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e0187b95f0
mbedtls/library/ecp.c
Manuel Pégourié-Gonnard c80555d835 Add public function generating private keys 
We need to separate the uses of the RNG for blinding and for key
generation for the sake of an upcoming security fix in deterministic
ECDSA.
2019-01-16 15:47:26 +00:00

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/*
* Elliptic curves over GF(p): generic functions
*
* Copyright (C) 2006-2015, ARM Limited, All Rights Reserved
* SPDX-License-Identifier: Apache-2.0
*
* Licensed under the Apache License, Version 2.0 (the "License"); you may
* not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* 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.
*
* This file is part of mbed TLS (https://tls.mbed.org)
*/
/*
* References:
*
* SEC1 http://www.secg.org/index.php?action=secg,docs_secg
* GECC = Guide to Elliptic Curve Cryptography - Hankerson, Menezes, Vanstone
* FIPS 186-3 http://csrc.nist.gov/publications/fips/fips186-3/fips_186-3.pdf
* RFC 4492 for the related TLS structures and constants
*
* [Curve25519] http://cr.yp.to/ecdh/curve25519-20060209.pdf
*
* [2] CORON, Jean-S'ebastien. Resistance against differential power analysis
* for elliptic curve cryptosystems. In : Cryptographic Hardware and
* Embedded Systems. Springer Berlin Heidelberg, 1999. p. 292-302.
* <http://link.springer.com/chapter/10.1007/3-540-48059-5_25>
*
* [3] HEDABOU, Mustapha, PINEL, Pierre, et B'EN'ETEAU, Lucien. A comb method to
* render ECC resistant against Side Channel Attacks. IACR Cryptology
* ePrint Archive, 2004, vol. 2004, p. 342.
* <http://eprint.iacr.org/2004/342.pdf>
*/
#if !defined(MBEDTLS_CONFIG_FILE)
#include "mbedtls/config.h"
#else
#include MBEDTLS_CONFIG_FILE
#endif
#if defined(MBEDTLS_ECP_C)
#include "mbedtls/ecp.h"
#include "mbedtls/threading.h"
#include <string.h>
#if !defined(MBEDTLS_ECP_ALT)
#if defined(MBEDTLS_PLATFORM_C)
#include "mbedtls/platform.h"
#else
#include <stdlib.h>
#include <stdio.h>
#define mbedtls_printf printf
#define mbedtls_calloc calloc
#define mbedtls_free free
#endif
#include "mbedtls/ecp_internal.h"
#if ( defined(__ARMCC_VERSION) || defined(_MSC_VER) ) && \
!defined(inline) && !defined(__cplusplus)
#define inline __inline
#endif
/* Implementation that should never be optimized out by the compiler */
static void mbedtls_zeroize( void *v, size_t n ) {
volatile unsigned char *p = v; while( n-- ) *p++ = 0;
}
#if defined(MBEDTLS_SELF_TEST)
/*
* Counts of point addition and doubling, and field multiplications.
* Used to test resistance of point multiplication to simple timing attacks.
*/
static unsigned long add_count, dbl_count, mul_count;
#endif
#if defined(MBEDTLS_ECP_DP_SECP192R1_ENABLED) || \
defined(MBEDTLS_ECP_DP_SECP224R1_ENABLED) || \
defined(MBEDTLS_ECP_DP_SECP256R1_ENABLED) || \
defined(MBEDTLS_ECP_DP_SECP384R1_ENABLED) || \
defined(MBEDTLS_ECP_DP_SECP521R1_ENABLED) || \
defined(MBEDTLS_ECP_DP_BP256R1_ENABLED) || \
defined(MBEDTLS_ECP_DP_BP384R1_ENABLED) || \
defined(MBEDTLS_ECP_DP_BP512R1_ENABLED) || \
defined(MBEDTLS_ECP_DP_SECP192K1_ENABLED) || \
defined(MBEDTLS_ECP_DP_SECP224K1_ENABLED) || \
defined(MBEDTLS_ECP_DP_SECP256K1_ENABLED)
#define ECP_SHORTWEIERSTRASS
#endif
#if defined(MBEDTLS_ECP_DP_CURVE25519_ENABLED)
#define ECP_MONTGOMERY
#endif
/*
* Curve types: internal for now, might be exposed later
*/
typedef enum
{
ECP_TYPE_NONE = 0,
ECP_TYPE_SHORT_WEIERSTRASS, /* y^2 = x^3 + a x + b */
ECP_TYPE_MONTGOMERY, /* y^2 = x^3 + a x^2 + x */
} ecp_curve_type;
/*
* List of supported curves:
* - internal ID
* - TLS NamedCurve ID (RFC 4492 sec. 5.1.1, RFC 7071 sec. 2)
* - size in bits
* - readable name
*
* Curves are listed in order: largest curves first, and for a given size,
* fastest curves first. This provides the default order for the SSL module.
*
* Reminder: update profiles in x509_crt.c when adding a new curves!
*/
static const mbedtls_ecp_curve_info ecp_supported_curves[] =
{
#if defined(MBEDTLS_ECP_DP_SECP521R1_ENABLED)
{ MBEDTLS_ECP_DP_SECP521R1, 25, 521, "secp521r1" },
#endif
#if defined(MBEDTLS_ECP_DP_BP512R1_ENABLED)
{ MBEDTLS_ECP_DP_BP512R1, 28, 512, "brainpoolP512r1" },
#endif
#if defined(MBEDTLS_ECP_DP_SECP384R1_ENABLED)
{ MBEDTLS_ECP_DP_SECP384R1, 24, 384, "secp384r1" },
#endif
#if defined(MBEDTLS_ECP_DP_BP384R1_ENABLED)
{ MBEDTLS_ECP_DP_BP384R1, 27, 384, "brainpoolP384r1" },
#endif
#if defined(MBEDTLS_ECP_DP_SECP256R1_ENABLED)
{ MBEDTLS_ECP_DP_SECP256R1, 23, 256, "secp256r1" },
#endif
#if defined(MBEDTLS_ECP_DP_SECP256K1_ENABLED)
{ MBEDTLS_ECP_DP_SECP256K1, 22, 256, "secp256k1" },
#endif
#if defined(MBEDTLS_ECP_DP_BP256R1_ENABLED)
{ MBEDTLS_ECP_DP_BP256R1, 26, 256, "brainpoolP256r1" },
#endif
#if defined(MBEDTLS_ECP_DP_SECP224R1_ENABLED)
{ MBEDTLS_ECP_DP_SECP224R1, 21, 224, "secp224r1" },
#endif
#if defined(MBEDTLS_ECP_DP_SECP224K1_ENABLED)
{ MBEDTLS_ECP_DP_SECP224K1, 20, 224, "secp224k1" },
#endif
#if defined(MBEDTLS_ECP_DP_SECP192R1_ENABLED)
{ MBEDTLS_ECP_DP_SECP192R1, 19, 192, "secp192r1" },
#endif
#if defined(MBEDTLS_ECP_DP_SECP192K1_ENABLED)
{ MBEDTLS_ECP_DP_SECP192K1, 18, 192, "secp192k1" },
#endif
{ MBEDTLS_ECP_DP_NONE, 0, 0, NULL },
};
#define ECP_NB_CURVES sizeof( ecp_supported_curves ) / \
sizeof( ecp_supported_curves[0] )
static mbedtls_ecp_group_id ecp_supported_grp_id[ECP_NB_CURVES];
/*
* List of supported curves and associated info
*/
const mbedtls_ecp_curve_info *mbedtls_ecp_curve_list( void )
{
return( ecp_supported_curves );
}
/*
* List of supported curves, group ID only
*/
const mbedtls_ecp_group_id *mbedtls_ecp_grp_id_list( void )
{
static int init_done = 0;
if( ! init_done )
{
size_t i = 0;
const mbedtls_ecp_curve_info *curve_info;
for( curve_info = mbedtls_ecp_curve_list();
curve_info->grp_id != MBEDTLS_ECP_DP_NONE;
curve_info++ )
{
ecp_supported_grp_id[i++] = curve_info->grp_id;
}
ecp_supported_grp_id[i] = MBEDTLS_ECP_DP_NONE;
init_done = 1;
}
return( ecp_supported_grp_id );
}
/*
* Get the curve info for the internal identifier
*/
const mbedtls_ecp_curve_info *mbedtls_ecp_curve_info_from_grp_id( mbedtls_ecp_group_id grp_id )
{
const mbedtls_ecp_curve_info *curve_info;
for( curve_info = mbedtls_ecp_curve_list();
curve_info->grp_id != MBEDTLS_ECP_DP_NONE;
curve_info++ )
{
if( curve_info->grp_id == grp_id )
return( curve_info );
}
return( NULL );
}
/*
* Get the curve info from the TLS identifier
*/
const mbedtls_ecp_curve_info *mbedtls_ecp_curve_info_from_tls_id( uint16_t tls_id )
{
const mbedtls_ecp_curve_info *curve_info;
for( curve_info = mbedtls_ecp_curve_list();
curve_info->grp_id != MBEDTLS_ECP_DP_NONE;
curve_info++ )
{
if( curve_info->tls_id == tls_id )
return( curve_info );
}
return( NULL );
}
/*
* Get the curve info from the name
*/
const mbedtls_ecp_curve_info *mbedtls_ecp_curve_info_from_name( const char *name )
{
const mbedtls_ecp_curve_info *curve_info;
for( curve_info = mbedtls_ecp_curve_list();
curve_info->grp_id != MBEDTLS_ECP_DP_NONE;
curve_info++ )
{
if( strcmp( curve_info->name, name ) == 0 )
return( curve_info );
}
return( NULL );
}
/*
* Get the type of a curve
*/
static inline ecp_curve_type ecp_get_type( const mbedtls_ecp_group *grp )
{
if( grp->G.X.p == NULL )
return( ECP_TYPE_NONE );
if( grp->G.Y.p == NULL )
return( ECP_TYPE_MONTGOMERY );
else
return( ECP_TYPE_SHORT_WEIERSTRASS );
}
/*
* Initialize (the components of) a point
*/
void mbedtls_ecp_point_init( mbedtls_ecp_point *pt )
{
if( pt == NULL )
return;
mbedtls_mpi_init( &pt->X );
mbedtls_mpi_init( &pt->Y );
mbedtls_mpi_init( &pt->Z );
}
/*
* Initialize (the components of) a group
*/
void mbedtls_ecp_group_init( mbedtls_ecp_group *grp )
{
if( grp == NULL )
return;
memset( grp, 0, sizeof( mbedtls_ecp_group ) );
}
/*
* Initialize (the components of) a key pair
*/
void mbedtls_ecp_keypair_init( mbedtls_ecp_keypair *key )
{
if( key == NULL )
return;
mbedtls_ecp_group_init( &key->grp );
mbedtls_mpi_init( &key->d );
mbedtls_ecp_point_init( &key->Q );
}
/*
* Unallocate (the components of) a point
*/
void mbedtls_ecp_point_free( mbedtls_ecp_point *pt )
{
if( pt == NULL )
return;
mbedtls_mpi_free( &( pt->X ) );
mbedtls_mpi_free( &( pt->Y ) );
mbedtls_mpi_free( &( pt->Z ) );
}
/*
* Unallocate (the components of) a group
*/
void mbedtls_ecp_group_free( mbedtls_ecp_group *grp )
{
size_t i;
if( grp == NULL )
return;
if( grp->h != 1 )
{
mbedtls_mpi_free( &grp->P );
mbedtls_mpi_free( &grp->A );
mbedtls_mpi_free( &grp->B );
mbedtls_ecp_point_free( &grp->G );
mbedtls_mpi_free( &grp->N );
}
if( grp->T != NULL )
{
for( i = 0; i < grp->T_size; i++ )
mbedtls_ecp_point_free( &grp->T[i] );
mbedtls_free( grp->T );
}
mbedtls_zeroize( grp, sizeof( mbedtls_ecp_group ) );
}
/*
* Unallocate (the components of) a key pair
*/
void mbedtls_ecp_keypair_free( mbedtls_ecp_keypair *key )
{
if( key == NULL )
return;
mbedtls_ecp_group_free( &key->grp );
mbedtls_mpi_free( &key->d );
mbedtls_ecp_point_free( &key->Q );
}
/*
* Copy the contents of a point
*/
int mbedtls_ecp_copy( mbedtls_ecp_point *P, const mbedtls_ecp_point *Q )
{
int ret;
MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &P->X, &Q->X ) );
MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &P->Y, &Q->Y ) );
MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &P->Z, &Q->Z ) );
cleanup:
return( ret );
}
/*
* Copy the contents of a group object
*/
int mbedtls_ecp_group_copy( mbedtls_ecp_group *dst, const mbedtls_ecp_group *src )
{
return mbedtls_ecp_group_load( dst, src->id );
}
/*
* Set point to zero
*/
int mbedtls_ecp_set_zero( mbedtls_ecp_point *pt )
{
int ret;
MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &pt->X , 1 ) );
MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &pt->Y , 1 ) );
MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &pt->Z , 0 ) );
cleanup:
return( ret );
}
/*
* Tell if a point is zero
*/
int mbedtls_ecp_is_zero( mbedtls_ecp_point *pt )
{
return( mbedtls_mpi_cmp_int( &pt->Z, 0 ) == 0 );
}
/*
* Compare two points lazily
*/
int mbedtls_ecp_point_cmp( const mbedtls_ecp_point *P,
const mbedtls_ecp_point *Q )
{
if( mbedtls_mpi_cmp_mpi( &P->X, &Q->X ) == 0 &&
mbedtls_mpi_cmp_mpi( &P->Y, &Q->Y ) == 0 &&
mbedtls_mpi_cmp_mpi( &P->Z, &Q->Z ) == 0 )
{
return( 0 );
}
return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );
}
/*
* Import a non-zero point from ASCII strings
*/
int mbedtls_ecp_point_read_string( mbedtls_ecp_point *P, int radix,
const char *x, const char *y )
{
int ret;
MBEDTLS_MPI_CHK( mbedtls_mpi_read_string( &P->X, radix, x ) );
MBEDTLS_MPI_CHK( mbedtls_mpi_read_string( &P->Y, radix, y ) );
MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &P->Z, 1 ) );
cleanup:
return( ret );
}
/*
* Export a point into unsigned binary data (SEC1 2.3.3)
*/
int mbedtls_ecp_point_write_binary( const mbedtls_ecp_group *grp, const mbedtls_ecp_point *P,
int format, size_t *olen,
unsigned char *buf, size_t buflen )
{
int ret = 0;
size_t plen;
if( format != MBEDTLS_ECP_PF_UNCOMPRESSED &&
format != MBEDTLS_ECP_PF_COMPRESSED )
return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );
/*
* Common case: P == 0
*/
if( mbedtls_mpi_cmp_int( &P->Z, 0 ) == 0 )
{
if( buflen < 1 )
return( MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL );
buf[0] = 0x00;
*olen = 1;
return( 0 );
}
plen = mbedtls_mpi_size( &grp->P );
if( format == MBEDTLS_ECP_PF_UNCOMPRESSED )
{
*olen = 2 * plen + 1;
if( buflen < *olen )
return( MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL );
buf[0] = 0x04;
MBEDTLS_MPI_CHK( mbedtls_mpi_write_binary( &P->X, buf + 1, plen ) );
MBEDTLS_MPI_CHK( mbedtls_mpi_write_binary( &P->Y, buf + 1 + plen, plen ) );
}
else if( format == MBEDTLS_ECP_PF_COMPRESSED )
{
*olen = plen + 1;
if( buflen < *olen )
return( MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL );
buf[0] = 0x02 + mbedtls_mpi_get_bit( &P->Y, 0 );
MBEDTLS_MPI_CHK( mbedtls_mpi_write_binary( &P->X, buf + 1, plen ) );
}
cleanup:
return( ret );
}
/*
* Import a point from unsigned binary data (SEC1 2.3.4)
*/
int mbedtls_ecp_point_read_binary( const mbedtls_ecp_group *grp, mbedtls_ecp_point *pt,
const unsigned char *buf, size_t ilen )
{
int ret;
size_t plen;
if( ilen < 1 )
return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );
if( buf[0] == 0x00 )
{
if( ilen == 1 )
return( mbedtls_ecp_set_zero( pt ) );
else
return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );
}
plen = mbedtls_mpi_size( &grp->P );
if( buf[0] != 0x04 )
return( MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE );
if( ilen != 2 * plen + 1 )
return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );
MBEDTLS_MPI_CHK( mbedtls_mpi_read_binary( &pt->X, buf + 1, plen ) );
MBEDTLS_MPI_CHK( mbedtls_mpi_read_binary( &pt->Y, buf + 1 + plen, plen ) );
MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &pt->Z, 1 ) );
cleanup:
return( ret );
}
/*
* Import a point from a TLS ECPoint record (RFC 4492)
* struct {
* opaque point <1..2^8-1>;
* } ECPoint;
*/
int mbedtls_ecp_tls_read_point( const mbedtls_ecp_group *grp, mbedtls_ecp_point *pt,
const unsigned char **buf, size_t buf_len )
{
unsigned char data_len;
const unsigned char *buf_start;
/*
* We must have at least two bytes (1 for length, at least one for data)
*/
if( buf_len < 2 )
return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );
data_len = *(*buf)++;
if( data_len < 1 || data_len > buf_len - 1 )
return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );
/*
* Save buffer start for read_binary and update buf
*/
buf_start = *buf;
*buf += data_len;
return mbedtls_ecp_point_read_binary( grp, pt, buf_start, data_len );
}
/*
* Export a point as a TLS ECPoint record (RFC 4492)
* struct {
* opaque point <1..2^8-1>;
* } ECPoint;
*/
int mbedtls_ecp_tls_write_point( const mbedtls_ecp_group *grp, const mbedtls_ecp_point *pt,
int format, size_t *olen,
unsigned char *buf, size_t blen )
{
int ret;
/*
* buffer length must be at least one, for our length byte
*/
if( blen < 1 )
return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );
if( ( ret = mbedtls_ecp_point_write_binary( grp, pt, format,
olen, buf + 1, blen - 1) ) != 0 )
return( ret );
/*
* write length to the first byte and update total length
*/
buf[0] = (unsigned char) *olen;
++*olen;
return( 0 );
}
/*
* Set a group from an ECParameters record (RFC 4492)
*/
int mbedtls_ecp_tls_read_group( mbedtls_ecp_group *grp, const unsigned char **buf, size_t len )
{
uint16_t tls_id;
const mbedtls_ecp_curve_info *curve_info;
/*
* We expect at least three bytes (see below)
*/
if( len < 3 )
return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );
/*
* First byte is curve_type; only named_curve is handled
*/
if( *(*buf)++ != MBEDTLS_ECP_TLS_NAMED_CURVE )
return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );
/*
* Next two bytes are the namedcurve value
*/
tls_id = *(*buf)++;
tls_id <<= 8;
tls_id |= *(*buf)++;
if( ( curve_info = mbedtls_ecp_curve_info_from_tls_id( tls_id ) ) == NULL )
return( MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE );
return mbedtls_ecp_group_load( grp, curve_info->grp_id );
}
/*
* Write the ECParameters record corresponding to a group (RFC 4492)
*/
int mbedtls_ecp_tls_write_group( const mbedtls_ecp_group *grp, size_t *olen,
unsigned char *buf, size_t blen )
{
const mbedtls_ecp_curve_info *curve_info;
if( ( curve_info = mbedtls_ecp_curve_info_from_grp_id( grp->id ) ) == NULL )
return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );
/*
* We are going to write 3 bytes (see below)
*/
*olen = 3;
if( blen < *olen )
return( MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL );
/*
* First byte is curve_type, always named_curve
*/
*buf++ = MBEDTLS_ECP_TLS_NAMED_CURVE;
/*
* Next two bytes are the namedcurve value
*/
buf[0] = curve_info->tls_id >> 8;
buf[1] = curve_info->tls_id & 0xFF;
return( 0 );
}
/*
* Wrapper around fast quasi-modp functions, with fall-back to mbedtls_mpi_mod_mpi.
* See the documentation of struct mbedtls_ecp_group.
*
* This function is in the critial loop for mbedtls_ecp_mul, so pay attention to perf.
*/
static int ecp_modp( mbedtls_mpi *N, const mbedtls_ecp_group *grp )
{
int ret;
if( grp->modp == NULL )
return( mbedtls_mpi_mod_mpi( N, N, &grp->P ) );
/* N->s < 0 is a much faster test, which fails only if N is 0 */
if( ( N->s < 0 && mbedtls_mpi_cmp_int( N, 0 ) != 0 ) ||
mbedtls_mpi_bitlen( N ) > 2 * grp->pbits )
{
return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );
}
MBEDTLS_MPI_CHK( grp->modp( N ) );
/* N->s < 0 is a much faster test, which fails only if N is 0 */
while( N->s < 0 && mbedtls_mpi_cmp_int( N, 0 ) != 0 )
MBEDTLS_MPI_CHK( mbedtls_mpi_add_mpi( N, N, &grp->P ) );
while( mbedtls_mpi_cmp_mpi( N, &grp->P ) >= 0 )
/* we known P, N and the result are positive */
MBEDTLS_MPI_CHK( mbedtls_mpi_sub_abs( N, N, &grp->P ) );
cleanup:
return( ret );
}
/*
* Fast mod-p functions expect their argument to be in the 0..p^2 range.
*
* In order to guarantee that, we need to ensure that operands of
* mbedtls_mpi_mul_mpi are in the 0..p range. So, after each operation we will
* bring the result back to this range.
*
* The following macros are shortcuts for doing that.
*/
/*
* Reduce a mbedtls_mpi mod p in-place, general case, to use after mbedtls_mpi_mul_mpi
*/
#if defined(MBEDTLS_SELF_TEST)
#define INC_MUL_COUNT mul_count++;
#else
#define INC_MUL_COUNT
#endif
#define MOD_MUL( N ) do { MBEDTLS_MPI_CHK( ecp_modp( &N, grp ) ); INC_MUL_COUNT } \
while( 0 )
/*
* Reduce a mbedtls_mpi mod p in-place, to use after mbedtls_mpi_sub_mpi
* N->s < 0 is a very fast test, which fails only if N is 0
*/
#define MOD_SUB( N ) \
while( N.s < 0 && mbedtls_mpi_cmp_int( &N, 0 ) != 0 ) \
MBEDTLS_MPI_CHK( mbedtls_mpi_add_mpi( &N, &N, &grp->P ) )
/*
* Reduce a mbedtls_mpi mod p in-place, to use after mbedtls_mpi_add_mpi and mbedtls_mpi_mul_int.
* We known P, N and the result are positive, so sub_abs is correct, and
* a bit faster.
*/
#define MOD_ADD( N ) \
while( mbedtls_mpi_cmp_mpi( &N, &grp->P ) >= 0 ) \
MBEDTLS_MPI_CHK( mbedtls_mpi_sub_abs( &N, &N, &grp->P ) )
#if defined(ECP_SHORTWEIERSTRASS)
/*
* For curves in short Weierstrass form, we do all the internal operations in
* Jacobian coordinates.
*
* For multiplication, we'll use a comb method with coutermeasueres against
* SPA, hence timing attacks.
*/
/*
* Normalize jacobian coordinates so that Z == 0 || Z == 1 (GECC 3.2.1)
* Cost: 1N := 1I + 3M + 1S
*/
static int ecp_normalize_jac( const mbedtls_ecp_group *grp, mbedtls_ecp_point *pt )
{
int ret;
mbedtls_mpi Zi, ZZi;
if( mbedtls_mpi_cmp_int( &pt->Z, 0 ) == 0 )
return( 0 );
#if defined(MBEDTLS_ECP_NORMALIZE_JAC_ALT)
if ( mbedtls_internal_ecp_grp_capable( grp ) )
{
return mbedtls_internal_ecp_normalize_jac( grp, pt );
}
#endif /* MBEDTLS_ECP_NORMALIZE_JAC_ALT */
mbedtls_mpi_init( &Zi ); mbedtls_mpi_init( &ZZi );
/*
* X = X / Z^2 mod p
*/
MBEDTLS_MPI_CHK( mbedtls_mpi_inv_mod( &Zi, &pt->Z, &grp->P ) );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &ZZi, &Zi, &Zi ) ); MOD_MUL( ZZi );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &pt->X, &pt->X, &ZZi ) ); MOD_MUL( pt->X );
/*
* Y = Y / Z^3 mod p
*/
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &pt->Y, &pt->Y, &ZZi ) ); MOD_MUL( pt->Y );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &pt->Y, &pt->Y, &Zi ) ); MOD_MUL( pt->Y );
/*
* Z = 1
*/
MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &pt->Z, 1 ) );
cleanup:
mbedtls_mpi_free( &Zi ); mbedtls_mpi_free( &ZZi );
return( ret );
}
/*
* Normalize jacobian coordinates of an array of (pointers to) points,
* using Montgomery's trick to perform only one inversion mod P.
* (See for example Cohen's "A Course in Computational Algebraic Number
* Theory", Algorithm 10.3.4.)
*
* Warning: fails (returning an error) if one of the points is zero!
* This should never happen, see choice of w in ecp_mul_comb().
*
* Cost: 1N(t) := 1I + (6t - 3)M + 1S
*/
static int ecp_normalize_jac_many( const mbedtls_ecp_group *grp,
mbedtls_ecp_point *T[], size_t t_len )
{
int ret;
size_t i;
mbedtls_mpi *c, u, Zi, ZZi;
if( t_len < 2 )
return( ecp_normalize_jac( grp, *T ) );
#if defined(MBEDTLS_ECP_NORMALIZE_JAC_MANY_ALT)
if ( mbedtls_internal_ecp_grp_capable( grp ) )
{
return mbedtls_internal_ecp_normalize_jac_many(grp, T, t_len);
}
#endif
if( ( c = mbedtls_calloc( t_len, sizeof( mbedtls_mpi ) ) ) == NULL )
return( MBEDTLS_ERR_ECP_ALLOC_FAILED );
mbedtls_mpi_init( &u ); mbedtls_mpi_init( &Zi ); mbedtls_mpi_init( &ZZi );
/*
* c[i] = Z_0 * ... * Z_i
*/
MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &c[0], &T[0]->Z ) );
for( i = 1; i < t_len; i++ )
{
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &c[i], &c[i-1], &T[i]->Z ) );
MOD_MUL( c[i] );
}
/*
* u = 1 / (Z_0 * ... * Z_n) mod P
*/
MBEDTLS_MPI_CHK( mbedtls_mpi_inv_mod( &u, &c[t_len-1], &grp->P ) );
for( i = t_len - 1; ; i-- )
{
/*
* Zi = 1 / Z_i mod p
* u = 1 / (Z_0 * ... * Z_i) mod P
*/
if( i == 0 ) {
MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &Zi, &u ) );
}
else
{
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &Zi, &u, &c[i-1] ) ); MOD_MUL( Zi );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &u, &u, &T[i]->Z ) ); MOD_MUL( u );
}
/*
* proceed as in normalize()
*/
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &ZZi, &Zi, &Zi ) ); MOD_MUL( ZZi );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T[i]->X, &T[i]->X, &ZZi ) ); MOD_MUL( T[i]->X );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T[i]->Y, &T[i]->Y, &ZZi ) ); MOD_MUL( T[i]->Y );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T[i]->Y, &T[i]->Y, &Zi ) ); MOD_MUL( T[i]->Y );
/*
* Post-precessing: reclaim some memory by shrinking coordinates
* - not storing Z (always 1)
* - shrinking other coordinates, but still keeping the same number of
* limbs as P, as otherwise it will too likely be regrown too fast.
*/
MBEDTLS_MPI_CHK( mbedtls_mpi_shrink( &T[i]->X, grp->P.n ) );
MBEDTLS_MPI_CHK( mbedtls_mpi_shrink( &T[i]->Y, grp->P.n ) );
mbedtls_mpi_free( &T[i]->Z );
if( i == 0 )
break;
}
cleanup:
mbedtls_mpi_free( &u ); mbedtls_mpi_free( &Zi ); mbedtls_mpi_free( &ZZi );
for( i = 0; i < t_len; i++ )
mbedtls_mpi_free( &c[i] );
mbedtls_free( c );
return( ret );
}
/*
* Conditional point inversion: Q -> -Q = (Q.X, -Q.Y, Q.Z) without leak.
* "inv" must be 0 (don't invert) or 1 (invert) or the result will be invalid
*/
static int ecp_safe_invert_jac( const mbedtls_ecp_group *grp,
mbedtls_ecp_point *Q,
unsigned char inv )
{
int ret;
unsigned char nonzero;
mbedtls_mpi mQY;
mbedtls_mpi_init( &mQY );
/* Use the fact that -Q.Y mod P = P - Q.Y unless Q.Y == 0 */
MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &mQY, &grp->P, &Q->Y ) );
nonzero = mbedtls_mpi_cmp_int( &Q->Y, 0 ) != 0;
MBEDTLS_MPI_CHK( mbedtls_mpi_safe_cond_assign( &Q->Y, &mQY, inv & nonzero ) );
cleanup:
mbedtls_mpi_free( &mQY );
return( ret );
}
/*
* Point doubling R = 2 P, Jacobian coordinates
*
* Based on http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian.html#doubling-dbl-1998-cmo-2 .
*
* We follow the variable naming fairly closely. The formula variations that trade a MUL for a SQR
* (plus a few ADDs) aren't useful as our bignum implementation doesn't distinguish squaring.
*
* Standard optimizations are applied when curve parameter A is one of { 0, -3 }.
*
* Cost: 1D := 3M + 4S (A == 0)
* 4M + 4S (A == -3)
* 3M + 6S + 1a otherwise
*/
static int ecp_double_jac( const mbedtls_ecp_group *grp, mbedtls_ecp_point *R,
const mbedtls_ecp_point *P )
{
int ret;
mbedtls_mpi M, S, T, U;
#if defined(MBEDTLS_SELF_TEST)
dbl_count++;
#endif
#if defined(MBEDTLS_ECP_DOUBLE_JAC_ALT)
if ( mbedtls_internal_ecp_grp_capable( grp ) )
{
return mbedtls_internal_ecp_double_jac( grp, R, P );
}
#endif /* MBEDTLS_ECP_DOUBLE_JAC_ALT */
mbedtls_mpi_init( &M ); mbedtls_mpi_init( &S ); mbedtls_mpi_init( &T ); mbedtls_mpi_init( &U );
/* Special case for A = -3 */
if( grp->A.p == NULL )
{
/* M = 3(X + Z^2)(X - Z^2) */
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &S, &P->Z, &P->Z ) ); MOD_MUL( S );
MBEDTLS_MPI_CHK( mbedtls_mpi_add_mpi( &T, &P->X, &S ) ); MOD_ADD( T );
MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &U, &P->X, &S ) ); MOD_SUB( U );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &S, &T, &U ) ); MOD_MUL( S );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_int( &M, &S, 3 ) ); MOD_ADD( M );
}
else
{
/* M = 3.X^2 */
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &S, &P->X, &P->X ) ); MOD_MUL( S );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_int( &M, &S, 3 ) ); MOD_ADD( M );
/* Optimize away for "koblitz" curves with A = 0 */
if( mbedtls_mpi_cmp_int( &grp->A, 0 ) != 0 )
{
/* M += A.Z^4 */
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &S, &P->Z, &P->Z ) ); MOD_MUL( S );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T, &S, &S ) ); MOD_MUL( T );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &S, &T, &grp->A ) ); MOD_MUL( S );
MBEDTLS_MPI_CHK( mbedtls_mpi_add_mpi( &M, &M, &S ) ); MOD_ADD( M );
}
}
/* S = 4.X.Y^2 */
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T, &P->Y, &P->Y ) ); MOD_MUL( T );
MBEDTLS_MPI_CHK( mbedtls_mpi_shift_l( &T, 1 ) ); MOD_ADD( T );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &S, &P->X, &T ) ); MOD_MUL( S );
MBEDTLS_MPI_CHK( mbedtls_mpi_shift_l( &S, 1 ) ); MOD_ADD( S );
/* U = 8.Y^4 */
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &U, &T, &T ) ); MOD_MUL( U );
MBEDTLS_MPI_CHK( mbedtls_mpi_shift_l( &U, 1 ) ); MOD_ADD( U );
/* T = M^2 - 2.S */
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T, &M, &M ) ); MOD_MUL( T );
MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &T, &T, &S ) ); MOD_SUB( T );
MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &T, &T, &S ) ); MOD_SUB( T );
/* S = M(S - T) - U */
MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &S, &S, &T ) ); MOD_SUB( S );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &S, &S, &M ) ); MOD_MUL( S );
MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &S, &S, &U ) ); MOD_SUB( S );
/* U = 2.Y.Z */
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &U, &P->Y, &P->Z ) ); MOD_MUL( U );
MBEDTLS_MPI_CHK( mbedtls_mpi_shift_l( &U, 1 ) ); MOD_ADD( U );
MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &R->X, &T ) );
MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &R->Y, &S ) );
MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &R->Z, &U ) );
cleanup:
mbedtls_mpi_free( &M ); mbedtls_mpi_free( &S ); mbedtls_mpi_free( &T ); mbedtls_mpi_free( &U );
return( ret );
}
/*
* Addition: R = P + Q, mixed affine-Jacobian coordinates (GECC 3.22)
*
* The coordinates of Q must be normalized (= affine),
* but those of P don't need to. R is not normalized.
*
* Special cases: (1) P or Q is zero, (2) R is zero, (3) P == Q.
* None of these cases can happen as intermediate step in ecp_mul_comb():
* - at each step, P, Q and R are multiples of the base point, the factor
* being less than its order, so none of them is zero;
* - Q is an odd multiple of the base point, P an even multiple,
* due to the choice of precomputed points in the modified comb method.
* So branches for these cases do not leak secret information.
*
* We accept Q->Z being unset (saving memory in tables) as meaning 1.
*
* Cost: 1A := 8M + 3S
*/
static int ecp_add_mixed( const mbedtls_ecp_group *grp, mbedtls_ecp_point *R,
const mbedtls_ecp_point *P, const mbedtls_ecp_point *Q )
{
int ret;
mbedtls_mpi T1, T2, T3, T4, X, Y, Z;
#if defined(MBEDTLS_SELF_TEST)
add_count++;
#endif
#if defined(MBEDTLS_ECP_ADD_MIXED_ALT)
if ( mbedtls_internal_ecp_grp_capable( grp ) )
{
return mbedtls_internal_ecp_add_mixed( grp, R, P, Q );
}
#endif /* MBEDTLS_ECP_ADD_MIXED_ALT */
/*
* Trivial cases: P == 0 or Q == 0 (case 1)
*/
if( mbedtls_mpi_cmp_int( &P->Z, 0 ) == 0 )
return( mbedtls_ecp_copy( R, Q ) );
if( Q->Z.p != NULL && mbedtls_mpi_cmp_int( &Q->Z, 0 ) == 0 )
return( mbedtls_ecp_copy( R, P ) );
/*
* Make sure Q coordinates are normalized
*/
if( Q->Z.p != NULL && mbedtls_mpi_cmp_int( &Q->Z, 1 ) != 0 )
return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );
mbedtls_mpi_init( &T1 ); mbedtls_mpi_init( &T2 ); mbedtls_mpi_init( &T3 ); mbedtls_mpi_init( &T4 );
mbedtls_mpi_init( &X ); mbedtls_mpi_init( &Y ); mbedtls_mpi_init( &Z );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T1, &P->Z, &P->Z ) ); MOD_MUL( T1 );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T2, &T1, &P->Z ) ); MOD_MUL( T2 );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T1, &T1, &Q->X ) ); MOD_MUL( T1 );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T2, &T2, &Q->Y ) ); MOD_MUL( T2 );
MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &T1, &T1, &P->X ) ); MOD_SUB( T1 );
MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &T2, &T2, &P->Y ) ); MOD_SUB( T2 );
/* Special cases (2) and (3) */
if( mbedtls_mpi_cmp_int( &T1, 0 ) == 0 )
{
if( mbedtls_mpi_cmp_int( &T2, 0 ) == 0 )
{
ret = ecp_double_jac( grp, R, P );
goto cleanup;
}
else
{
ret = mbedtls_ecp_set_zero( R );
goto cleanup;
}
}
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &Z, &P->Z, &T1 ) ); MOD_MUL( Z );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T3, &T1, &T1 ) ); MOD_MUL( T3 );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T4, &T3, &T1 ) ); MOD_MUL( T4 );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T3, &T3, &P->X ) ); MOD_MUL( T3 );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_int( &T1, &T3, 2 ) ); MOD_ADD( T1 );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &X, &T2, &T2 ) ); MOD_MUL( X );
MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &X, &X, &T1 ) ); MOD_SUB( X );
MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &X, &X, &T4 ) ); MOD_SUB( X );
MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &T3, &T3, &X ) ); MOD_SUB( T3 );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T3, &T3, &T2 ) ); MOD_MUL( T3 );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T4, &T4, &P->Y ) ); MOD_MUL( T4 );
MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &Y, &T3, &T4 ) ); MOD_SUB( Y );
MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &R->X, &X ) );
MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &R->Y, &Y ) );
MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &R->Z, &Z ) );
cleanup:
mbedtls_mpi_free( &T1 ); mbedtls_mpi_free( &T2 ); mbedtls_mpi_free( &T3 ); mbedtls_mpi_free( &T4 );
mbedtls_mpi_free( &X ); mbedtls_mpi_free( &Y ); mbedtls_mpi_free( &Z );
return( ret );
}
/*
* Randomize jacobian coordinates:
* (X, Y, Z) -> (l^2 X, l^3 Y, l Z) for random l
* This is sort of the reverse operation of ecp_normalize_jac().
*
* This countermeasure was first suggested in [2].
*/
static int ecp_randomize_jac( const mbedtls_ecp_group *grp, mbedtls_ecp_point *pt,
int (*f_rng)(void *, unsigned char *, size_t), void *p_rng )
{
int ret;
mbedtls_mpi l, ll;
size_t p_size;
int count = 0;
#if defined(MBEDTLS_ECP_RANDOMIZE_JAC_ALT)
if ( mbedtls_internal_ecp_grp_capable( grp ) )
{
return mbedtls_internal_ecp_randomize_jac( grp, pt, f_rng, p_rng );
}
#endif /* MBEDTLS_ECP_RANDOMIZE_JAC_ALT */
p_size = ( grp->pbits + 7 ) / 8;
mbedtls_mpi_init( &l ); mbedtls_mpi_init( &ll );
/* Generate l such that 1 < l < p */
do
{
MBEDTLS_MPI_CHK( mbedtls_mpi_fill_random( &l, p_size, f_rng, p_rng ) );
while( mbedtls_mpi_cmp_mpi( &l, &grp->P ) >= 0 )
MBEDTLS_MPI_CHK( mbedtls_mpi_shift_r( &l, 1 ) );
if( count++ > 10 )
return( MBEDTLS_ERR_ECP_RANDOM_FAILED );
}
while( mbedtls_mpi_cmp_int( &l, 1 ) <= 0 );
/* Z = l * Z */
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &pt->Z, &pt->Z, &l ) ); MOD_MUL( pt->Z );
/* X = l^2 * X */
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &ll, &l, &l ) ); MOD_MUL( ll );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &pt->X, &pt->X, &ll ) ); MOD_MUL( pt->X );
/* Y = l^3 * Y */
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &ll, &ll, &l ) ); MOD_MUL( ll );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &pt->Y, &pt->Y, &ll ) ); MOD_MUL( pt->Y );
cleanup:
mbedtls_mpi_free( &l ); mbedtls_mpi_free( &ll );
return( ret );
}
/*
* Check and define parameters used by the comb method (see below for details)
*/
#if MBEDTLS_ECP_WINDOW_SIZE < 2 || MBEDTLS_ECP_WINDOW_SIZE > 7
#error "MBEDTLS_ECP_WINDOW_SIZE out of bounds"
#endif
/* d = ceil( n / w ) */
#define COMB_MAX_D ( MBEDTLS_ECP_MAX_BITS + 1 ) / 2
/* number of precomputed points */
#define COMB_MAX_PRE ( 1 << ( MBEDTLS_ECP_WINDOW_SIZE - 1 ) )
/*
* Compute the representation of m that will be used with our comb method.
*
* The basic comb method is described in GECC 3.44 for example. We use a
* modified version that provides resistance to SPA by avoiding zero
* digits in the representation as in [3]. We modify the method further by
* requiring that all K_i be odd, which has the small cost that our
* representation uses one more K_i, due to carries.
*
* Also, for the sake of compactness, only the seven low-order bits of x[i]
* are used to represent K_i, and the msb of x[i] encodes the the sign (s_i in
* the paper): it is set if and only if if s_i == -1;
*
* Calling conventions:
* - x is an array of size d + 1
* - w is the size, ie number of teeth, of the comb, and must be between
* 2 and 7 (in practice, between 2 and MBEDTLS_ECP_WINDOW_SIZE)
* - m is the MPI, expected to be odd and such that bitlength(m) <= w * d
* (the result will be incorrect if these assumptions are not satisfied)
*/
static void ecp_comb_fixed( unsigned char x[], size_t d,
unsigned char w, const mbedtls_mpi *m )
{
size_t i, j;
unsigned char c, cc, adjust;
memset( x, 0, d+1 );
/* First get the classical comb values (except for x_d = 0) */
for( i = 0; i < d; i++ )
for( j = 0; j < w; j++ )
x[i] |= mbedtls_mpi_get_bit( m, i + d * j ) << j;
/* Now make sure x_1 .. x_d are odd */
c = 0;
for( i = 1; i <= d; i++ )
{
/* Add carry and update it */
cc = x[i] & c;
x[i] = x[i] ^ c;
c = cc;
/* Adjust if needed, avoiding branches */
adjust = 1 - ( x[i] & 0x01 );
c |= x[i] & ( x[i-1] * adjust );
x[i] = x[i] ^ ( x[i-1] * adjust );
x[i-1] |= adjust << 7;
}
}
/*
* Precompute points for the comb method
*
* If i = i_{w-1} ... i_1 is the binary representation of i, then
* T[i] = i_{w-1} 2^{(w-1)d} P + ... + i_1 2^d P + P
*
* T must be able to hold 2^{w - 1} elements
*
* Cost: d(w-1) D + (2^{w-1} - 1) A + 1 N(w-1) + 1 N(2^{w-1} - 1)
*/
static int ecp_precompute_comb( const mbedtls_ecp_group *grp,
mbedtls_ecp_point T[], const mbedtls_ecp_point *P,
unsigned char w, size_t d )
{
int ret;
unsigned char i, k;
size_t j;
mbedtls_ecp_point *cur, *TT[COMB_MAX_PRE - 1];
/*
* Set T[0] = P and
* T[2^{l-1}] = 2^{dl} P for l = 1 .. w-1 (this is not the final value)
*/
MBEDTLS_MPI_CHK( mbedtls_ecp_copy( &T[0], P ) );
k = 0;
for( i = 1; i < ( 1U << ( w - 1 ) ); i <<= 1 )
{
cur = T + i;
MBEDTLS_MPI_CHK( mbedtls_ecp_copy( cur, T + ( i >> 1 ) ) );
for( j = 0; j < d; j++ )
MBEDTLS_MPI_CHK( ecp_double_jac( grp, cur, cur ) );
TT[k++] = cur;
}
MBEDTLS_MPI_CHK( ecp_normalize_jac_many( grp, TT, k ) );
/*
* Compute the remaining ones using the minimal number of additions
* Be careful to update T[2^l] only after using it!
*/
k = 0;
for( i = 1; i < ( 1U << ( w - 1 ) ); i <<= 1 )
{
j = i;
while( j-- )
{
MBEDTLS_MPI_CHK( ecp_add_mixed( grp, &T[i + j], &T[j], &T[i] ) );
TT[k++] = &T[i + j];
}
}
MBEDTLS_MPI_CHK( ecp_normalize_jac_many( grp, TT, k ) );
cleanup:
return( ret );
}
/*
* Select precomputed point: R = sign(i) * T[ abs(i) / 2 ]
*/
static int ecp_select_comb( const mbedtls_ecp_group *grp, mbedtls_ecp_point *R,
const mbedtls_ecp_point T[], unsigned char t_len,
unsigned char i )
{
int ret;
unsigned char ii, j;
/* Ignore the "sign" bit and scale down */
ii = ( i & 0x7Fu ) >> 1;
/* Read the whole table to thwart cache-based timing attacks */
for( j = 0; j < t_len; j++ )
{
MBEDTLS_MPI_CHK( mbedtls_mpi_safe_cond_assign( &R->X, &T[j].X, j == ii ) );
MBEDTLS_MPI_CHK( mbedtls_mpi_safe_cond_assign( &R->Y, &T[j].Y, j == ii ) );
}
/* Safely invert result if i is "negative" */
MBEDTLS_MPI_CHK( ecp_safe_invert_jac( grp, R, i >> 7 ) );
cleanup:
return( ret );
}
/*
* Core multiplication algorithm for the (modified) comb method.
* This part is actually common with the basic comb method (GECC 3.44)
*
* Cost: d A + d D + 1 R
*/
static int ecp_mul_comb_core( const mbedtls_ecp_group *grp, mbedtls_ecp_point *R,
const mbedtls_ecp_point T[], unsigned char t_len,
const unsigned char x[], size_t d,
int (*f_rng)(void *, unsigned char *, size_t),
void *p_rng )
{
int ret;
mbedtls_ecp_point Txi;
size_t i;
mbedtls_ecp_point_init( &Txi );
/* Start with a non-zero point and randomize its coordinates */
i = d;
MBEDTLS_MPI_CHK( ecp_select_comb( grp, R, T, t_len, x[i] ) );
MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &R->Z, 1 ) );
if( f_rng != 0 )
MBEDTLS_MPI_CHK( ecp_randomize_jac( grp, R, f_rng, p_rng ) );
while( i-- != 0 )
{
MBEDTLS_MPI_CHK( ecp_double_jac( grp, R, R ) );
MBEDTLS_MPI_CHK( ecp_select_comb( grp, &Txi, T, t_len, x[i] ) );
MBEDTLS_MPI_CHK( ecp_add_mixed( grp, R, R, &Txi ) );
}
cleanup:
mbedtls_ecp_point_free( &Txi );
return( ret );
}
/*
* Multiplication using the comb method,
* for curves in short Weierstrass form
*/
static int ecp_mul_comb( mbedtls_ecp_group *grp, mbedtls_ecp_point *R,
const mbedtls_mpi *m, const mbedtls_ecp_point *P,
int (*f_rng)(void *, unsigned char *, size_t),
void *p_rng )
{
int ret;
unsigned char w, m_is_odd, p_eq_g, pre_len, i;
size_t d;
unsigned char k[COMB_MAX_D + 1];
mbedtls_ecp_point *T;
mbedtls_mpi M, mm;
mbedtls_mpi_init( &M );
mbedtls_mpi_init( &mm );
/* we need N to be odd to trnaform m in an odd number, check now */
if( mbedtls_mpi_get_bit( &grp->N, 0 ) != 1 )
return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );
/*
* Minimize the number of multiplications, that is minimize
* 10 * d * w + 18 * 2^(w-1) + 11 * d + 7 * w, with d = ceil( nbits / w )
* (see costs of the various parts, with 1S = 1M)
*/
w = grp->nbits >= 384 ? 5 : 4;
/*
* If P == G, pre-compute a bit more, since this may be re-used later.
* Just adding one avoids upping the cost of the first mul too much,
* and the memory cost too.
*/
#if MBEDTLS_ECP_FIXED_POINT_OPTIM == 1
p_eq_g = ( mbedtls_mpi_cmp_mpi( &P->Y, &grp->G.Y ) == 0 &&
mbedtls_mpi_cmp_mpi( &P->X, &grp->G.X ) == 0 );
if( p_eq_g )
w++;
#else
p_eq_g = 0;
#endif
/*
* Make sure w is within bounds.
* (The last test is useful only for very small curves in the test suite.)
*/
if( w > MBEDTLS_ECP_WINDOW_SIZE )
w = MBEDTLS_ECP_WINDOW_SIZE;
if( w >= grp->nbits )
w = 2;
/* Other sizes that depend on w */
pre_len = 1U << ( w - 1 );
d = ( grp->nbits + w - 1 ) / w;
/*
* Prepare precomputed points: if P == G we want to
* use grp->T if already initialized, or initialize it.
*/
T = p_eq_g ? grp->T : NULL;
if( T == NULL )
{
T = mbedtls_calloc( pre_len, sizeof( mbedtls_ecp_point ) );
if( T == NULL )
{
ret = MBEDTLS_ERR_ECP_ALLOC_FAILED;
goto cleanup;
}
MBEDTLS_MPI_CHK( ecp_precompute_comb( grp, T, P, w, d ) );
if( p_eq_g )
{
grp->T = T;
grp->T_size = pre_len;
}
}
/*
* Make sure M is odd (M = m or M = N - m, since N is odd)
* using the fact that m * P = - (N - m) * P
*/
m_is_odd = ( mbedtls_mpi_get_bit( m, 0 ) == 1 );
MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &M, m ) );
MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &mm, &grp->N, m ) );
MBEDTLS_MPI_CHK( mbedtls_mpi_safe_cond_assign( &M, &mm, ! m_is_odd ) );
/*
* Go for comb multiplication, R = M * P
*/
ecp_comb_fixed( k, d, w, &M );
MBEDTLS_MPI_CHK( ecp_mul_comb_core( grp, R, T, pre_len, k, d, f_rng, p_rng ) );
/*
* Now get m * P from M * P and normalize it
*/
MBEDTLS_MPI_CHK( ecp_safe_invert_jac( grp, R, ! m_is_odd ) );
MBEDTLS_MPI_CHK( ecp_normalize_jac( grp, R ) );
cleanup:
/* There are two cases where T is not stored in grp:
* - P != G
* - An intermediate operation failed before setting grp->T
* In either case, T must be freed.
*/
if( T != NULL && T != grp->T )
{
for( i = 0; i < pre_len; i++ )
mbedtls_ecp_point_free( &T[i] );
mbedtls_free( T );
}
mbedtls_mpi_free( &M );
mbedtls_mpi_free( &mm );
if( ret != 0 )
mbedtls_ecp_point_free( R );
return( ret );
}
#endif /* ECP_SHORTWEIERSTRASS */
#if defined(ECP_MONTGOMERY)
/*
* For Montgomery curves, we do all the internal arithmetic in projective
* coordinates. Import/export of points uses only the x coordinates, which is
* internaly represented as X / Z.
*
* For scalar multiplication, we'll use a Montgomery ladder.
*/
/*
* Normalize Montgomery x/z coordinates: X = X/Z, Z = 1
* Cost: 1M + 1I
*/
static int ecp_normalize_mxz( const mbedtls_ecp_group *grp, mbedtls_ecp_point *P )
{
int ret;
#if defined(MBEDTLS_ECP_NORMALIZE_MXZ_ALT)
if ( mbedtls_internal_ecp_grp_capable( grp ) )
{
return mbedtls_internal_ecp_normalize_mxz( grp, P );
}
#endif /* MBEDTLS_ECP_NORMALIZE_MXZ_ALT */
MBEDTLS_MPI_CHK( mbedtls_mpi_inv_mod( &P->Z, &P->Z, &grp->P ) );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &P->X, &P->X, &P->Z ) ); MOD_MUL( P->X );
MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &P->Z, 1 ) );
cleanup:
return( ret );
}
/*
* Randomize projective x/z coordinates:
* (X, Z) -> (l X, l Z) for random l
* This is sort of the reverse operation of ecp_normalize_mxz().
*
* This countermeasure was first suggested in [2].
* Cost: 2M
*/
static int ecp_randomize_mxz( const mbedtls_ecp_group *grp, mbedtls_ecp_point *P,
int (*f_rng)(void *, unsigned char *, size_t), void *p_rng )
{
int ret;
mbedtls_mpi l;
size_t p_size;
int count = 0;
#if defined(MBEDTLS_ECP_RANDOMIZE_MXZ_ALT)
if ( mbedtls_internal_ecp_grp_capable( grp ) )
{
return mbedtls_internal_ecp_randomize_mxz( grp, P, f_rng, p_rng );
}
#endif /* MBEDTLS_ECP_RANDOMIZE_MXZ_ALT */
p_size = ( grp->pbits + 7 ) / 8;
mbedtls_mpi_init( &l );
/* Generate l such that 1 < l < p */
do
{
MBEDTLS_MPI_CHK( mbedtls_mpi_fill_random( &l, p_size, f_rng, p_rng ) );
while( mbedtls_mpi_cmp_mpi( &l, &grp->P ) >= 0 )
MBEDTLS_MPI_CHK( mbedtls_mpi_shift_r( &l, 1 ) );
if( count++ > 10 )
return( MBEDTLS_ERR_ECP_RANDOM_FAILED );
}
while( mbedtls_mpi_cmp_int( &l, 1 ) <= 0 );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &P->X, &P->X, &l ) ); MOD_MUL( P->X );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &P->Z, &P->Z, &l ) ); MOD_MUL( P->Z );
cleanup:
mbedtls_mpi_free( &l );
return( ret );
}
/*
* Double-and-add: R = 2P, S = P + Q, with d = X(P - Q),
* for Montgomery curves in x/z coordinates.
*
* http://www.hyperelliptic.org/EFD/g1p/auto-code/montgom/xz/ladder/mladd-1987-m.op3
* with
* d = X1
* P = (X2, Z2)
* Q = (X3, Z3)
* R = (X4, Z4)
* S = (X5, Z5)
* and eliminating temporary variables tO, ..., t4.
*
* Cost: 5M + 4S
*/
static int ecp_double_add_mxz( const mbedtls_ecp_group *grp,
mbedtls_ecp_point *R, mbedtls_ecp_point *S,
const mbedtls_ecp_point *P, const mbedtls_ecp_point *Q,
const mbedtls_mpi *d )
{
int ret;
mbedtls_mpi A, AA, B, BB, E, C, D, DA, CB;
#if defined(MBEDTLS_ECP_DOUBLE_ADD_MXZ_ALT)
if ( mbedtls_internal_ecp_grp_capable( grp ) )
{
return mbedtls_internal_ecp_double_add_mxz( grp, R, S, P, Q, d );
}
#endif /* MBEDTLS_ECP_DOUBLE_ADD_MXZ_ALT */
mbedtls_mpi_init( &A ); mbedtls_mpi_init( &AA ); mbedtls_mpi_init( &B );
mbedtls_mpi_init( &BB ); mbedtls_mpi_init( &E ); mbedtls_mpi_init( &C );
mbedtls_mpi_init( &D ); mbedtls_mpi_init( &DA ); mbedtls_mpi_init( &CB );
MBEDTLS_MPI_CHK( mbedtls_mpi_add_mpi( &A, &P->X, &P->Z ) ); MOD_ADD( A );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &AA, &A, &A ) ); MOD_MUL( AA );
MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &B, &P->X, &P->Z ) ); MOD_SUB( B );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &BB, &B, &B ) ); MOD_MUL( BB );
MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &E, &AA, &BB ) ); MOD_SUB( E );
MBEDTLS_MPI_CHK( mbedtls_mpi_add_mpi( &C, &Q->X, &Q->Z ) ); MOD_ADD( C );
MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &D, &Q->X, &Q->Z ) ); MOD_SUB( D );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &DA, &D, &A ) ); MOD_MUL( DA );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &CB, &C, &B ) ); MOD_MUL( CB );
MBEDTLS_MPI_CHK( mbedtls_mpi_add_mpi( &S->X, &DA, &CB ) ); MOD_MUL( S->X );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &S->X, &S->X, &S->X ) ); MOD_MUL( S->X );
MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &S->Z, &DA, &CB ) ); MOD_SUB( S->Z );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &S->Z, &S->Z, &S->Z ) ); MOD_MUL( S->Z );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &S->Z, d, &S->Z ) ); MOD_MUL( S->Z );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &R->X, &AA, &BB ) ); MOD_MUL( R->X );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &R->Z, &grp->A, &E ) ); MOD_MUL( R->Z );
MBEDTLS_MPI_CHK( mbedtls_mpi_add_mpi( &R->Z, &BB, &R->Z ) ); MOD_ADD( R->Z );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &R->Z, &E, &R->Z ) ); MOD_MUL( R->Z );
cleanup:
mbedtls_mpi_free( &A ); mbedtls_mpi_free( &AA ); mbedtls_mpi_free( &B );
mbedtls_mpi_free( &BB ); mbedtls_mpi_free( &E ); mbedtls_mpi_free( &C );
mbedtls_mpi_free( &D ); mbedtls_mpi_free( &DA ); mbedtls_mpi_free( &CB );
return( ret );
}
/*
* Multiplication with Montgomery ladder in x/z coordinates,
* for curves in Montgomery form
*/
static int ecp_mul_mxz( mbedtls_ecp_group *grp, mbedtls_ecp_point *R,
const mbedtls_mpi *m, const mbedtls_ecp_point *P,
int (*f_rng)(void *, unsigned char *, size_t),
void *p_rng )
{
int ret;
size_t i;
unsigned char b;
mbedtls_ecp_point RP;
mbedtls_mpi PX;
mbedtls_ecp_point_init( &RP ); mbedtls_mpi_init( &PX );
/* Save PX and read from P before writing to R, in case P == R */
MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &PX, &P->X ) );
MBEDTLS_MPI_CHK( mbedtls_ecp_copy( &RP, P ) );
/* Set R to zero in modified x/z coordinates */
MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &R->X, 1 ) );
MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &R->Z, 0 ) );
mbedtls_mpi_free( &R->Y );
/* RP.X might be sligtly larger than P, so reduce it */
MOD_ADD( RP.X );
/* Randomize coordinates of the starting point */
if( f_rng != NULL )
MBEDTLS_MPI_CHK( ecp_randomize_mxz( grp, &RP, f_rng, p_rng ) );
/* Loop invariant: R = result so far, RP = R + P */
i = mbedtls_mpi_bitlen( m ); /* one past the (zero-based) most significant bit */
while( i-- > 0 )
{
b = mbedtls_mpi_get_bit( m, i );
/*
* if (b) R = 2R + P else R = 2R,
* which is:
* if (b) double_add( RP, R, RP, R )
* else double_add( R, RP, R, RP )
* but using safe conditional swaps to avoid leaks
*/
MBEDTLS_MPI_CHK( mbedtls_mpi_safe_cond_swap( &R->X, &RP.X, b ) );
MBEDTLS_MPI_CHK( mbedtls_mpi_safe_cond_swap( &R->Z, &RP.Z, b ) );
MBEDTLS_MPI_CHK( ecp_double_add_mxz( grp, R, &RP, R, &RP, &PX ) );
MBEDTLS_MPI_CHK( mbedtls_mpi_safe_cond_swap( &R->X, &RP.X, b ) );
MBEDTLS_MPI_CHK( mbedtls_mpi_safe_cond_swap( &R->Z, &RP.Z, b ) );
}
MBEDTLS_MPI_CHK( ecp_normalize_mxz( grp, R ) );
cleanup:
mbedtls_ecp_point_free( &RP ); mbedtls_mpi_free( &PX );
return( ret );
}
#endif /* ECP_MONTGOMERY */
/*
* Multiplication R = m * P
*/
int mbedtls_ecp_mul( mbedtls_ecp_group *grp, mbedtls_ecp_point *R,
const mbedtls_mpi *m, const mbedtls_ecp_point *P,
int (*f_rng)(void *, unsigned char *, size_t), void *p_rng )
{
int ret = MBEDTLS_ERR_ECP_BAD_INPUT_DATA;
#if defined(MBEDTLS_ECP_INTERNAL_ALT)
char is_grp_capable = 0;
#endif
/* Common sanity checks */
if( mbedtls_mpi_cmp_int( &P->Z, 1 ) != 0 )
return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );
if( ( ret = mbedtls_ecp_check_privkey( grp, m ) ) != 0 ||
( ret = mbedtls_ecp_check_pubkey( grp, P ) ) != 0 )
return( ret );
#if defined(MBEDTLS_ECP_INTERNAL_ALT)
if ( is_grp_capable = mbedtls_internal_ecp_grp_capable( grp ) )
{
MBEDTLS_MPI_CHK( mbedtls_internal_ecp_init( grp ) );
}
#endif /* MBEDTLS_ECP_INTERNAL_ALT */
#if defined(ECP_MONTGOMERY)
if( ecp_get_type( grp ) == ECP_TYPE_MONTGOMERY )
ret = ecp_mul_mxz( grp, R, m, P, f_rng, p_rng );
#endif
#if defined(ECP_SHORTWEIERSTRASS)
if( ecp_get_type( grp ) == ECP_TYPE_SHORT_WEIERSTRASS )
ret = ecp_mul_comb( grp, R, m, P, f_rng, p_rng );
#endif
#if defined(MBEDTLS_ECP_INTERNAL_ALT)
cleanup:
if ( is_grp_capable )
{
mbedtls_internal_ecp_free( grp );
}
#endif /* MBEDTLS_ECP_INTERNAL_ALT */
return( ret );
}
#if defined(ECP_SHORTWEIERSTRASS)
/*
* Check that an affine point is valid as a public key,
* short weierstrass curves (SEC1 3.2.3.1)
*/
static int ecp_check_pubkey_sw( const mbedtls_ecp_group *grp, const mbedtls_ecp_point *pt )
{
int ret;
mbedtls_mpi YY, RHS;
/* pt coordinates must be normalized for our checks */
if( mbedtls_mpi_cmp_int( &pt->X, 0 ) < 0 ||
mbedtls_mpi_cmp_int( &pt->Y, 0 ) < 0 ||
mbedtls_mpi_cmp_mpi( &pt->X, &grp->P ) >= 0 ||
mbedtls_mpi_cmp_mpi( &pt->Y, &grp->P ) >= 0 )
return( MBEDTLS_ERR_ECP_INVALID_KEY );
mbedtls_mpi_init( &YY ); mbedtls_mpi_init( &RHS );
/*
* YY = Y^2
* RHS = X (X^2 + A) + B = X^3 + A X + B
*/
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &YY, &pt->Y, &pt->Y ) ); MOD_MUL( YY );
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &RHS, &pt->X, &pt->X ) ); MOD_MUL( RHS );
/* Special case for A = -3 */
if( grp->A.p == NULL )
{
MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &RHS, &RHS, 3 ) ); MOD_SUB( RHS );
}
else
{
MBEDTLS_MPI_CHK( mbedtls_mpi_add_mpi( &RHS, &RHS, &grp->A ) ); MOD_ADD( RHS );
}
MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &RHS, &RHS, &pt->X ) ); MOD_MUL( RHS );
MBEDTLS_MPI_CHK( mbedtls_mpi_add_mpi( &RHS, &RHS, &grp->B ) ); MOD_ADD( RHS );
if( mbedtls_mpi_cmp_mpi( &YY, &RHS ) != 0 )
ret = MBEDTLS_ERR_ECP_INVALID_KEY;
cleanup:
mbedtls_mpi_free( &YY ); mbedtls_mpi_free( &RHS );
return( ret );
}
#endif /* ECP_SHORTWEIERSTRASS */
/*
* R = m * P with shortcuts for m == 1 and m == -1
* NOT constant-time - ONLY for short Weierstrass!
*/
static int mbedtls_ecp_mul_shortcuts( mbedtls_ecp_group *grp,
mbedtls_ecp_point *R,
const mbedtls_mpi *m,
const mbedtls_ecp_point *P )
{
int ret;
if( mbedtls_mpi_cmp_int( m, 1 ) == 0 )
{
MBEDTLS_MPI_CHK( mbedtls_ecp_copy( R, P ) );
}
else if( mbedtls_mpi_cmp_int( m, -1 ) == 0 )
{
MBEDTLS_MPI_CHK( mbedtls_ecp_copy( R, P ) );
if( mbedtls_mpi_cmp_int( &R->Y, 0 ) != 0 )
MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &R->Y, &grp->P, &R->Y ) );
}
else
{
MBEDTLS_MPI_CHK( mbedtls_ecp_mul( grp, R, m, P, NULL, NULL ) );
}
cleanup:
return( ret );
}
/*
* Linear combination
* NOT constant-time
*/
int mbedtls_ecp_muladd( mbedtls_ecp_group *grp, mbedtls_ecp_point *R,
const mbedtls_mpi *m, const mbedtls_ecp_point *P,
const mbedtls_mpi *n, const mbedtls_ecp_point *Q )
{
int ret;
mbedtls_ecp_point mP;
#if defined(MBEDTLS_ECP_INTERNAL_ALT)
char is_grp_capable = 0;
#endif
if( ecp_get_type( grp ) != ECP_TYPE_SHORT_WEIERSTRASS )
return( MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE );
mbedtls_ecp_point_init( &mP );
MBEDTLS_MPI_CHK( mbedtls_ecp_mul_shortcuts( grp, &mP, m, P ) );
MBEDTLS_MPI_CHK( mbedtls_ecp_mul_shortcuts( grp, R, n, Q ) );
#if defined(MBEDTLS_ECP_INTERNAL_ALT)
if ( is_grp_capable = mbedtls_internal_ecp_grp_capable( grp ) )
{
MBEDTLS_MPI_CHK( mbedtls_internal_ecp_init( grp ) );
}
#endif /* MBEDTLS_ECP_INTERNAL_ALT */
MBEDTLS_MPI_CHK( ecp_add_mixed( grp, R, &mP, R ) );
MBEDTLS_MPI_CHK( ecp_normalize_jac( grp, R ) );
cleanup:
#if defined(MBEDTLS_ECP_INTERNAL_ALT)
if ( is_grp_capable )
{
mbedtls_internal_ecp_free( grp );
}
#endif /* MBEDTLS_ECP_INTERNAL_ALT */
mbedtls_ecp_point_free( &mP );
return( ret );
}
#if defined(ECP_MONTGOMERY)
/*
* Check validity of a public key for Montgomery curves with x-only schemes
*/
static int ecp_check_pubkey_mx( const mbedtls_ecp_group *grp, const mbedtls_ecp_point *pt )
{
/* [Curve25519 p. 5] Just check X is the correct number of bytes */
if( mbedtls_mpi_size( &pt->X ) > ( grp->nbits + 7 ) / 8 )
return( MBEDTLS_ERR_ECP_INVALID_KEY );
return( 0 );
}
#endif /* ECP_MONTGOMERY */
/*
* Check that a point is valid as a public key
*/
int mbedtls_ecp_check_pubkey( const mbedtls_ecp_group *grp, const mbedtls_ecp_point *pt )
{
/* Must use affine coordinates */
if( mbedtls_mpi_cmp_int( &pt->Z, 1 ) != 0 )
return( MBEDTLS_ERR_ECP_INVALID_KEY );
#if defined(ECP_MONTGOMERY)
if( ecp_get_type( grp ) == ECP_TYPE_MONTGOMERY )
return( ecp_check_pubkey_mx( grp, pt ) );
#endif
#if defined(ECP_SHORTWEIERSTRASS)
if( ecp_get_type( grp ) == ECP_TYPE_SHORT_WEIERSTRASS )
return( ecp_check_pubkey_sw( grp, pt ) );
#endif
return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );
}
/*
* Check that an mbedtls_mpi is valid as a private key
*/
int mbedtls_ecp_check_privkey( const mbedtls_ecp_group *grp, const mbedtls_mpi *d )
{
#if defined(ECP_MONTGOMERY)
if( ecp_get_type( grp ) == ECP_TYPE_MONTGOMERY )
{
/* see [Curve25519] page 5 */
if( mbedtls_mpi_get_bit( d, 0 ) != 0 ||
mbedtls_mpi_get_bit( d, 1 ) != 0 ||
mbedtls_mpi_get_bit( d, 2 ) != 0 ||
mbedtls_mpi_bitlen( d ) - 1 != grp->nbits ) /* mbedtls_mpi_bitlen is one-based! */
return( MBEDTLS_ERR_ECP_INVALID_KEY );
else
return( 0 );
}
#endif /* ECP_MONTGOMERY */
#if defined(ECP_SHORTWEIERSTRASS)
if( ecp_get_type( grp ) == ECP_TYPE_SHORT_WEIERSTRASS )
{
/* see SEC1 3.2 */
if( mbedtls_mpi_cmp_int( d, 1 ) < 0 ||
mbedtls_mpi_cmp_mpi( d, &grp->N ) >= 0 )
return( MBEDTLS_ERR_ECP_INVALID_KEY );
else
return( 0 );
}
#endif /* ECP_SHORTWEIERSTRASS */
return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );
}
/*
* Generate a private key
*/
int mbedtls_ecp_gen_privkey( const mbedtls_ecp_group *grp,
mbedtls_mpi *d,
int (*f_rng)(void *, unsigned char *, size_t),
void *p_rng )
{
int ret = MBEDTLS_ERR_ECP_BAD_INPUT_DATA;
size_t n_size = ( grp->nbits + 7 ) / 8;
#if defined(ECP_MONTGOMERY)
if( ecp_get_type( grp ) == ECP_TYPE_MONTGOMERY )
{
/* [M225] page 5 */
size_t b;
do {
MBEDTLS_MPI_CHK( mbedtls_mpi_fill_random( d, n_size, f_rng, p_rng ) );
} while( mbedtls_mpi_bitlen( d ) == 0);
/* Make sure the most significant bit is nbits */
b = mbedtls_mpi_bitlen( d ) - 1; /* mbedtls_mpi_bitlen is one-based */
if( b > grp->nbits )
MBEDTLS_MPI_CHK( mbedtls_mpi_shift_r( d, b - grp->nbits ) );
else
MBEDTLS_MPI_CHK( mbedtls_mpi_set_bit( d, grp->nbits, 1 ) );
/* Make sure the last three bits are unset */
MBEDTLS_MPI_CHK( mbedtls_mpi_set_bit( d, 0, 0 ) );
MBEDTLS_MPI_CHK( mbedtls_mpi_set_bit( d, 1, 0 ) );
MBEDTLS_MPI_CHK( mbedtls_mpi_set_bit( d, 2, 0 ) );
}
#endif /* ECP_MONTGOMERY */
#if defined(ECP_SHORTWEIERSTRASS)
if( ecp_get_type( grp ) == ECP_TYPE_SHORT_WEIERSTRASS )
{
/* SEC1 3.2.1: Generate d such that 1 <= n < N */
int count = 0;
/*
* Match the procedure given in RFC 6979 (deterministic ECDSA):
* - use the same byte ordering;
* - keep the leftmost nbits bits of the generated octet string;
* - try until result is in the desired range.
* This also avoids any biais, which is especially important for ECDSA.
*/
do
{
MBEDTLS_MPI_CHK( mbedtls_mpi_fill_random( d, n_size, f_rng, p_rng ) );
MBEDTLS_MPI_CHK( mbedtls_mpi_shift_r( d, 8 * n_size - grp->nbits ) );
/*
* Each try has at worst a probability 1/2 of failing (the msb has
* a probability 1/2 of being 0, and then the result will be < N),
* so after 30 tries failure probability is a most 2**(-30).
*
* For most curves, 1 try is enough with overwhelming probability,
* since N starts with a lot of 1s in binary, but some curves
* such as secp224k1 are actually very close to the worst case.
*/
if( ++count > 30 )
return( MBEDTLS_ERR_ECP_RANDOM_FAILED );
}
while( mbedtls_mpi_cmp_int( d, 1 ) < 0 ||
mbedtls_mpi_cmp_mpi( d, &grp->N ) >= 0 );
}
#endif /* ECP_SHORTWEIERSTRASS */
cleanup:
return( ret );
}
/*
* Generate a keypair with configurable base point
*/
int mbedtls_ecp_gen_keypair_base( mbedtls_ecp_group *grp,
const mbedtls_ecp_point *G,
mbedtls_mpi *d, mbedtls_ecp_point *Q,
int (*f_rng)(void *, unsigned char *, size_t),
void *p_rng )
{
int ret;
MBEDTLS_MPI_CHK( mbedtls_ecp_gen_privkey( grp, d, f_rng, p_rng ) );
MBEDTLS_MPI_CHK( mbedtls_ecp_mul( grp, Q, d, G, f_rng, p_rng ) );
cleanup:
return( ret );
}
/*
* Generate key pair, wrapper for conventional base point
*/
int mbedtls_ecp_gen_keypair( mbedtls_ecp_group *grp,
mbedtls_mpi *d, mbedtls_ecp_point *Q,
int (*f_rng)(void *, unsigned char *, size_t),
void *p_rng )
{
return( mbedtls_ecp_gen_keypair_base( grp, &grp->G, d, Q, f_rng, p_rng ) );
}
/*
* Generate a keypair, prettier wrapper
*/
int mbedtls_ecp_gen_key( mbedtls_ecp_group_id grp_id, mbedtls_ecp_keypair *key,
int (*f_rng)(void *, unsigned char *, size_t), void *p_rng )
{
int ret;
if( ( ret = mbedtls_ecp_group_load( &key->grp, grp_id ) ) != 0 )
return( ret );
return( mbedtls_ecp_gen_keypair( &key->grp, &key->d, &key->Q, f_rng, p_rng ) );
}
/*
* Check a public-private key pair
*/
int mbedtls_ecp_check_pub_priv( const mbedtls_ecp_keypair *pub, const mbedtls_ecp_keypair *prv )
{
int ret;
mbedtls_ecp_point Q;
mbedtls_ecp_group grp;
if( pub->grp.id == MBEDTLS_ECP_DP_NONE ||
pub->grp.id != prv->grp.id ||
mbedtls_mpi_cmp_mpi( &pub->Q.X, &prv->Q.X ) ||
mbedtls_mpi_cmp_mpi( &pub->Q.Y, &prv->Q.Y ) ||
mbedtls_mpi_cmp_mpi( &pub->Q.Z, &prv->Q.Z ) )
{
return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );
}
mbedtls_ecp_point_init( &Q );
mbedtls_ecp_group_init( &grp );
/* mbedtls_ecp_mul() needs a non-const group... */
mbedtls_ecp_group_copy( &grp, &prv->grp );
/* Also checks d is valid */
MBEDTLS_MPI_CHK( mbedtls_ecp_mul( &grp, &Q, &prv->d, &prv->grp.G, NULL, NULL ) );
if( mbedtls_mpi_cmp_mpi( &Q.X, &prv->Q.X ) ||
mbedtls_mpi_cmp_mpi( &Q.Y, &prv->Q.Y ) ||
mbedtls_mpi_cmp_mpi( &Q.Z, &prv->Q.Z ) )
{
ret = MBEDTLS_ERR_ECP_BAD_INPUT_DATA;
goto cleanup;
}
cleanup:
mbedtls_ecp_point_free( &Q );
mbedtls_ecp_group_free( &grp );
return( ret );
}
#if defined(MBEDTLS_SELF_TEST)
/*
* Checkup routine
*/
int mbedtls_ecp_self_test( int verbose )
{
int ret;
size_t i;
mbedtls_ecp_group grp;
mbedtls_ecp_point R, P;
mbedtls_mpi m;
unsigned long add_c_prev, dbl_c_prev, mul_c_prev;
/* exponents especially adapted for secp192r1 */
const char *exponents[] =
{
"000000000000000000000000000000000000000000000001", /* one */
"FFFFFFFFFFFFFFFFFFFFFFFF99DEF836146BC9B1B4D22830", /* N - 1 */
"5EA6F389A38B8BC81E767753B15AA5569E1782E30ABE7D25", /* random */
"400000000000000000000000000000000000000000000000", /* one and zeros */
"7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF", /* all ones */
"555555555555555555555555555555555555555555555555", /* 101010... */
};
mbedtls_ecp_group_init( &grp );
mbedtls_ecp_point_init( &R );
mbedtls_ecp_point_init( &P );
mbedtls_mpi_init( &m );
/* Use secp192r1 if available, or any available curve */
#if defined(MBEDTLS_ECP_DP_SECP192R1_ENABLED)
MBEDTLS_MPI_CHK( mbedtls_ecp_group_load( &grp, MBEDTLS_ECP_DP_SECP192R1 ) );
#else
MBEDTLS_MPI_CHK( mbedtls_ecp_group_load( &grp, mbedtls_ecp_curve_list()->grp_id ) );
#endif
if( verbose != 0 )
mbedtls_printf( " ECP test #1 (constant op_count, base point G): " );
/* Do a dummy multiplication first to trigger precomputation */
MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &m, 2 ) );
MBEDTLS_MPI_CHK( mbedtls_ecp_mul( &grp, &P, &m, &grp.G, NULL, NULL ) );
add_count = 0;
dbl_count = 0;
mul_count = 0;
MBEDTLS_MPI_CHK( mbedtls_mpi_read_string( &m, 16, exponents[0] ) );
MBEDTLS_MPI_CHK( mbedtls_ecp_mul( &grp, &R, &m, &grp.G, NULL, NULL ) );
for( i = 1; i < sizeof( exponents ) / sizeof( exponents[0] ); i++ )
{
add_c_prev = add_count;
dbl_c_prev = dbl_count;
mul_c_prev = mul_count;
add_count = 0;
dbl_count = 0;
mul_count = 0;
MBEDTLS_MPI_CHK( mbedtls_mpi_read_string( &m, 16, exponents[i] ) );
MBEDTLS_MPI_CHK( mbedtls_ecp_mul( &grp, &R, &m, &grp.G, NULL, NULL ) );
if( add_count != add_c_prev ||
dbl_count != dbl_c_prev ||
mul_count != mul_c_prev )
{
if( verbose != 0 )
mbedtls_printf( "failed (%u)\n", (unsigned int) i );
ret = 1;
goto cleanup;
}
}
if( verbose != 0 )
mbedtls_printf( "passed\n" );
if( verbose != 0 )
mbedtls_printf( " ECP test #2 (constant op_count, other point): " );
/* We computed P = 2G last time, use it */
add_count = 0;
dbl_count = 0;
mul_count = 0;
MBEDTLS_MPI_CHK( mbedtls_mpi_read_string( &m, 16, exponents[0] ) );
MBEDTLS_MPI_CHK( mbedtls_ecp_mul( &grp, &R, &m, &P, NULL, NULL ) );
for( i = 1; i < sizeof( exponents ) / sizeof( exponents[0] ); i++ )
{
add_c_prev = add_count;
dbl_c_prev = dbl_count;
mul_c_prev = mul_count;
add_count = 0;
dbl_count = 0;
mul_count = 0;
MBEDTLS_MPI_CHK( mbedtls_mpi_read_string( &m, 16, exponents[i] ) );
MBEDTLS_MPI_CHK( mbedtls_ecp_mul( &grp, &R, &m, &P, NULL, NULL ) );
if( add_count != add_c_prev ||
dbl_count != dbl_c_prev ||
mul_count != mul_c_prev )
{
if( verbose != 0 )
mbedtls_printf( "failed (%u)\n", (unsigned int) i );
ret = 1;
goto cleanup;
}
}
if( verbose != 0 )
mbedtls_printf( "passed\n" );
cleanup:
if( ret < 0 && verbose != 0 )
mbedtls_printf( "Unexpected error, return code = %08X\n", ret );
mbedtls_ecp_group_free( &grp );
mbedtls_ecp_point_free( &R );
mbedtls_ecp_point_free( &P );
mbedtls_mpi_free( &m );
if( verbose != 0 )
mbedtls_printf( "\n" );
return( ret );
}
#endif /* MBEDTLS_SELF_TEST */
#endif /* !MBEDTLS_ECP_ALT */
#endif /* MBEDTLS_ECP_C */
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