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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
/**
* \brief Function level alternative implementation.
*
* The MBEDTLS_ECP_INTERNAL_ALT macro enables alternative implementations to
* replace certain functions in this module. The alternative implementations are
* typically hardware accelerators and need to activate the hardware before the
* computation starts and deactivate it after it finishes. The
* mbedtls_internal_ecp_init() and mbedtls_internal_ecp_free() functions serve
* this purpose.
*
* To preserve the correct functionality the following conditions must hold:
*
* - The alternative implementation must be activated by
* mbedtls_internal_ecp_init() before any of the replaceable functions is
* called.
* - mbedtls_internal_ecp_free() must \b only be called when the alternative
* implementation is activated.
* - mbedtls_internal_ecp_init() must \b not be called when the alternative
* implementation is activated.
* - Public functions must not return while the alternative implementation is
* activated.
* - Replaceable functions are guarded by \c MBEDTLS_ECP_XXX_ALT macros and
* before calling them an \code if( mbedtls_internal_ecp_grp_capable( grp ) )
* \endcode ensures that the alternative implementation supports the current
* group.
*/
#if defined(MBEDTLS_ECP_INTERNAL_ALT)
#endif
#if defined(MBEDTLS_ECP_C)
#include "mbedtls/ecp.h"
#include "mbedtls/threading.h"
#include "mbedtls/platform_util.h"
#include <string.h>
#if !defined(MBEDTLS_ECP_ALT)
/* Parameter validation macros based on platform_util.h */
#define ECP_VALIDATE_RET( cond ) \
MBEDTLS_INTERNAL_VALIDATE_RET( cond, MBEDTLS_ERR_ECP_BAD_INPUT_DATA )
#define ECP_VALIDATE( cond ) \
MBEDTLS_INTERNAL_VALIDATE( cond )
#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
#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_RESTARTABLE)
/*
* Maximum number of "basic operations" to be done in a row.
*
* Default value 0 means that ECC operations will not yield.
* Note that regardless of the value of ecp_max_ops, always at
* least one step is performed before yielding.
*
* Setting ecp_max_ops=1 can be suitable for testing purposes
* as it will interrupt computation at all possible points.
*/
static unsigned ecp_max_ops = 0;
/*
* Set ecp_max_ops
*/
void mbedtls_ecp_set_max_ops( unsigned max_ops )
{
ecp_max_ops = max_ops;
}
/*
* Check if restart is enabled
*/
int mbedtls_ecp_restart_is_enabled( void )
{
return( ecp_max_ops != 0 );
}
/*
* Restart sub-context for ecp_mul_comb()
*/
struct mbedtls_ecp_restart_mul
{
mbedtls_ecp_point R; /* current intermediate result */
size_t i; /* current index in various loops, 0 outside */
mbedtls_ecp_point *T; /* table for precomputed points */
unsigned char T_size; /* number of points in table T */
enum { /* what were we doing last time we returned? */
ecp_rsm_init = 0, /* nothing so far, dummy initial state */
ecp_rsm_pre_dbl, /* precompute 2^n multiples */
ecp_rsm_pre_norm_dbl, /* normalize precomputed 2^n multiples */
ecp_rsm_pre_add, /* precompute remaining points by adding */
ecp_rsm_pre_norm_add, /* normalize all precomputed points */
ecp_rsm_comb_core, /* ecp_mul_comb_core() */
ecp_rsm_final_norm, /* do the final normalization */
} state;
};
/*
* Init restart_mul sub-context
*/
static void ecp_restart_rsm_init( mbedtls_ecp_restart_mul_ctx *ctx )
{
mbedtls_ecp_point_init( &ctx->R );
ctx->i = 0;
ctx->T = NULL;
ctx->T_size = 0;
ctx->state = ecp_rsm_init;
}
/*
* Free the components of a restart_mul sub-context
*/
static void ecp_restart_rsm_free( mbedtls_ecp_restart_mul_ctx *ctx )
{
unsigned char i;
if( ctx == NULL )
return;
mbedtls_ecp_point_free( &ctx->R );
if( ctx->T != NULL )
{
for( i = 0; i < ctx->T_size; i++ )
mbedtls_ecp_point_free( ctx->T + i );
mbedtls_free( ctx->T );
}
ecp_restart_rsm_init( ctx );
}
/*
* Restart context for ecp_muladd()
*/
struct mbedtls_ecp_restart_muladd
{
mbedtls_ecp_point mP; /* mP value */
mbedtls_ecp_point R; /* R intermediate result */
enum { /* what should we do next? */
ecp_rsma_mul1 = 0, /* first multiplication */
ecp_rsma_mul2, /* second multiplication */
ecp_rsma_add, /* addition */
ecp_rsma_norm, /* normalization */
} state;
};
/*
* Init restart_muladd sub-context
*/
static void ecp_restart_ma_init( mbedtls_ecp_restart_muladd_ctx *ctx )
{
mbedtls_ecp_point_init( &ctx->mP );
mbedtls_ecp_point_init( &ctx->R );
ctx->state = ecp_rsma_mul1;
}
/*
* Free the components of a restart_muladd sub-context
*/
static void ecp_restart_ma_free( mbedtls_ecp_restart_muladd_ctx *ctx )
{
if( ctx == NULL )
return;
mbedtls_ecp_point_free( &ctx->mP );
mbedtls_ecp_point_free( &ctx->R );
ecp_restart_ma_init( ctx );
}
/*
* Initialize a restart context
*/
void mbedtls_ecp_restart_init( mbedtls_ecp_restart_ctx *ctx )
{
ECP_VALIDATE( ctx != NULL );
ctx->ops_done = 0;
ctx->depth = 0;
ctx->rsm = NULL;
ctx->ma = NULL;
}
/*
* Free the components of a restart context
*/
void mbedtls_ecp_restart_free( mbedtls_ecp_restart_ctx *ctx )
{
if( ctx == NULL )
return;
ecp_restart_rsm_free( ctx->rsm );
mbedtls_free( ctx->rsm );
ecp_restart_ma_free( ctx->ma );
mbedtls_free( ctx->ma );
mbedtls_ecp_restart_init( ctx );
}
/*
* Check if we can do the next step
*/
int mbedtls_ecp_check_budget( const mbedtls_ecp_group *grp,
mbedtls_ecp_restart_ctx *rs_ctx,
unsigned ops )
{
ECP_VALIDATE_RET( grp != NULL );
if( rs_ctx != NULL && ecp_max_ops != 0 )
{
/* scale depending on curve size: the chosen reference is 256-bit,
* and multiplication is quadratic. Round to the closest integer. */
if( grp->pbits >= 512 )
ops *= 4;
else if( grp->pbits >= 384 )
ops *= 2;
/* Avoid infinite loops: always allow first step.
* Because of that, however, it's not generally true
* that ops_done <= ecp_max_ops, so the check
* ops_done > ecp_max_ops below is mandatory. */
if( ( rs_ctx->ops_done != 0 ) &&
( rs_ctx->ops_done > ecp_max_ops ||
ops > ecp_max_ops - rs_ctx->ops_done ) )
{
return( MBEDTLS_ERR_ECP_IN_PROGRESS );
}
/* update running count */
rs_ctx->ops_done += ops;
}
return( 0 );
}
/* Call this when entering a function that needs its own sub-context */
#define ECP_RS_ENTER( SUB ) do { \
/* reset ops count for this call if top-level */ \
if( rs_ctx != NULL && rs_ctx->depth++ == 0 ) \
rs_ctx->ops_done = 0; \
\
/* set up our own sub-context if needed */ \
if( mbedtls_ecp_restart_is_enabled() && \
rs_ctx != NULL && rs_ctx->SUB == NULL ) \
{ \
rs_ctx->SUB = mbedtls_calloc( 1, sizeof( *rs_ctx->SUB ) ); \
if( rs_ctx->SUB == NULL ) \
return( MBEDTLS_ERR_ECP_ALLOC_FAILED ); \
\
ecp_restart_## SUB ##_init( rs_ctx->SUB ); \
} \
} while( 0 )
/* Call this when leaving a function that needs its own sub-context */
#define ECP_RS_LEAVE( SUB ) do { \
/* clear our sub-context when not in progress (done or error) */ \
if( rs_ctx != NULL && rs_ctx->SUB != NULL && \
ret != MBEDTLS_ERR_ECP_IN_PROGRESS ) \
{ \
ecp_restart_## SUB ##_free( rs_ctx->SUB ); \
mbedtls_free( rs_ctx->SUB ); \
rs_ctx->SUB = NULL; \
} \
\
if( rs_ctx != NULL ) \
rs_ctx->depth--; \
} while( 0 )
#else /* MBEDTLS_ECP_RESTARTABLE */
#define ECP_RS_ENTER( sub ) (void) rs_ctx;
#define ECP_RS_LEAVE( sub ) (void) rs_ctx;
#endif /* MBEDTLS_ECP_RESTARTABLE */
#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;
if( name == NULL )
return( NULL );
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 )
{
ECP_VALIDATE( pt != NULL );
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 )
{
ECP_VALIDATE( grp != NULL );
grp->id = MBEDTLS_ECP_DP_NONE;
mbedtls_mpi_init( &grp->P );
mbedtls_mpi_init( &grp->A );
mbedtls_mpi_init( &grp->B );
mbedtls_ecp_point_init( &grp->G );
mbedtls_mpi_init( &grp->N );
grp->pbits = 0;
grp->nbits = 0;
grp->h = 0;
grp->modp = NULL;
grp->t_pre = NULL;
grp->t_post = NULL;
grp->t_data = NULL;
grp->T = NULL;
grp->T_size = 0;
}
/*
* Initialize (the components of) a key pair
*/
void mbedtls_ecp_keypair_init( mbedtls_ecp_keypair *key )
{
ECP_VALIDATE( key != NULL );
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_platform_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;
ECP_VALIDATE_RET( P != NULL );
ECP_VALIDATE_RET( Q != NULL );
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 )
{
ECP_VALIDATE_RET( dst != NULL );
ECP_VALIDATE_RET( src != NULL );
return( mbedtls_ecp_group_load( dst, src->id ) );
}
/*
* Set point to zero
*/
int mbedtls_ecp_set_zero( mbedtls_ecp_point *pt )
{
int ret;
ECP_VALIDATE_RET( pt != NULL );
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 )
{
ECP_VALIDATE_RET( pt != NULL );
return( mbedtls_mpi_cmp_int( &pt->Z, 0 ) == 0 );
}
/*
* Compare two points lazily
*/
int mbedtls_ecp_point_cmp( const mbedtls_ecp_point *P,
const mbedtls_ecp_point *Q )
{
ECP_VALIDATE_RET( P != NULL );
ECP_VALIDATE_RET( Q != NULL );
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;
ECP_VALIDATE_RET( P != NULL );
ECP_VALIDATE_RET( x != NULL );
ECP_VALIDATE_RET( y != NULL );
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;
ECP_VALIDATE_RET( grp != NULL );
ECP_VALIDATE_RET( P != NULL );
ECP_VALIDATE_RET( olen != NULL );
ECP_VALIDATE_RET( buf != NULL );
ECP_VALIDATE_RET( format == MBEDTLS_ECP_PF_UNCOMPRESSED ||
format == MBEDTLS_ECP_PF_COMPRESSED );
/*
* 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;
ECP_VALIDATE_RET( grp != NULL );
ECP_VALIDATE_RET( pt != NULL );
ECP_VALIDATE_RET( buf != NULL );
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;
ECP_VALIDATE_RET( grp != NULL );
ECP_VALIDATE_RET( pt != NULL );
ECP_VALIDATE_RET( buf != NULL );
ECP_VALIDATE_RET( *buf != NULL );
/*
* 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;
ECP_VALIDATE_RET( grp != NULL );
ECP_VALIDATE_RET( pt != NULL );
ECP_VALIDATE_RET( olen != NULL );
ECP_VALIDATE_RET( buf != NULL );
ECP_VALIDATE_RET( format == MBEDTLS_ECP_PF_UNCOMPRESSED ||
format == MBEDTLS_ECP_PF_COMPRESSED );
/*
* 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 )
{
int ret;
mbedtls_ecp_group_id grp_id;
ECP_VALIDATE_RET( grp != NULL );
ECP_VALIDATE_RET( buf != NULL );
ECP_VALIDATE_RET( *buf != NULL );
if( ( ret = mbedtls_ecp_tls_read_group_id( &grp_id, buf, len ) ) != 0 )
return( ret );
return( mbedtls_ecp_group_load( grp, grp_id ) );
}
/*
* Read a group id from an ECParameters record (RFC 4492) and convert it to
* mbedtls_ecp_group_id.
*/
int mbedtls_ecp_tls_read_group_id( mbedtls_ecp_group_id *grp,
const unsigned char **buf, size_t len )
{
uint16_t tls_id;
const mbedtls_ecp_curve_info *curve_info;
ECP_VALIDATE_RET( grp != NULL );
ECP_VALIDATE_RET( buf != NULL );
ECP_VALIDATE_RET( *buf != NULL );
/*
* 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 );
*grp = curve_info->grp_id;
return( 0 );
}
/*
* 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;
ECP_VALIDATE_RET( grp != NULL );
ECP_VALIDATE_RET( buf != NULL );
ECP_VALIDATE_RET( olen != NULL );
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_size )
{
int ret;
size_t i;
mbedtls_mpi *c, u, Zi, ZZi;
if( T_size < 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_size ) );
#endif
if( ( c = mbedtls_calloc( T_size, sizeof( mbedtls_mpi ) ) ) == NULL )
return( MBEDTLS_ERR_ECP_ALLOC_FAILED );
for( i = 0; i < T_size; i++ )
mbedtls_mpi_init( &c[i] );
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_size; 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_size-1], &grp->P ) );
for( i = T_size - 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_size; 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, but saves on the size of
* the precomputed table.
*
* Summary of the comb method and its modifications:
*
* - The goal is to compute m*P for some w*d-bit integer m.
*
* - The basic comb method splits m into the w-bit integers
* x[0] .. x[d-1] where x[i] consists of the bits in m whose
* index has residue i modulo d, and computes m * P as
* S[x[0]] + 2 * S[x[1]] + .. + 2^(d-1) S[x[d-1]], where
* S[i_{w-1} .. i_0] := i_{w-1} 2^{(w-1)d} P + ... + i_1 2^d P + i_0 P.
*
* - If it happens that, say, x[i+1]=0 (=> S[x[i+1]]=0), one can replace the sum by
* .. + 2^{i-1} S[x[i-1]] - 2^i S[x[i]] + 2^{i+1} S[x[i]] + 2^{i+2} S[x[i+2]] ..,
* thereby successively converting it into a form where all summands
* are nonzero, at the cost of negative summands. This is the basic idea of [3].
*
* - More generally, even if x[i+1] != 0, we can first transform the sum as
* .. - 2^i S[x[i]] + 2^{i+1} ( S[x[i]] + S[x[i+1]] ) + 2^{i+2} S[x[i+2]] ..,
* and then replace S[x[i]] + S[x[i+1]] = S[x[i] ^ x[i+1]] + 2 S[x[i] & x[i+1]].
* Performing and iterating this procedure for those x[i] that are even
* (keeping track of carry), we can transform the original sum into one of the form
* S[x'[0]] +- 2 S[x'[1]] +- .. +- 2^{d-1} S[x'[d-1]] + 2^d S[x'[d]]
* with all x'[i] odd. It is therefore only necessary to know S at odd indices,
* which is why we are only computing half of it in the first place in
* ecp_precompute_comb and accessing it with index abs(i) / 2 in ecp_select_comb.
*
* - For the sake of compactness, only the seven low-order bits of x[i]
* are used to represent its absolute value (K_i in the paper), and the msb
* of x[i] encodes 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_recode_core( 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 adapted comb method
*
* Assumption: T must be able to hold 2^{w - 1} elements.
*
* Operation: If i = i_{w-1} ... i_1 is the binary representation of i,
* sets T[i] = i_{w-1} 2^{(w-1)d} P + ... + i_1 2^d P + P.
*
* Cost: d(w-1) D + (2^{w-1} - 1) A + 1 N(w-1) + 1 N(2^{w-1} - 1)
*
* Note: Even comb values (those where P would be omitted from the
* sum defining T[i] above) are not needed in our adaption
* the comb method. See ecp_comb_recode_core().
*
* This function currently works in four steps:
* (1) [dbl] Computation of intermediate T[i] for 2-power values of i
* (2) [norm_dbl] Normalization of coordinates of these T[i]
* (3) [add] Computation of all T[i]
* (4) [norm_add] Normalization of all T[i]
*
* Step 1 can be interrupted but not the others; together with the final
* coordinate normalization they are the largest steps done at once, depending
* on the window size. Here are operation counts for P-256:
*
* step (2) (3) (4)
* w = 5 142 165 208
* w = 4 136 77 160
* w = 3 130 33 136
* w = 2 124 11 124
*
* So if ECC operations are blocking for too long even with a low max_ops
* value, it's useful to set MBEDTLS_ECP_WINDOW_SIZE to a lower value in order
* to minimize maximum blocking time.
*/
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,
mbedtls_ecp_restart_ctx *rs_ctx )
{
int ret;
unsigned char i;
size_t j = 0;
const unsigned char T_size = 1U << ( w - 1 );
mbedtls_ecp_point *cur, *TT[COMB_MAX_PRE - 1];
#if defined(MBEDTLS_ECP_RESTARTABLE)
if( rs_ctx != NULL && rs_ctx->rsm != NULL )
{
if( rs_ctx->rsm->state == ecp_rsm_pre_dbl )
goto dbl;
if( rs_ctx->rsm->state == ecp_rsm_pre_norm_dbl )
goto norm_dbl;
if( rs_ctx->rsm->state == ecp_rsm_pre_add )
goto add;
if( rs_ctx->rsm->state == ecp_rsm_pre_norm_add )
goto norm_add;
}
#else
(void) rs_ctx;
#endif
#if defined(MBEDTLS_ECP_RESTARTABLE)
if( rs_ctx != NULL && rs_ctx->rsm != NULL )
{
rs_ctx->rsm->state = ecp_rsm_pre_dbl;
/* initial state for the loop */
rs_ctx->rsm->i = 0;
}
dbl:
#endif
/*
* 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 ) );
#if defined(MBEDTLS_ECP_RESTARTABLE)
if( rs_ctx != NULL && rs_ctx->rsm != NULL && rs_ctx->rsm->i != 0 )
j = rs_ctx->rsm->i;
else
#endif
j = 0;
for( ; j < d * ( w - 1 ); j++ )
{
MBEDTLS_ECP_BUDGET( MBEDTLS_ECP_OPS_DBL );
i = 1U << ( j / d );
cur = T + i;
if( j % d == 0 )
MBEDTLS_MPI_CHK( mbedtls_ecp_copy( cur, T + ( i >> 1 ) ) );
MBEDTLS_MPI_CHK( ecp_double_jac( grp, cur, cur ) );
}
#if defined(MBEDTLS_ECP_RESTARTABLE)
if( rs_ctx != NULL && rs_ctx->rsm != NULL )
rs_ctx->rsm->state = ecp_rsm_pre_norm_dbl;
norm_dbl:
#endif
/*
* Normalize current elements in T. As T has holes,
* use an auxiliary array of pointers to elements in T.
*/
j = 0;
for( i = 1; i < T_size; i <<= 1 )
TT[j++] = T + i;
MBEDTLS_ECP_BUDGET( MBEDTLS_ECP_OPS_INV + 6 * j - 2 );
MBEDTLS_MPI_CHK( ecp_normalize_jac_many( grp, TT, j ) );
#if defined(MBEDTLS_ECP_RESTARTABLE)
if( rs_ctx != NULL && rs_ctx->rsm != NULL )
rs_ctx->rsm->state = ecp_rsm_pre_add;
add:
#endif
/*
* Compute the remaining ones using the minimal number of additions
* Be careful to update T[2^l] only after using it!
*/
MBEDTLS_ECP_BUDGET( ( T_size - 1 ) * MBEDTLS_ECP_OPS_ADD );
for( i = 1; i < T_size; i <<= 1 )
{
j = i;
while( j-- )
MBEDTLS_MPI_CHK( ecp_add_mixed( grp, &T[i + j], &T[j], &T[i] ) );
}
#if defined(MBEDTLS_ECP_RESTARTABLE)
if( rs_ctx != NULL && rs_ctx->rsm != NULL )
rs_ctx->rsm->state = ecp_rsm_pre_norm_add;
norm_add:
#endif
/*
* Normalize final elements in T. Even though there are no holes now, we
* still need the auxiliary array for homogeneity with the previous
* call. Also, skip T[0] which is already normalised, being a copy of P.
*/
for( j = 0; j + 1 < T_size; j++ )
TT[j] = T + j + 1;
MBEDTLS_ECP_BUDGET( MBEDTLS_ECP_OPS_INV + 6 * j - 2 );
MBEDTLS_MPI_CHK( ecp_normalize_jac_many( grp, TT, j ) );
cleanup:
#if defined(MBEDTLS_ECP_RESTARTABLE)
if( rs_ctx != NULL && rs_ctx->rsm != NULL &&
ret == MBEDTLS_ERR_ECP_IN_PROGRESS )
{
if( rs_ctx->rsm->state == ecp_rsm_pre_dbl )
rs_ctx->rsm->i = j;
}
#endif
return( ret );
}
/*
* Select precomputed point: R = sign(i) * T[ abs(i) / 2 ]
*
* See ecp_comb_recode_core() for background
*/
static int ecp_select_comb( const mbedtls_ecp_group *grp, mbedtls_ecp_point *R,
const mbedtls_ecp_point T[], unsigned char T_size,
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_size; 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_size,
const unsigned char x[], size_t d,
int (*f_rng)(void *, unsigned char *, size_t),
void *p_rng,
mbedtls_ecp_restart_ctx *rs_ctx )
{
int ret;
mbedtls_ecp_point Txi;
size_t i;
mbedtls_ecp_point_init( &Txi );
#if !defined(MBEDTLS_ECP_RESTARTABLE)
(void) rs_ctx;
#endif
#if defined(MBEDTLS_ECP_RESTARTABLE)
if( rs_ctx != NULL && rs_ctx->rsm != NULL &&
rs_ctx->rsm->state != ecp_rsm_comb_core )
{
rs_ctx->rsm->i = 0;
rs_ctx->rsm->state = ecp_rsm_comb_core;
}
/* new 'if' instead of nested for the sake of the 'else' branch */
if( rs_ctx != NULL && rs_ctx->rsm != NULL && rs_ctx->rsm->i != 0 )
{
/* restore current index (R already pointing to rs_ctx->rsm->R) */
i = rs_ctx->rsm->i;
}
else
#endif
{
/* Start with a non-zero point and randomize its coordinates */
i = d;
MBEDTLS_MPI_CHK( ecp_select_comb( grp, R, T, T_size, 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_ECP_BUDGET( MBEDTLS_ECP_OPS_DBL + MBEDTLS_ECP_OPS_ADD );
--i;
MBEDTLS_MPI_CHK( ecp_double_jac( grp, R, R ) );
MBEDTLS_MPI_CHK( ecp_select_comb( grp, &Txi, T, T_size, x[i] ) );
MBEDTLS_MPI_CHK( ecp_add_mixed( grp, R, R, &Txi ) );
}
cleanup:
mbedtls_ecp_point_free( &Txi );
#if defined(MBEDTLS_ECP_RESTARTABLE)
if( rs_ctx != NULL && rs_ctx->rsm != NULL &&
ret == MBEDTLS_ERR_ECP_IN_PROGRESS )
{
rs_ctx->rsm->i = i;
/* no need to save R, already pointing to rs_ctx->rsm->R */
}
#endif
return( ret );
}
/*
* Recode the scalar to get constant-time comb multiplication
*
* As the actual scalar recoding needs an odd scalar as a starting point,
* this wrapper ensures that by replacing m by N - m if necessary, and
* informs the caller that the result of multiplication will be negated.
*
* This works because we only support large prime order for Short Weierstrass
* curves, so N is always odd hence either m or N - m is.
*
* See ecp_comb_recode_core() for background.
*/
static int ecp_comb_recode_scalar( const mbedtls_ecp_group *grp,
const mbedtls_mpi *m,
unsigned char k[COMB_MAX_D + 1],
size_t d,
unsigned char w,
unsigned char *parity_trick )
{
int ret;
mbedtls_mpi M, mm;
mbedtls_mpi_init( &M );
mbedtls_mpi_init( &mm );
/* N is always odd (see above), just make extra sure */
if( mbedtls_mpi_get_bit( &grp->N, 0 ) != 1 )
return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA );
/* do we need the parity trick? */
*parity_trick = ( mbedtls_mpi_get_bit( m, 0 ) == 0 );
/* execute parity fix in constant time */
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, *parity_trick ) );
/* actual scalar recoding */
ecp_comb_recode_core( k, d, w, &M );
cleanup:
mbedtls_mpi_free( &mm );
mbedtls_mpi_free( &M );
return( ret );
}
/*
* Perform comb multiplication (for short Weierstrass curves)
* once the auxiliary table has been pre-computed.
*
* Scalar recoding may use a parity trick that makes us compute -m * P,
* if that is the case we'll need to recover m * P at the end.
*/
static int ecp_mul_comb_after_precomp( const mbedtls_ecp_group *grp,
mbedtls_ecp_point *R,
const mbedtls_mpi *m,
const mbedtls_ecp_point *T,
unsigned char T_size,
unsigned char w,
size_t d,
int (*f_rng)(void *, unsigned char *, size_t),
void *p_rng,
mbedtls_ecp_restart_ctx *rs_ctx )
{
int ret;
unsigned char parity_trick;
unsigned char k[COMB_MAX_D + 1];
mbedtls_ecp_point *RR = R;
#if defined(MBEDTLS_ECP_RESTARTABLE)
if( rs_ctx != NULL && rs_ctx->rsm != NULL )
{
RR = &rs_ctx->rsm->R;
if( rs_ctx->rsm->state == ecp_rsm_final_norm )
goto final_norm;
}
#endif
MBEDTLS_MPI_CHK( ecp_comb_recode_scalar( grp, m, k, d, w,
&parity_trick ) );
MBEDTLS_MPI_CHK( ecp_mul_comb_core( grp, RR, T, T_size, k, d,
f_rng, p_rng, rs_ctx ) );
MBEDTLS_MPI_CHK( ecp_safe_invert_jac( grp, RR, parity_trick ) );
#if defined(MBEDTLS_ECP_RESTARTABLE)
if( rs_ctx != NULL && rs_ctx->rsm != NULL )
rs_ctx->rsm->state = ecp_rsm_final_norm;
final_norm:
#endif
MBEDTLS_ECP_BUDGET( MBEDTLS_ECP_OPS_INV );
MBEDTLS_MPI_CHK( ecp_normalize_jac( grp, RR ) );
#if defined(MBEDTLS_ECP_RESTARTABLE)
if( rs_ctx != NULL && rs_ctx->rsm != NULL )
MBEDTLS_MPI_CHK( mbedtls_ecp_copy( R, RR ) );
#endif
cleanup:
return( ret );
}
/*
* Pick window size based on curve size and whether we optimize for base point
*/
static unsigned char ecp_pick_window_size( const mbedtls_ecp_group *grp,
unsigned char p_eq_g )
{
unsigned char w;
/*
* 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( p_eq_g )
w++;
/*
* 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;
return( w );
}
/*
* Multiplication using the comb method - for curves in short Weierstrass form
*
* This function is mainly responsible for administrative work:
* - managing the restart context if enabled
* - managing the table of precomputed points (passed between the below two
* functions): allocation, computation, ownership tranfer, freeing.
*
* It delegates the actual arithmetic work to:
* ecp_precompute_comb() and ecp_mul_comb_with_precomp()
*
* See comments on ecp_comb_recode_core() regarding the computation strategy.
*/
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,
mbedtls_ecp_restart_ctx *rs_ctx )
{
int ret;
unsigned char w, p_eq_g, i;
size_t d;
unsigned char T_size, T_ok;
mbedtls_ecp_point *T;
ECP_RS_ENTER( rsm );
/* Is P the base point ? */
#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 );
#else
p_eq_g = 0;
#endif
/* Pick window size and deduce related sizes */
w = ecp_pick_window_size( grp, p_eq_g );
T_size = 1U << ( w - 1 );
d = ( grp->nbits + w - 1 ) / w;
/* Pre-computed table: do we have it already for the base point? */
if( p_eq_g && grp->T != NULL )
{
/* second pointer to the same table, will be deleted on exit */
T = grp->T;
T_ok = 1;
}
else
#if defined(MBEDTLS_ECP_RESTARTABLE)
/* Pre-computed table: do we have one in progress? complete? */
if( rs_ctx != NULL && rs_ctx->rsm != NULL && rs_ctx->rsm->T != NULL )
{
/* transfer ownership of T from rsm to local function */
T = rs_ctx->rsm->T;
rs_ctx->rsm->T = NULL;
rs_ctx->rsm->T_size = 0;
/* This effectively jumps to the call to mul_comb_after_precomp() */
T_ok = rs_ctx->rsm->state >= ecp_rsm_comb_core;
}
else
#endif
/* Allocate table if we didn't have any */
{
T = mbedtls_calloc( T_size, sizeof( mbedtls_ecp_point ) );
if( T == NULL )
{
ret = MBEDTLS_ERR_ECP_ALLOC_FAILED;
goto cleanup;
}
for( i = 0; i < T_size; i++ )
mbedtls_ecp_point_init( &T[i] );
T_ok = 0;
}
/* Compute table (or finish computing it) if not done already */
if( !T_ok )
{
MBEDTLS_MPI_CHK( ecp_precompute_comb( grp, T, P, w, d, rs_ctx ) );
if( p_eq_g )
{
/* almost transfer ownership of T to the group, but keep a copy of
* the pointer to use for calling the next function more easily */
grp->T = T;
grp->T_size = T_size;
}
}
/* Actual comb multiplication using precomputed points */
MBEDTLS_MPI_CHK( ecp_mul_comb_after_precomp( grp, R, m,
T, T_size, w, d,
f_rng, p_rng, rs_ctx ) );
cleanup:
/* does T belong to the group? */
if( T == grp->T )
T = NULL;
/* does T belong to the restart context? */
#if defined(MBEDTLS_ECP_RESTARTABLE)
if( rs_ctx != NULL && rs_ctx->rsm != NULL && ret == MBEDTLS_ERR_ECP_IN_PROGRESS && T != NULL )
{
/* transfer ownership of T from local function to rsm */
rs_ctx->rsm->T_size = T_size;
rs_ctx->rsm->T = T;
T = NULL;
}
#endif
/* did T belong to us? then let's destroy it! */
if( T != NULL )
{
for( i = 0; i < T_size; i++ )
mbedtls_ecp_point_free( &T[i] );
mbedtls_free( T );
}
/* don't free R while in progress in case R == P */
#if defined(MBEDTLS_ECP_RESTARTABLE)
if( ret != MBEDTLS_ERR_ECP_IN_PROGRESS )
#endif
/* prevent caller from using invalid value */
if( ret != 0 )
mbedtls_ecp_point_free( R );
ECP_RS_LEAVE( rsm );
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 */
/*
* Restartable multiplication R = m * P
*/
int mbedtls_ecp_mul_restartable( 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,
mbedtls_ecp_restart_ctx *rs_ctx )
{
int ret = MBEDTLS_ERR_ECP_BAD_INPUT_DATA;
#if defined(MBEDTLS_ECP_INTERNAL_ALT)
char is_grp_capable = 0;
#endif
ECP_VALIDATE_RET( grp != NULL );
ECP_VALIDATE_RET( R != NULL );
ECP_VALIDATE_RET( m != NULL );
ECP_VALIDATE_RET( P != NULL );
#if defined(MBEDTLS_ECP_RESTARTABLE)
/* reset ops count for this call if top-level */
if( rs_ctx != NULL && rs_ctx->depth++ == 0 )
rs_ctx->ops_done = 0;
#endif
#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(MBEDTLS_ECP_RESTARTABLE)
/* skip argument check when restarting */
if( rs_ctx == NULL || rs_ctx->rsm == NULL )
#endif
{
/* check_privkey is free */
MBEDTLS_ECP_BUDGET( MBEDTLS_ECP_OPS_CHK );
/* Common sanity checks */
MBEDTLS_MPI_CHK( mbedtls_ecp_check_privkey( grp, m ) );
MBEDTLS_MPI_CHK( mbedtls_ecp_check_pubkey( grp, P ) );
}
ret = MBEDTLS_ERR_ECP_BAD_INPUT_DATA;
#if defined(ECP_MONTGOMERY)
if( ecp_get_type( grp ) == ECP_TYPE_MONTGOMERY )
MBEDTLS_MPI_CHK( 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 )
MBEDTLS_MPI_CHK( ecp_mul_comb( grp, R, m, P, f_rng, p_rng, rs_ctx ) );
#endif
cleanup:
#if defined(MBEDTLS_ECP_INTERNAL_ALT)
if( is_grp_capable )
mbedtls_internal_ecp_free( grp );
#endif /* MBEDTLS_ECP_INTERNAL_ALT */
#if defined(MBEDTLS_ECP_RESTARTABLE)
if( rs_ctx != NULL )
rs_ctx->depth--;
#endif
return( ret );
}
/*
* 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 )
{
ECP_VALIDATE_RET( grp != NULL );
ECP_VALIDATE_RET( R != NULL );
ECP_VALIDATE_RET( m != NULL );
ECP_VALIDATE_RET( P != NULL );
return( mbedtls_ecp_mul_restartable( grp, R, m, P, f_rng, p_rng, NULL ) );
}
#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,
mbedtls_ecp_restart_ctx *rs_ctx )
{
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_restartable( grp, R, m, P,
NULL, NULL, rs_ctx ) );
}
cleanup:
return( ret );
}
/*
* Restartable linear combination
* NOT constant-time
*/
int mbedtls_ecp_muladd_restartable(
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,
mbedtls_ecp_restart_ctx *rs_ctx )
{
int ret;
mbedtls_ecp_point mP;
mbedtls_ecp_point *pmP = &mP;
mbedtls_ecp_point *pR = R;
#if defined(MBEDTLS_ECP_INTERNAL_ALT)
char is_grp_capable = 0;
#endif
ECP_VALIDATE_RET( grp != NULL );
ECP_VALIDATE_RET( R != NULL );
ECP_VALIDATE_RET( m != NULL );
ECP_VALIDATE_RET( P != NULL );
ECP_VALIDATE_RET( n != NULL );
ECP_VALIDATE_RET( Q != NULL );
if( ecp_get_type( grp ) != ECP_TYPE_SHORT_WEIERSTRASS )
return( MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE );
mbedtls_ecp_point_init( &mP );
ECP_RS_ENTER( ma );
#if defined(MBEDTLS_ECP_RESTARTABLE)
if( rs_ctx != NULL && rs_ctx->ma != NULL )
{
/* redirect intermediate results to restart context */
pmP = &rs_ctx->ma->mP;
pR = &rs_ctx->ma->R;
/* jump to next operation */
if( rs_ctx->ma->state == ecp_rsma_mul2 )
goto mul2;
if( rs_ctx->ma->state == ecp_rsma_add )
goto add;
if( rs_ctx->ma->state == ecp_rsma_norm )
goto norm;
}
#endif /* MBEDTLS_ECP_RESTARTABLE */
MBEDTLS_MPI_CHK( mbedtls_ecp_mul_shortcuts( grp, pmP, m, P, rs_ctx ) );
#if defined(MBEDTLS_ECP_RESTARTABLE)
if( rs_ctx != NULL && rs_ctx->ma != NULL )
rs_ctx->ma->state = ecp_rsma_mul2;
mul2:
#endif
MBEDTLS_MPI_CHK( mbedtls_ecp_mul_shortcuts( grp, pR, n, Q, rs_ctx ) );
#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(MBEDTLS_ECP_RESTARTABLE)
if( rs_ctx != NULL && rs_ctx->ma != NULL )
rs_ctx->ma->state = ecp_rsma_add;
add:
#endif
MBEDTLS_ECP_BUDGET( MBEDTLS_ECP_OPS_ADD );
MBEDTLS_MPI_CHK( ecp_add_mixed( grp, pR, pmP, pR ) );
#if defined(MBEDTLS_ECP_RESTARTABLE)
if( rs_ctx != NULL && rs_ctx->ma != NULL )
rs_ctx->ma->state = ecp_rsma_norm;
norm:
#endif
MBEDTLS_ECP_BUDGET( MBEDTLS_ECP_OPS_INV );
MBEDTLS_MPI_CHK( ecp_normalize_jac( grp, pR ) );
#if defined(MBEDTLS_ECP_RESTARTABLE)
if( rs_ctx != NULL && rs_ctx->ma != NULL )
MBEDTLS_MPI_CHK( mbedtls_ecp_copy( R, pR ) );
#endif
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 );
ECP_RS_LEAVE( ma );
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 )
{
ECP_VALIDATE_RET( grp != NULL );
ECP_VALIDATE_RET( R != NULL );
ECP_VALIDATE_RET( m != NULL );
ECP_VALIDATE_RET( P != NULL );
ECP_VALIDATE_RET( n != NULL );
ECP_VALIDATE_RET( Q != NULL );
return( mbedtls_ecp_muladd_restartable( grp, R, m, P, n, Q, NULL ) );
}
#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 )
{
ECP_VALIDATE_RET( grp != NULL );
ECP_VALIDATE_RET( pt != NULL );
/* 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 )
{
ECP_VALIDATE_RET( grp != NULL );
ECP_VALIDATE_RET( d != NULL );
#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 );
/* 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 private key
*/
int mbedtls_ecp_gen_privkey( const mbedtls_ecp_group *grp,
mbedtls_mpi *d,
int (*f_rng)(void *, unsigned char *, size_t),
void *p_rng )
{
int ret = MBEDTLS_ERR_ECP_BAD_INPUT_DATA;
size_t n_size;
ECP_VALIDATE_RET( grp != NULL );
ECP_VALIDATE_RET( d != NULL );
ECP_VALIDATE_RET( f_rng != NULL );
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 ) );
}
}
#endif /* ECP_MONTGOMERY */
#if defined(ECP_SHORTWEIERSTRASS)
if( ecp_get_type( grp ) == ECP_TYPE_SHORT_WEIERSTRASS )
{
/* SEC1 3.2.1: Generate d such that 1 <= n < N */
int count = 0;
/*
* Match the procedure given in RFC 6979 (deterministic ECDSA):
* - use the same byte ordering;
* - keep the leftmost nbits bits of the generated octet string;
* - try until result is in the desired range.
* This also avoids any biais, which is especially important for ECDSA.
*/
do
{
MBEDTLS_MPI_CHK( mbedtls_mpi_fill_random( d, n_size, f_rng, p_rng ) );
MBEDTLS_MPI_CHK( mbedtls_mpi_shift_r( d, 8 * n_size - grp->nbits ) );
/*
* Each try has at worst a probability 1/2 of failing (the msb has
* a probability 1/2 of being 0, and then the result will be < N),
* so after 30 tries failure probability is a most 2**(-30).
*
* For most curves, 1 try is enough with overwhelming probability,
* since N starts with a lot of 1s in binary, but some curves
* such as secp224k1 are actually very close to the worst case.
*/
if( ++count > 30 )
return( MBEDTLS_ERR_ECP_RANDOM_FAILED );
}
while( mbedtls_mpi_cmp_int( d, 1 ) < 0 ||
mbedtls_mpi_cmp_mpi( d, &grp->N ) >= 0 );
}
#endif /* ECP_SHORTWEIERSTRASS */
cleanup:
return( ret );
}
/*
* Generate a keypair with configurable base point
*/
int mbedtls_ecp_gen_keypair_base( mbedtls_ecp_group *grp,
const mbedtls_ecp_point *G,
mbedtls_mpi *d, mbedtls_ecp_point *Q,
int (*f_rng)(void *, unsigned char *, size_t),
void *p_rng )
{
int ret;
ECP_VALIDATE_RET( grp != NULL );
ECP_VALIDATE_RET( d != NULL );
ECP_VALIDATE_RET( G != NULL );
ECP_VALIDATE_RET( Q != NULL );
ECP_VALIDATE_RET( f_rng != NULL );
MBEDTLS_MPI_CHK( mbedtls_ecp_gen_privkey( grp, d, f_rng, p_rng ) );
MBEDTLS_MPI_CHK( mbedtls_ecp_mul( grp, Q, d, G, f_rng, p_rng ) );
cleanup:
return( ret );
}
/*
* Generate key pair, wrapper for conventional base point
*/
int mbedtls_ecp_gen_keypair( mbedtls_ecp_group *grp,
mbedtls_mpi *d, mbedtls_ecp_point *Q,
int (*f_rng)(void *, unsigned char *, size_t),
void *p_rng )
{
ECP_VALIDATE_RET( grp != NULL );
ECP_VALIDATE_RET( d != NULL );
ECP_VALIDATE_RET( Q != NULL );
ECP_VALIDATE_RET( f_rng != NULL );
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;
ECP_VALIDATE_RET( key != NULL );
ECP_VALIDATE_RET( f_rng != NULL );
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;
ECP_VALIDATE_RET( pub != NULL );
ECP_VALIDATE_RET( prv != NULL );
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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