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>

#ifdef OPENSSL_FIPS
#include <openssl/fips.h>
#endif

#include "ec_lcl.h"

const EC_METHOD *EC_GFp_simple_method(void)
	{
	static const EC_METHOD ret = {
		EC_FLAGS_DEFAULT_OCT,
		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,
		0,0,0,
		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 */ };

#ifdef OPENSSL_FIPS
	if (FIPS_mode())
		return fips_ec_gfp_simple_method();
#endif

	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_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 *tmp, *tmp_Z;
	BIGNUM **prod_Z = 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);
	tmp = BN_CTX_get(ctx);
	tmp_Z = BN_CTX_get(ctx);
	if (tmp == NULL || tmp_Z == NULL) goto err;

	prod_Z = OPENSSL_malloc(num * sizeof prod_Z[0]);
	if (prod_Z == NULL) goto err;
	for (i = 0; i < num; i++)
		{
		prod_Z[i] = BN_new();
		if (prod_Z[i] == NULL) goto err;
		}

	/* Set each prod_Z[i] to the product of points[0]->Z .. points[i]->Z,
	 * skipping any zero-valued inputs (pretend that they're 1). */

	if (!BN_is_zero(&points[0]->Z))
		{
		if (!BN_copy(prod_Z[0], &points[0]->Z)) goto err;
		}
	else
		{
		if (group->meth->field_set_to_one != 0)
			{
			if (!group->meth->field_set_to_one(group, prod_Z[0], ctx)) goto err;
			}
		else
			{
			if (!BN_one(prod_Z[0])) goto err;
			}
		}

	for (i = 1; i < num; i++)
		{
		if (!BN_is_zero(&points[i]->Z))
			{
			if (!group->meth->field_mul(group, prod_Z[i], prod_Z[i - 1], &points[i]->Z, ctx)) goto err;
			}
		else
			{
			if (!BN_copy(prod_Z[i], prod_Z[i - 1])) goto err;
			}
		}

	/* Now use a single explicit inversion to replace every
	 * non-zero points[i]->Z by its inverse. */

	if (!BN_mod_inverse(tmp, prod_Z[num - 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 need to multiply by the Montgomery factor twice. */
		if (!group->meth->field_encode(group, tmp, tmp, ctx)) goto err;
		if (!group->meth->field_encode(group, tmp, tmp, ctx)) goto err;
		}

	for (i = num - 1; i > 0; --i)
		{
		/* Loop invariant: tmp is the product of the inverses of
		 * points[0]->Z .. points[i]->Z (zero-valued inputs skipped). */
		if (!BN_is_zero(&points[i]->Z))
			{
			/* Set tmp_Z to the inverse of points[i]->Z (as product
			 * of Z inverses 0 .. i, Z values 0 .. i - 1). */
			if (!group->meth->field_mul(group, tmp_Z, prod_Z[i - 1], tmp, ctx)) goto err;
			/* Update tmp to satisfy the loop invariant for i - 1. */
			if (!group->meth->field_mul(group, tmp, tmp, &points[i]->Z, ctx)) goto err;
			/* Replace points[i]->Z by its inverse. */
			if (!BN_copy(&points[i]->Z, tmp_Z)) goto err;
			}
		}

	if (!BN_is_zero(&points[0]->Z))
		{
		/* Replace points[0]->Z by its inverse. */
		if (!BN_copy(&points[0]->Z, tmp)) goto err;
		}

	/* Finally, fix up the X and Y coordinates for all 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, tmp, &p->Z, ctx)) goto err;
			if (!group->meth->field_mul(group, &p->X, &p->X, tmp, ctx)) goto err;

			if (!group->meth->field_mul(group, tmp, tmp, &p->Z, ctx)) goto err;
			if (!group->meth->field_mul(group, &p->Y, &p->Y, tmp, 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 (prod_Z != NULL)
		{
		for (i = 0; i < num; i++)
			{
			if (prod_Z[i] == NULL) break;
			BN_clear_free(prod_Z[i]);
			}
		OPENSSL_free(prod_Z);
		}
	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);
	}