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mbedtls/tinycrypt/ecc.c

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/* ecc.c - TinyCrypt implementation of common ECC functions */
/*
* Copyright (c) 2019, Arm Limited (or its affiliates), All Rights Reserved.
* SPDX-License-Identifier: BSD-3-Clause
*/
/*
* Copyright (c) 2014, Kenneth MacKay
* All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions are met:
* * Redistributions of source code must retain the above copyright notice,
* this list of conditions and the following disclaimer.
* * Redistributions in binary form must reproduce the above copyright notice,
* this list of conditions and the following disclaimer in the documentation
* and/or other materials provided with the distribution.
*
* THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND
* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED
* WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE
* DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR
* ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES
* (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
* LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON
* ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
* (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS
* SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*
* Copyright (C) 2017 by Intel Corporation, All Rights Reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions are met:
*
* - Redistributions of source code must retain the above copyright notice,
* this list of conditions and the following disclaimer.
*
* - Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
*
* - Neither the name of Intel Corporation nor the names of its contributors
* may be used to endorse or promote products derived from this software
* without specific prior written permission.
*
* THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
* AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
* ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE
* LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
* CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
* SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
* INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
* CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
* ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
* POSSIBILITY OF SUCH DAMAGE.
*/
#if !defined(MBEDTLS_CONFIG_FILE)
#include "mbedtls/config.h"
#else
#include MBEDTLS_CONFIG_FILE
#endif
#include <tinycrypt/ecc.h>
#include "mbedtls/platform_util.h"
#include "mbedtls/sha256.h"
#include <string.h>
/* Parameters for curve NIST P-256 aka secp256r1 */
const uECC_word_t curve_p[NUM_ECC_WORDS] = {
BYTES_TO_WORDS_8(FF, FF, FF, FF, FF, FF, FF, FF),
BYTES_TO_WORDS_8(FF, FF, FF, FF, 00, 00, 00, 00),
BYTES_TO_WORDS_8(00, 00, 00, 00, 00, 00, 00, 00),
BYTES_TO_WORDS_8(01, 00, 00, 00, FF, FF, FF, FF)
};
const uECC_word_t curve_n[NUM_ECC_WORDS] = {
BYTES_TO_WORDS_8(51, 25, 63, FC, C2, CA, B9, F3),
BYTES_TO_WORDS_8(84, 9E, 17, A7, AD, FA, E6, BC),
BYTES_TO_WORDS_8(FF, FF, FF, FF, FF, FF, FF, FF),
BYTES_TO_WORDS_8(00, 00, 00, 00, FF, FF, FF, FF)
};
const uECC_word_t curve_G[2 * NUM_ECC_WORDS] = {
BYTES_TO_WORDS_8(96, C2, 98, D8, 45, 39, A1, F4),
BYTES_TO_WORDS_8(A0, 33, EB, 2D, 81, 7D, 03, 77),
BYTES_TO_WORDS_8(F2, 40, A4, 63, E5, E6, BC, F8),
BYTES_TO_WORDS_8(47, 42, 2C, E1, F2, D1, 17, 6B),
BYTES_TO_WORDS_8(F5, 51, BF, 37, 68, 40, B6, CB),
BYTES_TO_WORDS_8(CE, 5E, 31, 6B, 57, 33, CE, 2B),
BYTES_TO_WORDS_8(16, 9E, 0F, 7C, 4A, EB, E7, 8E),
BYTES_TO_WORDS_8(9B, 7F, 1A, FE, E2, 42, E3, 4F)
};
const uECC_word_t curve_b[NUM_ECC_WORDS] = {
BYTES_TO_WORDS_8(4B, 60, D2, 27, 3E, 3C, CE, 3B),
BYTES_TO_WORDS_8(F6, B0, 53, CC, B0, 06, 1D, 65),
BYTES_TO_WORDS_8(BC, 86, 98, 76, 55, BD, EB, B3),
BYTES_TO_WORDS_8(E7, 93, 3A, AA, D8, 35, C6, 5A)
};
static int uECC_update_param_sha256(mbedtls_sha256_context *ctx,
const uECC_word_t val[NUM_ECC_WORDS])
{
uint8_t bytes[NUM_ECC_BYTES];
uECC_vli_nativeToBytes(bytes, NUM_ECC_BYTES, val);
return mbedtls_sha256_update_ret(ctx, bytes, NUM_ECC_BYTES);
}
static int uECC_compute_param_sha256(unsigned char output[32])
{
int ret = UECC_FAILURE;
mbedtls_sha256_context ctx;
mbedtls_sha256_init( &ctx );
if (mbedtls_sha256_starts_ret(&ctx, 0) != 0) {
goto exit;
}
if (uECC_update_param_sha256(&ctx, curve_p) != 0 ||
uECC_update_param_sha256(&ctx, curve_n) != 0 ||
uECC_update_param_sha256(&ctx, curve_G) != 0 ||
uECC_update_param_sha256(&ctx, curve_G + NUM_ECC_WORDS) != 0 ||
uECC_update_param_sha256(&ctx, curve_b) != 0)
{
goto exit;
}
if (mbedtls_sha256_finish_ret(&ctx, output) != 0) {
goto exit;
}
ret = UECC_SUCCESS;
exit:
mbedtls_sha256_free( &ctx );
return ret;
}
/*
* Check integrity of curve parameters.
* Return 0 if everything's OK, non-zero otherwise.
*/
static int uECC_check_curve_integrity(void)
{
unsigned char computed[32];
static const unsigned char reference[32] = {
0x2d, 0xa1, 0xa4, 0x64, 0x45, 0x28, 0x0d, 0xe1,
0x93, 0xf9, 0x29, 0x2f, 0xac, 0x3e, 0xe2, 0x92,
0x76, 0x0a, 0xe2, 0xbc, 0xce, 0x2a, 0xa2, 0xc6,
0x38, 0xf2, 0x19, 0x1d, 0x76, 0x72, 0x93, 0x49,
};
unsigned char diff = 0;
unsigned char tmp1, tmp2;
volatile unsigned i;
if (uECC_compute_param_sha256(computed) != UECC_SUCCESS) {
return UECC_FAILURE;
}
for (i = 0; i < 32; i++) {
/* make sure the order of volatile accesses is well-defined */
tmp1 = computed[i];
tmp2 = reference[i];
diff |= tmp1 ^ tmp2;
}
/* i should be 32 */
mbedtls_platform_enforce_volatile_reads();
diff |= (unsigned char) i ^ 32;
return diff;
}
/* IMPORTANT: Make sure a cryptographically-secure PRNG is set and the platform
* has access to enough entropy in order to feed the PRNG regularly. */
#if default_RNG_defined
static uECC_RNG_Function g_rng_function = &default_CSPRNG;
#else
static uECC_RNG_Function g_rng_function = 0;
#endif
void uECC_set_rng(uECC_RNG_Function rng_function)
{
g_rng_function = rng_function;
}
uECC_RNG_Function uECC_get_rng(void)
{
return g_rng_function;
}
int uECC_curve_private_key_size(void)
{
return BITS_TO_BYTES(NUM_ECC_BITS);
}
int uECC_curve_public_key_size(void)
{
return 2 * NUM_ECC_BYTES;
}
void uECC_vli_clear(uECC_word_t *vli)
{
wordcount_t i;
for (i = 0; i < NUM_ECC_WORDS; ++i) {
vli[i] = 0;
}
}
uECC_word_t uECC_vli_isZero(const uECC_word_t *vli)
{
uECC_word_t bits = 0;
wordcount_t i;
for (i = 0; i < NUM_ECC_WORDS; ++i) {
bits |= vli[i];
}
return (bits == 0);
}
uECC_word_t uECC_vli_testBit(const uECC_word_t *vli, bitcount_t bit)
{
return (vli[bit >> uECC_WORD_BITS_SHIFT] &
((uECC_word_t)1 << (bit & uECC_WORD_BITS_MASK)));
}
/* Counts the number of words in vli. */
static wordcount_t vli_numDigits(const uECC_word_t *vli)
{
wordcount_t i;
/* Search from the end until we find a non-zero digit. We do it in reverse
* because we expect that most digits will be nonzero. */
for (i = NUM_ECC_WORDS - 1; i >= 0 && vli[i] == 0; --i) {
}
return (i + 1);
}
bitcount_t uECC_vli_numBits(const uECC_word_t *vli)
{
uECC_word_t i;
uECC_word_t digit;
wordcount_t num_digits = vli_numDigits(vli);
if (num_digits == 0) {
return 0;
}
digit = vli[num_digits - 1];
for (i = 0; digit; ++i) {
digit >>= 1;
}
return (((bitcount_t)(num_digits - 1) << uECC_WORD_BITS_SHIFT) + i);
}
void uECC_vli_set(uECC_word_t *dest, const uECC_word_t *src)
{
wordcount_t i;
for (i = 0; i < NUM_ECC_WORDS; ++i) {
dest[i] = src[i];
}
}
cmpresult_t uECC_vli_cmp_unsafe(const uECC_word_t *left,
const uECC_word_t *right)
{
wordcount_t i;
for (i = NUM_ECC_WORDS - 1; i >= 0; --i) {
if (left[i] > right[i]) {
return 1;
} else if (left[i] < right[i]) {
return -1;
}
}
return 0;
}
uECC_word_t uECC_vli_equal(const uECC_word_t *left, const uECC_word_t *right)
{
uECC_word_t diff = 0;
uECC_word_t tmp1, tmp2;
volatile int i;
for (i = NUM_ECC_WORDS - 1; i >= 0; --i) {
tmp1 = left[i];
tmp2 = right[i];
diff |= (tmp1 ^ tmp2);
}
/* i should be -1 now */
mbedtls_platform_enforce_volatile_reads();
diff |= i ^ -1;
return diff;
}
uECC_word_t cond_set(uECC_word_t p_true, uECC_word_t p_false, unsigned int cond)
{
return (p_true*(cond)) | (p_false*(!cond));
}
/* Computes result = left - right, returning borrow, in constant time.
* Can modify in place. */
uECC_word_t uECC_vli_sub(uECC_word_t *result, const uECC_word_t *left,
const uECC_word_t *right)
{
uECC_word_t borrow = 0;
wordcount_t i;
for (i = 0; i < NUM_ECC_WORDS; ++i) {
uECC_word_t diff = left[i] - right[i] - borrow;
uECC_word_t val = (diff > left[i]);
borrow = cond_set(val, borrow, (diff != left[i]));
result[i] = diff;
}
return borrow;
}
/* Computes result = left + right, returning carry, in constant time.
* Can modify in place. */
static uECC_word_t uECC_vli_add(uECC_word_t *result, const uECC_word_t *left,
const uECC_word_t *right)
{
uECC_word_t carry = 0;
wordcount_t i;
for (i = 0; i < NUM_ECC_WORDS; ++i) {
uECC_word_t sum = left[i] + right[i] + carry;
uECC_word_t val = (sum < left[i]);
carry = cond_set(val, carry, (sum != left[i]));
result[i] = sum;
}
return carry;
}
cmpresult_t uECC_vli_cmp(const uECC_word_t *left, const uECC_word_t *right)
{
uECC_word_t tmp[NUM_ECC_WORDS];
uECC_word_t neg = !!uECC_vli_sub(tmp, left, right);
uECC_word_t equal = uECC_vli_isZero(tmp);
return (!equal - 2 * neg);
}
/* Computes vli = vli >> 1. */
static void uECC_vli_rshift1(uECC_word_t *vli)
{
uECC_word_t *end = vli;
uECC_word_t carry = 0;
vli += NUM_ECC_WORDS;
while (vli-- > end) {
uECC_word_t temp = *vli;
*vli = (temp >> 1) | carry;
carry = temp << (uECC_WORD_BITS - 1);
}
}
/* Compute a * b + r, where r is a double-word with high-order word r1 and
* low-order word r0, and store the result in the same double-word (r1, r0),
* with the carry bit stored in r2.
*
* (r2, r1, r0) = a * b + (r1, r0):
* [in] a, b: operands to be multiplied
* [in] r0, r1: low and high-order words of operand to add
* [out] r0, r1: low and high-order words of the result
* [out] r2: carry
*/
static void muladd(uECC_word_t a, uECC_word_t b, uECC_word_t *r0,
uECC_word_t *r1, uECC_word_t *r2)
{
uECC_dword_t p = (uECC_dword_t)a * b;
uECC_dword_t r01 = ((uECC_dword_t)(*r1) << uECC_WORD_BITS) | *r0;
r01 += p;
*r2 += (r01 < p);
*r1 = r01 >> uECC_WORD_BITS;
*r0 = (uECC_word_t)r01;
}
/* State for implementing random delays in uECC_vli_mult_rnd().
*
* The state is initialized by randomizing delays and setting i = 0.
* Each call to uECC_vli_mult_rnd() uses one byte of delays and increments i.
*
* Randomized vli multiplication is used only for point operations
* (XYcZ_add_rnd() * and XYcZ_addC_rnd()) in scalar multiplication
* (ECCPoint_mult()). Those go in pair, and each pair does 14 calls to
* uECC_vli_mult_rnd() (6 in XYcZ_add_rnd() and 8 in XYcZ_addC_rnd(),
* indirectly through uECC_vli_modMult_rnd().
*
* Considering this, in order to minimize the number of calls to the RNG
* (which impact performance) while keeping the size of the structure low,
* make room for 14 randomized vli mults, which corresponds to one step in the
* scalar multiplication routine.
*/
typedef struct {
uint8_t i;
uint8_t delays[14];
} ecc_wait_state_t;
/*
* Reset wait_state so that it's ready to be used.
*/
void ecc_wait_state_reset(ecc_wait_state_t *ws)
{
if (ws == NULL)
return;
ws->i = 0;
g_rng_function(ws->delays, sizeof(ws->delays));
}
/* Computes result = left * right. Result must be 2 * num_words long.
*
* As a counter-measure against horizontal attacks, add noise by performing
* a random number of extra computations performing random additional accesses
* to limbs of the input.
*
* Each of the two actual computation loops is surrounded by two
* similar-looking waiting loops, to make the beginning and end of the actual
* computation harder to spot.
*
* We add 4 waiting loops of between 0 and 3 calls to muladd() each. That
* makes an average of 6 extra calls. Compared to the main computation which
* makes 64 such calls, this represents an average performance degradation of
* less than 10%.
*
* Compared to the original uECC_vli_mult(), loose the num_words argument as we
* know it's always 8. This saves a bit of code size and execution speed.
*/
static void uECC_vli_mult_rnd(uECC_word_t *result, const uECC_word_t *left,
const uECC_word_t *right, ecc_wait_state_t *s)
{
uECC_word_t r0 = 0;
uECC_word_t r1 = 0;
uECC_word_t r2 = 0;
wordcount_t i, k;
const uint8_t num_words = NUM_ECC_WORDS;
/* Fetch 8 bit worth of delay from the state; 0 if we have no state */
uint8_t delays = s ? s->delays[s->i++] : 0;
uECC_word_t rr0 = 0, rr1 = 0;
volatile uECC_word_t r;
/* Mimic start of next loop: k in [0, 3] */
k = 0 + (delays & 0x03);
delays >>= 2;
/* k = 0 -> i in [1, 0] -> 0 extra muladd;
* k = 3 -> i in [1, 3] -> 3 extra muladd */
for (i = 1; i <= k; ++i) {
muladd(left[i], right[k - i], &rr0, &rr1, &r2);
}
r = rr0;
rr0 = rr1;
rr1 = r2;
r2 = 0;
/* Compute each digit of result in sequence, maintaining the carries. */
for (k = 0; k < num_words; ++k) {
for (i = 0; i <= k; ++i) {
muladd(left[i], right[k - i], &r0, &r1, &r2);
}
result[k] = r0;
r0 = r1;
r1 = r2;
r2 = 0;
}
/* Mimic end of previous loop: k in [4, 7] */
k = 4 + (delays & 0x03);
delays >>= 2;
/* k = 4 -> i in [5, 4] -> 0 extra muladd;
* k = 7 -> i in [5, 7] -> 3 extra muladd */
for (i = 5; i <= k; ++i) {
muladd(left[i], right[k - i], &rr0, &rr1, &r2);
}
r = rr0;
rr0 = rr1;
rr1 = r2;
r2 = 0;
/* Mimic start of next loop: k in [8, 11] */
k = 11 - (delays & 0x03);
delays >>= 2;
/* k = 8 -> i in [5, 7] -> 3 extra muladd;
* k = 11 -> i in [8, 7] -> 0 extra muladd */
for (i = (k + 5) - num_words; i < num_words; ++i) {
muladd(left[i], right[k - i], &rr0, &rr1, &r2);
}
r = rr0;
rr0 = rr1;
rr1 = r2;
r2 = 0;
for (k = num_words; k < num_words * 2 - 1; ++k) {
for (i = (k + 1) - num_words; i < num_words; ++i) {
muladd(left[i], right[k - i], &r0, &r1, &r2);
}
result[k] = r0;
r0 = r1;
r1 = r2;
r2 = 0;
}
result[num_words * 2 - 1] = r0;
/* Mimic end of previous loop: k in [12, 15] */
k = 15 - (delays & 0x03);
delays >>= 2;
/* k = 12 -> i in [5, 7] -> 3 extra muladd;
* k = 15 -> i in [8, 7] -> 0 extra muladd */
for (i = (k + 1) - num_words; i < num_words; ++i) {
muladd(left[i], right[k - i], &rr0, &rr1, &r2);
}
r = rr0;
rr0 = rr1;
rr1 = r2;
r2 = 0;
/* avoid warning that r is set but not used */
(void) r;
}
void uECC_vli_modAdd(uECC_word_t *result, const uECC_word_t *left,
const uECC_word_t *right, const uECC_word_t *mod)
{
uECC_word_t carry = uECC_vli_add(result, left, right);
if (carry || uECC_vli_cmp_unsafe(mod, result) != 1) {
/* result > mod (result = mod + remainder), so subtract mod to get
* remainder. */
uECC_vli_sub(result, result, mod);
}
}
void uECC_vli_modSub(uECC_word_t *result, const uECC_word_t *left,
const uECC_word_t *right, const uECC_word_t *mod)
{
uECC_word_t l_borrow = uECC_vli_sub(result, left, right);
if (l_borrow) {
/* In this case, result == -diff == (max int) - diff. Since -x % d == d - x,
* we can get the correct result from result + mod (with overflow). */
uECC_vli_add(result, result, mod);
}
}
/* Computes result = product % mod, where product is 2N words long. */
/* Currently only designed to work for curve_p or curve_n. */
void uECC_vli_mmod(uECC_word_t *result, uECC_word_t *product,
const uECC_word_t *mod)
{
uECC_word_t mod_multiple[2 * NUM_ECC_WORDS];
uECC_word_t tmp[2 * NUM_ECC_WORDS];
uECC_word_t *v[2] = {tmp, product};
uECC_word_t index;
const wordcount_t num_words = NUM_ECC_WORDS;
/* Shift mod so its highest set bit is at the maximum position. */
bitcount_t shift = (num_words * 2 * uECC_WORD_BITS) -
uECC_vli_numBits(mod);
wordcount_t word_shift = shift / uECC_WORD_BITS;
wordcount_t bit_shift = shift % uECC_WORD_BITS;
uECC_word_t carry = 0;
uECC_vli_clear(mod_multiple);
if (bit_shift > 0) {
for(index = 0; index < (uECC_word_t)num_words; ++index) {
mod_multiple[word_shift + index] = (mod[index] << bit_shift) | carry;
carry = mod[index] >> (uECC_WORD_BITS - bit_shift);
}
} else {
uECC_vli_set(mod_multiple + word_shift, mod);
}
for (index = 1; shift >= 0; --shift) {
uECC_word_t borrow = 0;
wordcount_t i;
for (i = 0; i < num_words * 2; ++i) {
uECC_word_t diff = v[index][i] - mod_multiple[i] - borrow;
if (diff != v[index][i]) {
borrow = (diff > v[index][i]);
}
v[1 - index][i] = diff;
}
/* Swap the index if there was no borrow */
index = !(index ^ borrow);
uECC_vli_rshift1(mod_multiple);
mod_multiple[num_words - 1] |= mod_multiple[num_words] <<
(uECC_WORD_BITS - 1);
uECC_vli_rshift1(mod_multiple + num_words);
}
uECC_vli_set(result, v[index]);
}
void uECC_vli_modMult(uECC_word_t *result, const uECC_word_t *left,
const uECC_word_t *right, const uECC_word_t *mod)
{
uECC_word_t product[2 * NUM_ECC_WORDS];
uECC_vli_mult_rnd(product, left, right, NULL);
uECC_vli_mmod(result, product, mod);
}
static void uECC_vli_modMult_rnd(uECC_word_t *result, const uECC_word_t *left,
const uECC_word_t *right, ecc_wait_state_t *s)
{
uECC_word_t product[2 * NUM_ECC_WORDS];
uECC_vli_mult_rnd(product, left, right, s);
vli_mmod_fast_secp256r1(result, product);
}
void uECC_vli_modMult_fast(uECC_word_t *result, const uECC_word_t *left,
const uECC_word_t *right)
{
uECC_vli_modMult_rnd(result, left, right, NULL);
}
#define EVEN(vli) (!(vli[0] & 1))
static void vli_modInv_update(uECC_word_t *uv,
const uECC_word_t *mod)
{
uECC_word_t carry = 0;
if (!EVEN(uv)) {
carry = uECC_vli_add(uv, uv, mod);
}
uECC_vli_rshift1(uv);
if (carry) {
uv[NUM_ECC_WORDS - 1] |= HIGH_BIT_SET;
}
}
void uECC_vli_modInv(uECC_word_t *result, const uECC_word_t *input,
const uECC_word_t *mod)
{
uECC_word_t a[NUM_ECC_WORDS], b[NUM_ECC_WORDS];
uECC_word_t u[NUM_ECC_WORDS], v[NUM_ECC_WORDS];
cmpresult_t cmpResult;
if (uECC_vli_isZero(input)) {
uECC_vli_clear(result);
return;
}
uECC_vli_set(a, input);
uECC_vli_set(b, mod);
uECC_vli_clear(u);
u[0] = 1;
uECC_vli_clear(v);
while ((cmpResult = uECC_vli_cmp_unsafe(a, b)) != 0) {
if (EVEN(a)) {
uECC_vli_rshift1(a);
vli_modInv_update(u, mod);
} else if (EVEN(b)) {
uECC_vli_rshift1(b);
vli_modInv_update(v, mod);
} else if (cmpResult > 0) {
uECC_vli_sub(a, a, b);
uECC_vli_rshift1(a);
if (uECC_vli_cmp_unsafe(u, v) < 0) {
uECC_vli_add(u, u, mod);
}
uECC_vli_sub(u, u, v);
vli_modInv_update(u, mod);
} else {
uECC_vli_sub(b, b, a);
uECC_vli_rshift1(b);
if (uECC_vli_cmp_unsafe(v, u) < 0) {
uECC_vli_add(v, v, mod);
}
uECC_vli_sub(v, v, u);
vli_modInv_update(v, mod);
}
}
uECC_vli_set(result, u);
}
/* ------ Point operations ------ */
void double_jacobian_default(uECC_word_t * X1, uECC_word_t * Y1,
uECC_word_t * Z1)
{
/* t1 = X, t2 = Y, t3 = Z */
uECC_word_t t4[NUM_ECC_WORDS];
uECC_word_t t5[NUM_ECC_WORDS];
wordcount_t num_words = NUM_ECC_WORDS;
if (uECC_vli_isZero(Z1)) {
return;
}
uECC_vli_modMult_fast(t4, Y1, Y1); /* t4 = y1^2 */
uECC_vli_modMult_fast(t5, X1, t4); /* t5 = x1*y1^2 = A */
uECC_vli_modMult_fast(t4, t4, t4); /* t4 = y1^4 */
uECC_vli_modMult_fast(Y1, Y1, Z1); /* t2 = y1*z1 = z3 */
uECC_vli_modMult_fast(Z1, Z1, Z1); /* t3 = z1^2 */
uECC_vli_modAdd(X1, X1, Z1, curve_p); /* t1 = x1 + z1^2 */
uECC_vli_modAdd(Z1, Z1, Z1, curve_p); /* t3 = 2*z1^2 */
uECC_vli_modSub(Z1, X1, Z1, curve_p); /* t3 = x1 - z1^2 */
uECC_vli_modMult_fast(X1, X1, Z1); /* t1 = x1^2 - z1^4 */
uECC_vli_modAdd(Z1, X1, X1, curve_p); /* t3 = 2*(x1^2 - z1^4) */
uECC_vli_modAdd(X1, X1, Z1, curve_p); /* t1 = 3*(x1^2 - z1^4) */
if (uECC_vli_testBit(X1, 0)) {
uECC_word_t l_carry = uECC_vli_add(X1, X1, curve_p);
uECC_vli_rshift1(X1);
X1[num_words - 1] |= l_carry << (uECC_WORD_BITS - 1);
} else {
uECC_vli_rshift1(X1);
}
/* t1 = 3/2*(x1^2 - z1^4) = B */
uECC_vli_modMult_fast(Z1, X1, X1); /* t3 = B^2 */
uECC_vli_modSub(Z1, Z1, t5, curve_p); /* t3 = B^2 - A */
uECC_vli_modSub(Z1, Z1, t5, curve_p); /* t3 = B^2 - 2A = x3 */
uECC_vli_modSub(t5, t5, Z1, curve_p); /* t5 = A - x3 */
uECC_vli_modMult_fast(X1, X1, t5); /* t1 = B * (A - x3) */
/* t4 = B * (A - x3) - y1^4 = y3: */
uECC_vli_modSub(t4, X1, t4, curve_p);
uECC_vli_set(X1, Z1);
uECC_vli_set(Z1, Y1);
uECC_vli_set(Y1, t4);
}
/*
* @brief Computes x^3 + ax + b. result must not overlap x.
* @param result OUT -- x^3 + ax + b
* @param x IN -- value of x
* @param curve IN -- elliptic curve
*/
static void x_side_default(uECC_word_t *result,
const uECC_word_t *x)
{
uECC_word_t _3[NUM_ECC_WORDS] = {3}; /* -a = 3 */
uECC_vli_modMult_fast(result, x, x); /* r = x^2 */
uECC_vli_modSub(result, result, _3, curve_p); /* r = x^2 - 3 */
uECC_vli_modMult_fast(result, result, x); /* r = x^3 - 3x */
/* r = x^3 - 3x + b: */
uECC_vli_modAdd(result, result, curve_b, curve_p);
}
void vli_mmod_fast_secp256r1(unsigned int *result, unsigned int*product)
{
unsigned int tmp[NUM_ECC_WORDS];
int carry;
/* t */
uECC_vli_set(result, product);
/* s1 */
tmp[0] = tmp[1] = tmp[2] = 0;
tmp[3] = product[11];
tmp[4] = product[12];
tmp[5] = product[13];
tmp[6] = product[14];
tmp[7] = product[15];
carry = uECC_vli_add(tmp, tmp, tmp);
carry += uECC_vli_add(result, result, tmp);
/* s2 */
tmp[3] = product[12];
tmp[4] = product[13];
tmp[5] = product[14];
tmp[6] = product[15];
tmp[7] = 0;
carry += uECC_vli_add(tmp, tmp, tmp);
carry += uECC_vli_add(result, result, tmp);
/* s3 */
tmp[0] = product[8];
tmp[1] = product[9];
tmp[2] = product[10];
tmp[3] = tmp[4] = tmp[5] = 0;
tmp[6] = product[14];
tmp[7] = product[15];
carry += uECC_vli_add(result, result, tmp);
/* s4 */
tmp[0] = product[9];
tmp[1] = product[10];
tmp[2] = product[11];
tmp[3] = product[13];
tmp[4] = product[14];
tmp[5] = product[15];
tmp[6] = product[13];
tmp[7] = product[8];
carry += uECC_vli_add(result, result, tmp);
/* d1 */
tmp[0] = product[11];
tmp[1] = product[12];
tmp[2] = product[13];
tmp[3] = tmp[4] = tmp[5] = 0;
tmp[6] = product[8];
tmp[7] = product[10];
carry -= uECC_vli_sub(result, result, tmp);
/* d2 */
tmp[0] = product[12];
tmp[1] = product[13];
tmp[2] = product[14];
tmp[3] = product[15];
tmp[4] = tmp[5] = 0;
tmp[6] = product[9];
tmp[7] = product[11];
carry -= uECC_vli_sub(result, result, tmp);
/* d3 */
tmp[0] = product[13];
tmp[1] = product[14];
tmp[2] = product[15];
tmp[3] = product[8];
tmp[4] = product[9];
tmp[5] = product[10];
tmp[6] = 0;
tmp[7] = product[12];
carry -= uECC_vli_sub(result, result, tmp);
/* d4 */
tmp[0] = product[14];
tmp[1] = product[15];
tmp[2] = 0;
tmp[3] = product[9];
tmp[4] = product[10];
tmp[5] = product[11];
tmp[6] = 0;
tmp[7] = product[13];
carry -= uECC_vli_sub(result, result, tmp);
if (carry < 0) {
do {
carry += uECC_vli_add(result, result, curve_p);
}
while (carry < 0);
} else {
while (carry ||
uECC_vli_cmp_unsafe(curve_p, result) != 1) {
carry -= uECC_vli_sub(result, result, curve_p);
}
}
}
uECC_word_t EccPoint_isZero(const uECC_word_t *point)
{
return uECC_vli_isZero(point);
}
void apply_z(uECC_word_t * X1, uECC_word_t * Y1, const uECC_word_t * const Z)
{
uECC_word_t t1[NUM_ECC_WORDS];
uECC_vli_modMult_fast(t1, Z, Z); /* z^2 */
uECC_vli_modMult_fast(X1, X1, t1); /* x1 * z^2 */
uECC_vli_modMult_fast(t1, t1, Z); /* z^3 */
uECC_vli_modMult_fast(Y1, Y1, t1); /* y1 * z^3 */
}
/* P = (x1, y1) => 2P, (x2, y2) => P' */
static void XYcZ_initial_double(uECC_word_t * X1, uECC_word_t * Y1,
uECC_word_t * X2, uECC_word_t * Y2,
const uECC_word_t * const initial_Z)
{
uECC_word_t z[NUM_ECC_WORDS];
if (initial_Z) {
uECC_vli_set(z, initial_Z);
} else {
uECC_vli_clear(z);
z[0] = 1;
}
uECC_vli_set(X2, X1);
uECC_vli_set(Y2, Y1);
apply_z(X1, Y1, z);
double_jacobian_default(X1, Y1, z);
apply_z(X2, Y2, z);
}
static void XYcZ_add_rnd(uECC_word_t * X1, uECC_word_t * Y1,
uECC_word_t * X2, uECC_word_t * Y2,
ecc_wait_state_t *s)
{
/* t1 = X1, t2 = Y1, t3 = X2, t4 = Y2 */
uECC_word_t t5[NUM_ECC_WORDS];
uECC_vli_modSub(t5, X2, X1, curve_p); /* t5 = x2 - x1 */
uECC_vli_modMult_rnd(t5, t5, t5, s); /* t5 = (x2 - x1)^2 = A */
uECC_vli_modMult_rnd(X1, X1, t5, s); /* t1 = x1*A = B */
uECC_vli_modMult_rnd(X2, X2, t5, s); /* t3 = x2*A = C */
uECC_vli_modSub(Y2, Y2, Y1, curve_p); /* t4 = y2 - y1 */
uECC_vli_modMult_rnd(t5, Y2, Y2, s); /* t5 = (y2 - y1)^2 = D */
uECC_vli_modSub(t5, t5, X1, curve_p); /* t5 = D - B */
uECC_vli_modSub(t5, t5, X2, curve_p); /* t5 = D - B - C = x3 */
uECC_vli_modSub(X2, X2, X1, curve_p); /* t3 = C - B */
uECC_vli_modMult_rnd(Y1, Y1, X2, s); /* t2 = y1*(C - B) */
uECC_vli_modSub(X2, X1, t5, curve_p); /* t3 = B - x3 */
uECC_vli_modMult_rnd(Y2, Y2, X2, s); /* t4 = (y2 - y1)*(B - x3) */
uECC_vli_modSub(Y2, Y2, Y1, curve_p); /* t4 = y3 */
uECC_vli_set(X2, t5);
}
void XYcZ_add(uECC_word_t * X1, uECC_word_t * Y1,
uECC_word_t * X2, uECC_word_t * Y2)
{
XYcZ_add_rnd(X1, Y1, X2, Y2, NULL);
}
/* Input P = (x1, y1, Z), Q = (x2, y2, Z)
Output P + Q = (x3, y3, Z3), P - Q = (x3', y3', Z3)
or P => P - Q, Q => P + Q
*/
static void XYcZ_addC_rnd(uECC_word_t * X1, uECC_word_t * Y1,
uECC_word_t * X2, uECC_word_t * Y2,
ecc_wait_state_t *s)
{
/* t1 = X1, t2 = Y1, t3 = X2, t4 = Y2 */
uECC_word_t t5[NUM_ECC_WORDS];
uECC_word_t t6[NUM_ECC_WORDS];
uECC_word_t t7[NUM_ECC_WORDS];
uECC_vli_modSub(t5, X2, X1, curve_p); /* t5 = x2 - x1 */
uECC_vli_modMult_rnd(t5, t5, t5, s); /* t5 = (x2 - x1)^2 = A */
uECC_vli_modMult_rnd(X1, X1, t5, s); /* t1 = x1*A = B */
uECC_vli_modMult_rnd(X2, X2, t5, s); /* t3 = x2*A = C */
uECC_vli_modAdd(t5, Y2, Y1, curve_p); /* t5 = y2 + y1 */
uECC_vli_modSub(Y2, Y2, Y1, curve_p); /* t4 = y2 - y1 */
uECC_vli_modSub(t6, X2, X1, curve_p); /* t6 = C - B */
uECC_vli_modMult_rnd(Y1, Y1, t6, s); /* t2 = y1 * (C - B) = E */
uECC_vli_modAdd(t6, X1, X2, curve_p); /* t6 = B + C */
uECC_vli_modMult_rnd(X2, Y2, Y2, s); /* t3 = (y2 - y1)^2 = D */
uECC_vli_modSub(X2, X2, t6, curve_p); /* t3 = D - (B + C) = x3 */
uECC_vli_modSub(t7, X1, X2, curve_p); /* t7 = B - x3 */
uECC_vli_modMult_rnd(Y2, Y2, t7, s); /* t4 = (y2 - y1)*(B - x3) */
/* t4 = (y2 - y1)*(B - x3) - E = y3: */
uECC_vli_modSub(Y2, Y2, Y1, curve_p);
uECC_vli_modMult_rnd(t7, t5, t5, s); /* t7 = (y2 + y1)^2 = F */
uECC_vli_modSub(t7, t7, t6, curve_p); /* t7 = F - (B + C) = x3' */
uECC_vli_modSub(t6, t7, X1, curve_p); /* t6 = x3' - B */
uECC_vli_modMult_rnd(t6, t6, t5, s); /* t6 = (y2+y1)*(x3' - B) */
/* t2 = (y2+y1)*(x3' - B) - E = y3': */
uECC_vli_modSub(Y1, t6, Y1, curve_p);
uECC_vli_set(X1, t7);
}
static void EccPoint_mult(uECC_word_t * result, const uECC_word_t * point,
const uECC_word_t * scalar,
const uECC_word_t * initial_Z)
{
/* R0 and R1 */
uECC_word_t Rx[2][NUM_ECC_WORDS];
uECC_word_t Ry[2][NUM_ECC_WORDS];
uECC_word_t z[NUM_ECC_WORDS];
bitcount_t i;
uECC_word_t nb;
const wordcount_t num_words = NUM_ECC_WORDS;
const bitcount_t num_bits = NUM_ECC_BITS + 1; /* from regularize_k */
ecc_wait_state_t wait_state;
ecc_wait_state_t * const ws = g_rng_function ? &wait_state : NULL;
uECC_vli_set(Rx[1], point);
uECC_vli_set(Ry[1], point + num_words);
XYcZ_initial_double(Rx[1], Ry[1], Rx[0], Ry[0], initial_Z);
for (i = num_bits - 2; i > 0; --i) {
ecc_wait_state_reset(ws);
nb = !uECC_vli_testBit(scalar, i);
XYcZ_addC_rnd(Rx[1 - nb], Ry[1 - nb], Rx[nb], Ry[nb], ws);
XYcZ_add_rnd(Rx[nb], Ry[nb], Rx[1 - nb], Ry[1 - nb], ws);
}
ecc_wait_state_reset(ws);
nb = !uECC_vli_testBit(scalar, 0);
XYcZ_addC_rnd(Rx[1 - nb], Ry[1 - nb], Rx[nb], Ry[nb], ws);
/* Find final 1/Z value. */
uECC_vli_modSub(z, Rx[1], Rx[0], curve_p); /* X1 - X0 */
uECC_vli_modMult_fast(z, z, Ry[1 - nb]); /* Yb * (X1 - X0) */
uECC_vli_modMult_fast(z, z, point); /* xP * Yb * (X1 - X0) */
uECC_vli_modInv(z, z, curve_p); /* 1 / (xP * Yb * (X1 - X0))*/
/* yP / (xP * Yb * (X1 - X0)) */
uECC_vli_modMult_fast(z, z, point + num_words);
/* Xb * yP / (xP * Yb * (X1 - X0)) */
uECC_vli_modMult_fast(z, z, Rx[1 - nb]);
/* End 1/Z calculation */
XYcZ_add_rnd(Rx[nb], Ry[nb], Rx[1 - nb], Ry[1 - nb], ws);
apply_z(Rx[0], Ry[0], z);
uECC_vli_set(result, Rx[0]);
uECC_vli_set(result + num_words, Ry[0]);
}
static uECC_word_t regularize_k(const uECC_word_t * const k, uECC_word_t *k0,
uECC_word_t *k1)
{
wordcount_t num_n_words = NUM_ECC_WORDS;
bitcount_t num_n_bits = NUM_ECC_BITS;
uECC_word_t carry = uECC_vli_add(k0, k, curve_n) ||
(num_n_bits < ((bitcount_t)num_n_words * uECC_WORD_SIZE * 8) &&
uECC_vli_testBit(k0, num_n_bits));
uECC_vli_add(k1, k0, curve_n);
return carry;
}
int EccPoint_mult_safer(uECC_word_t * result, const uECC_word_t * point,
const uECC_word_t * scalar)
{
uECC_word_t tmp[NUM_ECC_WORDS];
uECC_word_t s[NUM_ECC_WORDS];
uECC_word_t *k2[2] = {tmp, s};
wordcount_t num_words = NUM_ECC_WORDS;
uECC_word_t carry;
uECC_word_t *initial_Z = 0;
int r = UECC_FAULT_DETECTED;
volatile int problem;
/* Protect against faults modifying curve paremeters in flash */
problem = -1;
problem = uECC_check_curve_integrity();
if (problem != 0) {
return UECC_FAULT_DETECTED;
}
mbedtls_platform_enforce_volatile_reads();
if (problem != 0) {
return UECC_FAULT_DETECTED;
}
/* Protects against invalid curve attacks */
problem = -1;
problem = uECC_valid_point(point);
if (problem != 0) {
/* invalid input, can happen without fault */
return UECC_FAILURE;
}
mbedtls_platform_enforce_volatile_reads();
if (problem != 0) {
/* failure on second check means fault, though */
return UECC_FAULT_DETECTED;
}
/* Regularize the bitcount for the private key so that attackers cannot use a
* side channel attack to learn the number of leading zeros. */
carry = regularize_k(scalar, tmp, s);
/* If an RNG function was specified, get a random initial Z value to
* protect against side-channel attacks such as Template SPA */
if (g_rng_function) {
if (!uECC_generate_random_int(k2[carry], curve_p, num_words)) {
r = UECC_FAILURE;
goto clear_and_out;
}
initial_Z = k2[carry];
}
EccPoint_mult(result, point, k2[!carry], initial_Z);
/* Protect against fault injections that would make the resulting
* point not lie on the intended curve */
problem = -1;
problem = uECC_valid_point(result);
if (problem != 0) {
r = UECC_FAULT_DETECTED;
goto clear_and_out;
}
mbedtls_platform_enforce_volatile_reads();
if (problem != 0) {
r = UECC_FAULT_DETECTED;
goto clear_and_out;
}
/* Protect against faults modifying curve paremeters in flash */
problem = -1;
problem = uECC_check_curve_integrity();
if (problem != 0) {
r = UECC_FAULT_DETECTED;
goto clear_and_out;
}
mbedtls_platform_enforce_volatile_reads();
if (problem != 0) {
r = UECC_FAULT_DETECTED;
goto clear_and_out;
}
r = UECC_SUCCESS;
clear_and_out:
/* erasing temporary buffer used to store secret: */
mbedtls_platform_zeroize(k2, sizeof(k2));
mbedtls_platform_zeroize(tmp, sizeof(tmp));
mbedtls_platform_zeroize(s, sizeof(s));
return r;
}
uECC_word_t EccPoint_compute_public_key(uECC_word_t *result,
uECC_word_t *private_key)
{
return EccPoint_mult_safer(result, curve_G, private_key);
}
/* Converts an integer in uECC native format to big-endian bytes. */
void uECC_vli_nativeToBytes(uint8_t *bytes, int num_bytes,
const unsigned int *native)
{
wordcount_t i;
for (i = 0; i < num_bytes; ++i) {
unsigned b = num_bytes - 1 - i;
bytes[i] = native[b / uECC_WORD_SIZE] >> (8 * (b % uECC_WORD_SIZE));
}
}
/* Converts big-endian bytes to an integer in uECC native format. */
void uECC_vli_bytesToNative(unsigned int *native, const uint8_t *bytes,
int num_bytes)
{
wordcount_t i;
uECC_vli_clear(native);
for (i = 0; i < num_bytes; ++i) {
unsigned b = num_bytes - 1 - i;
native[b / uECC_WORD_SIZE] |=
(uECC_word_t)bytes[i] << (8 * (b % uECC_WORD_SIZE));
}
}
int uECC_generate_random_int(uECC_word_t *random, const uECC_word_t *top,
wordcount_t num_words)
{
uECC_word_t mask = (uECC_word_t)-1;
uECC_word_t tries;
bitcount_t num_bits = uECC_vli_numBits(top);
if (!g_rng_function) {
return 0;
}
for (tries = 0; tries < uECC_RNG_MAX_TRIES; ++tries) {
if (!g_rng_function((uint8_t *)random, num_words * uECC_WORD_SIZE)) {
return 0;
}
random[num_words - 1] &=
mask >> ((bitcount_t)(num_words * uECC_WORD_SIZE * 8 - num_bits));
if (!uECC_vli_isZero(random) &&
uECC_vli_cmp(top, random) == 1) {
return 1;
}
}
return 0;
}
int uECC_valid_point(const uECC_word_t *point)
{
uECC_word_t tmp1[NUM_ECC_WORDS];
uECC_word_t tmp2[NUM_ECC_WORDS];
wordcount_t num_words = NUM_ECC_WORDS;
volatile uECC_word_t diff = 0xffffffff;
/* The point at infinity is invalid. */
if (EccPoint_isZero(point)) {
return -1;
}
/* x and y must be smaller than p. */
if (uECC_vli_cmp_unsafe(curve_p, point) != 1 ||
uECC_vli_cmp_unsafe(curve_p, point + num_words) != 1) {
return -2;
}
uECC_vli_modMult_fast(tmp1, point + num_words, point + num_words);
x_side_default(tmp2, point); /* tmp2 = x^3 + ax + b */
/* Make sure that y^2 == x^3 + ax + b */
diff = uECC_vli_equal(tmp1, tmp2);
if (diff == 0) {
mbedtls_platform_enforce_volatile_reads();
if (diff == 0) {
return 0;
}
}
return -3;
}
int uECC_valid_public_key(const uint8_t *public_key)
{
uECC_word_t _public[NUM_ECC_WORDS * 2];
uECC_vli_bytesToNative(_public, public_key, NUM_ECC_BYTES);
uECC_vli_bytesToNative(
_public + NUM_ECC_WORDS,
public_key + NUM_ECC_BYTES,
NUM_ECC_BYTES);
if (memcmp(_public, curve_G, NUM_ECC_WORDS * 2) == 0) {
return -4;
}
return uECC_valid_point(_public);
}
int uECC_compute_public_key(const uint8_t *private_key, uint8_t *public_key)
{
int ret;
uECC_word_t _private[NUM_ECC_WORDS];
uECC_word_t _public[NUM_ECC_WORDS * 2];
uECC_vli_bytesToNative(
_private,
private_key,
BITS_TO_BYTES(NUM_ECC_BITS));
/* Make sure the private key is in the range [1, n-1]. */
if (uECC_vli_isZero(_private)) {
return UECC_FAILURE;
}
if (uECC_vli_cmp(curve_n, _private) != 1) {
return UECC_FAILURE;
}
/* Compute public key. */
ret = EccPoint_compute_public_key(_public, _private);
if (ret != UECC_SUCCESS) {
return ret;
}
uECC_vli_nativeToBytes(public_key, NUM_ECC_BYTES, _public);
uECC_vli_nativeToBytes(
public_key +
NUM_ECC_BYTES, NUM_ECC_BYTES, _public + NUM_ECC_WORDS);
return UECC_SUCCESS;
}
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