First version of ecp_mul_comb()
diff --git a/library/ecp.c b/library/ecp.c
index 3a075c4..7aa98c0 100644
--- a/library/ecp.c
+++ b/library/ecp.c
@@ -41,6 +41,11 @@
* for elliptic curve cryptosystems. In : Cryptographic Hardware and
* Embedded Systems. Springer Berlin Heidelberg, 1999. p. 292-302.
* <http://link.springer.com/chapter/10.1007/3-540-48059-5_25>
+ *
+ * [3] HEDABOU, Mustapha, PINEL, Pierre, et BÉNÉTEAU, Lucien. A comb method to
+ * render ECC resistant against Side Channel Attacks. IACR Cryptology
+ * ePrint Archive, 2004, vol. 2004, p. 342.
+ * <http://eprint.iacr.org/2004/342.pdf>
*/
#include "polarssl/config.h"
@@ -902,7 +907,7 @@
}
/*
- * Normalize jacobian coordinates of an array of points,
+ * 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.)
@@ -911,14 +916,14 @@
* This should never happen, see choice of w in ecp_mul().
*/
static int ecp_normalize_many( const ecp_group *grp,
- ecp_point T[], size_t t_len )
+ ecp_point *T[], size_t t_len )
{
int ret;
size_t i;
mpi *c, u, Zi, ZZi;
if( t_len < 2 )
- return( ecp_normalize( grp, T ) );
+ return( ecp_normalize( grp, *T ) );
if( ( c = (mpi *) polarssl_malloc( t_len * sizeof( mpi ) ) ) == NULL )
return( POLARSSL_ERR_ECP_MALLOC_FAILED );
@@ -930,10 +935,10 @@
/*
* c[i] = Z_0 * ... * Z_i
*/
- MPI_CHK( mpi_copy( &c[0], &T[0].Z ) );
+ MPI_CHK( mpi_copy( &c[0], &T[0]->Z ) );
for( i = 1; i < t_len; i++ )
{
- MPI_CHK( mpi_mul_mpi( &c[i], &c[i-1], &T[i].Z ) );
+ MPI_CHK( mpi_mul_mpi( &c[i], &c[i-1], &T[i]->Z ) );
MOD_MUL( c[i] );
}
@@ -953,18 +958,18 @@
}
else
{
- MPI_CHK( mpi_mul_mpi( &Zi, &u, &c[i-1] ) ); MOD_MUL( Zi );
- MPI_CHK( mpi_mul_mpi( &u, &u, &T[i].Z ) ); MOD_MUL( u );
+ MPI_CHK( mpi_mul_mpi( &Zi, &u, &c[i-1] ) ); MOD_MUL( Zi );
+ MPI_CHK( mpi_mul_mpi( &u, &u, &T[i]->Z ) ); MOD_MUL( u );
}
/*
* proceed as in normalize()
*/
- MPI_CHK( mpi_mul_mpi( &ZZi, &Zi, &Zi ) ); MOD_MUL( ZZi );
- MPI_CHK( mpi_mul_mpi( &T[i].X, &T[i].X, &ZZi ) ); MOD_MUL( T[i].X );
- MPI_CHK( mpi_mul_mpi( &T[i].Y, &T[i].Y, &ZZi ) ); MOD_MUL( T[i].Y );
- MPI_CHK( mpi_mul_mpi( &T[i].Y, &T[i].Y, &Zi ) ); MOD_MUL( T[i].Y );
- MPI_CHK( mpi_lset( &T[i].Z, 1 ) );
+ MPI_CHK( mpi_mul_mpi( &ZZi, &Zi, &Zi ) ); MOD_MUL( ZZi );
+ MPI_CHK( mpi_mul_mpi( &T[i]->X, &T[i]->X, &ZZi ) ); MOD_MUL( T[i]->X );
+ MPI_CHK( mpi_mul_mpi( &T[i]->Y, &T[i]->Y, &ZZi ) ); MOD_MUL( T[i]->Y );
+ MPI_CHK( mpi_mul_mpi( &T[i]->Y, &T[i]->Y, &Zi ) ); MOD_MUL( T[i]->Y );
+ MPI_CHK( mpi_lset( &T[i]->Z, 1 ) );
if( i == 0 )
break;
@@ -1250,6 +1255,7 @@
int ret;
size_t i;
ecp_point PP;
+ ecp_point *TT[ 1 << ( POLARSSL_ECP_WINDOW_SIZE - 1 ) ];
ecp_point_init( &PP );
@@ -1261,9 +1267,11 @@
MPI_CHK( ecp_add_mixed( grp, &T[i], &T[i-1], &PP, +1 ) );
/*
- * T[0] = P already has normalized coordinates
+ * T[0] = P already has normalized coordinates, normalize others
*/
- MPI_CHK( ecp_normalize_many( grp, T + 1, t_len - 1 ) );
+ for( i = 1; i < t_len; i++ )
+ TT[i-1] = &T[i];
+ MPI_CHK( ecp_normalize_many( grp, TT, t_len - 1 ) );
cleanup:
@@ -1342,9 +1350,9 @@
* countermeasure against DPA in 5.3 of [2] (with the obvious adaptation that
* we use jacobian coordinates, not standard projective coordinates).
*/
-int ecp_mul( ecp_group *grp, ecp_point *R,
- const mpi *m, const ecp_point *P,
- int (*f_rng)(void *, unsigned char *, size_t), void *p_rng )
+int ecp_mul_wnaf( ecp_group *grp, ecp_point *R,
+ const mpi *m, const ecp_point *P,
+ int (*f_rng)(void *, unsigned char *, size_t), void *p_rng )
{
int ret;
unsigned char w, m_is_odd, p_eq_g;
@@ -1504,6 +1512,276 @@
}
/*
+ * Compute the representation of m that will be used with the comb method.
+ *
+ * The basic comb method is described in GECC 3.44 for example. We use a
+ * modified version [3] that provides resistance to SPA by avoiding zero
+ * digits in the representation. We represent (K_i, s_i) from the paper as a
+ * single signed char.
+ *
+ * Calling conventions:
+ * - x is an array of size d
+ * - w is the size, ie number of teeth, of the comb
+ * - m is the MPI, expected to be odd and such that, if l = bitlength(m):
+ * ceil( l / w ) <= d (these two assumptions are not checked, an incorrect
+ * result my be returned if they are not satisfied)
+ */
+static void ecp_comb_fixed( signed char x[], size_t d,
+ unsigned char w, const mpi *m )
+{
+ size_t i, j;
+
+ memset( x, 0, d );
+
+ /* For x[0] use the classical comb value without adjustement */
+ for( j = 0; j < w; j++ )
+ x[0] |= mpi_get_bit( m, d * j ) << j;
+
+ for( i = 1; i < d; i++ )
+ {
+ /* Get the classical comb value */
+ for( j = 0; j < w; j++ )
+ x[i] |= mpi_get_bit( m, i + d * j ) << j;
+
+ /* Adjust if it's zero */
+ if( x[i] == 0 )
+ {
+ x[i] = x[i-1];
+ x[i-1] *= -1;
+ }
+ }
+}
+
+/*
+ * Precompute points for the comb method
+ *
+ * If i = i_{w-1} ... i_0 is the binary representation of i, then
+ * T[i-1] = i_{w-1} 2^{(w-1)d} P + ... + i_1 2^d P + i_0 P
+ *
+ * T must be able to hold at least 2^w - 1 elements
+ */
+static int ecp_precompute_comb( const ecp_group *grp,
+ ecp_point T[], const ecp_point *P,
+ unsigned char w, size_t d )
+{
+ int ret;
+ unsigned char i, mask;
+ size_t j, t_len = ( 1U << w ) - 1;
+ ecp_point *cur, *TT[t_len - 1];
+
+ /*
+ * Compute the 2^{di}
+ */
+ MPI_CHK( ecp_copy( &T[0], P ) );
+
+ for( i = 1; i < w; i++ )
+ {
+ cur = T + ( 1 << i ) - 1;
+ ecp_copy( cur, T + ( 1 << (i-1) ) - 1 );
+ for( j = 0; j < d; j++ )
+ MPI_CHK( ecp_double_jac( grp, cur, cur ) );
+ TT[i-1] = cur;
+ }
+
+ /* P already normalized, so w - 1 points to do */
+ ecp_normalize_many( grp, TT, w - 1);
+
+ /*
+ * Compute the remaining ones using the minimal number of additions
+ */
+ j = 0;
+ for( i = 3; i < (1U << w); i++ )
+ {
+ if( T[i - 1].X.p != NULL )
+ continue;
+
+ /* Find the least significant non-zero bit of the index */
+ for( mask = 1; mask != 0; mask <<=1 )
+ if( ( i & mask ) != 0 )
+ break;
+
+ /* Use the previously computed values */
+ ecp_add_mixed( grp, &T[i - 1], &T[i - mask - 1], &T[mask - 1], +1 );
+
+ /* Register for normalisation */
+ TT[j++] = &T[i - 1];
+ }
+
+ ecp_normalize_many( grp, TT, j );
+
+cleanup:
+ return( ret );
+}
+
+/*
+ * Select precomputed point: R = sign(i) * T[ abs(i) ]
+ */
+static int ecp_select_comb( const ecp_group *grp, ecp_point *R,
+ const ecp_point T[], signed char i )
+{
+ int ret;
+
+ if( i > 0 )
+ return( ecp_copy( R, &T[i - 1] ) );
+
+ MPI_CHK( ecp_copy( R, &T[-i - 1] ) );
+
+ /*
+ * -R = (R.X, -R.Y, R.Z), and
+ * -R.Y mod P = P - R.Y unless R.Y == 0
+ */
+ if( mpi_cmp_int( &R->Y, 0 ) != 0 )
+ MPI_CHK( mpi_sub_mpi( &R->Y, &grp->P, &R->Y ) );
+
+cleanup:
+ return( ret );
+}
+
+/*
+ * Core multiplication algorithm for the (modified) comb method.
+ * This part is actually common with the basic comb method (GECC 3.44)
+ */
+static int ecp_mul_comb_core( const ecp_group *grp, ecp_point *R,
+ const ecp_point T[], const signed char x[],
+ size_t d )
+{
+ int ret;
+ ecp_point Txi;
+ size_t i;
+
+ ecp_point_init( &Txi );
+
+ /* Avoid useless doubling/addition of 0 by better initialisation */
+ i = d - 1;
+ MPI_CHK( ecp_select_comb( grp, R, T, x[i] ) );
+
+ while( i-- != 0 )
+ {
+ MPI_CHK( ecp_double_jac( grp, R, R ) );
+ MPI_CHK( ecp_select_comb( grp, &Txi, T, x[i] ) );
+ MPI_CHK( ecp_add_mixed( grp, R, R, &Txi, +1 ) );
+ }
+
+cleanup:
+ ecp_point_free( &Txi );
+
+ return( ret );
+}
+
+/*
+ * Multiplication using the comb method, WIP
+ */
+int ecp_mul_comb( ecp_group *grp, ecp_point *R,
+ const mpi *m, const ecp_point *P,
+ int (*f_rng)(void *, unsigned char *, size_t), void *p_rng )
+{
+ int ret;
+ unsigned char w, m_is_odd, p_eq_g;
+ size_t pre_len, d, i;
+ signed char k[100]; // TODO
+ ecp_point Q, *T = NULL, S[2];
+ mpi M;
+
+ (void) f_rng;
+ (void) p_rng; // TODO
+
+ if( mpi_cmp_int( m, 0 ) < 0 || mpi_msb( m ) > grp->nbits )
+ return( POLARSSL_ERR_ECP_BAD_INPUT_DATA );
+
+ mpi_init( &M );
+ ecp_point_init( &Q );
+ ecp_point_init( &S[0] );
+ ecp_point_init( &S[1] );
+
+ /*
+ * Check if P == G
+ */
+ p_eq_g = ( mpi_cmp_int( &P->Z, 1 ) == 0 &&
+ mpi_cmp_mpi( &P->Y, &grp->G.Y ) == 0 &&
+ mpi_cmp_mpi( &P->X, &grp->G.X ) == 0 );
+
+ /* TODO: adjust exact value */
+ w = grp->nbits >= 192 ? 5 : 2;
+
+ pre_len = 1U << w;
+ d = ( grp->nbits + w - 1 ) / w;
+
+ /*
+ * Prepare precomputed points: if P == G we want to
+ * use grp->T if already initialized, or initiliaze it.
+ */
+ if( ! p_eq_g || grp->T == NULL )
+ {
+ T = (ecp_point *) polarssl_malloc( pre_len * sizeof( ecp_point ) );
+ if( T == NULL )
+ {
+ ret = POLARSSL_ERR_ECP_MALLOC_FAILED;
+ goto cleanup;
+ }
+
+ for( i = 0; i < pre_len; i++ )
+ ecp_point_init( &T[i] );
+
+ MPI_CHK( ecp_precompute_comb( grp, T, P, w, d ) );
+
+ if( p_eq_g )
+ {
+ grp->T = T;
+ grp->T_size = pre_len;
+ }
+ }
+ else
+ {
+ T = grp->T;
+
+ /* Should never happen, but we want to be extra sure */
+ if( pre_len != grp->T_size )
+ {
+ ret = POLARSSL_ERR_ECP_BAD_INPUT_DATA;
+ goto cleanup;
+ }
+ }
+
+ /*
+ * Make sure M is odd (M = m + 1 or M = m + 2)
+ * later we'll get m * P by subtracting P or 2 * P to M * P.
+ */
+ m_is_odd = ( mpi_get_bit( m, 0 ) == 1 );
+
+ MPI_CHK( mpi_copy( &M, m ) );
+ MPI_CHK( mpi_add_int( &M, &M, 1 + m_is_odd ) );
+
+ /*
+ * Go for comb multiplication, Q = M * P
+ */
+ ecp_comb_fixed( k, d, w, &M );
+ ecp_mul_comb_core( grp, &Q, T, k, d );
+
+ /*
+ * Now get m * P from M * P
+ */
+ MPI_CHK( ecp_copy( &S[0], P ) );
+ MPI_CHK( ecp_add( grp, &S[1], P, P ) );
+ MPI_CHK( ecp_sub( grp, R, &Q, &S[m_is_odd] ) );
+
+cleanup:
+
+ if( T != NULL && ! p_eq_g )
+ {
+ for( i = 0; i < pre_len; i++ )
+ ecp_point_free( &T[i] );
+ polarssl_free( T );
+ }
+
+ ecp_point_free( &S[1] );
+ ecp_point_free( &S[0] );
+ ecp_point_free( &Q );
+ mpi_free( &M );
+
+ return( ret );
+}
+
+/*
* Check that a point is valid as a public key (SEC1 3.2.3.1)
*/
int ecp_check_pubkey( const ecp_group *grp, const ecp_point *pt )