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mbedtls/library/ecp.c
Manuel Pégourié-Gonnard 62aad14139 Added slot for fast modp, with mod_p521
2013-01-16 16:31:51 +01:00

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/*
* Elliptic curves over GF(p)
*
* Copyright (C) 2012, Brainspark B.V.
*
* This file is part of PolarSSL (http://www.polarssl.org)
* Lead Maintainer: Paul Bakker <polarssl_maintainer at polarssl.org>
*
* All rights reserved.
*
* This program is free software; you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation; either version 2 of the License, or
* (at your option) any later version.
*
* This program is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License along
* with this program; if not, write to the Free Software Foundation, Inc.,
* 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA.
*/
/*
* References:
*
* SEC1 http://www.secg.org/index.php?action=secg,docs_secg
* GECC = Guide to Elliptic Curve Cryptography - Hankerson, Menezes, Vanstone
* FIPS 186-3 http://csrc.nist.gov/publications/fips/fips186-3/fips_186-3.pdf
*/
#include "polarssl/config.h"
#if defined(POLARSSL_ECP_C)
#include "polarssl/ecp.h"
/*
* Initialize (the components of) a point
*/
void ecp_point_init( ecp_point *pt )
{
if( pt == NULL )
return;
pt->is_zero = 1;
mpi_init( &pt->X );
mpi_init( &pt->Y );
}
/*
* Initialize (the components of) a group
*/
void ecp_group_init( ecp_group *grp )
{
if( grp == NULL )
return;
mpi_init( &grp->P );
mpi_init( &grp->B );
ecp_point_init( &grp->G );
mpi_init( &grp->N );
grp->modp = NULL;
grp->pbits = 0;
}
/*
* Unallocate (the components of) a point
*/
void ecp_point_free( ecp_point *pt )
{
if( pt == NULL )
return;
pt->is_zero = 1;
mpi_free( &( pt->X ) );
mpi_free( &( pt->Y ) );
}
/*
* Unallocate (the components of) a group
*/
void ecp_group_free( ecp_group *grp )
{
if( grp == NULL )
return;
mpi_free( &grp->P );
mpi_free( &grp->B );
ecp_point_free( &grp->G );
mpi_free( &grp->N );
}
/*
* Set point to zero
*/
void ecp_set_zero( ecp_point *pt )
{
pt->is_zero = 1;
mpi_free( &pt->X );
mpi_free( &pt->Y );
}
/*
* Copy the contents of Q into P
*/
int ecp_copy( ecp_point *P, const ecp_point *Q )
{
int ret = 0;
if( Q->is_zero ) {
ecp_set_zero( P );
return( ret );
}
P->is_zero = Q->is_zero;
MPI_CHK( mpi_copy( &P->X, &Q->X ) );
MPI_CHK( mpi_copy( &P->Y, &Q->Y ) );
cleanup:
return( ret );
}
/*
* Import a non-zero point from ASCII strings
*/
int ecp_point_read_string( ecp_point *P, int radix,
const char *x, const char *y )
{
int ret = 0;
P->is_zero = 0;
MPI_CHK( mpi_read_string( &P->X, radix, x ) );
MPI_CHK( mpi_read_string( &P->Y, radix, y ) );
cleanup:
return( ret );
}
/*
* Import an ECP group from ASCII strings
*/
int ecp_group_read_string( ecp_group *grp, int radix,
const char *p, const char *b,
const char *gx, const char *gy, const char *n)
{
int ret = 0;
MPI_CHK( mpi_read_string( &grp->P, radix, p ) );
MPI_CHK( mpi_read_string( &grp->B, radix, b ) );
MPI_CHK( ecp_point_read_string( &grp->G, radix, gx, gy ) );
MPI_CHK( mpi_read_string( &grp->N, radix, n ) );
cleanup:
return( ret );
}
/*
* Wrapper around fast quasi-modp functions, with fallback to mpi_mod_mpi
*
* The quasi-modp functions expect an mpi N such that 0 <= N < 2^(2*pbits)
* and change it in-place so that it can easily be brought in the 0..P-1
* range by a few additions or substractions.
*/
static int ecp_modp( mpi *N, const ecp_group *grp )
{
int ret = 0;
if( grp->modp == NULL )
return( mpi_mod_mpi( N, N, &grp->P ) );
if( mpi_cmp_int( N, 0 ) < 0 || mpi_msb( N ) > 2 * grp->pbits )
return( POLARSSL_ERR_ECP_GENERIC );
MPI_CHK( grp->modp( N ) );
while( mpi_cmp_int( N, 0 ) < 0 )
MPI_CHK( mpi_add_mpi( N, N, &grp->P ) );
while( mpi_cmp_mpi( N, &grp->P ) >= 0 )
MPI_CHK( mpi_sub_mpi( N, N, &grp->P ) );
cleanup:
return( ret );
}
/*
* Size of p521 in terms of t_uint
*/
#define P521_SIZE_INT ( 521 / (sizeof( t_uint ) << 3) + 1 )
/*
* Bits to keep in the most significant t_uint
*/
#if defined(POLARSS_HAVE_INT8)
#define P521_MASK 0x01
#else
#define P521_MASK 0x01FF
#endif
/*
* Fast quasi-reduction modulo p521 (FIPS 186-3 D.2.5)
*
* It is required that 0 <= N < 2^(2*521) on entry.
* On exit, it is only guaranteed that 0 <= N < 2^(521+1).
*/
static int ecp_mod_p521( mpi *N )
{
int ret = 0;
t_uint Mp[P521_SIZE_INT];
mpi M;
if( N->n < P521_SIZE_INT )
return( 0 );
memset( Mp, 0, P521_SIZE_INT * sizeof( t_uint ) );
memcpy( Mp, N->p, P521_SIZE_INT * sizeof( t_uint ) );
Mp[P521_SIZE_INT - 1] &= P521_MASK;
M.s = 1;
M.n = P521_SIZE_INT;
M.p = Mp;
MPI_CHK( mpi_shift_r( N, 521 ) );
MPI_CHK( mpi_add_abs( N, N, &M ) );
cleanup:
return( ret );
}
/*
* Set a group using well-known domain parameters
*/
int ecp_use_known_dp( ecp_group *grp, size_t index )
{
switch( index )
{
case POLARSSL_ECP_DP_SECP192R1:
return( ecp_group_read_string( grp, 16,
POLARSSL_ECP_SECP192R1_P,
POLARSSL_ECP_SECP192R1_B,
POLARSSL_ECP_SECP192R1_GX,
POLARSSL_ECP_SECP192R1_GY,
POLARSSL_ECP_SECP192R1_N )
);
case POLARSSL_ECP_DP_SECP224R1:
return( ecp_group_read_string( grp, 16,
POLARSSL_ECP_SECP224R1_P,
POLARSSL_ECP_SECP224R1_B,
POLARSSL_ECP_SECP224R1_GX,
POLARSSL_ECP_SECP224R1_GY,
POLARSSL_ECP_SECP224R1_N )
);
case POLARSSL_ECP_DP_SECP256R1:
return( ecp_group_read_string( grp, 16,
POLARSSL_ECP_SECP256R1_P,
POLARSSL_ECP_SECP256R1_B,
POLARSSL_ECP_SECP256R1_GX,
POLARSSL_ECP_SECP256R1_GY,
POLARSSL_ECP_SECP256R1_N )
);
case POLARSSL_ECP_DP_SECP384R1:
return( ecp_group_read_string( grp, 16,
POLARSSL_ECP_SECP384R1_P,
POLARSSL_ECP_SECP384R1_B,
POLARSSL_ECP_SECP384R1_GX,
POLARSSL_ECP_SECP384R1_GY,
POLARSSL_ECP_SECP384R1_N )
);
case POLARSSL_ECP_DP_SECP521R1:
grp->modp = ecp_mod_p521;
grp->pbits = 521;
return( ecp_group_read_string( grp, 16,
POLARSSL_ECP_SECP521R1_P,
POLARSSL_ECP_SECP521R1_B,
POLARSSL_ECP_SECP521R1_GX,
POLARSSL_ECP_SECP521R1_GY,
POLARSSL_ECP_SECP521R1_N )
);
}
return( POLARSSL_ERR_ECP_GENERIC );
}
/*
* Reduce a mpi mod p in-place, general case, to use after mpi_mul_mpi
*/
#define MOD_MUL( N ) MPI_CHK( ecp_modp( &N, grp ) )
/*
* Reduce a mpi mod p in-place, to use after mpi_sub_mpi
*/
#define MOD_SUB( N ) \
while( mpi_cmp_int( &N, 0 ) < 0 ) \
MPI_CHK( mpi_add_mpi( &N, &N, &grp->P ) )
/*
* Reduce a mpi mod p in-place, to use after mpi_add_mpi and mpi_mul_int
*/
#define MOD_ADD( N ) \
while( mpi_cmp_mpi( &N, &grp->P ) >= 0 ) \
MPI_CHK( mpi_sub_mpi( &N, &N, &grp->P ) )
/*
* Internal point format used for fast addition/doubling/multiplication:
* Jacobian coordinates (GECC example 3.20)
*/
typedef struct
{
mpi X, Y, Z;
}
ecp_ptjac;
/*
* Initialize a point in Jacobian coordinates
*/
static void ecp_ptjac_init( ecp_ptjac *P )
{
mpi_init( &P->X ); mpi_init( &P->Y ); mpi_init( &P->Z );
}
/*
* Free a point in Jacobian coordinates
*/
static void ecp_ptjac_free( ecp_ptjac *P )
{
mpi_free( &P->X ); mpi_free( &P->Y ); mpi_free( &P->Z );
}
/*
* Copy P to R in Jacobian coordinates
*/
static int ecp_ptjac_copy( ecp_ptjac *R, const ecp_ptjac *P )
{
int ret = 0;
MPI_CHK( mpi_copy( &R->X, &P->X ) );
MPI_CHK( mpi_copy( &R->Y, &P->Y ) );
MPI_CHK( mpi_copy( &R->Z, &P->Z ) );
cleanup:
return( ret );
}
/*
* Set P to zero in Jacobian coordinates
*/
static int ecp_ptjac_set_zero( ecp_ptjac *P )
{
int ret = 0;
MPI_CHK( mpi_lset( &P->X, 1 ) );
MPI_CHK( mpi_lset( &P->Y, 1 ) );
MPI_CHK( mpi_lset( &P->Z, 0 ) );
cleanup:
return( ret );
}
/*
* Convert from affine to Jacobian coordinates
*/
static int ecp_aff_to_jac( ecp_ptjac *jac, const ecp_point *aff )
{
int ret = 0;
if( aff->is_zero )
return( ecp_ptjac_set_zero( jac ) );
MPI_CHK( mpi_copy( &jac->X, &aff->X ) );
MPI_CHK( mpi_copy( &jac->Y, &aff->Y ) );
MPI_CHK( mpi_lset( &jac->Z, 1 ) );
cleanup:
return( ret );
}
/*
* Convert from Jacobian to affine coordinates
*/
static int ecp_jac_to_aff( const ecp_group *grp,
ecp_point *aff, const ecp_ptjac *jac )
{
int ret = 0;
mpi Zi, ZZi, T;
if( mpi_cmp_int( &jac->Z, 0 ) == 0 ) {
ecp_set_zero( aff );
return( 0 );
}
mpi_init( &Zi ); mpi_init( &ZZi ); mpi_init( &T );
aff->is_zero = 0;
/*
* aff.X = jac.X / (jac.Z)^2 mod p
*/
MPI_CHK( mpi_inv_mod( &Zi, &jac->Z, &grp->P ) );
MPI_CHK( mpi_mul_mpi( &ZZi, &Zi, &Zi ) ); MOD_MUL( ZZi );
MPI_CHK( mpi_mul_mpi( &aff->X, &jac->X, &ZZi ) ); MOD_MUL( aff->X );
/*
* aff.Y = jac.Y / (jac.Z)^3 mod p
*/
MPI_CHK( mpi_mul_mpi( &aff->Y, &jac->Y, &ZZi ) ); MOD_MUL( aff->Y );
MPI_CHK( mpi_mul_mpi( &aff->Y, &aff->Y, &Zi ) ); MOD_MUL( aff->Y );
cleanup:
mpi_free( &Zi ); mpi_free( &ZZi ); mpi_free( &T );
return( ret );
}
/*
* Point doubling R = 2 P, Jacobian coordinates (GECC 3.21)
*/
static int ecp_double_jac( const ecp_group *grp, ecp_ptjac *R,
const ecp_ptjac *P )
{
int ret = 0;
mpi T1, T2, T3, X, Y, Z;
if( mpi_cmp_int( &P->Z, 0 ) == 0 )
return( ecp_ptjac_set_zero( R ) );
mpi_init( &T1 ); mpi_init( &T2 ); mpi_init( &T3 );
mpi_init( &X ); mpi_init( &Y ); mpi_init( &Z );
MPI_CHK( mpi_mul_mpi( &T1, &P->Z, &P->Z ) ); MOD_MUL( T1 );
MPI_CHK( mpi_sub_mpi( &T2, &P->X, &T1 ) ); MOD_SUB( T2 );
MPI_CHK( mpi_add_mpi( &T1, &P->X, &T1 ) ); MOD_ADD( T1 );
MPI_CHK( mpi_mul_mpi( &T2, &T2, &T1 ) ); MOD_MUL( T2 );
MPI_CHK( mpi_mul_int( &T2, &T2, 3 ) ); MOD_ADD( T2 );
MPI_CHK( mpi_copy ( &Y, &P->Y ) );
MPI_CHK( mpi_shift_l( &Y, 1 ) ); MOD_ADD( Y );
MPI_CHK( mpi_mul_mpi( &Z, &Y, &P->Z ) ); MOD_MUL( Z );
MPI_CHK( mpi_mul_mpi( &Y, &Y, &Y ) ); MOD_MUL( Y );
MPI_CHK( mpi_mul_mpi( &T3, &Y, &P->X ) ); MOD_MUL( T3 );
MPI_CHK( mpi_mul_mpi( &Y, &Y, &Y ) ); MOD_MUL( Y );
/*
* For Y = Y / 2 mod p, we must make sure that Y is even before
* using right-shift. No need to reduce mod p afterwards.
*/
if( mpi_get_bit( &Y, 0 ) == 1 )
MPI_CHK( mpi_add_mpi( &Y, &Y, &grp->P ) );
MPI_CHK( mpi_shift_r( &Y, 1 ) );
MPI_CHK( mpi_mul_mpi( &X, &T2, &T2 ) ); MOD_MUL( X );
MPI_CHK( mpi_copy ( &T1, &T3 ) );
MPI_CHK( mpi_shift_l( &T1, 1 ) ); MOD_ADD( T1 );
MPI_CHK( mpi_sub_mpi( &X, &X, &T1 ) ); MOD_SUB( X );
MPI_CHK( mpi_sub_mpi( &T1, &T3, &X ) ); MOD_SUB( T1 );
MPI_CHK( mpi_mul_mpi( &T1, &T1, &T2 ) ); MOD_MUL( T1 );
MPI_CHK( mpi_sub_mpi( &Y, &T1, &Y ) ); MOD_SUB( Y );
MPI_CHK( mpi_copy( &R->X, &X ) );
MPI_CHK( mpi_copy( &R->Y, &Y ) );
MPI_CHK( mpi_copy( &R->Z, &Z ) );
cleanup:
mpi_free( &T1 ); mpi_free( &T2 ); mpi_free( &T3 );
mpi_free( &X ); mpi_free( &Y ); mpi_free( &Z );
return( ret );
}
/*
* Addition: R = P + Q, mixed affine-Jacobian coordinates (GECC 3.22)
*/
static int ecp_add_mixed( const ecp_group *grp, ecp_ptjac *R,
const ecp_ptjac *P, const ecp_point *Q )
{
int ret = 0;
mpi T1, T2, T3, T4, X, Y, Z;
/*
* Trivial cases: P == 0 or Q == 0
*/
if( mpi_cmp_int( &P->Z, 0 ) == 0 )
return( ecp_aff_to_jac( R, Q ) );
if( Q->is_zero )
return( ecp_ptjac_copy( R, P ) );
mpi_init( &T1 ); mpi_init( &T2 ); mpi_init( &T3 ); mpi_init( &T4 );
mpi_init( &X ); mpi_init( &Y ); mpi_init( &Z );
MPI_CHK( mpi_mul_mpi( &T1, &P->Z, &P->Z ) ); MOD_MUL( T1 );
MPI_CHK( mpi_mul_mpi( &T2, &T1, &P->Z ) ); MOD_MUL( T2 );
MPI_CHK( mpi_mul_mpi( &T1, &T1, &Q->X ) ); MOD_MUL( T1 );
MPI_CHK( mpi_mul_mpi( &T2, &T2, &Q->Y ) ); MOD_MUL( T2 );
MPI_CHK( mpi_sub_mpi( &T1, &T1, &P->X ) ); MOD_SUB( T1 );
MPI_CHK( mpi_sub_mpi( &T2, &T2, &P->Y ) ); MOD_SUB( T2 );
if( mpi_cmp_int( &T1, 0 ) == 0 )
{
if( mpi_cmp_int( &T2, 0 ) == 0 )
{
ret = ecp_double_jac( grp, R, P );
goto cleanup;
}
else
{
ret = ecp_ptjac_set_zero( R );
goto cleanup;
}
}
MPI_CHK( mpi_mul_mpi( &Z, &P->Z, &T1 ) ); MOD_MUL( Z );
MPI_CHK( mpi_mul_mpi( &T3, &T1, &T1 ) ); MOD_MUL( T3 );
MPI_CHK( mpi_mul_mpi( &T4, &T3, &T1 ) ); MOD_MUL( T4 );
MPI_CHK( mpi_mul_mpi( &T3, &T3, &P->X ) ); MOD_MUL( T3 );
MPI_CHK( mpi_mul_int( &T1, &T3, 2 ) ); MOD_ADD( T1 );
MPI_CHK( mpi_mul_mpi( &X, &T2, &T2 ) ); MOD_MUL( X );
MPI_CHK( mpi_sub_mpi( &X, &X, &T1 ) ); MOD_SUB( X );
MPI_CHK( mpi_sub_mpi( &X, &X, &T4 ) ); MOD_SUB( X );
MPI_CHK( mpi_sub_mpi( &T3, &T3, &X ) ); MOD_SUB( T3 );
MPI_CHK( mpi_mul_mpi( &T3, &T3, &T2 ) ); MOD_MUL( T3 );
MPI_CHK( mpi_mul_mpi( &T4, &T4, &P->Y ) ); MOD_MUL( T4 );
MPI_CHK( mpi_sub_mpi( &Y, &T3, &T4 ) ); MOD_SUB( Y );
MPI_CHK( mpi_copy( &R->X, &X ) );
MPI_CHK( mpi_copy( &R->Y, &Y ) );
MPI_CHK( mpi_copy( &R->Z, &Z ) );
cleanup:
mpi_free( &T1 ); mpi_free( &T2 ); mpi_free( &T3 ); mpi_free( &T4 );
mpi_free( &X ); mpi_free( &Y ); mpi_free( &Z );
return( ret );
}
/*
* Addition: R = P + Q, affine wrapper
*/
int ecp_add( const ecp_group *grp, ecp_point *R,
const ecp_point *P, const ecp_point *Q )
{
int ret = 0;
ecp_ptjac J;
ecp_ptjac_init( &J );
MPI_CHK( ecp_aff_to_jac( &J, P ) );
MPI_CHK( ecp_add_mixed( grp, &J, &J, Q ) );
MPI_CHK( ecp_jac_to_aff( grp, R, &J ) );
cleanup:
ecp_ptjac_free( &J );
return( ret );
}
/*
* Integer multiplication: R = m * P (GECC 5.7, SPA-resistant variant)
*/
int ecp_mul( const ecp_group *grp, ecp_point *R,
const mpi *m, const ecp_point *P )
{
int ret = 0;
size_t pos;
ecp_ptjac Q[2];
ecp_ptjac_init( &Q[0] ); ecp_ptjac_init( &Q[1] );
/*
* The general method works only for m >= 1
*/
if( mpi_cmp_int( m, 0 ) == 0 ) {
ecp_set_zero( R );
goto cleanup;
}
ecp_ptjac_set_zero( &Q[0] );
for( pos = mpi_msb( m ) - 1 ; ; pos-- )
{
MPI_CHK( ecp_double_jac( grp, &Q[0], &Q[0] ) );
MPI_CHK( ecp_add_mixed( grp, &Q[1], &Q[0], P ) );
MPI_CHK( ecp_ptjac_copy( &Q[0], &Q[ mpi_get_bit( m, pos ) ] ) );
if( pos == 0 )
break;
}
MPI_CHK( ecp_jac_to_aff( grp, R, &Q[0] ) );
cleanup:
ecp_ptjac_free( &Q[0] ); ecp_ptjac_free( &Q[1] );
return( ret );
}
#if defined(POLARSSL_SELF_TEST)
/*
* Checkup routine
*/
int ecp_self_test( int verbose )
{
return( verbose++ );
}
#endif
#endif
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