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

/* -*-c-*-
 *
 * $Id: ec-prime.c,v 1.3.4.2 2004/03/20 00:13:31 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 1.3.4.2  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);
  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)) {
      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;
  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(void)
{
  field *f;
  ec_curve *c;
  ec g = EC_INIT, d = EC_INIT;
  mp *p, *a, *b, *r;

  printf("ec-prime: ");
  fflush(stdout);
  a = MP(-3);
  b = MP(0x64210519e59c80e70fa7e9ab72243049feb8deecc146b9b1);
  p = MP(6277101735386680763835789423207666416083908700390324961279);
  r = MP(6277101735386680763835789423176059013767194773182842284080);

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

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