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Improve comb method (less precomputed points)

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Manuel Pégourié-Gonnard 2013-11-20 14:47:19 +01:00
parent d1c1ba90ca
commit 101a39f55f
1 changed files with 66 additions and 55 deletions
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121
library/ecp.c
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@ -1515,27 +1515,33 @@ cleanup:
* Compute the representation of m that will be used with the comb method. * 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 * 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 * modified version that provides resistance to SPA by avoiding zero
* digits in the representation. We represent (K_i, s_i) from the paper as a * digits in the representation as in [3]. We modify the method further by
* single signed char. * requiring that all K_i be odd, which has the small cost that our
* representation uses on more K, due to carries.
*
* Also, for the sake of compactness, only the seven low-order bits of x[i]
* are used to represent K_i, and the msb of x[i] encodes the the sign (s_i in
* the paper): it is set if and only if if s_i == -1;
* *
* Calling conventions: * Calling conventions:
* - x is an array of size d * - x is an array of size d + 1
* - w is the size, ie number of teeth, of the comb * - 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): * - m is the MPI, expected to be odd and such that bitlength(m) <= w * d
* ceil( l / w ) <= d (these two assumptions are not checked, an incorrect * (the result will be incorrect if these assumptions are not satisfied)
* result my be returned if they are not satisfied)
*/ */
static void ecp_comb_fixed( signed char x[], size_t d, static void ecp_comb_fixed( unsigned char x[], size_t d,
unsigned char w, const mpi *m ) unsigned char w, const mpi *m )
{ {
size_t i, j; size_t i, j;
unsigned char c, cc, adjust;
memset( x, 0, d ); memset( x, 0, d+1 );
/* For x[0] use the classical comb value without adjustement */ /* For x[0] use the classical comb value without adjustement */
for( j = 0; j < w; j++ ) for( j = 0; j < w; j++ )
x[0] |= mpi_get_bit( m, d * j ) << j; x[0] |= mpi_get_bit( m, d * j ) << j;
c = 0;
for( i = 1; i < d; i++ ) for( i = 1; i < d; i++ )
{ {
@ -1543,95 +1549,100 @@ static void ecp_comb_fixed( signed char x[], size_t d,
for( j = 0; j < w; j++ ) for( j = 0; j < w; j++ )
x[i] |= mpi_get_bit( m, i + d * j ) << j; x[i] |= mpi_get_bit( m, i + d * j ) << j;
/* Adjust if it's zero */ /* Add carry and update it */
if( x[i] == 0 ) cc = x[i] & c;
{ x[i] = x[i] ^ c;
x[i] = x[i-1]; c = cc;
x[i-1] *= -1;
} /* Make sure x[i] is odd, avoiding if-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;
} }
/* Finish with the carry */
x[i] = c;
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 comb method * Precompute points for the comb method
* *
* If i = i_{w-1} ... i_0 is the binary representation of i, then * If i = i_{w-1} ... i_1 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[i] = i_{w-1} 2^{(w-1)d} P + ... + i_1 2^d P + P
* *
* T must be able to hold at least 2^w - 1 elements * T must be able to hold at least 2^{w - 1} elements
*/ */
static int ecp_precompute_comb( const ecp_group *grp, static int ecp_precompute_comb( const ecp_group *grp,
ecp_point T[], const ecp_point *P, ecp_point T[], const ecp_point *P,
unsigned char w, size_t d ) unsigned char w, size_t d )
{ {
int ret; int ret;
unsigned char i, mask; unsigned char i, j, k;
size_t j, t_len = ( 1U << w ) - 1; ecp_point *cur, *TT[200]; // TODO
ecp_point *cur, *TT[t_len - 1];
/* /*
* Compute the 2^{di} * Set T[0] = P and
* T[2^{l-1}] = 2^{dl} P for l = 1 .. w-1 (this is not the final value)
*/ */
MPI_CHK( ecp_copy( &T[0], P ) ); MPI_CHK( ecp_copy( &T[0], P ) );
for( i = 1; i < w; i++ ) k = 0;
for( i = 1; i < ( 1U << (w-1) ); i <<= 1 )
{ {
cur = T + ( 1 << i ) - 1; cur = T + i;
ecp_copy( cur, T + ( 1 << (i-1) ) - 1 ); MPI_CHK( ecp_copy( cur, T + ( i >> 1 ) ) );
for( j = 0; j < d; j++ ) for( j = 0; j < d; j++ )
MPI_CHK( ecp_double_jac( grp, cur, cur ) ); MPI_CHK( ecp_double_jac( grp, cur, cur ) );
TT[i-1] = cur;
TT[k++] = cur;
} }
/* P already normalized, so w - 1 points to do */ ecp_normalize_many( grp, TT, k );
ecp_normalize_many( grp, TT, w - 1);
/* /*
* Compute the remaining ones using the minimal number of additions * Compute the remaining ones using the minimal number of additions
* Be careful to update T[2^l] only after using it!
*/ */
j = 0; k = 0;
for( i = 3; i < (1U << w); i++ ) for( i = 1; i < ( 1U << (w-1) ); i <<= 1 )
{ {
if( T[i - 1].X.p != NULL ) j = i;
continue; while( j-- )
{
/* Find the least significant non-zero bit of the index */ ecp_add_mixed( grp, &T[i + j], &T[j], &T[i], +1 );
for( mask = 1; mask != 0; mask <<=1 ) TT[k++] = &T[i + j];
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 ); ecp_normalize_many( grp, TT, k );
cleanup: cleanup:
return( ret ); return( ret );
} }
/* /*
* Select precomputed point: R = sign(i) * T[ abs(i) ] * Select precomputed point: R = sign(i) * T[ abs(i) / 2 ]
*/ */
static int ecp_select_comb( const ecp_group *grp, ecp_point *R, static int ecp_select_comb( const ecp_group *grp, ecp_point *R,
const ecp_point T[], signed char i ) const ecp_point T[], unsigned char i )
{ {
int ret; int ret;
if( i > 0 ) /* Ignore the "sign" bit */
return( ecp_copy( R, &T[i - 1] ) ); MPI_CHK( ecp_copy( R, &T[ ( i & 0x7Fu ) >> 1 ] ) );
MPI_CHK( ecp_copy( R, &T[-i - 1] ) );
/* /*
* -R = (R.X, -R.Y, R.Z), and * -R = (R.X, -R.Y, R.Z), and
* -R.Y mod P = P - R.Y unless R.Y == 0 * -R.Y mod P = P - R.Y unless R.Y == 0
*/ */
if( mpi_cmp_int( &R->Y, 0 ) != 0 ) if( ( i & 0x80 ) != 0 )
MPI_CHK( mpi_sub_mpi( &R->Y, &grp->P, &R->Y ) ); if( mpi_cmp_int( &R->Y, 0 ) != 0 )
MPI_CHK( mpi_sub_mpi( &R->Y, &grp->P, &R->Y ) );
cleanup: cleanup:
return( ret ); return( ret );
@ -1642,7 +1653,7 @@ cleanup:
* This part is actually common with the basic comb method (GECC 3.44) * 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, static int ecp_mul_comb_core( const ecp_group *grp, ecp_point *R,
const ecp_point T[], const signed char x[], const ecp_point T[], const unsigned char x[],
size_t d ) size_t d )
{ {
int ret; int ret;
@ -1652,7 +1663,7 @@ static int ecp_mul_comb_core( const ecp_group *grp, ecp_point *R,
ecp_point_init( &Txi ); ecp_point_init( &Txi );
/* Avoid useless doubling/addition of 0 by better initialisation */ /* Avoid useless doubling/addition of 0 by better initialisation */
i = d - 1; i = d;
MPI_CHK( ecp_select_comb( grp, R, T, x[i] ) ); MPI_CHK( ecp_select_comb( grp, R, T, x[i] ) );
while( i-- != 0 ) while( i-- != 0 )
@ -1678,7 +1689,7 @@ int ecp_mul_comb( ecp_group *grp, ecp_point *R,
int ret; int ret;
unsigned char w, m_is_odd, p_eq_g; unsigned char w, m_is_odd, p_eq_g;
size_t pre_len, d, i; size_t pre_len, d, i;
signed char k[100]; // TODO unsigned char k[200]; // TODO
ecp_point Q, *T = NULL, S[2]; ecp_point Q, *T = NULL, S[2];
mpi M; mpi M;