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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"


00075 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;
            }
      
      if (!EC_POINT_is_on_curve(group, point, ctx)) /* test required by X9.62 */
            {
            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) goto err;

      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);
      }

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