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/* This Source Code Form is subject to the terms of the Mozilla Public
* License, v. 2.0. If a copy of the MPL was not distributed with this
#include "ecp.h"
#include "mplogic.h"
#include <stdlib.h>
#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));
MP_CHECKOK(mp_init(&z2));
MP_CHECKOK(mp_init(&z3));
/* 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));
MP_CHECKOK(mp_init(&B));
MP_CHECKOK(mp_init(&C));
MP_CHECKOK(mp_init(&D));
MP_CHECKOK(mp_init(&C2));
MP_CHECKOK(mp_init(&C3));
/* 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));
if (mp_cmp_z(&C) == 0) {
/* P == Q or P == -Q */
if (mp_cmp_z(&D) == 0) {
/* P == Q */
/* It is cheaper to double (qx, qy, 1) than (px, py, pz). */
MP_DIGIT(&D, 0) = 1; /* Set D to 1. */
MP_CHECKOK(ec_GFp_pt_dbl_jac(qx, qy, &D, rx, ry, rz, group));
} else {
/* P == -Q */
MP_CHECKOK(ec_GFp_pt_set_inf_jac(rx, ry, rz));
}
goto CLEANUP;
}
/* 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));
MP_CHECKOK(mp_init(&t1));
MP_CHECKOK(mp_init(&M));
MP_CHECKOK(mp_init(&S));
/* P == inf or P == -P */
if (ec_GFp_pt_is_inf_jac(px, py, pz) == MP_YES || mp_cmp_z(py) == 0) {
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_SIGN(&group->curvea) == MP_NEG &&
MP_USED(&group->curvea) == 1 &&
MP_DIGIT(&group->curvea, 0) == 3) {
/* 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;
unsigned 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]));
MP_CHECKOK(mp_init(&precomp[i][j][1]));
}
}
/* 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));
MP_CHECKOK(ec_GFp_pt_set_inf_jac(rx, ry, &rz));
for (i = d; i-- > 0;) {
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;
}