michael@0: /* This Source Code Form is subject to the terms of the Mozilla Public
michael@0:  * License, v. 2.0. If a copy of the MPL was not distributed with this
michael@0:  * file, You can obtain one at http://mozilla.org/MPL/2.0/. */
michael@0: 
michael@0: #include "ec2.h"
michael@0: #include "mplogic.h"
michael@0: #include "mp_gf2m.h"
michael@0: #include <stdlib.h>
michael@0: #ifdef ECL_DEBUG
michael@0: #include <assert.h>
michael@0: #endif
michael@0: 
michael@0: /* by default, these routines are unused and thus don't need to be compiled */
michael@0: #ifdef ECL_ENABLE_GF2M_PROJ
michael@0: /* Converts a point P(px, py) from affine coordinates to projective
michael@0:  * coordinates R(rx, ry, rz). Assumes input is already field-encoded using 
michael@0:  * field_enc, and returns output that is still field-encoded. */
michael@0: mp_err
michael@0: ec_GF2m_pt_aff2proj(const mp_int *px, const mp_int *py, mp_int *rx,
michael@0: 					mp_int *ry, mp_int *rz, const ECGroup *group)
michael@0: {
michael@0: 	mp_err res = MP_OKAY;
michael@0: 
michael@0: 	MP_CHECKOK(mp_copy(px, rx));
michael@0: 	MP_CHECKOK(mp_copy(py, ry));
michael@0: 	MP_CHECKOK(mp_set_int(rz, 1));
michael@0: 	if (group->meth->field_enc) {
michael@0: 		MP_CHECKOK(group->meth->field_enc(rz, rz, group->meth));
michael@0: 	}
michael@0:   CLEANUP:
michael@0: 	return res;
michael@0: }
michael@0: 
michael@0: /* Converts a point P(px, py, pz) from projective coordinates to affine
michael@0:  * coordinates R(rx, ry).  P and R can share x and y coordinates. Assumes
michael@0:  * input is already field-encoded using field_enc, and returns output that
michael@0:  * is still field-encoded. */
michael@0: mp_err
michael@0: ec_GF2m_pt_proj2aff(const mp_int *px, const mp_int *py, const mp_int *pz,
michael@0: 					mp_int *rx, mp_int *ry, const ECGroup *group)
michael@0: {
michael@0: 	mp_err res = MP_OKAY;
michael@0: 	mp_int z1, z2;
michael@0: 
michael@0: 	MP_DIGITS(&z1) = 0;
michael@0: 	MP_DIGITS(&z2) = 0;
michael@0: 	MP_CHECKOK(mp_init(&z1));
michael@0: 	MP_CHECKOK(mp_init(&z2));
michael@0: 
michael@0: 	/* if point at infinity, then set point at infinity and exit */
michael@0: 	if (ec_GF2m_pt_is_inf_proj(px, py, pz) == MP_YES) {
michael@0: 		MP_CHECKOK(ec_GF2m_pt_set_inf_aff(rx, ry));
michael@0: 		goto CLEANUP;
michael@0: 	}
michael@0: 
michael@0: 	/* transform (px, py, pz) into (px / pz, py / pz^2) */
michael@0: 	if (mp_cmp_d(pz, 1) == 0) {
michael@0: 		MP_CHECKOK(mp_copy(px, rx));
michael@0: 		MP_CHECKOK(mp_copy(py, ry));
michael@0: 	} else {
michael@0: 		MP_CHECKOK(group->meth->field_div(NULL, pz, &z1, group->meth));
michael@0: 		MP_CHECKOK(group->meth->field_sqr(&z1, &z2, group->meth));
michael@0: 		MP_CHECKOK(group->meth->field_mul(px, &z1, rx, group->meth));
michael@0: 		MP_CHECKOK(group->meth->field_mul(py, &z2, ry, group->meth));
michael@0: 	}
michael@0: 
michael@0:   CLEANUP:
michael@0: 	mp_clear(&z1);
michael@0: 	mp_clear(&z2);
michael@0: 	return res;
michael@0: }
michael@0: 
michael@0: /* Checks if point P(px, py, pz) is at infinity. Uses projective
michael@0:  * coordinates. */
michael@0: mp_err
michael@0: ec_GF2m_pt_is_inf_proj(const mp_int *px, const mp_int *py,
michael@0: 					   const mp_int *pz)
michael@0: {
michael@0: 	return mp_cmp_z(pz);
michael@0: }
michael@0: 
michael@0: /* Sets P(px, py, pz) to be the point at infinity.  Uses projective
michael@0:  * coordinates. */
michael@0: mp_err
michael@0: ec_GF2m_pt_set_inf_proj(mp_int *px, mp_int *py, mp_int *pz)
michael@0: {
michael@0: 	mp_zero(pz);
michael@0: 	return MP_OKAY;
michael@0: }
michael@0: 
michael@0: /* Computes R = P + Q where R is (rx, ry, rz), P is (px, py, pz) and Q is
michael@0:  * (qx, qy, 1).  Elliptic curve points P, Q, and R can all be identical.
michael@0:  * Uses mixed projective-affine coordinates. Assumes input is already
michael@0:  * field-encoded using field_enc, and returns output that is still
michael@0:  * field-encoded. Uses equation (3) from Hankerson, Hernandez, Menezes.
michael@0:  * Software Implementation of Elliptic Curve Cryptography Over Binary
michael@0:  * Fields. */
michael@0: mp_err
michael@0: ec_GF2m_pt_add_proj(const mp_int *px, const mp_int *py, const mp_int *pz,
michael@0: 					const mp_int *qx, const mp_int *qy, mp_int *rx,
michael@0: 					mp_int *ry, mp_int *rz, const ECGroup *group)
michael@0: {
michael@0: 	mp_err res = MP_OKAY;
michael@0: 	mp_int A, B, C, D, E, F, G;
michael@0: 
michael@0: 	/* If either P or Q is the point at infinity, then return the other
michael@0: 	 * point */
michael@0: 	if (ec_GF2m_pt_is_inf_proj(px, py, pz) == MP_YES) {
michael@0: 		return ec_GF2m_pt_aff2proj(qx, qy, rx, ry, rz, group);
michael@0: 	}
michael@0: 	if (ec_GF2m_pt_is_inf_aff(qx, qy) == MP_YES) {
michael@0: 		MP_CHECKOK(mp_copy(px, rx));
michael@0: 		MP_CHECKOK(mp_copy(py, ry));
michael@0: 		return mp_copy(pz, rz);
michael@0: 	}
michael@0: 
michael@0: 	MP_DIGITS(&A) = 0;
michael@0: 	MP_DIGITS(&B) = 0;
michael@0: 	MP_DIGITS(&C) = 0;
michael@0: 	MP_DIGITS(&D) = 0;
michael@0: 	MP_DIGITS(&E) = 0;
michael@0: 	MP_DIGITS(&F) = 0;
michael@0: 	MP_DIGITS(&G) = 0;
michael@0: 	MP_CHECKOK(mp_init(&A));
michael@0: 	MP_CHECKOK(mp_init(&B));
michael@0: 	MP_CHECKOK(mp_init(&C));
michael@0: 	MP_CHECKOK(mp_init(&D));
michael@0: 	MP_CHECKOK(mp_init(&E));
michael@0: 	MP_CHECKOK(mp_init(&F));
michael@0: 	MP_CHECKOK(mp_init(&G));
michael@0: 
michael@0: 	/* D = pz^2 */
michael@0: 	MP_CHECKOK(group->meth->field_sqr(pz, &D, group->meth));
michael@0: 
michael@0: 	/* A = qy * pz^2 + py */
michael@0: 	MP_CHECKOK(group->meth->field_mul(qy, &D, &A, group->meth));
michael@0: 	MP_CHECKOK(group->meth->field_add(&A, py, &A, group->meth));
michael@0: 
michael@0: 	/* B = qx * pz + px */
michael@0: 	MP_CHECKOK(group->meth->field_mul(qx, pz, &B, group->meth));
michael@0: 	MP_CHECKOK(group->meth->field_add(&B, px, &B, group->meth));
michael@0: 
michael@0: 	/* C = pz * B */
michael@0: 	MP_CHECKOK(group->meth->field_mul(pz, &B, &C, group->meth));
michael@0: 
michael@0: 	/* D = B^2 * (C + a * pz^2) (using E as a temporary variable) */
michael@0: 	MP_CHECKOK(group->meth->
michael@0: 			   field_mul(&group->curvea, &D, &D, group->meth));
michael@0: 	MP_CHECKOK(group->meth->field_add(&C, &D, &D, group->meth));
michael@0: 	MP_CHECKOK(group->meth->field_sqr(&B, &E, group->meth));
michael@0: 	MP_CHECKOK(group->meth->field_mul(&E, &D, &D, group->meth));
michael@0: 
michael@0: 	/* rz = C^2 */
michael@0: 	MP_CHECKOK(group->meth->field_sqr(&C, rz, group->meth));
michael@0: 
michael@0: 	/* E = A * C */
michael@0: 	MP_CHECKOK(group->meth->field_mul(&A, &C, &E, group->meth));
michael@0: 
michael@0: 	/* rx = A^2 + D + E */
michael@0: 	MP_CHECKOK(group->meth->field_sqr(&A, rx, group->meth));
michael@0: 	MP_CHECKOK(group->meth->field_add(rx, &D, rx, group->meth));
michael@0: 	MP_CHECKOK(group->meth->field_add(rx, &E, rx, group->meth));
michael@0: 
michael@0: 	/* F = rx + qx * rz */
michael@0: 	MP_CHECKOK(group->meth->field_mul(qx, rz, &F, group->meth));
michael@0: 	MP_CHECKOK(group->meth->field_add(rx, &F, &F, group->meth));
michael@0: 
michael@0: 	/* G = rx + qy * rz */
michael@0: 	MP_CHECKOK(group->meth->field_mul(qy, rz, &G, group->meth));
michael@0: 	MP_CHECKOK(group->meth->field_add(rx, &G, &G, group->meth));
michael@0: 
michael@0: 	/* ry = E * F + rz * G (using G as a temporary variable) */
michael@0: 	MP_CHECKOK(group->meth->field_mul(rz, &G, &G, group->meth));
michael@0: 	MP_CHECKOK(group->meth->field_mul(&E, &F, ry, group->meth));
michael@0: 	MP_CHECKOK(group->meth->field_add(ry, &G, ry, group->meth));
michael@0: 
michael@0:   CLEANUP:
michael@0: 	mp_clear(&A);
michael@0: 	mp_clear(&B);
michael@0: 	mp_clear(&C);
michael@0: 	mp_clear(&D);
michael@0: 	mp_clear(&E);
michael@0: 	mp_clear(&F);
michael@0: 	mp_clear(&G);
michael@0: 	return res;
michael@0: }
michael@0: 
michael@0: /* Computes R = 2P.  Elliptic curve points P and R can be identical.  Uses 
michael@0:  * projective coordinates.
michael@0:  *
michael@0:  * Assumes input is already field-encoded using field_enc, and returns 
michael@0:  * output that is still field-encoded.
michael@0:  *
michael@0:  * Uses equation (3) from Hankerson, Hernandez, Menezes. Software 
michael@0:  * Implementation of Elliptic Curve Cryptography Over Binary Fields.
michael@0:  */
michael@0: mp_err
michael@0: ec_GF2m_pt_dbl_proj(const mp_int *px, const mp_int *py, const mp_int *pz,
michael@0: 					mp_int *rx, mp_int *ry, mp_int *rz,
michael@0: 					const ECGroup *group)
michael@0: {
michael@0: 	mp_err res = MP_OKAY;
michael@0: 	mp_int t0, t1;
michael@0: 
michael@0: 	if (ec_GF2m_pt_is_inf_proj(px, py, pz) == MP_YES) {
michael@0: 		return ec_GF2m_pt_set_inf_proj(rx, ry, rz);
michael@0: 	}
michael@0: 
michael@0: 	MP_DIGITS(&t0) = 0;
michael@0: 	MP_DIGITS(&t1) = 0;
michael@0: 	MP_CHECKOK(mp_init(&t0));
michael@0: 	MP_CHECKOK(mp_init(&t1));
michael@0: 
michael@0: 	/* t0 = px^2 */
michael@0: 	/* t1 = pz^2 */
michael@0: 	MP_CHECKOK(group->meth->field_sqr(px, &t0, group->meth));
michael@0: 	MP_CHECKOK(group->meth->field_sqr(pz, &t1, group->meth));
michael@0: 
michael@0: 	/* rz = px^2 * pz^2 */
michael@0: 	MP_CHECKOK(group->meth->field_mul(&t0, &t1, rz, group->meth));
michael@0: 
michael@0: 	/* t0 = px^4 */
michael@0: 	/* t1 = b * pz^4 */
michael@0: 	MP_CHECKOK(group->meth->field_sqr(&t0, &t0, group->meth));
michael@0: 	MP_CHECKOK(group->meth->field_sqr(&t1, &t1, group->meth));
michael@0: 	MP_CHECKOK(group->meth->
michael@0: 			   field_mul(&group->curveb, &t1, &t1, group->meth));
michael@0: 
michael@0: 	/* rx = px^4 + b * pz^4 */
michael@0: 	MP_CHECKOK(group->meth->field_add(&t0, &t1, rx, group->meth));
michael@0: 
michael@0: 	/* ry = b * pz^4 * rz + rx * (a * rz + py^2 + b * pz^4) */
michael@0: 	MP_CHECKOK(group->meth->field_sqr(py, ry, group->meth));
michael@0: 	MP_CHECKOK(group->meth->field_add(ry, &t1, ry, group->meth));
michael@0: 	/* t0 = a * rz */
michael@0: 	MP_CHECKOK(group->meth->
michael@0: 			   field_mul(&group->curvea, rz, &t0, group->meth));
michael@0: 	MP_CHECKOK(group->meth->field_add(&t0, ry, ry, group->meth));
michael@0: 	MP_CHECKOK(group->meth->field_mul(rx, ry, ry, group->meth));
michael@0: 	/* t1 = b * pz^4 * rz */
michael@0: 	MP_CHECKOK(group->meth->field_mul(&t1, rz, &t1, group->meth));
michael@0: 	MP_CHECKOK(group->meth->field_add(&t1, ry, ry, group->meth));
michael@0: 
michael@0:   CLEANUP:
michael@0: 	mp_clear(&t0);
michael@0: 	mp_clear(&t1);
michael@0: 	return res;
michael@0: }
michael@0: 
michael@0: /* Computes R = nP where R is (rx, ry) and P is (px, py). The parameters
michael@0:  * a, b and p are the elliptic curve coefficients and the prime that
michael@0:  * determines the field GF2m.  Elliptic curve points P and R can be
michael@0:  * identical.  Uses mixed projective-affine coordinates. Assumes input is
michael@0:  * already field-encoded using field_enc, and returns output that is still
michael@0:  * field-encoded. Uses 4-bit window method. */
michael@0: mp_err
michael@0: ec_GF2m_pt_mul_proj(const mp_int *n, const mp_int *px, const mp_int *py,
michael@0: 					mp_int *rx, mp_int *ry, const ECGroup *group)
michael@0: {
michael@0: 	mp_err res = MP_OKAY;
michael@0: 	mp_int precomp[16][2], rz;
michael@0: 	mp_digit precomp_arr[ECL_MAX_FIELD_SIZE_DIGITS * 16 * 2], *t;
michael@0: 	int i, ni, d;
michael@0: 
michael@0: 	ARGCHK(group != NULL, MP_BADARG);
michael@0: 	ARGCHK((n != NULL) && (px != NULL) && (py != NULL), MP_BADARG);
michael@0: 
michael@0: 	/* initialize precomputation table */
michael@0: 	t = precomp_arr;
michael@0: 	for (i = 0; i < 16; i++) {
michael@0: 		/* x co-ord */
michael@0: 		MP_SIGN(&precomp[i][0]) = MP_ZPOS;
michael@0: 		MP_ALLOC(&precomp[i][0]) = ECL_MAX_FIELD_SIZE_DIGITS;
michael@0: 		MP_USED(&precomp[i][0]) = 1;
michael@0: 		*t = 0;
michael@0: 		MP_DIGITS(&precomp[i][0]) = t;
michael@0: 		t += ECL_MAX_FIELD_SIZE_DIGITS;
michael@0: 		/* y co-ord */
michael@0: 		MP_SIGN(&precomp[i][1]) = MP_ZPOS;
michael@0: 		MP_ALLOC(&precomp[i][1]) = ECL_MAX_FIELD_SIZE_DIGITS;
michael@0: 		MP_USED(&precomp[i][1]) = 1;
michael@0: 		*t = 0;
michael@0: 		MP_DIGITS(&precomp[i][1]) = t;
michael@0: 		t += ECL_MAX_FIELD_SIZE_DIGITS;
michael@0: 	}
michael@0: 
michael@0: 	/* fill precomputation table */
michael@0: 	mp_zero(&precomp[0][0]);
michael@0: 	mp_zero(&precomp[0][1]);
michael@0: 	MP_CHECKOK(mp_copy(px, &precomp[1][0]));
michael@0: 	MP_CHECKOK(mp_copy(py, &precomp[1][1]));
michael@0: 	for (i = 2; i < 16; i++) {
michael@0: 		MP_CHECKOK(group->
michael@0: 				   point_add(&precomp[1][0], &precomp[1][1],
michael@0: 							 &precomp[i - 1][0], &precomp[i - 1][1],
michael@0: 							 &precomp[i][0], &precomp[i][1], group));
michael@0: 	}
michael@0: 
michael@0: 	d = (mpl_significant_bits(n) + 3) / 4;
michael@0: 
michael@0: 	/* R = inf */
michael@0: 	MP_DIGITS(&rz) = 0;
michael@0: 	MP_CHECKOK(mp_init(&rz));
michael@0: 	MP_CHECKOK(ec_GF2m_pt_set_inf_proj(rx, ry, &rz));
michael@0: 
michael@0: 	for (i = d - 1; i >= 0; i--) {
michael@0: 		/* compute window ni */
michael@0: 		ni = MP_GET_BIT(n, 4 * i + 3);
michael@0: 		ni <<= 1;
michael@0: 		ni |= MP_GET_BIT(n, 4 * i + 2);
michael@0: 		ni <<= 1;
michael@0: 		ni |= MP_GET_BIT(n, 4 * i + 1);
michael@0: 		ni <<= 1;
michael@0: 		ni |= MP_GET_BIT(n, 4 * i);
michael@0: 		/* R = 2^4 * R */
michael@0: 		MP_CHECKOK(ec_GF2m_pt_dbl_proj(rx, ry, &rz, rx, ry, &rz, group));
michael@0: 		MP_CHECKOK(ec_GF2m_pt_dbl_proj(rx, ry, &rz, rx, ry, &rz, group));
michael@0: 		MP_CHECKOK(ec_GF2m_pt_dbl_proj(rx, ry, &rz, rx, ry, &rz, group));
michael@0: 		MP_CHECKOK(ec_GF2m_pt_dbl_proj(rx, ry, &rz, rx, ry, &rz, group));
michael@0: 		/* R = R + (ni * P) */
michael@0: 		MP_CHECKOK(ec_GF2m_pt_add_proj
michael@0: 				   (rx, ry, &rz, &precomp[ni][0], &precomp[ni][1], rx, ry,
michael@0: 					&rz, group));
michael@0: 	}
michael@0: 
michael@0: 	/* convert result S to affine coordinates */
michael@0: 	MP_CHECKOK(ec_GF2m_pt_proj2aff(rx, ry, &rz, rx, ry, group));
michael@0: 
michael@0:   CLEANUP:
michael@0: 	mp_clear(&rz);
michael@0: 	return res;
michael@0: }
michael@0: #endif