/* -*-c-*-
*
- * $Id: ec.c,v 1.1 2001/04/29 18:12:33 mdw Exp $
+ * $Id: ec.c,v 1.9 2004/04/01 21:28:41 mdw Exp $
*
* Elliptic curve definitions
*
/*----- Revision history --------------------------------------------------*
*
* $Log: ec.c,v $
+ * Revision 1.9 2004/04/01 21:28:41 mdw
+ * Normal basis support (translates to poly basis internally). Rewrite
+ * EC and prime group table generators in awk, so that they can reuse data
+ * for repeated constants.
+ *
+ * Revision 1.8 2004/04/01 12:50:09 mdw
+ * Add cyclic group abstraction, with test code. Separate off exponentation
+ * functions for better static linking. Fix a buttload of bugs on the way.
+ * Generally ensure that negative exponents do inversion correctly. Add
+ * table of standard prime-field subgroups. (Binary field subgroups are
+ * currently unimplemented but easy to add if anyone ever finds a good one.)
+ *
+ * Revision 1.7 2004/03/27 17:54:11 mdw
+ * Standard curves and curve checking.
+ *
+ * Revision 1.6 2004/03/23 15:19:32 mdw
+ * Test elliptic curves more thoroughly.
+ *
+ * Revision 1.5 2004/03/21 22:52:06 mdw
+ * Merge and close elliptic curve branch.
+ *
+ * Revision 1.4.4.2 2004/03/20 00:13:31 mdw
+ * Projective coordinates for prime curves
+ *
+ * Revision 1.4.4.1 2003/06/10 13:43:53 mdw
+ * Simple (non-projective) curves over prime fields now seem to work.
+ *
+ * Revision 1.4 2003/05/15 23:25:59 mdw
+ * Make elliptic curve stuff build.
+ *
+ * Revision 1.3 2002/01/13 13:48:44 mdw
+ * Further progress.
+ *
+ * Revision 1.2 2001/05/07 17:29:44 mdw
+ * Treat projective coordinates as an internal representation. Various
+ * minor interface changes.
+ *
* Revision 1.1 2001/04/29 18:12:33 mdw
* Prototype version.
*
/*----- Trivial wrappers --------------------------------------------------*/
+/* --- @ec_samep@ --- *
+ *
+ * Arguments: @ec_curve *c, *d@ = two elliptic curves
+ *
+ * Returns: Nonzero if the curves are identical (not just isomorphic).
+ *
+ * Use: Checks for sameness of curves. This function does the full
+ * check, not just the curve-type-specific check done by the
+ * @sampep@ field operation.
+ */
+
+int ec_samep(ec_curve *c, ec_curve *d)
+{
+ return (field_samep(c->f, d->f) && c->ops == d->ops && EC_SAMEP(c, d));
+}
+
/* --- @ec_create@ --- *
*
* Arguments: @ec *p@ = pointer to an elliptic-curve point
*
- * Returns: ---
+ * Returns: The argument @p@.
*
* Use: Initializes a new point. The initial value is the additive
* identity (which is universal for all curves).
*/
-void ec_create(ec *p) { EC_CREATE(p); }
+ec *ec_create(ec *p) { EC_CREATE(p); return (p); }
/* --- @ec_destroy@ --- *
*
*
* Arguments: @ec *p@ = pointer to a point
*
- * Returns: ---
+ * Returns: The argument @p@.
*
* Use: Sets the given point to be the point %$O$% at infinity.
*/
-void ec_setinf(ec *p) { EC_SETINF(p); }
+ec *ec_setinf(ec *p) { EC_SETINF(p); return (p); }
/* --- @ec_copy@ --- *
*
* Arguments: @ec *d@ = pointer to destination point
* @const ec *p@ = pointer to source point
*
- * Returns: ---
+ * Returns: The destination @d@.
*
* Use: Creates a copy of an elliptic curve point.
*/
-void ec_copy(ec *d, const ec *p) { EC_COPY(d, p); }
+ec *ec_copy(ec *d, const ec *p) { EC_COPY(d, p); return (d); }
-/*----- Real arithmetic ---------------------------------------------------*/
+/* --- @ec_eq@ --- *
+ *
+ * Arguments: @const ec *p, *q@ = two points
+ *
+ * Returns: Nonzero if the points are equal. Compares external-format
+ * points.
+ */
+
+int ec_eq(const ec *p, const ec *q) { return (EC_EQ(p, q)); }
+
+/*----- Standard curve operations -----------------------------------------*/
+
+/* --- @ec_stdsamep@ --- *
+ *
+ * Arguments: @ec_curve *c, *d@ = two elliptic curves
+ *
+ * Returns: Nonzero if the curves are identical (not just isomorphic).
+ *
+ * Use: Simple sameness check on @a@ and @b@ curve members.
+ */
+
+int ec_stdsamep(ec_curve *c, ec_curve *d)
+{
+ return (MP_EQ(c->a, d->a) && MP_EQ(c->b, d->b));
+}
-/* --- @ec_denorm@ --- *
+/* --- @ec_idin@, @ec_idout@, @ec_idfix@ --- *
*
* Arguments: @ec_curve *c@ = pointer to an elliptic curve
- * @ec *d@ = pointer to the destination point
- * @const ec *p@ = pointer to the source point
+ * @ec *d@ = pointer to the destination
+ * @const ec *p@ = pointer to a source point
*
- * Returns: ---
+ * Returns: The destination @d@.
*
- * Use: Denormalizes the given point, converting to internal
- * representations and setting the denominator to 1.
+ * Use: An identity operation if your curve has no internal
+ * representation. (The field internal representation is still
+ * used.)
*/
-void ec_denorm(ec_curve *c, ec *d, const ec *p)
+ec *ec_idin(ec_curve *c, ec *d, const ec *p)
{
if (EC_ATINF(p))
EC_SETINF(d);
field *f = c->f;
d->x = F_IN(f, d->x, p->x);
d->y = F_IN(f, d->y, p->y);
- mp_drop(d->z);
- d->z = MP_COPY(f->one);
+ mp_drop(d->z); d->z = 0;
}
+ return (d);
}
-/* --- @ec_norm@ --- *
+ec *ec_idout(ec_curve *c, ec *d, const ec *p)
+{
+ if (EC_ATINF(p))
+ EC_SETINF(d);
+ else {
+ field *f = c->f;
+ d->x = F_OUT(f, d->x, p->x);
+ d->y = F_OUT(f, d->y, p->y);
+ mp_drop(d->z); d->z = 0;
+ }
+ return (d);
+}
+
+ec *ec_idfix(ec_curve *c, ec *d, const ec *p)
+{
+ EC_COPY(d, p);
+ return (d);
+}
+
+/* --- @ec_projin@, @ec_projout@, @ec_projfix@ --- *
*
* Arguments: @ec_curve *c@ = pointer to an elliptic curve
- * @ec *d@ = pointer to the destination point
- * @const ec *p@ = pointer to the source point
+ * @ec *d@ = pointer to the destination
+ * @const ec *p@ = pointer to a source point
*
- * Returns: ---
+ * Returns: The destination @d@.
*
- * Use: Normalizes the given point, by dividing through by the
- * denominator and returning to external representation.
+ * Use: Conversion functions if your curve operations use a
+ * projective representation.
*/
-void ec_norm(ec_curve *c, ec *d, const ec *p)
+ec *ec_projin(ec_curve *c, ec *d, const ec *p)
{
if (EC_ATINF(p))
EC_SETINF(d);
else {
- mp *x, *y, *z;
+ field *f = c->f;
+ d->x = F_IN(f, d->x, p->x);
+ d->y = F_IN(f, d->y, p->y);
+ mp_drop(d->z); d->z = MP_COPY(f->one);
+ }
+ return (d);
+}
+
+ec *ec_projout(ec_curve *c, ec *d, const ec *p)
+{
+ if (EC_ATINF(p))
+ EC_SETINF(d);
+ else {
+ mp *x, *y, *z, *zz;
field *f = c->f;
z = F_INV(f, MP_NEW, p->z);
- x = F_MUL(f, d->x, p->x, z);
+ zz = F_SQR(f, MP_NEW, z);
+ z = F_MUL(f, z, zz, z);
+ x = F_MUL(f, d->x, p->x, zz);
y = F_MUL(f, d->y, p->y, z);
mp_drop(z);
+ mp_drop(zz);
mp_drop(d->z);
d->x = F_OUT(f, x, x);
d->y = F_OUT(f, y, y);
d->z = 0;
}
+ return (d);
+}
+
+ec *ec_projfix(ec_curve *c, ec *d, const ec *p)
+{
+ if (EC_ATINF(p))
+ EC_SETINF(d);
+ else if (d->z == c->f->one)
+ EC_COPY(d, p);
+ else {
+ mp *z, *zz;
+ field *f = c->f;
+ z = F_INV(f, MP_NEW, p->z);
+ zz = F_SQR(f, MP_NEW, z);
+ z = F_MUL(f, z, zz, z);
+ d->x = F_MUL(f, d->x, p->x, zz);
+ d->y = F_MUL(f, d->y, p->y, z);
+ mp_drop(z);
+ mp_drop(zz);
+ mp_drop(d->z);
+ d->z = MP_COPY(f->one);
+ }
+ return (d);
}
+/* --- @ec_stdsub@ --- *
+ *
+ * Arguments: @ec_curve *c@ = pointer to an elliptic curve
+ * @ec *d@ = pointer to the destination
+ * @const ec *p, *q@ = the operand points
+ *
+ * Returns: The destination @d@.
+ *
+ * Use: Standard point subtraction operation, in terms of negation
+ * and addition. This isn't as efficient as a ready-made
+ * subtraction operator.
+ */
+
+ec *ec_stdsub(ec_curve *c, ec *d, const ec *p, const ec *q)
+{
+ ec t = EC_INIT;
+ EC_NEG(c, &t, q);
+ EC_FIX(c, &t, &t);
+ EC_ADD(c, d, p, &t);
+ EC_DESTROY(&t);
+ return (d);
+}
+
+/*----- Creating curves ---------------------------------------------------*/
+
+/* --- @ec_destroycurve@ --- *
+ *
+ * Arguments: @ec_curve *c@ = pointer to an ellptic curve
+ *
+ * Returns: ---
+ *
+ * Use: Destroys a description of an elliptic curve.
+ */
+
+void ec_destroycurve(ec_curve *c) { c->ops->destroy(c); }
+
+/*----- Real arithmetic ---------------------------------------------------*/
+
/* --- @ec_find@ --- *
*
* Arguments: @ec_curve *c@ = pointer to an elliptic curve
* Use: Finds a point on an elliptic curve with a given x-coordinate.
*/
-void ec_find(ec_curve *c, ec *d, mp *x)
+ec *ec_find(ec_curve *c, ec *d, mp *x)
{
- int rc;
x = F_IN(c->f, MP_NEW, x);
- if ((rc = EC_FIND(c, d, x)) == 0)
- ec_norm(c, d, d);
- mp_drop(x);
- return (rc);
+ if ((d = EC_FIND(c, d, x)) != 0)
+ EC_OUT(c, d, d);
+ MP_DROP(x);
+ return (d);
+}
+
+/* --- @ec_neg@ --- *
+ *
+ * Arguments: @ec_curve *c@ = pointer to an elliptic curve
+ * @ec *d@ = pointer to the destination point
+ * @const ec *p@ = pointer to the operand point
+ *
+ * Returns: The destination point.
+ *
+ * Use: Computes the negation of the given point.
+ */
+
+ec *ec_neg(ec_curve *c, ec *d, const ec *p)
+{
+ EC_IN(c, d, p);
+ EC_NEG(c, d, d);
+ return (EC_OUT(c, d, d));
}
/* --- @ec_add@ --- *
* Use: Adds two points on an elliptic curve.
*/
-void ec_add(ec_curve *c, ec *d, const ec *p, const ec *q)
+ec *ec_add(ec_curve *c, ec *d, const ec *p, const ec *q)
{
ec pp = EC_INIT, qq = EC_INIT;
- ec_denorm(c, &pp, p);
- ec_denorm(c, &qq, q);
+ EC_IN(c, &pp, p);
+ EC_IN(c, &qq, q);
EC_ADD(c, d, &pp, &qq);
- ec_norm(c, d, d);
+ EC_OUT(c, d, d);
+ EC_DESTROY(&pp);
+ EC_DESTROY(&qq);
+ return (d);
+}
+
+/* --- @ec_sub@ --- *
+ *
+ * Arguments: @ec_curve *c@ = pointer to an elliptic curve
+ * @ec *d@ = pointer to the destination point
+ * @const ec *p, *q@ = pointers to the operand points
+ *
+ * Returns: The destination @d@.
+ *
+ * Use: Subtracts one point from another on an elliptic curve.
+ */
+
+ec *ec_sub(ec_curve *c, ec *d, const ec *p, const ec *q)
+{
+ ec pp = EC_INIT, qq = EC_INIT;
+ EC_IN(c, &pp, p);
+ EC_IN(c, &qq, q);
+ EC_SUB(c, d, &pp, &qq);
+ EC_OUT(c, d, d);
EC_DESTROY(&pp);
EC_DESTROY(&qq);
+ return (d);
}
/* --- @ec_dbl@ --- *
* Use: Doubles a point on an elliptic curve.
*/
-void ec_dbl(ec_curve *c, ec *d, const ec *p)
+ec *ec_dbl(ec_curve *c, ec *d, const ec *p)
{
- ec_denorm(c, d, p);
+ EC_IN(c, d, p);
EC_DBL(c, d, d);
- ec_norm(c, d, d);
+ return (EC_OUT(c, d, d));
}
-/* --- @ec_mul@ --- *
+/* --- @ec_check@ --- *
*
* Arguments: @ec_curve *c@ = pointer to an elliptic curve
- * @ec *d@ = pointer to the destination point
- * @const ec *p@ = pointer to the generator point
- * @mp *n@ = integer multiplier
+ * @const ec *p@ = pointer to the point
*
- * Returns: ---
+ * Returns: Zero if OK, nonzero if this is an invalid point.
*
- * Use: Multiplies a point by a scalar, returning %$n p$%.
+ * Use: Checks that a point is actually on an elliptic curve.
*/
-void ec_mul(ec_curve *c, ec *d, const ec *p, mp *n)
+int ec_check(ec_curve *c, const ec *p)
{
- mpscan sc;
- ec g = EC_INIT;
- unsigned sq = 0;
+ ec t = EC_INIT;
+ int rc;
- EC_SETINF(d);
if (EC_ATINF(p))
- return;
-
- mp_rscan(&sc, n);
- if (!MP_RSTEP(&sc))
- goto exit;
- while (!MP_RBIT(&sc))
- MP_RSTEP(&sc);
-
- ec_denorm(c, &g, p);
- if ((n->f & MP_BURN) && !(g.x->f & MP_BURN))
- MP_DEST(g.x, 0, MP_BURN);
- if ((n->f & MP_BURN) && !(g.y->f & MP_BURN))
- MP_DEST(g.y, 0, MP_BURN);
-
- for (;;) {
- EC_ADD(c, d, d, &g);
- sq = 0;
- for (;;) {
- if (!MP_RSTEP(&sc))
- goto done;
- if (MP_RBIT(&sc))
- break;
- sq++;
- }
- sq++;
- while (sq) {
- EC_DBL(c, d, d);
- sq--;
- }
- }
+ return (0);
+ EC_IN(c, &t, p);
+ rc = EC_CHECK(c, &t);
+ EC_DESTROY(&t);
+ return (rc);
+}
-done:
- while (sq) {
- EC_DBL(c, d, d);
- sq--;
- }
+/* --- @ec_rand@ --- *
+ *
+ * Arguments: @ec_curve *c@ = pointer to an elliptic curve
+ * @ec *d@ = pointer to the destination point
+ * @grand *r@ = random number source
+ *
+ * Returns: The destination @d@.
+ *
+ * Use: Finds a random point on the given curve.
+ */
- EC_DESTROY(&g);
-exit:
- ec_norm(c, d, d);
+ec *ec_rand(ec_curve *c, ec *d, grand *r)
+{
+ mp *x = MP_NEW;
+ do x = F_RAND(c->f, x, r); while (!EC_FIND(c, d, x));
+ mp_drop(x);
+ if (grand_range(r, 2)) EC_NEG(c, d, d);
+ return (EC_OUT(c, d, d));
}
/*----- That's all, folks -------------------------------------------------*/
projects / u / mdw / catacomb / blobdiff