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src.rivoreo.one / security / mbedtls / 855560758cfc8b844f4b4dbb69b56e6bfe8793dc / . / library / ecp.c
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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"
#include <limits.h>
#if defined(POLARSSL_SELF_TEST)
/*
* Counts of point addition and doubling operations.
* Used to test resistance of point multiplication to SPA/timing attacks.
*/
unsigned long add_count, dbl_count;
#endif
/*
* Initialize (the components of) a point
*/
void ecp_point_init( ecp_point *pt )
{
if( pt == NULL )
return;
mpi_init( &pt->X );
mpi_init( &pt->Y );
mpi_init( &pt->Z );
}
/*
* Initialize (the components of) a group
*/
void ecp_group_init( ecp_group *grp )
{
if( grp == NULL )
return;
mpi_init( &grp->P );
mpi_init( &grp->B );
ecp_point_init( &grp->G );
mpi_init( &grp->N );
grp->pbits = 0;
grp->nbits = 0;
grp->modp = NULL;
}
/*
* Unallocate (the components of) a point
*/
void ecp_point_free( ecp_point *pt )
{
if( pt == NULL )
return;
mpi_free( &( pt->X ) );
mpi_free( &( pt->Y ) );
mpi_free( &( pt->Z ) );
}
/*
* Unallocate (the components of) a group
*/
void ecp_group_free( ecp_group *grp )
{
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
*/
int ecp_set_zero( ecp_point *pt )
{
int ret;
MPI_CHK( mpi_lset( &pt->X , 1 ) );
MPI_CHK( mpi_lset( &pt->Y , 1 ) );
MPI_CHK( mpi_lset( &pt->Z , 0 ) );
cleanup:
return( ret );
}
/*
* Copy the contents of Q into P
*/
int ecp_copy( ecp_point *P, const ecp_point *Q )
{
int ret;
MPI_CHK( mpi_copy( &P->X, &Q->X ) );
MPI_CHK( mpi_copy( &P->Y, &Q->Y ) );
MPI_CHK( mpi_copy( &P->Z, &Q->Z ) );
cleanup:
return( ret );
}
/*
* Import a non-zero point from ASCII strings
*/
int ecp_point_read_string( ecp_point *P, int radix,
const char *x, const char *y )
{
int ret;
MPI_CHK( mpi_read_string( &P->X, radix, x ) );
MPI_CHK( mpi_read_string( &P->Y, radix, y ) );
MPI_CHK( mpi_lset( &P->Z, 1 ) );
cleanup:
return( ret );
}
/*
* 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;
MPI_CHK( mpi_read_string( &grp->P, radix, p ) );
MPI_CHK( mpi_read_string( &grp->B, radix, b ) );
MPI_CHK( ecp_point_read_string( &grp->G, radix, gx, gy ) );
MPI_CHK( mpi_read_string( &grp->N, radix, n ) );
grp->pbits = mpi_msb( &grp->P );
grp->nbits = mpi_msb( &grp->N );
cleanup:
return( ret );
}
/*
* Wrapper around fast quasi-modp functions, with fall-back to mpi_mod_mpi.
* See the documentation of struct ecp_group.
*/
static int ecp_modp( mpi *N, const ecp_group *grp )
{
int ret;
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 );
}
/*
* 192 bits in terms of t_uint
*/
#define P192_SIZE_INT ( 192 / CHAR_BIT / sizeof( t_uint ) )
/*
* Table to get S1, S2, S3 of FIPS 186-3 D.2.1:
* -1 means let this chunk be 0
* a positive value i means A_i.
*/
#define P192_CHUNKS 3
#define P192_CHUNK_CHAR ( 64 / CHAR_BIT )
#define P192_CHUNK_INT ( P192_CHUNK_CHAR / sizeof( t_uint ) )
const signed char p192_tbl[][P192_CHUNKS] = {
{ -1, 3, 3 }, /* S1 */
{ 4, 4, -1 }, /* S2 */
{ 5, 5, 5 }, /* S3 */
};
/*
* Fast quasi-reduction modulo p192 (FIPS 186-3 D.2.1)
*/
static int ecp_mod_p192( mpi *N )
{
int ret;
unsigned char i, j, offset;
signed char chunk;
mpi tmp, acc;
t_uint tmp_p[P192_SIZE_INT], acc_p[P192_SIZE_INT + 1];
tmp.s = 1;
tmp.n = sizeof( tmp_p ) / sizeof( tmp_p[0] );
tmp.p = tmp_p;
acc.s = 1;
acc.n = sizeof( acc_p ) / sizeof( acc_p[0] );
acc.p = acc_p;
MPI_CHK( mpi_grow( N, P192_SIZE_INT * 2 ) );
/*
* acc = T
*/
memset( acc_p, 0, sizeof( acc_p ) );
memcpy( acc_p, N->p, P192_CHUNK_CHAR * P192_CHUNKS );
for( i = 0; i < sizeof( p192_tbl ) / sizeof( p192_tbl[0] ); i++)
{
/*
* tmp = S_i
*/
memset( tmp_p, 0, sizeof( tmp_p ) );
for( j = 0, offset = P192_CHUNKS - 1; j < P192_CHUNKS; j++, offset-- )
{
chunk = p192_tbl[i][j];
if( chunk >= 0 )
memcpy( tmp_p + offset * P192_CHUNK_INT,
N->p + chunk * P192_CHUNK_INT,
P192_CHUNK_CHAR );
}
/*
* acc += tmp
*/
MPI_CHK( mpi_add_abs( &acc, &acc, &tmp ) );
}
MPI_CHK( mpi_copy( N, &acc ) );
cleanup:
return( ret );
}
/*
* Size of p521 in terms of t_uint
*/
#define P521_SIZE_INT ( 521 / CHAR_BIT / sizeof( t_uint ) + 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)
*/
static int ecp_mod_p521( mpi *N )
{
int ret;
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 );
}
/*
* Domain parameters for secp192r1
*/
#define SECP192R1_P \
"FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFFFFFFFFFFFF"
#define SECP192R1_B \
"64210519E59C80E70FA7E9AB72243049FEB8DEECC146B9B1"
#define SECP192R1_GX \
"188DA80EB03090F67CBF20EB43A18800F4FF0AFD82FF1012"
#define SECP192R1_GY \
"07192B95FFC8DA78631011ED6B24CDD573F977A11E794811"
#define SECP192R1_N \
"FFFFFFFFFFFFFFFFFFFFFFFF99DEF836146BC9B1B4D22831"
/*
* Domain parameters for secp224r1
*/
#define SECP224R1_P \
"FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF000000000000000000000001"
#define SECP224R1_B \
"B4050A850C04B3ABF54132565044B0B7D7BFD8BA270B39432355FFB4"
#define SECP224R1_GX \
"B70E0CBD6BB4BF7F321390B94A03C1D356C21122343280D6115C1D21"
#define SECP224R1_GY \
"BD376388B5F723FB4C22DFE6CD4375A05A07476444D5819985007E34"
#define SECP224R1_N \
"FFFFFFFFFFFFFFFFFFFFFFFFFFFF16A2E0B8F03E13DD29455C5C2A3D"
/*
* Domain parameters for secp256r1
*/
#define SECP256R1_P \
"FFFFFFFF00000001000000000000000000000000FFFFFFFFFFFFFFFFFFFFFFFF"
#define SECP256R1_B \
"5AC635D8AA3A93E7B3EBBD55769886BC651D06B0CC53B0F63BCE3C3E27D2604B"
#define SECP256R1_GX \
"6B17D1F2E12C4247F8BCE6E563A440F277037D812DEB33A0F4A13945D898C296"
#define SECP256R1_GY \
"4FE342E2FE1A7F9B8EE7EB4A7C0F9E162BCE33576B315ECECBB6406837BF51F5"
#define SECP256R1_N \
"FFFFFFFF00000000FFFFFFFFFFFFFFFFBCE6FAADA7179E84F3B9CAC2FC632551"
/*
* Domain parameters for secp384r1
*/
#define SECP384R1_P \
"FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF" \
"FFFFFFFFFFFFFFFEFFFFFFFF0000000000000000FFFFFFFF"
#define SECP384R1_B \
"B3312FA7E23EE7E4988E056BE3F82D19181D9C6EFE814112" \
"0314088F5013875AC656398D8A2ED19D2A85C8EDD3EC2AEF"
#define SECP384R1_GX \
"AA87CA22BE8B05378EB1C71EF320AD746E1D3B628BA79B98" \
"59F741E082542A385502F25DBF55296C3A545E3872760AB7"
#define SECP384R1_GY \
"3617DE4A96262C6F5D9E98BF9292DC29F8F41DBD289A147C" \
"E9DA3113B5F0B8C00A60B1CE1D7E819D7A431D7C90EA0E5F"
#define SECP384R1_N \
"FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF" \
"C7634D81F4372DDF581A0DB248B0A77AECEC196ACCC52973"
/*
* Domain parameters for secp521r1
*/
#define SECP521R1_P \
"000001FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF" \
"FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF" \
"FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF"
#define SECP521R1_B \
"00000051953EB9618E1C9A1F929A21A0B68540EEA2DA725B" \
"99B315F3B8B489918EF109E156193951EC7E937B1652C0BD" \
"3BB1BF073573DF883D2C34F1EF451FD46B503F00"
#define SECP521R1_GX \
"000000C6858E06B70404E9CD9E3ECB662395B4429C648139" \
"053FB521F828AF606B4D3DBAA14B5E77EFE75928FE1DC127" \
"A2FFA8DE3348B3C1856A429BF97E7E31C2E5BD66"
#define SECP521R1_GY \
"0000011839296A789A3BC0045C8A5FB42C7D1BD998F54449" \
"579B446817AFBD17273E662C97EE72995EF42640C550B901" \
"3FAD0761353C7086A272C24088BE94769FD16650"
#define SECP521R1_N \
"000001FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF" \
"FFFFFFFFFFFFFFFFFFFFFFFA51868783BF2F966B7FCC0148" \
"F709A5D03BB5C9B8899C47AEBB6FB71E91386409"
/*
* 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:
grp->modp = ecp_mod_p192;
return( ecp_group_read_string( grp, 16,
SECP192R1_P, SECP192R1_B,
SECP192R1_GX, SECP192R1_GY, SECP192R1_N ) );
case POLARSSL_ECP_DP_SECP224R1:
return( ecp_group_read_string( grp, 16,
SECP224R1_P, SECP224R1_B,
SECP224R1_GX, SECP224R1_GY, SECP224R1_N ) );
case POLARSSL_ECP_DP_SECP256R1:
return( ecp_group_read_string( grp, 16,
SECP256R1_P, SECP256R1_B,
SECP256R1_GX, SECP256R1_GY, SECP256R1_N ) );
case POLARSSL_ECP_DP_SECP384R1:
return( ecp_group_read_string( grp, 16,
SECP384R1_P, SECP384R1_B,
SECP384R1_GX, SECP384R1_GY, SECP384R1_N ) );
case POLARSSL_ECP_DP_SECP521R1:
grp->modp = ecp_mod_p521;
return( ecp_group_read_string( grp, 16,
SECP521R1_P, SECP521R1_B,
SECP521R1_GX, SECP521R1_GY, SECP521R1_N ) );
}
return( POLARSSL_ERR_ECP_GENERIC );
}
/*
* Fast mod-p functions expect their argument to be in the 0..p^2 range.
*
* In order to guarantee that, we need to ensure that operands of
* mpi_mul_mpi are in the 0..p range. So, after each operation we will
* bring the result back to this range.
*
* The following macros are shortcuts for doing that.
*/
/*
* Reduce a mpi mod p in-place, general case, to use after mpi_mul_mpi
*/
#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 ) )
/*
* Normalize jacobian coordinates so that Z == 0 || Z == 1 (GECC 3.2.1)
*/
static int ecp_normalize( const ecp_group *grp, ecp_point *pt )
{
int ret;
mpi Zi, ZZi, T;
if( mpi_cmp_int( &pt->Z, 0 ) == 0 )
return( 0 );
mpi_init( &Zi ); mpi_init( &ZZi ); mpi_init( &T );
/*
* X = X / Z^2 mod p
*/
MPI_CHK( mpi_inv_mod( &Zi, &pt->Z, &grp->P ) );
MPI_CHK( mpi_mul_mpi( &ZZi, &Zi, &Zi ) ); MOD_MUL( ZZi );
MPI_CHK( mpi_mul_mpi( &pt->X, &pt->X, &ZZi ) ); MOD_MUL( pt->X );
/*
* Y = Y / Z^3 mod p
*/
MPI_CHK( mpi_mul_mpi( &pt->Y, &pt->Y, &ZZi ) ); MOD_MUL( pt->Y );
MPI_CHK( mpi_mul_mpi( &pt->Y, &pt->Y, &Zi ) ); MOD_MUL( pt->Y );
/*
* Z = 1
*/
MPI_CHK( mpi_lset( &pt->Z, 1 ) );
cleanup:
mpi_free( &Zi ); mpi_free( &ZZi ); 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_point *R,
const ecp_point *P )
{
int ret;
mpi T1, T2, T3, X, Y, Z;
#if defined(POLARSSL_SELF_TEST)
dbl_count++;
#endif
if( mpi_cmp_int( &P->Z, 0 ) == 0 )
return( ecp_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_mul_int( &Y, &P->Y, 2 ) ); 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_mul_int( &T1, &T3, 2 ) ); 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 or subtraction: R = P + Q or 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.
*
* If sign >= 0, perform addition, otherwise perform subtraction,
* taking advantage of the fact that, for Q != 0, we have
* -Q = (Q.X, -Q.Y, Q.Z)
*/
static int ecp_add_mixed( const ecp_group *grp, ecp_point *R,
const ecp_point *P, const ecp_point *Q,
signed char sign )
{
int ret;
mpi T1, T2, T3, T4, X, Y, Z;
#if defined(POLARSSL_SELF_TEST)
add_count++;
#endif
/*
* Trivial cases: P == 0 or Q == 0
* (Check Q first, so that we know Q != 0 when we compute -Q.)
*/
if( mpi_cmp_int( &Q->Z, 0 ) == 0 )
return( ecp_copy( R, P ) );
if( mpi_cmp_int( &P->Z, 0 ) == 0 )
{
ret = ecp_copy( R, Q );
/*
* -R.Y mod P = P - R.Y unless R.Y == 0
*/
if( ret == 0 && sign < 0)
if( mpi_cmp_int( &R->Y, 0 ) != 0 )
ret = mpi_sub_mpi( &R->Y, &grp->P, &R->Y );
return( ret );
}
/*
* Make sure Q coordinates are normalized
*/
if( mpi_cmp_int( &Q->Z, 1 ) != 0 )
return( POLARSSL_ERR_ECP_GENERIC );
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 );
/*
* For subtraction, -Q.Y should have been used instead of Q.Y,
* so we replace T2 by -T2, which is P - T2 mod P
*/
if( sign < 0 )
{
MPI_CHK( mpi_sub_mpi( &T2, &grp->P, &T2 ) );
MOD_SUB( 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_set_zero( R );
goto cleanup;
}
}
MPI_CHK( mpi_mul_mpi( &Z, &P->Z, &T1 ) ); MOD_MUL( Z );
MPI_CHK( mpi_mul_mpi( &T3, &T1, &T1 ) ); MOD_MUL( T3 );
MPI_CHK( mpi_mul_mpi( &T4, &T3, &T1 ) ); MOD_MUL( T4 );
MPI_CHK( mpi_mul_mpi( &T3, &T3, &P->X ) ); MOD_MUL( T3 );
MPI_CHK( mpi_mul_int( &T1, &T3, 2 ) ); MOD_ADD( T1 );
MPI_CHK( mpi_mul_mpi( &X, &T2, &T2 ) ); MOD_MUL( X );
MPI_CHK( mpi_sub_mpi( &X, &X, &T1 ) ); MOD_SUB( X );
MPI_CHK( mpi_sub_mpi( &X, &X, &T4 ) ); MOD_SUB( X );
MPI_CHK( mpi_sub_mpi( &T3, &T3, &X ) ); MOD_SUB( T3 );
MPI_CHK( mpi_mul_mpi( &T3, &T3, &T2 ) ); MOD_MUL( T3 );
MPI_CHK( mpi_mul_mpi( &T4, &T4, &P->Y ) ); MOD_MUL( T4 );
MPI_CHK( mpi_sub_mpi( &Y, &T3, &T4 ) ); MOD_SUB( Y );
MPI_CHK( mpi_copy( &R->X, &X ) );
MPI_CHK( mpi_copy( &R->Y, &Y ) );
MPI_CHK( mpi_copy( &R->Z, &Z ) );
cleanup:
mpi_free( &T1 ); mpi_free( &T2 ); mpi_free( &T3 ); mpi_free( &T4 );
mpi_free( &X ); mpi_free( &Y ); mpi_free( &Z );
return( ret );
}
/*
* Addition: R = P + Q, result's coordinates normalized
*/
int ecp_add( const ecp_group *grp, ecp_point *R,
const ecp_point *P, const ecp_point *Q )
{
int ret;
MPI_CHK( ecp_add_mixed( grp, R, P, Q , 1 ) );
MPI_CHK( ecp_normalize( grp, R ) );
cleanup:
return( ret );
}
/*
* Subtraction: R = P - Q, result's coordinates normalized
*/
int ecp_sub( const ecp_group *grp, ecp_point *R,
const ecp_point *P, const ecp_point *Q )
{
int ret;
MPI_CHK( ecp_add_mixed( grp, R, P, Q, -1 ) );
MPI_CHK( ecp_normalize( grp, R ) );
cleanup:
return( ret );
}
/*
* Compute a modified width-w non-adjacent form (NAF) of a number,
* with a fixed pattern for resistance to SPA/timing attacks,
* see <http://rd.springer.com/chapter/10.1007/3-540-36563-X_23>.
* (The resulting multiplication algorithm can also been seen as a
* modification of 2^w-ary multiplication, with signed coefficients,
* all of them odd.)
*
* Input:
* m must be an odd positive mpi less than w * k bits long
* x must be an array of k elements
* w must be less than a certain maximum (currently 8)
*
* The result is a sequence x[0], ..., x[k-1] with x[i] in the range
* - 2^(width - 1) .. 2^(width - 1) - 1 such that
* m = (2 * x[0] + 1) + 2^width * (2 * x[1] + 1) + ...
* + 2^((k-1) * width) * (2 * x[k-1] + 1)
*
* Compared to "Algorithm SPA-resistant Width-w NAF with Odd Scalar"
* p. 335 of the cited reference, here we return only u, not d_w since
* it is known that the other d_w[j] will be 0. Moreover, the returned
* string doesn't actually store u_i but x_i = u_i / 2 since it is known
* that u_i is odd. Also, since we always select a positive value for d
* mod 2^w, we don't need to check the sign of u[i-1] when the reference
* does. Finally, there is an off-by-one error in the reference: the
* last index should be k-1, not k.
*/
static int ecp_w_naf_fixed( signed char x[], size_t k, unsigned char w,
const mpi *m )
{
int ret;
unsigned int i, u, mask, carry;
mpi M;
mpi_init( &M );
MPI_CHK( mpi_copy( &M, m ) );
mask = ( 1 << w ) - 1;
carry = 1 << ( w - 1 );
for( i = 0; i < k; i++ )
{
u = M.p[0] & mask;
if( ( u & 1 ) == 0 && i > 0 )
x[i - 1] -= carry;
x[i] = u >> 1;
mpi_shift_r( &M, w );
}
/*
* We should have consumed all the bits now
*/
if( mpi_cmp_int( &M, 0 ) != 0 )
ret = POLARSSL_ERR_ECP_GENERIC;
cleanup:
mpi_free( &M );
return( ret );
}
/*
* Integer multiplication: R = m * P (GECC 5.7, SPA-resistant)
*/
int ecp_mul( const ecp_group *grp, ecp_point *R,
const mpi *m, const ecp_point *P )
{
int ret, cmp;
size_t pos;
ecp_point Q[2];
cmp = mpi_cmp_int( m, 0 );
if( cmp < 0 )
return( POLARSSL_ERR_ECP_GENERIC );
/*
* The general method works only for m != 0
*/
if( cmp == 0 ) {
return( ecp_set_zero( R ) );
}
ecp_point_init( &Q[0] ); ecp_point_init( &Q[1] );
MPI_CHK( ecp_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, 1 ) );
MPI_CHK( ecp_copy( &Q[0], &Q[ mpi_get_bit( m, pos ) ] ) );
if( pos == 0 )
break;
}
MPI_CHK( ecp_copy( R, &Q[0] ) );
MPI_CHK( ecp_normalize( grp, R ) );
cleanup:
ecp_point_free( &Q[0] ); ecp_point_free( &Q[1] );
return( ret );
}
#if defined(POLARSSL_SELF_TEST)
/*
* Checkup routine
*/
int ecp_self_test( int verbose )
{
int ret;
size_t i;
int j, jj;
ecp_group grp;
ecp_point R;
mpi m;
unsigned long add_c_prev, dbl_c_prev;
char *exponents[] =
{
"400000000000000000000000000000000000000000000000",
"7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF",
"555555555555555555555555555555555555555555555555",
"5EA6F389A38B8BC81E767753B15AA5569E1782E30ABE7D25",
/* "000000000000000000000000000000000000000000000010", TODO */
};
signed char x[3];
ecp_group_init( &grp );
ecp_point_init( &R );
mpi_init( &m );
if( verbose != 0 )
printf( " ECP test #0 (naf): " );
for( j = 1; j < 32; j += 2 )
{
mpi_lset( &m, j );
x[0] = x[1] = x[2] = 0;
MPI_CHK( ecp_w_naf_fixed( x, 3, 2, &m ) );
jj = ( 2 * x[0] + 1 ) + 4 * ( 2 * x[1] + 1 ) + 16 * ( 2 * x[2] + 1 );
if( j != jj ||
x[0] > 1 || x[0] < -2 ||
x[1] > 1 || x[1] < -2 ||
x[2] > 1 || x[2] < -2 )
{
if( verbose != 0 )
printf( "failed\n" );
printf( "%i != %i (%i, %i, %i)\n", j, jj, x[0], x[1], x[2] );
ret = 1;
goto cleanup;
}
x[0] = x[1] = x[2] = 0;
MPI_CHK( ecp_w_naf_fixed( x, 2, 3, &m ) );
jj = ( 2 * x[0] + 1 ) + 8 * ( 2 * x[1] + 1 );
if( j != jj ||
x[0] > 3 || x[0] < -4 ||
x[1] > 3 || x[1] < -4 ||
x[2] != 0 )
{
if( verbose != 0 )
printf( "failed\n" );
printf( "%i != %i (%i, %i)\n", j, jj, x[0], x[1] );
ret = 1;
goto cleanup;
}
}
if( verbose != 0 )
printf( "passed\n" );
MPI_CHK( ecp_use_known_dp( &grp, POLARSSL_ECP_DP_SECP192R1 ) );
if( verbose != 0 )
printf( " ECP test #1 (SPA resistance): " );
add_count = 0;
dbl_count = 0;
MPI_CHK( mpi_read_string( &m, 16, exponents[0] ) );
MPI_CHK( ecp_mul( &grp, &R, &m, &grp.G ) );
for( i = 1; i < sizeof( exponents ) / sizeof( exponents[0] ); i++ )
{
add_c_prev = add_count;
dbl_c_prev = dbl_count;
add_count = 0;
dbl_count = 0;
MPI_CHK( mpi_read_string( &m, 16, exponents[i] ) );
MPI_CHK( ecp_mul( &grp, &R, &m, &grp.G ) );
if( add_count != add_c_prev || dbl_count != dbl_c_prev )
{
if( verbose != 0 )
printf( "failed (%zu)\n", i );
ret = 1;
goto cleanup;
}
}
if( verbose != 0 )
printf( "passed\n" );
cleanup:
if( ret < 0 && verbose != 0 )
printf( "Unexpected error, return code = %08X\n", ret );
ecp_group_free( &grp );
ecp_point_free( &R );
mpi_free( &m );
if( verbose != 0 )
printf( "\n" );
return( ret );
}
#endif
#endif