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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
*
* [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 <string.h>
#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
#if ( defined(__ARMCC_VERSION) || defined(_MSC_VER) ) && \
!defined(inline) && !defined(__cplusplus)
#define inline __inline
#endif
/* Implementation that should never be optimized out by the compiler */
static void mbedtls_zeroize( void *v, size_t n ) {
volatile unsigned char *p = v; while( n-- ) *p++ = 0;
}
#if defined(MBEDTLS_SELF_TEST)
/*
* Counts of point addition and doubling, and field multiplications.
* Used to test resistance of point multiplication to simple timing attacks.
*/
static unsigned long add_count, dbl_count, mul_count;
#endif
#if defined(MBEDTLS_ECP_DP_SECP192R1_ENABLED) || \
defined(MBEDTLS_ECP_DP_SECP224R1_ENABLED) || \
defined(MBEDTLS_ECP_DP_SECP256R1_ENABLED) || \
defined(MBEDTLS_ECP_DP_SECP384R1_ENABLED) || \
defined(MBEDTLS_ECP_DP_SECP521R1_ENABLED) || \
defined(MBEDTLS_ECP_DP_BP256R1_ENABLED) || \
defined(MBEDTLS_ECP_DP_BP384R1_ENABLED) || \
defined(MBEDTLS_ECP_DP_BP512R1_ENABLED) || \
defined(MBEDTLS_ECP_DP_SECP192K1_ENABLED) || \
defined(MBEDTLS_ECP_DP_SECP224K1_ENABLED) || \
defined(MBEDTLS_ECP_DP_SECP256K1_ENABLED)
#define ECP_SHORTWEIERSTRASS
#endif
#if defined(MBEDTLS_ECP_DP_CURVE25519_ENABLED)
#define ECP_MONTGOMERY
#endif
/*
* Curve types: internal for now, might be exposed later
*/
typedef enum
{
ECP_TYPE_NONE = 0,
ECP_TYPE_SHORT_WEIERSTRASS, /* y^2 = x^3 + a x + b */
ECP_TYPE_MONTGOMERY, /* y^2 = x^3 + a x^2 + x */
} ecp_curve_type;
/*
* List of supported curves:
* - internal ID
* - TLS NamedCurve ID (RFC 4492 sec. 5.1.1, RFC 7071 sec. 2)
* - size in bits
* - readable name
*
* Curves are listed in order: largest curves first, and for a given size,
* fastest curves first. This provides the default order for the SSL module.
*
* Reminder: update profiles in x509_crt.c when adding a new curves!
*/
static const mbedtls_ecp_curve_info ecp_supported_curves[] =
{
#if defined(MBEDTLS_ECP_DP_SECP521R1_ENABLED)
{ MBEDTLS_ECP_DP_SECP521R1, 25, 521, "secp521r1" },
#endif
#if defined(MBEDTLS_ECP_DP_BP512R1_ENABLED)
{ MBEDTLS_ECP_DP_BP512R1, 28, 512, "brainpoolP512r1" },
#endif
#if defined(MBEDTLS_ECP_DP_SECP384R1_ENABLED)
{ MBEDTLS_ECP_DP_SECP384R1, 24, 384, "secp384r1" },
#endif
#if defined(MBEDTLS_ECP_DP_BP384R1_ENABLED)
{ MBEDTLS_ECP_DP_BP384R1, 27, 384, "brainpoolP384r1" },
#endif
#if defined(MBEDTLS_ECP_DP_SECP256R1_ENABLED)
{ MBEDTLS_ECP_DP_SECP256R1, 23, 256, "secp256r1" },
#endif
#if defined(MBEDTLS_ECP_DP_SECP256K1_ENABLED)
{ MBEDTLS_ECP_DP_SECP256K1, 22, 256, "secp256k1" },
#endif
#if defined(MBEDTLS_ECP_DP_BP256R1_ENABLED)
{ MBEDTLS_ECP_DP_BP256R1, 26, 256, "brainpoolP256r1" },
#endif
#if defined(MBEDTLS_ECP_DP_SECP224R1_ENABLED)
{ MBEDTLS_ECP_DP_SECP224R1, 21, 224, "secp224r1" },
#endif
#if defined(MBEDTLS_ECP_DP_SECP224K1_ENABLED)
{ MBEDTLS_ECP_DP_SECP224K1, 20, 224, "secp224k1" },
#endif
#if defined(MBEDTLS_ECP_DP_SECP192R1_ENABLED)
{ MBEDTLS_ECP_DP_SECP192R1, 19, 192, "secp192r1" },
#endif
#if defined(MBEDTLS_ECP_DP_SECP192K1_ENABLED)
{ MBEDTLS_ECP_DP_SECP192K1, 18, 192, "secp192k1" },
#endif
{ MBEDTLS_ECP_DP_NONE, 0, 0, NULL },
};
#define ECP_NB_CURVES sizeof( ecp_supported_curves ) / \
sizeof( ecp_supported_curves[0] )
static mbedtls_ecp_group_id ecp_supported_grp_id[ECP_NB_CURVES];
/*
* List of supported curves and associated info
*/
const mbedtls_ecp_curve_info *mbedtls_ecp_curve_list( void )
{
return( ecp_supported_curves );
}
/*
* List of supported curves, group ID only
*/
const mbedtls_ecp_group_id *mbedtls_ecp_grp_id_list( void )
{
static int init_done = 0;
if( ! init_done )
{
size_t i = 0;
const mbedtls_ecp_curve_info *curve_info;
for( curve_info = mbedtls_ecp_curve_list();
curve_info->grp_id != MBEDTLS_ECP_DP_NONE;
curve_info++ )
{
ecp_supported_grp_id[i++] = curve_info->grp_id;
}
ecp_supported_grp_id[i] = MBEDTLS_ECP_DP_NONE;
init_done = 1;
}
return( ecp_supported_grp_id );
}
/*
* Get the curve info for the internal identifier
*/
const mbedtls_ecp_curve_info *mbedtls_ecp_curve_info_from_grp_id( mbedtls_ecp_group_id grp_id )
{
const mbedtls_ecp_curve_info *curve_info;
for( curve_info = mbedtls_ecp_curve_list();
curve_info->grp_id != MBEDTLS_ECP_DP_NONE;
curve_info++ )
{
if( curve_info->grp_id == grp_id )
return( curve_info );
}
return( NULL );
}
/*
* Get the curve info from the TLS identifier
*/
const mbedtls_ecp_curve_info *mbedtls_ecp_curve_info_from_tls_id( uint16_t tls_id )
{
const mbedtls_ecp_curve_info *curve_info;
for( curve_info = mbedtls_ecp_curve_list();
curve_info->grp_id != MBEDTLS_ECP_DP_NONE;
curve_info++ )
{
if( curve_info->tls_id == tls_id )
return( curve_info );
}
return( NULL );
}
/*
* Get the curve info from the name
*/
const mbedtls_ecp_curve_info *mbedtls_ecp_curve_info_from_name( const char *name )
{
const mbedtls_ecp_curve_info *curve_info;
for( curve_info = mbedtls_ecp_curve_list();
curve_info->grp_id != MBEDTLS_ECP_DP_NONE;
curve_info++ )
{
if( strcmp( curve_info->name, name ) == 0 )
return( curve_info );
}
return( NULL );
}
/*
* Get the type of a curve
*/
static inline ecp_curve_type ecp_get_type( const mbedtls_ecp_group *grp )
{
if( grp->G.X.p == NULL )
return( ECP_TYPE_NONE );
if( grp->G.Y.p == NULL )
return( ECP_TYPE_MONTGOMERY );
else
return( ECP_TYPE_SHORT_WEIERSTRASS );
}
/*
* Initialize (the components of) a point
*/
void mbedtls_ecp_point_init( mbedtls_ecp_point *pt )
{
if( pt == NULL )
return;
mbedtls_mpi_init( &pt->X );
mbedtls_mpi_init( &pt->Y );
mbedtls_mpi_init( &pt->Z );
}
/*
* Initialize (the components of) a group
*/
void mbedtls_ecp_group_init( mbedtls_ecp_group *grp )
{
if( grp == NULL )
return;
memset( grp, 0, sizeof( mbedtls_ecp_group ) );
}
/*
* Initialize (the components of) a key pair
*/
void mbedtls_ecp_keypair_init( mbedtls_ecp_keypair *key )
{
if( key == NULL )
return;
mbedtls_ecp_group_init( &key->grp );
mbedtls_mpi_init( &key->d );
mbedtls_ecp_point_init( &key->Q );
}
/*
* Unallocate (the components of) a point
*/
void mbedtls_ecp_point_free( mbedtls_ecp_point *pt )
{
if( pt == NULL )
return;
mbedtls_mpi_free( &( pt->X ) );
mbedtls_mpi_free( &( pt->Y ) );
mbedtls_mpi_free( &( pt->Z ) );
}
/*
* Unallocate (the components of) a group
*/
void mbedtls_ecp_group_free( mbedtls_ecp_group *grp )
{
size_t i;
if( grp == NULL )
return;
if( grp->h != 1 )
{
mbedtls_mpi_free( &grp->P );
mbedtls_mpi_free( &grp->A );
mbedtls_mpi_free( &grp->B );
mbedtls_ecp_point_free( &grp->G );
mbedtls_mpi_free( &grp->N );
}
if( grp->T != NULL )
{
for( i = 0; i < grp->T_size; i++ )
mbedtls_ecp_point_free( &grp->T[i] );
mbedtls_free( grp->T );
}
mbedtls_zeroize( grp, sizeof( mbedtls_ecp_group ) );
}
/*
* Unallocate (the components of) a key pair
*/
void mbedtls_ecp_keypair_free( mbedtls_ecp_keypair *key )
{
if( key == NULL )
return;
mbedtls_ecp_group_free( &key->grp );
mbedtls_mpi_free( &key->d );
mbedtls_ecp_point_free( &key->Q );
}
/*
* Copy the contents of a point
*/
int mbedtls_ecp_copy( mbedtls_ecp_point *P, const mbedtls_ecp_point *Q )
{
int ret;
MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &P->X, &Q->X ) );
MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &P->Y, &Q->Y ) );
MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &P->Z, &Q->Z ) );
cleanup:
return( ret );
}
/*
* Copy the contents of a group object
*/
int mbedtls_ecp_group_copy( mbedtls_ecp_group *dst, const mbedtls_ecp_group *src )
{
return mbedtls_ecp_group_load( dst, src->id );
}
/*
* Set point to zero
*/
int mbedtls_ecp_set_zero( mbedtls_ecp_point *pt )
{
int ret;
MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &pt->X , 1 ) );
MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &pt->Y , 1 ) );
MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &pt->Z , 0 ) );
cleanup:
return( ret );
}
/*
* Tell if a point is zero
*/
int mbedtls_ecp_is_zero( mbedtls_ecp_point *pt )
{
return( mbedtls_mpi_cmp_int( &pt->Z, 0 ) == 0 );
}
/*
* Compare two points 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 );
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( ( 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
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;