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ecp_smpl.c

/* crypto/ec/ecp_smpl.c */
/* Includes code written by Lenka Fibikova <fibikova@exp-math.uni-essen.de>
 * for the OpenSSL project. 
 * Includes code written by Bodo Moeller for the OpenSSL project.
*/
/* ====================================================================
 * Copyright (c) 1998-2002 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).
 *
 */
/* ====================================================================
 * Copyright 2002 Sun Microsystems, Inc. ALL RIGHTS RESERVED.
 * Portions of this software developed by SUN MICROSYSTEMS, INC.,
 * and contributed to the OpenSSL project.
 */

#include <openssl/err.h>
#include <openssl/symhacks.h>

#include "ec_lcl.h"

const EC_METHOD *EC_GFp_simple_method(void)
      {
      static const EC_METHOD ret = {
            NID_X9_62_prime_field,
            ec_GFp_simple_group_init,
            ec_GFp_simple_group_finish,
            ec_GFp_simple_group_clear_finish,
            ec_GFp_simple_group_copy,
            ec_GFp_simple_group_set_curve,
            ec_GFp_simple_group_get_curve,
            ec_GFp_simple_group_get_degree,
            ec_GFp_simple_group_check_discriminant,
            ec_GFp_simple_point_init,
            ec_GFp_simple_point_finish,
            ec_GFp_simple_point_clear_finish,
            ec_GFp_simple_point_copy,
            ec_GFp_simple_point_set_to_infinity,
            ec_GFp_simple_set_Jprojective_coordinates_GFp,
            ec_GFp_simple_get_Jprojective_coordinates_GFp,
            ec_GFp_simple_point_set_affine_coordinates,
            ec_GFp_simple_point_get_affine_coordinates,
            ec_GFp_simple_set_compressed_coordinates,
            ec_GFp_simple_point2oct,
            ec_GFp_simple_oct2point,
            ec_GFp_simple_add,
            ec_GFp_simple_dbl,
            ec_GFp_simple_invert,
            ec_GFp_simple_is_at_infinity,
            ec_GFp_simple_is_on_curve,
            ec_GFp_simple_cmp,
            ec_GFp_simple_make_affine,
            ec_GFp_simple_points_make_affine,
            0 /* mul */,
            0 /* precompute_mult */,
            0 /* have_precompute_mult */, 
            ec_GFp_simple_field_mul,
            ec_GFp_simple_field_sqr,
            0 /* field_div */,
            0 /* field_encode */,
            0 /* field_decode */,
            0 /* field_set_to_one */ };

      return &ret;
      }


/* Most method functions in this file are designed to work with
 * non-trivial representations of field elements if necessary
 * (see ecp_mont.c): while standard modular addition and subtraction
 * are used, the field_mul and field_sqr methods will be used for
 * multiplication, and field_encode and field_decode (if defined)
 * will be used for converting between representations.

 * Functions ec_GFp_simple_points_make_affine() and
 * ec_GFp_simple_point_get_affine_coordinates() specifically assume
 * that if a non-trivial representation is used, it is a Montgomery
 * representation (i.e. 'encoding' means multiplying by some factor R).
 */


int ec_GFp_simple_group_init(EC_GROUP *group)
      {
      BN_init(&group->field);
      BN_init(&group->a);
      BN_init(&group->b);
      group->a_is_minus3 = 0;
      return 1;
      }


void ec_GFp_simple_group_finish(EC_GROUP *group)
      {
      BN_free(&group->field);
      BN_free(&group->a);
      BN_free(&group->b);
      }


void ec_GFp_simple_group_clear_finish(EC_GROUP *group)
      {
      BN_clear_free(&group->field);
      BN_clear_free(&group->a);
      BN_clear_free(&group->b);
      }


int ec_GFp_simple_group_copy(EC_GROUP *dest, const EC_GROUP *src)
      {
      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->a_is_minus3 = src->a_is_minus3;

      return 1;
      }


int ec_GFp_simple_group_set_curve(EC_GROUP *group,
      const BIGNUM *p, const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx)
      {
      int ret = 0;
      BN_CTX *new_ctx = NULL;
      BIGNUM *tmp_a;
      
      /* p must be a prime > 3 */
      if (BN_num_bits(p) <= 2 || !BN_is_odd(p))
            {
            ECerr(EC_F_EC_GFP_SIMPLE_GROUP_SET_CURVE, EC_R_INVALID_FIELD);
            return 0;
            }

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

      BN_CTX_start(ctx);
      tmp_a = BN_CTX_get(ctx);
      if (tmp_a == NULL) goto err;

      /* group->field */
      if (!BN_copy(&group->field, p)) goto err;
      BN_set_negative(&group->field, 0);

      /* group->a */
      if (!BN_nnmod(tmp_a, a, p, ctx)) goto err;
      if (group->meth->field_encode)
            { if (!group->meth->field_encode(group, &group->a, tmp_a, ctx)) goto err; }   
      else
            if (!BN_copy(&group->a, tmp_a)) goto err;
      
      /* group->b */
      if (!BN_nnmod(&group->b, b, p, ctx)) goto err;
      if (group->meth->field_encode)
            if (!group->meth->field_encode(group, &group->b, &group->b, ctx)) goto err;
      
      /* group->a_is_minus3 */
      if (!BN_add_word(tmp_a, 3)) goto err;
      group->a_is_minus3 = (0 == BN_cmp(tmp_a, &group->field));

      ret = 1;

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


int ec_GFp_simple_group_get_curve(const EC_GROUP *group, BIGNUM *p, BIGNUM *a, BIGNUM *b, BN_CTX *ctx)
      {
      int ret = 0;
      BN_CTX *new_ctx = NULL;
      
      if (p != NULL)
            {
            if (!BN_copy(p, &group->field)) return 0;
            }

      if (a != NULL || b != NULL)
            {
            if (group->meth->field_decode)
                  {
                  if (ctx == NULL)
                        {
                        ctx = new_ctx = BN_CTX_new();
                        if (ctx == NULL)
                              return 0;
                        }
                  if (a != NULL)
                        {
                        if (!group->meth->field_decode(group, a, &group->a, ctx)) goto err;
                        }
                  if (b != NULL)
                        {
                        if (!group->meth->field_decode(group, b, &group->b, ctx)) goto err;
                        }
                  }
            else
                  {
                  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:
      if (new_ctx)
            BN_CTX_free(new_ctx);
      return ret;
      }


int ec_GFp_simple_group_get_degree(const EC_GROUP *group)
      {
      return BN_num_bits(&group->field);
      }


int ec_GFp_simple_group_check_discriminant(const EC_GROUP *group, BN_CTX *ctx)
      {
      int ret = 0;
      BIGNUM *a,*b,*order,*tmp_1,*tmp_2;
      const BIGNUM *p = &group->field;
      BN_CTX *new_ctx = NULL;

      if (ctx == NULL)
            {
            ctx = new_ctx = BN_CTX_new();
            if (ctx == NULL)
                  {
                  ECerr(EC_F_EC_GFP_SIMPLE_GROUP_CHECK_DISCRIMINANT, ERR_R_MALLOC_FAILURE);
                  goto err;
                  }
            }
      BN_CTX_start(ctx);
      a = BN_CTX_get(ctx);
      b = BN_CTX_get(ctx);
      tmp_1 = BN_CTX_get(ctx);
      tmp_2 = BN_CTX_get(ctx);
      order = BN_CTX_get(ctx);
      if (order == NULL) goto err;

      if (group->meth->field_decode)
            {
            if (!group->meth->field_decode(group, a, &group->a, ctx)) goto err;
            if (!group->meth->field_decode(group, b, &group->b, ctx)) goto err;
            }
      else
            {
            if (!BN_copy(a, &group->a)) goto err;
            if (!BN_copy(b, &group->b)) goto err;
            }
      
      /* check the discriminant:
       * y^2 = x^3 + a*x + b is an elliptic curve <=> 4*a^3 + 27*b^2 != 0 (mod p) 
         * 0 =< a, b < p */
      if (BN_is_zero(a))
            {
            if (BN_is_zero(b)) goto err;
            }
      else if (!BN_is_zero(b))
            {
            if (!BN_mod_sqr(tmp_1, a, p, ctx)) goto err;
            if (!BN_mod_mul(tmp_2, tmp_1, a, p, ctx)) goto err;
            if (!BN_lshift(tmp_1, tmp_2, 2)) goto err;
            /* tmp_1 = 4*a^3 */

            if (!BN_mod_sqr(tmp_2, b, p, ctx)) goto err;
            if (!BN_mul_word(tmp_2, 27)) goto err;
            /* tmp_2 = 27*b^2 */

            if (!BN_mod_add(a, tmp_1, tmp_2, p, ctx)) goto err;
            if (BN_is_zero(a)) goto err;
            }
      ret = 1;

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


int ec_GFp_simple_point_init(EC_POINT *point)
      {
      BN_init(&point->X);
      BN_init(&point->Y);
      BN_init(&point->Z);
      point->Z_is_one = 0;

      return 1;
      }


void ec_GFp_simple_point_finish(EC_POINT *point)
      {
      BN_free(&point->X);
      BN_free(&point->Y);
      BN_free(&point->Z);
      }


void ec_GFp_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;
      }


int ec_GFp_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;
      }


int ec_GFp_simple_point_set_to_infinity(const EC_GROUP *group, EC_POINT *point)
      {
      point->Z_is_one = 0;
      BN_zero(&point->Z);
      return 1;
      }


int ec_GFp_simple_set_Jprojective_coordinates_GFp(const EC_GROUP *group, EC_POINT *point,
      const BIGNUM *x, const BIGNUM *y, const BIGNUM *z, BN_CTX *ctx)
      {
      BN_CTX *new_ctx = NULL;
      int ret = 0;
      
      if (ctx == NULL)
            {
            ctx = new_ctx = BN_CTX_new();
            if (ctx == NULL)
                  return 0;
            }

      if (x != NULL)
            {
            if (!BN_nnmod(&point->X, x, &group->field, ctx)) goto err;
            if (group->meth->field_encode)
                  {
                  if (!group->meth->field_encode(group, &point->X, &point->X, ctx)) goto err;
                  }
            }
      
      if (y != NULL)
            {
            if (!BN_nnmod(&point->Y, y, &group->field, ctx)) goto err;
            if (group->meth->field_encode)
                  {
                  if (!group->meth->field_encode(group, &point->Y, &point->Y, ctx)) goto err;
                  }
            }
      
      if (z != NULL)
            {
            int Z_is_one;

            if (!BN_nnmod(&point->Z, z, &group->field, ctx)) goto err;
            Z_is_one = BN_is_one(&point->Z);
            if (group->meth->field_encode)
                  {
                  if (Z_is_one && (group->meth->field_set_to_one != 0))
                        {
                        if (!group->meth->field_set_to_one(group, &point->Z, ctx)) goto err;
                        }
                  else
                        {
                        if (!group->meth->field_encode(group, &point->Z, &point->Z, ctx)) goto err;
                        }
                  }
            point->Z_is_one = Z_is_one;
            }
      
      ret = 1;
      
 err:
      if (new_ctx != NULL)
            BN_CTX_free(new_ctx);
      return ret;
      }


int ec_GFp_simple_get_Jprojective_coordinates_GFp(const EC_GROUP *group, const EC_POINT *point,
      BIGNUM *x, BIGNUM *y, BIGNUM *z, BN_CTX *ctx)
      {
      BN_CTX *new_ctx = NULL;
      int ret = 0;
      
      if (group->meth->field_decode != 0)
            {
            if (ctx == NULL)
                  {
                  ctx = new_ctx = BN_CTX_new();
                  if (ctx == NULL)
                        return 0;
                  }

            if (x != NULL)
                  {
                  if (!group->meth->field_decode(group, x, &point->X, ctx)) goto err;
                  }
            if (y != NULL)
                  {
                  if (!group->meth->field_decode(group, y, &point->Y, ctx)) goto err;
                  }
            if (z != NULL)
                  {
                  if (!group->meth->field_decode(group, z, &point->Z, ctx)) goto err;
                  }
            }
      else  
            {
            if (x != NULL)
                  {
                  if (!BN_copy(x, &point->X)) goto err;
                  }
            if (y != NULL)
                  {
                  if (!BN_copy(y, &point->Y)) goto err;
                  }
            if (z != NULL)
                  {
                  if (!BN_copy(z, &point->Z)) goto err;
                  }
            }
      
      ret = 1;

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


int ec_GFp_simple_point_set_affine_coordinates(const EC_GROUP *group, EC_POINT *point,
      const BIGNUM *x, const BIGNUM *y, BN_CTX *ctx)
      {
      if (x == NULL || y == NULL)
            {
            /* unlike for projective coordinates, we do not tolerate this */
            ECerr(EC_F_EC_GFP_SIMPLE_POINT_SET_AFFINE_COORDINATES, ERR_R_PASSED_NULL_PARAMETER);
            return 0;
            }

      return EC_POINT_set_Jprojective_coordinates_GFp(group, point, x, y, BN_value_one(), ctx);
      }


int ec_GFp_simple_point_get_affine_coordinates(const EC_GROUP *group, const EC_POINT *point,
      BIGNUM *x, BIGNUM *y, BN_CTX *ctx)
      {
      BN_CTX *new_ctx = NULL;
      BIGNUM *Z, *Z_1, *Z_2, *Z_3;
      const BIGNUM *Z_;
      int ret = 0;

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

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

      BN_CTX_start(ctx);
      Z = BN_CTX_get(ctx);
      Z_1 = BN_CTX_get(ctx);
      Z_2 = BN_CTX_get(ctx);
      Z_3 = BN_CTX_get(ctx);
      if (Z_3 == NULL) goto err;

      /* transform  (X, Y, Z)  into  (x, y) := (X/Z^2, Y/Z^3) */
      
      if (group->meth->field_decode)
            {
            if (!group->meth->field_decode(group, Z, &point->Z, ctx)) goto err;
            Z_ = Z;
            }
      else
            {
            Z_ = &point->Z;
            }
      
      if (BN_is_one(Z_))
            {
            if (group->meth->field_decode)
                  {
                  if (x != NULL)
                        {
                        if (!group->meth->field_decode(group, x, &point->X, ctx)) goto err;
                        }
                  if (y != NULL)
                        {
                        if (!group->meth->field_decode(group, y, &point->Y, ctx)) goto err;
                        }
                  }
            else
                  {
                  if (x != NULL)
                        {
                        if (!BN_copy(x, &point->X)) goto err;
                        }
                  if (y != NULL)
                        {
                        if (!BN_copy(y, &point->Y)) goto err;
                        }
                  }
            }
      else
            {
            if (!BN_mod_inverse(Z_1, Z_, &group->field, ctx))
                  {
                  ECerr(EC_F_EC_GFP_SIMPLE_POINT_GET_AFFINE_COORDINATES, ERR_R_BN_LIB);
                  goto err;
                  }
            
            if (group->meth->field_encode == 0)
                  {
                  /* field_sqr works on standard representation */
                  if (!group->meth->field_sqr(group, Z_2, Z_1, ctx)) goto err;
                  }
            else
                  {
                  if (!BN_mod_sqr(Z_2, Z_1, &group->field, ctx)) goto err;
                  }
      
            if (x != NULL)
                  {
                  /* in the Montgomery case, field_mul will cancel out Montgomery factor in X: */
                  if (!group->meth->field_mul(group, x, &point->X, Z_2, ctx)) goto err;
                  }

            if (y != NULL)
                  {
                  if (group->meth->field_encode == 0)
                        {
                        /* field_mul works on standard representation */
                        if (!group->meth->field_mul(group, Z_3, Z_2, Z_1, ctx)) goto err;
                        }
                  else
                        {
                        if (!BN_mod_mul(Z_3, Z_2, Z_1, &group->field, ctx)) goto err;
                        }

                  /* in the Montgomery case, field_mul will cancel out Montgomery factor in Y: */
                  if (!group->meth->field_mul(group, y, &point->Y, Z_3, ctx)) goto err;
                  }
            }

      ret = 1;

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


int ec_GFp_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 *tmp1, *tmp2, *x, *y;
      int ret = 0;

      /* 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);

      BN_CTX_start(ctx);
      tmp1 = BN_CTX_get(ctx);
      tmp2 = BN_CTX_get(ctx);
      x = BN_CTX_get(ctx);
      y = BN_CTX_get(ctx);
      if (y == NULL) goto err;

      /* Recover y.  We have a Weierstrass equation
       *     y^2 = x^3 + a*x + b,
       * so  y  is one of the square roots of  x^3 + a*x + b.
       */

      /* tmp1 := x^3 */
      if (!BN_nnmod(x, x_, &group->field,ctx)) goto err;
      if (group->meth->field_decode == 0)
            {
            /* field_{sqr,mul} work on standard representation */
            if (!group->meth->field_sqr(group, tmp2, x_, ctx)) goto err;
            if (!group->meth->field_mul(group, tmp1, tmp2, x_, ctx)) goto err;
            }
      else
            {
            if (!BN_mod_sqr(tmp2, x_, &group->field, ctx)) goto err;
            if (!BN_mod_mul(tmp1, tmp2, x_, &group->field, ctx)) goto err;
            }
      
      /* tmp1 := tmp1 + a*x */
      if (group->a_is_minus3)
            {
            if (!BN_mod_lshift1_quick(tmp2, x, &group->field)) goto err;
            if (!BN_mod_add_quick(tmp2, tmp2, x, &group->field)) goto err;
            if (!BN_mod_sub_quick(tmp1, tmp1, tmp2, &group->field)) goto err;
            }
      else
            {
            if (group->meth->field_decode)
                  {
                  if (!group->meth->field_decode(group, tmp2, &group->a, ctx)) goto err;
                  if (!BN_mod_mul(tmp2, tmp2, x, &group->field, ctx)) goto err;
                  }
            else
                  {
                  /* field_mul works on standard representation */
                  if (!group->meth->field_mul(group, tmp2, &group->a, x, ctx)) goto err;
                  }
            
            if (!BN_mod_add_quick(tmp1, tmp1, tmp2, &group->field)) goto err;
            }
      
      /* tmp1 := tmp1 + b */
      if (group->meth->field_decode)
            {
            if (!group->meth->field_decode(group, tmp2, &group->b, ctx)) goto err;
            if (!BN_mod_add_quick(tmp1, tmp1, tmp2, &group->field)) goto err;
            }
      else
            {
            if (!BN_mod_add_quick(tmp1, tmp1, &group->b, &group->field)) goto err;
            }
      
      if (!BN_mod_sqrt(y, tmp1, &group->field, ctx))
            {
            unsigned long err = ERR_peek_last_error();
            
            if (ERR_GET_LIB(err) == ERR_LIB_BN && ERR_GET_REASON(err) == BN_R_NOT_A_SQUARE)
                  {
                  ERR_clear_error();
                  ECerr(EC_F_EC_GFP_SIMPLE_SET_COMPRESSED_COORDINATES, EC_R_INVALID_COMPRESSED_POINT);
                  }
            else
                  ECerr(EC_F_EC_GFP_SIMPLE_SET_COMPRESSED_COORDINATES, ERR_R_BN_LIB);
            goto err;
            }

      if (y_bit != BN_is_odd(y))
            {
            if (BN_is_zero(y))
                  {
                  int kron;

                  kron = BN_kronecker(x, &group->field, ctx);
                  if (kron == -2) goto err;

                  if (kron == 1)
                        ECerr(EC_F_EC_GFP_SIMPLE_SET_COMPRESSED_COORDINATES, EC_R_INVALID_COMPRESSION_BIT);
                  else
                        /* BN_mod_sqrt() should have cought this error (not a square) */
                        ECerr(EC_F_EC_GFP_SIMPLE_SET_COMPRESSED_COORDINATES, EC_R_INVALID_COMPRESSED_POINT);
                  goto err;
                  }
            if (!BN_usub(y, &group->field, y)) goto err;
            }
      if (y_bit != BN_is_odd(y))
            {
            ECerr(EC_F_EC_GFP_SIMPLE_SET_COMPRESSED_COORDINATES, ERR_R_INTERNAL_ERROR);
            goto err;
            }

      if (!EC_POINT_set_affine_coordinates_GFp(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;
      }


size_t ec_GFp_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;
      size_t field_len, i, skip;

      if ((form != POINT_CONVERSION_COMPRESSED)
            && (form != POINT_CONVERSION_UNCOMPRESSED)
            && (form != POINT_CONVERSION_HYBRID))
            {
            ECerr(EC_F_EC_GFP_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_GFP_SIMPLE_POINT2OCT, EC_R_BUFFER_TOO_SMALL);
                        return 0;
                        }
                  buf[0] = 0;
                  }
            return 1;
            }


      /* ret := required output buffer length */
      field_len = BN_num_bytes(&group->field);
      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_GFP_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);
            if (y == NULL) goto err;

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

            if ((form == POINT_CONVERSION_COMPRESSED || form == POINT_CONVERSION_HYBRID) && BN_is_odd(y))
                  buf[0] = form + 1;
            else
                  buf[0] = form;
      
            i = 1;
            
            skip = field_len - BN_num_bytes(x);
            if (skip > field_len)
                  {
                  ECerr(EC_F_EC_GFP_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_GFP_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_GFP_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_GFP_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;
      }


int ec_GFp_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;
      size_t field_len, enc_len;
      int ret = 0;

      if (len == 0)
            {
            ECerr(EC_F_EC_GFP_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_GFP_SIMPLE_OCT2POINT, EC_R_INVALID_ENCODING);
            return 0;
            }
      if ((form == 0 || form == POINT_CONVERSION_UNCOMPRESSED) && y_bit)
            {
            ECerr(EC_F_EC_GFP_SIMPLE_OCT2POINT, EC_R_INVALID_ENCODING);
            return 0;
            }

      if (form == 0)
            {
            if (len != 1)
                  {
                  ECerr(EC_F_EC_GFP_SIMPLE_OCT2POINT, EC_R_INVALID_ENCODING);
                  return 0;
                  }

            return EC_POINT_set_to_infinity(group, point);
            }
      
      field_len = BN_num_bytes(&group->field);
      enc_len = (form == POINT_CONVERSION_COMPRESSED) ? 1 + field_len : 1 + 2*field_len;

      if (len != enc_len)
            {
            ECerr(EC_F_EC_GFP_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);
      if (y == NULL) goto err;

      if (!BN_bin2bn(buf + 1, field_len, x)) goto err;
      if (BN_ucmp(x, &group->field) >= 0)
            {
            ECerr(EC_F_EC_GFP_SIMPLE_OCT2POINT, EC_R_INVALID_ENCODING);
            goto err;
            }

      if (form == POINT_CONVERSION_COMPRESSED)
            {
            if (!EC_POINT_set_compressed_coordinates_GFp(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_GFP_SIMPLE_OCT2POINT, EC_R_INVALID_ENCODING);
                  goto err;
                  }
            if (form == POINT_CONVERSION_HYBRID)
                  {
                  if (y_bit != BN_is_odd(y))
                        {
                        ECerr(EC_F_EC_GFP_SIMPLE_OCT2POINT, EC_R_INVALID_ENCODING);
                        goto err;
                        }
                  }

            if (!EC_POINT_set_affine_coordinates_GFp(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_GFP_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;
      }


int ec_GFp_simple_add(const EC_GROUP *group, EC_POINT *r, const EC_POINT *a, const EC_POINT *b, BN_CTX *ctx)
      {
      int (*field_mul)(const EC_GROUP *, BIGNUM *, const BIGNUM *, const BIGNUM *, BN_CTX *);
      int (*field_sqr)(const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
      const BIGNUM *p;
      BN_CTX *new_ctx = NULL;
      BIGNUM *n0, *n1, *n2, *n3, *n4, *n5, *n6;
      int ret = 0;
      
      if (a == b)
            return EC_POINT_dbl(group, r, a, ctx);
      if (EC_POINT_is_at_infinity(group, a))
            return EC_POINT_copy(r, b);
      if (EC_POINT_is_at_infinity(group, b))
            return EC_POINT_copy(r, a);
      
      field_mul = group->meth->field_mul;
      field_sqr = group->meth->field_sqr;
      p = &group->field;

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

      BN_CTX_start(ctx);
      n0 = BN_CTX_get(ctx);
      n1 = BN_CTX_get(ctx);
      n2 = BN_CTX_get(ctx);
      n3 = BN_CTX_get(ctx);
      n4 = BN_CTX_get(ctx);
      n5 = BN_CTX_get(ctx);
      n6 = BN_CTX_get(ctx);
      if (n6 == NULL) goto end;

      /* Note that in this function we must not read components of 'a' or 'b'
       * once we have written the corresponding components of 'r'.
       * ('r' might be one of 'a' or 'b'.)
       */

      /* n1, n2 */
      if (b->Z_is_one)
            {
            if (!BN_copy(n1, &a->X)) goto end;
            if (!BN_copy(n2, &a->Y)) goto end;
            /* n1 = X_a */
            /* n2 = Y_a */
            }
      else
            {
            if (!field_sqr(group, n0, &b->Z, ctx)) goto end;
            if (!field_mul(group, n1, &a->X, n0, ctx)) goto end;
            /* n1 = X_a * Z_b^2 */

            if (!field_mul(group, n0, n0, &b->Z, ctx)) goto end;
            if (!field_mul(group, n2, &a->Y, n0, ctx)) goto end;
            /* n2 = Y_a * Z_b^3 */
            }

      /* n3, n4 */
      if (a->Z_is_one)
            {
            if (!BN_copy(n3, &b->X)) goto end;
            if (!BN_copy(n4, &b->Y)) goto end;
            /* n3 = X_b */
            /* n4 = Y_b */
            }
      else
            {
            if (!field_sqr(group, n0, &a->Z, ctx)) goto end;
            if (!field_mul(group, n3, &b->X, n0, ctx)) goto end;
            /* n3 = X_b * Z_a^2 */

            if (!field_mul(group, n0, n0, &a->Z, ctx)) goto end;
            if (!field_mul(group, n4, &b->Y, n0, ctx)) goto end;
            /* n4 = Y_b * Z_a^3 */
            }

      /* n5, n6 */
      if (!BN_mod_sub_quick(n5, n1, n3, p)) goto end;
      if (!BN_mod_sub_quick(n6, n2, n4, p)) goto end;
      /* n5 = n1 - n3 */
      /* n6 = n2 - n4 */

      if (BN_is_zero(n5))
            {
            if (BN_is_zero(n6))
                  {
                  /* a is the same point as b */
                  BN_CTX_end(ctx);
                  ret = EC_POINT_dbl(group, r, a, ctx);
                  ctx = NULL;
                  goto end;
                  }
            else
                  {
                  /* a is the inverse of b */
                  BN_zero(&r->Z);
                  r->Z_is_one = 0;
                  ret = 1;
                  goto end;
                  }
            }

      /* 'n7', 'n8' */
      if (!BN_mod_add_quick(n1, n1, n3, p)) goto end;
      if (!BN_mod_add_quick(n2, n2, n4, p)) goto end;
      /* 'n7' = n1 + n3 */
      /* 'n8' = n2 + n4 */

      /* Z_r */
      if (a->Z_is_one && b->Z_is_one)
            {
            if (!BN_copy(&r->Z, n5)) goto end;
            }
      else
            {
            if (a->Z_is_one)
                  { if (!BN_copy(n0, &b->Z)) goto end; }
            else if (b->Z_is_one)
                  { if (!BN_copy(n0, &a->Z)) goto end; }
            else
                  { if (!field_mul(group, n0, &a->Z, &b->Z, ctx)) goto end; }
            if (!field_mul(group, &r->Z, n0, n5, ctx)) goto end;
            }
      r->Z_is_one = 0;
      /* Z_r = Z_a * Z_b * n5 */

      /* X_r */
      if (!field_sqr(group, n0, n6, ctx)) goto end;
      if (!field_sqr(group, n4, n5, ctx)) goto end;
      if (!field_mul(group, n3, n1, n4, ctx)) goto end;
      if (!BN_mod_sub_quick(&r->X, n0, n3, p)) goto end;
      /* X_r = n6^2 - n5^2 * 'n7' */
      
      /* 'n9' */
      if (!BN_mod_lshift1_quick(n0, &r->X, p)) goto end;
      if (!BN_mod_sub_quick(n0, n3, n0, p)) goto end;
      /* n9 = n5^2 * 'n7' - 2 * X_r */

      /* Y_r */
      if (!field_mul(group, n0, n0, n6, ctx)) goto end;
      if (!field_mul(group, n5, n4, n5, ctx)) goto end; /* now n5 is n5^3 */
      if (!field_mul(group, n1, n2, n5, ctx)) goto end;
      if (!BN_mod_sub_quick(n0, n0, n1, p)) goto end;
      if (BN_is_odd(n0))
            if (!BN_add(n0, n0, p)) goto end;
      /* now  0 <= n0 < 2*p,  and n0 is even */
      if (!BN_rshift1(&r->Y, n0)) goto end;
      /* Y_r = (n6 * 'n9' - 'n8' * 'n5^3') / 2 */

      ret = 1;

 end:
      if (ctx) /* otherwise we already called BN_CTX_end */
            BN_CTX_end(ctx);
      if (new_ctx != NULL)
            BN_CTX_free(new_ctx);
      return ret;
      }


int ec_GFp_simple_dbl(const EC_GROUP *group, EC_POINT *r, const EC_POINT *a, BN_CTX *ctx)
      {
      int (*field_mul)(const EC_GROUP *, BIGNUM *, const BIGNUM *, const BIGNUM *, BN_CTX *);
      int (*field_sqr)(const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
      const BIGNUM *p;
      BN_CTX *new_ctx = NULL;
      BIGNUM *n0, *n1, *n2, *n3;
      int ret = 0;
      
      if (EC_POINT_is_at_infinity(group, a))
            {
            BN_zero(&r->Z);
            r->Z_is_one = 0;
            return 1;
            }

      field_mul = group->meth->field_mul;
      field_sqr = group->meth->field_sqr;
      p = &group->field;

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

      BN_CTX_start(ctx);
      n0 = BN_CTX_get(ctx);
      n1 = BN_CTX_get(ctx);
      n2 = BN_CTX_get(ctx);
      n3 = BN_CTX_get(ctx);
      if (n3 == NULL) goto err;

      /* Note that in this function we must not read components of 'a'
       * once we have written the corresponding components of 'r'.
       * ('r' might the same as 'a'.)
       */

      /* n1 */
      if (a->Z_is_one)
            {
            if (!field_sqr(group, n0, &a->X, ctx)) goto err;
            if (!BN_mod_lshift1_quick(n1, n0, p)) goto err;
            if (!BN_mod_add_quick(n0, n0, n1, p)) goto err;
            if (!BN_mod_add_quick(n1, n0, &group->a, p)) goto err;
            /* n1 = 3 * X_a^2 + a_curve */
            }
      else if (group->a_is_minus3)
            {
            if (!field_sqr(group, n1, &a->Z, ctx)) goto err;
            if (!BN_mod_add_quick(n0, &a->X, n1, p)) goto err;
            if (!BN_mod_sub_quick(n2, &a->X, n1, p)) goto err;
            if (!field_mul(group, n1, n0, n2, ctx)) goto err;
            if (!BN_mod_lshift1_quick(n0, n1, p)) goto err;
            if (!BN_mod_add_quick(n1, n0, n1, p)) goto err;
            /* n1 = 3 * (X_a + Z_a^2) * (X_a - Z_a^2)
             *    = 3 * X_a^2 - 3 * Z_a^4 */
            }
      else
            {
            if (!field_sqr(group, n0, &a->X, ctx)) goto err;
            if (!BN_mod_lshift1_quick(n1, n0, p)) goto err;
            if (!BN_mod_add_quick(n0, n0, n1, p)) goto err;
            if (!field_sqr(group, n1, &a->Z, ctx)) goto err;
            if (!field_sqr(group, n1, n1, ctx)) goto err;
            if (!field_mul(group, n1, n1, &group->a, ctx)) goto err;
            if (!BN_mod_add_quick(n1, n1, n0, p)) goto err;
            /* n1 = 3 * X_a^2 + a_curve * Z_a^4 */
            }

      /* Z_r */
      if (a->Z_is_one)
            {
            if (!BN_copy(n0, &a->Y)) goto err;
            }
      else
            {
            if (!field_mul(group, n0, &a->Y, &a->Z, ctx)) goto err;
            }
      if (!BN_mod_lshift1_quick(&r->Z, n0, p)) goto err;
      r->Z_is_one = 0;
      /* Z_r = 2 * Y_a * Z_a */

      /* n2 */
      if (!field_sqr(group, n3, &a->Y, ctx)) goto err;
      if (!field_mul(group, n2, &a->X, n3, ctx)) goto err;
      if (!BN_mod_lshift_quick(n2, n2, 2, p)) goto err;
      /* n2 = 4 * X_a * Y_a^2 */

      /* X_r */
      if (!BN_mod_lshift1_quick(n0, n2, p)) goto err;
      if (!field_sqr(group, &r->X, n1, ctx)) goto err;
      if (!BN_mod_sub_quick(&r->X, &r->X, n0, p)) goto err;
      /* X_r = n1^2 - 2 * n2 */
      
      /* n3 */
      if (!field_sqr(group, n0, n3, ctx)) goto err;
      if (!BN_mod_lshift_quick(n3, n0, 3, p)) goto err;
      /* n3 = 8 * Y_a^4 */
      
      /* Y_r */
      if (!BN_mod_sub_quick(n0, n2, &r->X, p)) goto err;
      if (!field_mul(group, n0, n1, n0, ctx)) goto err;
      if (!BN_mod_sub_quick(&r->Y, n0, n3, p)) goto err;
      /* Y_r = n1 * (n2 - X_r) - n3 */

      ret = 1;

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


int ec_GFp_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;
      
      return BN_usub(&point->Y, &group->field, &point->Y);
      }


int ec_GFp_simple_is_at_infinity(const EC_GROUP *group, const EC_POINT *point)
      {
      return BN_is_zero(&point->Z);
      }


int ec_GFp_simple_is_on_curve(const EC_GROUP *group, const EC_POINT *point, BN_CTX *ctx)
      {
      int (*field_mul)(const EC_GROUP *, BIGNUM *, const BIGNUM *, const BIGNUM *, BN_CTX *);
      int (*field_sqr)(const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
      const BIGNUM *p;
      BN_CTX *new_ctx = NULL;
      BIGNUM *rh, *tmp, *Z4, *Z6;
      int ret = -1;

      if (EC_POINT_is_at_infinity(group, point))
            return 1;
      
      field_mul = group->meth->field_mul;
      field_sqr = group->meth->field_sqr;
      p = &group->field;

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

      BN_CTX_start(ctx);
      rh = BN_CTX_get(ctx);
      tmp = BN_CTX_get(ctx);
      Z4 = BN_CTX_get(ctx);
      Z6 = BN_CTX_get(ctx);
      if (Z6 == NULL) goto err;

      /* We have a curve defined by a Weierstrass equation
       *      y^2 = x^3 + a*x + b.
       * The point to consider is given in Jacobian projective coordinates
       * where  (X, Y, Z)  represents  (x, y) = (X/Z^2, Y/Z^3).
       * Substituting this and multiplying by  Z^6  transforms the above equation into
       *      Y^2 = X^3 + a*X*Z^4 + b*Z^6.
       * To test this, we add up the right-hand side in 'rh'.
       */

      /* rh := X^2 */
      if (!field_sqr(group, rh, &point->X, ctx)) goto err;

      if (!point->Z_is_one)
            {
            if (!field_sqr(group, tmp, &point->Z, ctx)) goto err;
            if (!field_sqr(group, Z4, tmp, ctx)) goto err;
            if (!field_mul(group, Z6, Z4, tmp, ctx)) goto err;

            /* rh := (rh + a*Z^4)*X */
            if (group->a_is_minus3)
                  {
                  if (!BN_mod_lshift1_quick(tmp, Z4, p)) goto err;
                  if (!BN_mod_add_quick(tmp, tmp, Z4, p)) goto err;
                  if (!BN_mod_sub_quick(rh, rh, tmp, p)) goto err;
                  if (!field_mul(group, rh, rh, &point->X, ctx)) goto err;
                  }
            else
                  {
                  if (!field_mul(group, tmp, Z4, &group->a, ctx)) goto err;
                  if (!BN_mod_add_quick(rh, rh, tmp, p)) goto err;
                  if (!field_mul(group, rh, rh, &point->X, ctx)) goto err;
                  }

            /* rh := rh + b*Z^6 */
            if (!field_mul(group, tmp, &group->b, Z6, ctx)) goto err;
            if (!BN_mod_add_quick(rh, rh, tmp, p)) goto err;
            }
      else
            {
            /* point->Z_is_one */

            /* rh := (rh + a)*X */
            if (!BN_mod_add_quick(rh, rh, &group->a, p)) goto err;
            if (!field_mul(group, rh, rh, &point->X, ctx)) goto err;
            /* rh := rh + b */
            if (!BN_mod_add_quick(rh, rh, &group->b, p)) goto err;
            }

      /* 'lh' := Y^2 */
      if (!field_sqr(group, tmp, &point->Y, ctx)) goto err;

      ret = (0 == BN_ucmp(tmp, rh));

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


int ec_GFp_simple_cmp(const EC_GROUP *group, const EC_POINT *a, const EC_POINT *b, BN_CTX *ctx)
      {
      /* return values:
       *  -1   error
       *   0   equal (in affine coordinates)
       *   1   not equal
       */

      int (*field_mul)(const EC_GROUP *, BIGNUM *, const BIGNUM *, const BIGNUM *, BN_CTX *);
      int (*field_sqr)(const EC_GROUP *, BIGNUM *, const BIGNUM *, BN_CTX *);
      BN_CTX *new_ctx = NULL;
      BIGNUM *tmp1, *tmp2, *Za23, *Zb23;
      const BIGNUM *tmp1_, *tmp2_;
      int ret = -1;
      
      if (EC_POINT_is_at_infinity(group, a))
            {
            return EC_POINT_is_at_infinity(group, b) ? 0 : 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;
            }

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

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

      BN_CTX_start(ctx);
      tmp1 = BN_CTX_get(ctx);
      tmp2 = BN_CTX_get(ctx);
      Za23 = BN_CTX_get(ctx);
      Zb23 = BN_CTX_get(ctx);
      if (Zb23 == NULL) goto end;

      /* We have to decide whether
       *     (X_a/Z_a^2, Y_a/Z_a^3) = (X_b/Z_b^2, Y_b/Z_b^3),
       * or equivalently, whether
       *     (X_a*Z_b^2, Y_a*Z_b^3) = (X_b*Z_a^2, Y_b*Z_a^3).
       */

      if (!b->Z_is_one)
            {
            if (!field_sqr(group, Zb23, &b->Z, ctx)) goto end;
            if (!field_mul(group, tmp1, &a->X, Zb23, ctx)) goto end;
            tmp1_ = tmp1;
            }
      else
            tmp1_ = &a->X;
      if (!a->Z_is_one)
            {
            if (!field_sqr(group, Za23, &a->Z, ctx)) goto end;
            if (!field_mul(group, tmp2, &b->X, Za23, ctx)) goto end;
            tmp2_ = tmp2;
            }
      else
            tmp2_ = &b->X;
      
      /* compare  X_a*Z_b^2  with  X_b*Z_a^2 */
      if (BN_cmp(tmp1_, tmp2_) != 0)
            {
            ret = 1; /* points differ */
            goto end;
            }


      if (!b->Z_is_one)
            {
            if (!field_mul(group, Zb23, Zb23, &b->Z, ctx)) goto end;
            if (!field_mul(group, tmp1, &a->Y, Zb23, ctx)) goto end;
            /* tmp1_ = tmp1 */
            }
      else
            tmp1_ = &a->Y;
      if (!a->Z_is_one)
            {
            if (!field_mul(group, Za23, Za23, &a->Z, ctx)) goto end;
            if (!field_mul(group, tmp2, &b->Y, Za23, ctx)) goto end;
            /* tmp2_ = tmp2 */
            }
      else
            tmp2_ = &b->Y;

      /* compare  Y_a*Z_b^3  with  Y_b*Z_a^3 */
      if (BN_cmp(tmp1_, tmp2_) != 0)
            {
            ret = 1; /* points differ */
            goto end;
            }

      /* points are equal */
      ret = 0;

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


int ec_GFp_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_GFp(group, point, x, y, ctx)) goto err;
      if (!EC_POINT_set_affine_coordinates_GFp(group, point, x, y, ctx)) goto err;
      if (!point->Z_is_one)
            {
            ECerr(EC_F_EC_GFP_SIMPLE_MAKE_AFFINE, ERR_R_INTERNAL_ERROR);
            goto err;
            }
      
      ret = 1;

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


int ec_GFp_simple_points_make_affine(const EC_GROUP *group, size_t num, EC_POINT *points[], BN_CTX *ctx)
      {
      BN_CTX *new_ctx = NULL;
      BIGNUM *tmp0, *tmp1;
      size_t pow2 = 0;
      BIGNUM **heap = NULL;
      size_t i;
      int ret = 0;

      if (num == 0)
            return 1;

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

      BN_CTX_start(ctx);
      tmp0 = BN_CTX_get(ctx);
      tmp1 = BN_CTX_get(ctx);
      if (tmp0  == NULL || tmp1 == NULL) goto err;

      /* Before converting the individual points, compute inverses of all Z values.
       * Modular inversion is rather slow, but luckily we can do with a single
       * explicit inversion, plus about 3 multiplications per input value.
       */

      pow2 = 1;
      while (num > pow2)
            pow2 <<= 1;
      /* Now pow2 is the smallest power of 2 satifsying pow2 >= num.
       * We need twice that. */
      pow2 <<= 1;

      heap = OPENSSL_malloc(pow2 * sizeof heap[0]);
      if (heap == NULL) goto err;
      
      /* The array is used as a binary tree, exactly as in heapsort:
       *
       *                               heap[1]
       *                 heap[2]                     heap[3]
       *          heap[4]       heap[5]       heap[6]       heap[7]
       *   heap[8]heap[9] heap[10]heap[11] heap[12]heap[13] heap[14] heap[15]
       *
       * We put the Z's in the last line;
       * then we set each other node to the product of its two child-nodes (where
       * empty or 0 entries are treated as ones);
       * then we invert heap[1];
       * then we invert each other node by replacing it by the product of its
       * parent (after inversion) and its sibling (before inversion).
       */
      heap[0] = NULL;
      for (i = pow2/2 - 1; i > 0; i--)
            heap[i] = NULL;
      for (i = 0; i < num; i++)
            heap[pow2/2 + i] = &points[i]->Z;
      for (i = pow2/2 + num; i < pow2; i++)
            heap[i] = NULL;
      
      /* set each node to the product of its children */
      for (i = pow2/2 - 1; i > 0; i--)
            {
            heap[i] = BN_new();
            if (heap[i] == NULL) goto err;
            
            if (heap[2*i] != NULL)
                  {
                  if ((heap[2*i + 1] == NULL) || BN_is_zero(heap[2*i + 1]))
                        {
                        if (!BN_copy(heap[i], heap[2*i])) goto err;
                        }
                  else
                        {
                        if (BN_is_zero(heap[2*i]))
                              {
                              if (!BN_copy(heap[i], heap[2*i + 1])) goto err;
                              }
                        else
                              {
                              if (!group->meth->field_mul(group, heap[i],
                                    heap[2*i], heap[2*i + 1], ctx)) goto err;
                              }
                        }
                  }
            }

      /* invert heap[1] */
      if (!BN_is_zero(heap[1]))
            {
            if (!BN_mod_inverse(heap[1], heap[1], &group->field, ctx))
                  {
                  ECerr(EC_F_EC_GFP_SIMPLE_POINTS_MAKE_AFFINE, ERR_R_BN_LIB);
                  goto err;
                  }
            }
      if (group->meth->field_encode != 0)
            {
            /* in the Montgomery case, we just turned  R*H  (representing H)
             * into  1/(R*H),  but we need  R*(1/H)  (representing 1/H);
             * i.e. we have need to multiply by the Montgomery factor twice */
            if (!group->meth->field_encode(group, heap[1], heap[1], ctx)) goto err;
            if (!group->meth->field_encode(group, heap[1], heap[1], ctx)) goto err;
            }

      /* set other heap[i]'s to their inverses */
      for (i = 2; i < pow2/2 + num; i += 2)
            {
            /* i is even */
            if ((heap[i + 1] != NULL) && !BN_is_zero(heap[i + 1]))
                  {
                  if (!group->meth->field_mul(group, tmp0, heap[i/2], heap[i + 1], ctx)) goto err;
                  if (!group->meth->field_mul(group, tmp1, heap[i/2], heap[i], ctx)) goto err;
                  if (!BN_copy(heap[i], tmp0)) goto err;
                  if (!BN_copy(heap[i + 1], tmp1)) goto err;
                  }
            else
                  {
                  if (!BN_copy(heap[i], heap[i/2])) goto err;
                  }
            }

      /* we have replaced all non-zero Z's by their inverses, now fix up all the points */
      for (i = 0; i < num; i++)
            {
            EC_POINT *p = points[i];
            
            if (!BN_is_zero(&p->Z))
                  {
                  /* turn  (X, Y, 1/Z)  into  (X/Z^2, Y/Z^3, 1) */

                  if (!group->meth->field_sqr(group, tmp1, &p->Z, ctx)) goto err;
                  if (!group->meth->field_mul(group, &p->X, &p->X, tmp1, ctx)) goto err;

                  if (!group->meth->field_mul(group, tmp1, tmp1, &p->Z, ctx)) goto err;
                  if (!group->meth->field_mul(group, &p->Y, &p->Y, tmp1, ctx)) goto err;
            
                  if (group->meth->field_set_to_one != 0)
                        {
                        if (!group->meth->field_set_to_one(group, &p->Z, ctx)) goto err;
                        }
                  else
                        {
                        if (!BN_one(&p->Z)) goto err;
                        }
                  p->Z_is_one = 1;
                  }
            }

      ret = 1;
            
 err:
      BN_CTX_end(ctx);
      if (new_ctx != NULL)
            BN_CTX_free(new_ctx);
      if (heap != NULL)
            {
            /* heap[pow2/2] .. heap[pow2-1] have not been allocated locally! */
            for (i = pow2/2 - 1; i > 0; i--)
                  {
                  if (heap[i] != NULL)
                        BN_clear_free(heap[i]);
                  }
            OPENSSL_free(heap);
            }
      return ret;
      }


int ec_GFp_simple_field_mul(const EC_GROUP *group, BIGNUM *r, const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx)
      {
      return BN_mod_mul(r, a, b, &group->field, ctx);
      }


int ec_GFp_simple_field_sqr(const EC_GROUP *group, BIGNUM *r, const BIGNUM *a, BN_CTX *ctx)
      {
      return BN_mod_sqr(r, a, &group->field, ctx);
      }

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