/*
* Elliptic curves over GF(p): generic functions
*
* Copyright (C) 2006-2015, ARM Limited, All Rights Reserved
* SPDX-License-Identifier: GPL-2.0
*
* 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.
*
* This file is part of mbed TLS (https://tls.mbed.org)
*/
/*
* 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
* RFC 7748 for the Curve448 and Curve25519 curve definitions
*
* [Curve25519] http://cr.yp.to/ecdh/curve25519-20060209.pdf
*
* [2] CORON, Jean-S'ebastien. Resistance against differential power analysis
* for elliptic curve cryptosystems. In : Cryptographic Hardware and
* Embedded Systems. Springer Berlin Heidelberg, 1999. p. 292-302.
* <http://link.springer.com/chapter/10.1007/3-540-48059-5_25>
*
* [3] HEDABOU, Mustapha, PINEL, Pierre, et B'EN'ETEAU, Lucien. A comb method to
* render ECC resistant against Side Channel Attacks. IACR Cryptology
* ePrint Archive, 2004, vol. 2004, p. 342.
* <http://eprint.iacr.org/2004/342.pdf>
*/
#if !defined(MBEDTLS_CONFIG_FILE)
#include "mbedtls/config.h"
#else
#include MBEDTLS_CONFIG_FILE
#endif
#if defined(MBEDTLS_ECP_C)
#include "mbedtls/ecp.h"
#include "mbedtls/threading.h"
#include "mbedtls/platform_util.h"
#include <string.h>
#if !defined(MBEDTLS_ECP_ALT)
#if defined(MBEDTLS_PLATFORM_C)
#include "mbedtls/platform.h"
#else
#include <stdlib.h>
#include <stdio.h>
#define mbedtls_printf printf
#define mbedtls_calloc calloc
#define mbedtls_free free
#endif
#include "mbedtls/ecp_internal.h"
#if ( defined(__ARMCC_VERSION) || defined(_MSC_VER) ) && \
!defined(inline) && !defined(__cplusplus)
#define inline __inline
#endif
#if defined(MBEDTLS_SELF_TEST)
/*
* Counts of point addition and doubling, and field multiplications.
* Used to test resistance of point multiplication to simple timing attacks.
*/
static unsigned long add_count, dbl_count, mul_count;
#endif
#if defined(MBEDTLS_ECP_DP_SECP192R1_ENABLED) || \
defined(MBEDTLS_ECP_DP_SECP224R1_ENABLED) || \
defined(MBEDTLS_ECP_DP_SECP256R1_ENABLED) || \
defined(MBEDTLS_ECP_DP_SECP384R1_ENABLED) || \
defined(MBEDTLS_ECP_DP_SECP521R1_ENABLED) || \
defined(MBEDTLS_ECP_DP_BP256R1_ENABLED) || \
defined(MBEDTLS_ECP_DP_BP384R1_ENABLED) || \
defined(MBEDTLS_ECP_DP_BP512R1_ENABLED) || \
defined(MBEDTLS_ECP_DP_SECP192K1_ENABLED) || \
defined(MBEDTLS_ECP_DP_SECP224K1_ENABLED) || \
defined(MBEDTLS_ECP_DP_SECP256K1_ENABLED)
#define ECP_SHORTWEIERSTRASS
#endif
#if defined(MBEDTLS_ECP_DP_CURVE25519_ENABLED) || \
defined(MBEDTLS_ECP_DP_CURVE448_ENABLED)
#define ECP_MONTGOMERY
#endif
/*
* Curve types: internal for now, might be exposed later
*/
typedef enum {
ECP_TYPE_NONE = 0,
ECP_TYPE_SHORT_WEIERSTRASS, /* y^2 = x^3 + a x + b */
ECP_TYPE_MONTGOMERY, /* y^2 = x^3 + a x^2 + x */
} ecp_curve_type;
/*
* List of supported curves:
* - internal ID
* - TLS NamedCurve ID (RFC 4492 sec. 5.1.1, RFC 7071 sec. 2)
* - size in bits
* - readable name
*
* Curves are listed in order: largest curves first, and for a given size,
* fastest curves first. This provides the default order for the SSL module.
*
* Reminder: update profiles in x509_crt.c when adding a new curves!
*/
static const mbedtls_ecp_curve_info ecp_supported_curves[] = {
#if defined(MBEDTLS_ECP_DP_SECP521R1_ENABLED)
{ MBEDTLS_ECP_DP_SECP521R1, 25, 521, "secp521r1" },
#endif
#if defined(MBEDTLS_ECP_DP_BP512R1_ENABLED)
{ MBEDTLS_ECP_DP_BP512R1, 28, 512, "brainpoolP512r1" },
#endif
#if defined(MBEDTLS_ECP_DP_SECP384R1_ENABLED)
{ MBEDTLS_ECP_DP_SECP384R1, 24, 384, "secp384r1" },
#endif
#if defined(MBEDTLS_ECP_DP_BP384R1_ENABLED)
{ MBEDTLS_ECP_DP_BP384R1, 27, 384, "brainpoolP384r1" },
#endif
#if defined(MBEDTLS_ECP_DP_SECP256R1_ENABLED)
{ MBEDTLS_ECP_DP_SECP256R1, 23, 256, "secp256r1" },
#endif
#if defined(MBEDTLS_ECP_DP_SECP256K1_ENABLED)
{ MBEDTLS_ECP_DP_SECP256K1, 22, 256, "secp256k1" },
#endif
#if defined(MBEDTLS_ECP_DP_BP256R1_ENABLED)
{ MBEDTLS_ECP_DP_BP256R1, 26, 256, "brainpoolP256r1" },
#endif
#if defined(MBEDTLS_ECP_DP_SECP224R1_ENABLED)
{ MBEDTLS_ECP_DP_SECP224R1, 21, 224, "secp224r1" },
#endif
#if defined(MBEDTLS_ECP_DP_SECP224K1_ENABLED)
{ MBEDTLS_ECP_DP_SECP224K1, 20, 224, "secp224k1" },
#endif
#if defined(MBEDTLS_ECP_DP_SECP192R1_ENABLED)
{ MBEDTLS_ECP_DP_SECP192R1, 19, 192, "secp192r1" },
#endif
#if defined(MBEDTLS_ECP_DP_SECP192K1_ENABLED)
{ MBEDTLS_ECP_DP_SECP192K1, 18, 192, "secp192k1" },
#endif
{ MBEDTLS_ECP_DP_NONE, 0, 0, NULL },
};
#define ECP_NB_CURVES sizeof( ecp_supported_curves ) / \
sizeof( ecp_supported_curves[0] )
static mbedtls_ecp_group_id ecp_supported_grp_id[ECP_NB_CURVES];
/*
* List of supported curves and associated info
*/
const mbedtls_ecp_curve_info *mbedtls_ecp_curve_list(void) {
return (ecp_supported_curves);
}
/*
* List of supported curves, group ID only
*/
const mbedtls_ecp_group_id *mbedtls_ecp_grp_id_list(void) {
static int init_done = 0;
if (! init_done) {
size_t i = 0;
const mbedtls_ecp_curve_info *curve_info;
for (curve_info = mbedtls_ecp_curve_list();
curve_info->grp_id != MBEDTLS_ECP_DP_NONE;
curve_info++) {
ecp_supported_grp_id[i++] = curve_info->grp_id;
}
ecp_supported_grp_id[i] = MBEDTLS_ECP_DP_NONE;
init_done = 1;
}
return (ecp_supported_grp_id);
}
/*
* Get the curve info for the internal identifier
*/
const mbedtls_ecp_curve_info *mbedtls_ecp_curve_info_from_grp_id(mbedtls_ecp_group_id grp_id) {
const mbedtls_ecp_curve_info *curve_info;
for (curve_info = mbedtls_ecp_curve_list();
curve_info->grp_id != MBEDTLS_ECP_DP_NONE;
curve_info++) {
if (curve_info->grp_id == grp_id)
return (curve_info);
}
return (NULL);
}
/*
* Get the curve info from the TLS identifier
*/
const mbedtls_ecp_curve_info *mbedtls_ecp_curve_info_from_tls_id(uint16_t tls_id) {
const mbedtls_ecp_curve_info *curve_info;
for (curve_info = mbedtls_ecp_curve_list();
curve_info->grp_id != MBEDTLS_ECP_DP_NONE;
curve_info++) {
if (curve_info->tls_id == tls_id)
return (curve_info);
}
return (NULL);
}
/*
* Get the curve info from the name
*/
const mbedtls_ecp_curve_info *mbedtls_ecp_curve_info_from_name(const char *name) {
const mbedtls_ecp_curve_info *curve_info;
for (curve_info = mbedtls_ecp_curve_list();
curve_info->grp_id != MBEDTLS_ECP_DP_NONE;
curve_info++) {
if (strcmp(curve_info->name, name) == 0)
return (curve_info);
}
return (NULL);
}
/*
* Get the type of a curve
*/
static inline ecp_curve_type ecp_get_type(const mbedtls_ecp_group *grp) {
if (grp->G.X.p == NULL)
return (ECP_TYPE_NONE);
if (grp->G.Y.p == NULL)
return (ECP_TYPE_MONTGOMERY);
else
return (ECP_TYPE_SHORT_WEIERSTRASS);
}
/*
* Initialize (the components of) a point
*/
void mbedtls_ecp_point_init(mbedtls_ecp_point *pt) {
if (pt == NULL)
return;
mbedtls_mpi_init(&pt->X);
mbedtls_mpi_init(&pt->Y);
mbedtls_mpi_init(&pt->Z);
}
/*
* Initialize (the components of) a group
*/
void mbedtls_ecp_group_init(mbedtls_ecp_group *grp) {
if (grp == NULL)
return;
memset(grp, 0, sizeof(mbedtls_ecp_group));
}
/*
* Initialize (the components of) a key pair
*/
void mbedtls_ecp_keypair_init(mbedtls_ecp_keypair *key) {
if (key == NULL)
return;
mbedtls_ecp_group_init(&key->grp);
mbedtls_mpi_init(&key->d);
mbedtls_ecp_point_init(&key->Q);
}
/*
* Unallocate (the components of) a point
*/
void mbedtls_ecp_point_free(mbedtls_ecp_point *pt) {
if (pt == NULL)
return;
mbedtls_mpi_free(&(pt->X));
mbedtls_mpi_free(&(pt->Y));
mbedtls_mpi_free(&(pt->Z));
}
/*
* Unallocate (the components of) a group
*/
void mbedtls_ecp_group_free(mbedtls_ecp_group *grp) {
size_t i;
if (grp == NULL)
return;
if (grp->h != 1) {
mbedtls_mpi_free(&grp->P);
mbedtls_mpi_free(&grp->A);
mbedtls_mpi_free(&grp->B);
mbedtls_ecp_point_free(&grp->G);
mbedtls_mpi_free(&grp->N);
}
if (grp->T != NULL) {
for (i = 0; i < grp->T_size; i++)
mbedtls_ecp_point_free(&grp->T[i]);
mbedtls_free(grp->T);
}
mbedtls_platform_zeroize(grp, sizeof(mbedtls_ecp_group));
}
/*
* Unallocate (the components of) a key pair
*/
void mbedtls_ecp_keypair_free(mbedtls_ecp_keypair *key) {
if (key == NULL)
return;
mbedtls_ecp_group_free(&key->grp);
mbedtls_mpi_free(&key->d);
mbedtls_ecp_point_free(&key->Q);
}
/*
* Copy the contents of a point
*/
int mbedtls_ecp_copy(mbedtls_ecp_point *P, const mbedtls_ecp_point *Q) {
int ret;
MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&P->X, &Q->X));
MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&P->Y, &Q->Y));
MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&P->Z, &Q->Z));
cleanup:
return (ret);
}
/*
* Copy the contents of a group object
*/
int mbedtls_ecp_group_copy(mbedtls_ecp_group *dst, const mbedtls_ecp_group *src) {
return mbedtls_ecp_group_load(dst, src->id);
}
/*
* Set point to zero
*/
int mbedtls_ecp_set_zero(mbedtls_ecp_point *pt) {
int ret;
MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&pt->X, 1));
MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&pt->Y, 1));
MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&pt->Z, 0));
cleanup:
return (ret);
}
/*
* Tell if a point is zero
*/
int mbedtls_ecp_is_zero(mbedtls_ecp_point *pt) {
return (mbedtls_mpi_cmp_int(&pt->Z, 0) == 0);
}
/*
* Compare two points lazyly
*/
int mbedtls_ecp_point_cmp(const mbedtls_ecp_point *P,
const mbedtls_ecp_point *Q) {
if (mbedtls_mpi_cmp_mpi(&P->X, &Q->X) == 0 &&
mbedtls_mpi_cmp_mpi(&P->Y, &Q->Y) == 0 &&
mbedtls_mpi_cmp_mpi(&P->Z, &Q->Z) == 0) {
return (0);
}
return (MBEDTLS_ERR_ECP_BAD_INPUT_DATA);
}
/*
* Import a non-zero point from ASCII strings
*/
int mbedtls_ecp_point_read_string(mbedtls_ecp_point *P, int radix,
const char *x, const char *y) {
int ret;
MBEDTLS_MPI_CHK(mbedtls_mpi_read_string(&P->X, radix, x));
MBEDTLS_MPI_CHK(mbedtls_mpi_read_string(&P->Y, radix, y));
MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&P->Z, 1));
cleanup:
return (ret);
}
/*
* Export a point into unsigned binary data (SEC1 2.3.3)
*/
int mbedtls_ecp_point_write_binary(const mbedtls_ecp_group *grp, const mbedtls_ecp_point *P,
int format, size_t *olen,
unsigned char *buf, size_t buflen) {
int ret = 0;
size_t plen;
if (format != MBEDTLS_ECP_PF_UNCOMPRESSED &&
format != MBEDTLS_ECP_PF_COMPRESSED)
return (MBEDTLS_ERR_ECP_BAD_INPUT_DATA);
/*
* Common case: P == 0
*/
if (mbedtls_mpi_cmp_int(&P->Z, 0) == 0) {
if (buflen < 1)
return (MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL);
buf[0] = 0x00;
*olen = 1;
return (0);
}
plen = mbedtls_mpi_size(&grp->P);
if (format == MBEDTLS_ECP_PF_UNCOMPRESSED) {
*olen = 2 * plen + 1;
if (buflen < *olen)
return (MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL);
buf[0] = 0x04;
MBEDTLS_MPI_CHK(mbedtls_mpi_write_binary(&P->X, buf + 1, plen));
MBEDTLS_MPI_CHK(mbedtls_mpi_write_binary(&P->Y, buf + 1 + plen, plen));
} else if (format == MBEDTLS_ECP_PF_COMPRESSED) {
*olen = plen + 1;
if (buflen < *olen)
return (MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL);
buf[0] = 0x02 + mbedtls_mpi_get_bit(&P->Y, 0);
MBEDTLS_MPI_CHK(mbedtls_mpi_write_binary(&P->X, buf + 1, plen));
}
cleanup:
return (ret);
}
/*
* Import a point from unsigned binary data (SEC1 2.3.4)
*/
int mbedtls_ecp_point_read_binary(const mbedtls_ecp_group *grp, mbedtls_ecp_point *pt,
const unsigned char *buf, size_t ilen) {
int ret;
size_t plen;
if (ilen < 1)
return (MBEDTLS_ERR_ECP_BAD_INPUT_DATA);
if (buf[0] == 0x00) {
if (ilen == 1)
return (mbedtls_ecp_set_zero(pt));
else
return (MBEDTLS_ERR_ECP_BAD_INPUT_DATA);
}
plen = mbedtls_mpi_size(&grp->P);
if (buf[0] != 0x04)
return (MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE);
if (ilen != 2 * plen + 1)
return (MBEDTLS_ERR_ECP_BAD_INPUT_DATA);
MBEDTLS_MPI_CHK(mbedtls_mpi_read_binary(&pt->X, buf + 1, plen));
MBEDTLS_MPI_CHK(mbedtls_mpi_read_binary(&pt->Y, buf + 1 + plen, plen));
MBEDTLS_MPI_CHK(mbedtls_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 mbedtls_ecp_tls_read_point(const mbedtls_ecp_group *grp, mbedtls_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 one for data)
*/
if (buf_len < 2)
return (MBEDTLS_ERR_ECP_BAD_INPUT_DATA);
data_len = *(*buf)++;
if (data_len < 1 || data_len > buf_len - 1)
return (MBEDTLS_ERR_ECP_BAD_INPUT_DATA);
/*
* Save buffer start for read_binary and update buf
*/
buf_start = *buf;
*buf += data_len;
return mbedtls_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 mbedtls_ecp_tls_write_point(const mbedtls_ecp_group *grp, const mbedtls_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 (MBEDTLS_ERR_ECP_BAD_INPUT_DATA);
if ((ret = mbedtls_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] = (unsigned char) * olen;
++*olen;
return (0);
}
/*
* Set a group from an ECParameters record (RFC 4492)
*/
int mbedtls_ecp_tls_read_group(mbedtls_ecp_group *grp, const unsigned char **buf, size_t len) {
uint16_t tls_id;
const mbedtls_ecp_curve_info *curve_info;
/*
* We expect at least three bytes (see below)
*/
if (len < 3)
return (MBEDTLS_ERR_ECP_BAD_INPUT_DATA);
/*
* First byte is curve_type; only named_curve is handled
*/
if (*(*buf)++ != MBEDTLS_ECP_TLS_NAMED_CURVE)
return (MBEDTLS_ERR_ECP_BAD_INPUT_DATA);
/*
* Next two bytes are the namedcurve value
*/
tls_id = *(*buf)++;
tls_id <<= 8;
tls_id |= *(*buf)++;
if ((curve_info = mbedtls_ecp_curve_info_from_tls_id(tls_id)) == NULL)
return (MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE);
return mbedtls_ecp_group_load(grp, curve_info->grp_id);
}
/*
* Write the ECParameters record corresponding to a group (RFC 4492)
*/
int mbedtls_ecp_tls_write_group(const mbedtls_ecp_group *grp, size_t *olen,
unsigned char *buf, size_t blen) {
const mbedtls_ecp_curve_info *curve_info;
if ((curve_info = mbedtls_ecp_curve_info_from_grp_id(grp->id)) == NULL)
return (MBEDTLS_ERR_ECP_BAD_INPUT_DATA);
/*
* We are going to write 3 bytes (see below)
*/
*olen = 3;
if (blen < *olen)
return (MBEDTLS_ERR_ECP_BUFFER_TOO_SMALL);
/*
* First byte is curve_type, always named_curve
*/
*buf++ = MBEDTLS_ECP_TLS_NAMED_CURVE;
/*
* Next two bytes are the namedcurve value
*/
buf[0] = curve_info->tls_id >> 8;
buf[1] = curve_info->tls_id & 0xFF;
return (0);
}
/*
* Wrapper around fast quasi-modp functions, with fall-back to mbedtls_mpi_mod_mpi.
* See the documentation of struct mbedtls_ecp_group.
*
* This function is in the critial loop for mbedtls_ecp_mul, so pay attention to perf.
*/
static int ecp_modp(mbedtls_mpi *N, const mbedtls_ecp_group *grp) {
int ret;
if (grp->modp == NULL)
return (mbedtls_mpi_mod_mpi(N, N, &grp->P));
/* N->s < 0 is a much faster test, which fails only if N is 0 */
if ((N->s < 0 && mbedtls_mpi_cmp_int(N, 0) != 0) ||
mbedtls_mpi_bitlen(N) > 2 * grp->pbits) {
return (MBEDTLS_ERR_ECP_BAD_INPUT_DATA);
}
MBEDTLS_MPI_CHK(grp->modp(N));
/* N->s < 0 is a much faster test, which fails only if N is 0 */
while (N->s < 0 && mbedtls_mpi_cmp_int(N, 0) != 0)
MBEDTLS_MPI_CHK(mbedtls_mpi_add_mpi(N, N, &grp->P));
while (mbedtls_mpi_cmp_mpi(N, &grp->P) >= 0)
/* we known P, N and the result are positive */
MBEDTLS_MPI_CHK(mbedtls_mpi_sub_abs(N, N, &grp->P));
cleanup:
return (ret);
}
/*
* 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
* mbedtls_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 mbedtls_mpi mod p in-place, general case, to use after mbedtls_mpi_mul_mpi
*/
#if defined(MBEDTLS_SELF_TEST)
#define INC_MUL_COUNT mul_count++;
#else
#define INC_MUL_COUNT
#endif
#define MOD_MUL( N ) do { MBEDTLS_MPI_CHK( ecp_modp( &N, grp ) ); INC_MUL_COUNT } \
while( 0 )
/*
* Reduce a mbedtls_mpi mod p in-place, to use after mbedtls_mpi_sub_mpi
* N->s < 0 is a very fast test, which fails only if N is 0
*/
#define MOD_SUB( N ) \
while( N.s < 0 && mbedtls_mpi_cmp_int( &N, 0 ) != 0 ) \
MBEDTLS_MPI_CHK( mbedtls_mpi_add_mpi( &N, &N, &grp->P ) )
/*
* Reduce a mbedtls_mpi mod p in-place, to use after mbedtls_mpi_add_mpi and mbedtls_mpi_mul_int.
* We known P, N and the result are positive, so sub_abs is correct, and
* a bit faster.
*/
#define MOD_ADD( N ) \
while( mbedtls_mpi_cmp_mpi( &N, &grp->P ) >= 0 ) \
MBEDTLS_MPI_CHK( mbedtls_mpi_sub_abs( &N, &N, &grp->P ) )
#if defined(ECP_SHORTWEIERSTRASS)
/*
* For curves in short Weierstrass form, we do all the internal operations in
* Jacobian coordinates.
*
* For multiplication, we'll use a comb method with coutermeasueres against
* SPA, hence timing attacks.
*/
/*
* Normalize jacobian coordinates so that Z == 0 || Z == 1 (GECC 3.2.1)
* Cost: 1N := 1I + 3M + 1S
*/
static int ecp_normalize_jac(const mbedtls_ecp_group *grp, mbedtls_ecp_point *pt) {
int ret;
mbedtls_mpi Zi, ZZi;
if (mbedtls_mpi_cmp_int(&pt->Z, 0) == 0)
return (0);
#if defined(MBEDTLS_ECP_NORMALIZE_JAC_ALT)
if (mbedtls_internal_ecp_grp_capable(grp)) {
return mbedtls_internal_ecp_normalize_jac(grp, pt);
}
#endif /* MBEDTLS_ECP_NORMALIZE_JAC_ALT */
mbedtls_mpi_init(&Zi);
mbedtls_mpi_init(&ZZi);
/*
* X = X / Z^2 mod p
*/
MBEDTLS_MPI_CHK(mbedtls_mpi_inv_mod(&Zi, &pt->Z, &grp->P));
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&ZZi, &Zi, &Zi));
MOD_MUL(ZZi);
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&pt->X, &pt->X, &ZZi));
MOD_MUL(pt->X);
/*
* Y = Y / Z^3 mod p
*/
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&pt->Y, &pt->Y, &ZZi));
MOD_MUL(pt->Y);
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&pt->Y, &pt->Y, &Zi));
MOD_MUL(pt->Y);
/*
* Z = 1
*/
MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&pt->Z, 1));
cleanup:
mbedtls_mpi_free(&Zi);
mbedtls_mpi_free(&ZZi);
return (ret);
}
/*
* Normalize jacobian coordinates of an array of (pointers to) points,
* using Montgomery's trick to perform only one inversion mod P.
* (See for example Cohen's "A Course in Computational Algebraic Number
* Theory", Algorithm 10.3.4.)
*
* Warning: fails (returning an error) if one of the points is zero!
* This should never happen, see choice of w in ecp_mul_comb().
*
* Cost: 1N(t) := 1I + (6t - 3)M + 1S
*/
static int ecp_normalize_jac_many(const mbedtls_ecp_group *grp,
mbedtls_ecp_point *T[], size_t t_len) {
int ret;
size_t i;
mbedtls_mpi *c, u, Zi, ZZi;
if (t_len < 2)
return (ecp_normalize_jac(grp, *T));
#if defined(MBEDTLS_ECP_NORMALIZE_JAC_MANY_ALT)
if (mbedtls_internal_ecp_grp_capable(grp)) {
return mbedtls_internal_ecp_normalize_jac_many(grp, T, t_len);
}
#endif
if ((c = mbedtls_calloc(t_len, sizeof(mbedtls_mpi))) == NULL)
return (MBEDTLS_ERR_ECP_ALLOC_FAILED);
mbedtls_mpi_init(&u);
mbedtls_mpi_init(&Zi);
mbedtls_mpi_init(&ZZi);
/*
* c[i] = Z_0 * ... * Z_i
*/
MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&c[0], &T[0]->Z));
for (i = 1; i < t_len; i++) {
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&c[i], &c[i - 1], &T[i]->Z));
MOD_MUL(c[i]);
}
/*
* u = 1 / (Z_0 * ... * Z_n) mod P
*/
MBEDTLS_MPI_CHK(mbedtls_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) {
MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&Zi, &u));
} else {
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&Zi, &u, &c[i - 1]));
MOD_MUL(Zi);
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&u, &u, &T[i]->Z));
MOD_MUL(u);
}
/*
* proceed as in normalize()
*/
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&ZZi, &Zi, &Zi));
MOD_MUL(ZZi);
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T[i]->X, &T[i]->X, &ZZi));
MOD_MUL(T[i]->X);
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T[i]->Y, &T[i]->Y, &ZZi));
MOD_MUL(T[i]->Y);
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T[i]->Y, &T[i]->Y, &Zi));
MOD_MUL(T[i]->Y);
/*
* Post-precessing: reclaim some memory by shrinking coordinates
* - not storing Z (always 1)
* - shrinking other coordinates, but still keeping the same number of
* limbs as P, as otherwise it will too likely be regrown too fast.
*/
MBEDTLS_MPI_CHK(mbedtls_mpi_shrink(&T[i]->X, grp->P.n));
MBEDTLS_MPI_CHK(mbedtls_mpi_shrink(&T[i]->Y, grp->P.n));
mbedtls_mpi_free(&T[i]->Z);
if (i == 0)
break;
}
cleanup:
mbedtls_mpi_free(&u);
mbedtls_mpi_free(&Zi);
mbedtls_mpi_free(&ZZi);
for (i = 0; i < t_len; i++)
mbedtls_mpi_free(&c[i]);
mbedtls_free(c);
return (ret);
}
/*
* Conditional point inversion: Q -> -Q = (Q.X, -Q.Y, Q.Z) without leak.
* "inv" must be 0 (don't invert) or 1 (invert) or the result will be invalid
*/
static int ecp_safe_invert_jac(const mbedtls_ecp_group *grp,
mbedtls_ecp_point *Q,
unsigned char inv) {
int ret;
unsigned char nonzero;
mbedtls_mpi mQY;
mbedtls_mpi_init(&mQY);
/* Use the fact that -Q.Y mod P = P - Q.Y unless Q.Y == 0 */
MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&mQY, &grp->P, &Q->Y));
nonzero = mbedtls_mpi_cmp_int(&Q->Y, 0) != 0;
MBEDTLS_MPI_CHK(mbedtls_mpi_safe_cond_assign(&Q->Y, &mQY, inv & nonzero));
cleanup:
mbedtls_mpi_free(&mQY);
return (ret);
}
/*
* Point doubling R = 2 P, Jacobian coordinates
*
* Based on http://www.hyperelliptic.org/EFD/g1p/auto-shortw-jacobian.html#doubling-dbl-1998-cmo-2 .
*
* We follow the variable naming fairly closely. The formula variations that trade a MUL for a SQR
* (plus a few ADDs) aren't useful as our bignum implementation doesn't distinguish squaring.
*
* Standard optimizations are applied when curve parameter A is one of { 0, -3 }.
*
* Cost: 1D := 3M + 4S (A == 0)
* 4M + 4S (A == -3)
* 3M + 6S + 1a otherwise
*/
static int ecp_double_jac(const mbedtls_ecp_group *grp, mbedtls_ecp_point *R,
const mbedtls_ecp_point *P) {
int ret;
mbedtls_mpi M, S, T, U;
#if defined(MBEDTLS_SELF_TEST)
dbl_count++;
#endif
#if defined(MBEDTLS_ECP_DOUBLE_JAC_ALT)
if (mbedtls_internal_ecp_grp_capable(grp)) {
return mbedtls_internal_ecp_double_jac(grp, R, P);
}
#endif /* MBEDTLS_ECP_DOUBLE_JAC_ALT */
mbedtls_mpi_init(&M);
mbedtls_mpi_init(&S);
mbedtls_mpi_init(&T);
mbedtls_mpi_init(&U);
/* Special case for A = -3 */
if (grp->A.p == NULL) {
/* M = 3(X + Z^2)(X - Z^2) */
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&S, &P->Z, &P->Z));
MOD_MUL(S);
MBEDTLS_MPI_CHK(mbedtls_mpi_add_mpi(&T, &P->X, &S));
MOD_ADD(T);
MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&U, &P->X, &S));
MOD_SUB(U);
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&S, &T, &U));
MOD_MUL(S);
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_int(&M, &S, 3));
MOD_ADD(M);
} else {
/* M = 3.X^2 */
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&S, &P->X, &P->X));
MOD_MUL(S);
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_int(&M, &S, 3));
MOD_ADD(M);
/* Optimize away for "koblitz" curves with A = 0 */
if (mbedtls_mpi_cmp_int(&grp->A, 0) != 0) {
/* M += A.Z^4 */
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&S, &P->Z, &P->Z));
MOD_MUL(S);
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T, &S, &S));
MOD_MUL(T);
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&S, &T, &grp->A));
MOD_MUL(S);
MBEDTLS_MPI_CHK(mbedtls_mpi_add_mpi(&M, &M, &S));
MOD_ADD(M);
}
}
/* S = 4.X.Y^2 */
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T, &P->Y, &P->Y));
MOD_MUL(T);
MBEDTLS_MPI_CHK(mbedtls_mpi_shift_l(&T, 1));
MOD_ADD(T);
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&S, &P->X, &T));
MOD_MUL(S);
MBEDTLS_MPI_CHK(mbedtls_mpi_shift_l(&S, 1));
MOD_ADD(S);
/* U = 8.Y^4 */
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&U, &T, &T));
MOD_MUL(U);
MBEDTLS_MPI_CHK(mbedtls_mpi_shift_l(&U, 1));
MOD_ADD(U);
/* T = M^2 - 2.S */
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T, &M, &M));
MOD_MUL(T);
MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&T, &T, &S));
MOD_SUB(T);
MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&T, &T, &S));
MOD_SUB(T);
/* S = M(S - T) - U */
MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&S, &S, &T));
MOD_SUB(S);
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&S, &S, &M));
MOD_MUL(S);
MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&S, &S, &U));
MOD_SUB(S);
/* U = 2.Y.Z */
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&U, &P->Y, &P->Z));
MOD_MUL(U);
MBEDTLS_MPI_CHK(mbedtls_mpi_shift_l(&U, 1));
MOD_ADD(U);
MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&R->X, &T));
MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&R->Y, &S));
MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&R->Z, &U));
cleanup:
mbedtls_mpi_free(&M);
mbedtls_mpi_free(&S);
mbedtls_mpi_free(&T);
mbedtls_mpi_free(&U);
return (ret);
}
/*
* Addition: 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.
*
* Special cases: (1) P or Q is zero, (2) R is zero, (3) P == Q.
* None of these cases can happen as intermediate step in ecp_mul_comb():
* - at each step, P, Q and R are multiples of the base point, the factor
* being less than its order, so none of them is zero;
* - Q is an odd multiple of the base point, P an even multiple,
* due to the choice of precomputed points in the modified comb method.
* So branches for these cases do not leak secret information.
*
* We accept Q->Z being unset (saving memory in tables) as meaning 1.
*
* Cost: 1A := 8M + 3S
*/
static int ecp_add_mixed(const mbedtls_ecp_group *grp, mbedtls_ecp_point *R,
const mbedtls_ecp_point *P, const mbedtls_ecp_point *Q) {
int ret;
mbedtls_mpi T1, T2, T3, T4, X, Y, Z;
#if defined(MBEDTLS_SELF_TEST)
add_count++;
#endif
#if defined(MBEDTLS_ECP_ADD_MIXED_ALT)
if (mbedtls_internal_ecp_grp_capable(grp)) {
return mbedtls_internal_ecp_add_mixed(grp, R, P, Q);
}
#endif /* MBEDTLS_ECP_ADD_MIXED_ALT */
/*
* Trivial cases: P == 0 or Q == 0 (case 1)
*/
if (mbedtls_mpi_cmp_int(&P->Z, 0) == 0)
return (mbedtls_ecp_copy(R, Q));
if (Q->Z.p != NULL && mbedtls_mpi_cmp_int(&Q->Z, 0) == 0)
return (mbedtls_ecp_copy(R, P));
/*
* Make sure Q coordinates are normalized
*/
if (Q->Z.p != NULL && mbedtls_mpi_cmp_int(&Q->Z, 1) != 0)
return (MBEDTLS_ERR_ECP_BAD_INPUT_DATA);
mbedtls_mpi_init(&T1);
mbedtls_mpi_init(&T2);
mbedtls_mpi_init(&T3);
mbedtls_mpi_init(&T4);
mbedtls_mpi_init(&X);
mbedtls_mpi_init(&Y);
mbedtls_mpi_init(&Z);
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T1, &P->Z, &P->Z));
MOD_MUL(T1);
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T2, &T1, &P->Z));
MOD_MUL(T2);
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T1, &T1, &Q->X));
MOD_MUL(T1);
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T2, &T2, &Q->Y));
MOD_MUL(T2);
MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&T1, &T1, &P->X));
MOD_SUB(T1);
MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&T2, &T2, &P->Y));
MOD_SUB(T2);
/* Special cases (2) and (3) */
if (mbedtls_mpi_cmp_int(&T1, 0) == 0) {
if (mbedtls_mpi_cmp_int(&T2, 0) == 0) {
ret = ecp_double_jac(grp, R, P);
goto cleanup;
} else {
ret = mbedtls_ecp_set_zero(R);
goto cleanup;
}
}
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&Z, &P->Z, &T1));
MOD_MUL(Z);
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T3, &T1, &T1));
MOD_MUL(T3);
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T4, &T3, &T1));
MOD_MUL(T4);
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T3, &T3, &P->X));
MOD_MUL(T3);
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_int(&T1, &T3, 2));
MOD_ADD(T1);
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&X, &T2, &T2));
MOD_MUL(X);
MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&X, &X, &T1));
MOD_SUB(X);
MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&X, &X, &T4));
MOD_SUB(X);
MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&T3, &T3, &X));
MOD_SUB(T3);
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T3, &T3, &T2));
MOD_MUL(T3);
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&T4, &T4, &P->Y));
MOD_MUL(T4);
MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&Y, &T3, &T4));
MOD_SUB(Y);
MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&R->X, &X));
MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&R->Y, &Y));
MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&R->Z, &Z));
cleanup:
mbedtls_mpi_free(&T1);
mbedtls_mpi_free(&T2);
mbedtls_mpi_free(&T3);
mbedtls_mpi_free(&T4);
mbedtls_mpi_free(&X);
mbedtls_mpi_free(&Y);
mbedtls_mpi_free(&Z);
return (ret);
}
/*
* Randomize jacobian coordinates:
* (X, Y, Z) -> (l^2 X, l^3 Y, l Z) for random l
* This is sort of the reverse operation of ecp_normalize_jac().
*
* This countermeasure was first suggested in [2].
*/
static int ecp_randomize_jac(const mbedtls_ecp_group *grp, mbedtls_ecp_point *pt,
int (*f_rng)(void *, unsigned char *, size_t), void *p_rng) {
int ret;
mbedtls_mpi l, ll;
size_t p_size;
int count = 0;
#if defined(MBEDTLS_ECP_RANDOMIZE_JAC_ALT)
if (mbedtls_internal_ecp_grp_capable(grp)) {
return mbedtls_internal_ecp_randomize_jac(grp, pt, f_rng, p_rng);
}
#endif /* MBEDTLS_ECP_RANDOMIZE_JAC_ALT */
p_size = (grp->pbits + 7) / 8;
mbedtls_mpi_init(&l);
mbedtls_mpi_init(&ll);
/* Generate l such that 1 < l < p */
do {
MBEDTLS_MPI_CHK(mbedtls_mpi_fill_random(&l, p_size, f_rng, p_rng));
while (mbedtls_mpi_cmp_mpi(&l, &grp->P) >= 0)
MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&l, 1));
if (count++ > 10)
return (MBEDTLS_ERR_ECP_RANDOM_FAILED);
} while (mbedtls_mpi_cmp_int(&l, 1) <= 0);
/* Z = l * Z */
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&pt->Z, &pt->Z, &l));
MOD_MUL(pt->Z);
/* X = l^2 * X */
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&ll, &l, &l));
MOD_MUL(ll);
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&pt->X, &pt->X, &ll));
MOD_MUL(pt->X);
/* Y = l^3 * Y */
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&ll, &ll, &l));
MOD_MUL(ll);
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&pt->Y, &pt->Y, &ll));
MOD_MUL(pt->Y);
cleanup:
mbedtls_mpi_free(&l);
mbedtls_mpi_free(&ll);
return (ret);
}
/*
* Check and define parameters used by the comb method (see below for details)
*/
#if MBEDTLS_ECP_WINDOW_SIZE < 2 || MBEDTLS_ECP_WINDOW_SIZE > 7
#error "MBEDTLS_ECP_WINDOW_SIZE out of bounds"
#endif
/* d = ceil( n / w ) */
#define COMB_MAX_D ( MBEDTLS_ECP_MAX_BITS + 1 ) / 2
/* number of precomputed points */
#define COMB_MAX_PRE ( 1 << ( MBEDTLS_ECP_WINDOW_SIZE - 1 ) )
/*
* Compute the representation of m that will be used with our comb method.
*
* The basic comb method is described in GECC 3.44 for example. We use a
* modified version that provides resistance to SPA by avoiding zero
* digits in the representation as in [3]. We modify the method further by
* requiring that all K_i be odd, which has the small cost that our
* representation uses one more K_i, 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:
* - x is an array of size d + 1
* - w is the size, ie number of teeth, of the comb, and must be between
* 2 and 7 (in practice, between 2 and MBEDTLS_ECP_WINDOW_SIZE)
* - m is the MPI, expected to be odd and such that bitlength(m) <= w * d
* (the result will be incorrect if these assumptions are not satisfied)
*/
static void ecp_comb_fixed(unsigned char x[], size_t d,
unsigned char w, const mbedtls_mpi *m) {
size_t i, j;
unsigned char c, cc, adjust;
memset(x, 0, d + 1);
/* First get the classical comb values (except for x_d = 0) */
for (i = 0; i < d; i++)
for (j = 0; j < w; j++)
x[i] |= mbedtls_mpi_get_bit(m, i + d * j) << j;
/* Now make sure x_1 .. x_d are odd */
c = 0;
for (i = 1; i <= d; i++) {
/* Add carry and update it */
cc = x[i] & c;
x[i] = x[i] ^ c;
c = cc;
/* Adjust if needed, avoiding 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;
}
}
/*
* Precompute points for the comb method
*
* If i = i_{w-1} ... i_1 is the binary representation of i, then
* T[i] = i_{w-1} 2^{(w-1)d} P + ... + i_1 2^d P + P
*
* T must be able to hold 2^{w - 1} elements
*
* Cost: d(w-1) D + (2^{w-1} - 1) A + 1 N(w-1) + 1 N(2^{w-1} - 1)
*/
static int ecp_precompute_comb(const mbedtls_ecp_group *grp,
mbedtls_ecp_point T[], const mbedtls_ecp_point *P,
unsigned char w, size_t d) {
int ret;
unsigned char i, k;
size_t j;
mbedtls_ecp_point *cur, *TT[COMB_MAX_PRE - 1];
/*
* Set T[0] = P and
* T[2^{l-1}] = 2^{dl} P for l = 1 .. w-1 (this is not the final value)
*/
MBEDTLS_MPI_CHK(mbedtls_ecp_copy(&T[0], P));
k = 0;
for (i = 1; i < (1U << (w - 1)); i <<= 1) {
cur = T + i;
MBEDTLS_MPI_CHK(mbedtls_ecp_copy(cur, T + (i >> 1)));
for (j = 0; j < d; j++)
MBEDTLS_MPI_CHK(ecp_double_jac(grp, cur, cur));
TT[k++] = cur;
}
MBEDTLS_MPI_CHK(ecp_normalize_jac_many(grp, TT, k));
/*
* Compute the remaining ones using the minimal number of additions
* Be careful to update T[2^l] only after using it!
*/
k = 0;
for (i = 1; i < (1U << (w - 1)); i <<= 1) {
j = i;
while (j--) {
MBEDTLS_MPI_CHK(ecp_add_mixed(grp, &T[i + j], &T[j], &T[i]));
TT[k++] = &T[i + j];
}
}
MBEDTLS_MPI_CHK(ecp_normalize_jac_many(grp, TT, k));
cleanup:
return (ret);
}
/*
* Select precomputed point: R = sign(i) * T[ abs(i) / 2 ]
*/
static int ecp_select_comb(const mbedtls_ecp_group *grp, mbedtls_ecp_point *R,
const mbedtls_ecp_point T[], unsigned char t_len,
unsigned char i) {
int ret;
unsigned char ii, j;
/* Ignore the "sign" bit and scale down */
ii = (i & 0x7Fu) >> 1;
/* Read the whole table to thwart cache-based timing attacks */
for (j = 0; j < t_len; j++) {
MBEDTLS_MPI_CHK(mbedtls_mpi_safe_cond_assign(&R->X, &T[j].X, j == ii));
MBEDTLS_MPI_CHK(mbedtls_mpi_safe_cond_assign(&R->Y, &T[j].Y, j == ii));
}
/* Safely invert result if i is "negative" */
MBEDTLS_MPI_CHK(ecp_safe_invert_jac(grp, R, i >> 7));
cleanup:
return (ret);
}
/*
* Core multiplication algorithm for the (modified) comb method.
* This part is actually common with the basic comb method (GECC 3.44)
*
* Cost: d A + d D + 1 R
*/
static int ecp_mul_comb_core(const mbedtls_ecp_group *grp, mbedtls_ecp_point *R,
const mbedtls_ecp_point T[], unsigned char t_len,
const unsigned char x[], size_t d,
int (*f_rng)(void *, unsigned char *, size_t),
void *p_rng) {
int ret;
mbedtls_ecp_point Txi;
size_t i;
mbedtls_ecp_point_init(&Txi);
/* Start with a non-zero point and randomize its coordinates */
i = d;
MBEDTLS_MPI_CHK(ecp_select_comb(grp, R, T, t_len, x[i]));
MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&R->Z, 1));
if (f_rng != 0)
MBEDTLS_MPI_CHK(ecp_randomize_jac(grp, R, f_rng, p_rng));
while (i-- != 0) {
MBEDTLS_MPI_CHK(ecp_double_jac(grp, R, R));
MBEDTLS_MPI_CHK(ecp_select_comb(grp, &Txi, T, t_len, x[i]));
MBEDTLS_MPI_CHK(ecp_add_mixed(grp, R, R, &Txi));
}
cleanup:
mbedtls_ecp_point_free(&Txi);
return (ret);
}
/*
* Multiplication using the comb method,
* for curves in short Weierstrass form
*/
static int ecp_mul_comb(mbedtls_ecp_group *grp, mbedtls_ecp_point *R,
const mbedtls_mpi *m, const mbedtls_ecp_point *P,
int (*f_rng)(void *, unsigned char *, size_t),
void *p_rng) {
int ret;
unsigned char w, m_is_odd, p_eq_g, pre_len, i;
size_t d;
unsigned char k[COMB_MAX_D + 1];
mbedtls_ecp_point *T;
mbedtls_mpi M, mm;
mbedtls_mpi_init(&M);
mbedtls_mpi_init(&mm);
/* we need N to be odd to trnaform m in an odd number, check now */
if (mbedtls_mpi_get_bit(&grp->N, 0) != 1)
return (MBEDTLS_ERR_ECP_BAD_INPUT_DATA);
/*
* Minimize the number of multiplications, that is minimize
* 10 * d * w + 18 * 2^(w-1) + 11 * d + 7 * w, with d = ceil( nbits / w )
* (see costs of the various parts, with 1S = 1M)
*/
w = grp->nbits >= 384 ? 5 : 4;
/*
* If P == G, pre-compute a bit more, since this may be re-used later.
* Just adding one avoids upping the cost of the first mul too much,
* and the memory cost too.
*/
#if MBEDTLS_ECP_FIXED_POINT_OPTIM == 1
p_eq_g = (mbedtls_mpi_cmp_mpi(&P->Y, &grp->G.Y) == 0 &&
mbedtls_mpi_cmp_mpi(&P->X, &grp->G.X) == 0);
if (p_eq_g)
w++;
#else
p_eq_g = 0;
#endif
/*
* Make sure w is within bounds.
* (The last test is useful only for very small curves in the test suite.)
*/
if (w > MBEDTLS_ECP_WINDOW_SIZE)
w = MBEDTLS_ECP_WINDOW_SIZE;
if (w >= grp->nbits)
w = 2;
/* Other sizes that depend on w */
pre_len = 1U << (w - 1);
d = (grp->nbits + w - 1) / w;
/*
* Prepare precomputed points: if P == G we want to
* use grp->T if already initialized, or initialize it.
*/
T = p_eq_g ? grp->T : NULL;
if (T == NULL) {
T = mbedtls_calloc(pre_len, sizeof(mbedtls_ecp_point));
if (T == NULL) {
ret = MBEDTLS_ERR_ECP_ALLOC_FAILED;
goto cleanup;
}
MBEDTLS_MPI_CHK(ecp_precompute_comb(grp, T, P, w, d));
if (p_eq_g) {
grp->T = T;
grp->T_size = pre_len;
}
}
/*
* Make sure M is odd (M = m or M = N - m, since N is odd)
* using the fact that m * P = - (N - m) * P
*/
m_is_odd = (mbedtls_mpi_get_bit(m, 0) == 1);
MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&M, m));
MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&mm, &grp->N, m));
MBEDTLS_MPI_CHK(mbedtls_mpi_safe_cond_assign(&M, &mm, ! m_is_odd));
/*
* Go for comb multiplication, R = M * P
*/
ecp_comb_fixed(k, d, w, &M);
MBEDTLS_MPI_CHK(ecp_mul_comb_core(grp, R, T, pre_len, k, d, f_rng, p_rng));
/*
* Now get m * P from M * P and normalize it
*/
MBEDTLS_MPI_CHK(ecp_safe_invert_jac(grp, R, ! m_is_odd));
MBEDTLS_MPI_CHK(ecp_normalize_jac(grp, R));
cleanup:
/* There are two cases where T is not stored in grp:
* - P != G
* - An intermediate operation failed before setting grp->T
* In either case, T must be freed.
*/
if (T != NULL && T != grp->T) {
for (i = 0; i < pre_len; i++)
mbedtls_ecp_point_free(&T[i]);
mbedtls_free(T);
}
mbedtls_mpi_free(&M);
mbedtls_mpi_free(&mm);
if (ret != 0)
mbedtls_ecp_point_free(R);
return (ret);
}
#endif /* ECP_SHORTWEIERSTRASS */
#if defined(ECP_MONTGOMERY)
/*
* For Montgomery curves, we do all the internal arithmetic in projective
* coordinates. Import/export of points uses only the x coordinates, which is
* internaly represented as X / Z.
*
* For scalar multiplication, we'll use a Montgomery ladder.
*/
/*
* Normalize Montgomery x/z coordinates: X = X/Z, Z = 1
* Cost: 1M + 1I
*/
static int ecp_normalize_mxz(const mbedtls_ecp_group *grp, mbedtls_ecp_point *P) {
int ret;
#if defined(MBEDTLS_ECP_NORMALIZE_MXZ_ALT)
if (mbedtls_internal_ecp_grp_capable(grp)) {
return mbedtls_internal_ecp_normalize_mxz(grp, P);
}
#endif /* MBEDTLS_ECP_NORMALIZE_MXZ_ALT */
MBEDTLS_MPI_CHK(mbedtls_mpi_inv_mod(&P->Z, &P->Z, &grp->P));
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&P->X, &P->X, &P->Z));
MOD_MUL(P->X);
MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&P->Z, 1));
cleanup:
return (ret);
}
/*
* Randomize projective x/z coordinates:
* (X, Z) -> (l X, l Z) for random l
* This is sort of the reverse operation of ecp_normalize_mxz().
*
* This countermeasure was first suggested in [2].
* Cost: 2M
*/
static int ecp_randomize_mxz(const mbedtls_ecp_group *grp, mbedtls_ecp_point *P,
int (*f_rng)(void *, unsigned char *, size_t), void *p_rng) {
int ret;
mbedtls_mpi l;
size_t p_size;
int count = 0;
#if defined(MBEDTLS_ECP_RANDOMIZE_MXZ_ALT)
if (mbedtls_internal_ecp_grp_capable(grp)) {
return mbedtls_internal_ecp_randomize_mxz(grp, P, f_rng, p_rng);
}
#endif /* MBEDTLS_ECP_RANDOMIZE_MXZ_ALT */
p_size = (grp->pbits + 7) / 8;
mbedtls_mpi_init(&l);
/* Generate l such that 1 < l < p */
do {
MBEDTLS_MPI_CHK(mbedtls_mpi_fill_random(&l, p_size, f_rng, p_rng));
while (mbedtls_mpi_cmp_mpi(&l, &grp->P) >= 0)
MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(&l, 1));
if (count++ > 10)
return (MBEDTLS_ERR_ECP_RANDOM_FAILED);
} while (mbedtls_mpi_cmp_int(&l, 1) <= 0);
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&P->X, &P->X, &l));
MOD_MUL(P->X);
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&P->Z, &P->Z, &l));
MOD_MUL(P->Z);
cleanup:
mbedtls_mpi_free(&l);
return (ret);
}
/*
* Double-and-add: R = 2P, S = P + Q, with d = X(P - Q),
* for Montgomery curves in x/z coordinates.
*
* http://www.hyperelliptic.org/EFD/g1p/auto-code/montgom/xz/ladder/mladd-1987-m.op3
* with
* d = X1
* P = (X2, Z2)
* Q = (X3, Z3)
* R = (X4, Z4)
* S = (X5, Z5)
* and eliminating temporary variables tO, ..., t4.
*
* Cost: 5M + 4S
*/
static int ecp_double_add_mxz(const mbedtls_ecp_group *grp,
mbedtls_ecp_point *R, mbedtls_ecp_point *S,
const mbedtls_ecp_point *P, const mbedtls_ecp_point *Q,
const mbedtls_mpi *d) {
int ret;
mbedtls_mpi A, AA, B, BB, E, C, D, DA, CB;
#if defined(MBEDTLS_ECP_DOUBLE_ADD_MXZ_ALT)
if (mbedtls_internal_ecp_grp_capable(grp)) {
return mbedtls_internal_ecp_double_add_mxz(grp, R, S, P, Q, d);
}
#endif /* MBEDTLS_ECP_DOUBLE_ADD_MXZ_ALT */
mbedtls_mpi_init(&A);
mbedtls_mpi_init(&AA);
mbedtls_mpi_init(&B);
mbedtls_mpi_init(&BB);
mbedtls_mpi_init(&E);
mbedtls_mpi_init(&C);
mbedtls_mpi_init(&D);
mbedtls_mpi_init(&DA);
mbedtls_mpi_init(&CB);
MBEDTLS_MPI_CHK(mbedtls_mpi_add_mpi(&A, &P->X, &P->Z));
MOD_ADD(A);
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&AA, &A, &A));
MOD_MUL(AA);
MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&B, &P->X, &P->Z));
MOD_SUB(B);
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&BB, &B, &B));
MOD_MUL(BB);
MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&E, &AA, &BB));
MOD_SUB(E);
MBEDTLS_MPI_CHK(mbedtls_mpi_add_mpi(&C, &Q->X, &Q->Z));
MOD_ADD(C);
MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&D, &Q->X, &Q->Z));
MOD_SUB(D);
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&DA, &D, &A));
MOD_MUL(DA);
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&CB, &C, &B));
MOD_MUL(CB);
MBEDTLS_MPI_CHK(mbedtls_mpi_add_mpi(&S->X, &DA, &CB));
MOD_MUL(S->X);
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&S->X, &S->X, &S->X));
MOD_MUL(S->X);
MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&S->Z, &DA, &CB));
MOD_SUB(S->Z);
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&S->Z, &S->Z, &S->Z));
MOD_MUL(S->Z);
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&S->Z, d, &S->Z));
MOD_MUL(S->Z);
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&R->X, &AA, &BB));
MOD_MUL(R->X);
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&R->Z, &grp->A, &E));
MOD_MUL(R->Z);
MBEDTLS_MPI_CHK(mbedtls_mpi_add_mpi(&R->Z, &BB, &R->Z));
MOD_ADD(R->Z);
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&R->Z, &E, &R->Z));
MOD_MUL(R->Z);
cleanup:
mbedtls_mpi_free(&A);
mbedtls_mpi_free(&AA);
mbedtls_mpi_free(&B);
mbedtls_mpi_free(&BB);
mbedtls_mpi_free(&E);
mbedtls_mpi_free(&C);
mbedtls_mpi_free(&D);
mbedtls_mpi_free(&DA);
mbedtls_mpi_free(&CB);
return (ret);
}
/*
* Multiplication with Montgomery ladder in x/z coordinates,
* for curves in Montgomery form
*/
static int ecp_mul_mxz(mbedtls_ecp_group *grp, mbedtls_ecp_point *R,
const mbedtls_mpi *m, const mbedtls_ecp_point *P,
int (*f_rng)(void *, unsigned char *, size_t),
void *p_rng) {
int ret;
size_t i;
unsigned char b;
mbedtls_ecp_point RP;
mbedtls_mpi PX;
mbedtls_ecp_point_init(&RP);
mbedtls_mpi_init(&PX);
/* Save PX and read from P before writing to R, in case P == R */
MBEDTLS_MPI_CHK(mbedtls_mpi_copy(&PX, &P->X));
MBEDTLS_MPI_CHK(mbedtls_ecp_copy(&RP, P));
/* Set R to zero in modified x/z coordinates */
MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&R->X, 1));
MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&R->Z, 0));
mbedtls_mpi_free(&R->Y);
/* RP.X might be sligtly larger than P, so reduce it */
MOD_ADD(RP.X);
/* Randomize coordinates of the starting point */
if (f_rng != NULL)
MBEDTLS_MPI_CHK(ecp_randomize_mxz(grp, &RP, f_rng, p_rng));
/* Loop invariant: R = result so far, RP = R + P */
i = mbedtls_mpi_bitlen(m); /* one past the (zero-based) most significant bit */
while (i-- > 0) {
b = mbedtls_mpi_get_bit(m, i);
/*
* if (b) R = 2R + P else R = 2R,
* which is:
* if (b) double_add( RP, R, RP, R )
* else double_add( R, RP, R, RP )
* but using safe conditional swaps to avoid leaks
*/
MBEDTLS_MPI_CHK(mbedtls_mpi_safe_cond_swap(&R->X, &RP.X, b));
MBEDTLS_MPI_CHK(mbedtls_mpi_safe_cond_swap(&R->Z, &RP.Z, b));
MBEDTLS_MPI_CHK(ecp_double_add_mxz(grp, R, &RP, R, &RP, &PX));
MBEDTLS_MPI_CHK(mbedtls_mpi_safe_cond_swap(&R->X, &RP.X, b));
MBEDTLS_MPI_CHK(mbedtls_mpi_safe_cond_swap(&R->Z, &RP.Z, b));
}
MBEDTLS_MPI_CHK(ecp_normalize_mxz(grp, R));
cleanup:
mbedtls_ecp_point_free(&RP);
mbedtls_mpi_free(&PX);
return (ret);
}
#endif /* ECP_MONTGOMERY */
/*
* Multiplication R = m * P
*/
int mbedtls_ecp_mul(mbedtls_ecp_group *grp, mbedtls_ecp_point *R,
const mbedtls_mpi *m, const mbedtls_ecp_point *P,
int (*f_rng)(void *, unsigned char *, size_t), void *p_rng) {
int ret = MBEDTLS_ERR_ECP_BAD_INPUT_DATA;
#if defined(MBEDTLS_ECP_INTERNAL_ALT)
char is_grp_capable = 0;
#endif
/* Common sanity checks */
if (mbedtls_mpi_cmp_int(&P->Z, 1) != 0)
return (MBEDTLS_ERR_ECP_BAD_INPUT_DATA);
if ((ret = mbedtls_ecp_check_privkey(grp, m)) != 0 ||
(ret = mbedtls_ecp_check_pubkey(grp, P)) != 0)
return (ret);
#if defined(MBEDTLS_ECP_INTERNAL_ALT)
if (is_grp_capable = mbedtls_internal_ecp_grp_capable(grp)) {
MBEDTLS_MPI_CHK(mbedtls_internal_ecp_init(grp));
}
#endif /* MBEDTLS_ECP_INTERNAL_ALT */
#if defined(ECP_MONTGOMERY)
if (ecp_get_type(grp) == ECP_TYPE_MONTGOMERY)
ret = ecp_mul_mxz(grp, R, m, P, f_rng, p_rng);
#endif
#if defined(ECP_SHORTWEIERSTRASS)
if (ecp_get_type(grp) == ECP_TYPE_SHORT_WEIERSTRASS)
ret = ecp_mul_comb(grp, R, m, P, f_rng, p_rng);
#endif
#if defined(MBEDTLS_ECP_INTERNAL_ALT)
cleanup:
if (is_grp_capable) {
mbedtls_internal_ecp_free(grp);
}
#endif /* MBEDTLS_ECP_INTERNAL_ALT */
return (ret);
}
#if defined(ECP_SHORTWEIERSTRASS)
/*
* Check that an affine point is valid as a public key,
* short weierstrass curves (SEC1 3.2.3.1)
*/
static int ecp_check_pubkey_sw(const mbedtls_ecp_group *grp, const mbedtls_ecp_point *pt) {
int ret;
mbedtls_mpi YY, RHS;
/* pt coordinates must be normalized for our checks */
if (mbedtls_mpi_cmp_int(&pt->X, 0) < 0 ||
mbedtls_mpi_cmp_int(&pt->Y, 0) < 0 ||
mbedtls_mpi_cmp_mpi(&pt->X, &grp->P) >= 0 ||
mbedtls_mpi_cmp_mpi(&pt->Y, &grp->P) >= 0)
return (MBEDTLS_ERR_ECP_INVALID_KEY);
mbedtls_mpi_init(&YY);
mbedtls_mpi_init(&RHS);
/*
* YY = Y^2
* RHS = X (X^2 + A) + B = X^3 + A X + B
*/
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&YY, &pt->Y, &pt->Y));
MOD_MUL(YY);
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&RHS, &pt->X, &pt->X));
MOD_MUL(RHS);
/* Special case for A = -3 */
if (grp->A.p == NULL) {
MBEDTLS_MPI_CHK(mbedtls_mpi_sub_int(&RHS, &RHS, 3));
MOD_SUB(RHS);
} else {
MBEDTLS_MPI_CHK(mbedtls_mpi_add_mpi(&RHS, &RHS, &grp->A));
MOD_ADD(RHS);
}
MBEDTLS_MPI_CHK(mbedtls_mpi_mul_mpi(&RHS, &RHS, &pt->X));
MOD_MUL(RHS);
MBEDTLS_MPI_CHK(mbedtls_mpi_add_mpi(&RHS, &RHS, &grp->B));
MOD_ADD(RHS);
if (mbedtls_mpi_cmp_mpi(&YY, &RHS) != 0)
ret = MBEDTLS_ERR_ECP_INVALID_KEY;
cleanup:
mbedtls_mpi_free(&YY);
mbedtls_mpi_free(&RHS);
return (ret);
}
#endif /* ECP_SHORTWEIERSTRASS */
/*
* R = m * P with shortcuts for m == 1 and m == -1
* NOT constant-time - ONLY for short Weierstrass!
*/
static int mbedtls_ecp_mul_shortcuts(mbedtls_ecp_group *grp,
mbedtls_ecp_point *R,
const mbedtls_mpi *m,
const mbedtls_ecp_point *P) {
int ret;
if (mbedtls_mpi_cmp_int(m, 1) == 0) {
MBEDTLS_MPI_CHK(mbedtls_ecp_copy(R, P));
} else if (mbedtls_mpi_cmp_int(m, -1) == 0) {
MBEDTLS_MPI_CHK(mbedtls_ecp_copy(R, P));
if (mbedtls_mpi_cmp_int(&R->Y, 0) != 0)
MBEDTLS_MPI_CHK(mbedtls_mpi_sub_mpi(&R->Y, &grp->P, &R->Y));
} else {
MBEDTLS_MPI_CHK(mbedtls_ecp_mul(grp, R, m, P, NULL, NULL));
}
cleanup:
return (ret);
}
/*
* Linear combination
* NOT constant-time
*/
int mbedtls_ecp_muladd(mbedtls_ecp_group *grp, mbedtls_ecp_point *R,
const mbedtls_mpi *m, const mbedtls_ecp_point *P,
const mbedtls_mpi *n, const mbedtls_ecp_point *Q) {
int ret;
mbedtls_ecp_point mP;
#if defined(MBEDTLS_ECP_INTERNAL_ALT)
char is_grp_capable = 0;
#endif
if (ecp_get_type(grp) != ECP_TYPE_SHORT_WEIERSTRASS)
return (MBEDTLS_ERR_ECP_FEATURE_UNAVAILABLE);
mbedtls_ecp_point_init(&mP);
MBEDTLS_MPI_CHK(mbedtls_ecp_mul_shortcuts(grp, &mP, m, P));
MBEDTLS_MPI_CHK(mbedtls_ecp_mul_shortcuts(grp, R, n, Q));
#if defined(MBEDTLS_ECP_INTERNAL_ALT)
if (is_grp_capable = mbedtls_internal_ecp_grp_capable(grp)) {
MBEDTLS_MPI_CHK(mbedtls_internal_ecp_init(grp));
}
#endif /* MBEDTLS_ECP_INTERNAL_ALT */
MBEDTLS_MPI_CHK(ecp_add_mixed(grp, R, &mP, R));
MBEDTLS_MPI_CHK(ecp_normalize_jac(grp, R));
cleanup:
#if defined(MBEDTLS_ECP_INTERNAL_ALT)
if (is_grp_capable) {
mbedtls_internal_ecp_free(grp);
}
#endif /* MBEDTLS_ECP_INTERNAL_ALT */
mbedtls_ecp_point_free(&mP);
return (ret);
}
#if defined(ECP_MONTGOMERY)
/*
* Check validity of a public key for Montgomery curves with x-only schemes
*/
static int ecp_check_pubkey_mx(const mbedtls_ecp_group *grp, const mbedtls_ecp_point *pt) {
/* [Curve25519 p. 5] Just check X is the correct number of bytes */
/* Allow any public value, if it's too big then we'll just reduce it mod p
* (RFC 7748 sec. 5 para. 3). */
if (mbedtls_mpi_size(&pt->X) > (grp->nbits + 7) / 8)
return (MBEDTLS_ERR_ECP_INVALID_KEY);
return (0);
}
#endif /* ECP_MONTGOMERY */
/*
* Check that a point is valid as a public key
*/
int mbedtls_ecp_check_pubkey(const mbedtls_ecp_group *grp, const mbedtls_ecp_point *pt) {
/* Must use affine coordinates */
if (mbedtls_mpi_cmp_int(&pt->Z, 1) != 0)
return (MBEDTLS_ERR_ECP_INVALID_KEY);
#if defined(ECP_MONTGOMERY)
if (ecp_get_type(grp) == ECP_TYPE_MONTGOMERY)
return (ecp_check_pubkey_mx(grp, pt));
#endif
#if defined(ECP_SHORTWEIERSTRASS)
if (ecp_get_type(grp) == ECP_TYPE_SHORT_WEIERSTRASS)
return (ecp_check_pubkey_sw(grp, pt));
#endif
return (MBEDTLS_ERR_ECP_BAD_INPUT_DATA);
}
/*
* Check that an mbedtls_mpi is valid as a private key
*/
int mbedtls_ecp_check_privkey(const mbedtls_ecp_group *grp, const mbedtls_mpi *d) {
#if defined(ECP_MONTGOMERY)
if (ecp_get_type(grp) == ECP_TYPE_MONTGOMERY) {
/* see RFC 7748 sec. 5 para. 5 */
if (mbedtls_mpi_get_bit(d, 0) != 0 ||
mbedtls_mpi_get_bit(d, 1) != 0 ||
mbedtls_mpi_bitlen(d) - 1 != grp->nbits) /* mbedtls_mpi_bitlen is one-based! */
return (MBEDTLS_ERR_ECP_INVALID_KEY);
/* see [Curve25519] page 5 */
if (grp->nbits == 254 && mbedtls_mpi_get_bit(d, 2) != 0)
return (MBEDTLS_ERR_ECP_INVALID_KEY);
return (0);
}
#endif /* ECP_MONTGOMERY */
#if defined(ECP_SHORTWEIERSTRASS)
if (ecp_get_type(grp) == ECP_TYPE_SHORT_WEIERSTRASS) {
/* see SEC1 3.2 */
if (mbedtls_mpi_cmp_int(d, 1) < 0 ||
mbedtls_mpi_cmp_mpi(d, &grp->N) >= 0)
return (MBEDTLS_ERR_ECP_INVALID_KEY);
else
return (0);
}
#endif /* ECP_SHORTWEIERSTRASS */
return (MBEDTLS_ERR_ECP_BAD_INPUT_DATA);
}
/*
* Generate a keypair with configurable base point
*/
int mbedtls_ecp_gen_keypair_base(mbedtls_ecp_group *grp,
const mbedtls_ecp_point *G,
mbedtls_mpi *d, mbedtls_ecp_point *Q,
int (*f_rng)(void *, unsigned char *, size_t),
void *p_rng) {
int ret;
size_t n_size = (grp->nbits + 7) / 8;
#if defined(ECP_MONTGOMERY)
if (ecp_get_type(grp) == ECP_TYPE_MONTGOMERY) {
/* [M225] page 5 */
size_t b;
do {
MBEDTLS_MPI_CHK(mbedtls_mpi_fill_random(d, n_size, f_rng, p_rng));
} while (mbedtls_mpi_bitlen(d) == 0);
/* Make sure the most significant bit is nbits */
b = mbedtls_mpi_bitlen(d) - 1; /* mbedtls_mpi_bitlen is one-based */
if (b > grp->nbits)
MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(d, b - grp->nbits));
else
MBEDTLS_MPI_CHK(mbedtls_mpi_set_bit(d, grp->nbits, 1));
/* Make sure the last two bits are unset for Curve448, three bits for
Curve25519 */
MBEDTLS_MPI_CHK(mbedtls_mpi_set_bit(d, 0, 0));
MBEDTLS_MPI_CHK(mbedtls_mpi_set_bit(d, 1, 0));
if (grp->nbits == 254) {
MBEDTLS_MPI_CHK(mbedtls_mpi_set_bit(d, 2, 0));
}
} else
#endif /* ECP_MONTGOMERY */
#if defined(ECP_SHORTWEIERSTRASS)
if (ecp_get_type(grp) == ECP_TYPE_SHORT_WEIERSTRASS) {
/* SEC1 3.2.1: Generate d such that 1 <= n < N */
int count = 0;
/*
* Match the procedure given in RFC 6979 (deterministic ECDSA):
* - use the same byte ordering;
* - keep the leftmost nbits bits of the generated octet string;
* - try until result is in the desired range.
* This also avoids any biais, which is especially important for ECDSA.
*/
do {
MBEDTLS_MPI_CHK(mbedtls_mpi_fill_random(d, n_size, f_rng, p_rng));
MBEDTLS_MPI_CHK(mbedtls_mpi_shift_r(d, 8 * n_size - grp->nbits));
/*
* Each try has at worst a probability 1/2 of failing (the msb has
* a probability 1/2 of being 0, and then the result will be < N),
* so after 30 tries failure probability is a most 2**(-30).
*
* For most curves, 1 try is enough with overwhelming probability,
* since N starts with a lot of 1s in binary, but some curves
* such as secp224k1 are actually very close to the worst case.
*/
if (++count > 30)
return (MBEDTLS_ERR_ECP_RANDOM_FAILED);
} while (mbedtls_mpi_cmp_int(d, 1) < 0 ||
mbedtls_mpi_cmp_mpi(d, &grp->N) >= 0);
} else
#endif /* ECP_SHORTWEIERSTRASS */
return (MBEDTLS_ERR_ECP_BAD_INPUT_DATA);
cleanup:
if (ret != 0)
return (ret);
return (mbedtls_ecp_mul(grp, Q, d, G, f_rng, p_rng));
}
/*
* Generate key pair, wrapper for conventional base point
*/
int mbedtls_ecp_gen_keypair(mbedtls_ecp_group *grp,
mbedtls_mpi *d, mbedtls_ecp_point *Q,
int (*f_rng)(void *, unsigned char *, size_t),
void *p_rng) {
return (mbedtls_ecp_gen_keypair_base(grp, &grp->G, d, Q, f_rng, p_rng));
}
/*
* Generate a keypair, prettier wrapper
*/
int mbedtls_ecp_gen_key(mbedtls_ecp_group_id grp_id, mbedtls_ecp_keypair *key,
int (*f_rng)(void *, unsigned char *, size_t), void *p_rng) {
int ret;
if ((ret = mbedtls_ecp_group_load(&key->grp, grp_id)) != 0)
return (ret);
return (mbedtls_ecp_gen_keypair(&key->grp, &key->d, &key->Q, f_rng, p_rng));
}
/*
* Check a public-private key pair
*/
int mbedtls_ecp_check_pub_priv(const mbedtls_ecp_keypair *pub, const mbedtls_ecp_keypair *prv) {
int ret;
mbedtls_ecp_point Q;
mbedtls_ecp_group grp;
if (pub->grp.id == MBEDTLS_ECP_DP_NONE ||
pub->grp.id != prv->grp.id ||
mbedtls_mpi_cmp_mpi(&pub->Q.X, &prv->Q.X) ||
mbedtls_mpi_cmp_mpi(&pub->Q.Y, &prv->Q.Y) ||
mbedtls_mpi_cmp_mpi(&pub->Q.Z, &prv->Q.Z)) {
return (MBEDTLS_ERR_ECP_BAD_INPUT_DATA);
}
mbedtls_ecp_point_init(&Q);
mbedtls_ecp_group_init(&grp);
/* mbedtls_ecp_mul() needs a non-const group... */
mbedtls_ecp_group_copy(&grp, &prv->grp);
/* Also checks d is valid */
MBEDTLS_MPI_CHK(mbedtls_ecp_mul(&grp, &Q, &prv->d, &prv->grp.G, NULL, NULL));
if (mbedtls_mpi_cmp_mpi(&Q.X, &prv->Q.X) ||
mbedtls_mpi_cmp_mpi(&Q.Y, &prv->Q.Y) ||
mbedtls_mpi_cmp_mpi(&Q.Z, &prv->Q.Z)) {
ret = MBEDTLS_ERR_ECP_BAD_INPUT_DATA;
goto cleanup;
}
cleanup:
mbedtls_ecp_point_free(&Q);
mbedtls_ecp_group_free(&grp);
return (ret);
}
#if defined(MBEDTLS_SELF_TEST)
/*
* Checkup routine
*/
int mbedtls_ecp_self_test(int verbose) {
int ret;
size_t i;
mbedtls_ecp_group grp;
mbedtls_ecp_point R, P;
mbedtls_mpi m;
unsigned long add_c_prev, dbl_c_prev, mul_c_prev;
/* exponents especially adapted for secp192r1 */
const char *exponents[] = {
"000000000000000000000000000000000000000000000001", /* one */
"FFFFFFFFFFFFFFFFFFFFFFFF99DEF836146BC9B1B4D22830", /* N - 1 */
"5EA6F389A38B8BC81E767753B15AA5569E1782E30ABE7D25", /* random */
"400000000000000000000000000000000000000000000000", /* one and zeros */
"7FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF", /* all ones */
"555555555555555555555555555555555555555555555555", /* 101010... */
};
mbedtls_ecp_group_init(&grp);
mbedtls_ecp_point_init(&R);
mbedtls_ecp_point_init(&P);
mbedtls_mpi_init(&m);
/* Use secp192r1 if available, or any available curve */
#if defined(MBEDTLS_ECP_DP_SECP192R1_ENABLED)
MBEDTLS_MPI_CHK(mbedtls_ecp_group_load(&grp, MBEDTLS_ECP_DP_SECP192R1));
#else
MBEDTLS_MPI_CHK(mbedtls_ecp_group_load(&grp, mbedtls_ecp_curve_list()->grp_id));
#endif
if (verbose != 0)
mbedtls_printf(" ECP test #1 (constant op_count, base point G): ");
/* Do a dummy multiplication first to trigger precomputation */
MBEDTLS_MPI_CHK(mbedtls_mpi_lset(&m, 2));
MBEDTLS_MPI_CHK(mbedtls_ecp_mul(&grp, &P, &m, &grp.G, NULL, NULL));
add_count = 0;
dbl_count = 0;
mul_count = 0;
MBEDTLS_MPI_CHK(mbedtls_mpi_read_string(&m, 16, exponents[0]));
MBEDTLS_MPI_CHK(mbedtls_ecp_mul(&grp, &R, &m, &grp.G, NULL, NULL));
for (i = 1; i < sizeof(exponents) / sizeof(exponents[0]); i++) {
add_c_prev = add_count;
dbl_c_prev = dbl_count;
mul_c_prev = mul_count;
add_count = 0;
dbl_count = 0;
mul_count = 0;
MBEDTLS_MPI_CHK(mbedtls_mpi_read_string(&m, 16, exponents[i]));
MBEDTLS_MPI_CHK(mbedtls_ecp_mul(&grp, &R, &m, &grp.G, NULL, NULL));
if (add_count != add_c_prev ||
dbl_count != dbl_c_prev ||
mul_count != mul_c_prev) {
if (verbose != 0)
mbedtls_printf("failed (%u)\n", (unsigned int) i);
ret = 1;
goto cleanup;
}
}
if (verbose != 0)
mbedtls_printf("passed\n");
if (verbose != 0)
mbedtls_printf(" ECP test #2 (constant op_count, other point): ");
/* We computed P = 2G last time, use it */
add_count = 0;
dbl_count = 0;
mul_count = 0;
MBEDTLS_MPI_CHK(mbedtls_mpi_read_string(&m, 16, exponents[0]));
MBEDTLS_MPI_CHK(mbedtls_ecp_mul(&grp, &R, &m, &P, NULL, NULL));
for (i = 1; i < sizeof(exponents) / sizeof(exponents[0]); i++) {
add_c_prev = add_count;
dbl_c_prev = dbl_count;
mul_c_prev = mul_count;
add_count = 0;
dbl_count = 0;
mul_count = 0;
MBEDTLS_MPI_CHK(mbedtls_mpi_read_string(&m, 16, exponents[i]));
MBEDTLS_MPI_CHK(mbedtls_ecp_mul(&grp, &R, &m, &P, NULL, NULL));
if (add_count != add_c_prev ||
dbl_count != dbl_c_prev ||
mul_count != mul_c_prev) {
if (verbose != 0)
mbedtls_printf("failed (%u)\n", (unsigned int) i);
ret = 1;
goto cleanup;
}
}
if (verbose != 0)
mbedtls_printf("passed\n");
cleanup:
if (ret < 0 && verbose != 0)
mbedtls_printf("Unexpected error, return code = %08X\n", ret);
mbedtls_ecp_group_free(&grp);
mbedtls_ecp_point_free(&R);
mbedtls_ecp_point_free(&P);
mbedtls_mpi_free(&m);
if (verbose != 0)
mbedtls_printf("\n");
return (ret);
}
#endif /* MBEDTLS_SELF_TEST */
#endif /* !MBEDTLS_ECP_ALT */
#endif /* MBEDTLS_ECP_C */