--- /dev/null
+/* -*-c-*-
+ *
+ * Arithmetic for elliptic curves over binary fields
+ *
+ * (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.
+ */
+
+/*----- Header files ------------------------------------------------------*/
+
+#include <mLib/sub.h>
+
+#include "ec.h"
+#include "ec-guts.h"
+
+/*----- Main code ---------------------------------------------------------*/
+
+static const ec_ops ec_binops, ec_binprojops;
+
+static ec *ecneg(ec_curve *c, ec *d, const ec *p)
+{
+ EC_COPY(d, p);
+ if (d->x)
+ d->y = F_ADD(c->f, d->y, d->y, d->x);
+ return (d);
+}
+
+static ec *ecprojneg(ec_curve *c, ec *d, const ec *p)
+{
+ EC_COPY(d, p);
+ if (d->x) {
+ mp *t = F_MUL(c->f, MP_NEW, d->x, d->z);
+ d->y = F_ADD(c->f, d->y, d->y, t);
+ MP_DROP(t);
+ }
+ return (d);
+}
+
+static ec *ecfind(ec_curve *c, ec *d, mp *x)
+{
+ field *f = c->f;
+ mp *y, *u, *v;
+
+ if (F_ZEROP(f, x))
+ y = F_SQRT(f, MP_NEW, c->b);
+ else {
+ u = F_SQR(f, MP_NEW, x); /* %$x^2$% */
+ y = F_MUL(f, MP_NEW, u, c->a); /* %$a x^2$% */
+ y = F_ADD(f, y, y, c->b); /* %$a x^2 + b$% */
+ v = F_MUL(f, MP_NEW, u, x); /* %$x^3$% */
+ y = F_ADD(f, y, y, v); /* %$A = x^3 + a x^2 + b$% */
+ if (!F_ZEROP(f, y)) {
+ u = F_INV(f, u, u); /* %$x^{-2}$% */
+ v = F_MUL(f, v, u, y); /* %$B = A x^{-2} = x + a + b x^{-2}$% */
+ y = F_QUADSOLVE(f, y, v); /* %$z^2 + z = B$% */
+ if (y) y = F_MUL(f, y, y, x); /* %$y = z x$% */
+ }
+ MP_DROP(u);
+ MP_DROP(v);
+ }
+ if (!y) return (0);
+ EC_DESTROY(d);
+ d->x = MP_COPY(x);
+ d->y = y;
+ d->z = MP_COPY(f->one);
+ return (d);
+}
+
+static ec *ecdbl(ec_curve *c, ec *d, const ec *a)
+{
+ if (EC_ATINF(a) || F_ZEROP(c->f, a->x))
+ EC_SETINF(d);
+ else {
+ field *f = c->f;
+ mp *lambda;
+ mp *dx, *dy;
+
+ dx = F_INV(f, MP_NEW, a->x); /* %$x^{-1}$% */
+ dy = F_MUL(f, MP_NEW, dx, a->y); /* %$y/x$% */
+ lambda = F_ADD(f, dy, dy, a->x); /* %$\lambda = x + y/x$% */
+
+ dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */
+ dx = F_ADD(f, dx, dx, lambda); /* %$\lambda^2 + \lambda$% */
+ dx = F_ADD(f, dx, dx, c->a); /* %$x' = a + \lambda^2 + \lambda$% */
+
+ dy = F_ADD(f, MP_NEW, a->x, dx); /* %$ x + x' $% */
+ dy = F_MUL(f, dy, dy, lambda); /* %$ (x + x') \lambda$% */
+ dy = F_ADD(f, dy, dy, a->y); /* %$ (x + x') \lambda + y$% */
+ dy = F_ADD(f, dy, dy, dx); /* %$ y' = (x + x') \lambda + y + x'$% */
+
+ EC_DESTROY(d);
+ d->x = dx;
+ d->y = dy;
+ d->z = 0;
+ MP_DROP(lambda);
+ }
+ return (d);
+}
+
+static ec *ecprojdbl(ec_curve *c, ec *d, const ec *a)
+{
+ if (EC_ATINF(a) || F_ZEROP(c->f, a->x))
+ EC_SETINF(d);
+ else {
+ field *f = c->f;
+ ecctx_bin *cc = (ecctx_bin *)c;
+ mp *dx, *dy, *dz, *u, *v;
+
+ dy = F_SQR(f, MP_NEW, a->z); /* %$z^2$% */
+ dx = F_MUL(f, MP_NEW, dy, cc->bb); /* %$c z^2$% */
+ dx = F_ADD(f, dx, dx, a->x); /* %$x + c z^2$% */
+ dz = F_SQR(f, MP_NEW, dx); /* %$(x + c z^2)^2$% */
+ dx = F_SQR(f, dx, dz); /* %$x' = (x + c z^2)^4$% */
+
+ dz = F_MUL(f, dz, dy, a->x); /* %$z' = x z^2$% */
+
+ dy = F_SQR(f, dy, a->x); /* %$x^2$% */
+ u = F_MUL(f, MP_NEW, a->y, a->z); /* %$y z$% */
+ u = F_ADD(f, u, u, dz); /* %$z' + y z$% */
+ u = F_ADD(f, u, u, dy); /* %$u = z' + x^2 + y z$% */
+
+ v = F_SQR(f, MP_NEW, dy); /* %$x^4$% */
+ dy = F_MUL(f, dy, v, dz); /* %$x^4 z'$% */
+ v = F_MUL(f, v, u, dx); /* %$u x'$% */
+ dy = F_ADD(f, dy, dy, v); /* %$y' = x^4 z' + u x'$% */
+
+ EC_DESTROY(d);
+ d->x = dx;
+ d->y = dy;
+ d->z = dz;
+ MP_DROP(u);
+ MP_DROP(v);
+ }
+ return (d);
+}
+
+static ec *ecadd(ec_curve *c, ec *d, const ec *a, const ec *b)
+{
+ if (a == b)
+ ecdbl(c, d, a);
+ else if (EC_ATINF(a))
+ EC_COPY(d, b);
+ else if (EC_ATINF(b))
+ EC_COPY(d, a);
+ else {
+ field *f = c->f;
+ mp *lambda;
+ mp *dx, *dy;
+
+ if (!MP_EQ(a->x, b->x)) {
+ dx = F_ADD(f, MP_NEW, a->x, b->x); /* %$x_0 + x_1$% */
+ dy = F_INV(f, MP_NEW, dx); /* %$(x_0 + x_1)^{-1}$% */
+ dx = F_ADD(f, dx, a->y, b->y); /* %$y_0 + y_1$% */
+ lambda = F_MUL(f, MP_NEW, dy, dx);
+ /* %$\lambda = (y_0 + y_1)/(x_0 + x_1)$% */
+
+ dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */
+ dx = F_ADD(f, dx, dx, lambda); /* %$\lambda^2 + \lambda$% */
+ dx = F_ADD(f, dx, dx, c->a); /* %$a + \lambda^2 + \lambda$% */
+ dx = F_ADD(f, dx, dx, a->x); /* %$a + \lambda^2 + \lambda + x_0$% */
+ dx = F_ADD(f, dx, dx, b->x);
+ /* %$x' = a + \lambda^2 + \lambda + x_0 + x_1$% */
+ } else if (!MP_EQ(a->y, b->y) || F_ZEROP(f, a->x)) {
+ EC_SETINF(d);
+ return (d);
+ } else {
+ dx = F_INV(f, MP_NEW, a->x); /* %$x^{-1}$% */
+ dy = F_MUL(f, MP_NEW, dx, a->y); /* %$y/x$% */
+ lambda = F_ADD(f, dy, dy, a->x); /* %$\lambda = x + y/x$% */
+
+ dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */
+ dx = F_ADD(f, dx, dx, lambda); /* %$\lambda^2 + \lambda$% */
+ dx = F_ADD(f, dx, dx, c->a); /* %$x' = a + \lambda^2 + \lambda$% */
+ dy = MP_NEW;
+ }
+
+ dy = F_ADD(f, dy, a->x, dx); /* %$ x + x' $% */
+ dy = F_MUL(f, dy, dy, lambda); /* %$ (x + x') \lambda$% */
+ dy = F_ADD(f, dy, dy, a->y); /* %$ (x + x') \lambda + y$% */
+ dy = F_ADD(f, dy, dy, dx); /* %$ y' = (x + x') \lambda + y + x'$% */
+
+ EC_DESTROY(d);
+ d->x = dx;
+ d->y = dy;
+ d->z = 0;
+ MP_DROP(lambda);
+ }
+ return (d);
+}
+
+static ec *ecprojadd(ec_curve *c, ec *d, const ec *a, const ec *b)
+{
+ if (a == b)
+ c->ops->dbl(c, d, a);
+ else if (EC_ATINF(a))
+ EC_COPY(d, b);
+ else if (EC_ATINF(b))
+ EC_COPY(d, a);
+ else {
+ field *f = c->f;
+ mp *dx, *dy, *dz, *u, *uu, *v, *t, *s, *ss, *r, *w, *l;
+
+ dz = F_SQR(f, MP_NEW, b->z); /* %$z_1^2$% */
+ u = F_MUL(f, MP_NEW, dz, a->x); /* %$u_0 = x_0 z_1^2$% */
+ t = F_MUL(f, MP_NEW, dz, b->z); /* %$z_1^3$% */
+ s = F_MUL(f, MP_NEW, t, a->y); /* %$s_0 = y_0 z_1^3$% */
+
+ dz = F_SQR(f, dz, a->z); /* %$z_0^2$% */
+ uu = F_MUL(f, MP_NEW, dz, b->x); /* %$u_1 = x_1 z_0^2$% */
+ t = F_MUL(f, t, dz, a->z); /* %$z_0^3$% */
+ ss = F_MUL(f, MP_NEW, t, b->y); /* %$s_1 = y_1 z_0^3$% */
+
+ w = F_ADD(f, u, u, uu); /* %$r = u_0 + u_1$% */
+ r = F_ADD(f, s, s, ss); /* %$w = s_0 + s_1$% */
+ if (F_ZEROP(f, w)) {
+ MP_DROP(w);
+ MP_DROP(uu);
+ MP_DROP(ss);
+ MP_DROP(t);
+ MP_DROP(dz);
+ if (F_ZEROP(f, r)) {
+ MP_DROP(r);
+ return (c->ops->dbl(c, d, a));
+ } else {
+ MP_DROP(r);
+ EC_SETINF(d);
+ return (d);
+ }
+ }
+
+ l = F_MUL(f, t, a->z, w); /* %$l = z_0 w$% */
+
+ dz = F_MUL(f, dz, l, b->z); /* %$z' = l z_1$% */
+
+ ss = F_MUL(f, ss, r, b->x); /* %$r x_1$% */
+ t = F_MUL(f, uu, l, b->y); /* %$l y_1$% */
+ v = F_ADD(f, ss, ss, t); /* %$v = r x_1 + l y_1$% */
+
+ t = F_ADD(f, t, r, dz); /* %$t = r + z'$% */
+
+ uu = F_SQR(f, MP_NEW, dz); /* %$z'^2$% */
+ dx = F_MUL(f, MP_NEW, uu, c->a); /* %$a z'^2$% */
+ uu = F_MUL(f, uu, t, r); /* %$t r$% */
+ dx = F_ADD(f, dx, dx, uu); /* %$a z'^2 + t r$% */
+ r = F_SQR(f, r, w); /* %$w^2$% */
+ uu = F_MUL(f, uu, r, w); /* %$w^3$% */
+ dx = F_ADD(f, dx, dx, uu); /* %$x' = a z'^2 + t r + w^3$% */
+
+ r = F_SQR(f, r, l); /* %$l^2$% */
+ dy = F_MUL(f, uu, v, r); /* %$v l^2$% */
+ l = F_MUL(f, l, t, dx); /* %$t x'$% */
+ dy = F_ADD(f, dy, dy, l); /* %$y' = t x' + v l^2$% */
+
+ EC_DESTROY(d);
+ d->x = dx;
+ d->y = dy;
+ d->z = dz;
+ MP_DROP(l);
+ MP_DROP(r);
+ MP_DROP(w);
+ MP_DROP(t);
+ MP_DROP(v);
+ }
+ return (d);
+}
+
+static int eccheck(ec_curve *c, const ec *p)
+{
+ field *f = c->f;
+ int rc;
+ mp *u, *v;
+
+ if (EC_ATINF(p)) return (0);
+ v = F_SQR(f, MP_NEW, p->x);
+ u = F_MUL(f, MP_NEW, v, p->x);
+ v = F_MUL(f, v, v, c->a);
+ u = F_ADD(f, u, u, v);
+ u = F_ADD(f, u, u, c->b);
+ v = F_MUL(f, v, p->x, p->y);
+ u = F_ADD(f, u, u, v);
+ v = F_SQR(f, v, p->y);
+ u = F_ADD(f, u, u, v);
+ rc = F_ZEROP(f, u) ? 0 : -1;
+ mp_drop(u);
+ mp_drop(v);
+ return (rc);
+}
+
+static int ecprojcheck(ec_curve *c, const ec *p)
+{
+ ec t = EC_INIT;
+ int rc;
+
+ c->ops->fix(c, &t, p);
+ rc = eccheck(c, &t);
+ EC_DESTROY(&t);
+ return (rc);
+}
+
+static void ecdestroy(ec_curve *c)
+{
+ ecctx_bin *cc = (ecctx_bin *)c;
+ MP_DROP(cc->c.a);
+ MP_DROP(cc->c.b);
+ if (cc->bb) MP_DROP(cc->bb);
+ DESTROY(cc);
+}
+
+/* --- @ec_bin@, @ec_binproj@ --- *
+ *
+ * Arguments: @field *f@ = the underlying field for this elliptic curve
+ * @mp *a, *b@ = the coefficients for this curve
+ *
+ * Returns: A pointer to the curve, or null.
+ *
+ * Use: Creates a curve structure for an elliptic curve defined over
+ * a binary field. The @binproj@ variant uses projective
+ * coordinates, which can be a win.
+ */
+
+ec_curve *ec_bin(field *f, mp *a, mp *b)
+{
+ ecctx_bin *cc = CREATE(ecctx_bin);
+ cc->c.ops = &ec_binops;
+ cc->c.f = f;
+ cc->c.a = F_IN(f, MP_NEW, a);
+ cc->c.b = F_IN(f, MP_NEW, b);
+ cc->bb = 0;
+ return (&cc->c);
+}
+
+ec_curve *ec_binproj(field *f, mp *a, mp *b)
+{
+ ecctx_bin *cc = CREATE(ecctx_bin);
+ int i;
+ mp *c, *d;
+
+ cc->c.ops = &ec_binprojops;
+ cc->c.f = f;
+ cc->c.a = F_IN(f, MP_NEW, a);
+ cc->c.b = F_IN(f, MP_NEW, b);
+
+ c = MP_COPY(cc->c.b);
+ for (i = 0; i < f->nbits - 2; i++)
+ c = F_SQR(f, c, c);
+ d = F_SQR(f, MP_NEW, c); d = F_SQR(f, d, d);
+ if (!MP_EQ(d, cc->c.b)) {
+ MP_DROP(c);
+ MP_DROP(d);
+ MP_DROP(cc->c.a);
+ MP_DROP(cc->c.b);
+ DESTROY(cc);
+ return (0);
+ }
+ cc->bb = c;
+ MP_DROP(d);
+ return (&cc->c);
+}
+
+static const ec_ops ec_binops = {
+ "bin",
+ ecdestroy, ec_stdsamep, ec_idin, ec_idout, ec_idfix,
+ ecfind, ecneg, ecadd, ec_stdsub, ecdbl, eccheck
+};
+
+static const ec_ops ec_binprojops = {
+ "binproj",
+ ecdestroy, ec_stdsamep, ec_projin, ec_projout, ec_projfix,
+ ecfind, ecprojneg, ecprojadd, ec_stdsub, ecprojdbl, ecprojcheck
+};
+
+/*----- Test rig ----------------------------------------------------------*/
+
+#ifdef TEST_RIG
+
+#define MP(x) mp_readstring(MP_NEW, #x, 0, 0)
+
+int main(int argc, char *argv[])
+{
+ field *f;
+ ec_curve *c;
+ ec g = EC_INIT, d = EC_INIT;
+ mp *p, *a, *b, *r, *beta;
+ int i, n = argc == 1 ? 1 : atoi(argv[1]);
+
+ printf("ec-bin: ");
+ fflush(stdout);
+ a = MP(0x7ffffffffffffffffffffffffffffffffffffffff);
+ b = MP(0x6645f3cacf1638e139c6cd13ef61734fbc9e3d9fb);
+ p = MP(0x800000000000000000000000000000000000000c9);
+ beta = MP(0x715169c109c612e390d347c748342bcd3b02a0bef);
+ r = MP(0x040000000000000000000292fe77e70c12a4234c32);
+
+ f = field_binnorm(p, beta);
+ c = ec_binproj(f, a, b);
+ g.x = MP(0x0311103c17167564ace77ccb09c681f886ba54ee8);
+ g.y = MP(0x333ac13c6447f2e67613bf7009daf98c87bb50c7f);
+
+ for (i = 0; i < n; i++) {
+ ec_mul(c, &d, &g, r);
+ if (EC_ATINF(&d)) {
+ fprintf(stderr, "zero too early\n");
+ return (1);
+ }
+ ec_add(c, &d, &d, &g);
+ if (!EC_ATINF(&d)) {
+ fprintf(stderr, "didn't reach zero\n");
+ MP_EPRINTX("d.x", d.x);
+ MP_EPRINTX("d.y", d.y);
+ return (1);
+ }
+ ec_destroy(&d);
+ }
+
+ ec_destroy(&g);
+ ec_destroycurve(c);
+ F_DESTROY(f);
+ MP_DROP(p); MP_DROP(a); MP_DROP(b); MP_DROP(r); MP_DROP(beta);
+ assert(!mparena_count(&mparena_global));
+ printf("ok\n");
+ return (0);
+}
+
+#endif
+
+/*----- That's all, folks -------------------------------------------------*/
projects / u / mdw / catacomb / blobdiff