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mbedtls/library/ecp.c

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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
* RFC 7748 for the Curve448 and Curve25519 curve definitions
*
* [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) || \
defined(MBEDTLS_ECP_DP_CURVE448_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 lazyly
*/
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:
if( T != NULL && ! p_eq_g )
{
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 */
/* Allow any public value, if it's too big then we'll just reduce it mod p
* (RFC 7748 sec. 5 para. 3). */
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 RFC 7748 sec. 5 para. 5 */
if( mbedtls_mpi_get_bit( d, 0 ) != 0 ||
mbedtls_mpi_get_bit( d, 1 ) != 0 ||
mbedtls_mpi_bitlen( d ) - 1 != grp->nbits ) /* mbedtls_mpi_bitlen is one-based! */
return( MBEDTLS_ERR_ECP_INVALID_KEY );
else
/* see [Curve25519] page 5 */
if( grp->nbits == 254 && mbedtls_mpi_get_bit( d, 2 ) != 0 )
return( MBEDTLS_ERR_ECP_INVALID_KEY );
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 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;
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 two bits are unset for Curve448, three bits for
Curve25519 */
MBEDTLS_MPI_CHK( mbedtls_mpi_set_bit( d, 0, 0 ) );
MBEDTLS_MPI_CHK( mbedtls_mpi_set_bit( d, 1, 0 ) );
if( grp->nbits == 254 )
{
MBEDTLS_MPI_CHK( mbedtls_mpi_set_bit( d, 2, 0 ) );
}
}
else
#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 );
}
else
#endif /* ECP_SHORTWEIERSTRASS */
return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );
cleanup:
if( ret != 0 )
return( ret );
return( mbedtls_ecp_mul( grp, Q, d, G, f_rng, p_rng ) );
}
/*
* 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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