/*
* Copyright (c) 2007, 2017, Oracle and/or its affiliates. All rights reserved.
* Use is subject to license terms.
*
* This library is free software; you can redistribute it and/or
* modify it under the terms of the GNU Lesser General Public
* License as published by the Free Software Foundation; either
* version 2.1 of the License, or (at your option) any later version.
*
* This library 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
* Lesser General Public License for more details.
*
* You should have received a copy of the GNU Lesser General Public License
* along with this library; if not, write to the Free Software Foundation,
* Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA.
*
* Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA
* or visit www.oracle.com if you need additional information or have any
* questions.
*/
/* *********************************************************************
*
* 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):
* Stephen Fung <fungstep@hotmail.com>, Sun Microsystems Laboratories
*
* Last Modified Date from the Original Code: May 2017
*********************************************************************** */
#include "ecp.h"
#include "ecl-priv.h"
#include "mplogic.h"
#ifndef _KERNEL
#include <stdlib.h>
#endif
#define MAX_SCRATCH 6
/* Computes R = 2P. Elliptic curve points P and R can be identical. Uses
* Modified Jacobian coordinates.
*
* Assumes input is already field-encoded using field_enc, and returns
* output that is still field-encoded.
*
*/
mp_err
ec_GFp_pt_dbl_jm(const mp_int *px, const mp_int *py, const mp_int *pz,
const mp_int *paz4, mp_int *rx, mp_int *ry, mp_int *rz,
mp_int *raz4, mp_int scratch[], const ECGroup *group)
{
mp_err res = MP_OKAY;
mp_int *t0, *t1, *M, *S;
t0 = &scratch[0];
t1 = &scratch[1];
M = &scratch[2];
S = &scratch[3];
#if MAX_SCRATCH < 4
#error "Scratch array defined too small "
#endif
/* Check for point at infinity */
if (ec_GFp_pt_is_inf_jac(px, py, pz) == MP_YES) {
/* Set r = pt at infinity by setting rz = 0 */
MP_CHECKOK(ec_GFp_pt_set_inf_jac(rx, ry, rz));
goto CLEANUP;
}
/* 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_add(t0, paz4, M, group->meth));
/* rz = 2 * py * pz */
MP_CHECKOK(group->meth->field_mul(py, pz, S, group->meth));
MP_CHECKOK(group->meth->field_add(S, S, rz, group->meth));
/* t0 = 2y^2 , t1 = 8y^4 */
MP_CHECKOK(group->meth->field_sqr(py, t0, group->meth));
MP_CHECKOK(group->meth->field_add(t0, t0, t0, group->meth));
MP_CHECKOK(group->meth->field_sqr(t0, t1, group->meth));
MP_CHECKOK(group->meth->field_add(t1, t1, t1, group->meth));
/* S = 4 * px * py^2 = 2 * px * t0 */
MP_CHECKOK(group->meth->field_mul(px, t0, S, group->meth));
MP_CHECKOK(group->meth->field_add(S, S, S, group->meth));
/* rx = M^2 - 2S */
MP_CHECKOK(group->meth->field_sqr(M, rx, group->meth));
MP_CHECKOK(group->meth->field_sub(rx, S, rx, group->meth));
MP_CHECKOK(group->meth->field_sub(rx, S, rx, group->meth));
/* ry = M * (S - rx) - t1 */
MP_CHECKOK(group->meth->field_sub(S, rx, S, group->meth));
MP_CHECKOK(group->meth->field_mul(S, M, ry, group->meth));
MP_CHECKOK(group->meth->field_sub(ry, t1, ry, group->meth));
/* ra*z^4 = 2*t1*(apz4) */
MP_CHECKOK(group->meth->field_mul(paz4, t1, raz4, group->meth));
MP_CHECKOK(group->meth->field_add(raz4, raz4, raz4, group->meth));
CLEANUP:
return res;
}
/* 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 Modified_Jacobian-affine coordinates. Assumes input is
* already field-encoded using field_enc, and returns output that is still
* field-encoded. */
mp_err
ec_GFp_pt_add_jm_aff(const mp_int *px, const mp_int *py, const mp_int *pz,
const mp_int *paz4, const mp_int *qx,
const mp_int *qy, mp_int *rx, mp_int *ry, mp_int *rz,
mp_int *raz4, mp_int scratch[], const ECGroup *group)
{
mp_err res = MP_OKAY;
mp_int *A, *B, *C, *D, *C2, *C3;
A = &scratch[0];
B = &scratch[1];
C = &scratch[2];
D = &scratch[3];
C2 = &scratch[4];
C3 = &scratch[5];
#if MAX_SCRATCH < 6
#error "Scratch array defined too small "
#endif
/* 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));
MP_CHECKOK(group->meth->field_sqr(rz, raz4, group->meth));
MP_CHECKOK(group->meth->field_sqr(raz4, raz4, group->meth));
MP_CHECKOK(group->meth->
field_mul(raz4, &group->curvea, raz4, group->meth));
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));
MP_CHECKOK(mp_copy(paz4, raz4));
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));
/*
* Additional checks for point equality and point at infinity
*/
if (mp_cmp(px, A) == 0 && mp_cmp(py, B) == 0) {
/* POINT_DOUBLE(P) */
MP_CHECKOK(ec_GFp_pt_dbl_jm(px, py, pz, paz4, rx, ry, rz, raz4,
scratch, group));
goto CLEANUP;
}
/* 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));
/* raz4 = a * rz^4 */
MP_CHECKOK(group->meth->field_sqr(rz, raz4, group->meth));
MP_CHECKOK(group->meth->field_sqr(raz4, raz4, group->meth));
MP_CHECKOK(group->meth->
field_mul(raz4, &group->curvea, raz4, group->meth));
CLEANUP:
return res;
}
/* Computes R = nP where R is (rx, ry) and P is the base point. Elliptic
* curve points P and R can be identical. Uses mixed Modified-Jacobian
* co-ordinates for doubling and Chudnovsky Jacobian coordinates for
* additions. Assumes input is already field-encoded using field_enc, and
* returns output that is still field-encoded. Uses 5-bit window NAF
* method (algorithm 11) for scalar-point multiplication from Brown,
* Hankerson, Lopez, Menezes. Software Implementation of the NIST Elliptic
* Curves Over Prime Fields. */
mp_err
ec_GFp_pt_mul_jm_wNAF(const mp_int *n, const mp_int *px, const mp_int *py,
mp_int *rx, mp_int *ry, const ECGroup *group,
int timing)
{
mp_err res = MP_OKAY;
mp_int precomp[16][2], rz, tpx, tpy, tpz;
mp_int raz4, tpaz4;
mp_int scratch[MAX_SCRATCH];
signed char *naf = NULL;
int i, orderBitSize;
int numDoubles, numAdds, extraDoubles, extraAdds;
MP_DIGITS(&rz) = 0;
MP_DIGITS(&raz4) = 0;
MP_DIGITS(&tpx) = 0;
MP_DIGITS(&tpy) = 0;
MP_DIGITS(&tpz) = 0;
MP_DIGITS(&tpaz4) = 0;
for (i = 0; i < 16; i++) {
MP_DIGITS(&precomp[i][0]) = 0;
MP_DIGITS(&precomp[i][1]) = 0;
}
for (i = 0; i < MAX_SCRATCH; i++) {
MP_DIGITS(&scratch[i]) = 0;
}
ARGCHK(group != NULL, MP_BADARG);
ARGCHK((n != NULL) && (px != NULL) && (py != NULL), MP_BADARG);
/* initialize precomputation table */
MP_CHECKOK(mp_init(&tpx, FLAG(n)));
MP_CHECKOK(mp_init(&tpy, FLAG(n)));
MP_CHECKOK(mp_init(&tpz, FLAG(n)));
MP_CHECKOK(mp_init(&tpaz4, FLAG(n)));
MP_CHECKOK(mp_init(&rz, FLAG(n)));
MP_CHECKOK(mp_init(&raz4, FLAG(n)));
for (i = 0; i < 16; i++) {
MP_CHECKOK(mp_init(&precomp[i][0], FLAG(n)));
MP_CHECKOK(mp_init(&precomp[i][1], FLAG(n)));
}
for (i = 0; i < MAX_SCRATCH; i++) {
MP_CHECKOK(mp_init(&scratch[i], FLAG(n)));
}
/* Set out[8] = P */
MP_CHECKOK(mp_copy(px, &precomp[8][0]));
MP_CHECKOK(mp_copy(py, &precomp[8][1]));
/* Set (tpx, tpy) = 2P */
MP_CHECKOK(group->
point_dbl(&precomp[8][0], &precomp[8][1], &tpx, &tpy,
group));
/* Set 3P, 5P, ..., 15P */
for (i = 8; i < 15; i++) {
MP_CHECKOK(group->
point_add(&precomp[i][0], &precomp[i][1], &tpx, &tpy,
&precomp[i + 1][0], &precomp[i + 1][1],
group));
}
/* Set -15P, -13P, ..., -P */
for (i = 0; i < 8; i++) {
MP_CHECKOK(mp_copy(&precomp[15 - i][0], &precomp[i][0]));
MP_CHECKOK(group->meth->
field_neg(&precomp[15 - i][1], &precomp[i][1],
group->meth));
}
/* R = inf */
MP_CHECKOK(ec_GFp_pt_set_inf_jac(rx, ry, &rz));
orderBitSize = mpl_significant_bits(&group->order);
/* Allocate memory for NAF */
#ifdef _KERNEL
naf = (signed char *) kmem_alloc((orderBitSize + 1), FLAG(n));
#else
naf = (signed char *) malloc(sizeof(signed char) * (orderBitSize + 1));
if (naf == NULL) {
res = MP_MEM;
goto CLEANUP;
}
#endif
/* Compute 5NAF */
ec_compute_wNAF(naf, orderBitSize, n, 5);
numAdds = 0;
numDoubles = orderBitSize;
/* wNAF method */
for (i = orderBitSize; i >= 0; i--) {
if (ec_GFp_pt_is_inf_jac(rx, ry, &rz) == MP_YES) {
numDoubles--;
}
/* R = 2R */
ec_GFp_pt_dbl_jm(rx, ry, &rz, &raz4, rx, ry, &rz,
&raz4, scratch, group);
if (naf[i] != 0) {
ec_GFp_pt_add_jm_aff(rx, ry, &rz, &raz4,
&precomp[(naf[i] + 15) / 2][0],
&precomp[(naf[i] + 15) / 2][1], rx, ry,
&rz, &raz4, scratch, group);
numAdds++;
}
}
/* extra operations to make timing less dependent on secrets */
if (timing) {
/* low-order bit of timing argument contains no entropy */
timing >>= 1;
MP_CHECKOK(ec_GFp_pt_set_inf_jac(&tpx, &tpy, &tpz));
mp_zero(&tpaz4);
/* Set the temp value to a non-infinite point */
ec_GFp_pt_add_jm_aff(&tpx, &tpy, &tpz, &tpaz4,
&precomp[8][0],
&precomp[8][1], &tpx, &tpy,
&tpz, &tpaz4, scratch, group);
/* two bits of extra adds */
extraAdds = timing & 0x3;
timing >>= 2;
/* Window size is 5, so the maximum number of additions is ceil(orderBitSize/5) */
/* This is the same as (orderBitSize + 4) / 5 */
for(i = numAdds; i <= (orderBitSize + 4) / 5 + extraAdds; i++) {
ec_GFp_pt_add_jm_aff(&tpx, &tpy, &tpz, &tpaz4,
&precomp[9 + (i % 3)][0],
&precomp[9 + (i % 3)][1], &tpx, &tpy,
&tpz, &tpaz4, scratch, group);
}
/* two bits of extra doubles */
extraDoubles = timing & 0x3;
timing >>= 2;
for(i = numDoubles; i <= orderBitSize + extraDoubles; i++) {
ec_GFp_pt_dbl_jm(&tpx, &tpy, &tpz, &tpaz4, &tpx, &tpy, &tpz,
&tpaz4, scratch, group);
}
}
/* convert result S to affine coordinates */
MP_CHECKOK(ec_GFp_pt_jac2aff(rx, ry, &rz, rx, ry, group));
CLEANUP:
for (i = 0; i < MAX_SCRATCH; i++) {
mp_clear(&scratch[i]);
}
for (i = 0; i < 16; i++) {
mp_clear(&precomp[i][0]);
mp_clear(&precomp[i][1]);
}
mp_clear(&tpx);
mp_clear(&tpy);
mp_clear(&tpz);
mp_clear(&tpaz4);
mp_clear(&rz);
mp_clear(&raz4);
#ifdef _KERNEL
kmem_free(naf, (orderBitSize + 1));
#else
free(naf);
#endif
return res;
}