[u/mdw/catacomb] / ec-prime.c

/* -*-c-*-
 *
 * $Id: ec-prime.c,v 1.4 2004/03/21 22:52:06 mdw Exp $
 *
 * Elliptic curves over prime fields
 *
 * (c) 2001 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: ec-prime.c,v $
 * Revision 1.4  2004/03/21 22:52:06  mdw
 * Merge and close elliptic curve branch.
 *
 * Revision 1.3.4.3  2004/03/21 22:39:46  mdw
 * Elliptic curves on binary fields work.
 *
 * 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.
 *
 * Revision 1.3  2003/05/15 23:25:59  mdw
 * Make elliptic curve stuff build.
 *
 * Revision 1.2  2002/01/13 13:48:44  mdw
 * Further progress.
 *
 * Revision 1.1  2001/04/29 18:12:33  mdw
 * Prototype version.
 *
 */

/*----- Header files ------------------------------------------------------*/

#include <mLib/sub.h>

#include "ec.h"

/*----- Data structures ---------------------------------------------------*/

typedef struct ecctx {
  ec_curve c;
  mp *a, *b;
} ecctx;

/*----- 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;
  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 (F_ZEROP(c->f, a->y))
    EC_COPY(d, a);
  else {
    field *f = c->f;
    ecctx *cc = (ecctx *)c;
    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, 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);		/* %$\lambda^2$% */
    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;
    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))
    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)
    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 *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);
      return (d);
    } else {
      ecctx *cc = (ecctx *)c;
      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);		/* %$\lambda^2$% */
    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, *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)) {
      MP_DROP(w);
      MP_DROP(u);
      MP_DROP(s);
      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, 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;
  MP_DROP(cc->a);
  MP_DROP(cc->b);
  DESTROY(cc);
}

/* --- @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.
 *
 * 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)
{
  ecctx *cc = CREATE(ecctx);
  cc->c.ops = &ec_primeops;
  cc->c.f = f;
  cc->a = F_IN(f, MP_NEW, a);
  cc->b = F_IN(f, MP_NEW, b);
  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, 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 ----------------------------------------------------------*/

#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(0x64210519e59c80e70fa7e9ab72243049feb8deecc146b9b1);
  p = MP(6277101735386680763835789423207666416083908700390324961279);
  r = MP(6277101735386680763835789423176059013767194773182842284080);

  f = field_prime(p);
  c = ec_primeproj(f, a, b);
  
  g.x = MP(0x188da80eb03090f67cbf20eb43a18800f4ff0afd82ff1012);
  g.y = MP(0x07192b95ffc8da78631011ed6b24cdd573f977a11e794811);

  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 -------------------------------------------------*/
Commit	Line	Data
b0ab12e6	1	/* --c--
b0ab12e6	2	*
c3caa2fa	3	* $Id: ec-prime.c,v 1.4 2004/03/21 22:52:06 mdw Exp $
b0ab12e6	4	*
	5	* Elliptic curves over prime fields
	6	*
	7	* (c) 2001 Straylight/Edgeware
	8	*/
	9
	10	/----- Licensing notice --------------------------------------------------
	11	*
	12	* This file is part of Catacomb.
	13	*
	14	* Catacomb is free software; you can redistribute it and/or modify
	15	* it under the terms of the GNU Library General Public License as
	16	* published by the Free Software Foundation; either version 2 of the
	17	* License, or (at your option) any later version.
	18	*
	19	* Catacomb is distributed in the hope that it will be useful,
	20	* but WITHOUT ANY WARRANTY; without even the implied warranty of
	21	* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
	22	* GNU Library General Public License for more details.
	23	*
	24	* You should have received a copy of the GNU Library General Public
	25	* License along with Catacomb; if not, write to the Free
	26	* Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
	27	* MA 02111-1307, USA.
	28	*/
	29
	30	/----- Revision history --------------------------------------------------
	31	*
	32	* $Log: ec-prime.c,v $
c3caa2fa	33	* Revision 1.4 2004/03/21 22:52:06 mdw
	34	* Merge and close elliptic curve branch.
	35	*
ceb3f0c0	36	* Revision 1.3.4.3 2004/03/21 22:39:46 mdw
	37	* Elliptic curves on binary fields work.
	38	*
8823192f	39	* Revision 1.3.4.2 2004/03/20 00:13:31 mdw
	40	* Projective coordinates for prime curves
	41	*
dbfee00a	42	* Revision 1.3.4.1 2003/06/10 13:43:53 mdw
	43	* Simple (non-projective) curves over prime fields now seem to work.
	44	*
41cb1beb	45	* Revision 1.3 2003/05/15 23:25:59 mdw
	46	* Make elliptic curve stuff build.
	47	*
b085fd91	48	* Revision 1.2 2002/01/13 13:48:44 mdw
	49	* Further progress.
	50	*
b0ab12e6	51	* Revision 1.1 2001/04/29 18:12:33 mdw
	52	* Prototype version.
	53	*
	54	*/
	55
	56	/----- Header files ------------------------------------------------------/
	57
41cb1beb	58	#include <mLib/sub.h>
41cb1beb	59
b0ab12e6	60	#include "ec.h"
	61
	62	/----- Data structures ---------------------------------------------------/
	63
	64	typedef struct ecctx {
	65	ec_curve c;
	66	mp a, b;
	67	} ecctx;
	68
dbfee00a	69	/----- Simple prime curves -----------------------------------------------/
b0ab12e6	70
8823192f	71	static const ec_ops ec_primeops, ec_primeprojops, ec_primeprojxops;
41cb1beb	72
41cb1beb	73	static ec ecneg(ec_curve c, ec d, const ec p)
b085fd91	74	{
b085fd91	75	EC_COPY(d, p);
ceb3f0c0	76	if (d->y)
ceb3f0c0	77	d->y = F_NEG(c->f, d->y, d->y);
b085fd91	78	return (d);
	79	}
	80
8823192f	81	static ec ecfind(ec_curve c, ec d, mp x)
	82	{
	83	mp p, q;
	84	ecctx cc = (ecctx )c;
	85	field *f = c->f;
	86
	87	q = F_SQR(f, MP_NEW, x);
	88	p = F_MUL(f, MP_NEW, x, q);
	89	q = F_MUL(f, q, x, cc->a);
	90	p = F_ADD(f, p, p, q);
	91	p = F_ADD(f, p, p, cc->b);
	92	MP_DROP(q);
	93	p = F_SQRT(f, p, p);
	94	if (!p)
	95	return (0);
	96	EC_DESTROY(d);
	97	d->x = MP_COPY(x);
	98	d->y = p;
	99	d->z = MP_COPY(f->one);
b085fd91	100	return (d);
	101	}
	102
	103	static ec ecdbl(ec_curve c, ec d, const ec a)
b0ab12e6	104	{
b085fd91	105	if (EC_ATINF(a))
b085fd91	106	EC_SETINF(d);
8823192f	107	else if (F_ZEROP(c->f, a->y))
b085fd91	108	EC_COPY(d, a);
	109	else {
	110	field *f = c->f;
	111	ecctx cc = (ecctx )c;
	112	mp *lambda;
	113	mp dy, dx;
	114
8823192f	115	dx = F_SQR(f, MP_NEW, a->x); /* %$x^2$% */
	116	dy = F_DBL(f, MP_NEW, a->y); /* %$2 y$% */
	117	dx = F_TPL(f, dx, dx); /* %$3 x^2$% */
	118	dx = F_ADD(f, dx, dx, cc->a); /* %$3 x^2 + A$% */
	119	dy = F_INV(f, dy, dy); /* %$(2 y)^{-1}$% */
	120	lambda = F_MUL(f, MP_NEW, dx, dy); /* %$\lambda = (3 x^2 + A)/(2 y)$% */
b085fd91	121
8823192f	122	dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */
	123	dy = F_DBL(f, dy, a->x); /* %$2 x$% */
	124	dx = F_SUB(f, dx, dx, dy); /* %$x' = \lambda^2 - 2 x */
	125	dy = F_SUB(f, dy, a->x, dx); /* %$x - x'$% */
	126	dy = F_MUL(f, dy, lambda, dy); /* %$\lambda (x - x')$% */
	127	dy = F_SUB(f, dy, dy, a->y); /* %$y' = \lambda (x - x') - y$% */
b0ab12e6	128
b085fd91	129	EC_DESTROY(d);
	130	d->x = dx;
	131	d->y = dy;
	132	d->z = 0;
	133	MP_DROP(lambda);
	134	}
	135	return (d);
	136	}
	137
8823192f	138	static ec ecprojdbl(ec_curve c, ec d, const ec a)
	139	{
	140	if (EC_ATINF(a))
	141	EC_SETINF(d);
	142	else if (F_ZEROP(c->f, a->y))
	143	EC_COPY(d, a);
	144	else {
	145	field *f = c->f;
	146	ecctx cc = (ecctx )c;
	147	mp p, q, m, s, dx, dy, *dz;
	148
	149	p = F_SQR(f, MP_NEW, a->z); /* %$z^2$% */
	150	q = F_SQR(f, MP_NEW, p); /* %$z^4$% */
	151	p = F_MUL(f, p, q, cc->a); /* %$A z^4$% */
	152	m = F_SQR(f, MP_NEW, a->x); /* %$x^2$% */
	153	m = F_TPL(f, m, m); /* %$3 x^2$% */
	154	m = F_ADD(f, m, m, p); /* %$m = 3 x^2 + A z^4$% */
	155
	156	q = F_DBL(f, q, a->y); /* %$2 y$% */
	157	dz = F_MUL(f, MP_NEW, q, a->z); /* %$z' = 2 y z$% */
	158
	159	p = F_SQR(f, p, q); /* %$4 y^2$% */
	160	s = F_MUL(f, MP_NEW, p, a->x); /* %$s = 4 x y^2$% */
	161	q = F_SQR(f, q, p); /* %$16 y^4$% */
	162	q = F_HLV(f, q, q); /* %$t = 8 y^4$% */
	163
	164	p = F_DBL(f, p, s); /* %$2 s$% */
	165	dx = F_SQR(f, MP_NEW, m); /* %$m^2$% */
	166	dx = F_SUB(f, dx, dx, p); /* %$x' = m^2 - 2 s$% */
	167
	168	s = F_SUB(f, s, s, dx); /* %$s - x'$% */
	169	dy = F_MUL(f, p, m, s); /* %$m (s - x')$% */
	170	dy = F_SUB(f, dy, dy, q); /* %$y' = m (s - x') - t$% */
	171
	172	EC_DESTROY(d);
	173	d->x = dx;
	174	d->y = dy;
	175	d->z = dz;
	176	MP_DROP(m);
	177	MP_DROP(q);
	178	MP_DROP(s);
	179	}
	180	return (d);
	181	}
	182
	183	static ec ecprojxdbl(ec_curve c, ec d, const ec a)
	184	{
	185	if (EC_ATINF(a))
	186	EC_SETINF(d);
	187	else if (F_ZEROP(c->f, a->y))
	188	EC_COPY(d, a);
	189	else {
	190	field *f = c->f;
	191	mp p, q, m, s, dx, dy, *dz;
	192
	193	m = F_SQR(f, MP_NEW, a->z); /* %$z^2$% */
	194	p = F_SUB(f, MP_NEW, a->x, m); /* %$x - z^2$% */
	195	q = F_ADD(f, MP_NEW, a->x, m); /* %$x + z^2$% */
	196	m = F_MUL(f, m, p, q); /* %$x^2 - z^4$% */
	197	m = F_TPL(f, m, m); /* %$m = 3 x^2 - 3 z^4$% */
	198
	199	q = F_DBL(f, q, a->y); /* %$2 y$% */
	200	dz = F_MUL(f, MP_NEW, q, a->z); /* %$z' = 2 y z$% */
	201
202	p = F_SQR(f, p, q); /* %$4 y^2$% */
203	s = F_MUL(f, MP_NEW, p, a->x); /* %$s = 4 x y^2$% */
204	q = F_SQR(f, q, p); /* %$16 y^4$% */
205	q = F_HLV(f, q, q); /* %$t = 8 y^4$% */
206
207	p = F_DBL(f, p, s); /* %$2 s$% */
208	dx = F_SQR(f, MP_NEW, m); /* %$m^2$% */
209	dx = F_SUB(f, dx, dx, p); /* %$x' = m^2 - 2 s$% */
210
211	s = F_SUB(f, s, s, dx); /* %$s - x'$% */
212	dy = F_MUL(f, p, m, s); /* %$m (s - x')$% */
213	dy = F_SUB(f, dy, dy, q); /* %$y' = m (s - x') - t$% */
214
215	EC_DESTROY(d);
216	d->x = dx;
217	d->y = dy;
218	d->z = dz;
219	MP_DROP(m);
220	MP_DROP(q);
221	MP_DROP(s);
222	}
223	return (d);
224	}
225
b085fd91	226	static ec ecadd(ec_curve c, ec d, const ec a, const ec *b)
b085fd91	227	{
b0ab12e6	228	if (a == b)
	229	ecdbl(c, d, a);
	230	else if (EC_ATINF(a))
	231	EC_COPY(d, b);
	232	else if (EC_ATINF(b))
	233	EC_COPY(d, a);
b085fd91	234	else {
	235	field *f = c->f;
	236	mp *lambda;
	237	mp dy, dx;
	238
	239	if (!MP_EQ(a->x, b->x)) {
8823192f	240	dy = F_SUB(f, MP_NEW, a->y, b->y); /* %$y_0 - y_1$% */
	241	dx = F_SUB(f, MP_NEW, a->x, b->x); /* %$x_0 - x_1$% */
	242	dx = F_INV(f, dx, dx); /* %$(x_0 - x_1)^{-1}$% */
b085fd91	243	lambda = F_MUL(f, MP_NEW, dy, dx);
8823192f	244	/* %$\lambda = (y_0 - y1)/(x_0 - x_1)$% */
8823192f	245	} else if (F_ZEROP(c->f, a->y) \|\| !MP_EQ(a->y, b->y)) {
b0ab12e6	246	EC_SETINF(d);
b085fd91	247	return (d);
	248	} else {
	249	ecctx cc = (ecctx )c;
8823192f	250	dx = F_SQR(f, MP_NEW, a->x); /* %$x_0^2$% */
	251	dx = F_TPL(f, dx, dx); /* %$3 x_0^2$% */
	252	dx = F_ADD(f, dx, dx, cc->a); /* %$3 x_0^2 + A$% */
	253	dy = F_DBL(f, MP_NEW, a->y); /* %$2 y_0$% */
	254	dy = F_INV(f, dy, dy); /* %$(2 y_0)^{-1}$% */
41cb1beb	255	lambda = F_MUL(f, MP_NEW, dx, dy);
8823192f	256	/* %$\lambda = (3 x_0^2 + A)/(2 y_0)$% */
b085fd91	257	}
b085fd91	258
8823192f	259	dx = F_SQR(f, dx, lambda); /* %$\lambda^2$% */
	260	dx = F_SUB(f, dx, dx, a->x); /* %$\lambda^2 - x_0$% */
	261	dx = F_SUB(f, dx, dx, b->x); /* %$x' = \lambda^2 - x_0 - x_1$% */
	262	dy = F_SUB(f, dy, b->x, dx); /* %$x_1 - x'$% */
	263	dy = F_MUL(f, dy, lambda, dy); /* %$\lambda (x_1 - x')$% */
ceb3f0c0	264	dy = F_SUB(f, dy, dy, b->y); /* %$y' = \lambda (x_1 - x') - y_1$% */
b0ab12e6	265
b085fd91	266	EC_DESTROY(d);
	267	d->x = dx;
	268	d->y = dy;
	269	d->z = 0;
	270	MP_DROP(lambda);
b0ab12e6	271	}
b085fd91	272	return (d);
b0ab12e6	273	}
b0ab12e6	274
8823192f	275	static ec ecprojadd(ec_curve c, ec d, const ec a, const ec *b)
	276	{
	277	if (a == b)
	278	c->ops->dbl(c, d, a);
	279	else if (EC_ATINF(a))
	280	EC_COPY(d, b);
	281	else if (EC_ATINF(b))
	282	EC_COPY(d, a);
	283	else {
	284	field *f = c->f;
	285	mp p, q, r, w, u, s, dx, dy, *dz;
	286
	287	q = F_SQR(f, MP_NEW, a->z); /* %$z_0^2$% */
	288	u = F_MUL(f, MP_NEW, q, b->x); /* %$u = x_1 z_0^2$% */
	289	p = F_MUL(f, MP_NEW, q, b->y); /* %$y_1 z_0^2$% */
	290	s = F_MUL(f, q, p, a->z); /* %$s = y_1 z_0^3$% */
	291
	292	w = F_SUB(f, p, a->x, u); /* %$w = x_0 - u$% */
	293	r = F_SUB(f, MP_NEW, a->y, s); /* %$r = y_0 - s$% */
	294	if (F_ZEROP(f, w)) {
ceb3f0c0	295	MP_DROP(w);
	296	MP_DROP(u);
	297	MP_DROP(s);
8823192f	298	if (F_ZEROP(f, r)) {
8823192f	299	MP_DROP(r);
8823192f	300	return (c->ops->dbl(c, d, a));
8823192f	301	} else {
8823192f	302	MP_DROP(r);
8823192f	303	EC_SETINF(d);
	304	return (d);
	305	}
	306	}
	307	u = F_ADD(f, u, u, a->x); /* %$t = x_0 + u$% */
	308	s = F_ADD(f, s, s, a->y); /* %$m = y_0 + r$% */
	309
	310	dz = F_MUL(f, MP_NEW, a->z, w); /* %$z' = z_0 w$% */
	311
	312	p = F_SQR(f, MP_NEW, w); /* %$w^2$% */
	313	q = F_MUL(f, MP_NEW, p, u); /* %$t w^2$% */
	314	u = F_MUL(f, u, p, w); /* %$w^3$% */
	315	p = F_MUL(f, p, u, s); /* %$m w^3$% */
	316
	317	dx = F_SQR(f, u, r); /* %$r^2$% */
	318	dx = F_SUB(f, dx, dx, q); /* %$x' = r^2 - t w^2$% */
	319
	320	s = F_DBL(f, s, dx); /* %$2 x'$% */
	321	q = F_SUB(f, q, q, s); /* %$v = t w^2 - 2 x'$% */
	322	dy = F_MUL(f, s, q, r); /* %$v r$% */
	323	dy = F_SUB(f, dy, dy, p); /* %$v r - m w^3$% */
	324	dy = F_HLV(f, dy, dy); /* %$y' = (v r - m w^3)/2$% */
	325
	326	EC_DESTROY(d);
	327	d->x = dx;
	328	d->y = dy;
	329	d->z = dz;
	330	MP_DROP(p);
	331	MP_DROP(q);
	332	MP_DROP(r);
	333	MP_DROP(w);
	334	}
	335	return (d);
	336	}
	337
	338	static int eccheck(ec_curve c, const ec p)
	339	{
	340	ecctx cc = (ecctx )c;
	341	field *f = c->f;
	342	int rc;
	343	mp *l = F_SQR(f, MP_NEW, p->y);
	344	mp *x = F_SQR(f, MP_NEW, p->x);
	345	mp *r = F_MUL(f, MP_NEW, x, p->x);
	346	x = F_MUL(f, x, cc->a, p->x);
	347	r = F_ADD(f, r, r, x);
	348	r = F_ADD(f, r, r, cc->b);
	349	rc = MP_EQ(l, r) ? 0 : -1;
	350	mp_drop(l);
	351	mp_drop(x);
	352	mp_drop(r);
	353	return (rc);
	354	}
	355
	356	static int ecprojcheck(ec_curve c, const ec p)
	357	{
	358	ec t = EC_INIT;
	359	int rc;
	360
	361	c->ops->fix(c, &t, p);
	362	rc = eccheck(c, &t);
	363	EC_DESTROY(&t);
	364	return (rc);
	365	}
	366
41cb1beb	367	static void ecdestroy(ec_curve *c)
	368	{
	369	ecctx cc = (ecctx )c;
	370	MP_DROP(cc->a);
	371	MP_DROP(cc->b);
	372	DESTROY(cc);
	373	}
	374
	375	/* --- @ec_prime@, @ec_primeproj@ --- *
	376	*
dbfee00a	377	* Arguments: @field *f@ = the underlying field for this elliptic curve
41cb1beb	378	* @mp a, b@ = the coefficients for this curve
	379	*
	380	* Returns: A pointer to the curve.
	381	*
	382	* Use: Creates a curve structure for an elliptic curve defined over
	383	* a prime field. The @primeproj@ variant uses projective
	384	* coordinates, which can be a win.
	385	*/
	386
	387	extern ec_curve ec_prime(field f, mp a, mp b)
	388	{
	389	ecctx *cc = CREATE(ecctx);
	390	cc->c.ops = &ec_primeops;
	391	cc->c.f = f;
dbfee00a	392	cc->a = F_IN(f, MP_NEW, a);
dbfee00a	393	cc->b = F_IN(f, MP_NEW, b);
41cb1beb	394	return (&cc->c);
	395	}
	396
8823192f	397	extern ec_curve ec_primeproj(field f, mp a, mp b)
	398	{
	399	ecctx *cc = CREATE(ecctx);
	400	mp *ax;
	401
	402	ax = mp_add(MP_NEW, a, MP_THREE);
	403	ax = F_IN(f, ax, ax);
	404	if (F_ZEROP(f, ax))
	405	cc->c.ops = &ec_primeprojxops;
	406	else
	407	cc->c.ops = &ec_primeprojops;
	408	MP_DROP(ax);
	409	cc->c.f = f;
	410	cc->a = F_IN(f, MP_NEW, a);
	411	cc->b = F_IN(f, MP_NEW, b);
41cb1beb	412	return (&cc->c);
	413	}
	414
	415	static const ec_ops ec_primeops = {
8823192f	416	ecdestroy, ec_idin, ec_idout, ec_idfix,
	417	0, ecneg, ecadd, ec_stdsub, ecdbl, eccheck
	418	};
	419
	420	static const ec_ops ec_primeprojops = {
	421	ecdestroy, ec_projin, ec_projout, ec_projfix,
	422	0, ecneg, ecprojadd, ec_stdsub, ecprojdbl, ecprojcheck
	423	};
	424
	425	static const ec_ops ec_primeprojxops = {
	426	ecdestroy, ec_projin, ec_projout, ec_projfix,
	427	0, ecneg, ecprojadd, ec_stdsub, ecprojxdbl, ecprojcheck
41cb1beb	428	};
	429
	430	/----- Test rig ----------------------------------------------------------/
	431
	432	#ifdef TEST_RIG
	433
	434	#define MP(x) mp_readstring(MP_NEW, #x, 0, 0)
	435
ceb3f0c0	436	int main(int argc, char *argv[])
41cb1beb	437	{
	438	field *f;
	439	ec_curve *c;
	440	ec g = EC_INIT, d = EC_INIT;
	441	mp p, a, b, r;
ceb3f0c0	442	int i, n = argc == 1 ? 1 : atoi(argv[1]);
41cb1beb	443
dbfee00a	444	printf("ec-prime: ");
dbfee00a	445	fflush(stdout);
41cb1beb	446	a = MP(-3);
	447	b = MP(0x64210519e59c80e70fa7e9ab72243049feb8deecc146b9b1);
	448	p = MP(6277101735386680763835789423207666416083908700390324961279);
dbfee00a	449	r = MP(6277101735386680763835789423176059013767194773182842284080);
41cb1beb	450
41cb1beb	451	f = field_prime(p);
ceb3f0c0	452	c = ec_primeproj(f, a, b);
41cb1beb	453
	454	g.x = MP(0x188da80eb03090f67cbf20eb43a18800f4ff0afd82ff1012);
	455	g.y = MP(0x07192b95ffc8da78631011ed6b24cdd573f977a11e794811);
	456
ceb3f0c0	457	for (i = 0; i < n; i++) {
	458	ec_mul(c, &d, &g, r);
	459	if (EC_ATINF(&d)) {
	460	fprintf(stderr, "zero too early\n");
	461	return (1);
	462	}
	463	ec_add(c, &d, &d, &g);
	464	if (!EC_ATINF(&d)) {
	465	fprintf(stderr, "didn't reach zero\n");
	466	MP_EPRINT("d.x", d.x);
	467	MP_EPRINT("d.y", d.y);
	468	return (1);
	469	}
	470	ec_destroy(&d);
dbfee00a	471	}
41cb1beb	472	ec_destroy(&g);
	473	ec_destroycurve(c);
	474	F_DESTROY(f);
dbfee00a	475	MP_DROP(p); MP_DROP(a); MP_DROP(b); MP_DROP(r);
	476	assert(!mparena_count(&mparena_global));
	477	printf("ok\n");
41cb1beb	478	return (0);
	479	}
	480
	481	#endif
	482
b0ab12e6	483	/----- That's all, folks -------------------------------------------------/