openssl/crypto/ec/ec2_smpl.c

/* crypto/ec/ec2_smpl.c */
/* ====================================================================
 * Copyright 2002 Sun Microsystems, Inc. ALL RIGHTS RESERVED.
 *
 * The Elliptic Curve Public-Key Crypto Library (ECC Code) included
 * herein is developed by SUN MICROSYSTEMS, INC., and is contributed
 * to the OpenSSL project.
 *
 * The ECC Code is licensed pursuant to the OpenSSL open source
 * license provided below.
 *
 * The software is originally written by Sheueling Chang Shantz and
 * Douglas Stebila of Sun Microsystems Laboratories.
 *
 */
/* ====================================================================
 * Copyright (c) 1998-2005 The OpenSSL Project.  All rights reserved.
 *
 * Redistribution and use in source and binary forms, with or without
 * modification, are permitted provided that the following conditions
 * are met:
 *
 * 1. Redistributions of source code must retain the above copyright
 *    notice, this list of conditions and the following disclaimer.
 *
 * 2. Redistributions in binary form must reproduce the above copyright
 *    notice, this list of conditions and the following disclaimer in
 *    the documentation and/or other materials provided with the
 *    distribution.
 *
 * 3. All advertising materials mentioning features or use of this
 *    software must display the following acknowledgment:
 *    "This product includes software developed by the OpenSSL Project
 *    for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
 *
 * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
 *    endorse or promote products derived from this software without
 *    prior written permission. For written permission, please contact
 *    openssl-core@openssl.org.
 *
 * 5. Products derived from this software may not be called "OpenSSL"
 *    nor may "OpenSSL" appear in their names without prior written
 *    permission of the OpenSSL Project.
 *
 * 6. Redistributions of any form whatsoever must retain the following
 *    acknowledgment:
 *    "This product includes software developed by the OpenSSL Project
 *    for use in the OpenSSL Toolkit (http://www.openssl.org/)"
 *
 * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
 * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
 * PURPOSE ARE DISCLAIMED.  IN NO EVENT SHALL THE OpenSSL PROJECT OR
 * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
 * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
 * OF THE POSSIBILITY OF SUCH DAMAGE.
 * ====================================================================
 *
 * This product includes cryptographic software written by Eric Young
 * (eay@cryptsoft.com).  This product includes software written by Tim
 * Hudson (tjh@cryptsoft.com).
 *
 */

#include <openssl/err.h>

#include "ec_lcl.h"

const EC_METHOD *EC_GF2m_simple_method(void)
{
    static const EC_METHOD ret = {
        NID_X9_62_characteristic_two_field,
        ec_GF2m_simple_group_init,
        ec_GF2m_simple_group_finish,
        ec_GF2m_simple_group_clear_finish,
        ec_GF2m_simple_group_copy,
        ec_GF2m_simple_group_set_curve,
        ec_GF2m_simple_group_get_curve,
        ec_GF2m_simple_group_get_degree,
        ec_GF2m_simple_group_check_discriminant,
        ec_GF2m_simple_point_init,
        ec_GF2m_simple_point_finish,
        ec_GF2m_simple_point_clear_finish,
        ec_GF2m_simple_point_copy,
        ec_GF2m_simple_point_set_to_infinity,
        0 /* set_Jprojective_coordinates_GFp */ ,
        0 /* get_Jprojective_coordinates_GFp */ ,
        ec_GF2m_simple_point_set_affine_coordinates,
        ec_GF2m_simple_point_get_affine_coordinates,
        ec_GF2m_simple_set_compressed_coordinates,
        ec_GF2m_simple_point2oct,
        ec_GF2m_simple_oct2point,
        ec_GF2m_simple_add,
        ec_GF2m_simple_dbl,
        ec_GF2m_simple_invert,
        ec_GF2m_simple_is_at_infinity,
        ec_GF2m_simple_is_on_curve,
        ec_GF2m_simple_cmp,
        ec_GF2m_simple_make_affine,
        ec_GF2m_simple_points_make_affine,

        /*
         * the following three method functions are defined in ec2_mult.c
         */
        ec_GF2m_simple_mul,
        ec_GF2m_precompute_mult,
        ec_GF2m_have_precompute_mult,

        ec_GF2m_simple_field_mul,
        ec_GF2m_simple_field_sqr,
        ec_GF2m_simple_field_div,
        0 /* field_encode */ ,
        0 /* field_decode */ ,
        0                       /* field_set_to_one */
    };

    return &ret;
}

/*
 * Initialize a GF(2^m)-based EC_GROUP structure. Note that all other members
 * are handled by EC_GROUP_new.
 */
int ec_GF2m_simple_group_init(EC_GROUP *group)
{
    BN_init(&group->field);
    BN_init(&group->a);
    BN_init(&group->b);
    return 1;
}

/*
 * Free a GF(2^m)-based EC_GROUP structure. Note that all other members are
 * handled by EC_GROUP_free.
 */
void ec_GF2m_simple_group_finish(EC_GROUP *group)
{
    BN_free(&group->field);
    BN_free(&group->a);
    BN_free(&group->b);
}

/*
 * Clear and free a GF(2^m)-based EC_GROUP structure. Note that all other
 * members are handled by EC_GROUP_clear_free.
 */
void ec_GF2m_simple_group_clear_finish(EC_GROUP *group)
{
    BN_clear_free(&group->field);
    BN_clear_free(&group->a);
    BN_clear_free(&group->b);
    group->poly[0] = 0;
    group->poly[1] = 0;
    group->poly[2] = 0;
    group->poly[3] = 0;
    group->poly[4] = 0;
    group->poly[5] = -1;
}

/*
 * Copy a GF(2^m)-based EC_GROUP structure. Note that all other members are
 * handled by EC_GROUP_copy.
 */
int ec_GF2m_simple_group_copy(EC_GROUP *dest, const EC_GROUP *src)
{
    int i;
    if (!BN_copy(&dest->field, &src->field))
        return 0;
    if (!BN_copy(&dest->a, &src->a))
        return 0;
    if (!BN_copy(&dest->b, &src->b))
        return 0;
    dest->poly[0] = src->poly[0];
    dest->poly[1] = src->poly[1];
    dest->poly[2] = src->poly[2];
    dest->poly[3] = src->poly[3];
    dest->poly[4] = src->poly[4];
    dest->poly[5] = src->poly[5];
    if (bn_wexpand(&dest->a, (int)(dest->poly[0] + BN_BITS2 - 1) / BN_BITS2)
        == NULL)
        return 0;
    if (bn_wexpand(&dest->b, (int)(dest->poly[0] + BN_BITS2 - 1) / BN_BITS2)
        == NULL)
        return 0;
    for (i = dest->a.top; i < dest->a.dmax; i++)
        dest->a.d[i] = 0;
    for (i = dest->b.top; i < dest->b.dmax; i++)
        dest->b.d[i] = 0;
    return 1;
}

/* Set the curve parameters of an EC_GROUP structure. */
int ec_GF2m_simple_group_set_curve(EC_GROUP *group,
                                   const BIGNUM *p, const BIGNUM *a,
                                   const BIGNUM *b, BN_CTX *ctx)
{
    int ret = 0, i;

    /* group->field */
    if (!BN_copy(&group->field, p))
        goto err;
    i = BN_GF2m_poly2arr(&group->field, group->poly, 6) - 1;
    if ((i != 5) && (i != 3)) {
        ECerr(EC_F_EC_GF2M_SIMPLE_GROUP_SET_CURVE, EC_R_UNSUPPORTED_FIELD);
        goto err;
    }

    /* group->a */
    if (!BN_GF2m_mod_arr(&group->a, a, group->poly))
        goto err;
    if (bn_wexpand(&group->a, (int)(group->poly[0] + BN_BITS2 - 1) / BN_BITS2)
        == NULL)
        goto err;
    for (i = group->a.top; i < group->a.dmax; i++)
        group->a.d[i] = 0;

    /* group->b */
    if (!BN_GF2m_mod_arr(&group->b, b, group->poly))
        goto err;
    if (bn_wexpand(&group->b, (int)(group->poly[0] + BN_BITS2 - 1) / BN_BITS2)
        == NULL)
        goto err;
    for (i = group->b.top; i < group->b.dmax; i++)
        group->b.d[i] = 0;

    ret = 1;
 err:
    return ret;
}

/*
 * Get the curve parameters of an EC_GROUP structure. If p, a, or b are NULL
 * then there values will not be set but the method will return with success.
 */
int ec_GF2m_simple_group_get_curve(const EC_GROUP *group, BIGNUM *p,
                                   BIGNUM *a, BIGNUM *b, BN_CTX *ctx)
{
    int ret = 0;

    if (p != NULL) {
        if (!BN_copy(p, &group->field))
            return 0;
    }

    if (a != NULL) {
        if (!BN_copy(a, &group->a))
            goto err;
    }

    if (b != NULL) {
        if (!BN_copy(b, &group->b))
            goto err;
    }

    ret = 1;

 err:
    return ret;
}

/*
 * Gets the degree of the field.  For a curve over GF(2^m) this is the value
 * m.
 */
int ec_GF2m_simple_group_get_degree(const EC_GROUP *group)
{
    return BN_num_bits(&group->field) - 1;
}

/*
 * Checks the discriminant of the curve. y^2 + x*y = x^3 + a*x^2 + b is an
 * elliptic curve <=> b != 0 (mod p)
 */
int ec_GF2m_simple_group_check_discriminant(const EC_GROUP *group,
                                            BN_CTX *ctx)
{
    int ret = 0;
    BIGNUM *b;
    BN_CTX *new_ctx = NULL;

    if (ctx == NULL) {
        ctx = new_ctx = BN_CTX_new();
        if (ctx == NULL) {
            ECerr(EC_F_EC_GF2M_SIMPLE_GROUP_CHECK_DISCRIMINANT,
                  ERR_R_MALLOC_FAILURE);
            goto err;
        }
    }
    BN_CTX_start(ctx);
    b = BN_CTX_get(ctx);
    if (b == NULL)
        goto err;

    if (!BN_GF2m_mod_arr(b, &group->b, group->poly))
        goto err;

    /*
     * check the discriminant: y^2 + x*y = x^3 + a*x^2 + b is an elliptic
     * curve <=> b != 0 (mod p)
     */
    if (BN_is_zero(b))
        goto err;

    ret = 1;

 err:
    if (ctx != NULL)
        BN_CTX_end(ctx);
    if (new_ctx != NULL)
        BN_CTX_free(new_ctx);
    return ret;
}

/* Initializes an EC_POINT. */
int ec_GF2m_simple_point_init(EC_POINT *point)
{
    BN_init(&point->X);
    BN_init(&point->Y);
    BN_init(&point->Z);
    return 1;
}

/* Frees an EC_POINT. */
void ec_GF2m_simple_point_finish(EC_POINT *point)
{
    BN_free(&point->X);
    BN_free(&point->Y);
    BN_free(&point->Z);
}

/* Clears and frees an EC_POINT. */
void ec_GF2m_simple_point_clear_finish(EC_POINT *point)
{
    BN_clear_free(&point->X);
    BN_clear_free(&point->Y);
    BN_clear_free(&point->Z);
    point->Z_is_one = 0;
}

/*
 * Copy the contents of one EC_POINT into another.  Assumes dest is
 * initialized.
 */
int ec_GF2m_simple_point_copy(EC_POINT *dest, const EC_POINT *src)
{
    if (!BN_copy(&dest->X, &src->X))
        return 0;
    if (!BN_copy(&dest->Y, &src->Y))
        return 0;
    if (!BN_copy(&dest->Z, &src->Z))
        return 0;
    dest->Z_is_one = src->Z_is_one;

    return 1;
}

/*
 * Set an EC_POINT to the point at infinity. A point at infinity is
 * represented by having Z=0.
 */
int ec_GF2m_simple_point_set_to_infinity(const EC_GROUP *group,
                                         EC_POINT *point)
{
    point->Z_is_one = 0;
    BN_zero(&point->Z);
    return 1;
}

/*
 * Set the coordinates of an EC_POINT using affine coordinates. Note that
 * the simple implementation only uses affine coordinates.
 */
int ec_GF2m_simple_point_set_affine_coordinates(const EC_GROUP *group,
                                                EC_POINT *point,
                                                const BIGNUM *x,
                                                const BIGNUM *y, BN_CTX *ctx)
{
    int ret = 0;
    if (x == NULL || y == NULL) {
        ECerr(EC_F_EC_GF2M_SIMPLE_POINT_SET_AFFINE_COORDINATES,
              ERR_R_PASSED_NULL_PARAMETER);
        return 0;
    }

    if (!BN_copy(&point->X, x))
        goto err;
    BN_set_negative(&point->X, 0);
    if (!BN_copy(&point->Y, y))
        goto err;
    BN_set_negative(&point->Y, 0);
    if (!BN_copy(&point->Z, BN_value_one()))
        goto err;
    BN_set_negative(&point->Z, 0);
    point->Z_is_one = 1;
    ret = 1;

 err:
    return ret;
}

/*
 * Gets the affine coordinates of an EC_POINT. Note that the simple
 * implementation only uses affine coordinates.
 */
int ec_GF2m_simple_point_get_affine_coordinates(const EC_GROUP *group,
                                                const EC_POINT *point,
                                                BIGNUM *x, BIGNUM *y,
                                                BN_CTX *ctx)
{
    int ret = 0;

    if (EC_POINT_is_at_infinity(group, point)) {
        ECerr(EC_F_EC_GF2M_SIMPLE_POINT_GET_AFFINE_COORDINATES,
              EC_R_POINT_AT_INFINITY);
        return 0;
    }

    if (BN_cmp(&point->Z, BN_value_one())) {
        ECerr(EC_F_EC_GF2M_SIMPLE_POINT_GET_AFFINE_COORDINATES,
              ERR_R_SHOULD_NOT_HAVE_BEEN_CALLED);
        return 0;
    }
    if (x != NULL) {
        if (!BN_copy(x, &point->X))
            goto err;
        BN_set_negative(x, 0);
    }
    if (y != NULL) {
        if (!BN_copy(y, &point->Y))
            goto err;
        BN_set_negative(y, 0);
    }
    ret = 1;

 err:
    return ret;
}

/*-
 * Calculates and sets the affine coordinates of an EC_POINT from the given
 * compressed coordinates.  Uses algorithm 2.3.4 of SEC 1.
 * Note that the simple implementation only uses affine coordinates.
 *
 * The method is from the following publication:
 *
 *     Harper, Menezes, Vanstone:
 *     "Public-Key Cryptosystems with Very Small Key Lengths",
 *     EUROCRYPT '92, Springer-Verlag LNCS 658,
 *     published February 1993
 *
 * US Patents 6,141,420 and 6,618,483 (Vanstone, Mullin, Agnew) describe
 * the same method, but claim no priority date earlier than July 29, 1994
 * (and additionally fail to cite the EUROCRYPT '92 publication as prior art).
 */
int ec_GF2m_simple_set_compressed_coordinates(const EC_GROUP *group,
                                              EC_POINT *point,
                                              const BIGNUM *x_, int y_bit,
                                              BN_CTX *ctx)
{
    BN_CTX *new_ctx = NULL;
    BIGNUM *tmp, *x, *y, *z;
    int ret = 0, z0;

    /* clear error queue */
    ERR_clear_error();

    if (ctx == NULL) {
        ctx = new_ctx = BN_CTX_new();
        if (ctx == NULL)
            return 0;
    }

    y_bit = (y_bit != 0) ? 1 : 0;

    BN_CTX_start(ctx);
    tmp = BN_CTX_get(ctx);
    x = BN_CTX_get(ctx);
    y = BN_CTX_get(ctx);
    z = BN_CTX_get(ctx);
    if (z == NULL)
        goto err;

    if (!BN_GF2m_mod_arr(x, x_, group->poly))
        goto err;
    if (BN_is_zero(x)) {
        if (!BN_GF2m_mod_sqrt_arr(y, &group->b, group->poly, ctx))
            goto err;
    } else {
        if (!group->meth->field_sqr(group, tmp, x, ctx))
            goto err;
        if (!group->meth->field_div(group, tmp, &group->b, tmp, ctx))
            goto err;
        if (!BN_GF2m_add(tmp, &group->a, tmp))
            goto err;
        if (!BN_GF2m_add(tmp, x, tmp))
            goto err;
        if (!BN_GF2m_mod_solve_quad_arr(z, tmp, group->poly, ctx)) {
            unsigned long err = ERR_peek_last_error();

            if (ERR_GET_LIB(err) == ERR_LIB_BN
                && ERR_GET_REASON(err) == BN_R_NO_SOLUTION) {
                ERR_clear_error();
                ECerr(EC_F_EC_GF2M_SIMPLE_SET_COMPRESSED_COORDINATES,
                      EC_R_INVALID_COMPRESSED_POINT);
            } else
                ECerr(EC_F_EC_GF2M_SIMPLE_SET_COMPRESSED_COORDINATES,
                      ERR_R_BN_LIB);
            goto err;
        }
        z0 = (BN_is_odd(z)) ? 1 : 0;
        if (!group->meth->field_mul(group, y, x, z, ctx))
            goto err;
        if (z0 != y_bit) {
            if (!BN_GF2m_add(y, y, x))
                goto err;
        }
    }

    if (!EC_POINT_set_affine_coordinates_GF2m(group, point, x, y, ctx))
        goto err;

    ret = 1;

 err:
    BN_CTX_end(ctx);
    if (new_ctx != NULL)
        BN_CTX_free(new_ctx);
    return ret;
}

/*
 * Converts an EC_POINT to an octet string. If buf is NULL, the encoded
 * length will be returned. If the length len of buf is smaller than required
 * an error will be returned.
 */
size_t ec_GF2m_simple_point2oct(const EC_GROUP *group, const EC_POINT *point,
                                point_conversion_form_t form,
                                unsigned char *buf, size_t len, BN_CTX *ctx)
{
    size_t ret;
    BN_CTX *new_ctx = NULL;
    int used_ctx = 0;
    BIGNUM *x, *y, *yxi;
    size_t field_len, i, skip;

    if ((form != POINT_CONVERSION_COMPRESSED)
        && (form != POINT_CONVERSION_UNCOMPRESSED)
        && (form != POINT_CONVERSION_HYBRID)) {
        ECerr(EC_F_EC_GF2M_SIMPLE_POINT2OCT, EC_R_INVALID_FORM);
        goto err;
    }

    if (EC_POINT_is_at_infinity(group, point)) {
        /* encodes to a single 0 octet */
        if (buf != NULL) {
            if (len < 1) {
                ECerr(EC_F_EC_GF2M_SIMPLE_POINT2OCT, EC_R_BUFFER_TOO_SMALL);
                return 0;
            }
            buf[0] = 0;
        }
        return 1;
    }

    /* ret := required output buffer length */
    field_len = (EC_GROUP_get_degree(group) + 7) / 8;
    ret =
        (form ==
         POINT_CONVERSION_COMPRESSED) ? 1 + field_len : 1 + 2 * field_len;

    /* if 'buf' is NULL, just return required length */
    if (buf != NULL) {
        if (len < ret) {
            ECerr(EC_F_EC_GF2M_SIMPLE_POINT2OCT, EC_R_BUFFER_TOO_SMALL);
            goto err;
        }

        if (ctx == NULL) {
            ctx = new_ctx = BN_CTX_new();
            if (ctx == NULL)
                return 0;
        }

        BN_CTX_start(ctx);
        used_ctx = 1;
        x = BN_CTX_get(ctx);
        y = BN_CTX_get(ctx);
        yxi = BN_CTX_get(ctx);
        if (yxi == NULL)
            goto err;

        if (!EC_POINT_get_affine_coordinates_GF2m(group, point, x, y, ctx))
            goto err;

        buf[0] = form;
        if ((form != POINT_CONVERSION_UNCOMPRESSED) && !BN_is_zero(x)) {
            if (!group->meth->field_div(group, yxi, y, x, ctx))
                goto err;
            if (BN_is_odd(yxi))
                buf[0]++;
        }

        i = 1;

        skip = field_len - BN_num_bytes(x);
        if (skip > field_len) {
            ECerr(EC_F_EC_GF2M_SIMPLE_POINT2OCT, ERR_R_INTERNAL_ERROR);
            goto err;
        }
        while (skip > 0) {
            buf[i++] = 0;
            skip--;
        }
        skip = BN_bn2bin(x, buf + i);
        i += skip;
        if (i != 1 + field_len) {
            ECerr(EC_F_EC_GF2M_SIMPLE_POINT2OCT, ERR_R_INTERNAL_ERROR);
            goto err;
        }

        if (form == POINT_CONVERSION_UNCOMPRESSED
            || form == POINT_CONVERSION_HYBRID) {
            skip = field_len - BN_num_bytes(y);
            if (skip > field_len) {
                ECerr(EC_F_EC_GF2M_SIMPLE_POINT2OCT, ERR_R_INTERNAL_ERROR);
                goto err;
            }
            while (skip > 0) {
                buf[i++] = 0;
                skip--;
            }
            skip = BN_bn2bin(y, buf + i);
            i += skip;
        }

        if (i != ret) {
            ECerr(EC_F_EC_GF2M_SIMPLE_POINT2OCT, ERR_R_INTERNAL_ERROR);
            goto err;
        }
    }

    if (used_ctx)
        BN_CTX_end(ctx);
    if (new_ctx != NULL)
        BN_CTX_free(new_ctx);
    return ret;

 err:
    if (used_ctx)
        BN_CTX_end(ctx);
    if (new_ctx != NULL)
        BN_CTX_free(new_ctx);
    return 0;
}

/*
 * Converts an octet string representation to an EC_POINT. Note that the
 * simple implementation only uses affine coordinates.
 */
int ec_GF2m_simple_oct2point(const EC_GROUP *group, EC_POINT *point,
                             const unsigned char *buf, size_t len,
                             BN_CTX *ctx)
{
    point_conversion_form_t form;
    int y_bit;
    BN_CTX *new_ctx = NULL;
    BIGNUM *x, *y, *yxi;
    size_t field_len, enc_len;
    int ret = 0;

    if (len == 0) {
        ECerr(EC_F_EC_GF2M_SIMPLE_OCT2POINT, EC_R_BUFFER_TOO_SMALL);
        return 0;
    }
    form = buf[0];
    y_bit = form & 1;
    form = form & ~1U;
    if ((form != 0) && (form != POINT_CONVERSION_COMPRESSED)
        && (form != POINT_CONVERSION_UNCOMPRESSED)
        && (form != POINT_CONVERSION_HYBRID)) {
        ECerr(EC_F_EC_GF2M_SIMPLE_OCT2POINT, EC_R_INVALID_ENCODING);
        return 0;
    }
    if ((form == 0 || form == POINT_CONVERSION_UNCOMPRESSED) && y_bit) {
        ECerr(EC_F_EC_GF2M_SIMPLE_OCT2POINT, EC_R_INVALID_ENCODING);
        return 0;
    }

    if (form == 0) {
        if (len != 1) {
            ECerr(EC_F_EC_GF2M_SIMPLE_OCT2POINT, EC_R_INVALID_ENCODING);
            return 0;
        }

        return EC_POINT_set_to_infinity(group, point);
    }

    field_len = (EC_GROUP_get_degree(group) + 7) / 8;
    enc_len =
        (form ==
         POINT_CONVERSION_COMPRESSED) ? 1 + field_len : 1 + 2 * field_len;

    if (len != enc_len) {
        ECerr(EC_F_EC_GF2M_SIMPLE_OCT2POINT, EC_R_INVALID_ENCODING);
        return 0;
    }

    if (ctx == NULL) {
        ctx = new_ctx = BN_CTX_new();
        if (ctx == NULL)
            return 0;
    }

    BN_CTX_start(ctx);
    x = BN_CTX_get(ctx);
    y = BN_CTX_get(ctx);
    yxi = BN_CTX_get(ctx);
    if (yxi == NULL)
        goto err;

    if (!BN_bin2bn(buf + 1, field_len, x))
        goto err;
    if (BN_ucmp(x, &group->field) >= 0) {
        ECerr(EC_F_EC_GF2M_SIMPLE_OCT2POINT, EC_R_INVALID_ENCODING);
        goto err;
    }

    if (form == POINT_CONVERSION_COMPRESSED) {
        if (!EC_POINT_set_compressed_coordinates_GF2m
            (group, point, x, y_bit, ctx))
            goto err;
    } else {
        if (!BN_bin2bn(buf + 1 + field_len, field_len, y))
            goto err;
        if (BN_ucmp(y, &group->field) >= 0) {
            ECerr(EC_F_EC_GF2M_SIMPLE_OCT2POINT, EC_R_INVALID_ENCODING);
            goto err;
        }
        if (form == POINT_CONVERSION_HYBRID) {
            if (!group->meth->field_div(group, yxi, y, x, ctx))
                goto err;
            if (y_bit != BN_is_odd(yxi)) {
                ECerr(EC_F_EC_GF2M_SIMPLE_OCT2POINT, EC_R_INVALID_ENCODING);
                goto err;
            }
        }

        if (!EC_POINT_set_affine_coordinates_GF2m(group, point, x, y, ctx))
            goto err;
    }

    /* test required by X9.62 */
    if (EC_POINT_is_on_curve(group, point, ctx) <= 0) {
        ECerr(EC_F_EC_GF2M_SIMPLE_OCT2POINT, EC_R_POINT_IS_NOT_ON_CURVE);
        goto err;
    }

    ret = 1;

 err:
    BN_CTX_end(ctx);
    if (new_ctx != NULL)
        BN_CTX_free(new_ctx);
    return ret;
}

/*
 * Computes a + b and stores the result in r.  r could be a or b, a could be
 * b. Uses algorithm A.10.2 of IEEE P1363.
 */
int ec_GF2m_simple_add(const EC_GROUP *group, EC_POINT *r, const EC_POINT *a,
                       const EC_POINT *b, BN_CTX *ctx)
{
    BN_CTX *new_ctx = NULL;
    BIGNUM *x0, *y0, *x1, *y1, *x2, *y2, *s, *t;
    int ret = 0;

    if (EC_POINT_is_at_infinity(group, a)) {
        if (!EC_POINT_copy(r, b))
            return 0;
        return 1;
    }

    if (EC_POINT_is_at_infinity(group, b)) {
        if (!EC_POINT_copy(r, a))
            return 0;
        return 1;
    }

    if (ctx == NULL) {
        ctx = new_ctx = BN_CTX_new();
        if (ctx == NULL)
            return 0;
    }

    BN_CTX_start(ctx);
    x0 = BN_CTX_get(ctx);
    y0 = BN_CTX_get(ctx);
    x1 = BN_CTX_get(ctx);
    y1 = BN_CTX_get(ctx);
    x2 = BN_CTX_get(ctx);
    y2 = BN_CTX_get(ctx);
    s = BN_CTX_get(ctx);
    t = BN_CTX_get(ctx);
    if (t == NULL)
        goto err;

    if (a->Z_is_one) {
        if (!BN_copy(x0, &a->X))
            goto err;
        if (!BN_copy(y0, &a->Y))
            goto err;
    } else {
        if (!EC_POINT_get_affine_coordinates_GF2m(group, a, x0, y0, ctx))
            goto err;
    }
    if (b->Z_is_one) {
        if (!BN_copy(x1, &b->X))
            goto err;
        if (!BN_copy(y1, &b->Y))
            goto err;
    } else {
        if (!EC_POINT_get_affine_coordinates_GF2m(group, b, x1, y1, ctx))
            goto err;
    }

    if (BN_GF2m_cmp(x0, x1)) {
        if (!BN_GF2m_add(t, x0, x1))
            goto err;
        if (!BN_GF2m_add(s, y0, y1))
            goto err;
        if (!group->meth->field_div(group, s, s, t, ctx))
            goto err;
        if (!group->meth->field_sqr(group, x2, s, ctx))
            goto err;
        if (!BN_GF2m_add(x2, x2, &group->a))
            goto err;
        if (!BN_GF2m_add(x2, x2, s))
            goto err;
        if (!BN_GF2m_add(x2, x2, t))
            goto err;
    } else {
        if (BN_GF2m_cmp(y0, y1) || BN_is_zero(x1)) {
            if (!EC_POINT_set_to_infinity(group, r))
                goto err;
            ret = 1;
            goto err;
        }
        if (!group->meth->field_div(group, s, y1, x1, ctx))
            goto err;
        if (!BN_GF2m_add(s, s, x1))
            goto err;

        if (!group->meth->field_sqr(group, x2, s, ctx))
            goto err;
        if (!BN_GF2m_add(x2, x2, s))
            goto err;
        if (!BN_GF2m_add(x2, x2, &group->a))
            goto err;
    }

    if (!BN_GF2m_add(y2, x1, x2))
        goto err;
    if (!group->meth->field_mul(group, y2, y2, s, ctx))
        goto err;
    if (!BN_GF2m_add(y2, y2, x2))
        goto err;
    if (!BN_GF2m_add(y2, y2, y1))
        goto err;

    if (!EC_POINT_set_affine_coordinates_GF2m(group, r, x2, y2, ctx))
        goto err;

    ret = 1;

 err:
    BN_CTX_end(ctx);
    if (new_ctx != NULL)
        BN_CTX_free(new_ctx);
    return ret;
}

/*
 * Computes 2 * a and stores the result in r.  r could be a. Uses algorithm
 * A.10.2 of IEEE P1363.
 */
int ec_GF2m_simple_dbl(const EC_GROUP *group, EC_POINT *r, const EC_POINT *a,
                       BN_CTX *ctx)
{
    return ec_GF2m_simple_add(group, r, a, a, ctx);
}

int ec_GF2m_simple_invert(const EC_GROUP *group, EC_POINT *point, BN_CTX *ctx)
{
    if (EC_POINT_is_at_infinity(group, point) || BN_is_zero(&point->Y))
        /* point is its own inverse */
        return 1;

    if (!EC_POINT_make_affine(group, point, ctx))
        return 0;
    return BN_GF2m_add(&point->Y, &point->X, &point->Y);
}

/* Indicates whether the given point is the point at infinity. */
int ec_GF2m_simple_is_at_infinity(const EC_GROUP *group,
                                  const EC_POINT *point)
{
    return BN_is_zero(&point->Z);
}

/*-
 * Determines whether the given EC_POINT is an actual point on the curve defined
 * in the EC_GROUP.  A point is valid if it satisfies the Weierstrass equation:
 *      y^2 + x*y = x^3 + a*x^2 + b.
 */
int ec_GF2m_simple_is_on_curve(const EC_GROUP *group, const EC_POINT *point,
                               BN_CTX *ctx)
{
    int ret = -1;
    BN_CTX *new_ctx = NULL;
    BIGNUM *lh, *y2;
    int (*field_mul) (const EC_GROUP *, BIGNUM *, const BIGNUM *,
                      const BIGNUM *, BN_CTX *);
    int (*field_sqr) (const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);

    if (EC_POINT_is_at_infinity(group, point))
        return 1;

    field_mul = group->meth->field_mul;
    field_sqr = group->meth->field_sqr;

    /* only support affine coordinates */
    if (!point->Z_is_one)
        return -1;

    if (ctx == NULL) {
        ctx = new_ctx = BN_CTX_new();
        if (ctx == NULL)
            return -1;
    }

    BN_CTX_start(ctx);
    y2 = BN_CTX_get(ctx);
    lh = BN_CTX_get(ctx);
    if (lh == NULL)
        goto err;

    /*-
     * We have a curve defined by a Weierstrass equation
     *      y^2 + x*y = x^3 + a*x^2 + b.
     *  <=> x^3 + a*x^2 + x*y + b + y^2 = 0
     *  <=> ((x + a) * x + y ) * x + b + y^2 = 0
     */
    if (!BN_GF2m_add(lh, &point->X, &group->a))
        goto err;
    if (!field_mul(group, lh, lh, &point->X, ctx))
        goto err;
    if (!BN_GF2m_add(lh, lh, &point->Y))
        goto err;
    if (!field_mul(group, lh, lh, &point->X, ctx))
        goto err;
    if (!BN_GF2m_add(lh, lh, &group->b))
        goto err;
    if (!field_sqr(group, y2, &point->Y, ctx))
        goto err;
    if (!BN_GF2m_add(lh, lh, y2))
        goto err;
    ret = BN_is_zero(lh);
 err:
    if (ctx)
        BN_CTX_end(ctx);
    if (new_ctx)
        BN_CTX_free(new_ctx);
    return ret;
}

/*-
 * Indicates whether two points are equal.
 * Return values:
 *  -1   error
 *   0   equal (in affine coordinates)
 *   1   not equal
 */
int ec_GF2m_simple_cmp(const EC_GROUP *group, const EC_POINT *a,
                       const EC_POINT *b, BN_CTX *ctx)
{
    BIGNUM *aX, *aY, *bX, *bY;
    BN_CTX *new_ctx = NULL;
    int ret = -1;

    if (EC_POINT_is_at_infinity(group, a)) {
        return EC_POINT_is_at_infinity(group, b) ? 0 : 1;
    }

    if (EC_POINT_is_at_infinity(group, b))
        return 1;

    if (a->Z_is_one && b->Z_is_one) {
        return ((BN_cmp(&a->X, &b->X) == 0)
                && BN_cmp(&a->Y, &b->Y) == 0) ? 0 : 1;
    }

    if (ctx == NULL) {
        ctx = new_ctx = BN_CTX_new();
        if (ctx == NULL)
            return -1;
    }

    BN_CTX_start(ctx);
    aX = BN_CTX_get(ctx);
    aY = BN_CTX_get(ctx);
    bX = BN_CTX_get(ctx);
    bY = BN_CTX_get(ctx);
    if (bY == NULL)
        goto err;

    if (!EC_POINT_get_affine_coordinates_GF2m(group, a, aX, aY, ctx))
        goto err;
    if (!EC_POINT_get_affine_coordinates_GF2m(group, b, bX, bY, ctx))
        goto err;
    ret = ((BN_cmp(aX, bX) == 0) && BN_cmp(aY, bY) == 0) ? 0 : 1;

 err:
    if (ctx)
        BN_CTX_end(ctx);
    if (new_ctx)
        BN_CTX_free(new_ctx);
    return ret;
}

/* Forces the given EC_POINT to internally use affine coordinates. */
int ec_GF2m_simple_make_affine(const EC_GROUP *group, EC_POINT *point,
                               BN_CTX *ctx)
{
    BN_CTX *new_ctx = NULL;
    BIGNUM *x, *y;
    int ret = 0;

    if (point->Z_is_one || EC_POINT_is_at_infinity(group, point))
        return 1;

    if (ctx == NULL) {
        ctx = new_ctx = BN_CTX_new();
        if (ctx == NULL)
            return 0;
    }

    BN_CTX_start(ctx);
    x = BN_CTX_get(ctx);
    y = BN_CTX_get(ctx);
    if (y == NULL)
        goto err;

    if (!EC_POINT_get_affine_coordinates_GF2m(group, point, x, y, ctx))
        goto err;
    if (!BN_copy(&point->X, x))
        goto err;
    if (!BN_copy(&point->Y, y))
        goto err;
    if (!BN_one(&point->Z))
        goto err;

    ret = 1;

 err:
    if (ctx)
        BN_CTX_end(ctx);
    if (new_ctx)
        BN_CTX_free(new_ctx);
    return ret;
}

/*
 * Forces each of the EC_POINTs in the given array to use affine coordinates.
 */
int ec_GF2m_simple_points_make_affine(const EC_GROUP *group, size_t num,
                                      EC_POINT *points[], BN_CTX *ctx)
{
    size_t i;

    for (i = 0; i < num; i++) {
        if (!group->meth->make_affine(group, points[i], ctx))
            return 0;
    }

    return 1;
}

/* Wrapper to simple binary polynomial field multiplication implementation. */
int ec_GF2m_simple_field_mul(const EC_GROUP *group, BIGNUM *r,
                             const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx)
{
    return BN_GF2m_mod_mul_arr(r, a, b, group->poly, ctx);
}

/* Wrapper to simple binary polynomial field squaring implementation. */
int ec_GF2m_simple_field_sqr(const EC_GROUP *group, BIGNUM *r,
                             const BIGNUM *a, BN_CTX *ctx)
{
    return BN_GF2m_mod_sqr_arr(r, a, group->poly, ctx);
}

/* Wrapper to simple binary polynomial field division implementation. */
int ec_GF2m_simple_field_div(const EC_GROUP *group, BIGNUM *r,
                             const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx)
{
    return BN_GF2m_mod_div(r, a, b, &group->field, ctx);
}