/* -*-c-*-
*
- * $Id: ec-prime.c,v 1.1 2001/04/29 18:12:33 mdw Exp $
+ * $Id: ec-prime.c,v 1.11 2004/04/08 01:36:15 mdw Exp $
*
* Elliptic curves over prime fields
*
* MA 02111-1307, USA.
*/
-/*----- Revision history --------------------------------------------------*
- *
- * $Log: ec-prime.c,v $
- * Revision 1.1 2001/04/29 18:12:33 mdw
- * Prototype version.
- *
- */
-
/*----- Header files ------------------------------------------------------*/
+#include <mLib/sub.h>
+
#include "ec.h"
-/*----- Data structures ---------------------------------------------------*/
+/*----- Simple prime curves -----------------------------------------------*/
+
+static const ec_ops ec_primeops, ec_primeprojops, ec_primeprojxops;
+
+static ec *ecneg(ec_curve *c, ec *d, const ec *p)
+{
+ EC_COPY(d, p);
+ if (d->y)
+ d->y = F_NEG(c->f, d->y, d->y);
+ return (d);
+}
+
+static ec *ecfind(ec_curve *c, ec *d, mp *x)
+{
+ mp *p, *q;
+ 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, c->a);
+ p = F_ADD(f, p, p, q);
+ p = F_ADD(f, p, p, c->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 (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); /* %$x^2$% */
+ 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, c->a); /* %$3 x^2 + A$% */
+ dy = F_INV(f, dy, dy); /* %$(2 y)^{-1}$% */
+ lambda = F_MUL(f, MP_NEW, dx, dy); /* %$\lambda = (3 x^2 + A)/(2 y)$% */
-typedef struct ecctx {
- ec_curve c;
- mp *a, *b;
-} ecctx;
+ 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$% */
-/*----- Main code ---------------------------------------------------------*/
+ EC_DESTROY(d);
+ d->x = dx;
+ d->y = dy;
+ d->z = 0;
+ MP_DROP(lambda);
+ }
+ return (d);
+}
-static void ecadd(ec_curve *c, ec *d, const ec *a, const ec *b)
+static ec *ecprojdbl(ec_curve *c, ec *d, const ec *a)
{
- /* --- Deal with the simple cases --- */
+ if (EC_ATINF(a))
+ EC_SETINF(d);
+ else if (F_ZEROP(c->f, a->y))
+ EC_COPY(d, a);
+ else {
+ field *f = c->f;
+ mp *p, *q, *m, *s, *dx, *dy, *dz;
+
+ p = F_SQR(f, MP_NEW, a->z); /* %$z^2$% */
+ q = F_SQR(f, MP_NEW, p); /* %$z^4$% */
+ p = F_MUL(f, p, q, c->a); /* %$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)
ecdbl(c, d, a);
else if (EC_ATINF(a))
EC_COPY(d, b);
else if (EC_ATINF(b))
EC_COPY(d, a);
- else if (MP_EQ(a->x, b->x) && MP_EQ(a->z, b->z)) {
- if ((a->y->f ^ b->y->f) & MP_NEG)
+ else {
+ field *f = c->f;
+ mp *lambda;
+ mp *dy, *dx;
+
+ if (!MP_EQ(a->x, b->x)) {
+ 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);
+ /* %$\lambda = (y_0 - y1)/(x_0 - x_1)$% */
+ } else if (F_ZEROP(c->f, a->y) || !MP_EQ(a->y, b->y)) {
EC_SETINF(d);
- else
- ecdbl(c, d, a);
- } else {
+ return (d);
+ } else {
+ 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, c->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); /* %$\lambda^2$% */
+ 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;
+ 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 *p, *q, *r, *w, *u, *uu, *s, *ss, *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); /* %$s = y_1 z_0^3$% */
+
+ q = F_SQR(f, MP_NEW, b->z); /* %$z_1^2$% */
+ uu = F_MUL(f, MP_NEW, q, a->x); /* %$uu = x_0 z_1^2$%*/
+ p = F_MUL(f, p, q, a->y); /* %$y_0 z_1^2$% */
+ ss = F_MUL(f, q, p, b->z); /* %$ss = y_0 z_1^3$% */
+
+ w = F_SUB(f, p, uu, u); /* %$w = uu - u$% */
+ r = F_SUB(f, MP_NEW, ss, s); /* %$r = ss - s$% */
+ if (F_ZEROP(f, w)) {
+ MP_DROP(w);
+ MP_DROP(u);
+ MP_DROP(s);
+ MP_DROP(uu);
+ MP_DROP(ss);
+ if (F_ZEROP(f, r)) {
+ MP_DROP(r);
+ return (c->ops->dbl(c, d, a));
+ } else {
+ MP_DROP(r);
+ EC_SETINF(d);
+ return (d);
+ }
+ }
+ u = F_ADD(f, u, u, uu); /* %$t = uu + u$% */
+ s = F_ADD(f, s, s, ss); /* %$m = ss + r$% */
+
+ uu = F_MUL(f, uu, a->z, w); /* %$z_0 w$% */
+ dz = F_MUL(f, ss, uu, b->z); /* %$z' = z_0 z_1 w$% */
+
+ p = F_SQR(f, uu, 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)
+{
+ field *f = c->f;
+ mp *l, *x, *r;
+ int rc;
+ if (EC_ATINF(p)) return (0);
+ l = F_SQR(f, MP_NEW, p->y);
+ x = F_SQR(f, MP_NEW, p->x);
+ r = F_MUL(f, MP_NEW, x, p->x);
+ x = F_MUL(f, x, c->a, p->x);
+ r = F_ADD(f, r, r, x);
+ r = F_ADD(f, r, r, c->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)
+{
+ MP_DROP(c->a);
+ MP_DROP(c->b);
+ DESTROY(c);
+}
+
+/* --- @ec_prime@, @ec_primeproj@ --- *
+ *
+ * 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 prime field. The @primeproj@ variant uses projective
+ * coordinates, which can be a win.
+ */
+
+extern ec_curve *ec_prime(field *f, mp *a, mp *b)
+{
+ ec_curve *c = CREATE(ec_curve);
+ c->ops = &ec_primeops;
+ c->f = f;
+ c->a = F_IN(f, MP_NEW, a);
+ c->b = F_IN(f, MP_NEW, b);
+ return (c);
+}
+
+extern ec_curve *ec_primeproj(field *f, mp *a, mp *b)
+{
+ ec_curve *c = CREATE(ec_curve);
+ mp *ax;
+
+ ax = mp_add(MP_NEW, a, MP_THREE);
+ ax = F_IN(f, ax, ax);
+ if (F_ZEROP(f, ax))
+ c->ops = &ec_primeprojxops;
+ else
+ c->ops = &ec_primeprojops;
+ MP_DROP(ax);
+ c->f = f;
+ c->a = F_IN(f, MP_NEW, a);
+ c->b = F_IN(f, MP_NEW, b);
+ return (c);
+}
+
+static const ec_ops ec_primeops = {
+ ecdestroy, ec_stdsamep, ec_idin, ec_idout, ec_idfix,
+ ecfind, ecneg, ecadd, ec_stdsub, ecdbl, eccheck
+};
+
+static const ec_ops ec_primeprojops = {
+ ecdestroy, ec_stdsamep, ec_projin, ec_projout, ec_projfix,
+ ecfind, ecneg, ecprojadd, ec_stdsub, ecprojdbl, ecprojcheck
+};
+
+static const ec_ops ec_primeprojxops = {
+ ecdestroy, ec_stdsamep, ec_projin, ec_projout, ec_projfix,
+ ecfind, ecneg, ecprojadd, ec_stdsub, ecprojxdbl, 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;
+ int i, n = argc == 1 ? 1 : atoi(argv[1]);
+
+ printf("ec-prime: ");
+ fflush(stdout);
+ a = MP(-3);
+ b = MP(0xb3312fa7e23ee7e4988e056be3f82d19181d9c6efe8141120314088f5013875ac656398d8a2ed19d2a85c8edd3ec2aef);
+ p = MP(39402006196394479212279040100143613805079739270465446667948293404245721771496870329047266088258938001861606973112319);
+ r = MP(39402006196394479212279040100143613805079739270465446667946905279627659399113263569398956308152294913554433653942642);
+
+ f = field_niceprime(p);
+ c = ec_primeproj(f, a, b);
+
+ g.x = MP(0xaa87ca22be8b05378eb1c71ef320ad746e1d3b628ba79b9859f741e082542a385502f25dbf55296c3a545e3872760ab7);
+ g.y = MP(0x3617de4a96262c6f5d9e98bf9292dc29f8f41dbd289a147ce9da3113b5f0b8c00a60b1ce1d7e819d7a431d7c90ea0e5f);
+
+ 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_EPRINT("d.x", d.x);
+ MP_EPRINT("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);
+ assert(!mparena_count(&mparena_global));
+ printf("ok\n");
+ return (0);
+}
+
+#endif
+
/*----- That's all, folks -------------------------------------------------*/
projects / u / mdw / catacomb / blobdiff