path: root/nss/lib/freebl/ecl/ecp_jac.c

/* 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
 * file, You can obtain one at http://mozilla.org/MPL/2.0/. */

#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;
}