xref: /illumos-gate/usr/src/common/crypto/ecc/ecp_jac.c (revision c40a6cd785e883b3f052b122c332e21174fc1871)

/*
 * ***** BEGIN LICENSE BLOCK *****
 * Version: MPL 1.1/GPL 2.0/LGPL 2.1
 *
 * The contents of this file are subject to the Mozilla Public License Version
 * 1.1 (the "License"); you may not use this file except in compliance with
 * the License. You may obtain a copy of the License at
 * http://www.mozilla.org/MPL/
 *
 * Software distributed under the License is distributed on an "AS IS" basis,
 * WITHOUT WARRANTY OF ANY KIND, either express or implied. See the License
 * for the specific language governing rights and limitations under the
 * License.
 *
 * The Original Code is the elliptic curve math library for prime field curves.
 *
 * The Initial Developer of the Original Code is
 * Sun Microsystems, Inc.
 * Portions created by the Initial Developer are Copyright (C) 2003
 * the Initial Developer. All Rights Reserved.
 *
 * Contributor(s):
 *   Sheueling Chang-Shantz <sheueling.chang@sun.com>,
 *   Stephen Fung <fungstep@hotmail.com>, and
 *   Douglas Stebila <douglas@stebila.ca>, Sun Microsystems Laboratories.
 *   Bodo Moeller <moeller@cdc.informatik.tu-darmstadt.de>,
 *   Nils Larsch <nla@trustcenter.de>, and
 *   Lenka Fibikova <fibikova@exp-math.uni-essen.de>, the OpenSSL Project
 *
 * Alternatively, the contents of this file may be used under the terms of
 * either the GNU General Public License Version 2 or later (the "GPL"), or
 * the GNU Lesser General Public License Version 2.1 or later (the "LGPL"),
 * in which case the provisions of the GPL or the LGPL are applicable instead
 * of those above. If you wish to allow use of your version of this file only
 * under the terms of either the GPL or the LGPL, and not to allow others to
 * use your version of this file under the terms of the MPL, indicate your
 * decision by deleting the provisions above and replace them with the notice
 * and other provisions required by the GPL or the LGPL. If you do not delete
 * the provisions above, a recipient may use your version of this file under
 * the terms of any one of the MPL, the GPL or the LGPL.
 *
 * ***** END LICENSE BLOCK ***** */
/*
 * Copyright 2007 Sun Microsystems, Inc.  All rights reserved.
 * Use is subject to license terms.
 *
 * Sun elects to use this software under the MPL license.
 */

#include "ecp.h"
#include "mplogic.h"
#ifndef _KERNEL
#include <stdlib.h>
#endif
#ifdef ECL_DEBUG
#include <assert.h>
#endif

/* Converts a point P(px, py) from affine coordinates to Jacobian
 * projective coordinates R(rx, ry, rz). Assumes input is already
 * field-encoded using field_enc, and returns output that is still
 * field-encoded. */
mp_err
ec_GFp_pt_aff2jac(const mp_int *px, const mp_int *py, mp_int *rx,
				  mp_int *ry, mp_int *rz, const ECGroup *group)
{
	mp_err res = MP_OKAY;

	if (ec_GFp_pt_is_inf_aff(px, py) == MP_YES) {
		MP_CHECKOK(ec_GFp_pt_set_inf_jac(rx, ry, rz));
	} else {
		MP_CHECKOK(mp_copy(px, rx));
		MP_CHECKOK(mp_copy(py, ry));
		MP_CHECKOK(mp_set_int(rz, 1));
		if (group->meth->field_enc) {
			MP_CHECKOK(group->meth->field_enc(rz, rz, group->meth));
		}
	}
  CLEANUP:
	return res;
}

/* Converts a point P(px, py, pz) from Jacobian projective coordinates to
 * affine coordinates R(rx, ry).  P and R can share x and y coordinates.
 * Assumes input is already field-encoded using field_enc, and returns
 * output that is still field-encoded. */
mp_err
ec_GFp_pt_jac2aff(const mp_int *px, const mp_int *py, const mp_int *pz,
				  mp_int *rx, mp_int *ry, const ECGroup *group)
{
	mp_err res = MP_OKAY;
	mp_int z1, z2, z3;

	MP_DIGITS(&z1) = 0;
	MP_DIGITS(&z2) = 0;
	MP_DIGITS(&z3) = 0;
	MP_CHECKOK(mp_init(&z1, FLAG(px)));
	MP_CHECKOK(mp_init(&z2, FLAG(px)));
	MP_CHECKOK(mp_init(&z3, FLAG(px)));

	/* if point at infinity, then set point at infinity and exit */
	if (ec_GFp_pt_is_inf_jac(px, py, pz) == MP_YES) {
		MP_CHECKOK(ec_GFp_pt_set_inf_aff(rx, ry));
		goto CLEANUP;
	}

	/* transform (px, py, pz) into (px / pz^2, py / pz^3) */
	if (mp_cmp_d(pz, 1) == 0) {
		MP_CHECKOK(mp_copy(px, rx));
		MP_CHECKOK(mp_copy(py, ry));
	} else {
		MP_CHECKOK(group->meth->field_div(NULL, pz, &z1, group->meth));
		MP_CHECKOK(group->meth->field_sqr(&z1, &z2, group->meth));
		MP_CHECKOK(group->meth->field_mul(&z1, &z2, &z3, group->meth));
		MP_CHECKOK(group->meth->field_mul(px, &z2, rx, group->meth));
		MP_CHECKOK(group->meth->field_mul(py, &z3, ry, group->meth));
	}

  CLEANUP:
	mp_clear(&z1);
	mp_clear(&z2);
	mp_clear(&z3);
	return res;
}

/* Checks if point P(px, py, pz) is at infinity. Uses Jacobian
 * coordinates. */
mp_err
ec_GFp_pt_is_inf_jac(const mp_int *px, const mp_int *py, const mp_int *pz)
{
	return mp_cmp_z(pz);
}

/* Sets P(px, py, pz) to be the point at infinity.  Uses Jacobian
 * coordinates. */
mp_err
ec_GFp_pt_set_inf_jac(mp_int *px, mp_int *py, mp_int *pz)
{
	mp_zero(pz);
	return MP_OKAY;
}

/* Computes R = P + Q where R is (rx, ry, rz), P is (px, py, pz) and Q is
 * (qx, qy, 1).  Elliptic curve points P, Q, and R can all be identical.
 * Uses mixed Jacobian-affine coordinates. Assumes input is already
 * field-encoded using field_enc, and returns output that is still
 * field-encoded. Uses equation (2) from Brown, Hankerson, Lopez, and
 * Menezes. Software Implementation of the NIST Elliptic Curves Over Prime
 * Fields. */
mp_err
ec_GFp_pt_add_jac_aff(const mp_int *px, const mp_int *py, const mp_int *pz,
					  const mp_int *qx, const mp_int *qy, mp_int *rx,
					  mp_int *ry, mp_int *rz, const ECGroup *group)
{
	mp_err res = MP_OKAY;
	mp_int A, B, C, D, C2, C3;

	MP_DIGITS(&A) = 0;
	MP_DIGITS(&B) = 0;
	MP_DIGITS(&C) = 0;
	MP_DIGITS(&D) = 0;
	MP_DIGITS(&C2) = 0;
	MP_DIGITS(&C3) = 0;
	MP_CHECKOK(mp_init(&A, FLAG(px)));
	MP_CHECKOK(mp_init(&B, FLAG(px)));
	MP_CHECKOK(mp_init(&C, FLAG(px)));
	MP_CHECKOK(mp_init(&D, FLAG(px)));
	MP_CHECKOK(mp_init(&C2, FLAG(px)));
	MP_CHECKOK(mp_init(&C3, FLAG(px)));

	/* If either P or Q is the point at infinity, then return the other
	 * point */
	if (ec_GFp_pt_is_inf_jac(px, py, pz) == MP_YES) {
		MP_CHECKOK(ec_GFp_pt_aff2jac(qx, qy, rx, ry, rz, group));
		goto CLEANUP;
	}
	if (ec_GFp_pt_is_inf_aff(qx, qy) == MP_YES) {
		MP_CHECKOK(mp_copy(px, rx));
		MP_CHECKOK(mp_copy(py, ry));
		MP_CHECKOK(mp_copy(pz, rz));
		goto CLEANUP;
	}

	/* A = qx * pz^2, B = qy * pz^3 */
	MP_CHECKOK(group->meth->field_sqr(pz, &A, group->meth));
	MP_CHECKOK(group->meth->field_mul(&A, pz, &B, group->meth));
	MP_CHECKOK(group->meth->field_mul(&A, qx, &A, group->meth));
	MP_CHECKOK(group->meth->field_mul(&B, qy, &B, group->meth));

	/* C = A - px, D = B - py */
	MP_CHECKOK(group->meth->field_sub(&A, px, &C, group->meth));
	MP_CHECKOK(group->meth->field_sub(&B, py, &D, group->meth));

	/* C2 = C^2, C3 = C^3 */
	MP_CHECKOK(group->meth->field_sqr(&C, &C2, group->meth));
	MP_CHECKOK(group->meth->field_mul(&C, &C2, &C3, group->meth));

	/* rz = pz * C */
	MP_CHECKOK(group->meth->field_mul(pz, &C, rz, group->meth));

	/* C = px * C^2 */
	MP_CHECKOK(group->meth->field_mul(px, &C2, &C, group->meth));
	/* A = D^2 */
	MP_CHECKOK(group->meth->field_sqr(&D, &A, group->meth));

	/* rx = D^2 - (C^3 + 2 * (px * C^2)) */
	MP_CHECKOK(group->meth->field_add(&C, &C, rx, group->meth));
	MP_CHECKOK(group->meth->field_add(&C3, rx, rx, group->meth));
	MP_CHECKOK(group->meth->field_sub(&A, rx, rx, group->meth));

	/* C3 = py * C^3 */
	MP_CHECKOK(group->meth->field_mul(py, &C3, &C3, group->meth));

	/* ry = D * (px * C^2 - rx) - py * C^3 */
	MP_CHECKOK(group->meth->field_sub(&C, rx, ry, group->meth));
	MP_CHECKOK(group->meth->field_mul(&D, ry, ry, group->meth));
	MP_CHECKOK(group->meth->field_sub(ry, &C3, ry, group->meth));

  CLEANUP:
	mp_clear(&A);
	mp_clear(&B);
	mp_clear(&C);
	mp_clear(&D);
	mp_clear(&C2);
	mp_clear(&C3);
	return res;
}

/* Computes R = 2P.  Elliptic curve points P and R can be identical.  Uses
 * Jacobian coordinates.
 *
 * Assumes input is already field-encoded using field_enc, and returns
 * output that is still field-encoded.
 *
 * This routine implements Point Doubling in the Jacobian Projective
 * space as described in the paper "Efficient elliptic curve exponentiation
 * using mixed coordinates", by H. Cohen, A Miyaji, T. Ono.
 */
mp_err
ec_GFp_pt_dbl_jac(const mp_int *px, const mp_int *py, const mp_int *pz,
				  mp_int *rx, mp_int *ry, mp_int *rz, const ECGroup *group)
{
	mp_err res = MP_OKAY;
	mp_int t0, t1, M, S;

	MP_DIGITS(&t0) = 0;
	MP_DIGITS(&t1) = 0;
	MP_DIGITS(&M) = 0;
	MP_DIGITS(&S) = 0;
	MP_CHECKOK(mp_init(&t0, FLAG(px)));
	MP_CHECKOK(mp_init(&t1, FLAG(px)));
	MP_CHECKOK(mp_init(&M, FLAG(px)));
	MP_CHECKOK(mp_init(&S, FLAG(px)));

	if (ec_GFp_pt_is_inf_jac(px, py, pz) == MP_YES) {
		MP_CHECKOK(ec_GFp_pt_set_inf_jac(rx, ry, rz));
		goto CLEANUP;
	}

	if (mp_cmp_d(pz, 1) == 0) {
		/* M = 3 * px^2 + a */
		MP_CHECKOK(group->meth->field_sqr(px, &t0, group->meth));
		MP_CHECKOK(group->meth->field_add(&t0, &t0, &M, group->meth));
		MP_CHECKOK(group->meth->field_add(&t0, &M, &t0, group->meth));
		MP_CHECKOK(group->meth->
				   field_add(&t0, &group->curvea, &M, group->meth));
	} else if (mp_cmp_int(&group->curvea, -3, FLAG(px)) == 0) {
		/* M = 3 * (px + pz^2) * (px - pz^2) */
		MP_CHECKOK(group->meth->field_sqr(pz, &M, group->meth));
		MP_CHECKOK(group->meth->field_add(px, &M, &t0, group->meth));
		MP_CHECKOK(group->meth->field_sub(px, &M, &t1, group->meth));
		MP_CHECKOK(group->meth->field_mul(&t0, &t1, &M, group->meth));
		MP_CHECKOK(group->meth->field_add(&M, &M, &t0, group->meth));
		MP_CHECKOK(group->meth->field_add(&t0, &M, &M, group->meth));
	} else {
		/* M = 3 * (px^2) + a * (pz^4) */
		MP_CHECKOK(group->meth->field_sqr(px, &t0, group->meth));
		MP_CHECKOK(group->meth->field_add(&t0, &t0, &M, group->meth));
		MP_CHECKOK(group->meth->field_add(&t0, &M, &t0, group->meth));
		MP_CHECKOK(group->meth->field_sqr(pz, &M, group->meth));
		MP_CHECKOK(group->meth->field_sqr(&M, &M, group->meth));
		MP_CHECKOK(group->meth->
				   field_mul(&M, &group->curvea, &M, group->meth));
		MP_CHECKOK(group->meth->field_add(&M, &t0, &M, group->meth));
	}

	/* rz = 2 * py * pz */
	/* t0 = 4 * py^2 */
	if (mp_cmp_d(pz, 1) == 0) {
		MP_CHECKOK(group->meth->field_add(py, py, rz, group->meth));
		MP_CHECKOK(group->meth->field_sqr(rz, &t0, group->meth));
	} else {
		MP_CHECKOK(group->meth->field_add(py, py, &t0, group->meth));
		MP_CHECKOK(group->meth->field_mul(&t0, pz, rz, group->meth));
		MP_CHECKOK(group->meth->field_sqr(&t0, &t0, group->meth));
	}

	/* S = 4 * px * py^2 = px * (2 * py)^2 */
	MP_CHECKOK(group->meth->field_mul(px, &t0, &S, group->meth));

	/* rx = M^2 - 2 * S */
	MP_CHECKOK(group->meth->field_add(&S, &S, &t1, group->meth));
	MP_CHECKOK(group->meth->field_sqr(&M, rx, group->meth));
	MP_CHECKOK(group->meth->field_sub(rx, &t1, rx, group->meth));

	/* ry = M * (S - rx) - 8 * py^4 */
	MP_CHECKOK(group->meth->field_sqr(&t0, &t1, group->meth));
	if (mp_isodd(&t1)) {
		MP_CHECKOK(mp_add(&t1, &group->meth->irr, &t1));
	}
	MP_CHECKOK(mp_div_2(&t1, &t1));
	MP_CHECKOK(group->meth->field_sub(&S, rx, &S, group->meth));
	MP_CHECKOK(group->meth->field_mul(&M, &S, &M, group->meth));
	MP_CHECKOK(group->meth->field_sub(&M, &t1, ry, group->meth));

  CLEANUP:
	mp_clear(&t0);
	mp_clear(&t1);
	mp_clear(&M);
	mp_clear(&S);
	return res;
}

/* by default, this routine is unused and thus doesn't need to be compiled */
#ifdef ECL_ENABLE_GFP_PT_MUL_JAC
/* Computes R = nP where R is (rx, ry) and P is (px, py). The parameters
 * a, b and p are the elliptic curve coefficients and the prime that
 * determines the field GFp.  Elliptic curve points P and R can be
 * identical.  Uses mixed Jacobian-affine coordinates. Assumes input is
 * already field-encoded using field_enc, and returns output that is still
 * field-encoded. Uses 4-bit window method. */
mp_err
ec_GFp_pt_mul_jac(const mp_int *n, const mp_int *px, const mp_int *py,
				  mp_int *rx, mp_int *ry, const ECGroup *group)
{
	mp_err res = MP_OKAY;
	mp_int precomp[16][2], rz;
	int i, ni, d;

	MP_DIGITS(&rz) = 0;
	for (i = 0; i < 16; i++) {
		MP_DIGITS(&precomp[i][0]) = 0;
		MP_DIGITS(&precomp[i][1]) = 0;
	}

	ARGCHK(group != NULL, MP_BADARG);
	ARGCHK((n != NULL) && (px != NULL) && (py != NULL), MP_BADARG);

	/* initialize precomputation table */
	for (i = 0; i < 16; i++) {
		MP_CHECKOK(mp_init(&precomp[i][0]));
		MP_CHECKOK(mp_init(&precomp[i][1]));
	}

	/* fill precomputation table */
	mp_zero(&precomp[0][0]);
	mp_zero(&precomp[0][1]);
	MP_CHECKOK(mp_copy(px, &precomp[1][0]));
	MP_CHECKOK(mp_copy(py, &precomp[1][1]));
	for (i = 2; i < 16; i++) {
		MP_CHECKOK(group->
				   point_add(&precomp[1][0], &precomp[1][1],
							 &precomp[i - 1][0], &precomp[i - 1][1],
							 &precomp[i][0], &precomp[i][1], group));
	}

	d = (mpl_significant_bits(n) + 3) / 4;

	/* R = inf */
	MP_CHECKOK(mp_init(&rz));
	MP_CHECKOK(ec_GFp_pt_set_inf_jac(rx, ry, &rz));

	for (i = d - 1; i >= 0; i--) {
		/* compute window ni */
		ni = MP_GET_BIT(n, 4 * i + 3);
		ni <<= 1;
		ni |= MP_GET_BIT(n, 4 * i + 2);
		ni <<= 1;
		ni |= MP_GET_BIT(n, 4 * i + 1);
		ni <<= 1;
		ni |= MP_GET_BIT(n, 4 * i);
		/* R = 2^4 * R */
		MP_CHECKOK(ec_GFp_pt_dbl_jac(rx, ry, &rz, rx, ry, &rz, group));
		MP_CHECKOK(ec_GFp_pt_dbl_jac(rx, ry, &rz, rx, ry, &rz, group));
		MP_CHECKOK(ec_GFp_pt_dbl_jac(rx, ry, &rz, rx, ry, &rz, group));
		MP_CHECKOK(ec_GFp_pt_dbl_jac(rx, ry, &rz, rx, ry, &rz, group));
		/* R = R + (ni * P) */
		MP_CHECKOK(ec_GFp_pt_add_jac_aff
				   (rx, ry, &rz, &precomp[ni][0], &precomp[ni][1], rx, ry,
					&rz, group));
	}

	/* convert result S to affine coordinates */
	MP_CHECKOK(ec_GFp_pt_jac2aff(rx, ry, &rz, rx, ry, group));

  CLEANUP:
	mp_clear(&rz);
	for (i = 0; i < 16; i++) {
		mp_clear(&precomp[i][0]);
		mp_clear(&precomp[i][1]);
	}
	return res;
}
#endif

/* Elliptic curve scalar-point multiplication. Computes R(x, y) = k1 * G +
 * k2 * P(x, y), where G is the generator (base point) of the group of
 * points on the elliptic curve. Allows k1 = NULL or { k2, P } = NULL.
 * Uses mixed Jacobian-affine coordinates. Input and output values are
 * assumed to be NOT field-encoded. Uses algorithm 15 (simultaneous
 * multiple point multiplication) from Brown, Hankerson, Lopez, Menezes.
 * Software Implementation of the NIST Elliptic Curves over Prime Fields. */
mp_err
ec_GFp_pts_mul_jac(const mp_int *k1, const mp_int *k2, const mp_int *px,
				   const mp_int *py, mp_int *rx, mp_int *ry,
				   const ECGroup *group)
{
	mp_err res = MP_OKAY;
	mp_int precomp[4][4][2];
	mp_int rz;
	const mp_int *a, *b;
	int i, j;
	int ai, bi, d;

	for (i = 0; i < 4; i++) {
		for (j = 0; j < 4; j++) {
			MP_DIGITS(&precomp[i][j][0]) = 0;
			MP_DIGITS(&precomp[i][j][1]) = 0;
		}
	}
	MP_DIGITS(&rz) = 0;

	ARGCHK(group != NULL, MP_BADARG);
	ARGCHK(!((k1 == NULL)
			 && ((k2 == NULL) || (px == NULL)
				 || (py == NULL))), MP_BADARG);

	/* if some arguments are not defined used ECPoint_mul */
	if (k1 == NULL) {
		return ECPoint_mul(group, k2, px, py, rx, ry);
	} else if ((k2 == NULL) || (px == NULL) || (py == NULL)) {
		return ECPoint_mul(group, k1, NULL, NULL, rx, ry);
	}

	/* initialize precomputation table */
	for (i = 0; i < 4; i++) {
		for (j = 0; j < 4; j++) {
			MP_CHECKOK(mp_init(&precomp[i][j][0], FLAG(k1)));
			MP_CHECKOK(mp_init(&precomp[i][j][1], FLAG(k1)));
		}
	}

	/* fill precomputation table */
	/* assign {k1, k2} = {a, b} such that len(a) >= len(b) */
	if (mpl_significant_bits(k1) < mpl_significant_bits(k2)) {
		a = k2;
		b = k1;
		if (group->meth->field_enc) {
			MP_CHECKOK(group->meth->
					   field_enc(px, &precomp[1][0][0], group->meth));
			MP_CHECKOK(group->meth->
					   field_enc(py, &precomp[1][0][1], group->meth));
		} else {
			MP_CHECKOK(mp_copy(px, &precomp[1][0][0]));
			MP_CHECKOK(mp_copy(py, &precomp[1][0][1]));
		}
		MP_CHECKOK(mp_copy(&group->genx, &precomp[0][1][0]));
		MP_CHECKOK(mp_copy(&group->geny, &precomp[0][1][1]));
	} else {
		a = k1;
		b = k2;
		MP_CHECKOK(mp_copy(&group->genx, &precomp[1][0][0]));
		MP_CHECKOK(mp_copy(&group->geny, &precomp[1][0][1]));
		if (group->meth->field_enc) {
			MP_CHECKOK(group->meth->
					   field_enc(px, &precomp[0][1][0], group->meth));
			MP_CHECKOK(group->meth->
					   field_enc(py, &precomp[0][1][1], group->meth));
		} else {
			MP_CHECKOK(mp_copy(px, &precomp[0][1][0]));
			MP_CHECKOK(mp_copy(py, &precomp[0][1][1]));
		}
	}
	/* precompute [*][0][*] */
	mp_zero(&precomp[0][0][0]);
	mp_zero(&precomp[0][0][1]);
	MP_CHECKOK(group->
			   point_dbl(&precomp[1][0][0], &precomp[1][0][1],
						 &precomp[2][0][0], &precomp[2][0][1], group));
	MP_CHECKOK(group->
			   point_add(&precomp[1][0][0], &precomp[1][0][1],
						 &precomp[2][0][0], &precomp[2][0][1],
						 &precomp[3][0][0], &precomp[3][0][1], group));
	/* precompute [*][1][*] */
	for (i = 1; i < 4; i++) {
		MP_CHECKOK(group->
				   point_add(&precomp[0][1][0], &precomp[0][1][1],
							 &precomp[i][0][0], &precomp[i][0][1],
							 &precomp[i][1][0], &precomp[i][1][1], group));
	}
	/* precompute [*][2][*] */
	MP_CHECKOK(group->
			   point_dbl(&precomp[0][1][0], &precomp[0][1][1],
						 &precomp[0][2][0], &precomp[0][2][1], group));
	for (i = 1; i < 4; i++) {
		MP_CHECKOK(group->
				   point_add(&precomp[0][2][0], &precomp[0][2][1],
							 &precomp[i][0][0], &precomp[i][0][1],
							 &precomp[i][2][0], &precomp[i][2][1], group));
	}
	/* precompute [*][3][*] */
	MP_CHECKOK(group->
			   point_add(&precomp[0][1][0], &precomp[0][1][1],
						 &precomp[0][2][0], &precomp[0][2][1],
						 &precomp[0][3][0], &precomp[0][3][1], group));
	for (i = 1; i < 4; i++) {
		MP_CHECKOK(group->
				   point_add(&precomp[0][3][0], &precomp[0][3][1],
							 &precomp[i][0][0], &precomp[i][0][1],
							 &precomp[i][3][0], &precomp[i][3][1], group));
	}

	d = (mpl_significant_bits(a) + 1) / 2;

	/* R = inf */
	MP_CHECKOK(mp_init(&rz, FLAG(k1)));
	MP_CHECKOK(ec_GFp_pt_set_inf_jac(rx, ry, &rz));

	for (i = d - 1; i >= 0; i--) {
		ai = MP_GET_BIT(a, 2 * i + 1);
		ai <<= 1;
		ai |= MP_GET_BIT(a, 2 * i);
		bi = MP_GET_BIT(b, 2 * i + 1);
		bi <<= 1;
		bi |= MP_GET_BIT(b, 2 * i);
		/* R = 2^2 * R */
		MP_CHECKOK(ec_GFp_pt_dbl_jac(rx, ry, &rz, rx, ry, &rz, group));
		MP_CHECKOK(ec_GFp_pt_dbl_jac(rx, ry, &rz, rx, ry, &rz, group));
		/* R = R + (ai * A + bi * B) */
		MP_CHECKOK(ec_GFp_pt_add_jac_aff
				   (rx, ry, &rz, &precomp[ai][bi][0], &precomp[ai][bi][1],
					rx, ry, &rz, group));
	}

	MP_CHECKOK(ec_GFp_pt_jac2aff(rx, ry, &rz, rx, ry, group));

	if (group->meth->field_dec) {
		MP_CHECKOK(group->meth->field_dec(rx, rx, group->meth));
		MP_CHECKOK(group->meth->field_dec(ry, ry, group->meth));
	}

  CLEANUP:
	mp_clear(&rz);
	for (i = 0; i < 4; i++) {
		for (j = 0; j < 4; j++) {
			mp_clear(&precomp[i][j][0]);
			mp_clear(&precomp[i][j][1]);
		}
	}
	return res;
}