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mbedtls/library/ecp.c

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/*
* Elliptic curves over GF(p)
*
* Copyright (C) 2006-2013, 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
* RFC 4492 for the related TLS structures and constants
*/
#include "polarssl/config.h"
#if defined(POLARSSL_ECP_C)
#include "polarssl/ecp.h"
#include <limits.h>
#include <stdlib.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;
grp->id = 0;
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 );
}
/*
* Tell if a point is zero
*/
int ecp_is_zero( ecp_point *pt )
{
return( mpi_cmp_int( &pt->Z, 0 ) == 0 );
}
/*
* 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 );
}
/*
* Export a point into unsigned binary data (SEC1 2.3.3)
*/
int ecp_point_write_binary( const ecp_group *grp, const ecp_point *P,
int format, size_t *olen,
unsigned char *buf, size_t buflen )
{
int ret = 0;
size_t plen;
if( format != POLARSSL_ECP_PF_UNCOMPRESSED &&
format != POLARSSL_ECP_PF_COMPRESSED )
return( POLARSSL_ERR_ECP_BAD_INPUT_DATA );
/*
* Common case: P == 0
*/
if( mpi_cmp_int( &P->Z, 0 ) == 0 )
{
if( buflen < 1 )
return( POLARSSL_ERR_ECP_BUFFER_TOO_SMALL );
buf[0] = 0x00;
*olen = 1;
return( 0 );
}
plen = mpi_size( &grp->P );
if( format == POLARSSL_ECP_PF_UNCOMPRESSED )
{
*olen = 2 * plen + 1;
if( buflen < *olen )
return( POLARSSL_ERR_ECP_BUFFER_TOO_SMALL );
buf[0] = 0x04;
MPI_CHK( mpi_write_binary( &P->X, buf + 1, plen ) );
MPI_CHK( mpi_write_binary( &P->Y, buf + 1 + plen, plen ) );
}
else if( format == POLARSSL_ECP_PF_COMPRESSED )
{
*olen = plen + 1;
if( buflen < *olen )
return( POLARSSL_ERR_ECP_BUFFER_TOO_SMALL );
buf[0] = 0x02 + mpi_get_bit( &P->Y, 0 );
MPI_CHK( mpi_write_binary( &P->X, buf + 1, plen ) );
}
cleanup:
return( ret );
}
/*
* Import a point from unsigned binary data (SEC1 2.3.4)
*/
int ecp_point_read_binary( const ecp_group *grp, ecp_point *pt,
const unsigned char *buf, size_t ilen ) {
int ret;
size_t plen;
if( ilen == 1 && buf[0] == 0x00 )
return( ecp_set_zero( pt ) );
plen = mpi_size( &grp->P );
if( ilen != 2 * plen + 1 || buf[0] != 0x04 )
return( POLARSSL_ERR_ECP_BAD_INPUT_DATA );
MPI_CHK( mpi_read_binary( &pt->X, buf + 1, plen ) );
MPI_CHK( mpi_read_binary( &pt->Y, buf + 1 + plen, plen ) );
MPI_CHK( 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 ecp_tls_read_point( const ecp_group *grp, ecp_point *pt,
const unsigned char **buf, size_t buf_len )
{
unsigned char data_len;
const unsigned char *buf_start;
/*
* We must have at least two bytes (1 for length, at least of for data)
*/
if( buf_len < 2 )
return( POLARSSL_ERR_ECP_BAD_INPUT_DATA );
data_len = *(*buf)++;
if( data_len < 1 || data_len > buf_len - 1 )
return( POLARSSL_ERR_ECP_BAD_INPUT_DATA );
/*
* Save buffer start for read_binary and update buf
*/
buf_start = *buf;
*buf += data_len;
return 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 ecp_tls_write_point( const ecp_group *grp, const ecp_point *pt,
int format, size_t *olen,
unsigned char *buf, size_t blen )
{
int ret;
/*
* buffer length must be at least one, for our length byte
*/
if( blen < 1 )
return( POLARSSL_ERR_ECP_BAD_INPUT_DATA );
if( ( ret = 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] = *olen;
++*olen;
return 0;
}
/*
* 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, ecp_group_id id )
{
grp->id = id;
switch( id )
{
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_BAD_INPUT_DATA );
}
/*
* Set a group from an ECParameters record (RFC 4492)
*/
int ecp_tls_read_group( ecp_group *grp, const unsigned char **buf, size_t len )
{
ecp_group_id id;
/*
* We expect at least three bytes (see below)
*/
if( len < 3 )
return( POLARSSL_ERR_ECP_BAD_INPUT_DATA );
/*
* First byte is curve_type; only named_curve is handled
*/
if( *(*buf)++ != POLARSSL_ECP_TLS_NAMED_CURVE )
return( POLARSSL_ERR_ECP_BAD_INPUT_DATA );
/*
* Next two bytes are the namedcurve value
*/
id = *(*buf)++;
id <<= 8;
id |= *(*buf)++;
return ecp_use_known_dp( grp, id );
}
/*
* Write the ECParameters record corresponding to a group (RFC 4492)
*/
int ecp_tls_write_group( const ecp_group *grp, size_t *olen,
unsigned char *buf, size_t blen )
{
/*
* We are going to write 3 bytes (see below)
*/
*olen = 3;
if( blen < *olen )
return( POLARSSL_ERR_ECP_BUFFER_TOO_SMALL );
/*
* First byte is curve_type, always named_curve
*/
*buf++ = POLARSSL_ECP_TLS_NAMED_CURVE;
/*
* Next two bytes are the namedcurve value
*/
buf[0] = grp->id >> 8;
buf[1] = grp->id & 0xFF;
return 0;
}
/*
* 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 ) )
/*
* 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 )
{
int ret;
mpi YY, RHS;
if( mpi_cmp_int( &pt->Z, 0 ) == 0 )
return( POLARSSL_ERR_ECP_GENERIC );
/*
* pt coordinates must be normalized for our checks
*/
if( mpi_cmp_int( &pt->Z, 1 ) != 0 )
return( POLARSSL_ERR_ECP_GENERIC );
if( mpi_cmp_int( &pt->X, 0 ) < 0 ||
mpi_cmp_int( &pt->Y, 0 ) < 0 ||
mpi_cmp_mpi( &pt->X, &grp->P ) >= 0 ||
mpi_cmp_mpi( &pt->Y, &grp->P ) >= 0 )
return( POLARSSL_ERR_ECP_GENERIC );
mpi_init( &YY ); mpi_init( &RHS );
/*
* YY = Y^2
* RHS = X (X^2 - 3) + B = X^3 - 3X + B
*/
MPI_CHK( mpi_mul_mpi( &YY, &pt->Y, &pt->Y ) ); MOD_MUL( YY );
MPI_CHK( mpi_mul_mpi( &RHS, &pt->X, &pt->X ) ); MOD_MUL( RHS );
MPI_CHK( mpi_sub_int( &RHS, &RHS, 3 ) ); MOD_SUB( RHS );
MPI_CHK( mpi_mul_mpi( &RHS, &RHS, &pt->X ) ); MOD_MUL( RHS );
MPI_CHK( mpi_add_mpi( &RHS, &RHS, &grp->B ) ); MOD_ADD( RHS );
if( mpi_cmp_mpi( &YY, &RHS ) != 0 )
ret = POLARSSL_ERR_ECP_GENERIC;
cleanup:
mpi_free( &YY ); mpi_free( &RHS );
return( ret );
}
/*
* 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;
if( mpi_cmp_int( &pt->Z, 0 ) == 0 )
return( 0 );
mpi_init( &Zi ); mpi_init( &ZZi );
/*
* 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 );
return( ret );
}
/*
* Normalize jacobian coordinates of an array of 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 if one of the points is zero!
* 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 )
{
int ret;
size_t i;
mpi *c, u, Zi, ZZi;
if( t_len < 2 )
return( ecp_normalize( grp, T ) );
if( ( c = (mpi *) malloc( t_len * sizeof( mpi ) ) ) == NULL )
return( POLARSSL_ERR_ECP_GENERIC );
mpi_init( &u ); mpi_init( &Zi ); mpi_init( &ZZi );
for( i = 0; i < t_len; i++ )
mpi_init( &c[i] );
/*
* c[i] = Z_0 * ... * Z_i
*/
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 ) );
MOD_MUL( c[i] );
}
/*
* u = 1 / (Z_0 * ... * Z_n) mod P
*/
MPI_CHK( mpi_inv_mod( &u, &c[t_len-1], &grp->P ) );
for( i = t_len - 1; ; i-- )
{
/*
* Zi = 1 / Z_i mod p
* u = 1 / (Z_0 * ... * Z_i) mod P
*/
if( i == 0 ) {
MPI_CHK( mpi_copy( &Zi, &u ) );
}
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 );
}
/*
* 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 ) );
if( i == 0 )
break;
}
cleanup:
mpi_free( &u ); mpi_free( &Zi ); mpi_free( &ZZi );
for( i = 0; i < t_len; i++ )
mpi_free( &c[i] );
free( c );
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 );
}
/*
* Precompute odd multiples of P up to (2 * t_len - 1) P.
* The table is filled with T[i] = (2 * i + 1) P.
*/
static int ecp_precompute( const ecp_group *grp,
ecp_point T[], size_t t_len,
const ecp_point *P )
{
int ret;
size_t i;
ecp_point PP;
ecp_point_init( &PP );
MPI_CHK( ecp_add( grp, &PP, P, P ) );
MPI_CHK( ecp_copy( &T[0], P ) );
for( i = 1; i < t_len; i++ )
MPI_CHK( ecp_add_mixed( grp, &T[i], &T[i-1], &PP, +1 ) );
/*
* T[0] = P already has normalized coordinates
*/
MPI_CHK( ecp_normalize_many( grp, T + 1, t_len - 1 ) );
cleanup:
ecp_point_free( &PP );
return( ret );
}
/*
* Maximum length of the precomputed table
*/
#define MAX_PRE_LEN ( 1 << (POLARSSL_ECP_WINDOW_SIZE - 1) )
/*
* Maximum length of the NAF: ceil( grp->nbits + 1 ) / w
* (that is: grp->nbits / w + 1)
* Allow p_bits + 1 bits in case M = grp->N + 1 is one bit longer than N.
*/
#define MAX_NAF_LEN ( POLARSSL_ECP_MAX_N_BITS / 2 + 1 )
/*
* Integer multiplication: R = m * P
*
* Based on fixed-pattern width-w NAF, see comments of ecp_w_naf_fixed()
* and <http://rd.springer.com/chapter/10.1007/3-540-36563-X_23>.
*
* This function executes a fixed number of operations for
* random m in the range 0 .. 2^nbits - 1.
*/
int ecp_mul( const ecp_group *grp, ecp_point *R,
const mpi *m, const ecp_point *P )
{
int ret;
unsigned char w, m_is_odd;
size_t pre_len, naf_len, i, j;
signed char naf[ MAX_NAF_LEN ];
ecp_point Q, T[ MAX_PRE_LEN ];
mpi M;
if( mpi_cmp_int( m, 0 ) < 0 || mpi_msb( m ) > grp->nbits )
return( POLARSSL_ERR_ECP_GENERIC );
w = grp->nbits >= 521 ? 6 :
grp->nbits >= 224 ? 5 :
4;
/*
* Make sure w is within the limits.
* The last test ensures that none of the precomputed points is zero,
* which wouldn't be handled correctly by ecp_normalize_many().
* It is only useful for small curves, as used in the test suite.
*/
if( w > POLARSSL_ECP_WINDOW_SIZE )
w = POLARSSL_ECP_WINDOW_SIZE;
if( w < 2 || w >= grp->nbits )
w = 2;
pre_len = 1 << ( w - 1 );
naf_len = grp->nbits / w + 1;
mpi_init( &M );
ecp_point_init( &Q );
for( i = 0; i < pre_len; i++ )
ecp_point_init( &T[i] );
m_is_odd = ( mpi_get_bit( m, 0 ) == 1 );
/*
* Make sure M is odd:
* later we'll get m * P by subtracting * P or 2 * P to M * P.
*/
MPI_CHK( mpi_copy( &M, m ) );
MPI_CHK( mpi_add_int( &M, &M, 1 + m_is_odd ) );
/*
* Compute the fixed-pattern NAF and precompute odd multiples
*/
MPI_CHK( ecp_w_naf_fixed( naf, naf_len, w, &M ) );
MPI_CHK( ecp_precompute( grp, T, pre_len, P ) );
/*
* Compute M * P, using a variant of left-to-right 2^w-ary multiplication:
* at each step we add (2 * naf[i] + 1) P, then multiply by 2^w.
*
* If naf[i] >= 0, we have (2 * naf[i] + 1) P == T[ naf[i] ]
* Otherwise, (2 * naf[i] + 1) P == - ( 2 * ( - naf[i] - 1 ) + 1) P
* == T[ - naf[i] - 1 ]
*/
MPI_CHK( ecp_set_zero( &Q ) );
i = naf_len - 1;
while( 1 )
{
if( naf[i] < 0 )
{
MPI_CHK( ecp_add_mixed( grp, &Q, &Q, &T[ - naf[i] - 1 ], -1 ) );
}
else
{
MPI_CHK( ecp_add_mixed( grp, &Q, &Q, &T[ naf[i] ], +1 ) );
}
if( i == 0 )
break;
i--;
for( j = 0; j < w; j++ )
{
MPI_CHK( ecp_double_jac( grp, &Q, &Q ) );
}
}
/*
* Now get m * P from M * P.
* Since we don't need T[] any more, we can recycle it:
* we already have T[0] = P, now set T[1] = 2 * P.
*/
MPI_CHK( ecp_add( grp, &T[1], P, P ) );
MPI_CHK( ecp_sub( grp, R, &Q, &T[m_is_odd] ) );
cleanup:
mpi_free( &M );
ecp_point_free( &Q );
for( i = 0; i < pre_len; i++ )
ecp_point_free( &T[i] );
return( ret );
}
/*
* Generate a keypair (SEC1 3.2.1)
*/
int ecp_gen_keypair( const ecp_group *grp, mpi *d, ecp_point *Q,
int (*f_rng)(void *, unsigned char *, size_t),
void *p_rng )
{
int count = 0;
size_t n_size = (grp->nbits + 7) / 8;
/*
* Generate d such that 1 <= n < N
*/
do
{
mpi_fill_random( d, n_size, f_rng, p_rng );
while( mpi_cmp_mpi( d, &grp->N ) >= 0 )
mpi_shift_r( d, 1 );
if( count++ > 10 )
return( POLARSSL_ERR_ECP_GENERIC );
}
while( mpi_cmp_int( d, 1 ) < 0 );
return( ecp_mul( grp, Q, d, &grp->G ) );
}
#if defined(POLARSSL_SELF_TEST)
/*
* Checkup routine
*/
int ecp_self_test( int verbose )
{
int ret;
size_t i;
ecp_group grp;
ecp_point R;
mpi m;
unsigned long add_c_prev, dbl_c_prev;
char *exponents[] =
{
"000000000000000000000000000000000000000000000000", /* zero */
"000000000000000000000000000000000000000000000001", /* one */
"FFFFFFFFFFFFFFFFFFFFFFFF99DEF836146BC9B1B4D22831", /* N */
"5EA6F389A38B8BC81E767753B15AA5569E1782E30ABE7D25", /* random */
"400000000000000000000000000000000000000000000000",
"7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF",
"555555555555555555555555555555555555555555555555",
};
ecp_group_init( &grp );
ecp_point_init( &R );
mpi_init( &m );
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
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