/* -*-c-*-
*
- * $Id: ec-prime.c,v 1.3.4.1 2003/06/10 13:43:53 mdw Exp $
+ * $Id: ec-prime.c,v 1.3.4.2 2004/03/20 00:13:31 mdw Exp $
*
* Elliptic curves over prime fields
*
/*----- Revision history --------------------------------------------------*
*
* $Log: ec-prime.c,v $
+ * Revision 1.3.4.2 2004/03/20 00:13:31 mdw
+ * Projective coordinates for prime curves
+ *
* Revision 1.3.4.1 2003/06/10 13:43:53 mdw
* Simple (non-projective) curves over prime fields now seem to work.
*
/*----- Simple prime curves -----------------------------------------------*/
-static const ec_ops ec_primeops;
+static const ec_ops ec_primeops, ec_primeprojops, ec_primeprojxops;
static ec *ecneg(ec_curve *c, ec *d, const ec *p)
{
return (d);
}
+static ec *ecfind(ec_curve *c, ec *d, mp *x)
+{
+ mp *p, *q;
+ ecctx *cc = (ecctx *)c;
+ field *f = c->f;
+
+ q = F_SQR(f, MP_NEW, x);
+ p = F_MUL(f, MP_NEW, x, q);
+ q = F_MUL(f, q, x, cc->a);
+ p = F_ADD(f, p, p, q);
+ p = F_ADD(f, p, p, cc->b);
+ MP_DROP(q);
+ p = F_SQRT(f, p, p);
+ if (!p)
+ return (0);
+ EC_DESTROY(d);
+ d->x = MP_COPY(x);
+ d->y = p;
+ d->z = MP_COPY(f->one);
+ return (d);
+}
+
static ec *ecdbl(ec_curve *c, ec *d, const ec *a)
{
if (EC_ATINF(a))
EC_SETINF(d);
- else if (!MP_LEN(a->y))
+ else if (F_ZEROP(c->f, a->y))
EC_COPY(d, a);
else {
field *f = c->f;
mp *lambda;
mp *dy, *dx;
- dx = F_SQR(f, MP_NEW, a->x);
- dy = F_DBL(f, MP_NEW, a->y);
- dx = F_TPL(f, dx, dx);
- dx = F_ADD(f, dx, dx, cc->a);
- dy = F_INV(f, dy, dy);
- lambda = F_MUL(f, MP_NEW, dx, dy);
+ dx = F_SQR(f, MP_NEW, a->x); /* %$x^2$% */
+ dy = F_DBL(f, MP_NEW, a->y); /* %$2 y$% */
+ dx = F_TPL(f, dx, dx); /* %$3 x^2$% */
+ dx = F_ADD(f, dx, dx, cc->a); /* %$3 x^2 + A$% */
+ dy = F_INV(f, dy, dy); /* %$(2 y)^{-1}$% */
+ lambda = F_MUL(f, MP_NEW, dx, dy); /* %$\lambda = (3 x^2 + A)/(2 y)$% */
- dx = F_SQR(f, dx, lambda);
- dy = F_DBL(f, dy, a->x);
- dx = F_SUB(f, dx, dx, dy);
- dy = F_SUB(f, dy, a->x, dx);
- dy = F_MUL(f, dy, lambda, dy);
- dy = F_SUB(f, dy, dy, a->y);
+ dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */
+ dy = F_DBL(f, dy, a->x); /* %$2 x$% */
+ dx = F_SUB(f, dx, dx, dy); /* %$x' = \lambda^2 - 2 x */
+ dy = F_SUB(f, dy, a->x, dx); /* %$x - x'$% */
+ dy = F_MUL(f, dy, lambda, dy); /* %$\lambda (x - x')$% */
+ dy = F_SUB(f, dy, dy, a->y); /* %$y' = \lambda (x - x') - y$% */
EC_DESTROY(d);
d->x = dx;
return (d);
}
+static ec *ecprojdbl(ec_curve *c, ec *d, const ec *a)
+{
+ if (EC_ATINF(a))
+ EC_SETINF(d);
+ else if (F_ZEROP(c->f, a->y))
+ EC_COPY(d, a);
+ else {
+ field *f = c->f;
+ ecctx *cc = (ecctx *)c;
+ mp *p, *q, *m, *s, *dx, *dy, *dz;
+
+ p = F_SQR(f, MP_NEW, a->z); /* %$z^2$% */
+ q = F_SQR(f, MP_NEW, p); /* %$z^4$% */
+ p = F_MUL(f, p, q, cc->a); /* %$A z^4$% */
+ m = F_SQR(f, MP_NEW, a->x); /* %$x^2$% */
+ m = F_TPL(f, m, m); /* %$3 x^2$% */
+ m = F_ADD(f, m, m, p); /* %$m = 3 x^2 + A z^4$% */
+
+ q = F_DBL(f, q, a->y); /* %$2 y$% */
+ dz = F_MUL(f, MP_NEW, q, a->z); /* %$z' = 2 y z$% */
+
+ p = F_SQR(f, p, q); /* %$4 y^2$% */
+ s = F_MUL(f, MP_NEW, p, a->x); /* %$s = 4 x y^2$% */
+ q = F_SQR(f, q, p); /* %$16 y^4$% */
+ q = F_HLV(f, q, q); /* %$t = 8 y^4$% */
+
+ p = F_DBL(f, p, s); /* %$2 s$% */
+ dx = F_SQR(f, MP_NEW, m); /* %$m^2$% */
+ dx = F_SUB(f, dx, dx, p); /* %$x' = m^2 - 2 s$% */
+
+ s = F_SUB(f, s, s, dx); /* %$s - x'$% */
+ dy = F_MUL(f, p, m, s); /* %$m (s - x')$% */
+ dy = F_SUB(f, dy, dy, q); /* %$y' = m (s - x') - t$% */
+
+ EC_DESTROY(d);
+ d->x = dx;
+ d->y = dy;
+ d->z = dz;
+ MP_DROP(m);
+ MP_DROP(q);
+ MP_DROP(s);
+ }
+ return (d);
+}
+
+static ec *ecprojxdbl(ec_curve *c, ec *d, const ec *a)
+{
+ if (EC_ATINF(a))
+ EC_SETINF(d);
+ else if (F_ZEROP(c->f, a->y))
+ EC_COPY(d, a);
+ else {
+ field *f = c->f;
+ mp *p, *q, *m, *s, *dx, *dy, *dz;
+
+ m = F_SQR(f, MP_NEW, a->z); /* %$z^2$% */
+ p = F_SUB(f, MP_NEW, a->x, m); /* %$x - z^2$% */
+ q = F_ADD(f, MP_NEW, a->x, m); /* %$x + z^2$% */
+ m = F_MUL(f, m, p, q); /* %$x^2 - z^4$% */
+ m = F_TPL(f, m, m); /* %$m = 3 x^2 - 3 z^4$% */
+
+ q = F_DBL(f, q, a->y); /* %$2 y$% */
+ dz = F_MUL(f, MP_NEW, q, a->z); /* %$z' = 2 y z$% */
+
+ p = F_SQR(f, p, q); /* %$4 y^2$% */
+ s = F_MUL(f, MP_NEW, p, a->x); /* %$s = 4 x y^2$% */
+ q = F_SQR(f, q, p); /* %$16 y^4$% */
+ q = F_HLV(f, q, q); /* %$t = 8 y^4$% */
+
+ p = F_DBL(f, p, s); /* %$2 s$% */
+ dx = F_SQR(f, MP_NEW, m); /* %$m^2$% */
+ dx = F_SUB(f, dx, dx, p); /* %$x' = m^2 - 2 s$% */
+
+ s = F_SUB(f, s, s, dx); /* %$s - x'$% */
+ dy = F_MUL(f, p, m, s); /* %$m (s - x')$% */
+ dy = F_SUB(f, dy, dy, q); /* %$y' = m (s - x') - t$% */
+
+ EC_DESTROY(d);
+ d->x = dx;
+ d->y = dy;
+ d->z = dz;
+ MP_DROP(m);
+ MP_DROP(q);
+ MP_DROP(s);
+ }
+ return (d);
+}
+
static ec *ecadd(ec_curve *c, ec *d, const ec *a, const ec *b)
{
if (a == b)
mp *dy, *dx;
if (!MP_EQ(a->x, b->x)) {
- dy = F_SUB(f, MP_NEW, a->y, b->y);
- dx = F_SUB(f, MP_NEW, a->x, b->x);
- dx = F_INV(f, dx, dx);
+ dy = F_SUB(f, MP_NEW, a->y, b->y); /* %$y_0 - y_1$% */
+ dx = F_SUB(f, MP_NEW, a->x, b->x); /* %$x_0 - x_1$% */
+ dx = F_INV(f, dx, dx); /* %$(x_0 - x_1)^{-1}$% */
lambda = F_MUL(f, MP_NEW, dy, dx);
- } else if (!MP_LEN(a->y) || !MP_EQ(a->y, b->y)) {
+ /* %$\lambda = (y_0 - y1)/(x_0 - x_1)$% */
+ } else if (F_ZEROP(c->f, a->y) || !MP_EQ(a->y, b->y)) {
EC_SETINF(d);
return (d);
} else {
ecctx *cc = (ecctx *)c;
- dx = F_SQR(f, MP_NEW, a->x);
- dx = F_TPL(f, dx, dx);
- dx = F_ADD(f, dx, dx, cc->a);
- dy = F_DBL(f, MP_NEW, a->y);
- dy = F_INV(f, dy, dy);
+ dx = F_SQR(f, MP_NEW, a->x); /* %$x_0^2$% */
+ dx = F_TPL(f, dx, dx); /* %$3 x_0^2$% */
+ dx = F_ADD(f, dx, dx, cc->a); /* %$3 x_0^2 + A$% */
+ dy = F_DBL(f, MP_NEW, a->y); /* %$2 y_0$% */
+ dy = F_INV(f, dy, dy); /* %$(2 y_0)^{-1}$% */
lambda = F_MUL(f, MP_NEW, dx, dy);
+ /* %$\lambda = (3 x_0^2 + A)/(2 y_0)$% */
}
- dx = F_SQR(f, dx, lambda);
- dx = F_SUB(f, dx, dx, a->x);
- dx = F_SUB(f, dx, dx, b->x);
- dy = F_SUB(f, dy, b->x, dx);
- dy = F_MUL(f, dy, lambda, dy);
+ dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */
+ dx = F_SUB(f, dx, dx, a->x); /* %$\lambda^2 - x_0$% */
+ dx = F_SUB(f, dx, dx, b->x); /* %$x' = \lambda^2 - x_0 - x_1$% */
+ dy = F_SUB(f, dy, b->x, dx); /* %$x_1 - x'$% */
+ dy = F_MUL(f, dy, lambda, dy); /* %$\lambda (x_1 - x')$% */
dy = F_SUB(f, dy, dy, b->y);
+ /* %$y' = \lambda (x_1 - x') - y_1$% */
EC_DESTROY(d);
d->x = dx;
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 *p, *q, *r, *w, *u, *s, *dx, *dy, *dz;
+
+ q = F_SQR(f, MP_NEW, a->z); /* %$z_0^2$% */
+ u = F_MUL(f, MP_NEW, q, b->x); /* %$u = x_1 z_0^2$% */
+ p = F_MUL(f, MP_NEW, q, b->y); /* %$y_1 z_0^2$% */
+ s = F_MUL(f, q, p, a->z); /* %$s = y_1 z_0^3$% */
+
+ w = F_SUB(f, p, a->x, u); /* %$w = x_0 - u$% */
+ r = F_SUB(f, MP_NEW, a->y, s); /* %$r = y_0 - s$% */
+ if (F_ZEROP(f, w)) {
+ if (F_ZEROP(f, r)) {
+ MP_DROP(w);
+ MP_DROP(r);
+ MP_DROP(u);
+ MP_DROP(s);
+ return (c->ops->dbl(c, d, a));
+ } else {
+ MP_DROP(w);
+ MP_DROP(r);
+ MP_DROP(u);
+ MP_DROP(s);
+ EC_SETINF(d);
+ return (d);
+ }
+ }
+ u = F_ADD(f, u, u, a->x); /* %$t = x_0 + u$% */
+ s = F_ADD(f, s, s, a->y); /* %$m = y_0 + r$% */
+
+ dz = F_MUL(f, MP_NEW, a->z, w); /* %$z' = z_0 w$% */
+
+ p = F_SQR(f, MP_NEW, w); /* %$w^2$% */
+ q = F_MUL(f, MP_NEW, p, u); /* %$t w^2$% */
+ u = F_MUL(f, u, p, w); /* %$w^3$% */
+ p = F_MUL(f, p, u, s); /* %$m w^3$% */
+
+ dx = F_SQR(f, u, r); /* %$r^2$% */
+ dx = F_SUB(f, dx, dx, q); /* %$x' = r^2 - t w^2$% */
+
+ s = F_DBL(f, s, dx); /* %$2 x'$% */
+ q = F_SUB(f, q, q, s); /* %$v = t w^2 - 2 x'$% */
+ dy = F_MUL(f, s, q, r); /* %$v r$% */
+ dy = F_SUB(f, dy, dy, p); /* %$v r - m w^3$% */
+ dy = F_HLV(f, dy, dy); /* %$y' = (v r - m w^3)/2$% */
+
+ EC_DESTROY(d);
+ d->x = dx;
+ d->y = dy;
+ d->z = dz;
+ MP_DROP(p);
+ MP_DROP(q);
+ MP_DROP(r);
+ MP_DROP(w);
+ }
+ return (d);
+}
+
+static int eccheck(ec_curve *c, const ec *p)
+{
+ ecctx *cc = (ecctx *)c;
+ field *f = c->f;
+ int rc;
+ mp *l = F_SQR(f, MP_NEW, p->y);
+ mp *x = F_SQR(f, MP_NEW, p->x);
+ mp *r = F_MUL(f, MP_NEW, x, p->x);
+ x = F_MUL(f, x, cc->a, p->x);
+ r = F_ADD(f, r, r, x);
+ r = F_ADD(f, r, r, cc->b);
+ rc = MP_EQ(l, r) ? 0 : -1;
+ mp_drop(l);
+ mp_drop(x);
+ mp_drop(r);
+ 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 *cc = (ecctx *)c;
return (&cc->c);
}
+extern ec_curve *ec_primeproj(field *f, mp *a, mp *b)
+{
+ ecctx *cc = CREATE(ecctx);
+ mp *ax;
+
+ ax = mp_add(MP_NEW, a, MP_THREE);
+ ax = F_IN(f, ax, ax);
+ if (F_ZEROP(f, ax))
+ cc->c.ops = &ec_primeprojxops;
+ else
+ cc->c.ops = &ec_primeprojops;
+ MP_DROP(ax);
+ cc->c.f = f;
+ cc->a = F_IN(f, MP_NEW, a);
+ cc->b = F_IN(f, MP_NEW, b);
+ return (&cc->c);
+}
+
static const ec_ops ec_primeops = {
- ecdestroy, ec_idin, ec_idout, 0, ecneg, ecadd, ec_stdsub, ecdbl
+ ecdestroy, ec_idin, ec_idout, ec_idfix,
+ 0, ecneg, ecadd, ec_stdsub, ecdbl, eccheck
+};
+
+static const ec_ops ec_primeprojops = {
+ ecdestroy, ec_projin, ec_projout, ec_projfix,
+ 0, ecneg, ecprojadd, ec_stdsub, ecprojdbl, ecprojcheck
+};
+
+static const ec_ops ec_primeprojxops = {
+ ecdestroy, ec_projin, ec_projout, ec_projfix,
+ 0, ecneg, ecprojadd, ec_stdsub, ecprojxdbl, ecprojcheck
};
/*----- Test rig ----------------------------------------------------------*/
projects / u / mdw / catacomb / blobdiff