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/*
* Elliptic curves over GF(p): generic functions
*
* Copyright (C) 2006-2014, ARM Limited, All Rights Reserved
*
* This file is part of mbed TLS (http://www.polarssl.org)
* Lead Maintainer: Paul Bakker <polarssl_maintainer at polarssl.org>
*
* All rights reserved.
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 2 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License along
* with this program; if not, write to the Free Software Foundation, Inc.,
* 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA.
*/
/*
* 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
*
* [M255] http://cr.yp.to/ecdh/curve25519-20060209.pdf
*
* [2] CORON, Jean-Sébastien. 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 BÉNÉTEAU, 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(POLARSSL_CONFIG_FILE)
#include "polarssl/config.h"
#else
#include POLARSSL_CONFIG_FILE
#endif
#if defined(POLARSSL_ECP_C)
#include "polarssl/ecp.h"
#if defined(POLARSSL_PLATFORM_C)
#include "polarssl/platform.h"
#else
#define polarssl_printf printf
#define polarssl_malloc malloc
#define polarssl_free free
#endif
#include <stdlib.h>
#if defined(_MSC_VER) && !defined strcasecmp && !defined(EFIX64) && \
!defined(EFI32)
#define strcasecmp _stricmp
#endif
#if defined(_MSC_VER) && !defined(inline)
#define inline _inline
#else
#if defined(__ARMCC_VERSION) && !defined(inline)
#define inline __inline
#endif /* __ARMCC_VERSION */
#endif /*_MSC_VER */
/* Implementation that should never be optimized out by the compiler */
static void polarssl_zeroize( void *v, size_t n ) {
volatile unsigned char *p = v; while( n-- ) *p++ = 0;
}
#if defined(POLARSSL_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(POLARSSL_ECP_DP_SECP192R1_ENABLED) || \
defined(POLARSSL_ECP_DP_SECP224R1_ENABLED) || \
defined(POLARSSL_ECP_DP_SECP256R1_ENABLED) || \
defined(POLARSSL_ECP_DP_SECP384R1_ENABLED) || \
defined(POLARSSL_ECP_DP_SECP521R1_ENABLED) || \
defined(POLARSSL_ECP_DP_BP256R1_ENABLED) || \
defined(POLARSSL_ECP_DP_BP384R1_ENABLED) || \
defined(POLARSSL_ECP_DP_BP512R1_ENABLED) || \
defined(POLARSSL_ECP_DP_SECP192K1_ENABLED) || \
defined(POLARSSL_ECP_DP_SECP224K1_ENABLED) || \
defined(POLARSSL_ECP_DP_SECP256K1_ENABLED)
#define POLARSSL_ECP_SHORT_WEIERSTRASS
#endif
#if defined(POLARSSL_ECP_DP_M221_ENABLED) || \
defined(POLARSSL_ECP_DP_M255_ENABLED) || \
defined(POLARSSL_ECP_DP_M383_ENABLED) || \
defined(POLARSSL_ECP_DP_M511_ENABLED)
#define POLARSSL_ECP_MONTGOMERY
#endif
/*
* Curve types: internal for now, might be exposed later
*/
typedef enum
{
POLARSSL_ECP_TYPE_NONE = 0,
POLARSSL_ECP_TYPE_SHORT_WEIERSTRASS, /* y^2 = x^3 + a x + b */
POLARSSL_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.
*/
static const ecp_curve_info ecp_supported_curves[] =
{
#if defined(POLARSSL_ECP_DP_SECP521R1_ENABLED)
{ POLARSSL_ECP_DP_SECP521R1, 25, 521, "secp521r1" },
#endif
#if defined(POLARSSL_ECP_DP_BP512R1_ENABLED)
{ POLARSSL_ECP_DP_BP512R1, 28, 512, "brainpoolP512r1" },
#endif
#if defined(POLARSSL_ECP_DP_SECP384R1_ENABLED)
{ POLARSSL_ECP_DP_SECP384R1, 24, 384, "secp384r1" },
#endif
#if defined(POLARSSL_ECP_DP_BP384R1_ENABLED)
{ POLARSSL_ECP_DP_BP384R1, 27, 384, "brainpoolP384r1" },
#endif
#if defined(POLARSSL_ECP_DP_SECP256R1_ENABLED)
{ POLARSSL_ECP_DP_SECP256R1, 23, 256, "secp256r1" },
#endif
#if defined(POLARSSL_ECP_DP_SECP256K1_ENABLED)
{ POLARSSL_ECP_DP_SECP256K1, 22, 256, "secp256k1" },
#endif
#if defined(POLARSSL_ECP_DP_BP256R1_ENABLED)
{ POLARSSL_ECP_DP_BP256R1, 26, 256, "brainpoolP256r1" },
#endif
#if defined(POLARSSL_ECP_DP_SECP224R1_ENABLED)
{ POLARSSL_ECP_DP_SECP224R1, 21, 224, "secp224r1" },
#endif
#if defined(POLARSSL_ECP_DP_SECP224K1_ENABLED)
{ POLARSSL_ECP_DP_SECP224K1, 20, 224, "secp224k1" },
#endif
#if defined(POLARSSL_ECP_DP_SECP192R1_ENABLED)
{ POLARSSL_ECP_DP_SECP192R1, 19, 192, "secp192r1" },
#endif
#if defined(POLARSSL_ECP_DP_SECP192K1_ENABLED)
{ POLARSSL_ECP_DP_SECP192K1, 18, 192, "secp192k1" },
#endif
{ POLARSSL_ECP_DP_NONE, 0, 0, NULL },
};
#define ECP_NB_CURVES sizeof( ecp_supported_curves ) / \
sizeof( ecp_supported_curves[0] )
static ecp_group_id ecp_supported_grp_id[ECP_NB_CURVES];
/*
* List of supported curves and associated info
*/
const ecp_curve_info *ecp_curve_list( void )
{
return( ecp_supported_curves );
}
/*
* List of supported curves, group ID only
*/
const ecp_group_id *ecp_grp_id_list( void )
{
static int init_done = 0;
if( ! init_done )
{
size_t i = 0;
const ecp_curve_info *curve_info;
for( curve_info = ecp_curve_list();
curve_info->grp_id != POLARSSL_ECP_DP_NONE;
curve_info++ )
{
ecp_supported_grp_id[i++] = curve_info->grp_id;
}
ecp_supported_grp_id[i] = POLARSSL_ECP_DP_NONE;
init_done = 1;
}
return( ecp_supported_grp_id );
}
/*
* Get the curve info for the internal identifier
*/
const ecp_curve_info *ecp_curve_info_from_grp_id( ecp_group_id grp_id )
{
const ecp_curve_info *curve_info;
for( curve_info = ecp_curve_list();
curve_info->grp_id != POLARSSL_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 ecp_curve_info *ecp_curve_info_from_tls_id( uint16_t tls_id )
{
const ecp_curve_info *curve_info;
for( curve_info = ecp_curve_list();
curve_info->grp_id != POLARSSL_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 ecp_curve_info *ecp_curve_info_from_name( const char *name )
{
const ecp_curve_info *curve_info;
for( curve_info = ecp_curve_list();
curve_info->grp_id != POLARSSL_ECP_DP_NONE;
curve_info++ )
{
if( strcasecmp( 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 ecp_group *grp )
{
if( grp->G.X.p == NULL )
return( POLARSSL_ECP_TYPE_NONE );
if( grp->G.Y.p == NULL )
return( POLARSSL_ECP_TYPE_MONTGOMERY );
else
return( POLARSSL_ECP_TYPE_SHORT_WEIERSTRASS );
}
/*
* Initialize (the components of) a point
*/
void ecp_point_init( ecp_point *pt )
{
if( pt == NULL )
return;
mpi_init( &pt->X );
mpi_init( &pt->Y );
mpi_init( &pt->Z );
}
/*
* Initialize (the components of) a group
*/
void ecp_group_init( ecp_group *grp )
{
if( grp == NULL )
return;
memset( grp, 0, sizeof( ecp_group ) );
}
/*
* Initialize (the components of) a key pair
*/
void ecp_keypair_init( ecp_keypair *key )
{
if( key == NULL )
return;
ecp_group_init( &key->grp );
mpi_init( &key->d );
ecp_point_init( &key->Q );
}
/*
* Unallocate (the components of) a point
*/
void ecp_point_free( ecp_point *pt )
{
if( pt == NULL )
return;
mpi_free( &( pt->X ) );
mpi_free( &( pt->Y ) );
mpi_free( &( pt->Z ) );
}
/*
* Unallocate (the components of) a group
*/
void ecp_group_free( ecp_group *grp )
{
size_t i;
if( grp == NULL )
return;
if( grp->h != 1 )
{
mpi_free( &grp->P );
mpi_free( &grp->A );
mpi_free( &grp->B );
ecp_point_free( &grp->G );
mpi_free( &grp->N );
}
if( grp->T != NULL )
{
for( i = 0; i < grp->T_size; i++ )
ecp_point_free( &grp->T[i] );
polarssl_free( grp->T );
}
polarssl_zeroize( grp, sizeof( ecp_group ) );
}
/*
* Unallocate (the components of) a key pair
*/
void ecp_keypair_free( ecp_keypair *key )
{
if( key == NULL )
return;
ecp_group_free( &key->grp );
mpi_free( &key->d );
ecp_point_free( &key->Q );
}
/*
* Copy the contents of a point
*/
int ecp_copy( ecp_point *P, const ecp_point *Q )
{
int ret;
MPI_CHK( mpi_copy( &P->X, &Q->X ) );
MPI_CHK( mpi_copy( &P->Y, &Q->Y ) );
MPI_CHK( mpi_copy( &P->Z, &Q->Z ) );
cleanup:
return( ret );
}
/*
* Copy the contents of a group object
*/
int ecp_group_copy( ecp_group *dst, const ecp_group *src )
{
return ecp_use_known_dp( dst, src->id );
}
/*
* Set point to zero
*/
int ecp_set_zero( ecp_point *pt )
{
int ret;
MPI_CHK( mpi_lset( &pt->X , 1 ) );
MPI_CHK( mpi_lset( &pt->Y , 1 ) );
MPI_CHK( mpi_lset( &pt->Z , 0 ) );
cleanup:
return( ret );
}
/*
* Tell if a point is zero
*/
int ecp_is_zero( ecp_point *pt )
{
return( mpi_cmp_int( &pt->Z, 0 ) == 0 );
}
/*
* Import a non-zero point from ASCII strings
*/
int ecp_point_read_string( ecp_point *P, int radix,
const char *x, const char *y )
{
int ret;
MPI_CHK( mpi_read_string( &P->X, radix, x ) );
MPI_CHK( mpi_read_string( &P->Y, radix, y ) );
MPI_CHK( mpi_lset( &P->Z, 1 ) );
cleanup:
return( ret );
}
/*
* Export a point into unsigned binary data (SEC1 2.3.3)
*/
int ecp_point_write_binary( const ecp_group *grp, const ecp_point *P,
int format, size_t *olen,
unsigned char *buf, size_t buflen )
{
int ret = 0;
size_t plen;
if( format != POLARSSL_ECP_PF_UNCOMPRESSED &&
format != POLARSSL_ECP_PF_COMPRESSED )
return( POLARSSL_ERR_ECP_BAD_INPUT_DATA );
/*
* Common case: P == 0
*/
if( mpi_cmp_int( &P->Z, 0 ) == 0 )
{
if( buflen < 1 )
return( POLARSSL_ERR_ECP_BUFFER_TOO_SMALL );
buf[0] = 0x00;
*olen = 1;
return( 0 );
}
plen = mpi_size( &grp->P );
if( format == POLARSSL_ECP_PF_UNCOMPRESSED )
{
*olen = 2 * plen + 1;
if( buflen < *olen )
return( POLARSSL_ERR_ECP_BUFFER_TOO_SMALL );
buf[0] = 0x04;
MPI_CHK( mpi_write_binary( &P->X, buf + 1, plen ) );
MPI_CHK( mpi_write_binary( &P->Y, buf + 1 + plen, plen ) );
}
else if( format == POLARSSL_ECP_PF_COMPRESSED )
{
*olen = plen + 1;
if( buflen < *olen )
return( POLARSSL_ERR_ECP_BUFFER_TOO_SMALL );
buf[0] = 0x02 + mpi_get_bit( &P->Y, 0 );
MPI_CHK( mpi_write_binary( &P->X, buf + 1, plen ) );
}
cleanup:
return( ret );
}
/*
* Import a point from unsigned binary data (SEC1 2.3.4)
*/
int ecp_point_read_binary( const ecp_group *grp, ecp_point *pt,
const unsigned char *buf, size_t ilen )
{
int ret;
size_t plen;
if( ilen < 1 )
return( POLARSSL_ERR_ECP_BAD_INPUT_DATA );
if( buf[0] == 0x00 )
{
if( ilen == 1 )
return( ecp_set_zero( pt ) );
else
return( POLARSSL_ERR_ECP_BAD_INPUT_DATA );
}
plen = mpi_size( &grp->P );
if( buf[0] != 0x04 )
return( POLARSSL_ERR_ECP_FEATURE_UNAVAILABLE );
if( ilen != 2 * plen + 1 )
return( POLARSSL_ERR_ECP_BAD_INPUT_DATA );
MPI_CHK( mpi_read_binary( &pt->X, buf + 1, plen ) );
MPI_CHK( mpi_read_binary( &pt->Y, buf + 1 + plen, plen ) );
MPI_CHK( 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 ecp_tls_read_point( const ecp_group *grp, 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( POLARSSL_ERR_ECP_BAD_INPUT_DATA );
data_len = *(*buf)++;
if( data_len < 1 || data_len > buf_len - 1 )
return( POLARSSL_ERR_ECP_BAD_INPUT_DATA );
/*
* Save buffer start for read_binary and update buf
*/
buf_start = *buf;
*buf += data_len;
return 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 ecp_tls_write_point( const ecp_group *grp, const 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( POLARSSL_ERR_ECP_BAD_INPUT_DATA );
if( ( ret = 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 );
}
/*
* Import an ECP group from ASCII strings, case A == -3
*/
int ecp_group_read_string( ecp_group *grp, int radix,
const char *p, const char *b,
const char *gx, const char *gy, const char *n)
{
int ret;
MPI_CHK( mpi_read_string( &grp->P, radix, p ) );
MPI_CHK( mpi_read_string( &grp->B, radix, b ) );
MPI_CHK( ecp_point_read_string( &grp->G, radix, gx, gy ) );
MPI_CHK( mpi_read_string( &grp->N, radix, n ) );
grp->pbits = mpi_msb( &grp->P );
grp->nbits = mpi_msb( &grp->N );
cleanup:
if( ret != 0 )
ecp_group_free( grp );
return( ret );
}
/*
* Set a group from an ECParameters record (RFC 4492)
*/
int ecp_tls_read_group( ecp_group *grp, const unsigned char **buf, size_t len )
{
uint16_t tls_id;
const ecp_curve_info *curve_info;
/*
* We expect at least three bytes (see below)
*/
if( len < 3 )
return( POLARSSL_ERR_ECP_BAD_INPUT_DATA );
/*
* First byte is curve_type; only named_curve is handled
*/
if( *(*buf)++ != POLARSSL_ECP_TLS_NAMED_CURVE )
return( POLARSSL_ERR_ECP_BAD_INPUT_DATA );
/*
* Next two bytes are the namedcurve value
*/
tls_id = *(*buf)++;
tls_id <<= 8;
tls_id |= *(*buf)++;
if( ( curve_info = ecp_curve_info_from_tls_id( tls_id ) ) == NULL )
return( POLARSSL_ERR_ECP_FEATURE_UNAVAILABLE );
return ecp_use_known_dp( grp, curve_info->grp_id );
}
/*
* Write the ECParameters record corresponding to a group (RFC 4492)
*/
int ecp_tls_write_group( const ecp_group *grp, size_t *olen,
unsigned char *buf, size_t blen )
{
const ecp_curve_info *curve_info;
if( ( curve_info = ecp_curve_info_from_grp_id( grp->id ) ) == NULL )
return( POLARSSL_ERR_ECP_BAD_INPUT_DATA );
/*
* We are going to write 3 bytes (see below)
*/
*olen = 3;
if( blen < *olen )
return( POLARSSL_ERR_ECP_BUFFER_TOO_SMALL );
/*
* First byte is curve_type, always named_curve
*/
*buf++ = POLARSSL_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 mpi_mod_mpi.
* See the documentation of struct ecp_group.
*
* This function is in the critial loop for ecp_mul, so pay attention to perf.
*/
static int ecp_modp( mpi *N, const ecp_group *grp )
{
int ret;
if( grp->modp == NULL )
return( 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 && mpi_cmp_int( N, 0 ) != 0 ) ||
mpi_msb( N ) > 2 * grp->pbits )
{
return( POLARSSL_ERR_ECP_BAD_INPUT_DATA );
}
MPI_CHK( grp->modp( N ) );
/* N->s < 0 is a much faster test, which fails only if N is 0 */
while( N->s < 0 && mpi_cmp_int( N, 0 ) != 0 )
MPI_CHK( mpi_add_mpi( N, N, &grp->P ) );
while( mpi_cmp_mpi( N, &grp->P ) >= 0 )
/* we known P, N and the result are positive */
MPI_CHK( 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
* 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 mpi mod p in-place, general case, to use after mpi_mul_mpi
*/
#if defined(POLARSSL_SELF_TEST)
#define INC_MUL_COUNT mul_count++;
#else
#define INC_MUL_COUNT
#endif
#define MOD_MUL( N ) do { MPI_CHK( ecp_modp( &N, grp ) ); INC_MUL_COUNT } \
while( 0 )
/*
* Reduce a mpi mod p in-place, to use after 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 && mpi_cmp_int( &N, 0 ) != 0 ) \
MPI_CHK( mpi_add_mpi( &N, &N, &grp->P ) )
/*
* Reduce a mpi mod p in-place, to use after mpi_add_mpi and 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( mpi_cmp_mpi( &N, &grp->P ) >= 0 ) \
MPI_CHK( mpi_sub_abs( &N, &N, &grp->P ) )
#if defined(POLARSSL_ECP_SHORT_WEIERSTRASS)
/*
* 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 ecp_group *grp, ecp_point *pt )
{
int ret;
mpi Zi, ZZi;
if( mpi_cmp_int( &pt->Z, 0 ) == 0 )
return( 0 );
mpi_init( &Zi ); mpi_init( &ZZi );
/*
* X = X / Z^2 mod p
*/
MPI_CHK( mpi_inv_mod( &Zi, &pt->Z, &grp->P ) );
MPI_CHK( mpi_mul_mpi( &ZZi, &Zi, &Zi ) ); MOD_MUL( ZZi );
MPI_CHK( mpi_mul_mpi( &pt->X, &pt->X, &ZZi ) ); MOD_MUL( pt->X );
/*
* Y = Y / Z^3 mod p
*/
MPI_CHK( mpi_mul_mpi( &pt->Y, &pt->Y, &ZZi ) ); MOD_MUL( pt->Y );
MPI_CHK( mpi_mul_mpi( &pt->Y, &pt->Y, &Zi ) ); MOD_MUL( pt->Y );
/*
* Z = 1
*/
MPI_CHK( mpi_lset( &pt->Z, 1 ) );
cleanup:
mpi_free( &Zi ); 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 ecp_group *grp,
ecp_point *T[], size_t t_len )
{
int ret;
size_t i;
mpi *c, u, Zi, ZZi;
if( t_len < 2 )
return( ecp_normalize_jac( grp, *T ) );
if( ( c = (mpi *) polarssl_malloc( t_len * sizeof( mpi ) ) ) == NULL )
return( POLARSSL_ERR_ECP_MALLOC_FAILED );
mpi_init( &u ); mpi_init( &Zi ); mpi_init( &ZZi );
for( i = 0; i < t_len; i++ )
mpi_init( &c[i] );
/*
* c[i] = Z_0 * ... * Z_i
*/
MPI_CHK( mpi_copy( &c[0], &T[0]->Z ) );
for( i = 1; i < t_len; i++ )
{
MPI_CHK( mpi_mul_mpi( &c[i], &c[i-1], &T[i]->Z ) );
MOD_MUL( c[i] );
}
/*
* u = 1 / (Z_0 * ... * Z_n) mod P
*/
MPI_CHK( 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 ) {
MPI_CHK( mpi_copy( &Zi, &u ) );
}
else
{
MPI_CHK( mpi_mul_mpi( &Zi, &u, &c[i-1] ) ); MOD_MUL( Zi );
MPI_CHK( mpi_mul_mpi( &u, &u, &T[i]->Z ) ); MOD_MUL( u );
}
/*
* proceed as in normalize()
*/
MPI_CHK( mpi_mul_mpi( &ZZi, &Zi, &Zi ) ); MOD_MUL( ZZi );
MPI_CHK( mpi_mul_mpi( &T[i]->X, &T[i]->X, &ZZi ) ); MOD_MUL( T[i]->X );
MPI_CHK( mpi_mul_mpi( &T[i]->Y, &T[i]->Y, &ZZi ) ); MOD_MUL( T[i]->Y );
MPI_CHK( 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.
*/
MPI_CHK( mpi_shrink( &T[i]->X, grp->P.n ) );
MPI_CHK( mpi_shrink( &T[i]->Y, grp->P.n ) );
mpi_free( &T[i]->Z );
if( i == 0 )
break;
}
cleanup:
mpi_free( &u ); mpi_free( &Zi ); mpi_free( &ZZi );
for( i = 0; i < t_len; i++ )
mpi_free( &c[i] );
polarssl_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 ecp_group *grp,
ecp_point *Q,
unsigned char inv )
{
int ret;
unsigned char nonzero;
mpi mQY;
mpi_init( &mQY );
/* Use the fact that -Q.Y mod P = P - Q.Y unless Q.Y == 0 */
MPI_CHK( mpi_sub_mpi( &mQY, &grp->P, &Q->Y ) );
nonzero = mpi_cmp_int( &Q->Y, 0 ) != 0;
MPI_CHK( mpi_safe_cond_assign( &Q->Y, &mQY, inv & nonzero ) );
cleanup:
mpi_free( &mQY );
return( ret );
}
/*
* Point doubling R = 2 P, Jacobian coordinates
*
* http://www.hyperelliptic.org/EFD/g1p/auto-code/shortw/jacobian/doubling/dbl-2007-bl.op3
* with heavy variable renaming, some reordering and one minor modification
* (a = 2 * b, c = d - 2a replaced with c = d, c = c - b, c = c - b)
* in order to use a lot less intermediate variables (6 vs 25).
*
* Cost: 1D := 2M + 8S
*/
static int ecp_double_jac( const ecp_group *grp, ecp_point *R,
const ecp_point *P )
{
int ret;
mpi T1, T2, T3, X3, Y3, Z3;
#if defined(POLARSSL_SELF_TEST)
dbl_count++;
#endif
mpi_init( &T1 ); mpi_init( &T2 ); mpi_init( &T3 );
mpi_init( &X3 ); mpi_init( &Y3 ); mpi_init( &Z3 );
MPI_CHK( mpi_mul_mpi( &T3, &P->X, &P->X ) ); MOD_MUL( T3 );
MPI_CHK( mpi_mul_mpi( &T2, &P->Y, &P->Y ) ); MOD_MUL( T2 );
MPI_CHK( mpi_mul_mpi( &Y3, &T2, &T2 ) ); MOD_MUL( Y3 );
MPI_CHK( mpi_add_mpi( &X3, &P->X, &T2 ) ); MOD_ADD( X3 );
MPI_CHK( mpi_mul_mpi( &X3, &X3, &X3 ) ); MOD_MUL( X3 );
MPI_CHK( mpi_sub_mpi( &X3, &X3, &Y3 ) ); MOD_SUB( X3 );
MPI_CHK( mpi_sub_mpi( &X3, &X3, &T3 ) ); MOD_SUB( X3 );
MPI_CHK( mpi_mul_int( &T1, &X3, 2 ) ); MOD_ADD( T1 );
MPI_CHK( mpi_mul_mpi( &Z3, &P->Z, &P->Z ) ); MOD_MUL( Z3 );
MPI_CHK( mpi_mul_mpi( &X3, &Z3, &Z3 ) ); MOD_MUL( X3 );
MPI_CHK( mpi_mul_int( &T3, &T3, 3 ) ); MOD_ADD( T3 );
/* Special case for A = -3 */
if( grp->A.p == NULL )
{
MPI_CHK( mpi_mul_int( &X3, &X3, 3 ) );
X3.s = -1; /* mpi_mul_int doesn't handle negative numbers */
MOD_SUB( X3 );
}
else
{
MPI_CHK( mpi_mul_mpi( &X3, &X3, &grp->A ) ); MOD_MUL( X3 );
}
MPI_CHK( mpi_add_mpi( &T3, &T3, &X3 ) ); MOD_ADD( T3 );
MPI_CHK( mpi_mul_mpi( &X3, &T3, &T3 ) ); MOD_MUL( X3 );
MPI_CHK( mpi_sub_mpi( &X3, &X3, &T1 ) ); MOD_SUB( X3 );
MPI_CHK( mpi_sub_mpi( &X3, &X3, &T1 ) ); MOD_SUB( X3 );
MPI_CHK( mpi_sub_mpi( &T1, &T1, &X3 ) ); MOD_SUB( T1 );
MPI_CHK( mpi_mul_mpi( &T1, &T3, &T1 ) ); MOD_MUL( T1 );
MPI_CHK( mpi_mul_int( &T3, &Y3, 8 ) ); MOD_ADD( T3 );
MPI_CHK( mpi_sub_mpi( &Y3, &T1, &T3 ) ); MOD_SUB( Y3 );
MPI_CHK( mpi_add_mpi( &T1, &P->Y, &P->Z ) ); MOD_ADD( T1 );
MPI_CHK( mpi_mul_mpi( &T1, &T1, &T1 ) ); MOD_MUL( T1 );
MPI_CHK( mpi_sub_mpi( &T1, &T1, &T2 ) ); MOD_SUB( T1 );
MPI_CHK( mpi_sub_mpi( &Z3, &T1, &Z3 ) ); MOD_SUB( Z3 );
MPI_CHK( mpi_copy( &R->X, &X3 ) );
MPI_CHK( mpi_copy( &R->Y, &Y3 ) );
MPI_CHK( mpi_copy( &R->Z, &Z3 ) );
cleanup:
mpi_free( &T1 ); mpi_free( &T2 ); mpi_free( &T3 );
mpi_free( &X3 ); mpi_free( &Y3 ); mpi_free( &Z3 );
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 ecp_group *grp, ecp_point *R,
const ecp_point *P, const ecp_point *Q )
{
int ret;
mpi T1, T2, T3, T4, X, Y, Z;
#if defined(POLARSSL_SELF_TEST)
add_count++;
#endif
/*
* Trivial cases: P == 0 or Q == 0 (case 1)
*/
if( mpi_cmp_int( &P->Z, 0 ) == 0 )
return( ecp_copy( R, Q ) );
if( Q->Z.p != NULL && mpi_cmp_int( &Q->Z, 0 ) == 0 )
return( ecp_copy( R, P ) );
/*
* Make sure Q coordinates are normalized
*/
if( Q->Z.p != NULL && mpi_cmp_int( &Q->Z, 1 ) != 0 )
return( POLARSSL_ERR_ECP_BAD_INPUT_DATA );
mpi_init( &T1 ); mpi_init( &T2 ); mpi_init( &T3 ); mpi_init( &T4 );
mpi_init( &X ); mpi_init( &Y ); mpi_init( &Z );
MPI_CHK( mpi_mul_mpi( &T1, &P->Z, &P->Z ) ); MOD_MUL( T1 );
MPI_CHK( mpi_mul_mpi( &T2, &T1, &P->Z ) ); MOD_MUL( T2 );
MPI_CHK( mpi_mul_mpi( &T1, &T1, &Q->X ) ); MOD_MUL( T1 );
MPI_CHK( mpi_mul_mpi( &T2, &T2, &Q->Y ) ); MOD_MUL( T2 );
MPI_CHK( mpi_sub_mpi( &T1, &T1, &P->X ) ); MOD_SUB( T1 );
MPI_CHK( mpi_sub_mpi( &T2, &T2, &P->Y ) ); MOD_SUB( T2 );
/* Special cases (2) and (3) */
if( mpi_cmp_int( &T1, 0 ) == 0 )
{
if( mpi_cmp_int( &T2, 0 ) == 0 )
{
ret = ecp_double_jac( grp, R, P );
goto cleanup;
}
else
{
ret = ecp_set_zero( R );
goto cleanup;
}
}
MPI_CHK( mpi_mul_mpi( &Z, &P->Z, &T1 ) ); MOD_MUL( Z );
MPI_CHK( mpi_mul_mpi( &T3, &T1, &T1 ) ); MOD_MUL( T3 );
MPI_CHK( mpi_mul_mpi( &T4, &T3, &T1 ) ); MOD_MUL( T4 );
MPI_CHK( mpi_mul_mpi( &T3, &T3, &P->X ) ); MOD_MUL( T3 );
MPI_CHK( mpi_mul_int( &T1, &T3, 2 ) ); MOD_ADD( T1 );
MPI_CHK( mpi_mul_mpi( &X, &T2, &T2 ) ); MOD_MUL( X );
MPI_CHK( mpi_sub_mpi( &X, &X, &T1 ) ); MOD_SUB( X );
MPI_CHK( mpi_sub_mpi( &X, &X, &T4 ) ); MOD_SUB( X );
MPI_CHK( mpi_sub_mpi( &T3, &T3, &X ) ); MOD_SUB( T3 );
MPI_CHK( mpi_mul_mpi( &T3, &T3, &T2 ) ); MOD_MUL( T3 );
MPI_CHK( mpi_mul_mpi( &T4, &T4, &P->Y ) ); MOD_MUL( T4 );
MPI_CHK( mpi_sub_mpi( &Y, &T3, &T4 ) ); MOD_SUB( Y );
MPI_CHK( mpi_copy( &R->X, &X ) );
MPI_CHK( mpi_copy( &R->Y, &Y ) );
MPI_CHK( mpi_copy( &R->Z, &Z ) );
cleanup:
mpi_free( &T1 ); mpi_free( &T2 ); mpi_free( &T3 ); mpi_free( &T4 );
mpi_free( &X ); mpi_free( &Y ); mpi_free( &Z );
return( ret );
}
/*
* Addition: R = P + Q, result's coordinates normalized
*/
int ecp_add( const ecp_group *grp, ecp_point *R,
const ecp_point *P, const ecp_point *Q )
{
int ret;
if( ecp_get_type( grp ) != POLARSSL_ECP_TYPE_SHORT_WEIERSTRASS )
return( POLARSSL_ERR_ECP_FEATURE_UNAVAILABLE );
MPI_CHK( ecp_add_mixed( grp, R, P, Q ) );
MPI_CHK( ecp_normalize_jac( grp, R ) );
cleanup:
return( ret );
}
/*
* Subtraction: R = P - Q, result's coordinates normalized
*/
int ecp_sub( const ecp_group *grp, ecp_point *R,
const ecp_point *P, const ecp_point *Q )
{
int ret;
ecp_point mQ;
ecp_point_init( &mQ );
if( ecp_get_type( grp ) != POLARSSL_ECP_TYPE_SHORT_WEIERSTRASS )
return( POLARSSL_ERR_ECP_FEATURE_UNAVAILABLE );
/* mQ = - Q */
MPI_CHK( ecp_copy( &mQ, Q ) );
if( mpi_cmp_int( &mQ.Y, 0 ) != 0 )
MPI_CHK( mpi_sub_mpi( &mQ.Y, &grp->P, &mQ.Y ) );
MPI_CHK( ecp_add_mixed( grp, R, P, &mQ ) );
MPI_CHK( ecp_normalize_jac( grp, R ) );
cleanup:
ecp_point_free( &mQ );
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 ecp_group *grp, ecp_point *pt,
int (*f_rng)(void *, unsigned char *, size_t), void *p_rng )
{
int ret;
mpi l, ll;
size_t p_size = ( grp->pbits + 7 ) / 8;
int count = 0;
mpi_init( &l ); mpi_init( &ll );
/* Generate l such that 1 < l < p */
do
{
mpi_fill_random( &l, p_size, f_rng, p_rng );
while( mpi_cmp_mpi( &l, &grp->P ) >= 0 )
MPI_CHK( mpi_shift_r( &l, 1 ) );
if( count++ > 10 )
return( POLARSSL_ERR_ECP_RANDOM_FAILED );
}
while( mpi_cmp_int( &l, 1 ) <= 0 );
/* Z = l * Z */
MPI_CHK( mpi_mul_mpi( &pt->Z, &pt->Z, &l ) ); MOD_MUL( pt->Z );
/* X = l^2 * X */
MPI_CHK( mpi_mul_mpi( &ll, &l, &l ) ); MOD_MUL( ll );
MPI_CHK( mpi_mul_mpi( &pt->X, &pt->X, &ll ) ); MOD_MUL( pt->X );
/* Y = l^3 * Y */
MPI_CHK( mpi_mul_mpi( &ll, &ll, &l ) ); MOD_MUL( ll );
MPI_CHK( mpi_mul_mpi( &pt->Y, &pt->Y, &ll ) ); MOD_MUL( pt->Y );
cleanup:
mpi_free( &l ); mpi_free( &ll );
return( ret );
}
/*
* Check and define parameters used by the comb method (see below for details)
*/
#if POLARSSL_ECP_WINDOW_SIZE < 2 || POLARSSL_ECP_WINDOW_SIZE > 7
#error "POLARSSL_ECP_WINDOW_SIZE out of bounds"
#endif
/* d = ceil( n / w ) */
#define COMB_MAX_D ( POLARSSL_ECP_MAX_BITS + 1 ) / 2
/* number of precomputed points */
#define COMB_MAX_PRE ( 1 << ( POLARSSL_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 POLARSSL_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 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] |= 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 ecp_group *grp,
ecp_point T[], const ecp_point *P,
unsigned char w, size_t d )
{
int ret;
unsigned char i, k;
size_t j;
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)
*/
MPI_CHK( ecp_copy( &T[0], P ) );
k = 0;
for( i = 1; i < ( 1U << ( w - 1 ) ); i <<= 1 )
{
cur = T + i;
MPI_CHK( ecp_copy( cur, T + ( i >> 1 ) ) );
for( j = 0; j < d; j++ )
MPI_CHK( ecp_double_jac( grp, cur, cur ) );
TT[k++] = cur;
}
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-- )
{
MPI_CHK( ecp_add_mixed( grp, &T[i + j], &T[j], &T[i] ) );
TT[k++] = &T[i + j];
}
}
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 ecp_group *grp, ecp_point *R,
const 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++ )
{
MPI_CHK( mpi_safe_cond_assign( &R->X, &T[j].X, j == ii ) );
MPI_CHK( mpi_safe_cond_assign( &R->Y, &T[j].Y, j == ii ) );
}
/* Safely invert result if i is "negative" */
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 ecp_group *grp, ecp_point *R,
const 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;
ecp_point Txi;
size_t i;
ecp_point_init( &Txi );
/* Start with a non-zero point and randomize its coordinates */
i = d;
MPI_CHK( ecp_select_comb( grp, R, T, t_len, x[i] ) );
MPI_CHK( mpi_lset( &R->Z, 1 ) );
if( f_rng != 0 )
MPI_CHK( ecp_randomize_jac( grp, R, f_rng, p_rng ) );
while( i-- != 0 )
{
MPI_CHK( ecp_double_jac( grp, R, R ) );
MPI_CHK( ecp_select_comb( grp, &Txi, T, t_len, x[i] ) );
MPI_CHK( ecp_add_mixed( grp, R, R, &Txi ) );
}
cleanup:
ecp_point_free( &Txi );
return( ret );
}
/*
* Multiplication using the comb method,
* for curves in short Weierstrass form
*/
static int ecp_mul_comb( ecp_group *grp, ecp_point *R,
const mpi *m, const 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];
ecp_point *T;
mpi M, mm;
mpi_init( &M );
mpi_init( &mm );
/* we need N to be odd to trnaform m in an odd number, check now */
if( mpi_get_bit( &grp->N, 0 ) != 1 )
return( POLARSSL_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 POLARSSL_ECP_FIXED_POINT_OPTIM == 1
p_eq_g = ( mpi_cmp_mpi( &P->Y, &grp->G.Y ) == 0 &&
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 > POLARSSL_ECP_WINDOW_SIZE )
w = POLARSSL_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 = (ecp_point *) polarssl_malloc( pre_len * sizeof( ecp_point ) );
if( T == NULL )
{
ret = POLARSSL_ERR_ECP_MALLOC_FAILED;
goto cleanup;
}
for( i = 0; i < pre_len; i++ )
ecp_point_init( &T[i] );
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 = ( mpi_get_bit( m, 0 ) == 1 );
MPI_CHK( mpi_copy( &M, m ) );
MPI_CHK( mpi_sub_mpi( &mm, &grp->N, m ) );
MPI_CHK( mpi_safe_cond_assign( &M, &mm, ! m_is_odd ) );
/*
* Go for comb multiplication, R = M * P
*/
ecp_comb_fixed( k, d, w, &M );
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
*/
MPI_CHK( ecp_safe_invert_jac( grp, R, ! m_is_odd ) );
MPI_CHK( ecp_normalize_jac( grp, R ) );
cleanup:
if( T != NULL && ! p_eq_g )
{
for( i = 0; i < pre_len; i++ )
ecp_point_free( &T[i] );
polarssl_free( T );
}
mpi_free( &M );
mpi_free( &mm );
if( ret != 0 )
ecp_point_free( R );
return( ret );
}
#endif /* POLARSSL_ECP_SHORT_WEIERSTRASS */
#if defined(POLARSSL_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 ecp_group *grp, ecp_point *P )
{
int ret;
MPI_CHK( mpi_inv_mod( &P->Z, &P->Z, &grp->P ) );
MPI_CHK( mpi_mul_mpi( &P->X, &P->X, &P->Z ) ); MOD_MUL( P->X );
MPI_CHK( 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 ecp_group *grp, ecp_point *P,
int (*f_rng)(void *, unsigned char *, size_t), void *p_rng )
{
int ret;
mpi l;
size_t p_size = ( grp->pbits + 7 ) / 8;
int count = 0;
mpi_init( &l );
/* Generate l such that 1 < l < p */
do
{
mpi_fill_random( &l, p_size, f_rng, p_rng );
while( mpi_cmp_mpi( &l, &grp->P ) >= 0 )
MPI_CHK( mpi_shift_r( &l, 1 ) );
if( count++ > 10 )
return( POLARSSL_ERR_ECP_RANDOM_FAILED );
}
while( mpi_cmp_int( &l, 1 ) <= 0 );
MPI_CHK( mpi_mul_mpi( &P->X, &P->X, &l ) ); MOD_MUL( P->X );
MPI_CHK( mpi_mul_mpi( &P->Z, &P->Z, &l ) ); MOD_MUL( P->Z );
cleanup:
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 ecp_group *grp,
ecp_point *R, ecp_point *S,
const ecp_point *P, const ecp_point *Q,
const mpi *d )
{
int ret;
mpi A, AA, B, BB, E, C, D, DA, CB;
mpi_init( &A ); mpi_init( &AA ); mpi_init( &B );
mpi_init( &BB ); mpi_init( &E ); mpi_init( &C );
mpi_init( &D ); mpi_init( &DA ); mpi_init( &CB );
MPI_CHK( mpi_add_mpi( &A, &P->X, &P->Z ) ); MOD_ADD( A );
MPI_CHK( mpi_mul_mpi( &AA, &A, &A ) ); MOD_MUL( AA );
MPI_CHK( mpi_sub_mpi( &B, &P->X, &P->Z ) ); MOD_SUB( B );
MPI_CHK( mpi_mul_mpi( &BB, &B, &B ) ); MOD_MUL( BB );
MPI_CHK( mpi_sub_mpi( &E, &AA, &BB ) ); MOD_SUB( E );
MPI_CHK( mpi_add_mpi( &C, &Q->X, &Q->Z ) ); MOD_ADD( C );
MPI_CHK( mpi_sub_mpi( &D, &Q->X, &Q->Z ) ); MOD_SUB( D );
MPI_CHK( mpi_mul_mpi( &DA, &D, &A ) ); MOD_MUL( DA );
MPI_CHK( mpi_mul_mpi( &CB, &C, &B ) ); MOD_MUL( CB );
MPI_CHK( mpi_add_mpi( &S->X, &DA, &CB ) ); MOD_MUL( S->X );
MPI_CHK( mpi_mul_mpi( &S->X, &S->X, &S->X ) ); MOD_MUL( S->X );
MPI_CHK( mpi_sub_mpi( &S->Z, &DA, &CB ) ); MOD_SUB( S->Z );
MPI_CHK( mpi_mul_mpi( &S->Z, &S->Z, &S->Z ) ); MOD_MUL( S->Z );
MPI_CHK( mpi_mul_mpi( &S->Z, d, &S->Z ) ); MOD_MUL( S->Z );
MPI_CHK( mpi_mul_mpi( &R->X, &AA, &BB ) ); MOD_MUL( R->X );
MPI_CHK( mpi_mul_mpi( &R->Z, &grp->A, &E ) ); MOD_MUL( R->Z );
MPI_CHK( mpi_add_mpi( &R->Z, &BB, &R->Z ) ); MOD_ADD( R->Z );
MPI_CHK( mpi_mul_mpi( &R->Z, &E, &R->Z ) ); MOD_MUL( R->Z );
cleanup:
mpi_free( &A ); mpi_free( &AA ); mpi_free( &B );
mpi_free( &BB ); mpi_free( &E ); mpi_free( &C );
mpi_free( &D ); mpi_free( &DA ); mpi_free( &CB );
return( ret );
}
/*
* Multiplication with Montgomery ladder in x/z coordinates,
* for curves in Montgomery form
*/
static int ecp_mul_mxz( ecp_group *grp, ecp_point *R,
const mpi *m, const ecp_point *P,
int (*f_rng)(void *, unsigned char *, size_t),
void *p_rng )
{
int ret;
size_t i;
unsigned char b;
ecp_point RP;
mpi PX;
ecp_point_init( &RP ); mpi_init( &PX );
/* Save PX and read from P before writing to R, in case P == R */
MPI_CHK( mpi_copy( &PX, &P->X ) );
MPI_CHK( ecp_copy( &RP, P ) );
/* Set R to zero in modified x/z coordinates */
MPI_CHK( mpi_lset( &R->X, 1 ) );
MPI_CHK( mpi_lset( &R->Z, 0 ) );
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 )
MPI_CHK( ecp_randomize_mxz( grp, &RP, f_rng, p_rng ) );
/* Loop invariant: R = result so far, RP = R + P */
i = mpi_msb( m ); /* one past the (zero-based) most significant bit */
while( i-- > 0 )
{
b = 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
*/
MPI_CHK( mpi_safe_cond_swap( &R->X, &RP.X, b ) );
MPI_CHK( mpi_safe_cond_swap( &R->Z, &RP.Z, b ) );
MPI_CHK( ecp_double_add_mxz( grp, R, &RP, R, &RP, &PX ) );
MPI_CHK( mpi_safe_cond_swap( &R->X, &RP.X, b ) );
MPI_CHK( mpi_safe_cond_swap( &R->Z, &RP.Z, b ) );
}
MPI_CHK( ecp_normalize_mxz( grp, R ) );
cleanup:
ecp_point_free( &RP ); mpi_free( &PX );
return( ret );
}
#endif /* POLARSSL_ECP_MONTGOMERY */
/*
* Multiplication R = m * P
*/
int ecp_mul( ecp_group *grp, ecp_point *R,
const mpi *m, const ecp_point *P,
int (*f_rng)(void *, unsigned char *, size_t), void *p_rng )
{
int ret;
/* Common sanity checks */
if( mpi_cmp_int( &P->Z, 1 ) != 0 )
return( POLARSSL_ERR_ECP_BAD_INPUT_DATA );
if( ( ret = ecp_check_privkey( grp, m ) ) != 0 ||
( ret = ecp_check_pubkey( grp, P ) ) != 0 )
return( ret );
#if defined(POLARSSL_ECP_MONTGOMERY)
if( ecp_get_type( grp ) == POLARSSL_ECP_TYPE_MONTGOMERY )
return( ecp_mul_mxz( grp, R, m, P, f_rng, p_rng ) );
#endif
#if defined(POLARSSL_ECP_SHORT_WEIERSTRASS)
if( ecp_get_type( grp ) == POLARSSL_ECP_TYPE_SHORT_WEIERSTRASS )
return( ecp_mul_comb( grp, R, m, P, f_rng, p_rng ) );
#endif
return( POLARSSL_ERR_ECP_BAD_INPUT_DATA );
}
#if defined(POLARSSL_ECP_SHORT_WEIERSTRASS)
/*
* 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 ecp_group *grp, const ecp_point *pt )
{
int ret;
mpi YY, RHS;
/* pt coordinates must be normalized for our checks */
if( mpi_cmp_int( &pt->X, 0 ) < 0 ||
mpi_cmp_int( &pt->Y, 0 ) < 0 ||
mpi_cmp_mpi( &pt->X, &grp->P ) >= 0 ||
mpi_cmp_mpi( &pt->Y, &grp->P ) >= 0 )
return( POLARSSL_ERR_ECP_INVALID_KEY );
mpi_init( &YY ); mpi_init( &RHS );
/*
* YY = Y^2
* RHS = X (X^2 + A) + B = X^3 + A X + B
*/
MPI_CHK( mpi_mul_mpi( &YY, &pt->Y, &pt->Y ) ); MOD_MUL( YY );
MPI_CHK( mpi_mul_mpi( &RHS, &pt->X, &pt->X ) ); MOD_MUL( RHS );
/* Special case for A = -3 */
if( grp->A.p == NULL )
{
MPI_CHK( mpi_sub_int( &RHS, &RHS, 3 ) ); MOD_SUB( RHS );
}
else
{
MPI_CHK( mpi_add_mpi( &RHS, &RHS, &grp->A ) ); MOD_ADD( RHS );
}
MPI_CHK( mpi_mul_mpi( &RHS, &RHS, &pt->X ) ); MOD_MUL( RHS );
MPI_CHK( mpi_add_mpi( &RHS, &RHS, &grp->B ) ); MOD_ADD( RHS );
if( mpi_cmp_mpi( &YY, &RHS ) != 0 )
ret = POLARSSL_ERR_ECP_INVALID_KEY;
cleanup:
mpi_free( &YY ); mpi_free( &RHS );
return( ret );
}
#endif /* POLARSSL_ECP_SHORT_WEIERSTRASS */
#if defined(POLARSSL_ECP_MONTGOMERY)
/*
* Check validity of a public key for Montgomery curves with x-only schemes
*/
static int ecp_check_pubkey_mx( const ecp_group *grp, const ecp_point *pt )
{
/* [M255 p. 5] Just check X is the correct number of bytes */
if( mpi_size( &pt->X ) > ( grp->nbits + 7 ) / 8 )
return( POLARSSL_ERR_ECP_INVALID_KEY );
return( 0 );
}
#endif /* POLARSSL_ECP_MONTGOMERY */
/*
* Check that a point is valid as a public key
*/
int ecp_check_pubkey( const ecp_group *grp, const ecp_point *pt )
{
/* Must use affine coordinates */
if( mpi_cmp_int( &pt->Z, 1 ) != 0 )
return( POLARSSL_ERR_ECP_INVALID_KEY );
#if defined(POLARSSL_ECP_MONTGOMERY)
if( ecp_get_type( grp ) == POLARSSL_ECP_TYPE_MONTGOMERY )
return( ecp_check_pubkey_mx( grp, pt ) );
#endif
#if defined(POLARSSL_ECP_SHORT_WEIERSTRASS)
if( ecp_get_type( grp ) == POLARSSL_ECP_TYPE_SHORT_WEIERSTRASS )
return( ecp_check_pubkey_sw( grp, pt ) );
#endif
return( POLARSSL_ERR_ECP_BAD_INPUT_DATA );
}
/*
* Check that an mpi is valid as a private key
*/
int ecp_check_privkey( const ecp_group *grp, const mpi *d )
{
#if defined(POLARSSL_ECP_MONTGOMERY)
if( ecp_get_type( grp ) == POLARSSL_ECP_TYPE_MONTGOMERY )
{
/* see [M255] page 5 */
if( mpi_get_bit( d, 0 ) != 0 ||
mpi_get_bit( d, 1 ) != 0 ||
mpi_get_bit( d, 2 ) != 0 ||
mpi_msb( d ) - 1 != grp->nbits ) /* mpi_msb is one-based! */
return( POLARSSL_ERR_ECP_INVALID_KEY );
else
return( 0 );
}
#endif /* POLARSSL_ECP_MONTGOMERY */
#if defined(POLARSSL_ECP_SHORT_WEIERSTRASS)
if( ecp_get_type( grp ) == POLARSSL_ECP_TYPE_SHORT_WEIERSTRASS )
{
/* see SEC1 3.2 */
if( mpi_cmp_int( d, 1 ) < 0 ||
mpi_cmp_mpi( d, &grp->N ) >= 0 )
return( POLARSSL_ERR_ECP_INVALID_KEY );
else
return( 0 );
}
#endif /* POLARSSL_ECP_SHORT_WEIERSTRASS */
return( POLARSSL_ERR_ECP_BAD_INPUT_DATA );
}
/*
* Generate a keypair
*/
int ecp_gen_keypair( ecp_group *grp, mpi *d, 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(POLARSSL_ECP_MONTGOMERY)
if( ecp_get_type( grp ) == POLARSSL_ECP_TYPE_MONTGOMERY )
{
/* [M225] page 5 */
size_t b;
MPI_CHK( mpi_fill_random( d, n_size, f_rng, p_rng ) );
/* Make sure the most significant bit is nbits */
b = mpi_msb( d ) - 1; /* mpi_msb is one-based */
if( b > grp->nbits )
MPI_CHK( mpi_shift_r( d, b - grp->nbits ) );
else
MPI_CHK( mpi_set_bit( d, grp->nbits, 1 ) );
/* Make sure the last three bits are unset */
MPI_CHK( mpi_set_bit( d, 0, 0 ) );
MPI_CHK( mpi_set_bit( d, 1, 0 ) );
MPI_CHK( mpi_set_bit( d, 2, 0 ) );
}
else
#endif /* POLARSSL_ECP_MONTGOMERY */
#if defined(POLARSSL_ECP_SHORT_WEIERSTRASS)
if( ecp_get_type( grp ) == POLARSSL_ECP_TYPE_SHORT_WEIERSTRASS )
{
/* SEC1 3.2.1: Generate d such that 1 <= n < N */
int count = 0;
unsigned char rnd[POLARSSL_ECP_MAX_BYTES];
/*
* 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
{
MPI_CHK( f_rng( p_rng, rnd, n_size ) );
MPI_CHK( mpi_read_binary( d, rnd, n_size ) );
MPI_CHK( 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( POLARSSL_ERR_ECP_RANDOM_FAILED );
}
while( mpi_cmp_int( d, 1 ) < 0 ||
mpi_cmp_mpi( d, &grp->N ) >= 0 );
}
else
#endif /* POLARSSL_ECP_SHORT_WEIERSTRASS */
return( POLARSSL_ERR_ECP_BAD_INPUT_DATA );
cleanup:
if( ret != 0 )
return( ret );
return( ecp_mul( grp, Q, d, &grp->G, f_rng, p_rng ) );
}
/*
* Generate a keypair, prettier wrapper
*/
int ecp_gen_key( ecp_group_id grp_id, ecp_keypair *key,
int (*f_rng)(void *, unsigned char *, size_t), void *p_rng )
{
int ret;
if( ( ret = ecp_use_known_dp( &key->grp, grp_id ) ) != 0 )
return( ret );
return( ecp_gen_keypair( &key->grp, &key->d, &key->Q, f_rng, p_rng ) );
}
/*
* Check a public-private key pair
*/
int ecp_check_pub_priv( const ecp_keypair *pub, const ecp_keypair *prv )
{
int ret;
ecp_point Q;
ecp_group grp;
if( pub->grp.id == POLARSSL_ECP_DP_NONE ||
pub->grp.id != prv->grp.id ||
mpi_cmp_mpi( &pub->Q.X, &prv->Q.X ) ||
mpi_cmp_mpi( &pub->Q.Y, &prv->Q.Y ) ||
mpi_cmp_mpi( &pub->Q.Z, &prv->Q.Z ) )
{
return( POLARSSL_ERR_ECP_BAD_INPUT_DATA );
}
ecp_point_init( &Q );
ecp_group_init( &grp );
/* ecp_mul() needs a non-const group... */
ecp_group_copy( &grp, &prv->grp );
/* Also checks d is valid */
MPI_CHK( ecp_mul( &grp, &Q, &prv->d, &prv->grp.G, NULL, NULL ) );
if( mpi_cmp_mpi( &Q.X, &prv->Q.X ) ||
mpi_cmp_mpi( &Q.Y, &prv->Q.Y ) ||
mpi_cmp_mpi( &Q.Z, &prv->Q.Z ) )
{
ret = POLARSSL_ERR_ECP_BAD_INPUT_DATA;
goto cleanup;
}
cleanup:
ecp_point_free( &Q );
ecp_group_free( &grp );
return( ret );
}
#if defined(POLARSSL_SELF_TEST)
/*
* Checkup routine
*/
int ecp_self_test( int verbose )
{
int ret;
size_t i;
ecp_group grp;
ecp_point R, P;
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... */
};
ecp_group_init( &grp );
ecp_point_init( &R );
ecp_point_init( &P );
mpi_init( &m );
/* Use secp192r1 if available, or any available curve */
#if defined(POLARSSL_ECP_DP_SECP192R1_ENABLED)
MPI_CHK( ecp_use_known_dp( &grp, POLARSSL_ECP_DP_SECP192R1 ) );
#else
MPI_CHK( ecp_use_known_dp( &grp, ecp_curve_list()->grp_id ) );
#endif
if( verbose != 0 )
polarssl_printf( " ECP test #1 (constant op_count, base point G): " );
/* Do a dummy multiplication first to trigger precomputation */
MPI_CHK( mpi_lset( &m, 2 ) );
MPI_CHK( ecp_mul( &grp, &P, &m, &grp.G, NULL, NULL ) );
add_count = 0;
dbl_count = 0;
mul_count = 0;
MPI_CHK( mpi_read_string( &m, 16, exponents[0] ) );
MPI_CHK( 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;
MPI_CHK( mpi_read_string( &m, 16, exponents[i] ) );
MPI_CHK( 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 )
polarssl_printf( "failed (%u)\n", (unsigned int) i );
ret = 1;
goto cleanup;
}
}
if( verbose != 0 )
polarssl_printf( "passed\n" );
if( verbose != 0 )
polarssl_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;
MPI_CHK( mpi_read_string( &m, 16, exponents[0] ) );
MPI_CHK( 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;
MPI_CHK( mpi_read_string( &m, 16, exponents[i] ) );
MPI_CHK( 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 )
polarssl_printf( "failed (%u)\n", (unsigned int) i );
ret = 1;
goto cleanup;
}
}
if( verbose != 0 )
polarssl_printf( "passed\n" );
cleanup:
if( ret < 0 && verbose != 0 )
polarssl_printf( "Unexpected error, return code = %08X\n", ret );
ecp_group_free( &grp );
ecp_point_free( &R );
ecp_point_free( &P );
mpi_free( &m );
if( verbose != 0 )
polarssl_printf( "\n" );
return( ret );
}
#endif /* POLARSSL_SELF_TEST */
#endif /* POLARSSL_ECP_C */