/* 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 #include "mbedtls/platform_util.h" #include "mbedtls/sha256.h" #include /* 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_random_delay(); 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 flow_monitor = 0; uECC_word_t tmp1, tmp2; volatile int i; /* Start from a random location and check the correct number of iterations */ int start_offset = mbedtls_platform_random_in_range(NUM_ECC_WORDS); for (i = start_offset; i < NUM_ECC_WORDS; ++i) { tmp1 = left[i]; tmp2 = right[i]; flow_monitor++; diff |= (tmp1 ^ tmp2); } for (i = 0; i < start_offset; ++i) { tmp1 = left[i]; tmp2 = right[i]; flow_monitor++; diff |= (tmp1 ^ tmp2); } /* Random delay to increase security */ mbedtls_platform_random_delay(); /* Return 0 only when diff is 0 and flow_counter is equal to NUM_ECC_WORDS */ return (diff | (flow_monitor ^ NUM_ECC_WORDS)); } 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) { bitcount_t num_n_bits = NUM_ECC_BITS; uECC_word_t carry = uECC_vli_add(k0, k, curve_n) || 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_random_delay(); 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_random_delay(); 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_random_delay(); 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_random_delay(); 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_random_delay(); 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; }