },
+ { "quadsolve", vquad, { &type_mp, &type_mp, &type_mp, 0 } },
+ { 0, 0, { 0 } }
+};
+
+int main(int argc, char *argv[])
+{
+ test_run(argc, argv, defs, SRCDIR"/tests/gfreduce");
+ return (0);
+}
+
+#endif
+
+/*----- That's all, folks -------------------------------------------------*/
projects / u / mdw / catacomb / blobdiff