text: /* crypto/ec/ec2_mult.c */
  /* ====================================================================
   * Copyright 2002 Sun Microsystems, Inc. ALL RIGHTS RESERVED.
   *
   * The Elliptic Curve Public-Key Crypto Library (ECC Code) included
   * herein is developed by SUN MICROSYSTEMS, INC., and is contributed
   * to the OpenSSL project.
   *
   * The ECC Code is licensed pursuant to the OpenSSL open source
   * license provided below.
   *
   * The software is originally written by Sheueling Chang Shantz and
   * Douglas Stebila of Sun Microsystems Laboratories.
   *
   */
  /* ====================================================================
   * Copyright (c) 1998-2003 The OpenSSL Project.  All rights reserved.
   *
   * Redistribution and use in source and binary forms, with or without
   * modification, are permitted provided that the following conditions
   * are met:
   *
   * 1. Redistributions of source code must retain the above copyright
   *    notice, this list of conditions and the following disclaimer. 
   *
   * 2. Redistributions in binary form must reproduce the above copyright
   *    notice, this list of conditions and the following disclaimer in
   *    the documentation and/or other materials provided with the
   *    distribution.
   *
   * 3. All advertising materials mentioning features or use of this
   *    software must display the following acknowledgment:
   *    "This product includes software developed by the OpenSSL Project
   *    for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
   *
   * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
   *    endorse or promote products derived from this software without
   *    prior written permission. For written permission, please contact
   *    openssl-core@openssl.org.
   *
   * 5. Products derived from this software may not be called "OpenSSL"
   *    nor may "OpenSSL" appear in their names without prior written
   *    permission of the OpenSSL Project.
   *
   * 6. Redistributions of any form whatsoever must retain the following
   *    acknowledgment:
   *    "This product includes software developed by the OpenSSL Project
   *    for use in the OpenSSL Toolkit (http://www.openssl.org/)"
   *
   * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
   * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
   * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
   * PURPOSE ARE DISCLAIMED.  IN NO EVENT SHALL THE OpenSSL PROJECT OR
   * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
   * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
   * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
   * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
   * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
   * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
   * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
   * OF THE POSSIBILITY OF SUCH DAMAGE.
   * ====================================================================
   *
   * This product includes cryptographic software written by Eric Young
   * (eay@cryptsoft.com).  This product includes software written by Tim
   * Hudson (tjh@cryptsoft.com).
   *
   */
  
  #include <openssl/err.h>
  
  #include "ec_lcl.h"
  
  
  /* Compute the x-coordinate x/z for the point 2*(x/z) in Montgomery projective 
   * coordinates.
   * Uses algorithm Mdouble in appendix of 
   *     Lopez, J. and Dahab, R.  "Fast multiplication on elliptic curves over 
   *     GF(2^m) without precomputation" (CHES '99, LNCS 1717).
   * modified to not require precomputation of c=b^{2^{m-1}}.
   */
  static int gf2m_Mdouble(const EC_GROUP *group, BIGNUM *x, BIGNUM *z, BN_CTX *ctx)
  	{
  	BIGNUM *t1;
  	int ret = 0;
  	
  	/* Since Mdouble is static we can guarantee that ctx != NULL. */
  	BN_CTX_start(ctx);
  	t1 = BN_CTX_get(ctx);
  	if (t1 == NULL) goto err;
  
  	if (!group->meth->field_sqr(group, x, x, ctx)) goto err;
  	if (!group->meth->field_sqr(group, t1, z, ctx)) goto err;
  	if (!group->meth->field_mul(group, z, x, t1, ctx)) goto err;
  	if (!group->meth->field_sqr(group, x, x, ctx)) goto err;
  	if (!group->meth->field_sqr(group, t1, t1, ctx)) goto err;
  	if (!group->meth->field_mul(group, t1, &group->b, t1, ctx)) goto err;
  	if (!BN_GF2m_add(x, x, t1)) goto err;
  
  	ret = 1;
  
   err:
  	BN_CTX_end(ctx);
  	return ret;
  	}
  
  /* Compute the x-coordinate x1/z1 for the point (x1/z1)+(x2/x2) in Montgomery 
   * projective coordinates.
   * Uses algorithm Madd in appendix of 
   *     Lopez, J. and Dahab, R.  "Fast multiplication on elliptic curves over 
   *     GF(2^m) without precomputation" (CHES '99, LNCS 1717).
   */
  static int gf2m_Madd(const EC_GROUP *group, const BIGNUM *x, BIGNUM *x1, BIGNUM *z1, 
  	const BIGNUM *x2, const BIGNUM *z2, BN_CTX *ctx)
  	{
  	BIGNUM *t1, *t2;
  	int ret = 0;
  	
  	/* Since Madd is static we can guarantee that ctx != NULL. */
  	BN_CTX_start(ctx);
  	t1 = BN_CTX_get(ctx);
  	t2 = BN_CTX_get(ctx);
  	if (t2 == NULL) goto err;
  
  	if (!BN_copy(t1, x)) goto err;
  	if (!group->meth->field_mul(group, x1, x1, z2, ctx)) goto err;
  	if (!group->meth->field_mul(group, z1, z1, x2, ctx)) goto err;
  	if (!group->meth->field_mul(group, t2, x1, z1, ctx)) goto err;
  	if (!BN_GF2m_add(z1, z1, x1)) goto err;
  	if (!group->meth->field_sqr(group, z1, z1, ctx)) goto err;
  	if (!group->meth->field_mul(group, x1, z1, t1, ctx)) goto err;
  	if (!BN_GF2m_add(x1, x1, t2)) goto err;
  
  	ret = 1;
  
   err:
  	BN_CTX_end(ctx);
  	return ret;
  	}
  
  /* Compute the x, y affine coordinates from the point (x1, z1) (x2, z2) 
   * using Montgomery point multiplication algorithm Mxy() in appendix of 
   *     Lopez, J. and Dahab, R.  "Fast multiplication on elliptic curves over 
   *     GF(2^m) without precomputation" (CHES '99, LNCS 1717).
   * Returns:
   *     0 on error
   *     1 if return value should be the point at infinity
   *     2 otherwise
   */
  static int gf2m_Mxy(const EC_GROUP *group, const BIGNUM *x, const BIGNUM *y, BIGNUM *x1, 
  	BIGNUM *z1, BIGNUM *x2, BIGNUM *z2, BN_CTX *ctx)
  	{
  	BIGNUM *t3, *t4, *t5;
  	int ret = 0;
  	
  	if (BN_is_zero(z1))
  		{
  		BN_zero(x2);
  		BN_zero(z2);
  		return 1;
  		}
  	
  	if (BN_is_zero(z2))
  		{
  		if (!BN_copy(x2, x)) return 0;
  		if (!BN_GF2m_add(z2, x, y)) return 0;
  		return 2;
  		}
  		
  	/* Since Mxy is static we can guarantee that ctx != NULL. */
  	BN_CTX_start(ctx);
  	t3 = BN_CTX_get(ctx);
  	t4 = BN_CTX_get(ctx);
  	t5 = BN_CTX_get(ctx);
  	if (t5 == NULL) goto err;
  
  	if (!BN_one(t5)) goto err;
  
  	if (!group->meth->field_mul(group, t3, z1, z2, ctx)) goto err;
  
  	if (!group->meth->field_mul(group, z1, z1, x, ctx)) goto err;
  	if (!BN_GF2m_add(z1, z1, x1)) goto err;
  	if (!group->meth->field_mul(group, z2, z2, x, ctx)) goto err;
  	if (!group->meth->field_mul(group, x1, z2, x1, ctx)) goto err;
  	if (!BN_GF2m_add(z2, z2, x2)) goto err;
  
  	if (!group->meth->field_mul(group, z2, z2, z1, ctx)) goto err;
  	if (!group->meth->field_sqr(group, t4, x, ctx)) goto err;
  	if (!BN_GF2m_add(t4, t4, y)) goto err;
  	if (!group->meth->field_mul(group, t4, t4, t3, ctx)) goto err;
  	if (!BN_GF2m_add(t4, t4, z2)) goto err;
  
  	if (!group->meth->field_mul(group, t3, t3, x, ctx)) goto err;
  	if (!group->meth->field_div(group, t3, t5, t3, ctx)) goto err;
  	if (!group->meth->field_mul(group, t4, t3, t4, ctx)) goto err;
  	if (!group->meth->field_mul(group, x2, x1, t3, ctx)) goto err;
  	if (!BN_GF2m_add(z2, x2, x)) goto err;
  
  	if (!group->meth->field_mul(group, z2, z2, t4, ctx)) goto err;
  	if (!BN_GF2m_add(z2, z2, y)) goto err;
  
  	ret = 2;
  
   err:
  	BN_CTX_end(ctx);
  	return ret;
  	}
  
  /* Computes scalar*point and stores the result in r.
   * point can not equal r.
   * Uses algorithm 2P of
   *     Lopez, J. and Dahab, R.  "Fast multiplication on elliptic curves over 
   *     GF(2^m) without precomputation" (CHES '99, LNCS 1717).
   */
  static int ec_GF2m_montgomery_point_multiply(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar,
  	const EC_POINT *point, BN_CTX *ctx)
  	{
  	BIGNUM *x1, *x2, *z1, *z2;
  	int ret = 0, i;
  	BN_ULONG mask,word;
  
  	if (r == point)
  		{
  		ECerr(EC_F_EC_GF2M_MONTGOMERY_POINT_MULTIPLY, EC_R_INVALID_ARGUMENT);
  		return 0;
  		}
  	
  	/* if result should be point at infinity */
  	if ((scalar == NULL) || BN_is_zero(scalar) || (point == NULL) || 
  		EC_POINT_is_at_infinity(group, point))
  		{
  		return EC_POINT_set_to_infinity(group, r);
  		}
  
  	/* only support affine coordinates */
  	if (!point->Z_is_one) return 0;
  
  	/* Since point_multiply is static we can guarantee that ctx != NULL. */
  	BN_CTX_start(ctx);
  	x1 = BN_CTX_get(ctx);
  	z1 = BN_CTX_get(ctx);
  	if (z1 == NULL) goto err;
  
  	x2 = &r->X;
  	z2 = &r->Y;
  
  	if (!BN_GF2m_mod_arr(x1, &point->X, group->poly)) goto err; /* x1 = x */
  	if (!BN_one(z1)) goto err; /* z1 = 1 */
  	if (!group->meth->field_sqr(group, z2, x1, ctx)) goto err; /* z2 = x1^2 = x^2 */
  	if (!group->meth->field_sqr(group, x2, z2, ctx)) goto err;
  	if (!BN_GF2m_add(x2, x2, &group->b)) goto err; /* x2 = x^4 + b */
  
  	/* find top most bit and go one past it */
  	i = scalar->top - 1;
  	mask = BN_TBIT;
  	word = scalar->d[i];
  	while (!(word & mask)) mask >>= 1;
  	mask >>= 1;
  	/* if top most bit was at word break, go to next word */
  	if (!mask) 
  		{
  		i--;
  		mask = BN_TBIT;
  		}
  
  	for (; i >= 0; i--)
  		{
  		word = scalar->d[i];
  		while (mask)
  			{
  			if (word & mask)
  				{
  				if (!gf2m_Madd(group, &point->X, x1, z1, x2, z2, ctx)) goto err;
  				if (!gf2m_Mdouble(group, x2, z2, ctx)) goto err;
  				}
  			else
  				{
  				if (!gf2m_Madd(group, &point->X, x2, z2, x1, z1, ctx)) goto err;
  				if (!gf2m_Mdouble(group, x1, z1, ctx)) goto err;
  				}
  			mask >>= 1;
  			}
  		mask = BN_TBIT;
  		}
  
  	/* convert out of "projective" coordinates */
  	i = gf2m_Mxy(group, &point->X, &point->Y, x1, z1, x2, z2, ctx);
  	if (i == 0) goto err;
  	else if (i == 1) 
  		{
  		if (!EC_POINT_set_to_infinity(group, r)) goto err;
  		}
  	else
  		{
  		if (!BN_one(&r->Z)) goto err;
  		r->Z_is_one = 1;
  		}
  
  	/* GF(2^m) field elements should always have BIGNUM::neg = 0 */
  	BN_set_negative(&r->X, 0);
  	BN_set_negative(&r->Y, 0);
  
  	ret = 1;
  
   err:
  	BN_CTX_end(ctx);
  	return ret;
  	}
  
  
  /* Computes the sum
   *     scalar*group->generator + scalars[0]*points[0] + ... + scalars[num-1]*points[num-1]
   * gracefully ignoring NULL scalar values.
   */
  int ec_GF2m_simple_mul(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar,
  	size_t num, const EC_POINT *points[], const BIGNUM *scalars[], BN_CTX *ctx)
  	{
  	BN_CTX *new_ctx = NULL;
  	int ret = 0;
  	size_t i;
  	EC_POINT *p=NULL;
  	EC_POINT *acc = NULL;
  
  	if (ctx == NULL)
  		{
  		ctx = new_ctx = BN_CTX_new();
  		if (ctx == NULL)
  			return 0;
  		}
  
  	/* This implementation is more efficient than the wNAF implementation for 2
  	 * or fewer points.  Use the ec_wNAF_mul implementation for 3 or more points,
  	 * or if we can perform a fast multiplication based on precomputation.
  	 */
  	if ((scalar && (num > 1)) || (num > 2) || (num == 0 && EC_GROUP_have_precompute_mult(group)))
  		{
  		ret = ec_wNAF_mul(group, r, scalar, num, points, scalars, ctx);
  		goto err;
  		}
  
  	if ((p = EC_POINT_new(group)) == NULL) goto err;
  	if ((acc = EC_POINT_new(group)) == NULL) goto err;
  
  	if (!EC_POINT_set_to_infinity(group, acc)) goto err;
  
  	if (scalar)
  		{
  		if (!ec_GF2m_montgomery_point_multiply(group, p, scalar, group->generator, ctx)) goto err;
  		if (BN_is_negative(scalar))
  			if (!group->meth->invert(group, p, ctx)) goto err;
  		if (!group->meth->add(group, acc, acc, p, ctx)) goto err;
  		}
  
  	for (i = 0; i < num; i++)
  		{
  		if (!ec_GF2m_montgomery_point_multiply(group, p, scalars[i], points[i], ctx)) goto err;
  		if (BN_is_negative(scalars[i]))
  			if (!group->meth->invert(group, p, ctx)) goto err;
  		if (!group->meth->add(group, acc, acc, p, ctx)) goto err;
  		}
  
  	if (!EC_POINT_copy(r, acc)) goto err;
  
  	ret = 1;
  
    err:
  	if (p) EC_POINT_free(p);
  	if (acc) EC_POINT_free(acc);
  	if (new_ctx != NULL)
  		BN_CTX_free(new_ctx);
  	return ret;
  	}
  
  
  /* Precomputation for point multiplication: fall back to wNAF methods
   * because ec_GF2m_simple_mul() uses ec_wNAF_mul() if appropriate */
  
  int ec_GF2m_precompute_mult(EC_GROUP *group, BN_CTX *ctx)
  	{
  	return ec_wNAF_precompute_mult(group, ctx);
   	}
  
  int ec_GF2m_have_precompute_mult(const EC_GROUP *group)
  	{
  	return ec_wNAF_have_precompute_mult(group);
   	}
  