text: /* 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:
  	if (ctx != NULL)
  		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 (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;
  		}
  
  	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);
  	}
  