X-Git-Url: https://git.distorted.org.uk/u/mdw/catacomb/blobdiff_plain/b0ab12e6a6cb035df2b6312df7ad1736af0a6128..578a86d91941a0f722b87973d88e84ec2cf9a608:/ec-prime.c diff --git a/ec-prime.c b/ec-prime.c index 5a806da..8f3c731 100644 --- a/ec-prime.c +++ b/ec-prime.c @@ -1,6 +1,6 @@ /* -*-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 * @@ -27,46 +27,428 @@ * 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 + #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 -------------------------------------------------*/