2 * Elliptic curves over GF(p): generic functions
4 * Copyright (C) 2006-2015, ARM Limited, All Rights Reserved
5 * SPDX-License-Identifier: GPL-2.0
7 * This program is free software; you can redistribute it and/or modify
8 * it under the terms of the GNU General Public License as published by
9 * the Free Software Foundation; either version 2 of the License, or
10 * (at your option) any later version.
12 * This program is distributed in the hope that it will be useful,
13 * but WITHOUT ANY WARRANTY; without even the implied warranty of
14 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
15 * GNU General Public License for more details.
17 * You should have received a copy of the GNU General Public License along
18 * with this program; if not, write to the Free Software Foundation, Inc.,
19 * 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA.
21 * This file is part of mbed TLS (https://tls.mbed.org)
27 * SEC1 http://www.secg.org/index.php?action=secg,docs_secg
28 * GECC = Guide to Elliptic Curve Cryptography - Hankerson, Menezes, Vanstone
29 * FIPS 186-3 http://csrc.nist.gov/publications/fips/fips186-3/fips_186-3.pdf
30 * RFC 4492 for the related TLS structures and constants
32 * [Curve25519] http://cr.yp.to/ecdh/curve25519-20060209.pdf
34 * [2] CORON, Jean-S'ebastien. Resistance against differential power analysis
35 * for elliptic curve cryptosystems. In : Cryptographic Hardware and
36 * Embedded Systems. Springer Berlin Heidelberg, 1999. p. 292-302.
37 * <http://link.springer.com/chapter/10.1007/3-540-48059-5_25>
39 * [3] HEDABOU, Mustapha, PINEL, Pierre, et B'EN'ETEAU, Lucien. A comb method to
40 * render ECC resistant against Side Channel Attacks. IACR Cryptology
41 * ePrint Archive, 2004, vol. 2004, p. 342.
42 * <http://eprint.iacr.org/2004/342.pdf>
45 #if !defined(MBEDTLS_CONFIG_FILE)
46 #include "mbedtls/config.h"
48 #include MBEDTLS_CONFIG_FILE
51 #if defined(MBEDTLS_ECP_C)
53 #include "mbedtls/ecp.h"
54 #include "mbedtls/threading.h"
58 #if !defined(MBEDTLS_ECP_ALT)
60 #if defined(MBEDTLS_PLATFORM_C)
61 #include "mbedtls/platform.h"
65 #define mbedtls_printf printf
66 #define mbedtls_calloc calloc
67 #define mbedtls_free free
70 #include "mbedtls/ecp_internal.h"
72 #if ( defined(__ARMCC_VERSION) || defined(_MSC_VER) ) && \
73 !defined(inline) && !defined(__cplusplus)
74 #define inline __inline
77 /* Implementation that should never be optimized out by the compiler */
78 static void mbedtls_zeroize( void *v
, size_t n
) {
79 volatile unsigned char *p
= v
; while( n
-- ) *p
++ = 0;
82 #if defined(MBEDTLS_SELF_TEST)
84 * Counts of point addition and doubling, and field multiplications.
85 * Used to test resistance of point multiplication to simple timing attacks.
87 static unsigned long add_count
, dbl_count
, mul_count
;
90 #if defined(MBEDTLS_ECP_DP_SECP192R1_ENABLED) || \
91 defined(MBEDTLS_ECP_DP_SECP224R1_ENABLED) || \
92 defined(MBEDTLS_ECP_DP_SECP256R1_ENABLED) || \
93 defined(MBEDTLS_ECP_DP_SECP384R1_ENABLED) || \
94 defined(MBEDTLS_ECP_DP_SECP521R1_ENABLED) || \
95 defined(MBEDTLS_ECP_DP_BP256R1_ENABLED) || \
96 defined(MBEDTLS_ECP_DP_BP384R1_ENABLED) || \
97 defined(MBEDTLS_ECP_DP_BP512R1_ENABLED) || \
98 defined(MBEDTLS_ECP_DP_SECP192K1_ENABLED) || \
99 defined(MBEDTLS_ECP_DP_SECP224K1_ENABLED) || \
100 defined(MBEDTLS_ECP_DP_SECP256K1_ENABLED)
101 #define ECP_SHORTWEIERSTRASS
104 #if defined(MBEDTLS_ECP_DP_CURVE25519_ENABLED)
105 #define ECP_MONTGOMERY
109 * Curve types: internal for now, might be exposed later
114 ECP_TYPE_SHORT_WEIERSTRASS
, /* y^2 = x^3 + a x + b */
115 ECP_TYPE_MONTGOMERY
, /* y^2 = x^3 + a x^2 + x */
119 * List of supported curves:
121 * - TLS NamedCurve ID (RFC 4492 sec. 5.1.1, RFC 7071 sec. 2)
125 * Curves are listed in order: largest curves first, and for a given size,
126 * fastest curves first. This provides the default order for the SSL module.
128 * Reminder: update profiles in x509_crt.c when adding a new curves!
130 static const mbedtls_ecp_curve_info ecp_supported_curves
[] =
132 #if defined(MBEDTLS_ECP_DP_SECP521R1_ENABLED)
133 { MBEDTLS_ECP_DP_SECP521R1
, 25, 521, "secp521r1" },
135 #if defined(MBEDTLS_ECP_DP_BP512R1_ENABLED)
136 { MBEDTLS_ECP_DP_BP512R1
, 28, 512, "brainpoolP512r1" },
138 #if defined(MBEDTLS_ECP_DP_SECP384R1_ENABLED)
139 { MBEDTLS_ECP_DP_SECP384R1
, 24, 384, "secp384r1" },
141 #if defined(MBEDTLS_ECP_DP_BP384R1_ENABLED)
142 { MBEDTLS_ECP_DP_BP384R1
, 27, 384, "brainpoolP384r1" },
144 #if defined(MBEDTLS_ECP_DP_SECP256R1_ENABLED)
145 { MBEDTLS_ECP_DP_SECP256R1
, 23, 256, "secp256r1" },
147 #if defined(MBEDTLS_ECP_DP_SECP256K1_ENABLED)
148 { MBEDTLS_ECP_DP_SECP256K1
, 22, 256, "secp256k1" },
150 #if defined(MBEDTLS_ECP_DP_BP256R1_ENABLED)
151 { MBEDTLS_ECP_DP_BP256R1
, 26, 256, "brainpoolP256r1" },
153 #if defined(MBEDTLS_ECP_DP_SECP224R1_ENABLED)
154 { MBEDTLS_ECP_DP_SECP224R1
, 21, 224, "secp224r1" },
156 #if defined(MBEDTLS_ECP_DP_SECP224K1_ENABLED)
157 { MBEDTLS_ECP_DP_SECP224K1
, 20, 224, "secp224k1" },
159 #if defined(MBEDTLS_ECP_DP_SECP192R1_ENABLED)
160 { MBEDTLS_ECP_DP_SECP192R1
, 19, 192, "secp192r1" },
162 #if defined(MBEDTLS_ECP_DP_SECP192K1_ENABLED)
163 { MBEDTLS_ECP_DP_SECP192K1
, 18, 192, "secp192k1" },
165 { MBEDTLS_ECP_DP_NONE
, 0, 0, NULL
},
168 #define ECP_NB_CURVES sizeof( ecp_supported_curves ) / \
169 sizeof( ecp_supported_curves[0] )
171 static mbedtls_ecp_group_id ecp_supported_grp_id
[ECP_NB_CURVES
];
174 * List of supported curves and associated info
176 const mbedtls_ecp_curve_info
*mbedtls_ecp_curve_list( void )
178 return( ecp_supported_curves
);
182 * List of supported curves, group ID only
184 const mbedtls_ecp_group_id
*mbedtls_ecp_grp_id_list( void )
186 static int init_done
= 0;
191 const mbedtls_ecp_curve_info
*curve_info
;
193 for( curve_info
= mbedtls_ecp_curve_list();
194 curve_info
->grp_id
!= MBEDTLS_ECP_DP_NONE
;
197 ecp_supported_grp_id
[i
++] = curve_info
->grp_id
;
199 ecp_supported_grp_id
[i
] = MBEDTLS_ECP_DP_NONE
;
204 return( ecp_supported_grp_id
);
208 * Get the curve info for the internal identifier
210 const mbedtls_ecp_curve_info
*mbedtls_ecp_curve_info_from_grp_id( mbedtls_ecp_group_id grp_id
)
212 const mbedtls_ecp_curve_info
*curve_info
;
214 for( curve_info
= mbedtls_ecp_curve_list();
215 curve_info
->grp_id
!= MBEDTLS_ECP_DP_NONE
;
218 if( curve_info
->grp_id
== grp_id
)
219 return( curve_info
);
226 * Get the curve info from the TLS identifier
228 const mbedtls_ecp_curve_info
*mbedtls_ecp_curve_info_from_tls_id( uint16_t tls_id
)
230 const mbedtls_ecp_curve_info
*curve_info
;
232 for( curve_info
= mbedtls_ecp_curve_list();
233 curve_info
->grp_id
!= MBEDTLS_ECP_DP_NONE
;
236 if( curve_info
->tls_id
== tls_id
)
237 return( curve_info
);
244 * Get the curve info from the name
246 const mbedtls_ecp_curve_info
*mbedtls_ecp_curve_info_from_name( const char *name
)
248 const mbedtls_ecp_curve_info
*curve_info
;
250 for( curve_info
= mbedtls_ecp_curve_list();
251 curve_info
->grp_id
!= MBEDTLS_ECP_DP_NONE
;
254 if( strcmp( curve_info
->name
, name
) == 0 )
255 return( curve_info
);
262 * Get the type of a curve
264 static inline ecp_curve_type
ecp_get_type( const mbedtls_ecp_group
*grp
)
266 if( grp
->G
.X
.p
== NULL
)
267 return( ECP_TYPE_NONE
);
269 if( grp
->G
.Y
.p
== NULL
)
270 return( ECP_TYPE_MONTGOMERY
);
272 return( ECP_TYPE_SHORT_WEIERSTRASS
);
276 * Initialize (the components of) a point
278 void mbedtls_ecp_point_init( mbedtls_ecp_point
*pt
)
283 mbedtls_mpi_init( &pt
->X
);
284 mbedtls_mpi_init( &pt
->Y
);
285 mbedtls_mpi_init( &pt
->Z
);
289 * Initialize (the components of) a group
291 void mbedtls_ecp_group_init( mbedtls_ecp_group
*grp
)
296 memset( grp
, 0, sizeof( mbedtls_ecp_group
) );
300 * Initialize (the components of) a key pair
302 void mbedtls_ecp_keypair_init( mbedtls_ecp_keypair
*key
)
307 mbedtls_ecp_group_init( &key
->grp
);
308 mbedtls_mpi_init( &key
->d
);
309 mbedtls_ecp_point_init( &key
->Q
);
313 * Unallocate (the components of) a point
315 void mbedtls_ecp_point_free( mbedtls_ecp_point
*pt
)
320 mbedtls_mpi_free( &( pt
->X
) );
321 mbedtls_mpi_free( &( pt
->Y
) );
322 mbedtls_mpi_free( &( pt
->Z
) );
326 * Unallocate (the components of) a group
328 void mbedtls_ecp_group_free( mbedtls_ecp_group
*grp
)
337 mbedtls_mpi_free( &grp
->P
);
338 mbedtls_mpi_free( &grp
->A
);
339 mbedtls_mpi_free( &grp
->B
);
340 mbedtls_ecp_point_free( &grp
->G
);
341 mbedtls_mpi_free( &grp
->N
);
346 for( i
= 0; i
< grp
->T_size
; i
++ )
347 mbedtls_ecp_point_free( &grp
->T
[i
] );
348 mbedtls_free( grp
->T
);
351 mbedtls_zeroize( grp
, sizeof( mbedtls_ecp_group
) );
355 * Unallocate (the components of) a key pair
357 void mbedtls_ecp_keypair_free( mbedtls_ecp_keypair
*key
)
362 mbedtls_ecp_group_free( &key
->grp
);
363 mbedtls_mpi_free( &key
->d
);
364 mbedtls_ecp_point_free( &key
->Q
);
368 * Copy the contents of a point
370 int mbedtls_ecp_copy( mbedtls_ecp_point
*P
, const mbedtls_ecp_point
*Q
)
374 MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &P
->X
, &Q
->X
) );
375 MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &P
->Y
, &Q
->Y
) );
376 MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &P
->Z
, &Q
->Z
) );
383 * Copy the contents of a group object
385 int mbedtls_ecp_group_copy( mbedtls_ecp_group
*dst
, const mbedtls_ecp_group
*src
)
387 return mbedtls_ecp_group_load( dst
, src
->id
);
393 int mbedtls_ecp_set_zero( mbedtls_ecp_point
*pt
)
397 MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &pt
->X
, 1 ) );
398 MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &pt
->Y
, 1 ) );
399 MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &pt
->Z
, 0 ) );
406 * Tell if a point is zero
408 int mbedtls_ecp_is_zero( mbedtls_ecp_point
*pt
)
410 return( mbedtls_mpi_cmp_int( &pt
->Z
, 0 ) == 0 );
414 * Compare two points lazily
416 int mbedtls_ecp_point_cmp( const mbedtls_ecp_point
*P
,
417 const mbedtls_ecp_point
*Q
)
419 if( mbedtls_mpi_cmp_mpi( &P
->X
, &Q
->X
) == 0 &&
420 mbedtls_mpi_cmp_mpi( &P
->Y
, &Q
->Y
) == 0 &&
421 mbedtls_mpi_cmp_mpi( &P
->Z
, &Q
->Z
) == 0 )
426 return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA
);
430 * Import a non-zero point from ASCII strings
432 int mbedtls_ecp_point_read_string( mbedtls_ecp_point
*P
, int radix
,
433 const char *x
, const char *y
)
437 MBEDTLS_MPI_CHK( mbedtls_mpi_read_string( &P
->X
, radix
, x
) );
438 MBEDTLS_MPI_CHK( mbedtls_mpi_read_string( &P
->Y
, radix
, y
) );
439 MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &P
->Z
, 1 ) );
446 * Export a point into unsigned binary data (SEC1 2.3.3)
448 int mbedtls_ecp_point_write_binary( const mbedtls_ecp_group
*grp
, const mbedtls_ecp_point
*P
,
449 int format
, size_t *olen
,
450 unsigned char *buf
, size_t buflen
)
455 if( format
!= MBEDTLS_ECP_PF_UNCOMPRESSED
&&
456 format
!= MBEDTLS_ECP_PF_COMPRESSED
)
457 return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA
);
460 * Common case: P == 0
462 if( mbedtls_mpi_cmp_int( &P
->Z
, 0 ) == 0 )
465 return( MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL
);
473 plen
= mbedtls_mpi_size( &grp
->P
);
475 if( format
== MBEDTLS_ECP_PF_UNCOMPRESSED
)
477 *olen
= 2 * plen
+ 1;
480 return( MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL
);
483 MBEDTLS_MPI_CHK( mbedtls_mpi_write_binary( &P
->X
, buf
+ 1, plen
) );
484 MBEDTLS_MPI_CHK( mbedtls_mpi_write_binary( &P
->Y
, buf
+ 1 + plen
, plen
) );
486 else if( format
== MBEDTLS_ECP_PF_COMPRESSED
)
491 return( MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL
);
493 buf
[0] = 0x02 + mbedtls_mpi_get_bit( &P
->Y
, 0 );
494 MBEDTLS_MPI_CHK( mbedtls_mpi_write_binary( &P
->X
, buf
+ 1, plen
) );
502 * Import a point from unsigned binary data (SEC1 2.3.4)
504 int mbedtls_ecp_point_read_binary( const mbedtls_ecp_group
*grp
, mbedtls_ecp_point
*pt
,
505 const unsigned char *buf
, size_t ilen
)
511 return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA
);
516 return( mbedtls_ecp_set_zero( pt
) );
518 return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA
);
521 plen
= mbedtls_mpi_size( &grp
->P
);
524 return( MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE
);
526 if( ilen
!= 2 * plen
+ 1 )
527 return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA
);
529 MBEDTLS_MPI_CHK( mbedtls_mpi_read_binary( &pt
->X
, buf
+ 1, plen
) );
530 MBEDTLS_MPI_CHK( mbedtls_mpi_read_binary( &pt
->Y
, buf
+ 1 + plen
, plen
) );
531 MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &pt
->Z
, 1 ) );
538 * Import a point from a TLS ECPoint record (RFC 4492)
540 * opaque point <1..2^8-1>;
543 int mbedtls_ecp_tls_read_point( const mbedtls_ecp_group
*grp
, mbedtls_ecp_point
*pt
,
544 const unsigned char **buf
, size_t buf_len
)
546 unsigned char data_len
;
547 const unsigned char *buf_start
;
550 * We must have at least two bytes (1 for length, at least one for data)
553 return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA
);
555 data_len
= *(*buf
)++;
556 if( data_len
< 1 || data_len
> buf_len
- 1 )
557 return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA
);
560 * Save buffer start for read_binary and update buf
565 return mbedtls_ecp_point_read_binary( grp
, pt
, buf_start
, data_len
);
569 * Export a point as a TLS ECPoint record (RFC 4492)
571 * opaque point <1..2^8-1>;
574 int mbedtls_ecp_tls_write_point( const mbedtls_ecp_group
*grp
, const mbedtls_ecp_point
*pt
,
575 int format
, size_t *olen
,
576 unsigned char *buf
, size_t blen
)
581 * buffer length must be at least one, for our length byte
584 return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA
);
586 if( ( ret
= mbedtls_ecp_point_write_binary( grp
, pt
, format
,
587 olen
, buf
+ 1, blen
- 1) ) != 0 )
591 * write length to the first byte and update total length
593 buf
[0] = (unsigned char) *olen
;
600 * Set a group from an ECParameters record (RFC 4492)
602 int mbedtls_ecp_tls_read_group( mbedtls_ecp_group
*grp
, const unsigned char **buf
, size_t len
)
605 const mbedtls_ecp_curve_info
*curve_info
;
608 * We expect at least three bytes (see below)
611 return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA
);
614 * First byte is curve_type; only named_curve is handled
616 if( *(*buf
)++ != MBEDTLS_ECP_TLS_NAMED_CURVE
)
617 return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA
);
620 * Next two bytes are the namedcurve value
626 if( ( curve_info
= mbedtls_ecp_curve_info_from_tls_id( tls_id
) ) == NULL
)
627 return( MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE
);
629 return mbedtls_ecp_group_load( grp
, curve_info
->grp_id
);
633 * Write the ECParameters record corresponding to a group (RFC 4492)
635 int mbedtls_ecp_tls_write_group( const mbedtls_ecp_group
*grp
, size_t *olen
,
636 unsigned char *buf
, size_t blen
)
638 const mbedtls_ecp_curve_info
*curve_info
;
640 if( ( curve_info
= mbedtls_ecp_curve_info_from_grp_id( grp
->id
) ) == NULL
)
641 return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA
);
644 * We are going to write 3 bytes (see below)
648 return( MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL
);
651 * First byte is curve_type, always named_curve
653 *buf
++ = MBEDTLS_ECP_TLS_NAMED_CURVE
;
656 * Next two bytes are the namedcurve value
658 buf
[0] = curve_info
->tls_id
>> 8;
659 buf
[1] = curve_info
->tls_id
& 0xFF;
665 * Wrapper around fast quasi-modp functions, with fall-back to mbedtls_mpi_mod_mpi.
666 * See the documentation of struct mbedtls_ecp_group.
668 * This function is in the critial loop for mbedtls_ecp_mul, so pay attention to perf.
670 static int ecp_modp( mbedtls_mpi
*N
, const mbedtls_ecp_group
*grp
)
674 if( grp
->modp
== NULL
)
675 return( mbedtls_mpi_mod_mpi( N
, N
, &grp
->P
) );
677 /* N->s < 0 is a much faster test, which fails only if N is 0 */
678 if( ( N
->s
< 0 && mbedtls_mpi_cmp_int( N
, 0 ) != 0 ) ||
679 mbedtls_mpi_bitlen( N
) > 2 * grp
->pbits
)
681 return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA
);
684 MBEDTLS_MPI_CHK( grp
->modp( N
) );
686 /* N->s < 0 is a much faster test, which fails only if N is 0 */
687 while( N
->s
< 0 && mbedtls_mpi_cmp_int( N
, 0 ) != 0 )
688 MBEDTLS_MPI_CHK( mbedtls_mpi_add_mpi( N
, N
, &grp
->P
) );
690 while( mbedtls_mpi_cmp_mpi( N
, &grp
->P
) >= 0 )
691 /* we known P, N and the result are positive */
692 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_abs( N
, N
, &grp
->P
) );
699 * Fast mod-p functions expect their argument to be in the 0..p^2 range.
701 * In order to guarantee that, we need to ensure that operands of
702 * mbedtls_mpi_mul_mpi are in the 0..p range. So, after each operation we will
703 * bring the result back to this range.
705 * The following macros are shortcuts for doing that.
709 * Reduce a mbedtls_mpi mod p in-place, general case, to use after mbedtls_mpi_mul_mpi
711 #if defined(MBEDTLS_SELF_TEST)
712 #define INC_MUL_COUNT mul_count++;
714 #define INC_MUL_COUNT
717 #define MOD_MUL( N ) do { MBEDTLS_MPI_CHK( ecp_modp( &N, grp ) ); INC_MUL_COUNT } \
721 * Reduce a mbedtls_mpi mod p in-place, to use after mbedtls_mpi_sub_mpi
722 * N->s < 0 is a very fast test, which fails only if N is 0
724 #define MOD_SUB( N ) \
725 while( N.s < 0 && mbedtls_mpi_cmp_int( &N, 0 ) != 0 ) \
726 MBEDTLS_MPI_CHK( mbedtls_mpi_add_mpi( &N, &N, &grp->P ) )
729 * Reduce a mbedtls_mpi mod p in-place, to use after mbedtls_mpi_add_mpi and mbedtls_mpi_mul_int.
730 * We known P, N and the result are positive, so sub_abs is correct, and
733 #define MOD_ADD( N ) \
734 while( mbedtls_mpi_cmp_mpi( &N, &grp->P ) >= 0 ) \
735 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_abs( &N, &N, &grp->P ) )
737 #if defined(ECP_SHORTWEIERSTRASS)
739 * For curves in short Weierstrass form, we do all the internal operations in
740 * Jacobian coordinates.
742 * For multiplication, we'll use a comb method with coutermeasueres against
743 * SPA, hence timing attacks.
747 * Normalize jacobian coordinates so that Z == 0 || Z == 1 (GECC 3.2.1)
748 * Cost: 1N := 1I + 3M + 1S
750 static int ecp_normalize_jac( const mbedtls_ecp_group
*grp
, mbedtls_ecp_point
*pt
)
755 if( mbedtls_mpi_cmp_int( &pt
->Z
, 0 ) == 0 )
758 #if defined(MBEDTLS_ECP_NORMALIZE_JAC_ALT)
759 if ( mbedtls_internal_ecp_grp_capable( grp
) )
761 return mbedtls_internal_ecp_normalize_jac( grp
, pt
);
763 #endif /* MBEDTLS_ECP_NORMALIZE_JAC_ALT */
764 mbedtls_mpi_init( &Zi
); mbedtls_mpi_init( &ZZi
);
769 MBEDTLS_MPI_CHK( mbedtls_mpi_inv_mod( &Zi
, &pt
->Z
, &grp
->P
) );
770 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &ZZi
, &Zi
, &Zi
) ); MOD_MUL( ZZi
);
771 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &pt
->X
, &pt
->X
, &ZZi
) ); MOD_MUL( pt
->X
);
776 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &pt
->Y
, &pt
->Y
, &ZZi
) ); MOD_MUL( pt
->Y
);
777 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &pt
->Y
, &pt
->Y
, &Zi
) ); MOD_MUL( pt
->Y
);
782 MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &pt
->Z
, 1 ) );
786 mbedtls_mpi_free( &Zi
); mbedtls_mpi_free( &ZZi
);
792 * Normalize jacobian coordinates of an array of (pointers to) points,
793 * using Montgomery's trick to perform only one inversion mod P.
794 * (See for example Cohen's "A Course in Computational Algebraic Number
795 * Theory", Algorithm 10.3.4.)
797 * Warning: fails (returning an error) if one of the points is zero!
798 * This should never happen, see choice of w in ecp_mul_comb().
800 * Cost: 1N(t) := 1I + (6t - 3)M + 1S
802 static int ecp_normalize_jac_many( const mbedtls_ecp_group
*grp
,
803 mbedtls_ecp_point
*T
[], size_t t_len
)
807 mbedtls_mpi
*c
, u
, Zi
, ZZi
;
810 return( ecp_normalize_jac( grp
, *T
) );
812 #if defined(MBEDTLS_ECP_NORMALIZE_JAC_MANY_ALT)
813 if ( mbedtls_internal_ecp_grp_capable( grp
) )
815 return mbedtls_internal_ecp_normalize_jac_many(grp
, T
, t_len
);
819 if( ( c
= mbedtls_calloc( t_len
, sizeof( mbedtls_mpi
) ) ) == NULL
)
820 return( MBEDTLS_ERR_ECP_ALLOC_FAILED
);
822 mbedtls_mpi_init( &u
); mbedtls_mpi_init( &Zi
); mbedtls_mpi_init( &ZZi
);
825 * c[i] = Z_0 * ... * Z_i
827 MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &c
[0], &T
[0]->Z
) );
828 for( i
= 1; i
< t_len
; i
++ )
830 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &c
[i
], &c
[i
-1], &T
[i
]->Z
) );
835 * u = 1 / (Z_0 * ... * Z_n) mod P
837 MBEDTLS_MPI_CHK( mbedtls_mpi_inv_mod( &u
, &c
[t_len
-1], &grp
->P
) );
839 for( i
= t_len
- 1; ; i
-- )
843 * u = 1 / (Z_0 * ... * Z_i) mod P
846 MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &Zi
, &u
) );
850 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &Zi
, &u
, &c
[i
-1] ) ); MOD_MUL( Zi
);
851 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &u
, &u
, &T
[i
]->Z
) ); MOD_MUL( u
);
855 * proceed as in normalize()
857 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &ZZi
, &Zi
, &Zi
) ); MOD_MUL( ZZi
);
858 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T
[i
]->X
, &T
[i
]->X
, &ZZi
) ); MOD_MUL( T
[i
]->X
);
859 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T
[i
]->Y
, &T
[i
]->Y
, &ZZi
) ); MOD_MUL( T
[i
]->Y
);
860 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T
[i
]->Y
, &T
[i
]->Y
, &Zi
) ); MOD_MUL( T
[i
]->Y
);
863 * Post-precessing: reclaim some memory by shrinking coordinates
864 * - not storing Z (always 1)
865 * - shrinking other coordinates, but still keeping the same number of
866 * limbs as P, as otherwise it will too likely be regrown too fast.
868 MBEDTLS_MPI_CHK( mbedtls_mpi_shrink( &T
[i
]->X
, grp
->P
.n
) );
869 MBEDTLS_MPI_CHK( mbedtls_mpi_shrink( &T
[i
]->Y
, grp
->P
.n
) );
870 mbedtls_mpi_free( &T
[i
]->Z
);
878 mbedtls_mpi_free( &u
); mbedtls_mpi_free( &Zi
); mbedtls_mpi_free( &ZZi
);
879 for( i
= 0; i
< t_len
; i
++ )
880 mbedtls_mpi_free( &c
[i
] );
887 * Conditional point inversion: Q -> -Q = (Q.X, -Q.Y, Q.Z) without leak.
888 * "inv" must be 0 (don't invert) or 1 (invert) or the result will be invalid
890 static int ecp_safe_invert_jac( const mbedtls_ecp_group
*grp
,
891 mbedtls_ecp_point
*Q
,
895 unsigned char nonzero
;
898 mbedtls_mpi_init( &mQY
);
900 /* Use the fact that -Q.Y mod P = P - Q.Y unless Q.Y == 0 */
901 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &mQY
, &grp
->P
, &Q
->Y
) );
902 nonzero
= mbedtls_mpi_cmp_int( &Q
->Y
, 0 ) != 0;
903 MBEDTLS_MPI_CHK( mbedtls_mpi_safe_cond_assign( &Q
->Y
, &mQY
, inv
& nonzero
) );
906 mbedtls_mpi_free( &mQY
);
912 * Point doubling R = 2 P, Jacobian coordinates
914 * Based on http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian.html#doubling-dbl-1998-cmo-2 .
916 * We follow the variable naming fairly closely. The formula variations that trade a MUL for a SQR
917 * (plus a few ADDs) aren't useful as our bignum implementation doesn't distinguish squaring.
919 * Standard optimizations are applied when curve parameter A is one of { 0, -3 }.
921 * Cost: 1D := 3M + 4S (A == 0)
923 * 3M + 6S + 1a otherwise
925 static int ecp_double_jac( const mbedtls_ecp_group
*grp
, mbedtls_ecp_point
*R
,
926 const mbedtls_ecp_point
*P
)
929 mbedtls_mpi M
, S
, T
, U
;
931 #if defined(MBEDTLS_SELF_TEST)
935 #if defined(MBEDTLS_ECP_DOUBLE_JAC_ALT)
936 if ( mbedtls_internal_ecp_grp_capable( grp
) )
938 return mbedtls_internal_ecp_double_jac( grp
, R
, P
);
940 #endif /* MBEDTLS_ECP_DOUBLE_JAC_ALT */
942 mbedtls_mpi_init( &M
); mbedtls_mpi_init( &S
); mbedtls_mpi_init( &T
); mbedtls_mpi_init( &U
);
944 /* Special case for A = -3 */
945 if( grp
->A
.p
== NULL
)
947 /* M = 3(X + Z^2)(X - Z^2) */
948 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &S
, &P
->Z
, &P
->Z
) ); MOD_MUL( S
);
949 MBEDTLS_MPI_CHK( mbedtls_mpi_add_mpi( &T
, &P
->X
, &S
) ); MOD_ADD( T
);
950 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &U
, &P
->X
, &S
) ); MOD_SUB( U
);
951 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &S
, &T
, &U
) ); MOD_MUL( S
);
952 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_int( &M
, &S
, 3 ) ); MOD_ADD( M
);
957 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &S
, &P
->X
, &P
->X
) ); MOD_MUL( S
);
958 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_int( &M
, &S
, 3 ) ); MOD_ADD( M
);
960 /* Optimize away for "koblitz" curves with A = 0 */
961 if( mbedtls_mpi_cmp_int( &grp
->A
, 0 ) != 0 )
964 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &S
, &P
->Z
, &P
->Z
) ); MOD_MUL( S
);
965 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T
, &S
, &S
) ); MOD_MUL( T
);
966 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &S
, &T
, &grp
->A
) ); MOD_MUL( S
);
967 MBEDTLS_MPI_CHK( mbedtls_mpi_add_mpi( &M
, &M
, &S
) ); MOD_ADD( M
);
972 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T
, &P
->Y
, &P
->Y
) ); MOD_MUL( T
);
973 MBEDTLS_MPI_CHK( mbedtls_mpi_shift_l( &T
, 1 ) ); MOD_ADD( T
);
974 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &S
, &P
->X
, &T
) ); MOD_MUL( S
);
975 MBEDTLS_MPI_CHK( mbedtls_mpi_shift_l( &S
, 1 ) ); MOD_ADD( S
);
978 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &U
, &T
, &T
) ); MOD_MUL( U
);
979 MBEDTLS_MPI_CHK( mbedtls_mpi_shift_l( &U
, 1 ) ); MOD_ADD( U
);
982 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T
, &M
, &M
) ); MOD_MUL( T
);
983 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &T
, &T
, &S
) ); MOD_SUB( T
);
984 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &T
, &T
, &S
) ); MOD_SUB( T
);
986 /* S = M(S - T) - U */
987 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &S
, &S
, &T
) ); MOD_SUB( S
);
988 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &S
, &S
, &M
) ); MOD_MUL( S
);
989 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &S
, &S
, &U
) ); MOD_SUB( S
);
992 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &U
, &P
->Y
, &P
->Z
) ); MOD_MUL( U
);
993 MBEDTLS_MPI_CHK( mbedtls_mpi_shift_l( &U
, 1 ) ); MOD_ADD( U
);
995 MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &R
->X
, &T
) );
996 MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &R
->Y
, &S
) );
997 MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &R
->Z
, &U
) );
1000 mbedtls_mpi_free( &M
); mbedtls_mpi_free( &S
); mbedtls_mpi_free( &T
); mbedtls_mpi_free( &U
);
1006 * Addition: R = P + Q, mixed affine-Jacobian coordinates (GECC 3.22)
1008 * The coordinates of Q must be normalized (= affine),
1009 * but those of P don't need to. R is not normalized.
1011 * Special cases: (1) P or Q is zero, (2) R is zero, (3) P == Q.
1012 * None of these cases can happen as intermediate step in ecp_mul_comb():
1013 * - at each step, P, Q and R are multiples of the base point, the factor
1014 * being less than its order, so none of them is zero;
1015 * - Q is an odd multiple of the base point, P an even multiple,
1016 * due to the choice of precomputed points in the modified comb method.
1017 * So branches for these cases do not leak secret information.
1019 * We accept Q->Z being unset (saving memory in tables) as meaning 1.
1021 * Cost: 1A := 8M + 3S
1023 static int ecp_add_mixed( const mbedtls_ecp_group
*grp
, mbedtls_ecp_point
*R
,
1024 const mbedtls_ecp_point
*P
, const mbedtls_ecp_point
*Q
)
1027 mbedtls_mpi T1
, T2
, T3
, T4
, X
, Y
, Z
;
1029 #if defined(MBEDTLS_SELF_TEST)
1033 #if defined(MBEDTLS_ECP_ADD_MIXED_ALT)
1034 if ( mbedtls_internal_ecp_grp_capable( grp
) )
1036 return mbedtls_internal_ecp_add_mixed( grp
, R
, P
, Q
);
1038 #endif /* MBEDTLS_ECP_ADD_MIXED_ALT */
1041 * Trivial cases: P == 0 or Q == 0 (case 1)
1043 if( mbedtls_mpi_cmp_int( &P
->Z
, 0 ) == 0 )
1044 return( mbedtls_ecp_copy( R
, Q
) );
1046 if( Q
->Z
.p
!= NULL
&& mbedtls_mpi_cmp_int( &Q
->Z
, 0 ) == 0 )
1047 return( mbedtls_ecp_copy( R
, P
) );
1050 * Make sure Q coordinates are normalized
1052 if( Q
->Z
.p
!= NULL
&& mbedtls_mpi_cmp_int( &Q
->Z
, 1 ) != 0 )
1053 return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA
);
1055 mbedtls_mpi_init( &T1
); mbedtls_mpi_init( &T2
); mbedtls_mpi_init( &T3
); mbedtls_mpi_init( &T4
);
1056 mbedtls_mpi_init( &X
); mbedtls_mpi_init( &Y
); mbedtls_mpi_init( &Z
);
1058 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T1
, &P
->Z
, &P
->Z
) ); MOD_MUL( T1
);
1059 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T2
, &T1
, &P
->Z
) ); MOD_MUL( T2
);
1060 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T1
, &T1
, &Q
->X
) ); MOD_MUL( T1
);
1061 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T2
, &T2
, &Q
->Y
) ); MOD_MUL( T2
);
1062 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &T1
, &T1
, &P
->X
) ); MOD_SUB( T1
);
1063 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &T2
, &T2
, &P
->Y
) ); MOD_SUB( T2
);
1065 /* Special cases (2) and (3) */
1066 if( mbedtls_mpi_cmp_int( &T1
, 0 ) == 0 )
1068 if( mbedtls_mpi_cmp_int( &T2
, 0 ) == 0 )
1070 ret
= ecp_double_jac( grp
, R
, P
);
1075 ret
= mbedtls_ecp_set_zero( R
);
1080 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &Z
, &P
->Z
, &T1
) ); MOD_MUL( Z
);
1081 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T3
, &T1
, &T1
) ); MOD_MUL( T3
);
1082 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T4
, &T3
, &T1
) ); MOD_MUL( T4
);
1083 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T3
, &T3
, &P
->X
) ); MOD_MUL( T3
);
1084 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_int( &T1
, &T3
, 2 ) ); MOD_ADD( T1
);
1085 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &X
, &T2
, &T2
) ); MOD_MUL( X
);
1086 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &X
, &X
, &T1
) ); MOD_SUB( X
);
1087 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &X
, &X
, &T4
) ); MOD_SUB( X
);
1088 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &T3
, &T3
, &X
) ); MOD_SUB( T3
);
1089 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T3
, &T3
, &T2
) ); MOD_MUL( T3
);
1090 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &T4
, &T4
, &P
->Y
) ); MOD_MUL( T4
);
1091 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &Y
, &T3
, &T4
) ); MOD_SUB( Y
);
1093 MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &R
->X
, &X
) );
1094 MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &R
->Y
, &Y
) );
1095 MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &R
->Z
, &Z
) );
1099 mbedtls_mpi_free( &T1
); mbedtls_mpi_free( &T2
); mbedtls_mpi_free( &T3
); mbedtls_mpi_free( &T4
);
1100 mbedtls_mpi_free( &X
); mbedtls_mpi_free( &Y
); mbedtls_mpi_free( &Z
);
1106 * Randomize jacobian coordinates:
1107 * (X, Y, Z) -> (l^2 X, l^3 Y, l Z) for random l
1108 * This is sort of the reverse operation of ecp_normalize_jac().
1110 * This countermeasure was first suggested in [2].
1112 static int ecp_randomize_jac( const mbedtls_ecp_group
*grp
, mbedtls_ecp_point
*pt
,
1113 int (*f_rng
)(void *, unsigned char *, size_t), void *p_rng
)
1120 #if defined(MBEDTLS_ECP_RANDOMIZE_JAC_ALT)
1121 if ( mbedtls_internal_ecp_grp_capable( grp
) )
1123 return mbedtls_internal_ecp_randomize_jac( grp
, pt
, f_rng
, p_rng
);
1125 #endif /* MBEDTLS_ECP_RANDOMIZE_JAC_ALT */
1127 p_size
= ( grp
->pbits
+ 7 ) / 8;
1128 mbedtls_mpi_init( &l
); mbedtls_mpi_init( &ll
);
1130 /* Generate l such that 1 < l < p */
1133 MBEDTLS_MPI_CHK( mbedtls_mpi_fill_random( &l
, p_size
, f_rng
, p_rng
) );
1135 while( mbedtls_mpi_cmp_mpi( &l
, &grp
->P
) >= 0 )
1136 MBEDTLS_MPI_CHK( mbedtls_mpi_shift_r( &l
, 1 ) );
1139 return( MBEDTLS_ERR_ECP_RANDOM_FAILED
);
1141 while( mbedtls_mpi_cmp_int( &l
, 1 ) <= 0 );
1144 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &pt
->Z
, &pt
->Z
, &l
) ); MOD_MUL( pt
->Z
);
1147 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &ll
, &l
, &l
) ); MOD_MUL( ll
);
1148 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &pt
->X
, &pt
->X
, &ll
) ); MOD_MUL( pt
->X
);
1151 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &ll
, &ll
, &l
) ); MOD_MUL( ll
);
1152 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &pt
->Y
, &pt
->Y
, &ll
) ); MOD_MUL( pt
->Y
);
1155 mbedtls_mpi_free( &l
); mbedtls_mpi_free( &ll
);
1161 * Check and define parameters used by the comb method (see below for details)
1163 #if MBEDTLS_ECP_WINDOW_SIZE < 2 || MBEDTLS_ECP_WINDOW_SIZE > 7
1164 #error "MBEDTLS_ECP_WINDOW_SIZE out of bounds"
1167 /* d = ceil( n / w ) */
1168 #define COMB_MAX_D ( MBEDTLS_ECP_MAX_BITS + 1 ) / 2
1170 /* number of precomputed points */
1171 #define COMB_MAX_PRE ( 1 << ( MBEDTLS_ECP_WINDOW_SIZE - 1 ) )
1174 * Compute the representation of m that will be used with our comb method.
1176 * The basic comb method is described in GECC 3.44 for example. We use a
1177 * modified version that provides resistance to SPA by avoiding zero
1178 * digits in the representation as in [3]. We modify the method further by
1179 * requiring that all K_i be odd, which has the small cost that our
1180 * representation uses one more K_i, due to carries.
1182 * Also, for the sake of compactness, only the seven low-order bits of x[i]
1183 * are used to represent K_i, and the msb of x[i] encodes the the sign (s_i in
1184 * the paper): it is set if and only if if s_i == -1;
1186 * Calling conventions:
1187 * - x is an array of size d + 1
1188 * - w is the size, ie number of teeth, of the comb, and must be between
1189 * 2 and 7 (in practice, between 2 and MBEDTLS_ECP_WINDOW_SIZE)
1190 * - m is the MPI, expected to be odd and such that bitlength(m) <= w * d
1191 * (the result will be incorrect if these assumptions are not satisfied)
1193 static void ecp_comb_fixed( unsigned char x
[], size_t d
,
1194 unsigned char w
, const mbedtls_mpi
*m
)
1197 unsigned char c
, cc
, adjust
;
1199 memset( x
, 0, d
+1 );
1201 /* First get the classical comb values (except for x_d = 0) */
1202 for( i
= 0; i
< d
; i
++ )
1203 for( j
= 0; j
< w
; j
++ )
1204 x
[i
] |= mbedtls_mpi_get_bit( m
, i
+ d
* j
) << j
;
1206 /* Now make sure x_1 .. x_d are odd */
1208 for( i
= 1; i
<= d
; i
++ )
1210 /* Add carry and update it */
1215 /* Adjust if needed, avoiding branches */
1216 adjust
= 1 - ( x
[i
] & 0x01 );
1217 c
|= x
[i
] & ( x
[i
-1] * adjust
);
1218 x
[i
] = x
[i
] ^ ( x
[i
-1] * adjust
);
1219 x
[i
-1] |= adjust
<< 7;
1224 * Precompute points for the comb method
1226 * If i = i_{w-1} ... i_1 is the binary representation of i, then
1227 * T[i] = i_{w-1} 2^{(w-1)d} P + ... + i_1 2^d P + P
1229 * T must be able to hold 2^{w - 1} elements
1231 * Cost: d(w-1) D + (2^{w-1} - 1) A + 1 N(w-1) + 1 N(2^{w-1} - 1)
1233 static int ecp_precompute_comb( const mbedtls_ecp_group
*grp
,
1234 mbedtls_ecp_point T
[], const mbedtls_ecp_point
*P
,
1235 unsigned char w
, size_t d
)
1240 mbedtls_ecp_point
*cur
, *TT
[COMB_MAX_PRE
- 1];
1244 * T[2^{l-1}] = 2^{dl} P for l = 1 .. w-1 (this is not the final value)
1246 MBEDTLS_MPI_CHK( mbedtls_ecp_copy( &T
[0], P
) );
1249 for( i
= 1; i
< ( 1U << ( w
- 1 ) ); i
<<= 1 )
1252 MBEDTLS_MPI_CHK( mbedtls_ecp_copy( cur
, T
+ ( i
>> 1 ) ) );
1253 for( j
= 0; j
< d
; j
++ )
1254 MBEDTLS_MPI_CHK( ecp_double_jac( grp
, cur
, cur
) );
1259 MBEDTLS_MPI_CHK( ecp_normalize_jac_many( grp
, TT
, k
) );
1262 * Compute the remaining ones using the minimal number of additions
1263 * Be careful to update T[2^l] only after using it!
1266 for( i
= 1; i
< ( 1U << ( w
- 1 ) ); i
<<= 1 )
1271 MBEDTLS_MPI_CHK( ecp_add_mixed( grp
, &T
[i
+ j
], &T
[j
], &T
[i
] ) );
1272 TT
[k
++] = &T
[i
+ j
];
1276 MBEDTLS_MPI_CHK( ecp_normalize_jac_many( grp
, TT
, k
) );
1284 * Select precomputed point: R = sign(i) * T[ abs(i) / 2 ]
1286 static int ecp_select_comb( const mbedtls_ecp_group
*grp
, mbedtls_ecp_point
*R
,
1287 const mbedtls_ecp_point T
[], unsigned char t_len
,
1291 unsigned char ii
, j
;
1293 /* Ignore the "sign" bit and scale down */
1294 ii
= ( i
& 0x7Fu
) >> 1;
1296 /* Read the whole table to thwart cache-based timing attacks */
1297 for( j
= 0; j
< t_len
; j
++ )
1299 MBEDTLS_MPI_CHK( mbedtls_mpi_safe_cond_assign( &R
->X
, &T
[j
].X
, j
== ii
) );
1300 MBEDTLS_MPI_CHK( mbedtls_mpi_safe_cond_assign( &R
->Y
, &T
[j
].Y
, j
== ii
) );
1303 /* Safely invert result if i is "negative" */
1304 MBEDTLS_MPI_CHK( ecp_safe_invert_jac( grp
, R
, i
>> 7 ) );
1311 * Core multiplication algorithm for the (modified) comb method.
1312 * This part is actually common with the basic comb method (GECC 3.44)
1314 * Cost: d A + d D + 1 R
1316 static int ecp_mul_comb_core( const mbedtls_ecp_group
*grp
, mbedtls_ecp_point
*R
,
1317 const mbedtls_ecp_point T
[], unsigned char t_len
,
1318 const unsigned char x
[], size_t d
,
1319 int (*f_rng
)(void *, unsigned char *, size_t),
1323 mbedtls_ecp_point Txi
;
1326 mbedtls_ecp_point_init( &Txi
);
1328 /* Start with a non-zero point and randomize its coordinates */
1330 MBEDTLS_MPI_CHK( ecp_select_comb( grp
, R
, T
, t_len
, x
[i
] ) );
1331 MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &R
->Z
, 1 ) );
1333 MBEDTLS_MPI_CHK( ecp_randomize_jac( grp
, R
, f_rng
, p_rng
) );
1337 MBEDTLS_MPI_CHK( ecp_double_jac( grp
, R
, R
) );
1338 MBEDTLS_MPI_CHK( ecp_select_comb( grp
, &Txi
, T
, t_len
, x
[i
] ) );
1339 MBEDTLS_MPI_CHK( ecp_add_mixed( grp
, R
, R
, &Txi
) );
1344 mbedtls_ecp_point_free( &Txi
);
1350 * Multiplication using the comb method,
1351 * for curves in short Weierstrass form
1353 static int ecp_mul_comb( mbedtls_ecp_group
*grp
, mbedtls_ecp_point
*R
,
1354 const mbedtls_mpi
*m
, const mbedtls_ecp_point
*P
,
1355 int (*f_rng
)(void *, unsigned char *, size_t),
1359 unsigned char w
, m_is_odd
, p_eq_g
, pre_len
, i
;
1361 unsigned char k
[COMB_MAX_D
+ 1];
1362 mbedtls_ecp_point
*T
;
1365 mbedtls_mpi_init( &M
);
1366 mbedtls_mpi_init( &mm
);
1368 /* we need N to be odd to trnaform m in an odd number, check now */
1369 if( mbedtls_mpi_get_bit( &grp
->N
, 0 ) != 1 )
1370 return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA
);
1373 * Minimize the number of multiplications, that is minimize
1374 * 10 * d * w + 18 * 2^(w-1) + 11 * d + 7 * w, with d = ceil( nbits / w )
1375 * (see costs of the various parts, with 1S = 1M)
1377 w
= grp
->nbits
>= 384 ? 5 : 4;
1380 * If P == G, pre-compute a bit more, since this may be re-used later.
1381 * Just adding one avoids upping the cost of the first mul too much,
1382 * and the memory cost too.
1384 #if MBEDTLS_ECP_FIXED_POINT_OPTIM == 1
1385 p_eq_g
= ( mbedtls_mpi_cmp_mpi( &P
->Y
, &grp
->G
.Y
) == 0 &&
1386 mbedtls_mpi_cmp_mpi( &P
->X
, &grp
->G
.X
) == 0 );
1394 * Make sure w is within bounds.
1395 * (The last test is useful only for very small curves in the test suite.)
1397 if( w
> MBEDTLS_ECP_WINDOW_SIZE
)
1398 w
= MBEDTLS_ECP_WINDOW_SIZE
;
1399 if( w
>= grp
->nbits
)
1402 /* Other sizes that depend on w */
1403 pre_len
= 1U << ( w
- 1 );
1404 d
= ( grp
->nbits
+ w
- 1 ) / w
;
1407 * Prepare precomputed points: if P == G we want to
1408 * use grp->T if already initialized, or initialize it.
1410 T
= p_eq_g
? grp
->T
: NULL
;
1414 T
= mbedtls_calloc( pre_len
, sizeof( mbedtls_ecp_point
) );
1417 ret
= MBEDTLS_ERR_ECP_ALLOC_FAILED
;
1421 MBEDTLS_MPI_CHK( ecp_precompute_comb( grp
, T
, P
, w
, d
) );
1426 grp
->T_size
= pre_len
;
1431 * Make sure M is odd (M = m or M = N - m, since N is odd)
1432 * using the fact that m * P = - (N - m) * P
1434 m_is_odd
= ( mbedtls_mpi_get_bit( m
, 0 ) == 1 );
1435 MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &M
, m
) );
1436 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &mm
, &grp
->N
, m
) );
1437 MBEDTLS_MPI_CHK( mbedtls_mpi_safe_cond_assign( &M
, &mm
, ! m_is_odd
) );
1440 * Go for comb multiplication, R = M * P
1442 ecp_comb_fixed( k
, d
, w
, &M
);
1443 MBEDTLS_MPI_CHK( ecp_mul_comb_core( grp
, R
, T
, pre_len
, k
, d
, f_rng
, p_rng
) );
1446 * Now get m * P from M * P and normalize it
1448 MBEDTLS_MPI_CHK( ecp_safe_invert_jac( grp
, R
, ! m_is_odd
) );
1449 MBEDTLS_MPI_CHK( ecp_normalize_jac( grp
, R
) );
1453 /* There are two cases where T is not stored in grp:
1455 * - An intermediate operation failed before setting grp->T
1456 * In either case, T must be freed.
1458 if( T
!= NULL
&& T
!= grp
->T
)
1460 for( i
= 0; i
< pre_len
; i
++ )
1461 mbedtls_ecp_point_free( &T
[i
] );
1465 mbedtls_mpi_free( &M
);
1466 mbedtls_mpi_free( &mm
);
1469 mbedtls_ecp_point_free( R
);
1474 #endif /* ECP_SHORTWEIERSTRASS */
1476 #if defined(ECP_MONTGOMERY)
1478 * For Montgomery curves, we do all the internal arithmetic in projective
1479 * coordinates. Import/export of points uses only the x coordinates, which is
1480 * internaly represented as X / Z.
1482 * For scalar multiplication, we'll use a Montgomery ladder.
1486 * Normalize Montgomery x/z coordinates: X = X/Z, Z = 1
1489 static int ecp_normalize_mxz( const mbedtls_ecp_group
*grp
, mbedtls_ecp_point
*P
)
1493 #if defined(MBEDTLS_ECP_NORMALIZE_MXZ_ALT)
1494 if ( mbedtls_internal_ecp_grp_capable( grp
) )
1496 return mbedtls_internal_ecp_normalize_mxz( grp
, P
);
1498 #endif /* MBEDTLS_ECP_NORMALIZE_MXZ_ALT */
1500 MBEDTLS_MPI_CHK( mbedtls_mpi_inv_mod( &P
->Z
, &P
->Z
, &grp
->P
) );
1501 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &P
->X
, &P
->X
, &P
->Z
) ); MOD_MUL( P
->X
);
1502 MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &P
->Z
, 1 ) );
1509 * Randomize projective x/z coordinates:
1510 * (X, Z) -> (l X, l Z) for random l
1511 * This is sort of the reverse operation of ecp_normalize_mxz().
1513 * This countermeasure was first suggested in [2].
1516 static int ecp_randomize_mxz( const mbedtls_ecp_group
*grp
, mbedtls_ecp_point
*P
,
1517 int (*f_rng
)(void *, unsigned char *, size_t), void *p_rng
)
1524 #if defined(MBEDTLS_ECP_RANDOMIZE_MXZ_ALT)
1525 if ( mbedtls_internal_ecp_grp_capable( grp
) )
1527 return mbedtls_internal_ecp_randomize_mxz( grp
, P
, f_rng
, p_rng
);
1529 #endif /* MBEDTLS_ECP_RANDOMIZE_MXZ_ALT */
1531 p_size
= ( grp
->pbits
+ 7 ) / 8;
1532 mbedtls_mpi_init( &l
);
1534 /* Generate l such that 1 < l < p */
1537 MBEDTLS_MPI_CHK( mbedtls_mpi_fill_random( &l
, p_size
, f_rng
, p_rng
) );
1539 while( mbedtls_mpi_cmp_mpi( &l
, &grp
->P
) >= 0 )
1540 MBEDTLS_MPI_CHK( mbedtls_mpi_shift_r( &l
, 1 ) );
1543 return( MBEDTLS_ERR_ECP_RANDOM_FAILED
);
1545 while( mbedtls_mpi_cmp_int( &l
, 1 ) <= 0 );
1547 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &P
->X
, &P
->X
, &l
) ); MOD_MUL( P
->X
);
1548 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &P
->Z
, &P
->Z
, &l
) ); MOD_MUL( P
->Z
);
1551 mbedtls_mpi_free( &l
);
1557 * Double-and-add: R = 2P, S = P + Q, with d = X(P - Q),
1558 * for Montgomery curves in x/z coordinates.
1560 * http://www.hyperelliptic.org/EFD/g1p/auto-code/montgom/xz/ladder/mladd-1987-m.op3
1567 * and eliminating temporary variables tO, ..., t4.
1571 static int ecp_double_add_mxz( const mbedtls_ecp_group
*grp
,
1572 mbedtls_ecp_point
*R
, mbedtls_ecp_point
*S
,
1573 const mbedtls_ecp_point
*P
, const mbedtls_ecp_point
*Q
,
1574 const mbedtls_mpi
*d
)
1577 mbedtls_mpi A
, AA
, B
, BB
, E
, C
, D
, DA
, CB
;
1579 #if defined(MBEDTLS_ECP_DOUBLE_ADD_MXZ_ALT)
1580 if ( mbedtls_internal_ecp_grp_capable( grp
) )
1582 return mbedtls_internal_ecp_double_add_mxz( grp
, R
, S
, P
, Q
, d
);
1584 #endif /* MBEDTLS_ECP_DOUBLE_ADD_MXZ_ALT */
1586 mbedtls_mpi_init( &A
); mbedtls_mpi_init( &AA
); mbedtls_mpi_init( &B
);
1587 mbedtls_mpi_init( &BB
); mbedtls_mpi_init( &E
); mbedtls_mpi_init( &C
);
1588 mbedtls_mpi_init( &D
); mbedtls_mpi_init( &DA
); mbedtls_mpi_init( &CB
);
1590 MBEDTLS_MPI_CHK( mbedtls_mpi_add_mpi( &A
, &P
->X
, &P
->Z
) ); MOD_ADD( A
);
1591 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &AA
, &A
, &A
) ); MOD_MUL( AA
);
1592 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &B
, &P
->X
, &P
->Z
) ); MOD_SUB( B
);
1593 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &BB
, &B
, &B
) ); MOD_MUL( BB
);
1594 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &E
, &AA
, &BB
) ); MOD_SUB( E
);
1595 MBEDTLS_MPI_CHK( mbedtls_mpi_add_mpi( &C
, &Q
->X
, &Q
->Z
) ); MOD_ADD( C
);
1596 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &D
, &Q
->X
, &Q
->Z
) ); MOD_SUB( D
);
1597 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &DA
, &D
, &A
) ); MOD_MUL( DA
);
1598 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &CB
, &C
, &B
) ); MOD_MUL( CB
);
1599 MBEDTLS_MPI_CHK( mbedtls_mpi_add_mpi( &S
->X
, &DA
, &CB
) ); MOD_MUL( S
->X
);
1600 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &S
->X
, &S
->X
, &S
->X
) ); MOD_MUL( S
->X
);
1601 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &S
->Z
, &DA
, &CB
) ); MOD_SUB( S
->Z
);
1602 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &S
->Z
, &S
->Z
, &S
->Z
) ); MOD_MUL( S
->Z
);
1603 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &S
->Z
, d
, &S
->Z
) ); MOD_MUL( S
->Z
);
1604 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &R
->X
, &AA
, &BB
) ); MOD_MUL( R
->X
);
1605 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &R
->Z
, &grp
->A
, &E
) ); MOD_MUL( R
->Z
);
1606 MBEDTLS_MPI_CHK( mbedtls_mpi_add_mpi( &R
->Z
, &BB
, &R
->Z
) ); MOD_ADD( R
->Z
);
1607 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &R
->Z
, &E
, &R
->Z
) ); MOD_MUL( R
->Z
);
1610 mbedtls_mpi_free( &A
); mbedtls_mpi_free( &AA
); mbedtls_mpi_free( &B
);
1611 mbedtls_mpi_free( &BB
); mbedtls_mpi_free( &E
); mbedtls_mpi_free( &C
);
1612 mbedtls_mpi_free( &D
); mbedtls_mpi_free( &DA
); mbedtls_mpi_free( &CB
);
1618 * Multiplication with Montgomery ladder in x/z coordinates,
1619 * for curves in Montgomery form
1621 static int ecp_mul_mxz( mbedtls_ecp_group
*grp
, mbedtls_ecp_point
*R
,
1622 const mbedtls_mpi
*m
, const mbedtls_ecp_point
*P
,
1623 int (*f_rng
)(void *, unsigned char *, size_t),
1629 mbedtls_ecp_point RP
;
1632 mbedtls_ecp_point_init( &RP
); mbedtls_mpi_init( &PX
);
1634 /* Save PX and read from P before writing to R, in case P == R */
1635 MBEDTLS_MPI_CHK( mbedtls_mpi_copy( &PX
, &P
->X
) );
1636 MBEDTLS_MPI_CHK( mbedtls_ecp_copy( &RP
, P
) );
1638 /* Set R to zero in modified x/z coordinates */
1639 MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &R
->X
, 1 ) );
1640 MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &R
->Z
, 0 ) );
1641 mbedtls_mpi_free( &R
->Y
);
1643 /* RP.X might be sligtly larger than P, so reduce it */
1646 /* Randomize coordinates of the starting point */
1648 MBEDTLS_MPI_CHK( ecp_randomize_mxz( grp
, &RP
, f_rng
, p_rng
) );
1650 /* Loop invariant: R = result so far, RP = R + P */
1651 i
= mbedtls_mpi_bitlen( m
); /* one past the (zero-based) most significant bit */
1654 b
= mbedtls_mpi_get_bit( m
, i
);
1656 * if (b) R = 2R + P else R = 2R,
1658 * if (b) double_add( RP, R, RP, R )
1659 * else double_add( R, RP, R, RP )
1660 * but using safe conditional swaps to avoid leaks
1662 MBEDTLS_MPI_CHK( mbedtls_mpi_safe_cond_swap( &R
->X
, &RP
.X
, b
) );
1663 MBEDTLS_MPI_CHK( mbedtls_mpi_safe_cond_swap( &R
->Z
, &RP
.Z
, b
) );
1664 MBEDTLS_MPI_CHK( ecp_double_add_mxz( grp
, R
, &RP
, R
, &RP
, &PX
) );
1665 MBEDTLS_MPI_CHK( mbedtls_mpi_safe_cond_swap( &R
->X
, &RP
.X
, b
) );
1666 MBEDTLS_MPI_CHK( mbedtls_mpi_safe_cond_swap( &R
->Z
, &RP
.Z
, b
) );
1669 MBEDTLS_MPI_CHK( ecp_normalize_mxz( grp
, R
) );
1672 mbedtls_ecp_point_free( &RP
); mbedtls_mpi_free( &PX
);
1677 #endif /* ECP_MONTGOMERY */
1680 * Multiplication R = m * P
1682 int mbedtls_ecp_mul( mbedtls_ecp_group
*grp
, mbedtls_ecp_point
*R
,
1683 const mbedtls_mpi
*m
, const mbedtls_ecp_point
*P
,
1684 int (*f_rng
)(void *, unsigned char *, size_t), void *p_rng
)
1686 int ret
= MBEDTLS_ERR_ECP_BAD_INPUT_DATA
;
1687 #if defined(MBEDTLS_ECP_INTERNAL_ALT)
1688 char is_grp_capable
= 0;
1691 /* Common sanity checks */
1692 if( mbedtls_mpi_cmp_int( &P
->Z
, 1 ) != 0 )
1693 return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA
);
1695 if( ( ret
= mbedtls_ecp_check_privkey( grp
, m
) ) != 0 ||
1696 ( ret
= mbedtls_ecp_check_pubkey( grp
, P
) ) != 0 )
1699 #if defined(MBEDTLS_ECP_INTERNAL_ALT)
1700 if ( is_grp_capable
= mbedtls_internal_ecp_grp_capable( grp
) )
1702 MBEDTLS_MPI_CHK( mbedtls_internal_ecp_init( grp
) );
1705 #endif /* MBEDTLS_ECP_INTERNAL_ALT */
1706 #if defined(ECP_MONTGOMERY)
1707 if( ecp_get_type( grp
) == ECP_TYPE_MONTGOMERY
)
1708 ret
= ecp_mul_mxz( grp
, R
, m
, P
, f_rng
, p_rng
);
1711 #if defined(ECP_SHORTWEIERSTRASS)
1712 if( ecp_get_type( grp
) == ECP_TYPE_SHORT_WEIERSTRASS
)
1713 ret
= ecp_mul_comb( grp
, R
, m
, P
, f_rng
, p_rng
);
1716 #if defined(MBEDTLS_ECP_INTERNAL_ALT)
1719 if ( is_grp_capable
)
1721 mbedtls_internal_ecp_free( grp
);
1724 #endif /* MBEDTLS_ECP_INTERNAL_ALT */
1728 #if defined(ECP_SHORTWEIERSTRASS)
1730 * Check that an affine point is valid as a public key,
1731 * short weierstrass curves (SEC1 3.2.3.1)
1733 static int ecp_check_pubkey_sw( const mbedtls_ecp_group
*grp
, const mbedtls_ecp_point
*pt
)
1736 mbedtls_mpi YY
, RHS
;
1738 /* pt coordinates must be normalized for our checks */
1739 if( mbedtls_mpi_cmp_int( &pt
->X
, 0 ) < 0 ||
1740 mbedtls_mpi_cmp_int( &pt
->Y
, 0 ) < 0 ||
1741 mbedtls_mpi_cmp_mpi( &pt
->X
, &grp
->P
) >= 0 ||
1742 mbedtls_mpi_cmp_mpi( &pt
->Y
, &grp
->P
) >= 0 )
1743 return( MBEDTLS_ERR_ECP_INVALID_KEY
);
1745 mbedtls_mpi_init( &YY
); mbedtls_mpi_init( &RHS
);
1749 * RHS = X (X^2 + A) + B = X^3 + A X + B
1751 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &YY
, &pt
->Y
, &pt
->Y
) ); MOD_MUL( YY
);
1752 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &RHS
, &pt
->X
, &pt
->X
) ); MOD_MUL( RHS
);
1754 /* Special case for A = -3 */
1755 if( grp
->A
.p
== NULL
)
1757 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_int( &RHS
, &RHS
, 3 ) ); MOD_SUB( RHS
);
1761 MBEDTLS_MPI_CHK( mbedtls_mpi_add_mpi( &RHS
, &RHS
, &grp
->A
) ); MOD_ADD( RHS
);
1764 MBEDTLS_MPI_CHK( mbedtls_mpi_mul_mpi( &RHS
, &RHS
, &pt
->X
) ); MOD_MUL( RHS
);
1765 MBEDTLS_MPI_CHK( mbedtls_mpi_add_mpi( &RHS
, &RHS
, &grp
->B
) ); MOD_ADD( RHS
);
1767 if( mbedtls_mpi_cmp_mpi( &YY
, &RHS
) != 0 )
1768 ret
= MBEDTLS_ERR_ECP_INVALID_KEY
;
1772 mbedtls_mpi_free( &YY
); mbedtls_mpi_free( &RHS
);
1776 #endif /* ECP_SHORTWEIERSTRASS */
1779 * R = m * P with shortcuts for m == 1 and m == -1
1780 * NOT constant-time - ONLY for short Weierstrass!
1782 static int mbedtls_ecp_mul_shortcuts( mbedtls_ecp_group
*grp
,
1783 mbedtls_ecp_point
*R
,
1784 const mbedtls_mpi
*m
,
1785 const mbedtls_ecp_point
*P
)
1789 if( mbedtls_mpi_cmp_int( m
, 1 ) == 0 )
1791 MBEDTLS_MPI_CHK( mbedtls_ecp_copy( R
, P
) );
1793 else if( mbedtls_mpi_cmp_int( m
, -1 ) == 0 )
1795 MBEDTLS_MPI_CHK( mbedtls_ecp_copy( R
, P
) );
1796 if( mbedtls_mpi_cmp_int( &R
->Y
, 0 ) != 0 )
1797 MBEDTLS_MPI_CHK( mbedtls_mpi_sub_mpi( &R
->Y
, &grp
->P
, &R
->Y
) );
1801 MBEDTLS_MPI_CHK( mbedtls_ecp_mul( grp
, R
, m
, P
, NULL
, NULL
) );
1809 * Linear combination
1812 int mbedtls_ecp_muladd( mbedtls_ecp_group
*grp
, mbedtls_ecp_point
*R
,
1813 const mbedtls_mpi
*m
, const mbedtls_ecp_point
*P
,
1814 const mbedtls_mpi
*n
, const mbedtls_ecp_point
*Q
)
1817 mbedtls_ecp_point mP
;
1818 #if defined(MBEDTLS_ECP_INTERNAL_ALT)
1819 char is_grp_capable
= 0;
1822 if( ecp_get_type( grp
) != ECP_TYPE_SHORT_WEIERSTRASS
)
1823 return( MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE
);
1825 mbedtls_ecp_point_init( &mP
);
1827 MBEDTLS_MPI_CHK( mbedtls_ecp_mul_shortcuts( grp
, &mP
, m
, P
) );
1828 MBEDTLS_MPI_CHK( mbedtls_ecp_mul_shortcuts( grp
, R
, n
, Q
) );
1830 #if defined(MBEDTLS_ECP_INTERNAL_ALT)
1831 if ( is_grp_capable
= mbedtls_internal_ecp_grp_capable( grp
) )
1833 MBEDTLS_MPI_CHK( mbedtls_internal_ecp_init( grp
) );
1836 #endif /* MBEDTLS_ECP_INTERNAL_ALT */
1837 MBEDTLS_MPI_CHK( ecp_add_mixed( grp
, R
, &mP
, R
) );
1838 MBEDTLS_MPI_CHK( ecp_normalize_jac( grp
, R
) );
1842 #if defined(MBEDTLS_ECP_INTERNAL_ALT)
1843 if ( is_grp_capable
)
1845 mbedtls_internal_ecp_free( grp
);
1848 #endif /* MBEDTLS_ECP_INTERNAL_ALT */
1849 mbedtls_ecp_point_free( &mP
);
1855 #if defined(ECP_MONTGOMERY)
1857 * Check validity of a public key for Montgomery curves with x-only schemes
1859 static int ecp_check_pubkey_mx( const mbedtls_ecp_group
*grp
, const mbedtls_ecp_point
*pt
)
1861 /* [Curve25519 p. 5] Just check X is the correct number of bytes */
1862 if( mbedtls_mpi_size( &pt
->X
) > ( grp
->nbits
+ 7 ) / 8 )
1863 return( MBEDTLS_ERR_ECP_INVALID_KEY
);
1867 #endif /* ECP_MONTGOMERY */
1870 * Check that a point is valid as a public key
1872 int mbedtls_ecp_check_pubkey( const mbedtls_ecp_group
*grp
, const mbedtls_ecp_point
*pt
)
1874 /* Must use affine coordinates */
1875 if( mbedtls_mpi_cmp_int( &pt
->Z
, 1 ) != 0 )
1876 return( MBEDTLS_ERR_ECP_INVALID_KEY
);
1878 #if defined(ECP_MONTGOMERY)
1879 if( ecp_get_type( grp
) == ECP_TYPE_MONTGOMERY
)
1880 return( ecp_check_pubkey_mx( grp
, pt
) );
1882 #if defined(ECP_SHORTWEIERSTRASS)
1883 if( ecp_get_type( grp
) == ECP_TYPE_SHORT_WEIERSTRASS
)
1884 return( ecp_check_pubkey_sw( grp
, pt
) );
1886 return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA
);
1890 * Check that an mbedtls_mpi is valid as a private key
1892 int mbedtls_ecp_check_privkey( const mbedtls_ecp_group
*grp
, const mbedtls_mpi
*d
)
1894 #if defined(ECP_MONTGOMERY)
1895 if( ecp_get_type( grp
) == ECP_TYPE_MONTGOMERY
)
1897 /* see [Curve25519] page 5 */
1898 if( mbedtls_mpi_get_bit( d
, 0 ) != 0 ||
1899 mbedtls_mpi_get_bit( d
, 1 ) != 0 ||
1900 mbedtls_mpi_get_bit( d
, 2 ) != 0 ||
1901 mbedtls_mpi_bitlen( d
) - 1 != grp
->nbits
) /* mbedtls_mpi_bitlen is one-based! */
1902 return( MBEDTLS_ERR_ECP_INVALID_KEY
);
1906 #endif /* ECP_MONTGOMERY */
1907 #if defined(ECP_SHORTWEIERSTRASS)
1908 if( ecp_get_type( grp
) == ECP_TYPE_SHORT_WEIERSTRASS
)
1911 if( mbedtls_mpi_cmp_int( d
, 1 ) < 0 ||
1912 mbedtls_mpi_cmp_mpi( d
, &grp
->N
) >= 0 )
1913 return( MBEDTLS_ERR_ECP_INVALID_KEY
);
1917 #endif /* ECP_SHORTWEIERSTRASS */
1919 return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA
);
1923 * Generate a keypair with configurable base point
1925 int mbedtls_ecp_gen_keypair_base( mbedtls_ecp_group
*grp
,
1926 const mbedtls_ecp_point
*G
,
1927 mbedtls_mpi
*d
, mbedtls_ecp_point
*Q
,
1928 int (*f_rng
)(void *, unsigned char *, size_t),
1932 size_t n_size
= ( grp
->nbits
+ 7 ) / 8;
1934 #if defined(ECP_MONTGOMERY)
1935 if( ecp_get_type( grp
) == ECP_TYPE_MONTGOMERY
)
1941 MBEDTLS_MPI_CHK( mbedtls_mpi_fill_random( d
, n_size
, f_rng
, p_rng
) );
1942 } while( mbedtls_mpi_bitlen( d
) == 0);
1944 /* Make sure the most significant bit is nbits */
1945 b
= mbedtls_mpi_bitlen( d
) - 1; /* mbedtls_mpi_bitlen is one-based */
1946 if( b
> grp
->nbits
)
1947 MBEDTLS_MPI_CHK( mbedtls_mpi_shift_r( d
, b
- grp
->nbits
) );
1949 MBEDTLS_MPI_CHK( mbedtls_mpi_set_bit( d
, grp
->nbits
, 1 ) );
1951 /* Make sure the last three bits are unset */
1952 MBEDTLS_MPI_CHK( mbedtls_mpi_set_bit( d
, 0, 0 ) );
1953 MBEDTLS_MPI_CHK( mbedtls_mpi_set_bit( d
, 1, 0 ) );
1954 MBEDTLS_MPI_CHK( mbedtls_mpi_set_bit( d
, 2, 0 ) );
1957 #endif /* ECP_MONTGOMERY */
1958 #if defined(ECP_SHORTWEIERSTRASS)
1959 if( ecp_get_type( grp
) == ECP_TYPE_SHORT_WEIERSTRASS
)
1961 /* SEC1 3.2.1: Generate d such that 1 <= n < N */
1965 * Match the procedure given in RFC 6979 (deterministic ECDSA):
1966 * - use the same byte ordering;
1967 * - keep the leftmost nbits bits of the generated octet string;
1968 * - try until result is in the desired range.
1969 * This also avoids any biais, which is especially important for ECDSA.
1973 MBEDTLS_MPI_CHK( mbedtls_mpi_fill_random( d
, n_size
, f_rng
, p_rng
) );
1974 MBEDTLS_MPI_CHK( mbedtls_mpi_shift_r( d
, 8 * n_size
- grp
->nbits
) );
1977 * Each try has at worst a probability 1/2 of failing (the msb has
1978 * a probability 1/2 of being 0, and then the result will be < N),
1979 * so after 30 tries failure probability is a most 2**(-30).
1981 * For most curves, 1 try is enough with overwhelming probability,
1982 * since N starts with a lot of 1s in binary, but some curves
1983 * such as secp224k1 are actually very close to the worst case.
1986 return( MBEDTLS_ERR_ECP_RANDOM_FAILED
);
1988 while( mbedtls_mpi_cmp_int( d
, 1 ) < 0 ||
1989 mbedtls_mpi_cmp_mpi( d
, &grp
->N
) >= 0 );
1992 #endif /* ECP_SHORTWEIERSTRASS */
1993 return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA
);
1999 return( mbedtls_ecp_mul( grp
, Q
, d
, G
, f_rng
, p_rng
) );
2003 * Generate key pair, wrapper for conventional base point
2005 int mbedtls_ecp_gen_keypair( mbedtls_ecp_group
*grp
,
2006 mbedtls_mpi
*d
, mbedtls_ecp_point
*Q
,
2007 int (*f_rng
)(void *, unsigned char *, size_t),
2010 return( mbedtls_ecp_gen_keypair_base( grp
, &grp
->G
, d
, Q
, f_rng
, p_rng
) );
2014 * Generate a keypair, prettier wrapper
2016 int mbedtls_ecp_gen_key( mbedtls_ecp_group_id grp_id
, mbedtls_ecp_keypair
*key
,
2017 int (*f_rng
)(void *, unsigned char *, size_t), void *p_rng
)
2021 if( ( ret
= mbedtls_ecp_group_load( &key
->grp
, grp_id
) ) != 0 )
2024 return( mbedtls_ecp_gen_keypair( &key
->grp
, &key
->d
, &key
->Q
, f_rng
, p_rng
) );
2028 * Check a public-private key pair
2030 int mbedtls_ecp_check_pub_priv( const mbedtls_ecp_keypair
*pub
, const mbedtls_ecp_keypair
*prv
)
2033 mbedtls_ecp_point Q
;
2034 mbedtls_ecp_group grp
;
2036 if( pub
->grp
.id
== MBEDTLS_ECP_DP_NONE
||
2037 pub
->grp
.id
!= prv
->grp
.id
||
2038 mbedtls_mpi_cmp_mpi( &pub
->Q
.X
, &prv
->Q
.X
) ||
2039 mbedtls_mpi_cmp_mpi( &pub
->Q
.Y
, &prv
->Q
.Y
) ||
2040 mbedtls_mpi_cmp_mpi( &pub
->Q
.Z
, &prv
->Q
.Z
) )
2042 return( MBEDTLS_ERR_ECP_BAD_INPUT_DATA
);
2045 mbedtls_ecp_point_init( &Q
);
2046 mbedtls_ecp_group_init( &grp
);
2048 /* mbedtls_ecp_mul() needs a non-const group... */
2049 mbedtls_ecp_group_copy( &grp
, &prv
->grp
);
2051 /* Also checks d is valid */
2052 MBEDTLS_MPI_CHK( mbedtls_ecp_mul( &grp
, &Q
, &prv
->d
, &prv
->grp
.G
, NULL
, NULL
) );
2054 if( mbedtls_mpi_cmp_mpi( &Q
.X
, &prv
->Q
.X
) ||
2055 mbedtls_mpi_cmp_mpi( &Q
.Y
, &prv
->Q
.Y
) ||
2056 mbedtls_mpi_cmp_mpi( &Q
.Z
, &prv
->Q
.Z
) )
2058 ret
= MBEDTLS_ERR_ECP_BAD_INPUT_DATA
;
2063 mbedtls_ecp_point_free( &Q
);
2064 mbedtls_ecp_group_free( &grp
);
2069 #if defined(MBEDTLS_SELF_TEST)
2074 int mbedtls_ecp_self_test( int verbose
)
2078 mbedtls_ecp_group grp
;
2079 mbedtls_ecp_point R
, P
;
2081 unsigned long add_c_prev
, dbl_c_prev
, mul_c_prev
;
2082 /* exponents especially adapted for secp192r1 */
2083 const char *exponents
[] =
2085 "000000000000000000000000000000000000000000000001", /* one */
2086 "FFFFFFFFFFFFFFFFFFFFFFFF99DEF836146BC9B1B4D22830", /* N - 1 */
2087 "5EA6F389A38B8BC81E767753B15AA5569E1782E30ABE7D25", /* random */
2088 "400000000000000000000000000000000000000000000000", /* one and zeros */
2089 "7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF", /* all ones */
2090 "555555555555555555555555555555555555555555555555", /* 101010... */
2093 mbedtls_ecp_group_init( &grp
);
2094 mbedtls_ecp_point_init( &R
);
2095 mbedtls_ecp_point_init( &P
);
2096 mbedtls_mpi_init( &m
);
2098 /* Use secp192r1 if available, or any available curve */
2099 #if defined(MBEDTLS_ECP_DP_SECP192R1_ENABLED)
2100 MBEDTLS_MPI_CHK( mbedtls_ecp_group_load( &grp
, MBEDTLS_ECP_DP_SECP192R1
) );
2102 MBEDTLS_MPI_CHK( mbedtls_ecp_group_load( &grp
, mbedtls_ecp_curve_list()->grp_id
) );
2106 mbedtls_printf( " ECP test #1 (constant op_count, base point G): " );
2108 /* Do a dummy multiplication first to trigger precomputation */
2109 MBEDTLS_MPI_CHK( mbedtls_mpi_lset( &m
, 2 ) );
2110 MBEDTLS_MPI_CHK( mbedtls_ecp_mul( &grp
, &P
, &m
, &grp
.G
, NULL
, NULL
) );
2115 MBEDTLS_MPI_CHK( mbedtls_mpi_read_string( &m
, 16, exponents
[0] ) );
2116 MBEDTLS_MPI_CHK( mbedtls_ecp_mul( &grp
, &R
, &m
, &grp
.G
, NULL
, NULL
) );
2118 for( i
= 1; i
< sizeof( exponents
) / sizeof( exponents
[0] ); i
++ )
2120 add_c_prev
= add_count
;
2121 dbl_c_prev
= dbl_count
;
2122 mul_c_prev
= mul_count
;
2127 MBEDTLS_MPI_CHK( mbedtls_mpi_read_string( &m
, 16, exponents
[i
] ) );
2128 MBEDTLS_MPI_CHK( mbedtls_ecp_mul( &grp
, &R
, &m
, &grp
.G
, NULL
, NULL
) );
2130 if( add_count
!= add_c_prev
||
2131 dbl_count
!= dbl_c_prev
||
2132 mul_count
!= mul_c_prev
)
2135 mbedtls_printf( "failed (%u)\n", (unsigned int) i
);
2143 mbedtls_printf( "passed\n" );
2146 mbedtls_printf( " ECP test #2 (constant op_count, other point): " );
2147 /* We computed P = 2G last time, use it */
2152 MBEDTLS_MPI_CHK( mbedtls_mpi_read_string( &m
, 16, exponents
[0] ) );
2153 MBEDTLS_MPI_CHK( mbedtls_ecp_mul( &grp
, &R
, &m
, &P
, NULL
, NULL
) );
2155 for( i
= 1; i
< sizeof( exponents
) / sizeof( exponents
[0] ); i
++ )
2157 add_c_prev
= add_count
;
2158 dbl_c_prev
= dbl_count
;
2159 mul_c_prev
= mul_count
;
2164 MBEDTLS_MPI_CHK( mbedtls_mpi_read_string( &m
, 16, exponents
[i
] ) );
2165 MBEDTLS_MPI_CHK( mbedtls_ecp_mul( &grp
, &R
, &m
, &P
, NULL
, NULL
) );
2167 if( add_count
!= add_c_prev
||
2168 dbl_count
!= dbl_c_prev
||
2169 mul_count
!= mul_c_prev
)
2172 mbedtls_printf( "failed (%u)\n", (unsigned int) i
);
2180 mbedtls_printf( "passed\n" );
2184 if( ret
< 0 && verbose
!= 0 )
2185 mbedtls_printf( "Unexpected error, return code = %08X\n", ret
);
2187 mbedtls_ecp_group_free( &grp
);
2188 mbedtls_ecp_point_free( &R
);
2189 mbedtls_ecp_point_free( &P
);
2190 mbedtls_mpi_free( &m
);
2193 mbedtls_printf( "\n" );
2198 #endif /* MBEDTLS_SELF_TEST */
2200 #endif /* !MBEDTLS_ECP_ALT */
2202 #endif /* MBEDTLS_ECP_C */