src/crypto/fipsmodule/ec/simple.c - boringssl - Git at Google

 /* Originally written by Bodo Moeller for the OpenSSL project.
  * ====================================================================
  * Copyright (c) 1998-2005 The OpenSSL Project.  All rights reserved.
  *
  * Redistribution and use in source and binary forms, with or without
  * modification, are permitted provided that the following conditions
  * are met:
  *
  * 1. Redistributions of source code must retain the above copyright
  *    notice, this list of conditions and the following disclaimer.
  *
  * 2. Redistributions in binary form must reproduce the above copyright
  *    notice, this list of conditions and the following disclaimer in
  *    the documentation and/or other materials provided with the
  *    distribution.
  *
  * 3. All advertising materials mentioning features or use of this
  *    software must display the following acknowledgment:
  *    "This product includes software developed by the OpenSSL Project
  *    for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
  *
  * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
  *    endorse or promote products derived from this software without
  *    prior written permission. For written permission, please contact
  *    openssl-core@openssl.org.
  *
  * 5. Products derived from this software may not be called "OpenSSL"
  *    nor may "OpenSSL" appear in their names without prior written
  *    permission of the OpenSSL Project.
  *
  * 6. Redistributions of any form whatsoever must retain the following
  *    acknowledgment:
  *    "This product includes software developed by the OpenSSL Project
  *    for use in the OpenSSL Toolkit (http://www.openssl.org/)"
  *
  * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
  * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
  * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
  * PURPOSE ARE DISCLAIMED.  IN NO EVENT SHALL THE OpenSSL PROJECT OR
  * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
  * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
  * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
  * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
  * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
  * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
  * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
  * OF THE POSSIBILITY OF SUCH DAMAGE.
  * ====================================================================
  *
  * This product includes cryptographic software written by Eric Young
  * (eay@cryptsoft.com).  This product includes software written by Tim
  * Hudson (tjh@cryptsoft.com).
  *
  */
 /* ====================================================================
  * Copyright 2002 Sun Microsystems, Inc. ALL RIGHTS RESERVED.
  *
  * Portions of the attached software ("Contribution") are developed by
  * SUN MICROSYSTEMS, INC., and are contributed to the OpenSSL project.
  *
  * The Contribution is licensed pursuant to the OpenSSL open source
  * license provided above.
  *
  * The elliptic curve binary polynomial software is originally written by
  * Sheueling Chang Shantz and Douglas Stebila of Sun Microsystems
  * Laboratories. */

 #include <openssl/ec.h>

 #include <string.h>

 #include <openssl/bn.h>
 #include <openssl/err.h>
 #include <openssl/mem.h>

 #include "internal.h"
 #include "../../internal.h"


 // 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 here 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_set_curve(EC_GROUP *group, const BIGNUM *p,
                                   const BIGNUM *a, const BIGNUM *b,
                                   BN_CTX *ctx) {
   // p must be a prime > 3
   if (BN_num_bits(p) <= 2 || !BN_is_odd(p)) {
     OPENSSL_PUT_ERROR(EC, EC_R_INVALID_FIELD);
     return 0;
   }

   int ret = 0;
   BN_CTX_start(ctx);
   BIGNUM *tmp = BN_CTX_get(ctx);
   if (tmp == NULL) {
     goto err;
   }

   if (!BN_MONT_CTX_set(&group->field, p, ctx) ||
       !ec_bignum_to_felem(group, &group->a, a) ||
       !ec_bignum_to_felem(group, &group->b, b) ||
       // Reuse Z from the generator to cache the value one.
       !ec_bignum_to_felem(group, &group->generator.raw.Z, BN_value_one())) {
     goto err;
   }

   // group->a_is_minus3
   if (!BN_copy(tmp, a) ||
       !BN_add_word(tmp, 3)) {
     goto err;
   }
   group->a_is_minus3 = (0 == BN_cmp(tmp, &group->field.N));

   ret = 1;

 err:
   BN_CTX_end(ctx);
   return ret;
 }

 int ec_GFp_simple_group_get_curve(const EC_GROUP *group, BIGNUM *p, BIGNUM *a,
                                   BIGNUM *b) {
   if ((p != NULL && !BN_copy(p, &group->field.N)) ||
       (a != NULL && !ec_felem_to_bignum(group, a, &group->a)) ||
       (b != NULL && !ec_felem_to_bignum(group, b, &group->b))) {
     return 0;
   }
   return 1;
 }

 void ec_GFp_simple_point_init(EC_JACOBIAN *point) {
   OPENSSL_memset(&point->X, 0, sizeof(EC_FELEM));
   OPENSSL_memset(&point->Y, 0, sizeof(EC_FELEM));
   OPENSSL_memset(&point->Z, 0, sizeof(EC_FELEM));
 }

 void ec_GFp_simple_point_copy(EC_JACOBIAN *dest, const EC_JACOBIAN *src) {
   OPENSSL_memcpy(&dest->X, &src->X, sizeof(EC_FELEM));
   OPENSSL_memcpy(&dest->Y, &src->Y, sizeof(EC_FELEM));
   OPENSSL_memcpy(&dest->Z, &src->Z, sizeof(EC_FELEM));
 }

 void ec_GFp_simple_point_set_to_infinity(const EC_GROUP *group,
                                          EC_JACOBIAN *point) {
   // Although it is strictly only necessary to zero Z, we zero the entire point
   // in case |point| was stack-allocated and yet to be initialized.
   ec_GFp_simple_point_init(point);
 }

 void ec_GFp_simple_invert(const EC_GROUP *group, EC_JACOBIAN *point) {
   ec_felem_neg(group, &point->Y, &point->Y);
 }

 int ec_GFp_simple_is_at_infinity(const EC_GROUP *group,
                                  const EC_JACOBIAN *point) {
   return ec_felem_non_zero_mask(group, &point->Z) == 0;
 }

 int ec_GFp_simple_is_on_curve(const EC_GROUP *group,
                               const EC_JACOBIAN *point) {
   // 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'.
   //
   // This function may be used when double-checking the secret result of a point
   // multiplication, so we proceed in constant-time.

   void (*const felem_mul)(const EC_GROUP *, EC_FELEM *r, const EC_FELEM *a,
                           const EC_FELEM *b) = group->meth->felem_mul;
   void (*const felem_sqr)(const EC_GROUP *, EC_FELEM *r, const EC_FELEM *a) =
       group->meth->felem_sqr;

   // rh := X^2
   EC_FELEM rh;
   felem_sqr(group, &rh, &point->X);

   EC_FELEM tmp, Z4, Z6;
   felem_sqr(group, &tmp, &point->Z);
   felem_sqr(group, &Z4, &tmp);
   felem_mul(group, &Z6, &Z4, &tmp);

   // rh := rh + a*Z^4
   if (group->a_is_minus3) {
     ec_felem_add(group, &tmp, &Z4, &Z4);
     ec_felem_add(group, &tmp, &tmp, &Z4);
     ec_felem_sub(group, &rh, &rh, &tmp);
   } else {
     felem_mul(group, &tmp, &Z4, &group->a);
     ec_felem_add(group, &rh, &rh, &tmp);
   }

   // rh := (rh + a*Z^4)*X
   felem_mul(group, &rh, &rh, &point->X);

   // rh := rh + b*Z^6
   felem_mul(group, &tmp, &group->b, &Z6);
   ec_felem_add(group, &rh, &rh, &tmp);

   // 'lh' := Y^2
   felem_sqr(group, &tmp, &point->Y);

   ec_felem_sub(group, &tmp, &tmp, &rh);
   BN_ULONG not_equal = ec_felem_non_zero_mask(group, &tmp);

   // If Z = 0, the point is infinity, which is always on the curve.
   BN_ULONG not_infinity = ec_felem_non_zero_mask(group, &point->Z);

   return 1 & ~(not_infinity & not_equal);
 }

 int ec_GFp_simple_points_equal(const EC_GROUP *group, const EC_JACOBIAN *a,
                                const EC_JACOBIAN *b) {
   // This function is implemented in constant-time for two reasons. First,
   // although EC points are usually public, their Jacobian Z coordinates may be
   // secret, or at least are not obviously public. Second, more complex
   // protocols will sometimes manipulate secret points.
   //
   // This does mean that we pay a 6M+2S Jacobian comparison when comparing two
   // publicly affine points costs no field operations at all. If needed, we can
   // restore this optimization by keeping better track of affine vs. Jacobian
   // forms. See https://crbug.com/boringssl/326.

   // If neither |a| or |b| is infinity, 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).

   void (*const felem_mul)(const EC_GROUP *, EC_FELEM *r, const EC_FELEM *a,
                           const EC_FELEM *b) = group->meth->felem_mul;
   void (*const felem_sqr)(const EC_GROUP *, EC_FELEM *r, const EC_FELEM *a) =
       group->meth->felem_sqr;

   EC_FELEM tmp1, tmp2, Za23, Zb23;
   felem_sqr(group, &Zb23, &b->Z);         // Zb23 = Z_b^2
   felem_mul(group, &tmp1, &a->X, &Zb23);  // tmp1 = X_a * Z_b^2
   felem_sqr(group, &Za23, &a->Z);         // Za23 = Z_a^2
   felem_mul(group, &tmp2, &b->X, &Za23);  // tmp2 = X_b * Z_a^2
   ec_felem_sub(group, &tmp1, &tmp1, &tmp2);
   const BN_ULONG x_not_equal = ec_felem_non_zero_mask(group, &tmp1);

   felem_mul(group, &Zb23, &Zb23, &b->Z);  // Zb23 = Z_b^3
   felem_mul(group, &tmp1, &a->Y, &Zb23);  // tmp1 = Y_a * Z_b^3
   felem_mul(group, &Za23, &Za23, &a->Z);  // Za23 = Z_a^3
   felem_mul(group, &tmp2, &b->Y, &Za23);  // tmp2 = Y_b * Z_a^3
   ec_felem_sub(group, &tmp1, &tmp1, &tmp2);
   const BN_ULONG y_not_equal = ec_felem_non_zero_mask(group, &tmp1);
   const BN_ULONG x_and_y_equal = ~(x_not_equal | y_not_equal);

   const BN_ULONG a_not_infinity = ec_felem_non_zero_mask(group, &a->Z);
   const BN_ULONG b_not_infinity = ec_felem_non_zero_mask(group, &b->Z);
   const BN_ULONG a_and_b_infinity = ~(a_not_infinity | b_not_infinity);

   const BN_ULONG equal =
       a_and_b_infinity | (a_not_infinity & b_not_infinity & x_and_y_equal);
   return equal & 1;
 }

 int ec_affine_jacobian_equal(const EC_GROUP *group, const EC_AFFINE *a,
                              const EC_JACOBIAN *b) {
   // If |b| is not infinity, we have to decide whether
   //     (X_a, Y_a) = (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, Y_b).

   void (*const felem_mul)(const EC_GROUP *, EC_FELEM *r, const EC_FELEM *a,
                           const EC_FELEM *b) = group->meth->felem_mul;
   void (*const felem_sqr)(const EC_GROUP *, EC_FELEM *r, const EC_FELEM *a) =
       group->meth->felem_sqr;

   EC_FELEM tmp, Zb2;
   felem_sqr(group, &Zb2, &b->Z);        // Zb2 = Z_b^2
   felem_mul(group, &tmp, &a->X, &Zb2);  // tmp = X_a * Z_b^2
   ec_felem_sub(group, &tmp, &tmp, &b->X);
   const BN_ULONG x_not_equal = ec_felem_non_zero_mask(group, &tmp);

   felem_mul(group, &tmp, &a->Y, &Zb2);  // tmp = Y_a * Z_b^2
   felem_mul(group, &tmp, &tmp, &b->Z);  // tmp = Y_a * Z_b^3
   ec_felem_sub(group, &tmp, &tmp, &b->Y);
   const BN_ULONG y_not_equal = ec_felem_non_zero_mask(group, &tmp);
   const BN_ULONG x_and_y_equal = ~(x_not_equal | y_not_equal);

   const BN_ULONG b_not_infinity = ec_felem_non_zero_mask(group, &b->Z);

   const BN_ULONG equal = b_not_infinity & x_and_y_equal;
   return equal & 1;
 }

 int ec_GFp_simple_cmp_x_coordinate(const EC_GROUP *group, const EC_JACOBIAN *p,
                                    const EC_SCALAR *r) {
   if (ec_GFp_simple_is_at_infinity(group, p)) {
     // |ec_get_x_coordinate_as_scalar| will check this internally, but this way
     // we do not push to the error queue.
     return 0;
   }

   EC_SCALAR x;
   return ec_get_x_coordinate_as_scalar(group, &x, p) &&
          ec_scalar_equal_vartime(group, &x, r);
 }

 void ec_GFp_simple_felem_to_bytes(const EC_GROUP *group, uint8_t *out,
                                   size_t *out_len, const EC_FELEM *in) {
   size_t len = BN_num_bytes(&group->field.N);
   bn_words_to_big_endian(out, len, in->words, group->field.N.width);
   *out_len = len;
 }

 int ec_GFp_simple_felem_from_bytes(const EC_GROUP *group, EC_FELEM *out,
                                    const uint8_t *in, size_t len) {
   if (len != BN_num_bytes(&group->field.N)) {
     OPENSSL_PUT_ERROR(EC, EC_R_DECODE_ERROR);
     return 0;
   }

   bn_big_endian_to_words(out->words, group->field.N.width, in, len);

   if (!bn_less_than_words(out->words, group->field.N.d, group->field.N.width)) {
     OPENSSL_PUT_ERROR(EC, EC_R_DECODE_ERROR);
     return 0;
   }

   return 1;
 }
	/* Originally written by Bodo Moeller for the OpenSSL project.
	* ====================================================================
	* Copyright (c) 1998-2005 The OpenSSL Project. All rights reserved.
	*
	* Redistribution and use in source and binary forms, with or without
	* modification, are permitted provided that the following conditions
	* are met:
	*
	* 1. Redistributions of source code must retain the above copyright
	* notice, this list of conditions and the following disclaimer.
	*
	* 2. Redistributions in binary form must reproduce the above copyright
	* notice, this list of conditions and the following disclaimer in
	* the documentation and/or other materials provided with the
	* distribution.
	*
	* 3. All advertising materials mentioning features or use of this
	* software must display the following acknowledgment:
	* "This product includes software developed by the OpenSSL Project
	* for use in the OpenSSL Toolkit. (http://www.openssl.org/)"
	*
	* 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to
	* endorse or promote products derived from this software without
	* prior written permission. For written permission, please contact
	* openssl-core@openssl.org.
	*
	* 5. Products derived from this software may not be called "OpenSSL"
	* nor may "OpenSSL" appear in their names without prior written
	* permission of the OpenSSL Project.
	*
	* 6. Redistributions of any form whatsoever must retain the following
	* acknowledgment:
	* "This product includes software developed by the OpenSSL Project
	* for use in the OpenSSL Toolkit (http://www.openssl.org/)"
	*
	* THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY
	* EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
	* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR
	* PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR
	* ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
	* SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
	* NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
	* LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
	* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
	* STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
	* ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED
	* OF THE POSSIBILITY OF SUCH DAMAGE.
	* ====================================================================
	*
	* This product includes cryptographic software written by Eric Young
	* (eay@cryptsoft.com). This product includes software written by Tim
	* Hudson (tjh@cryptsoft.com).
	*
	*/
	/* ====================================================================
	* Copyright 2002 Sun Microsystems, Inc. ALL RIGHTS RESERVED.
	*
	* Portions of the attached software ("Contribution") are developed by
	* SUN MICROSYSTEMS, INC., and are contributed to the OpenSSL project.
	*
	* The Contribution is licensed pursuant to the OpenSSL open source
	* license provided above.
	*
	* The elliptic curve binary polynomial software is originally written by
	* Sheueling Chang Shantz and Douglas Stebila of Sun Microsystems
	* Laboratories. */

	#include <openssl/ec.h>

	#include <string.h>

	#include <openssl/bn.h>
	#include <openssl/err.h>
	#include <openssl/mem.h>

	#include "internal.h"
	#include "../../internal.h"


	// 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 here 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_set_curve(EC_GROUP group, const BIGNUM p,
	const BIGNUM a, const BIGNUM b,
	BN_CTX *ctx) {
	// p must be a prime > 3
	if (BN_num_bits(p) <= 2 \|\| !BN_is_odd(p)) {
	OPENSSL_PUT_ERROR(EC, EC_R_INVALID_FIELD);
	return 0;
	}

	int ret = 0;
	BN_CTX_start(ctx);
	BIGNUM *tmp = BN_CTX_get(ctx);
	if (tmp == NULL) {
	goto err;
	}

	if (!BN_MONT_CTX_set(&group->field, p, ctx) \|\|
	!ec_bignum_to_felem(group, &group->a, a) \|\|
	!ec_bignum_to_felem(group, &group->b, b) \|\|
	// Reuse Z from the generator to cache the value one.
	!ec_bignum_to_felem(group, &group->generator.raw.Z, BN_value_one())) {
	goto err;
	}

	// group->a_is_minus3
	if (!BN_copy(tmp, a) \|\|
	!BN_add_word(tmp, 3)) {
	goto err;
	}
	group->a_is_minus3 = (0 == BN_cmp(tmp, &group->field.N));

	ret = 1;

	err:
	BN_CTX_end(ctx);
	return ret;
	}

	int ec_GFp_simple_group_get_curve(const EC_GROUP group, BIGNUM p, BIGNUM *a,
	BIGNUM *b) {
	if ((p != NULL && !BN_copy(p, &group->field.N)) \|\|
	(a != NULL && !ec_felem_to_bignum(group, a, &group->a)) \|\|
	(b != NULL && !ec_felem_to_bignum(group, b, &group->b))) {
	return 0;
	}
	return 1;
	}

	void ec_GFp_simple_point_init(EC_JACOBIAN *point) {
	OPENSSL_memset(&point->X, 0, sizeof(EC_FELEM));
	OPENSSL_memset(&point->Y, 0, sizeof(EC_FELEM));
	OPENSSL_memset(&point->Z, 0, sizeof(EC_FELEM));
	}

	void ec_GFp_simple_point_copy(EC_JACOBIAN dest, const EC_JACOBIAN src) {
	OPENSSL_memcpy(&dest->X, &src->X, sizeof(EC_FELEM));
	OPENSSL_memcpy(&dest->Y, &src->Y, sizeof(EC_FELEM));
	OPENSSL_memcpy(&dest->Z, &src->Z, sizeof(EC_FELEM));
	}

	void ec_GFp_simple_point_set_to_infinity(const EC_GROUP *group,
	EC_JACOBIAN *point) {
	// Although it is strictly only necessary to zero Z, we zero the entire point
	// in case \|point\| was stack-allocated and yet to be initialized.
	ec_GFp_simple_point_init(point);
	}

	void ec_GFp_simple_invert(const EC_GROUP group, EC_JACOBIAN point) {
	ec_felem_neg(group, &point->Y, &point->Y);
	}

	int ec_GFp_simple_is_at_infinity(const EC_GROUP *group,
	const EC_JACOBIAN *point) {
	return ec_felem_non_zero_mask(group, &point->Z) == 0;
	}

	int ec_GFp_simple_is_on_curve(const EC_GROUP *group,
	const EC_JACOBIAN *point) {
	// 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 + aXZ^4 + b*Z^6.
	// To test this, we add up the right-hand side in 'rh'.
	//
	// This function may be used when double-checking the secret result of a point
	// multiplication, so we proceed in constant-time.

	void (const felem_mul)(const EC_GROUP , EC_FELEM r, const EC_FELEM a,
	const EC_FELEM *b) = group->meth->felem_mul;
	void (const felem_sqr)(const EC_GROUP , EC_FELEM r, const EC_FELEM a) =
	group->meth->felem_sqr;

	// rh := X^2
	EC_FELEM rh;
	felem_sqr(group, &rh, &point->X);

	EC_FELEM tmp, Z4, Z6;
	felem_sqr(group, &tmp, &point->Z);
	felem_sqr(group, &Z4, &tmp);
	felem_mul(group, &Z6, &Z4, &tmp);

	// rh := rh + a*Z^4
	if (group->a_is_minus3) {
	ec_felem_add(group, &tmp, &Z4, &Z4);
	ec_felem_add(group, &tmp, &tmp, &Z4);
	ec_felem_sub(group, &rh, &rh, &tmp);
	} else {
	felem_mul(group, &tmp, &Z4, &group->a);
	ec_felem_add(group, &rh, &rh, &tmp);
	}

	// rh := (rh + aZ^4)X
	felem_mul(group, &rh, &rh, &point->X);

	// rh := rh + b*Z^6
	felem_mul(group, &tmp, &group->b, &Z6);
	ec_felem_add(group, &rh, &rh, &tmp);

	// 'lh' := Y^2
	felem_sqr(group, &tmp, &point->Y);

	ec_felem_sub(group, &tmp, &tmp, &rh);
	BN_ULONG not_equal = ec_felem_non_zero_mask(group, &tmp);

	// If Z = 0, the point is infinity, which is always on the curve.
	BN_ULONG not_infinity = ec_felem_non_zero_mask(group, &point->Z);

	return 1 & ~(not_infinity & not_equal);
	}

	int ec_GFp_simple_points_equal(const EC_GROUP group, const EC_JACOBIAN a,
	const EC_JACOBIAN *b) {
	// This function is implemented in constant-time for two reasons. First,
	// although EC points are usually public, their Jacobian Z coordinates may be
	// secret, or at least are not obviously public. Second, more complex
	// protocols will sometimes manipulate secret points.
	//
	// This does mean that we pay a 6M+2S Jacobian comparison when comparing two
	// publicly affine points costs no field operations at all. If needed, we can
	// restore this optimization by keeping better track of affine vs. Jacobian
	// forms. See https://crbug.com/boringssl/326.

	// If neither \|a\| or \|b\| is infinity, 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_aZ_b^2, Y_aZ_b^3) = (X_bZ_a^2, Y_bZ_a^3).

	void (const felem_mul)(const EC_GROUP , EC_FELEM r, const EC_FELEM a,
	const EC_FELEM *b) = group->meth->felem_mul;
	void (const felem_sqr)(const EC_GROUP , EC_FELEM r, const EC_FELEM a) =
	group->meth->felem_sqr;

	EC_FELEM tmp1, tmp2, Za23, Zb23;
	felem_sqr(group, &Zb23, &b->Z); // Zb23 = Z_b^2
	felem_mul(group, &tmp1, &a->X, &Zb23); // tmp1 = X_a * Z_b^2
	felem_sqr(group, &Za23, &a->Z); // Za23 = Z_a^2
	felem_mul(group, &tmp2, &b->X, &Za23); // tmp2 = X_b * Z_a^2
	ec_felem_sub(group, &tmp1, &tmp1, &tmp2);
	const BN_ULONG x_not_equal = ec_felem_non_zero_mask(group, &tmp1);

	felem_mul(group, &Zb23, &Zb23, &b->Z); // Zb23 = Z_b^3
	felem_mul(group, &tmp1, &a->Y, &Zb23); // tmp1 = Y_a * Z_b^3
	felem_mul(group, &Za23, &Za23, &a->Z); // Za23 = Z_a^3
	felem_mul(group, &tmp2, &b->Y, &Za23); // tmp2 = Y_b * Z_a^3
	ec_felem_sub(group, &tmp1, &tmp1, &tmp2);
	const BN_ULONG y_not_equal = ec_felem_non_zero_mask(group, &tmp1);
	const BN_ULONG x_and_y_equal = ~(x_not_equal \| y_not_equal);

	const BN_ULONG a_not_infinity = ec_felem_non_zero_mask(group, &a->Z);
	const BN_ULONG b_not_infinity = ec_felem_non_zero_mask(group, &b->Z);
	const BN_ULONG a_and_b_infinity = ~(a_not_infinity \| b_not_infinity);

	const BN_ULONG equal =
	a_and_b_infinity \| (a_not_infinity & b_not_infinity & x_and_y_equal);
	return equal & 1;
	}

	int ec_affine_jacobian_equal(const EC_GROUP group, const EC_AFFINE a,
	const EC_JACOBIAN *b) {
	// If \|b\| is not infinity, we have to decide whether
	// (X_a, Y_a) = (X_b/Z_b^2, Y_b/Z_b^3),
	// or equivalently, whether
	// (X_aZ_b^2, Y_aZ_b^3) = (X_b, Y_b).

	void (const felem_mul)(const EC_GROUP , EC_FELEM r, const EC_FELEM a,
	const EC_FELEM *b) = group->meth->felem_mul;
	void (const felem_sqr)(const EC_GROUP , EC_FELEM r, const EC_FELEM a) =
	group->meth->felem_sqr;

	EC_FELEM tmp, Zb2;
	felem_sqr(group, &Zb2, &b->Z); // Zb2 = Z_b^2
	felem_mul(group, &tmp, &a->X, &Zb2); // tmp = X_a * Z_b^2
	ec_felem_sub(group, &tmp, &tmp, &b->X);
	const BN_ULONG x_not_equal = ec_felem_non_zero_mask(group, &tmp);

	felem_mul(group, &tmp, &a->Y, &Zb2); // tmp = Y_a * Z_b^2
	felem_mul(group, &tmp, &tmp, &b->Z); // tmp = Y_a * Z_b^3
	ec_felem_sub(group, &tmp, &tmp, &b->Y);
	const BN_ULONG y_not_equal = ec_felem_non_zero_mask(group, &tmp);
	const BN_ULONG x_and_y_equal = ~(x_not_equal \| y_not_equal);

	const BN_ULONG b_not_infinity = ec_felem_non_zero_mask(group, &b->Z);

	const BN_ULONG equal = b_not_infinity & x_and_y_equal;
	return equal & 1;
	}

	int ec_GFp_simple_cmp_x_coordinate(const EC_GROUP group, const EC_JACOBIAN p,
	const EC_SCALAR *r) {
	if (ec_GFp_simple_is_at_infinity(group, p)) {
	// \|ec_get_x_coordinate_as_scalar\| will check this internally, but this way
	// we do not push to the error queue.
	return 0;
	}

	EC_SCALAR x;
	return ec_get_x_coordinate_as_scalar(group, &x, p) &&
	ec_scalar_equal_vartime(group, &x, r);
	}

	void ec_GFp_simple_felem_to_bytes(const EC_GROUP group, uint8_t out,
	size_t out_len, const EC_FELEM in) {
	size_t len = BN_num_bytes(&group->field.N);
	bn_words_to_big_endian(out, len, in->words, group->field.N.width);
	*out_len = len;
	}

	int ec_GFp_simple_felem_from_bytes(const EC_GROUP group, EC_FELEM out,
	const uint8_t *in, size_t len) {
	if (len != BN_num_bytes(&group->field.N)) {
	OPENSSL_PUT_ERROR(EC, EC_R_DECODE_ERROR);
	return 0;
	}

	bn_big_endian_to_words(out->words, group->field.N.width, in, len);

	if (!bn_less_than_words(out->words, group->field.N.d, group->field.N.width)) {
	OPENSSL_PUT_ERROR(EC, EC_R_DECODE_ERROR);
	return 0;
	}

	return 1;
	}