ecp_jm.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 "ecl-priv.h"
#include "mplogic.h"
#include <stdlib.h>

#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.
 *
 */
static 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. */
static 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));

    /* Check P == Q */
    if (mp_cmp(A, px) == 0) {
        if (mp_cmp(B, py) == 0) {
            /* If Px == Qx && Py == Qy, double P. */
            return ec_GFp_pt_dbl_jm(px, py, pz, paz4, rx, ry, rz, raz4,
                                    scratch, group);
        }
        /* If Px == Qx && Py != Qy, return point at infinity. */
        return ec_GFp_pt_set_inf_jac(rx, ry, rz);
    }

    /* 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)
{
    mp_err res = MP_OKAY;
    mp_int precomp[16][2], rz, tpx, tpy;
    mp_int raz4;
    mp_int scratch[MAX_SCRATCH];
    signed char *naf = NULL;
    int i, orderBitSize;

    MP_DIGITS(&rz) = 0;
    MP_DIGITS(&raz4) = 0;
    MP_DIGITS(&tpx) = 0;
    MP_DIGITS(&tpy) = 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));
    MP_CHECKOK(mp_init(&tpy));
    ;
    MP_CHECKOK(mp_init(&rz));
    MP_CHECKOK(mp_init(&raz4));

    for (i = 0; i < 16; i++) {
        MP_CHECKOK(mp_init(&precomp[i][0]));
        MP_CHECKOK(mp_init(&precomp[i][1]));
    }
    for (i = 0; i < MAX_SCRATCH; i++) {
        MP_CHECKOK(mp_init(&scratch[i]));
    }

    /* 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 */
    naf = (signed char *)malloc(sizeof(signed char) * (orderBitSize + 1));
    if (naf == NULL) {
        res = MP_MEM;
        goto CLEANUP;
    }

    /* Compute 5NAF */
    ec_compute_wNAF(naf, orderBitSize, n, 5);

    /* wNAF method */
    for (i = orderBitSize; i >= 0; i--) {
        /* 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);
        }
    }

    /* 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(&rz);
    mp_clear(&raz4);
    free(naf);
    return res;
}