From 3e00b4c9db42818c621f609e70569c7d9ae85717 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Bodo=20M=C3=B6ller?= Date: Tue, 18 Oct 2011 19:43:16 +0000 Subject: Improve optional 64-bit NIST-P224 implementation, and add NIST-P256 and NIST-P521. (Now -DEC_NISTP_64_GCC_128 enables all three of these; -DEC_NISTP224_64_GCC_128 no longer works.) Submitted by: Google Inc. --- crypto/ec/Makefile | 9 +- crypto/ec/ec.h | 22 +- crypto/ec/ec_curve.c | 13 +- crypto/ec/ec_err.c | 10 +- crypto/ec/ec_lcl.h | 48 +- crypto/ec/ecp_nistp224.c | 1254 ++++++++++++++------------ crypto/ec/ecp_nistp256.c | 2158 +++++++++++++++++++++++++++++++++++++++++++++ crypto/ec/ecp_nistp521.c | 2017 ++++++++++++++++++++++++++++++++++++++++++ crypto/ec/ecp_nistputil.c | 196 ++++ crypto/ec/ectest.c | 173 ++-- 10 files changed, 5286 insertions(+), 614 deletions(-) create mode 100644 crypto/ec/ecp_nistp256.c create mode 100644 crypto/ec/ecp_nistp521.c create mode 100644 crypto/ec/ecp_nistputil.c (limited to 'crypto') diff --git a/crypto/ec/Makefile b/crypto/ec/Makefile index 40262475ca..3eb04c0cfd 100644 --- a/crypto/ec/Makefile +++ b/crypto/ec/Makefile @@ -20,12 +20,14 @@ LIB=$(TOP)/libcrypto.a LIBSRC= ec_lib.c ecp_smpl.c ecp_mont.c ecp_nist.c ec_cvt.c ec_mult.c\ ec_err.c ec_curve.c ec_check.c ec_print.c ec_asn1.c ec_key.c\ ec2_smpl.c ec2_mult.c ec_ameth.c ec_pmeth.c eck_prn.c \ - ecp_nistp224.c ecp_oct.c ec2_oct.c ec_oct.c + ecp_nistp224.c ecp_nistp256.c ecp_nistp521.c ecp_nistputil.c \ + ecp_oct.c ec2_oct.c ec_oct.c LIBOBJ= ec_lib.o ecp_smpl.o ecp_mont.o ecp_nist.o ec_cvt.o ec_mult.o\ ec_err.o ec_curve.o ec_check.o ec_print.o ec_asn1.o ec_key.o\ ec2_smpl.o ec2_mult.o ec_ameth.o ec_pmeth.o eck_prn.o \ - ecp_nistp224.o ecp_oct.o ec2_oct.o ec_oct.o + ecp_nistp224.o ecp_nistp256.o ecp_nistp521.o ecp_nistputil.o \ + ecp_oct.o ec2_oct.o ec_oct.o SRC= $(LIBSRC) @@ -240,6 +242,9 @@ ecp_nist.o: ../../include/openssl/opensslv.h ../../include/openssl/ossl_typ.h ecp_nist.o: ../../include/openssl/safestack.h ../../include/openssl/stack.h ecp_nist.o: ../../include/openssl/symhacks.h ec_lcl.h ecp_nist.c ecp_nistp224.o: ../../include/openssl/opensslconf.h ecp_nistp224.c +ecp_nistp256.o: ../../include/openssl/opensslconf.h ecp_nistp256.c +ecp_nistp521.o: ../../include/openssl/opensslconf.h ecp_nistp521.c +ecp_nistputil.o: ../../include/openssl/opensslconf.h ecp_nistputil.c ecp_oct.o: ../../include/openssl/asn1.h ../../include/openssl/bio.h ecp_oct.o: ../../include/openssl/bn.h ../../include/openssl/crypto.h ecp_oct.o: ../../include/openssl/e_os2.h ../../include/openssl/ec.h diff --git a/crypto/ec/ec.h b/crypto/ec/ec.h index 0a380d61a3..1f46b5233b 100644 --- a/crypto/ec/ec.h +++ b/crypto/ec/ec.h @@ -151,12 +151,22 @@ const EC_METHOD *EC_GFp_mont_method(void); */ const EC_METHOD *EC_GFp_nist_method(void); -#ifndef OPENSSL_NO_EC_NISTP224_64_GCC_128 +#ifdef EC_NISTP_64_GCC_128 +#ifndef OPENSSL_SYS_WIN32 /** Returns 64-bit optimized methods for nistp224 * \return EC_METHOD object */ -#ifndef OPENSSL_SYS_WIN32 const EC_METHOD *EC_GFp_nistp224_method(void); + +/** Returns 64-bit optimized methods for nistp256 + * \return EC_METHOD object + */ +const EC_METHOD *EC_GFp_nistp256_method(void); + +/** Returns 64-bit optimized methods for nistp521 + * \return EC_METHOD object + */ +const EC_METHOD *EC_GFp_nistp521_method(void); #endif #endif @@ -1005,6 +1015,12 @@ void ERR_load_EC_strings(void); #define EC_F_EC_GFP_NISTP224_GROUP_SET_CURVE 225 #define EC_F_EC_GFP_NISTP224_POINTS_MUL 228 #define EC_F_EC_GFP_NISTP224_POINT_GET_AFFINE_COORDINATES 226 +#define EC_F_EC_GFP_NISTP256_GROUP_SET_CURVE 230 +#define EC_F_EC_GFP_NISTP256_POINTS_MUL 231 +#define EC_F_EC_GFP_NISTP256_POINT_GET_AFFINE_COORDINATES 232 +#define EC_F_EC_GFP_NISTP521_GROUP_SET_CURVE 233 +#define EC_F_EC_GFP_NISTP521_POINTS_MUL 234 +#define EC_F_EC_GFP_NISTP521_POINT_GET_AFFINE_COORDINATES 235 #define EC_F_EC_GFP_NIST_FIELD_MUL 200 #define EC_F_EC_GFP_NIST_FIELD_SQR 201 #define EC_F_EC_GFP_NIST_GROUP_SET_CURVE 202 @@ -1079,6 +1095,8 @@ void ERR_load_EC_strings(void); #define EC_F_I2D_ECPRIVATEKEY 192 #define EC_F_I2O_ECPUBLICKEY 151 #define EC_F_NISTP224_PRE_COMP_NEW 227 +#define EC_F_NISTP256_PRE_COMP_NEW 236 +#define EC_F_NISTP521_PRE_COMP_NEW 237 #define EC_F_O2I_ECPUBLICKEY 152 #define EC_F_OLD_EC_PRIV_DECODE 222 #define EC_F_PKEY_EC_CTRL 197 diff --git a/crypto/ec/ec_curve.c b/crypto/ec/ec_curve.c index a51a545500..f8722f8f51 100644 --- a/crypto/ec/ec_curve.c +++ b/crypto/ec/ec_curve.c @@ -1903,16 +1903,19 @@ static const ec_list_element curve_list[] = { /* SECG secp192r1 is the same as X9.62 prime192v1 and hence omitted */ { NID_secp192k1, &_EC_SECG_PRIME_192K1.h, 0, "SECG curve over a 192 bit prime field" }, { NID_secp224k1, &_EC_SECG_PRIME_224K1.h, 0, "SECG curve over a 224 bit prime field" }, -#ifdef EC_NISTP224_64_GCC_128 - { NID_secp224r1, &_EC_NIST_PRIME_224.h, EC_GFp_nistp224_method, "NIST/SECG curve over a 224 bit prime field,\n" - "\t\t64-bit optimized implementation." }, +#ifdef EC_NISTP_64_GCC_128 + { NID_secp224r1, &_EC_NIST_PRIME_224.h, EC_GFp_nistp224_method, "NIST/SECG curve over a 224 bit prime field" }, #else { NID_secp224r1, &_EC_NIST_PRIME_224.h, 0, "NIST/SECG curve over a 224 bit prime field" }, #endif { NID_secp256k1, &_EC_SECG_PRIME_256K1.h, 0, "SECG curve over a 256 bit prime field" }, /* SECG secp256r1 is the same as X9.62 prime256v1 and hence omitted */ { NID_secp384r1, &_EC_NIST_PRIME_384.h, 0, "NIST/SECG curve over a 384 bit prime field" }, +#ifdef EC_NISTP_64_GCC_128 + { NID_secp521r1, &_EC_NIST_PRIME_521.h, EC_GFp_nistp521_method, "NIST/SECG curve over a 521 bit prime field" }, +#else { NID_secp521r1, &_EC_NIST_PRIME_521.h, 0, "NIST/SECG curve over a 521 bit prime field" }, +#endif /* X9.62 curves */ { NID_X9_62_prime192v1, &_EC_NIST_PRIME_192.h, 0, "NIST/X9.62/SECG curve over a 192 bit prime field" }, { NID_X9_62_prime192v2, &_EC_X9_62_PRIME_192V2.h, 0, "X9.62 curve over a 192 bit prime field" }, @@ -1920,7 +1923,11 @@ static const ec_list_element curve_list[] = { { NID_X9_62_prime239v1, &_EC_X9_62_PRIME_239V1.h, 0, "X9.62 curve over a 239 bit prime field" }, { NID_X9_62_prime239v2, &_EC_X9_62_PRIME_239V2.h, 0, "X9.62 curve over a 239 bit prime field" }, { NID_X9_62_prime239v3, &_EC_X9_62_PRIME_239V3.h, 0, "X9.62 curve over a 239 bit prime field" }, +#ifdef EC_NISTP_64_GCC_128 + { NID_X9_62_prime256v1, &_EC_X9_62_PRIME_256V1.h, EC_GFp_nistp256_method, "X9.62/SECG curve over a 256 bit prime field" }, +#else { NID_X9_62_prime256v1, &_EC_X9_62_PRIME_256V1.h, 0, "X9.62/SECG curve over a 256 bit prime field" }, +#endif #ifndef OPENSSL_NO_EC2M /* characteristic two field curves */ /* NIST/SECG curves */ diff --git a/crypto/ec/ec_err.c b/crypto/ec/ec_err.c index 15d539cbf4..0d19398731 100644 --- a/crypto/ec/ec_err.c +++ b/crypto/ec/ec_err.c @@ -1,6 +1,6 @@ /* crypto/ec/ec_err.c */ /* ==================================================================== - * Copyright (c) 1999-2010 The OpenSSL Project. All rights reserved. + * Copyright (c) 1999-2011 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 @@ -116,6 +116,12 @@ static ERR_STRING_DATA EC_str_functs[]= {ERR_FUNC(EC_F_EC_GFP_NISTP224_GROUP_SET_CURVE), "ec_GFp_nistp224_group_set_curve"}, {ERR_FUNC(EC_F_EC_GFP_NISTP224_POINTS_MUL), "ec_GFp_nistp224_points_mul"}, {ERR_FUNC(EC_F_EC_GFP_NISTP224_POINT_GET_AFFINE_COORDINATES), "ec_GFp_nistp224_point_get_affine_coordinates"}, +{ERR_FUNC(EC_F_EC_GFP_NISTP256_GROUP_SET_CURVE), "ec_GFp_nistp256_group_set_curve"}, +{ERR_FUNC(EC_F_EC_GFP_NISTP256_POINTS_MUL), "ec_GFp_nistp256_points_mul"}, +{ERR_FUNC(EC_F_EC_GFP_NISTP256_POINT_GET_AFFINE_COORDINATES), "ec_GFp_nistp256_point_get_affine_coordinates"}, +{ERR_FUNC(EC_F_EC_GFP_NISTP521_GROUP_SET_CURVE), "ec_GFp_nistp521_group_set_curve"}, +{ERR_FUNC(EC_F_EC_GFP_NISTP521_POINTS_MUL), "ec_GFp_nistp521_points_mul"}, +{ERR_FUNC(EC_F_EC_GFP_NISTP521_POINT_GET_AFFINE_COORDINATES), "ec_GFp_nistp521_point_get_affine_coordinates"}, {ERR_FUNC(EC_F_EC_GFP_NIST_FIELD_MUL), "ec_GFp_nist_field_mul"}, {ERR_FUNC(EC_F_EC_GFP_NIST_FIELD_SQR), "ec_GFp_nist_field_sqr"}, {ERR_FUNC(EC_F_EC_GFP_NIST_GROUP_SET_CURVE), "ec_GFp_nist_group_set_curve"}, @@ -190,6 +196,8 @@ static ERR_STRING_DATA EC_str_functs[]= {ERR_FUNC(EC_F_I2D_ECPRIVATEKEY), "i2d_ECPrivateKey"}, {ERR_FUNC(EC_F_I2O_ECPUBLICKEY), "i2o_ECPublicKey"}, {ERR_FUNC(EC_F_NISTP224_PRE_COMP_NEW), "NISTP224_PRE_COMP_NEW"}, +{ERR_FUNC(EC_F_NISTP256_PRE_COMP_NEW), "NISTP256_PRE_COMP_NEW"}, +{ERR_FUNC(EC_F_NISTP521_PRE_COMP_NEW), "NISTP521_PRE_COMP_NEW"}, {ERR_FUNC(EC_F_O2I_ECPUBLICKEY), "o2i_ECPublicKey"}, {ERR_FUNC(EC_F_OLD_EC_PRIV_DECODE), "OLD_EC_PRIV_DECODE"}, {ERR_FUNC(EC_F_PKEY_EC_CTRL), "PKEY_EC_CTRL"}, diff --git a/crypto/ec/ec_lcl.h b/crypto/ec/ec_lcl.h index afa1efa4bc..72c0638a18 100644 --- a/crypto/ec/ec_lcl.h +++ b/crypto/ec/ec_lcl.h @@ -398,15 +398,49 @@ int ec_GF2m_simple_mul(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar, int ec_GF2m_precompute_mult(EC_GROUP *group, BN_CTX *ctx); int ec_GF2m_have_precompute_mult(const EC_GROUP *group); -#ifdef EC_NISTP224_64_GCC_128 +#ifdef EC_NISTP_64_GCC_128 +/* method functions in ec2_mult.c */ +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 *); +int ec_GF2m_precompute_mult(EC_GROUP *group, BN_CTX *ctx); +int ec_GF2m_have_precompute_mult(const EC_GROUP *group); + /* method functions in ecp_nistp224.c */ int ec_GFp_nistp224_group_init(EC_GROUP *group); -int ec_GFp_nistp224_group_set_curve(EC_GROUP *group, const BIGNUM *p, - const BIGNUM *a, const BIGNUM *n, BN_CTX *); -int ec_GFp_nistp224_point_get_affine_coordinates(const EC_GROUP *group, - const EC_POINT *point, BIGNUM *x, BIGNUM *y, BN_CTX *ctx); -int ec_GFp_nistp224_points_mul(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar, - size_t num, const EC_POINT *points[], const BIGNUM *scalars[], BN_CTX *); +int ec_GFp_nistp224_group_set_curve(EC_GROUP *group, const BIGNUM *p, const BIGNUM *a, const BIGNUM *n, BN_CTX *); +int ec_GFp_nistp224_point_get_affine_coordinates(const EC_GROUP *group, const EC_POINT *point, BIGNUM *x, BIGNUM *y, BN_CTX *ctx); +int ec_GFp_nistp224_mul(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar, size_t num, const EC_POINT *points[], const BIGNUM *scalars[], BN_CTX *); +int ec_GFp_nistp224_points_mul(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar, size_t num, const EC_POINT *points[], const BIGNUM *scalars[], BN_CTX *ctx); int ec_GFp_nistp224_precompute_mult(EC_GROUP *group, BN_CTX *ctx); int ec_GFp_nistp224_have_precompute_mult(const EC_GROUP *group); + +/* method functions in ecp_nistp256.c */ +int ec_GFp_nistp256_group_init(EC_GROUP *group); +int ec_GFp_nistp256_group_set_curve(EC_GROUP *group, const BIGNUM *p, const BIGNUM *a, const BIGNUM *n, BN_CTX *); +int ec_GFp_nistp256_point_get_affine_coordinates(const EC_GROUP *group, const EC_POINT *point, BIGNUM *x, BIGNUM *y, BN_CTX *ctx); +int ec_GFp_nistp256_mul(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar, size_t num, const EC_POINT *points[], const BIGNUM *scalars[], BN_CTX *); +int ec_GFp_nistp256_points_mul(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar, size_t num, const EC_POINT *points[], const BIGNUM *scalars[], BN_CTX *ctx); +int ec_GFp_nistp256_precompute_mult(EC_GROUP *group, BN_CTX *ctx); +int ec_GFp_nistp256_have_precompute_mult(const EC_GROUP *group); + +/* method functions in ecp_nistp521.c */ +int ec_GFp_nistp521_group_init(EC_GROUP *group); +int ec_GFp_nistp521_group_set_curve(EC_GROUP *group, const BIGNUM *p, const BIGNUM *a, const BIGNUM *n, BN_CTX *); +int ec_GFp_nistp521_point_get_affine_coordinates(const EC_GROUP *group, const EC_POINT *point, BIGNUM *x, BIGNUM *y, BN_CTX *ctx); +int ec_GFp_nistp521_mul(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar, size_t num, const EC_POINT *points[], const BIGNUM *scalars[], BN_CTX *); +int ec_GFp_nistp521_points_mul(const EC_GROUP *group, EC_POINT *r, const BIGNUM *scalar, size_t num, const EC_POINT *points[], const BIGNUM *scalars[], BN_CTX *ctx); +int ec_GFp_nistp521_precompute_mult(EC_GROUP *group, BN_CTX *ctx); +int ec_GFp_nistp521_have_precompute_mult(const EC_GROUP *group); + +/* utility functions in ecp_nistputil.c */ +void ec_GFp_nistp_points_make_affine_internal(size_t num, void *point_array, + size_t felem_size, void *tmp_felems, + void (*felem_one)(void *out), + int (*felem_is_zero)(const void *in), + void (*felem_assign)(void *out, const void *in), + void (*felem_square)(void *out, const void *in), + void (*felem_mul)(void *out, const void *in1, const void *in2), + void (*felem_inv)(void *out, const void *in), + void (*felem_contract)(void *out, const void *in)); +void ec_GFp_nistp_recode_scalar_bits(unsigned char *sign, unsigned char *digit, unsigned char in); #endif diff --git a/crypto/ec/ecp_nistp224.c b/crypto/ec/ecp_nistp224.c index 90c3589bdf..8b2c6d39c8 100644 --- a/crypto/ec/ecp_nistp224.c +++ b/crypto/ec/ecp_nistp224.c @@ -2,58 +2,20 @@ /* * Written by Emilia Kasper (Google) for the OpenSSL project. */ -/* ==================================================================== - * Copyright (c) 2000-2010 The OpenSSL Project. All rights reserved. +/* Copyright 2011 Google Inc. * - * Redistribution and use in source and binary forms, with or without - * modification, are permitted provided that the following conditions - * are met: + * Licensed under the Apache License, Version 2.0 (the "License"); * - * 1. Redistributions of source code must retain the above copyright - * notice, this list of conditions and the following disclaimer. + * you may not use this file except in compliance with the License. + * You may obtain a copy of the License at * - * 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 - * licensing@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). + * http://www.apache.org/licenses/LICENSE-2.0 * + * Unless required by applicable law or agreed to in writing, software + * distributed under the License is distributed on an "AS IS" BASIS, + * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. + * See the License for the specific language governing permissions and + * limitations under the License. */ /* @@ -62,8 +24,7 @@ * Inspired by Daniel J. Bernstein's public domain nistp224 implementation * and Adam Langley's public domain 64-bit C implementation of curve25519 */ -#include -#ifndef OPENSSL_NO_EC_NISTP224_64_GCC_128 +#ifdef EC_NISTP_64_GCC_128 #include #include #include @@ -77,28 +38,39 @@ #endif typedef uint8_t u8; +typedef uint64_t u64; +typedef int64_t s64; /******************************************************************************/ /* INTERNAL REPRESENTATION OF FIELD ELEMENTS * * Field elements are represented as a_0 + 2^56*a_1 + 2^112*a_2 + 2^168*a_3 - * where each slice a_i is a 64-bit word, i.e., a field element is an fslice - * array a with 4 elements, where a[i] = a_i. - * Outputs from multiplications are represented as unreduced polynomials + * using 64-bit coefficients called 'limbs', + * and sometimes (for multiplication results) as * b_0 + 2^56*b_1 + 2^112*b_2 + 2^168*b_3 + 2^224*b_4 + 2^280*b_5 + 2^336*b_6 - * where each b_i is a 128-bit word. We ensure that inputs to each field + * using 128-bit coefficients called 'widelimbs'. + * A 4-limb representation is an 'felem'; + * a 7-widelimb representation is a 'widefelem'. + * Even within felems, bits of adjacent limbs overlap, and we don't always + * reduce the representations: we ensure that inputs to each felem * multiplication satisfy a_i < 2^60, so outputs satisfy b_i < 4*2^60*2^60, * and fit into a 128-bit word without overflow. The coefficients are then - * again partially reduced to a_i < 2^57. We only reduce to the unique minimal - * representation at the end of the computation. - * + * again partially reduced to obtain an felem satisfying a_i < 2^57. + * We only reduce to the unique minimal representation at the end of the + * computation. */ -typedef uint64_t fslice; +typedef uint64_t limb; +typedef uint128_t widelimb; + +typedef limb felem[4]; +typedef widelimb widefelem[7]; /* Field element represented as a byte arrary. - * 28*8 = 224 bits is also the group order size for the elliptic curve. */ + * 28*8 = 224 bits is also the group order size for the elliptic curve, + * and we also use this type for scalars for point multiplication. + */ typedef u8 felem_bytearray[28]; static const felem_bytearray nistp224_curve_params[5] = { @@ -120,72 +92,143 @@ static const felem_bytearray nistp224_curve_params[5] = { }; /* Precomputed multiples of the standard generator - * b_0*G + b_1*2^56*G + b_2*2^112*G + b_3*2^168*G for - * (b_3, b_2, b_1, b_0) in [0,15], i.e., gmul[0] = point_at_infinity, - * gmul[1] = G, gmul[2] = 2^56*G, gmul[3] = 2^56*G + G, etc. - * Points are given in Jacobian projective coordinates: words 0-3 represent the - * X-coordinate (slice a_0 is word 0, etc.), words 4-7 represent the - * Y-coordinate and words 8-11 represent the Z-coordinate. */ -static const fslice gmul[16][3][4] = { - {{0x00000000000000, 0x00000000000000, 0x00000000000000, 0x00000000000000}, - {0x00000000000000, 0x00000000000000, 0x00000000000000, 0x00000000000000}, - {0x00000000000000, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, - {{0x3280d6115c1d21, 0xc1d356c2112234, 0x7f321390b94a03, 0xb70e0cbd6bb4bf}, - {0xd5819985007e34, 0x75a05a07476444, 0xfb4c22dfe6cd43, 0xbd376388b5f723}, - {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, - {{0xfd9675666ebbe9, 0xbca7664d40ce5e, 0x2242df8d8a2a43, 0x1f49bbb0f99bc5}, - {0x29e0b892dc9c43, 0xece8608436e662, 0xdc858f185310d0, 0x9812dd4eb8d321}, - {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, - {{0x6d3e678d5d8eb8, 0x559eed1cb362f1, 0x16e9a3bbce8a3f, 0xeedcccd8c2a748}, - {0xf19f90ed50266d, 0xabf2b4bf65f9df, 0x313865468fafec, 0x5cb379ba910a17}, - {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, - {{0x0641966cab26e3, 0x91fb2991fab0a0, 0xefec27a4e13a0b, 0x0499aa8a5f8ebe}, - {0x7510407766af5d, 0x84d929610d5450, 0x81d77aae82f706, 0x6916f6d4338c5b}, - {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, - {{0xea95ac3b1f15c6, 0x086000905e82d4, 0xdd323ae4d1c8b1, 0x932b56be7685a3}, - {0x9ef93dea25dbbf, 0x41665960f390f0, 0xfdec76dbe2a8a7, 0x523e80f019062a}, - {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, - {{0x822fdd26732c73, 0xa01c83531b5d0f, 0x363f37347c1ba4, 0xc391b45c84725c}, - {0xbbd5e1b2d6ad24, 0xddfbcde19dfaec, 0xc393da7e222a7f, 0x1efb7890ede244}, - {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, - {{0x4c9e90ca217da1, 0xd11beca79159bb, 0xff8d33c2c98b7c, 0x2610b39409f849}, - {0x44d1352ac64da0, 0xcdbb7b2c46b4fb, 0x966c079b753c89, 0xfe67e4e820b112}, - {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, - {{0xe28cae2df5312d, 0xc71b61d16f5c6e, 0x79b7619a3e7c4c, 0x05c73240899b47}, - {0x9f7f6382c73e3a, 0x18615165c56bda, 0x641fab2116fd56, 0x72855882b08394}, - {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, - {{0x0469182f161c09, 0x74a98ca8d00fb5, 0xb89da93489a3e0, 0x41c98768fb0c1d}, - {0xe5ea05fb32da81, 0x3dce9ffbca6855, 0x1cfe2d3fbf59e6, 0x0e5e03408738a7}, - {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, - {{0xdab22b2333e87f, 0x4430137a5dd2f6, 0xe03ab9f738beb8, 0xcb0c5d0dc34f24}, - {0x764a7df0c8fda5, 0x185ba5c3fa2044, 0x9281d688bcbe50, 0xc40331df893881}, - {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, - {{0xb89530796f0f60, 0xade92bd26909a3, 0x1a0c83fb4884da, 0x1765bf22a5a984}, - {0x772a9ee75db09e, 0x23bc6c67cec16f, 0x4c1edba8b14e2f, 0xe2a215d9611369}, - {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, - {{0x571e509fb5efb3, 0xade88696410552, 0xc8ae85fada74fe, 0x6c7e4be83bbde3}, - {0xff9f51160f4652, 0xb47ce2495a6539, 0xa2946c53b582f4, 0x286d2db3ee9a60}, - {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, - {{0x40bbd5081a44af, 0x0995183b13926c, 0xbcefba6f47f6d0, 0x215619e9cc0057}, - {0x8bc94d3b0df45e, 0xf11c54a3694f6f, 0x8631b93cdfe8b5, 0xe7e3f4b0982db9}, - {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, - {{0xb17048ab3e1c7b, 0xac38f36ff8a1d8, 0x1c29819435d2c6, 0xc813132f4c07e9}, - {0x2891425503b11f, 0x08781030579fea, 0xf5426ba5cc9674, 0x1e28ebf18562bc}, - {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}, - {{0x9f31997cc864eb, 0x06cd91d28b5e4c, 0xff17036691a973, 0xf1aef351497c58}, - {0xdd1f2d600564ff, 0xdead073b1402db, 0x74a684435bd693, 0xeea7471f962558}, - {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}} -}; + * Points are given in coordinates (X, Y, Z) where Z normally is 1 + * (0 for the point at infinity). + * For each field element, slice a_0 is word 0, etc. + * + * The table has 2 * 16 elements, starting with the following: + * index | bits | point + * ------+---------+------------------------------ + * 0 | 0 0 0 0 | 0G + * 1 | 0 0 0 1 | 1G + * 2 | 0 0 1 0 | 2^56G + * 3 | 0 0 1 1 | (2^56 + 1)G + * 4 | 0 1 0 0 | 2^112G + * 5 | 0 1 0 1 | (2^112 + 1)G + * 6 | 0 1 1 0 | (2^112 + 2^56)G + * 7 | 0 1 1 1 | (2^112 + 2^56 + 1)G + * 8 | 1 0 0 0 | 2^168G + * 9 | 1 0 0 1 | (2^168 + 1)G + * 10 | 1 0 1 0 | (2^168 + 2^56)G + * 11 | 1 0 1 1 | (2^168 + 2^56 + 1)G + * 12 | 1 1 0 0 | (2^168 + 2^112)G + * 13 | 1 1 0 1 | (2^168 + 2^112 + 1)G + * 14 | 1 1 1 0 | (2^168 + 2^112 + 2^56)G + * 15 | 1 1 1 1 | (2^168 + 2^112 + 2^56 + 1)G + * followed by a copy of this with each element multiplied by 2^28. + * + * The reason for this is so that we can clock bits into four different + * locations when doing simple scalar multiplies against the base point, + * and then another four locations using the second 16 elements. + */ +static const felem gmul[2][16][3] = +{{{{0, 0, 0, 0}, + {0, 0, 0, 0}, + {0, 0, 0, 0}}, + {{0x3280d6115c1d21, 0xc1d356c2112234, 0x7f321390b94a03, 0xb70e0cbd6bb4bf}, + {0xd5819985007e34, 0x75a05a07476444, 0xfb4c22dfe6cd43, 0xbd376388b5f723}, + {1, 0, 0, 0}}, + {{0xfd9675666ebbe9, 0xbca7664d40ce5e, 0x2242df8d8a2a43, 0x1f49bbb0f99bc5}, + {0x29e0b892dc9c43, 0xece8608436e662, 0xdc858f185310d0, 0x9812dd4eb8d321}, + {1, 0, 0, 0}}, + {{0x6d3e678d5d8eb8, 0x559eed1cb362f1, 0x16e9a3bbce8a3f, 0xeedcccd8c2a748}, + {0xf19f90ed50266d, 0xabf2b4bf65f9df, 0x313865468fafec, 0x5cb379ba910a17}, + {1, 0, 0, 0}}, + {{0x0641966cab26e3, 0x91fb2991fab0a0, 0xefec27a4e13a0b, 0x0499aa8a5f8ebe}, + {0x7510407766af5d, 0x84d929610d5450, 0x81d77aae82f706, 0x6916f6d4338c5b}, + {1, 0, 0, 0}}, + {{0xea95ac3b1f15c6, 0x086000905e82d4, 0xdd323ae4d1c8b1, 0x932b56be7685a3}, + {0x9ef93dea25dbbf, 0x41665960f390f0, 0xfdec76dbe2a8a7, 0x523e80f019062a}, + {1, 0, 0, 0}}, + {{0x822fdd26732c73, 0xa01c83531b5d0f, 0x363f37347c1ba4, 0xc391b45c84725c}, + {0xbbd5e1b2d6ad24, 0xddfbcde19dfaec, 0xc393da7e222a7f, 0x1efb7890ede244}, + {1, 0, 0, 0}}, + {{0x4c9e90ca217da1, 0xd11beca79159bb, 0xff8d33c2c98b7c, 0x2610b39409f849}, + {0x44d1352ac64da0, 0xcdbb7b2c46b4fb, 0x966c079b753c89, 0xfe67e4e820b112}, + {1, 0, 0, 0}}, + {{0xe28cae2df5312d, 0xc71b61d16f5c6e, 0x79b7619a3e7c4c, 0x05c73240899b47}, + {0x9f7f6382c73e3a, 0x18615165c56bda, 0x641fab2116fd56, 0x72855882b08394}, + {1, 0, 0, 0}}, + {{0x0469182f161c09, 0x74a98ca8d00fb5, 0xb89da93489a3e0, 0x41c98768fb0c1d}, + {0xe5ea05fb32da81, 0x3dce9ffbca6855, 0x1cfe2d3fbf59e6, 0x0e5e03408738a7}, + {1, 0, 0, 0}}, + {{0xdab22b2333e87f, 0x4430137a5dd2f6, 0xe03ab9f738beb8, 0xcb0c5d0dc34f24}, + {0x764a7df0c8fda5, 0x185ba5c3fa2044, 0x9281d688bcbe50, 0xc40331df893881}, + {1, 0, 0, 0}}, + {{0xb89530796f0f60, 0xade92bd26909a3, 0x1a0c83fb4884da, 0x1765bf22a5a984}, + {0x772a9ee75db09e, 0x23bc6c67cec16f, 0x4c1edba8b14e2f, 0xe2a215d9611369}, + {1, 0, 0, 0}}, + {{0x571e509fb5efb3, 0xade88696410552, 0xc8ae85fada74fe, 0x6c7e4be83bbde3}, + {0xff9f51160f4652, 0xb47ce2495a6539, 0xa2946c53b582f4, 0x286d2db3ee9a60}, + {1, 0, 0, 0}}, + {{0x40bbd5081a44af, 0x0995183b13926c, 0xbcefba6f47f6d0, 0x215619e9cc0057}, + {0x8bc94d3b0df45e, 0xf11c54a3694f6f, 0x8631b93cdfe8b5, 0xe7e3f4b0982db9}, + {1, 0, 0, 0}}, + {{0xb17048ab3e1c7b, 0xac38f36ff8a1d8, 0x1c29819435d2c6, 0xc813132f4c07e9}, + {0x2891425503b11f, 0x08781030579fea, 0xf5426ba5cc9674, 0x1e28ebf18562bc}, + {1, 0, 0, 0}}, + {{0x9f31997cc864eb, 0x06cd91d28b5e4c, 0xff17036691a973, 0xf1aef351497c58}, + {0xdd1f2d600564ff, 0xdead073b1402db, 0x74a684435bd693, 0xeea7471f962558}, + {1, 0, 0, 0}}}, + {{{0, 0, 0, 0}, + {0, 0, 0, 0}, + {0, 0, 0, 0}}, + {{0x9665266dddf554, 0x9613d78b60ef2d, 0xce27a34cdba417, 0xd35ab74d6afc31}, + {0x85ccdd22deb15e, 0x2137e5783a6aab, 0xa141cffd8c93c6, 0x355a1830e90f2d}, + {1, 0, 0, 0}}, + {{0x1a494eadaade65, 0xd6da4da77fe53c, 0xe7992996abec86, 0x65c3553c6090e3}, + {0xfa610b1fb09346, 0xf1c6540b8a4aaf, 0xc51a13ccd3cbab, 0x02995b1b18c28a}, + {1, 0, 0, 0}}, + {{0x7874568e7295ef, 0x86b419fbe38d04, 0xdc0690a7550d9a, 0xd3966a44beac33}, + {0x2b7280ec29132f, 0xbeaa3b6a032df3, 0xdc7dd88ae41200, 0xd25e2513e3a100}, + {1, 0, 0, 0}}, + {{0x924857eb2efafd, 0xac2bce41223190, 0x8edaa1445553fc, 0x825800fd3562d5}, + {0x8d79148ea96621, 0x23a01c3dd9ed8d, 0xaf8b219f9416b5, 0xd8db0cc277daea}, + {1, 0, 0, 0}}, + {{0x76a9c3b1a700f0, 0xe9acd29bc7e691, 0x69212d1a6b0327, 0x6322e97fe154be}, + {0x469fc5465d62aa, 0x8d41ed18883b05, 0x1f8eae66c52b88, 0xe4fcbe9325be51}, + {1, 0, 0, 0}}, + {{0x825fdf583cac16, 0x020b857c7b023a, 0x683c17744b0165, 0x14ffd0a2daf2f1}, + {0x323b36184218f9, 0x4944ec4e3b47d4, 0xc15b3080841acf, 0x0bced4b01a28bb}, + {1, 0, 0, 0}}, + {{0x92ac22230df5c4, 0x52f33b4063eda8, 0xcb3f19870c0c93, 0x40064f2ba65233}, + {0xfe16f0924f8992, 0x012da25af5b517, 0x1a57bb24f723a6, 0x06f8bc76760def}, + {1, 0, 0, 0}}, + {{0x4a7084f7817cb9, 0xbcab0738ee9a78, 0x3ec11e11d9c326, 0xdc0fe90e0f1aae}, + {0xcf639ea5f98390, 0x5c350aa22ffb74, 0x9afae98a4047b7, 0x956ec2d617fc45}, + {1, 0, 0, 0}}, + {{0x4306d648c1be6a, 0x9247cd8bc9a462, 0xf5595e377d2f2e, 0xbd1c3caff1a52e}, + {0x045e14472409d0, 0x29f3e17078f773, 0x745a602b2d4f7d, 0x191837685cdfbb}, + {1, 0, 0, 0}}, + {{0x5b6ee254a8cb79, 0x4953433f5e7026, 0xe21faeb1d1def4, 0xc4c225785c09de}, + {0x307ce7bba1e518, 0x31b125b1036db8, 0x47e91868839e8f, 0xc765866e33b9f3}, + {1, 0, 0, 0}}, + {{0x3bfece24f96906, 0x4794da641e5093, 0xde5df64f95db26, 0x297ecd89714b05}, + {0x701bd3ebb2c3aa, 0x7073b4f53cb1d5, 0x13c5665658af16, 0x9895089d66fe58}, + {1, 0, 0, 0}}, + {{0x0fef05f78c4790, 0x2d773633b05d2e, 0x94229c3a951c94, 0xbbbd70df4911bb}, + {0xb2c6963d2c1168, 0x105f47a72b0d73, 0x9fdf6111614080, 0x7b7e94b39e67b0}, + {1, 0, 0, 0}}, + {{0xad1a7d6efbe2b3, 0xf012482c0da69d, 0x6b3bdf12438345, 0x40d7558d7aa4d9}, + {0x8a09fffb5c6d3d, 0x9a356e5d9ffd38, 0x5973f15f4f9b1c, 0xdcd5f59f63c3ea}, + {1, 0, 0, 0}}, + {{0xacf39f4c5ca7ab, 0x4c8071cc5fd737, 0xc64e3602cd1184, 0x0acd4644c9abba}, + {0x6c011a36d8bf6e, 0xfecd87ba24e32a, 0x19f6f56574fad8, 0x050b204ced9405}, + {1, 0, 0, 0}}, + {{0xed4f1cae7d9a96, 0x5ceef7ad94c40a, 0x778e4a3bf3ef9b, 0x7405783dc3b55e}, + {0x32477c61b6e8c6, 0xb46a97570f018b, 0x91176d0a7e95d1, 0x3df90fbc4c7d0e}, + {1, 0, 0, 0}}}}; /* Precomputation for the group generator. */ typedef struct { - fslice g_pre_comp[16][3][4]; + felem g_pre_comp[2][16][3]; int references; } NISTP224_PRE_COMP; const EC_METHOD *EC_GFp_nistp224_method(void) { static const EC_METHOD ret = { + EC_FLAGS_DEFAULT_OCT, NID_X9_62_prime_field, ec_GFp_nistp224_group_init, ec_GFp_simple_group_finish, @@ -204,9 +247,9 @@ const EC_METHOD *EC_GFp_nistp224_method(void) ec_GFp_simple_get_Jprojective_coordinates_GFp, ec_GFp_simple_point_set_affine_coordinates, ec_GFp_nistp224_point_get_affine_coordinates, - ec_GFp_simple_set_compressed_coordinates, - ec_GFp_simple_point2oct, - ec_GFp_simple_oct2point, + 0 /* point_set_compressed_coordinates */, + 0 /* point2oct */, + 0 /* oct2point */, ec_GFp_simple_add, ec_GFp_simple_dbl, ec_GFp_simple_invert, @@ -229,7 +272,7 @@ const EC_METHOD *EC_GFp_nistp224_method(void) } /* Helper functions to convert field elements to/from internal representation */ -static void bin28_to_felem(fslice out[4], const u8 in[28]) +static void bin28_to_felem(felem out, const u8 in[28]) { out[0] = *((const uint64_t *)(in)) & 0x00ffffffffffffff; out[1] = (*((const uint64_t *)(in+7))) & 0x00ffffffffffffff; @@ -237,7 +280,7 @@ static void bin28_to_felem(fslice out[4], const u8 in[28]) out[3] = (*((const uint64_t *)(in+21))) & 0x00ffffffffffffff; } -static void felem_to_bin28(u8 out[28], const fslice in[4]) +static void felem_to_bin28(u8 out[28], const felem in) { unsigned i; for (i = 0; i < 7; ++i) @@ -258,9 +301,9 @@ static void flip_endian(u8 *out, const u8 *in, unsigned len) } /* From OpenSSL BIGNUM to internal representation */ -static int BN_to_felem(fslice out[4], const BIGNUM *bn) +static int BN_to_felem(felem out, const BIGNUM *bn) { - felem_bytearray b_in; + felem_bytearray b_in; felem_bytearray b_out; unsigned num_bytes; @@ -284,7 +327,7 @@ static int BN_to_felem(fslice out[4], const BIGNUM *bn) } /* From internal representation to OpenSSL BIGNUM */ -static BIGNUM *felem_to_BN(BIGNUM *out, const fslice in[4]) +static BIGNUM *felem_to_BN(BIGNUM *out, const felem in) { felem_bytearray b_in, b_out; felem_to_bin28(b_in, in); @@ -302,8 +345,24 @@ static BIGNUM *felem_to_BN(BIGNUM *out, const fslice in[4]) * */ +static void felem_one(felem out) + { + out[0] = 1; + out[1] = 0; + out[2] = 0; + out[3] = 0; + } + +static void felem_assign(felem out, const felem in) + { + out[0] = in[0]; + out[1] = in[1]; + out[2] = in[2]; + out[3] = in[3]; + } + /* Sum two field elements: out += in */ -static void felem_sum64(fslice out[4], const fslice in[4]) +static void felem_sum(felem out, const felem in) { out[0] += in[0]; out[1] += in[1]; @@ -311,14 +370,30 @@ static void felem_sum64(fslice out[4], const fslice in[4]) out[3] += in[3]; } +/* Get negative value: out = -in */ +/* Assumes in[i] < 2^57 */ +static void felem_neg(felem out, const felem in) + { + static const limb two58p2 = (((limb) 1) << 58) + (((limb) 1) << 2); + static const limb two58m2 = (((limb) 1) << 58) - (((limb) 1) << 2); + static const limb two58m42m2 = (((limb) 1) << 58) - + (((limb) 1) << 42) - (((limb) 1) << 2); + + /* Set to 0 mod 2^224-2^96+1 to ensure out > in */ + out[0] = two58p2 - in[0]; + out[1] = two58m42m2 - in[1]; + out[2] = two58m2 - in[2]; + out[3] = two58m2 - in[3]; + } + /* Subtract field elements: out -= in */ /* Assumes in[i] < 2^57 */ -static void felem_diff64(fslice out[4], const fslice in[4]) +static void felem_diff(felem out, const felem in) { - static const uint64_t two58p2 = (((uint64_t) 1) << 58) + (((uint64_t) 1) << 2); - static const uint64_t two58m2 = (((uint64_t) 1) << 58) - (((uint64_t) 1) << 2); - static const uint64_t two58m42m2 = (((uint64_t) 1) << 58) - - (((uint64_t) 1) << 42) - (((uint64_t) 1) << 2); + static const limb two58p2 = (((limb) 1) << 58) + (((limb) 1) << 2); + static const limb two58m2 = (((limb) 1) << 58) - (((limb) 1) << 2); + static const limb two58m42m2 = (((limb) 1) << 58) - + (((limb) 1) << 42) - (((limb) 1) << 2); /* Add 0 mod 2^224-2^96+1 to ensure out > in */ out[0] += two58p2; @@ -332,15 +407,15 @@ static void felem_diff64(fslice out[4], const fslice in[4]) out[3] -= in[3]; } -/* Subtract in unreduced 128-bit mode: out128 -= in128 */ +/* Subtract in unreduced 128-bit mode: out -= in */ /* Assumes in[i] < 2^119 */ -static void felem_diff128(uint128_t out[7], const uint128_t in[4]) +static void widefelem_diff(widefelem out, const widefelem in) { - static const uint128_t two120 = ((uint128_t) 1) << 120; - static const uint128_t two120m64 = (((uint128_t) 1) << 120) - - (((uint128_t) 1) << 64); - static const uint128_t two120m104m64 = (((uint128_t) 1) << 120) - - (((uint128_t) 1) << 104) - (((uint128_t) 1) << 64); + static const widelimb two120 = ((widelimb) 1) << 120; + static const widelimb two120m64 = (((widelimb) 1) << 120) - + (((widelimb) 1) << 64); + static const widelimb two120m104m64 = (((widelimb) 1) << 120) - + (((widelimb) 1) << 104) - (((widelimb) 1) << 64); /* Add 0 mod 2^224-2^96+1 to ensure out > in */ out[0] += two120; @@ -362,14 +437,14 @@ static void felem_diff128(uint128_t out[7], const uint128_t in[4]) /* Subtract in mixed mode: out128 -= in64 */ /* in[i] < 2^63 */ -static void felem_diff_128_64(uint128_t out[7], const fslice in[4]) +static void felem_diff_128_64(widefelem out, const felem in) { - static const uint128_t two64p8 = (((uint128_t) 1) << 64) + - (((uint128_t) 1) << 8); - static const uint128_t two64m8 = (((uint128_t) 1) << 64) - - (((uint128_t) 1) << 8); - static const uint128_t two64m48m8 = (((uint128_t) 1) << 64) - - (((uint128_t) 1) << 48) - (((uint128_t) 1) << 8); + static const widelimb two64p8 = (((widelimb) 1) << 64) + + (((widelimb) 1) << 8); + static const widelimb two64m8 = (((widelimb) 1) << 64) - + (((widelimb) 1) << 8); + static const widelimb two64m48m8 = (((widelimb) 1) << 64) - + (((widelimb) 1) << 48) - (((widelimb) 1) << 8); /* Add 0 mod 2^224-2^96+1 to ensure out > in */ out[0] += two64p8; @@ -383,9 +458,9 @@ static void felem_diff_128_64(uint128_t out[7], const fslice in[4]) out[3] -= in[3]; } -/* Multiply a field element by a scalar: out64 = out64 * scalar +/* Multiply a field element by a scalar: out = out * scalar * The scalars we actually use are small, so results fit without overflow */ -static void felem_scalar64(fslice out[4], const fslice scalar) +static void felem_scalar(felem out, const limb scalar) { out[0] *= scalar; out[1] *= scalar; @@ -393,9 +468,9 @@ static void felem_scalar64(fslice out[4], const fslice scalar) out[3] *= scalar; } -/* Multiply an unreduced field element by a scalar: out128 = out128 * scalar +/* Multiply an unreduced field element by a scalar: out = out * scalar * The scalars we actually use are small, so results fit without overflow */ -static void felem_scalar128(uint128_t out[7], const uint128_t scalar) +static void widefelem_scalar(widefelem out, const widelimb scalar) { out[0] *= scalar; out[1] *= scalar; @@ -407,44 +482,47 @@ static void felem_scalar128(uint128_t out[7], const uint128_t scalar) } /* Square a field element: out = in^2 */ -static void felem_square(uint128_t out[7], const fslice in[4]) +static void felem_square(widefelem out, const felem in) { - out[0] = ((uint128_t) in[0]) * in[0]; - out[1] = ((uint128_t) in[0]) * in[1] * 2; - out[2] = ((uint128_t) in[0]) * in[2] * 2 + ((uint128_t) in[1]) * in[1]; - out[3] = ((uint128_t) in[0]) * in[3] * 2 + - ((uint128_t) in[1]) * in[2] * 2; - out[4] = ((uint128_t) in[1]) * in[3] * 2 + ((uint128_t) in[2]) * in[2]; - out[5] = ((uint128_t) in[2]) * in[3] * 2; - out[6] = ((uint128_t) in[3]) * in[3]; + limb tmp0, tmp1, tmp2; + tmp0 = 2 * in[0]; tmp1 = 2 * in[1]; tmp2 = 2 * in[2]; + out[0] = ((widelimb) in[0]) * in[0]; + out[1] = ((widelimb) in[0]) * tmp1; + out[2] = ((widelimb) in[0]) * tmp2 + ((widelimb) in[1]) * in[1]; + out[3] = ((widelimb) in[3]) * tmp0 + + ((widelimb) in[1]) * tmp2; + out[4] = ((widelimb) in[3]) * tmp1 + ((widelimb) in[2]) * in[2]; + out[5] = ((widelimb) in[3]) * tmp2; + out[6] = ((widelimb) in[3]) * in[3]; } /* Multiply two field elements: out = in1 * in2 */ -static void felem_mul(uint128_t out[7], const fslice in1[4], const fslice in2[4]) +static void felem_mul(widefelem out, const felem in1, const felem in2) { - out[0] = ((uint128_t) in1[0]) * in2[0]; - out[1] = ((uint128_t) in1[0]) * in2[1] + ((uint128_t) in1[1]) * in2[0]; - out[2] = ((uint128_t) in1[0]) * in2[2] + ((uint128_t) in1[1]) * in2[1] + - ((uint128_t) in1[2]) * in2[0]; - out[3] = ((uint128_t) in1[0]) * in2[3] + ((uint128_t) in1[1]) * in2[2] + - ((uint128_t) in1[2]) * in2[1] + ((uint128_t) in1[3]) * in2[0]; - out[4] = ((uint128_t) in1[1]) * in2[3] + ((uint128_t) in1[2]) * in2[2] + - ((uint128_t) in1[3]) * in2[1]; - out[5] = ((uint128_t) in1[2]) * in2[3] + ((uint128_t) in1[3]) * in2[2]; - out[6] = ((uint128_t) in1[3]) * in2[3]; + out[0] = ((widelimb) in1[0]) * in2[0]; + out[1] = ((widelimb) in1[0]) * in2[1] + ((widelimb) in1[1]) * in2[0]; + out[2] = ((widelimb) in1[0]) * in2[2] + ((widelimb) in1[1]) * in2[1] + + ((widelimb) in1[2]) * in2[0]; + out[3] = ((widelimb) in1[0]) * in2[3] + ((widelimb) in1[1]) * in2[2] + + ((widelimb) in1[2]) * in2[1] + ((widelimb) in1[3]) * in2[0]; + out[4] = ((widelimb) in1[1]) * in2[3] + ((widelimb) in1[2]) * in2[2] + + ((widelimb) in1[3]) * in2[1]; + out[5] = ((widelimb) in1[2]) * in2[3] + ((widelimb) in1[3]) * in2[2]; + out[6] = ((widelimb) in1[3]) * in2[3]; } -/* Reduce 128-bit coefficients to 64-bit coefficients. Requires in[i] < 2^126, - * ensures out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, out[3] < 2^57 */ -static void felem_reduce(fslice out[4], const uint128_t in[7]) +/* Reduce seven 128-bit coefficients to four 64-bit coefficients. + * Requires in[i] < 2^126, + * ensures out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, out[3] <= 2^56 + 2^16 */ +static void felem_reduce(felem out, const widefelem in) { - static const uint128_t two127p15 = (((uint128_t) 1) << 127) + - (((uint128_t) 1) << 15); - static const uint128_t two127m71 = (((uint128_t) 1) << 127) - - (((uint128_t) 1) << 71); - static const uint128_t two127m71m55 = (((uint128_t) 1) << 127) - - (((uint128_t) 1) << 71) - (((uint128_t) 1) << 55); - uint128_t output[5]; + static const widelimb two127p15 = (((widelimb) 1) << 127) + + (((widelimb) 1) << 15); + static const widelimb two127m71 = (((widelimb) 1) << 127) - + (((widelimb) 1) << 71); + static const widelimb two127m71m55 = (((widelimb) 1) << 127) - + (((widelimb) 1) << 71) - (((widelimb) 1) << 55); + widelimb output[5]; /* Add 0 mod 2^224-2^96+1 to ensure all differences are positive */ output[0] = in[0] + two127p15; @@ -455,30 +533,30 @@ static void felem_reduce(fslice out[4], const uint128_t in[7]) /* Eliminate in[4], in[5], in[6] */ output[4] += in[6] >> 16; - output[3] += (in[6]&0xffff) << 40; + output[3] += (in[6] & 0xffff) << 40; output[2] -= in[6]; output[3] += in[5] >> 16; - output[2] += (in[5]&0xffff) << 40; + output[2] += (in[5] & 0xffff) << 40; output[1] -= in[5]; output[2] += output[4] >> 16; - output[1] += (output[4]&0xffff) << 40; + output[1] += (output[4] & 0xffff) << 40; output[0] -= output[4]; - output[4] = 0; /* Carry 2 -> 3 -> 4 */ output[3] += output[2] >> 56; output[2] &= 0x00ffffffffffffff; - output[4] += output[3] >> 56; + output[4] = output[3] >> 56; output[3] &= 0x00ffffffffffffff; - /* Now output[2] < 2^56, output[3] < 2^56 */ + /* Now output[2] < 2^56, output[3] < 2^56, output[4] < 2^72 */ /* Eliminate output[4] */ output[2] += output[4] >> 16; - output[1] += (output[4]&0xffff) << 40; + /* output[2] < 2^56 + 2^56 = 2^57 */ + output[1] += (output[4] & 0xffff) << 40; output[0] -= output[4]; /* Carry 0 -> 1 -> 2 -> 3 */ @@ -486,76 +564,68 @@ static void felem_reduce(fslice out[4], const uint128_t in[7]) out[0] = output[0] & 0x00ffffffffffffff; output[2] += output[1] >> 56; + /* output[2] < 2^57 + 2^72 */ out[1] = output[1] & 0x00ffffffffffffff; output[3] += output[2] >> 56; + /* output[3] <= 2^56 + 2^16 */ out[2] = output[2] & 0x00ffffffffffffff; /* out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, - * out[3] < 2^57 (due to final carry) */ + * out[3] <= 2^56 + 2^16 (due to final carry), + * so out < 2*p */ out[3] = output[3]; } -/* Reduce to unique minimal representation */ -static void felem_contract(fslice out[4], const fslice in[4]) +static void felem_square_reduce(felem out, const felem in) { - static const int64_t two56 = ((uint64_t) 1) << 56; - /* 0 <= in < 2^225 */ - /* if in > 2^224 , reduce in = in - 2^224 + 2^96 - 1 */ - int64_t tmp[4], a; - tmp[0] = (int64_t) in[0] - (in[3] >> 56); - tmp[1] = (int64_t) in[1] + ((in[3] >> 16) & 0x0000010000000000); - tmp[2] = (int64_t) in[2]; - tmp[3] = (int64_t) in[3] & 0x00ffffffffffffff; - - /* eliminate negative coefficients */ - a = tmp[0] >> 63; - tmp[0] += two56 & a; - tmp[1] -= 1 & a; - - a = tmp[1] >> 63; - tmp[1] += two56 & a; - tmp[2] -= 1 & a; - - a = tmp[2] >> 63; - tmp[2] += two56 & a; - tmp[3] -= 1 & a; - - a = tmp[3] >> 63; - tmp[3] += two56 & a; - tmp[0] += 1 & a; - tmp[1] -= (1 & a) << 40; - - /* carry 1 -> 2 -> 3 */ - tmp[2] += tmp[1] >> 56; - tmp[1] &= 0x00ffffffffffffff; + widefelem tmp; + felem_square(tmp, in); + felem_reduce(out, tmp); + } - tmp[3] += tmp[2] >> 56; - tmp[2] &= 0x00ffffffffffffff; +static void felem_mul_reduce(felem out, const felem in1, const felem in2) + { + widefelem tmp; + felem_mul(tmp, in1, in2); + felem_reduce(out, tmp); + } - /* 0 <= in < 2^224 + 2^96 - 1 */ - /* if in > 2^224 , reduce in = in - 2^224 + 2^96 - 1 */ - tmp[0] -= (tmp[3] >> 56); - tmp[1] += ((tmp[3] >> 16) & 0x0000010000000000); +/* Reduce to unique minimal representation. + * Requires 0 <= in < 2*p (always call felem_reduce first) */ +static void felem_contract(felem out, const felem in) + { + static const int64_t two56 = ((limb) 1) << 56; + /* 0 <= in < 2*p, p = 2^224 - 2^96 + 1 */ + /* if in > p , reduce in = in - 2^224 + 2^96 - 1 */ + int64_t tmp[4], a; + tmp[0] = in[0]; + tmp[1] = in[1]; + tmp[2] = in[2]; + tmp[3] = in[3]; + /* Case 1: a = 1 iff in >= 2^224 */ + a = (in[3] >> 56); + tmp[0] -= a; + tmp[1] += a << 40; tmp[3] &= 0x00ffffffffffffff; + /* Case 2: a = 0 iff p <= in < 2^224, i.e., + * the high 128 bits are all 1 and the lower part is non-zero */ + a = ((in[3] & in[2] & (in[1] | 0x000000ffffffffff)) + 1) | + (((int64_t)(in[0] + (in[1] & 0x000000ffffffffff)) - 1) >> 63); + a &= 0x00ffffffffffffff; + /* turn a into an all-one mask (if a = 0) or an all-zero mask */ + a = (a - 1) >> 63; + /* subtract 2^224 - 2^96 + 1 if a is all-one*/ + tmp[3] &= a ^ 0xffffffffffffffff; + tmp[2] &= a ^ 0xffffffffffffffff; + tmp[1] &= (a ^ 0xffffffffffffffff) | 0x000000ffffffffff; + tmp[0] -= 1 & a; - /* eliminate negative coefficients */ + /* eliminate negative coefficients: if tmp[0] is negative, tmp[1] must + * be non-zero, so we only need one step */ a = tmp[0] >> 63; tmp[0] += two56 & a; tmp[1] -= 1 & a; - a = tmp[1] >> 63; - tmp[1] += two56 & a; - tmp[2] -= 1 & a; - - a = tmp[2] >> 63; - tmp[2] += two56 & a; - tmp[3] -= 1 & a; - - a = tmp[3] >> 63; - tmp[3] += two56 & a; - tmp[0] += 1 & a; - tmp[1] -= (1 & a) << 40; - /* carry 1 -> 2 -> 3 */ tmp[2] += tmp[1] >> 56; tmp[1] &= 0x00ffffffffffffff; @@ -563,27 +633,7 @@ static void felem_contract(fslice out[4], const fslice in[4]) tmp[3] += tmp[2] >> 56; tmp[2] &= 0x00ffffffffffffff; - /* Now 0 <= in < 2^224 */ - - /* if in > 2^224 - 2^96, reduce */ - /* a = 0 iff in > 2^224 - 2^96, i.e., - * the high 128 bits are all 1 and the lower part is non-zero */ - a = (tmp[3] + 1) | (tmp[2] + 1) | - ((tmp[1] | 0x000000ffffffffff) + 1) | - ((((tmp[1] & 0xffff) - 1) >> 63) & ((tmp[0] - 1) >> 63)); - /* turn a into an all-one mask (if a = 0) or an all-zero mask */ - a = ((a & 0x00ffffffffffffff) - 1) >> 63; - /* subtract 2^224 - 2^96 + 1 if a is all-one*/ - tmp[3] &= a ^ 0xffffffffffffffff; - tmp[2] &= a ^ 0xffffffffffffffff; - tmp[1] &= (a ^ 0xffffffffffffffff) | 0x000000ffffffffff; - tmp[0] -= 1 & a; - /* eliminate negative coefficients: if tmp[0] is negative, tmp[1] must be - * non-zero, so we only need one step */ - a = tmp[0] >> 63; - tmp[0] += two56 & a; - tmp[1] -= 1 & a; - + /* Now 0 <= out < p */ out[0] = tmp[0]; out[1] = tmp[1]; out[2] = tmp[2]; @@ -594,9 +644,9 @@ static void felem_contract(fslice out[4], const fslice in[4]) * We know that field elements are reduced to in < 2^225, * so we only need to check three cases: 0, 2^224 - 2^96 + 1, * and 2^225 - 2^97 + 2 */ -static fslice felem_is_zero(const fslice in[4]) +static limb felem_is_zero(const felem in) { - fslice zero, two224m96p1, two225m97p2; + limb zero, two224m96p1, two225m97p2; zero = in[0] | in[1] | in[2] | in[3]; zero = (((int64_t)(zero) - 1) >> 63) & 1; @@ -609,12 +659,17 @@ static fslice felem_is_zero(const fslice in[4]) return (zero | two224m96p1 | two225m97p2); } +static limb felem_is_zero_int(const felem in) + { + return (int) (felem_is_zero(in) & ((limb)1)); + } + /* Invert a field element */ /* Computation chain copied from djb's code */ -static void felem_inv(fslice out[4], const fslice in[4]) +static void felem_inv(felem out, const felem in) { - fslice ftmp[4], ftmp2[4], ftmp3[4], ftmp4[4]; - uint128_t tmp[7]; + felem ftmp, ftmp2, ftmp3, ftmp4; + widefelem tmp; unsigned i; felem_square(tmp, in); felem_reduce(ftmp, tmp); /* 2 */ @@ -673,34 +728,18 @@ static void felem_inv(fslice out[4], const fslice in[4]) * if icopy == 1, copy in to out, * if icopy == 0, copy out to itself. */ static void -copy_conditional(fslice *out, const fslice *in, unsigned len, fslice icopy) +copy_conditional(felem out, const felem in, limb icopy) { unsigned i; /* icopy is a (64-bit) 0 or 1, so copy is either all-zero or all-one */ - const fslice copy = -icopy; - for (i = 0; i < len; ++i) + const limb copy = -icopy; + for (i = 0; i < 4; ++i) { - const fslice tmp = copy & (in[i] ^ out[i]); + const limb tmp = copy & (in[i] ^ out[i]); out[i] ^= tmp; } } -/* Copy in constant time: - * if isel == 1, copy in2 to out, - * if isel == 0, copy in1 to out. */ -static void select_conditional(fslice *out, const fslice *in1, const fslice *in2, - unsigned len, fslice isel) - { - unsigned i; - /* isel is a (64-bit) 0 or 1, so sel is either all-zero or all-one */ - const fslice sel = -isel; - for (i = 0; i < len; ++i) - { - const fslice tmp = sel & (in1[i] ^ in2[i]); - out[i] = in1[i] ^ tmp; - } -} - /******************************************************************************/ /* ELLIPTIC CURVE POINT OPERATIONS * @@ -718,17 +757,14 @@ static void select_conditional(fslice *out, const fslice *in1, const fslice *in2 * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed, * while x_out == y_in is not (maybe this works, but it's not tested). */ static void -point_double(fslice x_out[4], fslice y_out[4], fslice z_out[4], - const fslice x_in[4], const fslice y_in[4], const fslice z_in[4]) +point_double(felem x_out, felem y_out, felem z_out, + const felem x_in, const felem y_in, const felem z_in) { - uint128_t tmp[7], tmp2[7]; - fslice delta[4]; - fslice gamma[4]; - fslice beta[4]; - fslice alpha[4]; - fslice ftmp[4], ftmp2[4]; - memcpy(ftmp, x_in, 4 * sizeof(fslice)); - memcpy(ftmp2, x_in, 4 * sizeof(fslice)); + widefelem tmp, tmp2; + felem delta, gamma, beta, alpha, ftmp, ftmp2; + + felem_assign(ftmp, x_in); + felem_assign(ftmp2, x_in); /* delta = z^2 */ felem_square(tmp, z_in); @@ -743,11 +779,11 @@ point_double(fslice x_out[4], fslice y_out[4], fslice z_out[4], felem_reduce(beta, tmp); /* alpha = 3*(x-delta)*(x+delta) */ - felem_diff64(ftmp, delta); + felem_diff(ftmp, delta); /* ftmp[i] < 2^57 + 2^58 + 2 < 2^59 */ - felem_sum64(ftmp2, delta); + felem_sum(ftmp2, delta); /* ftmp2[i] < 2^57 + 2^57 = 2^58 */ - felem_scalar64(ftmp2, 3); + felem_scalar(ftmp2, 3); /* ftmp2[i] < 3 * 2^58 < 2^60 */ felem_mul(tmp, ftmp, ftmp2); /* tmp[i] < 2^60 * 2^59 * 4 = 2^121 */ @@ -756,18 +792,18 @@ point_double(fslice x_out[4], fslice y_out[4], fslice z_out[4], /* x' = alpha^2 - 8*beta */ felem_square(tmp, alpha); /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */ - memcpy(ftmp, beta, 4 * sizeof(fslice)); - felem_scalar64(ftmp, 8); + felem_assign(ftmp, beta); + felem_scalar(ftmp, 8); /* ftmp[i] < 8 * 2^57 = 2^60 */ felem_diff_128_64(tmp, ftmp); /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */ felem_reduce(x_out, tmp); /* z' = (y + z)^2 - gamma - delta */ - felem_sum64(delta, gamma); + felem_sum(delta, gamma); /* delta[i] < 2^57 + 2^57 = 2^58 */ - memcpy(ftmp, y_in, 4 * sizeof(fslice)); - felem_sum64(ftmp, z_in); + felem_assign(ftmp, y_in); + felem_sum(ftmp, z_in); /* ftmp[i] < 2^57 + 2^57 = 2^58 */ felem_square(tmp, ftmp); /* tmp[i] < 4 * 2^58 * 2^58 = 2^118 */ @@ -776,17 +812,17 @@ point_double(fslice x_out[4], fslice y_out[4], fslice z_out[4], felem_reduce(z_out, tmp); /* y' = alpha*(4*beta - x') - 8*gamma^2 */ - felem_scalar64(beta, 4); + felem_scalar(beta, 4); /* beta[i] < 4 * 2^57 = 2^59 */ - felem_diff64(beta, x_out); + felem_diff(beta, x_out); /* beta[i] < 2^59 + 2^58 + 2 < 2^60 */ felem_mul(tmp, alpha, beta); /* tmp[i] < 4 * 2^57 * 2^60 = 2^119 */ felem_square(tmp2, gamma); /* tmp2[i] < 4 * 2^57 * 2^57 = 2^116 */ - felem_scalar128(tmp2, 8); + widefelem_scalar(tmp2, 8); /* tmp2[i] < 8 * 2^116 = 2^119 */ - felem_diff128(tmp, tmp2); + widefelem_diff(tmp, tmp2); /* tmp[i] < 2^119 + 2^120 < 2^121 */ felem_reduce(y_out, tmp); } @@ -797,60 +833,76 @@ point_double(fslice x_out[4], fslice y_out[4], fslice z_out[4], * 2 * Z_2^2 * X_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^2 * Y_3 = (Z_1^3 * Y_2 - Z_2^3 * Y_1) * (Z_2^2 * X_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^2 - X_3) - * Z_2^3 * Y_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^3 - * Z_3 = (Z_1^2 * X_2 - Z_2^2 * X_1) * (Z_1 * Z_2) */ + * Z_3 = (Z_1^2 * X_2 - Z_2^2 * X_1) * (Z_1 * Z_2) + * + * This runs faster if 'mixed' is set, which requires Z_2 = 1 or Z_2 = 0. + */ /* This function is not entirely constant-time: * it includes a branch for checking whether the two input points are equal, * (while not equal to the point at infinity). * This case never happens during single point multiplication, * so there is no timing leak for ECDH or ECDSA signing. */ -static void point_add(fslice x3[4], fslice y3[4], fslice z3[4], - const fslice x1[4], const fslice y1[4], const fslice z1[4], - const fslice x2[4], const fslice y2[4], const fslice z2[4]) +static void point_add(felem x3, felem y3, felem z3, + const felem x1, const felem y1, const felem z1, + const int mixed, const felem x2, const felem y2, const felem z2) { - fslice ftmp[4], ftmp2[4], ftmp3[4], ftmp4[4], ftmp5[4]; - uint128_t tmp[7], tmp2[7]; - fslice z1_is_zero, z2_is_zero, x_equal, y_equal; + felem ftmp, ftmp2, ftmp3, ftmp4, ftmp5, x_out, y_out, z_out; + widefelem tmp, tmp2; + limb z1_is_zero, z2_is_zero, x_equal, y_equal; + + if (!mixed) + { + /* ftmp2 = z2^2 */ + felem_square(tmp, z2); + felem_reduce(ftmp2, tmp); + + /* ftmp4 = z2^3 */ + felem_mul(tmp, ftmp2, z2); + felem_reduce(ftmp4, tmp); + + /* ftmp4 = z2^3*y1 */ + felem_mul(tmp2, ftmp4, y1); + felem_reduce(ftmp4, tmp2); + + /* ftmp2 = z2^2*x1 */ + felem_mul(tmp2, ftmp2, x1); + felem_reduce(ftmp2, tmp2); + } + else + { + /* We'll assume z2 = 1 (special case z2 = 0 is handled later) */ + + /* ftmp4 = z2^3*y1 */ + felem_assign(ftmp4, y1); + + /* ftmp2 = z2^2*x1 */ + felem_assign(ftmp2, x1); + } /* ftmp = z1^2 */ felem_square(tmp, z1); felem_reduce(ftmp, tmp); - /* ftmp2 = z2^2 */ - felem_square(tmp, z2); - felem_reduce(ftmp2, tmp); - /* ftmp3 = z1^3 */ felem_mul(tmp, ftmp, z1); felem_reduce(ftmp3, tmp); - /* ftmp4 = z2^3 */ - felem_mul(tmp, ftmp2, z2); - felem_reduce(ftmp4, tmp); - - /* ftmp3 = z1^3*y2 */ + /* tmp = z1^3*y2 */ felem_mul(tmp, ftmp3, y2); /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */ - /* ftmp4 = z2^3*y1 */ - felem_mul(tmp2, ftmp4, y1); - felem_reduce(ftmp4, tmp2); - /* ftmp3 = z1^3*y2 - z2^3*y1 */ felem_diff_128_64(tmp, ftmp4); /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */ felem_reduce(ftmp3, tmp); - /* ftmp = z1^2*x2 */ + /* tmp = z1^2*x2 */ felem_mul(tmp, ftmp, x2); /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */ - /* ftmp2 =z2^2*x1 */ - felem_mul(tmp2, ftmp2, x1); - felem_reduce(ftmp2, tmp2); - /* ftmp = z1^2*x2 - z2^2*x1 */ - felem_diff128(tmp, tmp2); + felem_diff_128_64(tmp, ftmp2); /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */ felem_reduce(ftmp, tmp); @@ -868,15 +920,23 @@ static void point_add(fslice x3[4], fslice y3[4], fslice z3[4], } /* ftmp5 = z1*z2 */ - felem_mul(tmp, z1, z2); - felem_reduce(ftmp5, tmp); + if (!mixed) + { + felem_mul(tmp, z1, z2); + felem_reduce(ftmp5, tmp); + } + else + { + /* special case z2 = 0 is handled later */ + felem_assign(ftmp5, z1); + } - /* z3 = (z1^2*x2 - z2^2*x1)*(z1*z2) */ + /* z_out = (z1^2*x2 - z2^2*x1)*(z1*z2) */ felem_mul(tmp, ftmp, ftmp5); - felem_reduce(z3, tmp); + felem_reduce(z_out, tmp); /* ftmp = (z1^2*x2 - z2^2*x1)^2 */ - memcpy(ftmp5, ftmp, 4 * sizeof(fslice)); + felem_assign(ftmp5, ftmp); felem_square(tmp, ftmp); felem_reduce(ftmp, tmp); @@ -888,7 +948,7 @@ static void point_add(fslice x3[4], fslice y3[4], fslice z3[4], felem_mul(tmp, ftmp2, ftmp); felem_reduce(ftmp2, tmp); - /* ftmp4 = z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */ + /* tmp = z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */ felem_mul(tmp, ftmp4, ftmp5); /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */ @@ -901,131 +961,176 @@ static void point_add(fslice x3[4], fslice y3[4], fslice z3[4], /* tmp2[i] < 2^116 + 2^64 + 8 < 2^117 */ /* ftmp5 = 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */ - memcpy(ftmp5, ftmp2, 4 * sizeof(fslice)); - felem_scalar64(ftmp5, 2); + felem_assign(ftmp5, ftmp2); + felem_scalar(ftmp5, 2); /* ftmp5[i] < 2 * 2^57 = 2^58 */ - /* x3 = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 - + /* x_out = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 - 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */ felem_diff_128_64(tmp2, ftmp5); /* tmp2[i] < 2^117 + 2^64 + 8 < 2^118 */ - felem_reduce(x3, tmp2); + felem_reduce(x_out, tmp2); - /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x3 */ - felem_diff64(ftmp2, x3); + /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out */ + felem_diff(ftmp2, x_out); /* ftmp2[i] < 2^57 + 2^58 + 2 < 2^59 */ - /* tmp2 = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x3) */ + /* tmp2 = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out) */ felem_mul(tmp2, ftmp3, ftmp2); /* tmp2[i] < 4 * 2^57 * 2^59 = 2^118 */ - /* y3 = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x3) - + /* y_out = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x_out) - z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */ - felem_diff128(tmp2, tmp); + widefelem_diff(tmp2, tmp); /* tmp2[i] < 2^118 + 2^120 < 2^121 */ - felem_reduce(y3, tmp2); + felem_reduce(y_out, tmp2); - /* the result (x3, y3, z3) is incorrect if one of the inputs is the - * point at infinity, so we need to check for this separately */ + /* the result (x_out, y_out, z_out) is incorrect if one of the inputs is + * the point at infinity, so we need to check for this separately */ /* if point 1 is at infinity, copy point 2 to output, and vice versa */ - copy_conditional(x3, x2, 4, z1_is_zero); - copy_conditional(x3, x1, 4, z2_is_zero); - copy_conditional(y3, y2, 4, z1_is_zero); - copy_conditional(y3, y1, 4, z2_is_zero); - copy_conditional(z3, z2, 4, z1_is_zero); - copy_conditional(z3, z1, 4, z2_is_zero); + copy_conditional(x_out, x2, z1_is_zero); + copy_conditional(x_out, x1, z2_is_zero); + copy_conditional(y_out, y2, z1_is_zero); + copy_conditional(y_out, y1, z2_is_zero); + copy_conditional(z_out, z2, z1_is_zero); + copy_conditional(z_out, z1, z2_is_zero); + felem_assign(x3, x_out); + felem_assign(y3, y_out); + felem_assign(z3, z_out); } -/* Select a point from an array of 16 precomputed point multiples, - * in constant time: for bits = {b_0, b_1, b_2, b_3}, return the point - * pre_comp[8*b_3 + 4*b_2 + 2*b_1 + b_0] */ -static void select_point(const fslice bits[4], const fslice pre_comp[16][3][4], - fslice out[12]) +/* select_point selects the |index|th point from a precomputation table and + * copies it to out. */ +static void select_point(const u64 index, unsigned int size, const felem pre_comp[/*size*/][3], felem out[3]) { - fslice tmp[5][12]; - select_conditional(tmp[0], pre_comp[7][0], pre_comp[15][0], 12, bits[3]); - select_conditional(tmp[1], pre_comp[3][0], pre_comp[11][0], 12, bits[3]); - select_conditional(tmp[2], tmp[1], tmp[0], 12, bits[2]); - select_conditional(tmp[0], pre_comp[5][0], pre_comp[13][0], 12, bits[3]); - select_conditional(tmp[1], pre_comp[1][0], pre_comp[9][0], 12, bits[3]); - select_conditional(tmp[3], tmp[1], tmp[0], 12, bits[2]); - select_conditional(tmp[4], tmp[3], tmp[2], 12, bits[1]); - select_conditional(tmp[0], pre_comp[6][0], pre_comp[14][0], 12, bits[3]); - select_conditional(tmp[1], pre_comp[2][0], pre_comp[10][0], 12, bits[3]); - select_conditional(tmp[2], tmp[1], tmp[0], 12, bits[2]); - select_conditional(tmp[0], pre_comp[4][0], pre_comp[12][0], 12, bits[3]); - select_conditional(tmp[1], pre_comp[0][0], pre_comp[8][0], 12, bits[3]); - select_conditional(tmp[3], tmp[1], tmp[0], 12, bits[2]); - select_conditional(tmp[1], tmp[3], tmp[2], 12, bits[1]); - select_conditional(out, tmp[1], tmp[4], 12, bits[0]); + unsigned i, j; + limb *outlimbs = &out[0][0]; + memset(outlimbs, 0, 3 * sizeof(felem)); + + for (i = 0; i < size; i++) + { + const limb *inlimbs = &pre_comp[i][0][0]; + u64 mask = i ^ index; + mask |= mask >> 4; + mask |= mask >> 2; + mask |= mask >> 1; + mask &= 1; + mask--; + for (j = 0; j < 4 * 3; j++) + outlimbs[j] |= inlimbs[j] & mask; + } + } + +/* get_bit returns the |i|th bit in |in| */ +static char get_bit(const felem_bytearray in, unsigned i) + { + if (i >= 224) + return 0; + return (in[i >> 3] >> (i & 7)) & 1; } /* Interleaved point multiplication using precomputed point multiples: - * The small point multiples 0*P, 1*P, ..., 15*P are in pre_comp[], + * The small point multiples 0*P, 1*P, ..., 16*P are in pre_comp[], * the scalars in scalars[]. If g_scalar is non-NULL, we also add this multiple * of the generator, using certain (large) precomputed multiples in g_pre_comp. * Output point (X, Y, Z) is stored in x_out, y_out, z_out */ -static void batch_mul(fslice x_out[4], fslice y_out[4], fslice z_out[4], +static void batch_mul(felem x_out, felem y_out, felem z_out, const felem_bytearray scalars[], const unsigned num_points, const u8 *g_scalar, - const fslice pre_comp[][16][3][4], const fslice g_pre_comp[16][3][4]) + const int mixed, const felem pre_comp[][17][3], const felem g_pre_comp[2][16][3]) { - unsigned i, j, num; + int i, skip; + unsigned num; unsigned gen_mul = (g_scalar != NULL); - fslice nq[12], nqt[12], tmp[12]; - fslice bits[4]; - u8 byte; + felem nq[3], tmp[4]; + u64 bits; + u8 sign, digit; /* set nq to the point at infinity */ - memset(nq, 0, 12 * sizeof(fslice)); - - /* Loop over all scalars msb-to-lsb, 4 bits at a time: for each nibble, - * double 4 times, then add the precomputed point multiples. - * If we are also adding multiples of the generator, then interleave - * these additions with the last 56 doublings. */ - for (i = (num_points ? 28 : 7); i > 0; --i) + memset(nq, 0, 3 * sizeof(felem)); + + /* Loop over all scalars msb-to-lsb, interleaving additions + * of multiples of the generator (two in each of the last 28 rounds) + * and additions of other points multiples (every 5th round). + */ + skip = 1; /* save two point operations in the first round */ + for (i = (num_points ? 220 : 27); i >= 0; --i) { - for (j = 0; j < 8; ++j) + /* double */ + if (!skip) + point_double(nq[0], nq[1], nq[2], nq[0], nq[1], nq[2]); + + /* add multiples of the generator */ + if (gen_mul && (i <= 27)) { - /* double once */ - point_double(nq, nq+4, nq+8, nq, nq+4, nq+8); - /* add multiples of the generator */ - if ((gen_mul) && (i <= 7)) + /* first, look 28 bits upwards */ + bits = get_bit(g_scalar, i + 196) << 3; + bits |= get_bit(g_scalar, i + 140) << 2; + bits |= get_bit(g_scalar, i + 84) << 1; + bits |= get_bit(g_scalar, i + 28); + /* select the point to add, in constant time */ + select_point(bits, 16, g_pre_comp[1], tmp); + + if (!skip) + { + point_add(nq[0], nq[1], nq[2], + nq[0], nq[1], nq[2], + 1 /* mixed */, tmp[0], tmp[1], tmp[2]); + } + else { - bits[3] = (g_scalar[i+20] >> (7-j)) & 1; - bits[2] = (g_scalar[i+13] >> (7-j)) & 1; - bits[1] = (g_scalar[i+6] >> (7-j)) & 1; - bits[0] = (g_scalar[i-1] >> (7-j)) & 1; - /* select the point to add, in constant time */ - select_point(bits, g_pre_comp, tmp); - memcpy(nqt, nq, 12 * sizeof(fslice)); - point_add(nq, nq+4, nq+8, nqt, nqt+4, nqt+8, - tmp, tmp+4, tmp+8); + memcpy(nq, tmp, 3 * sizeof(felem)); + skip = 0; } - /* do an addition after every 4 doublings */ - if (j % 4 == 3) + + /* second, look at the current position */ + bits = get_bit(g_scalar, i + 168) << 3; + bits |= get_bit(g_scalar, i + 112) << 2; + bits |= get_bit(g_scalar, i + 56) << 1; + bits |= get_bit(g_scalar, i); + /* select the point to add, in constant time */ + select_point(bits, 16, g_pre_comp[0], tmp); + point_add(nq[0], nq[1], nq[2], + nq[0], nq[1], nq[2], + 1 /* mixed */, tmp[0], tmp[1], tmp[2]); + } + + /* do other additions every 5 doublings */ + if (num_points && (i % 5 == 0)) + { + /* loop over all scalars */ + for (num = 0; num < num_points; ++num) { - /* loop over all scalars */ - for (num = 0; num < num_points; ++num) + bits = get_bit(scalars[num], i + 4) << 5; + bits |= get_bit(scalars[num], i + 3) << 4; + bits |= get_bit(scalars[num], i + 2) << 3; + bits |= get_bit(scalars[num], i + 1) << 2; + bits |= get_bit(scalars[num], i) << 1; + bits |= get_bit(scalars[num], i - 1); + ec_GFp_nistp_recode_scalar_bits(&sign, &digit, bits); + + /* select the point to add or subtract */ + select_point(digit, 17, pre_comp[num], tmp); + felem_neg(tmp[3], tmp[1]); /* (X, -Y, Z) is the negative point */ + copy_conditional(tmp[1], tmp[3], sign); + + if (!skip) { - byte = scalars[num][i-1]; - bits[3] = (byte >> (10-j)) & 1; - bits[2] = (byte >> (9-j)) & 1; - bits[1] = (byte >> (8-j)) & 1; - bits[0] = (byte >> (7-j)) & 1; - /* select the point to add */ - select_point(bits, - pre_comp[num], tmp); - memcpy(nqt, nq, 12 * sizeof(fslice)); - point_add(nq, nq+4, nq+8, nqt, nqt+4, - nqt+8, tmp, tmp+4, tmp+8); + point_add(nq[0], nq[1], nq[2], + nq[0], nq[1], nq[2], + mixed, tmp[0], tmp[1], tmp[2]); + } + else + { + memcpy(nq, tmp, 3 * sizeof(felem)); + skip = 0; } } } } - memcpy(x_out, nq, 4 * sizeof(fslice)); - memcpy(y_out, nq+4, 4 * sizeof(fslice))