/**************************************************************************** * crypto/ecc.c * * SPDX-License-Identifier: BSD-2-Clause * * Copyright (c) 2013, 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 * SPECIAL, HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, * INCIDENTAL, 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. ****************************************************************************/ /**************************************************************************** * Included Files ****************************************************************************/ #include #include #include #include #include #include #include /**************************************************************************** * Pre-processor Definitions ****************************************************************************/ #define NUM_ECC_DIGITS (ECC_BYTES / 8) #define MAX_TRIES 16 #define EVEN(vli) (!(vli[0] & 1)) #define curve_p_16 { 0xFFFFFFFFFFFFFFFF, 0xFFFFFFFDFFFFFFFF } #define curve_p_24 { 0xFFFFFFFFFFFFFFFFull, 0xFFFFFFFFFFFFFFFEull, \ 0xFFFFFFFFFFFFFFFFull } #define curve_p_32 { 0xFFFFFFFFFFFFFFFFull, 0x00000000FFFFFFFFull, \ 0x0000000000000000ull, 0xFFFFFFFF00000001ull } #define curve_p_48 { 0x00000000FFFFFFFF, 0xFFFFFFFF00000000, \ 0xFFFFFFFFFFFFFFFE, 0xFFFFFFFFFFFFFFFF, \ 0xFFFFFFFFFFFFFFFF, 0xFFFFFFFFFFFFFFFF } #define curve_b_16 { 0xD824993C2CEE5ED3, 0xE87579C11079F43D } #define curve_b_24 { 0xFEB8DEECC146B9B1ull, 0x0FA7E9AB72243049ull, \ 0x64210519E59C80E7ull } #define curve_b_32 { 0x3BCE3C3E27D2604Bull, 0x651D06B0CC53B0F6ull, \ 0xB3EBBD55769886BCull, 0x5AC635D8AA3A93E7ull } #define curve_b_48 { 0x2A85C8EDD3EC2AEF, 0xC656398D8A2ED19D, \ 0x0314088F5013875A, 0x181D9C6EFE814112, \ 0x988E056BE3F82D19, 0xB3312FA7E23EE7E4 } #define curve_g_16 { \ { 0x0C28607CA52C5B86, 0x161FF7528B899B2D }, \ { 0xC02DA292DDED7A83, 0xCF5AC8395BAFEB13 }} #define curve_g_24 { \ { 0xF4FF0AFD82FF1012ull, 0x7CBF20EB43A18800ull, 0x188DA80EB03090F6ull }, \ { 0x73F977A11E794811ull, 0x631011ED6B24CDD5ull, 0x07192B95FFC8DA78ull }} #define curve_g_32 { \ { 0xF4A13945D898C296ull, 0x77037D812DEB33A0ull, \ 0xF8BCE6E563A440F2ull, 0x6B17D1F2E12C4247ull }, \ { 0xCBB6406837BF51F5ull, 0x2BCE33576B315ECEull, \ 0x8EE7EB4A7C0F9E16ull, 0x4FE342E2FE1A7F9Bull }} #define curve_g_48 { \ { 0x3A545E3872760AB7, 0x5502F25DBF55296C, 0x59F741E082542A38, \ 0x6E1D3B628BA79B98, 0x8EB1C71EF320AD74, 0xAA87CA22BE8B0537}, \ { 0x7A431D7C90EA0E5F, 0x0A60B1CE1D7E819D, 0xE9DA3113B5F0B8C0, \ 0xF8F41DBD289A147C, 0x5D9E98BF9292DC29, 0x3617DE4A96262C6F }} #define curve_n_16 { 0x75A30D1B9038A115, 0xFFFFFFFE00000000 } #define curve_n_24 { 0x146BC9B1B4D22831ull, 0xFFFFFFFF99DEF836ull, \ 0xFFFFFFFFFFFFFFFFull } #define curve_n_32 { 0xF3B9CAC2FC632551ull, 0xBCE6FAADA7179E84ull, \ 0xFFFFFFFFFFFFFFFFull, 0xFFFFFFFF00000000ull } #define curve_n_48 { 0xECEC196ACCC52973, 0x581A0DB248B0A77A, \ 0xC7634D81F4372DDF, 0xFFFFFFFFFFFFFFFF, \ 0xFFFFFFFFFFFFFFFF, 0xFFFFFFFFFFFFFFFF } #if defined(__SIZEOF_INT128__) # define SUPPORTS_INT128 1 #else # define SUPPORTS_INT128 0 #endif /**************************************************************************** * Private Type Definitions ****************************************************************************/ #if SUPPORTS_INT128 typedef unsigned __int128 uint128_t; #else typedef struct { uint64_t m_low; uint64_t m_high; } uint128_t; #endif typedef struct { uint64_t x[NUM_ECC_DIGITS]; uint64_t y[NUM_ECC_DIGITS]; } eccpoint_t; /**************************************************************************** * Private Data ****************************************************************************/ static uint64_t g_curve_p[NUM_ECC_DIGITS] = CONCATENATE(curve_p_, ECC_CURVE); static uint64_t g_curve_b[NUM_ECC_DIGITS] = CONCATENATE(curve_b_, ECC_CURVE); static uint64_t g_curve_n[NUM_ECC_DIGITS] = CONCATENATE(curve_n_, ECC_CURVE); static eccpoint_t g_curve_g = CONCATENATE(curve_g_, ECC_CURVE); /**************************************************************************** * Private Functions ****************************************************************************/ static void vli_clear(FAR uint64_t *vli) { uint i; for (i = 0; i < NUM_ECC_DIGITS; ++i) { vli[i] = 0; } } /* Returns 1 if vli == 0, 0 otherwise. */ static int vli_iszero(FAR uint64_t *vli) { uint i; for (i = 0; i < NUM_ECC_DIGITS; ++i) { if (vli[i]) { return 0; } } return 1; } /* Returns nonzero if bit bit of vli is set. */ static uint64_t vli_testbit(FAR uint64_t *vli, uint bit) { return vli[bit / 64] & ((uint64_t)1 << (bit % 64)); } /* Counts the number of 64-bit "digits" in vli. */ static uint vli_numdigits(FAR uint64_t *vli) { int 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_DIGITS - 1; i >= 0 && vli[i] == 0; --i) { } return i + 1; } /* Counts the number of bits required for vli. */ static uint vli_numbits(FAR uint64_t *vli) { uint64_t l_digit; uint l_numdigits = vli_numdigits(vli); uint i; if (l_numdigits == 0) { return 0; } l_digit = vli[l_numdigits - 1]; for (i = 0; l_digit; ++i) { l_digit >>= 1; } return (l_numdigits - 1) * 64 + i; } /* Sets dest = src. */ static void vli_set(FAR uint64_t *dest, FAR uint64_t *src) { uint i; for (i = 0; i < NUM_ECC_DIGITS; ++i) { dest[i] = src[i]; } } /* Returns sign of left - right. */ static int vli_cmp(FAR uint64_t *left, FAR uint64_t *right) { int i; for (i = NUM_ECC_DIGITS - 1; i >= 0; --i) { if (left[i] > right[i]) { return 1; } else if (left[i] < right[i]) { return -1; } } return 0; } /* Computes result = in << c, returning carry. * Can modify in place (if result == in). 0 < shift < 64. */ static uint64_t vli_lshift(FAR uint64_t *result, FAR uint64_t *in, uint shift) { uint64_t l_carry = 0; uint64_t l_temp; uint i; for (i = 0; i < NUM_ECC_DIGITS; ++i) { l_temp = in[i]; result[i] = (l_temp << shift) | l_carry; l_carry = l_temp >> (64 - shift); } return l_carry; } /* Computes vli = vli >> 1. */ static void vli_rshift1(FAR uint64_t *vli) { FAR uint64_t *l_end = vli; uint64_t l_carry = 0; uint64_t l_temp; vli += NUM_ECC_DIGITS; while (vli-- > l_end) { l_temp = *vli; *vli = (l_temp >> 1) | l_carry; l_carry = l_temp << 63; } } /* Computes result = left + right, returning carry. Can modify in place. */ static uint64_t vli_add(FAR uint64_t *result, FAR uint64_t *left, FAR uint64_t *right) { uint64_t l_carry = 0; uint64_t l_sum; uint i; for (i = 0; i < NUM_ECC_DIGITS; ++i) { l_sum = left[i] + right[i] + l_carry; if (l_sum != left[i]) { l_carry = (l_sum < left[i]); } result[i] = l_sum; } return l_carry; } /* Computes result = left - right, returning borrow. Can modify in place. */ static uint64_t vli_sub(FAR uint64_t *result, FAR uint64_t *left, FAR uint64_t *right) { uint64_t l_borrow = 0; uint64_t l_diff; uint i; for (i = 0; i < NUM_ECC_DIGITS; ++i) { l_diff = left[i] - right[i] - l_borrow; if (l_diff != left[i]) { l_borrow = (l_diff > left[i]); } result[i] = l_diff; } return l_borrow; } #if SUPPORTS_INT128 /* Computes result = left * right. */ static void vli_mult(FAR uint64_t *result, FAR uint64_t *left, FAR uint64_t *right) { uint128_t l_product; uint128_t r01 = 0; uint64_t r2 = 0; uint l_min; uint i; uint k; /* Compute each digit of result in sequence, maintaining the carries. */ for (k = 0; k < NUM_ECC_DIGITS * 2 - 1; ++k) { l_min = (k < NUM_ECC_DIGITS ? 0 : (k + 1) - NUM_ECC_DIGITS); for (i = l_min; i <= k && i < NUM_ECC_DIGITS; ++i) { l_product = (uint128_t)left[i] * right[k - i]; r01 += l_product; r2 += (r01 < l_product); } result[k] = (uint64_t)r01; r01 = (r01 >> 64) | (((uint128_t)r2) << 64); r2 = 0; } result[NUM_ECC_DIGITS * 2 - 1] = (uint64_t)r01; } /* Computes result = left^2. */ static void vli_square(FAR uint64_t *result, FAR uint64_t *left) { uint128_t l_product; uint128_t r01 = 0; uint64_t r2 = 0; uint l_min; uint i; uint k; for (k = 0; k < NUM_ECC_DIGITS * 2 - 1; ++k) { l_min = (k < NUM_ECC_DIGITS ? 0 : (k + 1) - NUM_ECC_DIGITS); for (i = l_min; i <= k && i <= k - i; ++i) { l_product = (uint128_t)left[i] * left[k - i]; if (i < k - i) { r2 += l_product >> 127; l_product *= 2; } r01 += l_product; r2 += (r01 < l_product); } result[k] = (uint64_t)r01; r01 = (r01 >> 64) | (((uint128_t)r2) << 64); r2 = 0; } result[NUM_ECC_DIGITS * 2 - 1] = (uint64_t)r01; } #else /* #if SUPPORTS_INT128 */ static uint128_t mul_64_64(uint64_t left, uint64_t right) { uint128_t l_result; uint64_t a0 = left & 0xffffffffull; uint64_t a1 = left >> 32; uint64_t b0 = right & 0xffffffffull; uint64_t b1 = right >> 32; uint64_t m0 = a0 * b0; uint64_t m1 = a0 * b1; uint64_t m2 = a1 * b0; uint64_t m3 = a1 * b1; m2 += (m0 >> 32); m2 += m1; if (m2 < m1) { m3 += 0x100000000ull; } l_result.m_low = (m0 & 0xffffffffull) | (m2 << 32); l_result.m_high = m3 + (m2 >> 32); return l_result; } static uint128_t add_128_128(uint128_t a, uint128_t b) { uint128_t l_result; l_result.m_low = a.m_low + b.m_low; l_result.m_high = a.m_high + b.m_high + (l_result.m_low < a.m_low); return l_result; } static void vli_mult(FAR uint64_t *result, FAR uint64_t *left, FAR uint64_t *right) { uint64_t r2 = 0; uint i; uint k; uint128_t l_product; uint128_t r01 = { 0, 0 }; /* Compute each digit of result in sequence, maintaining the carries. */ for (k = 0; k < NUM_ECC_DIGITS * 2 - 1; ++k) { uint l_min = (k < NUM_ECC_DIGITS ? 0 : (k + 1) - NUM_ECC_DIGITS); for (i = l_min; i <= k && i < NUM_ECC_DIGITS; ++i) { l_product = mul_64_64(left[i], right[k - i]); r01 = add_128_128(r01, l_product); r2 += (r01.m_high < l_product.m_high); } result[k] = r01.m_low; r01.m_low = r01.m_high; r01.m_high = r2; r2 = 0; } result[NUM_ECC_DIGITS * 2 - 1] = r01.m_low; } static void vli_square(FAR uint64_t *result, FAR uint64_t *left) { uint64_t r2 = 0; uint l_min; uint i; uint k; uint128_t l_product; uint128_t r01 = { 0, 0 }; for (k = 0; k < NUM_ECC_DIGITS * 2 - 1; ++k) { l_min = (k < NUM_ECC_DIGITS ? 0 : (k + 1) - NUM_ECC_DIGITS); for (i = l_min; i <= k && i <= k - i; ++i) { l_product = mul_64_64(left[i], left[k - i]); if (i < k - i) { r2 += l_product.m_high >> 63; l_product.m_high = (l_product.m_high << 1) | (l_product.m_low >> 63); l_product.m_low <<= 1; } r01 = add_128_128(r01, l_product); r2 += (r01.m_high < l_product.m_high); } result[k] = r01.m_low; r01.m_low = r01.m_high; r01.m_high = r2; r2 = 0; } result[NUM_ECC_DIGITS * 2 - 1] = r01.m_low; } #endif /* SUPPORTS_INT128 */ /* Computes result = (left + right) % mod. * Assumes that left < mod and right < mod, result != mod. */ static void vli_modadd(FAR uint64_t *result, FAR uint64_t *left, FAR uint64_t *right, FAR uint64_t *mod) { uint64_t l_carry = vli_add(result, left, right); if (l_carry || vli_cmp(result, mod) >= 0) { /* result > mod (result = mod + remainder), * so subtract mod to get remainder. */ vli_sub(result, result, mod); } } /* Computes result = (left - right) % mod. * Assumes that left < mod and right < mod, result != mod. */ static void vli_modsub(FAR uint64_t *result, FAR uint64_t *left, FAR uint64_t *right, FAR uint64_t *mod) { uint64_t l_borrow = 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). */ vli_add(result, result, mod); } } #if ECC_CURVE == secp128r1 /* Computes result = product % g_curve_p. * See algorithm 5 and 6 from * http://www.isys.uni-klu.ac.at/PDF/2001-0126-MT.pdf */ static void vli_mmod_fast(FAR uint64_t *result, FAR uint64_t *product) { uint64_t l_tmp[NUM_ECC_DIGITS]; int l_carry; vli_set(result, product); l_tmp[0] = product[2]; l_tmp[1] = (product[3] & 0x1ffffffffull) | (product[2] << 33); l_carry = vli_add(result, result, l_tmp); l_tmp[0] = (product[2] >> 31) | (product[3] << 33); l_tmp[1] = (product[3] >> 31) | ((product[2] & 0xffffffff80000000ull) << 2); l_carry += vli_add(result, result, l_tmp); l_tmp[0] = (product[2] >> 62) | (product[3] << 2); l_tmp[1] = (product[3] >> 62) | ((product[2] & 0xc000000000000000ull) >> 29) | (product[3] << 35); l_carry += vli_add(result, result, l_tmp); l_tmp[0] = (product[3] >> 29); l_tmp[1] = ((product[3] & 0xffffffffe0000000ull) << 4); l_carry += vli_add(result, result, l_tmp); l_tmp[0] = (product[3] >> 60); l_tmp[1] = (product[3] & 0xfffffffe00000000ull); l_carry += vli_add(result, result, l_tmp); l_tmp[0] = 0; l_tmp[1] = ((product[3] & 0xf000000000000000ull) >> 27); l_carry += vli_add(result, result, l_tmp); while (l_carry || vli_cmp(g_curve_p, result) != 1) { l_carry -= vli_sub(result, result, g_curve_p); } } #elif ECC_CURVE == secp192r1 /* Computes result = product % g_curve_p. * See algorithm 5 and 6 from * http://www.isys.uni-klu.ac.at/PDF/2001-0126-MT.pdf */ static void vli_mmod_fast(FAR uint64_t *result, FAR uint64_t *product) { uint64_t l_tmp[NUM_ECC_DIGITS]; int l_carry; vli_set(result, product); vli_set(l_tmp, &product[3]); l_carry = vli_add(result, result, l_tmp); l_tmp[0] = 0; l_tmp[1] = product[3]; l_tmp[2] = product[4]; l_carry += vli_add(result, result, l_tmp); l_tmp[0] = l_tmp[1] = product[5]; l_tmp[2] = 0; l_carry += vli_add(result, result, l_tmp); while (l_carry || vli_cmp(g_curve_p, result) != 1) { l_carry -= vli_sub(result, result, g_curve_p); } } #elif ECC_CURVE == secp256r1 /* Computes result = product % g_curve_p * from http://www.nsa.gov/ia/_files/nist-routines.pdf */ static void vli_mmod_fast(FAR uint64_t *result, FAR uint64_t *product) { uint64_t l_tmp[NUM_ECC_DIGITS]; int l_carry; /* t */ vli_set(result, product); /* s1 */ l_tmp[0] = 0; l_tmp[1] = product[5] & 0xffffffff00000000ull; l_tmp[2] = product[6]; l_tmp[3] = product[7]; l_carry = vli_lshift(l_tmp, l_tmp, 1); l_carry += vli_add(result, result, l_tmp); /* s2 */ l_tmp[1] = product[6] << 32; l_tmp[2] = (product[6] >> 32) | (product[7] << 32); l_tmp[3] = product[7] >> 32; l_carry += vli_lshift(l_tmp, l_tmp, 1); l_carry += vli_add(result, result, l_tmp); /* s3 */ l_tmp[0] = product[4]; l_tmp[1] = product[5] & 0xffffffff; l_tmp[2] = 0; l_tmp[3] = product[7]; l_carry += vli_add(result, result, l_tmp); /* s4 */ l_tmp[0] = (product[4] >> 32) | (product[5] << 32); l_tmp[1] = (product[5] >> 32) | (product[6] & 0xffffffff00000000ull); l_tmp[2] = product[7]; l_tmp[3] = (product[6] >> 32) | (product[4] << 32); l_carry += vli_add(result, result, l_tmp); /* d1 */ l_tmp[0] = (product[5] >> 32) | (product[6] << 32); l_tmp[1] = (product[6] >> 32); l_tmp[2] = 0; l_tmp[3] = (product[4] & 0xffffffff) | (product[5] << 32); l_carry -= vli_sub(result, result, l_tmp); /* d2 */ l_tmp[0] = product[6]; l_tmp[1] = product[7]; l_tmp[2] = 0; l_tmp[3] = (product[4] >> 32) | (product[5] & 0xffffffff00000000ull); l_carry -= vli_sub(result, result, l_tmp); /* d3 */ l_tmp[0] = (product[6] >> 32) | (product[7] << 32); l_tmp[1] = (product[7] >> 32) | (product[4] << 32); l_tmp[2] = (product[4] >> 32) | (product[5] << 32); l_tmp[3] = (product[6] << 32); l_carry -= vli_sub(result, result, l_tmp); /* d4 */ l_tmp[0] = product[7]; l_tmp[1] = product[4] & 0xffffffff00000000ull; l_tmp[2] = product[5]; l_tmp[3] = product[6] & 0xffffffff00000000ull; l_carry -= vli_sub(result, result, l_tmp); if (l_carry < 0) { do { l_carry += vli_add(result, result, g_curve_p); } while (l_carry < 0); } else { while (l_carry || vli_cmp(g_curve_p, result) != 1) { l_carry -= vli_sub(result, result, g_curve_p); } } } #elif ECC_CURVE == secp384r1 static void omega_mult(uint64_t *result, uint64_t *right) { uint64_t l_tmp[NUM_ECC_DIGITS]; uint64_t l_carry; uint64_t l_diff; uint i; /* Multiply by (2^128 + 2^96 - 2^32 + 1). */ vli_set(result, right); /* 1 */ l_carry = vli_lshift(l_tmp, right, 32); result[1 + NUM_ECC_DIGITS] = l_carry + vli_add(result + 1, result + 1, l_tmp); /* 2^96 + 1 */ /* 2^128 + 2^96 + 1 */ result[2 + NUM_ECC_DIGITS] = vli_add(result + 2, result + 2, right); /* 2^128 + 2^96 - 2^32 + 1 */ l_carry += vli_sub(result, result, l_tmp); l_diff = result[NUM_ECC_DIGITS] - l_carry; if (l_diff > result[NUM_ECC_DIGITS]) { /* Propagate borrow if necessary. */ for (i = 1 + NUM_ECC_DIGITS; ; ++i) { --result[i]; if (result[i] != (uint64_t)-1) { break; } } } result[NUM_ECC_DIGITS] = l_diff; } /* Computes result = product % g_curve_p * see PDF "Comparing Elliptic Curve Cryptography and RSA on 8-bit CPUs" * section "Curve-Specific Optimizations" */ static void vli_mmod_fast(uint64_t *result, uint64_t *product) { uint64_t l_tmp[2 * NUM_ECC_DIGITS]; uint64_t l_carry; uint64_t l_sum; uint i; while (!vli_iszero(product + NUM_ECC_DIGITS)) /* While c1 != 0 */ { l_carry = 0; vli_clear(l_tmp); vli_clear(l_tmp + NUM_ECC_DIGITS); omega_mult(l_tmp, product + NUM_ECC_DIGITS); /* tmp = w * c1 */ /* p = c0 */ vli_clear(product + NUM_ECC_DIGITS); /* (c1, c0) = c0 + w * c1 */ for (i = 0; i < NUM_ECC_DIGITS + 3; ++i) { l_sum = product[i] + l_tmp[i] + l_carry; if (l_sum != product[i]) { l_carry = (l_sum < product[i]); } product[i] = l_sum; } } while (vli_cmp(product, g_curve_p) > 0) { vli_sub(product, product, g_curve_p); } vli_set(result, product); } #endif /* Computes result = (left * right) % g_curve_p. */ static void vli_modmult_fast(FAR uint64_t *result, FAR uint64_t *left, FAR uint64_t *right) { uint64_t l_product[2 * NUM_ECC_DIGITS]; vli_mult(l_product, left, right); vli_mmod_fast(result, l_product); } /* Computes result = left^2 % g_curve_p. */ static void vli_modsquare_fast(FAR uint64_t *result, FAR uint64_t *left) { uint64_t l_product[2 * NUM_ECC_DIGITS]; vli_square(l_product, left); vli_mmod_fast(result, l_product); } /* Computes result = (1 / input) % mod. All VLIs are the same size. * See "From Euclid's GCD to Montgomery Multiplication to the Great Divide" * https://labs.oracle.com/techrep/2001/smli_tr-2001-95.pdf */ static void vli_modinv(FAR uint64_t *result, FAR uint64_t *input, FAR uint64_t *mod) { uint64_t a[NUM_ECC_DIGITS]; uint64_t b[NUM_ECC_DIGITS]; uint64_t u[NUM_ECC_DIGITS]; uint64_t v[NUM_ECC_DIGITS]; uint64_t l_carry; int l_cmpresult; if (vli_iszero(input)) { vli_clear(result); return; } vli_set(a, input); vli_set(b, mod); vli_clear(u); u[0] = 1; vli_clear(v); while ((l_cmpresult = vli_cmp(a, b)) != 0) { l_carry = 0; if (EVEN(a)) { vli_rshift1(a); if (!EVEN(u)) { l_carry = vli_add(u, u, mod); } vli_rshift1(u); if (l_carry) { u[NUM_ECC_DIGITS - 1] |= 0x8000000000000000ull; } } else if (EVEN(b)) { vli_rshift1(b); if (!EVEN(v)) { l_carry = vli_add(v, v, mod); } vli_rshift1(v); if (l_carry) { v[NUM_ECC_DIGITS - 1] |= 0x8000000000000000ull; } } else if (l_cmpresult > 0) { vli_sub(a, a, b); vli_rshift1(a); if (vli_cmp(u, v) < 0) { vli_add(u, u, mod); } vli_sub(u, u, v); if (!EVEN(u)) { l_carry = vli_add(u, u, mod); } vli_rshift1(u); if (l_carry) { u[NUM_ECC_DIGITS - 1] |= 0x8000000000000000ull; } } else { vli_sub(b, b, a); vli_rshift1(b); if (vli_cmp(v, u) < 0) { vli_add(v, v, mod); } vli_sub(v, v, u); if (!EVEN(v)) { l_carry = vli_add(v, v, mod); } vli_rshift1(v); if (l_carry) { v[NUM_ECC_DIGITS - 1] |= 0x8000000000000000ull; } } } vli_set(result, u); } /* ------ Point operations ------ */ /* Returns 1 if point is the point at infinity, 0 otherwise. */ static int eccpoint_iszero(FAR eccpoint_t *point) { return vli_iszero(point->x) && vli_iszero(point->y); } /* Point multiplication algorithm using Montgomery's ladder with * co-Z coordinates. From http://eprint.iacr.org/2011/338.pdf */ /* Double in place */ static void eccpoint_double_jacobian(FAR uint64_t *X1, FAR uint64_t *Y1, FAR uint64_t *Z1) { /* t1 = X, t2 = Y, t3 = Z */ uint64_t t4[NUM_ECC_DIGITS]; uint64_t t5[NUM_ECC_DIGITS]; uint64_t l_carry; if (vli_iszero(Z1)) { return; } vli_modsquare_fast(t4, Y1); /* t4 = y1^2 */ vli_modmult_fast(t5, X1, t4); /* t5 = x1*y1^2 = A */ vli_modsquare_fast(t4, t4); /* t4 = y1^4 */ vli_modmult_fast(Y1, Y1, Z1); /* t2 = y1*z1 = z3 */ vli_modsquare_fast(Z1, Z1); /* t3 = z1^2 */ vli_modadd(X1, X1, Z1, g_curve_p); /* t1 = x1 + z1^2 */ vli_modadd(Z1, Z1, Z1, g_curve_p); /* t3 = 2*z1^2 */ vli_modsub(Z1, X1, Z1, g_curve_p); /* t3 = x1 - z1^2 */ /* t1 = x1^2 - z1^4 */ vli_modmult_fast(X1, X1, Z1); vli_modadd(Z1, X1, X1, g_curve_p); /* t3 = 2*(x1^2 - z1^4) */ vli_modadd(X1, X1, Z1, g_curve_p); /* t1 = 3*(x1^2 - z1^4) */ if (vli_testbit(X1, 0)) { l_carry = vli_add(X1, X1, g_curve_p); vli_rshift1(X1); X1[NUM_ECC_DIGITS - 1] |= l_carry << 63; } else { vli_rshift1(X1); } /* t1 = 3/2*(x1^2 - z1^4) = B */ /* t3 = B^2 */ vli_modsquare_fast(Z1, X1); /* t3 = B^2 - A */ vli_modsub(Z1, Z1, t5, g_curve_p); /* t3 = B^2 - 2A = x3 */ vli_modsub(Z1, Z1, t5, g_curve_p); /* t5 = A - x3 */ vli_modsub(t5, t5, Z1, g_curve_p); /* t1 = B * (A - x3) */ vli_modmult_fast(X1, X1, t5); /* t4 = B * (A - x3) - y1^4 = y3 */ vli_modsub(t4, X1, t4, g_curve_p); vli_set(X1, Z1); vli_set(Z1, Y1); vli_set(Y1, t4); } /* Modify (x1, y1) => (x1 * z^2, y1 * z^3) */ static void apply_z(FAR uint64_t *X1, FAR uint64_t *Y1, FAR uint64_t *Z) { uint64_t t1[NUM_ECC_DIGITS]; /* z^2 */ vli_modsquare_fast(t1, Z); /* x1 * z^2 */ vli_modmult_fast(X1, X1, t1); /* z^3 */ vli_modmult_fast(t1, t1, Z); /* y1 * z^3 */ vli_modmult_fast(Y1, Y1, t1); } /* P = (x1, y1) => 2P, (x2, y2) => P' */ static void xycz_initial_double(FAR uint64_t *X1, FAR uint64_t *Y1, FAR uint64_t *X2, FAR uint64_t *Y2, FAR uint64_t *initialz) { uint64_t z[NUM_ECC_DIGITS]; vli_set(X2, X1); vli_set(Y2, Y1); vli_clear(z); z[0] = 1; if (initialz) { vli_set(z, initialz); } apply_z(X1, Y1, z); eccpoint_double_jacobian(X1, Y1, z); apply_z(X2, Y2, z); } /* Input P = (x1, y1, Z), Q = (x2, y2, Z) * Output P' = (x1', y1', Z3), P + Q = (x3, y3, Z3) * or P => P', Q => P + Q */ static void xycz_add(FAR uint64_t *X1, FAR uint64_t *Y1, FAR uint64_t *X2, FAR uint64_t *Y2) { /* t1 = X1, t2 = Y1, t3 = X2, t4 = Y2 */ uint64_t t5[NUM_ECC_DIGITS]; /* t5 = x2 - x1 */ vli_modsub(t5, X2, X1, g_curve_p); /* t5 = (x2 - x1)^2 = A */ vli_modsquare_fast(t5, t5); /* t1 = x1*A = B */ vli_modmult_fast(X1, X1, t5); /* t3 = x2*A = C */ vli_modmult_fast(X2, X2, t5); /* t4 = y2 - y1 */ vli_modsub(Y2, Y2, Y1, g_curve_p); /* t5 = (y2 - y1)^2 = D */ vli_modsquare_fast(t5, Y2); /* t5 = D - B */ vli_modsub(t5, t5, X1, g_curve_p); /* t5 = D - B - C = x3 */ vli_modsub(t5, t5, X2, g_curve_p); /* t3 = C - B */ vli_modsub(X2, X2, X1, g_curve_p); /* t2 = y1*(C - B) */ vli_modmult_fast(Y1, Y1, X2); /* t3 = B - x3 */ vli_modsub(X2, X1, t5, g_curve_p); /* t4 = (y2 - y1)*(B - x3) */ vli_modmult_fast(Y2, Y2, X2); /* t4 = y3 */ vli_modsub(Y2, Y2, Y1, g_curve_p); vli_set(X2, t5); } /* 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(FAR uint64_t *X1, FAR uint64_t *Y1, FAR uint64_t *X2, FAR uint64_t *Y2) { /* t1 = X1, t2 = Y1, t3 = X2, t4 = Y2 */ uint64_t t5[NUM_ECC_DIGITS]; uint64_t t6[NUM_ECC_DIGITS]; uint64_t t7[NUM_ECC_DIGITS]; /* t5 = x2 - x1 */ vli_modsub(t5, X2, X1, g_curve_p); /* t5 = (x2 - x1)^2 = A */ vli_modsquare_fast(t5, t5); /* t1 = x1*A = B */ vli_modmult_fast(X1, X1, t5); /* t3 = x2*A = C */ vli_modmult_fast(X2, X2, t5); /* t4 = y2 + y1 */ vli_modadd(t5, Y2, Y1, g_curve_p); /* t4 = y2 - y1 */ vli_modsub(Y2, Y2, Y1, g_curve_p); /* t6 = C - B */ vli_modsub(t6, X2, X1, g_curve_p); /* t2 = y1 * (C - B) */ vli_modmult_fast(Y1, Y1, t6); /* t6 = B + C */ vli_modadd(t6, X1, X2, g_curve_p); /* t3 = (y2 - y1)^2 */ vli_modsquare_fast(X2, Y2); /* t3 = x3 */ vli_modsub(X2, X2, t6, g_curve_p); /* t7 = B - x3 */ vli_modsub(t7, X1, X2, g_curve_p); /* t4 = (y2 - y1)*(B - x3) */ vli_modmult_fast(Y2, Y2, t7); /* t4 = y3 */ vli_modsub(Y2, Y2, Y1, g_curve_p); /* t7 = (y2 + y1)^2 = F */ vli_modsquare_fast(t7, t5); /* t7 = x3' */ vli_modsub(t7, t7, t6, g_curve_p); /* t6 = x3' - B */ vli_modsub(t6, t7, X1, g_curve_p); /* t6 = (y2 + y1)*(x3' - B) */ vli_modmult_fast(t6, t6, t5); /* t2 = y3' */ vli_modsub(Y1, t6, Y1, g_curve_p); vli_set(X1, t7); } static void eccpoint_mult(FAR eccpoint_t *result, FAR eccpoint_t *point, FAR uint64_t *scalar, FAR uint64_t *initialz) { /* R0 and R1 */ uint64_t rx[2][NUM_ECC_DIGITS]; uint64_t ry[2][NUM_ECC_DIGITS]; uint64_t z[NUM_ECC_DIGITS]; int nb; int i; vli_set(rx[1], point->x); vli_set(ry[1], point->y); xycz_initial_double(rx[1], ry[1], rx[0], ry[0], initialz); for (i = vli_numbits(scalar) - 2; i > 0; --i) { nb = !vli_testbit(scalar, i); xycz_addc(rx[1 - nb], ry[1 - nb], rx[nb], ry[nb]); xycz_add(rx[nb], ry[nb], rx[1 - nb], ry[1 - nb]); } nb = !vli_testbit(scalar, 0); xycz_addc(rx[1 - nb], ry[1 - nb], rx[nb], ry[nb]); /* Find final 1/Z value. */ /* X1 - X0 */ vli_modsub(z, rx[1], rx[0], g_curve_p); /* Yb * (X1 - X0) */ vli_modmult_fast(z, z, ry[1 - nb]); /* xP * Yb * (X1 - X0) */ vli_modmult_fast(z, z, point->x); /* 1 / (xP * Yb * (X1 - X0)) */ vli_modinv(z, z, g_curve_p); /* yP / (xP * Yb * (X1 - X0)) */ vli_modmult_fast(z, z, point->y); /* Xb * yP / (xP * Yb * (X1 - X0)) */ vli_modmult_fast(z, z, rx[1 - nb]); /* End 1/Z calculation */ xycz_add(rx[nb], ry[nb], rx[1 - nb], ry[1 - nb]); apply_z(rx[0], ry[0], z); vli_set(result->x, rx[0]); vli_set(result->y, ry[0]); } static void ecc_bytes2native(uint64_t native[NUM_ECC_DIGITS], const uint8_t bytes[ECC_BYTES]) { FAR const uint8_t *digit; unsigned i; for (i = 0; i < NUM_ECC_DIGITS; ++i) { digit = bytes + 8 * (NUM_ECC_DIGITS - 1 - i); native[i] = ((uint64_t)digit[0] << 56) | ((uint64_t)digit[1] << 48) | ((uint64_t)digit[2] << 40) | ((uint64_t)digit[3] << 32) | ((uint64_t)digit[4] << 24) | ((uint64_t)digit[5] << 16) | ((uint64_t)digit[6] << 8) | (uint64_t)digit[7]; } } static void ecc_native2bytes(uint8_t bytes[ECC_BYTES], const uint64_t native[NUM_ECC_DIGITS]) { FAR uint8_t *digit; unsigned i; for (i = 0; i < NUM_ECC_DIGITS; ++i) { digit = bytes + 8 * (NUM_ECC_DIGITS - 1 - i); digit[0] = native[i] >> 56; digit[1] = native[i] >> 48; digit[2] = native[i] >> 40; digit[3] = native[i] >> 32; digit[4] = native[i] >> 24; digit[5] = native[i] >> 16; digit[6] = native[i] >> 8; digit[7] = native[i]; } } /* Compute a = sqrt(a) (mod g_curve_p). */ static void mod_sqrt(uint64_t a[NUM_ECC_DIGITS]) { unsigned i; uint64_t l_result[NUM_ECC_DIGITS] = { 1 }; uint64_t p1[NUM_ECC_DIGITS] = { 1 }; /* Since g_curve_p == 3 (mod 4) for all supported curves, we can * compute sqrt(a) = a^((g_curve_p + 1) / 4) (mod g_curve_p). */ vli_add(p1, g_curve_p, p1); /* p1 = g_curve_p + 1 */ for (i = vli_numbits(p1) - 1; i > 1; --i) { vli_modsquare_fast(l_result, l_result); if (vli_testbit(p1, i)) { vli_modmult_fast(l_result, l_result, a); } } vli_set(a, l_result); } static void ecc_point_decompress(FAR eccpoint_t *point, const uint8_t compressed[ECC_BYTES + 1]) { /* -a = 3 */ uint64_t _3[NUM_ECC_DIGITS] = { 3 }; ecc_bytes2native(point->x, compressed + 1); /* y = x^2 */ vli_modsquare_fast(point->y, point->x); /* y = x^2 - 3 */ vli_modsub(point->y, point->y, _3, g_curve_p); /* y = x^3 - 3x */ vli_modmult_fast(point->y, point->y, point->x); /* y = x^3 - 3x + b */ vli_modadd(point->y, point->y, g_curve_b, g_curve_p); mod_sqrt(point->y); if ((point->y[0] & 0x01) != (compressed[0] & 0x01)) { vli_sub(point->y, g_curve_p, point->y); } } /* -------- ECDSA code -------- */ /* Computes result = (left * right) % mod. */ static void vli_modmult(FAR uint64_t *result, FAR uint64_t *left, FAR uint64_t *right, FAR uint64_t *mod) { uint64_t l_product[2 * NUM_ECC_DIGITS]; uint64_t l_modmultiple[2 * NUM_ECC_DIGITS]; uint64_t l_carry; uint l_modbits = vli_numbits(mod); uint l_productbits; uint l_digitshift; uint l_bitshift; int l_cmp; vli_mult(l_product, left, right); l_productbits = vli_numbits(l_product + NUM_ECC_DIGITS); if (l_productbits) { l_productbits += NUM_ECC_DIGITS * 64; } else { l_productbits = vli_numbits(l_product); } if (l_productbits < l_modbits) { /* l_product < mod. */ vli_set(result, l_product); return; } /* Shift mod by (l_leftBits - l_modbits). * This multiplies mod by the largest power of two possible * while still resulting in a number less than left. */ vli_clear(l_modmultiple); vli_clear(l_modmultiple + NUM_ECC_DIGITS); l_digitshift = (l_productbits - l_modbits) / 64; l_bitshift = (l_productbits - l_modbits) % 64; if (l_bitshift) { l_modmultiple[l_digitshift + NUM_ECC_DIGITS] = vli_lshift(l_modmultiple + l_digitshift, mod, l_bitshift); } else { vli_set(l_modmultiple + l_digitshift, mod); } /* Subtract all multiples of mod to get the remainder. */ vli_clear(result); /* Use result as a temp var to store 1 (for subtraction) */ result[0] = 1; while (l_productbits > NUM_ECC_DIGITS * 64 || vli_cmp(l_modmultiple, mod) >= 0) { l_cmp = vli_cmp(l_modmultiple + NUM_ECC_DIGITS, l_product + NUM_ECC_DIGITS); if (l_cmp < 0 || (l_cmp == 0 && vli_cmp(l_modmultiple, l_product) <= 0)) { if (vli_sub(l_product, l_product, l_modmultiple)) { /* borrow */ vli_sub(l_product + NUM_ECC_DIGITS, l_product + NUM_ECC_DIGITS, result); } vli_sub(l_product + NUM_ECC_DIGITS, l_product + NUM_ECC_DIGITS, l_modmultiple + NUM_ECC_DIGITS); } l_carry = (l_modmultiple[NUM_ECC_DIGITS] & 0x01) << 63; vli_rshift1(l_modmultiple + NUM_ECC_DIGITS); vli_rshift1(l_modmultiple); l_modmultiple[NUM_ECC_DIGITS - 1] |= l_carry; --l_productbits; } vli_set(result, l_product); } static uint umax(uint a, uint b) { return a > b ? a : b; } /**************************************************************************** * Public Functions ****************************************************************************/ int ecc_make_key(uint8_t publickey[ECC_BYTES + 1], uint8_t privatekey[ECC_BYTES]) { uint64_t l_private[NUM_ECC_DIGITS]; eccpoint_t l_public; unsigned l_tries = 0; memset(&l_public, 0, sizeof(eccpoint_t)); do { if (l_tries++ >= MAX_TRIES) { return 0; } arc4random_buf(l_private, NUM_ECC_DIGITS); if (vli_iszero(l_private)) { continue; } /* Make sure the private key is in the range [1, n-1]. * For the supported curves, n is always large enough that we only * need to subtract once at most. */ if (vli_cmp(g_curve_n, l_private) != 1) { vli_sub(l_private, l_private, g_curve_n); } eccpoint_mult(&l_public, &g_curve_g, l_private, NULL); } while (eccpoint_iszero(&l_public)); ecc_native2bytes(privatekey, l_private); ecc_native2bytes(publickey + 1, l_public.x); publickey[0] = 2 + (l_public.y[0] & 0x01); return 1; } int ecc_make_key_uncomp(uint8_t publickey_x[ECC_BYTES], uint8_t publickey_y[ECC_BYTES], uint8_t privatekey[ECC_BYTES]) { uint64_t l_private[NUM_ECC_DIGITS]; eccpoint_t l_public; unsigned l_tries = 0; do { if (l_tries++ >= MAX_TRIES) { return 0; } arc4random_buf(l_private, NUM_ECC_DIGITS); if (vli_iszero(l_private)) { continue; } /* Make sure the private key is in the range [1, n-1]. * For the supported curves, n is always large enough that we only * need to subtract once at most. */ if (vli_cmp(g_curve_n, l_private) != 1) { vli_sub(l_private, l_private, g_curve_n); } eccpoint_mult(&l_public, &g_curve_g, l_private, NULL); } while (eccpoint_iszero(&l_public)); ecc_native2bytes(privatekey, l_private); ecc_native2bytes(publickey_x, l_public.x); ecc_native2bytes(publickey_y, l_public.y); return 1; } int ecdh_shared_secret(const uint8_t publickey[ECC_BYTES + 1], const uint8_t privatekey[ECC_BYTES], uint8_t secret[ECC_BYTES]) { eccpoint_t l_product; eccpoint_t l_public; uint64_t l_private[NUM_ECC_DIGITS]; uint64_t l_random[NUM_ECC_DIGITS]; memset(&l_product, 0, sizeof(eccpoint_t)); memset(&l_public, 0, sizeof(eccpoint_t)); arc4random_buf(l_random, NUM_ECC_DIGITS); ecc_point_decompress(&l_public, publickey); ecc_bytes2native(l_private, privatekey); eccpoint_mult(&l_product, &l_public, l_private, l_random); ecc_native2bytes(secret, l_product.x); return !eccpoint_iszero(&l_product); } int ecdsa_sign(const uint8_t privatekey[ECC_BYTES], const uint8_t hash[ECC_BYTES], uint8_t signature[ECC_BYTES * 2]) { uint64_t k[NUM_ECC_DIGITS]; uint64_t l_tmp[NUM_ECC_DIGITS]; uint64_t l_s[NUM_ECC_DIGITS]; unsigned l_tries = 0; eccpoint_t p; memset(&p, 0, sizeof(eccpoint_t)); do { if (l_tries++ >= MAX_TRIES) { return 0; } arc4random_buf(k, NUM_ECC_DIGITS); if (vli_iszero(k)) { continue; } if (vli_cmp(g_curve_n, k) != 1) { vli_sub(k, k, g_curve_n); } /* tmp = k * G */ eccpoint_mult(&p, &g_curve_g, k, NULL); /* r = x1 (mod n) */ if (vli_cmp(g_curve_n, p.x) != 1) { vli_sub(p.x, p.x, g_curve_n); } } while (vli_iszero(p.x)); ecc_native2bytes(signature, p.x); ecc_bytes2native(l_tmp, privatekey); vli_modmult(l_s, p.x, l_tmp, g_curve_n); /* s = r*d */ ecc_bytes2native(l_tmp, hash); vli_modadd(l_s, l_tmp, l_s, g_curve_n); /* s = e + r*d */ /* k = 1 / k */ vli_modinv(k, k, g_curve_n); /* s = (e + r*d) / k */ vli_modmult(l_s, l_s, k, g_curve_n); ecc_native2bytes(signature + ECC_BYTES, l_s); return 1; } int ecdsa_verify(const uint8_t publickey[ECC_BYTES + 1], const uint8_t hash[ECC_BYTES], const uint8_t signature[ECC_BYTES * 2]) { uint64_t u1[NUM_ECC_DIGITS]; uint64_t u2[NUM_ECC_DIGITS]; uint64_t z[NUM_ECC_DIGITS]; uint64_t rx[NUM_ECC_DIGITS]; uint64_t ry[NUM_ECC_DIGITS]; uint64_t tx[NUM_ECC_DIGITS]; uint64_t ty[NUM_ECC_DIGITS]; uint64_t tz[NUM_ECC_DIGITS]; uint64_t l_r[NUM_ECC_DIGITS]; uint64_t l_s[NUM_ECC_DIGITS]; uint l_numbits; eccpoint_t *l_point; eccpoint_t l_public; eccpoint_t l_sum; int l_index; int i; /* Use Shamir's trick to calculate u1*G + u2*Q */ eccpoint_t *l_points[4] = { NULL, &g_curve_g, &l_public, &l_sum }; ecc_point_decompress(&l_public, publickey); ecc_bytes2native(l_r, signature); ecc_bytes2native(l_s, signature + ECC_BYTES); if (vli_iszero(l_r) || vli_iszero(l_s)) { /* r, s must not be 0. */ return 0; } if (vli_cmp(g_curve_n, l_r) != 1 || vli_cmp(g_curve_n, l_s) != 1) { /* r, s must be < n. */ return 0; } /* Calculate u1 and u2. */ vli_modinv(z, l_s, g_curve_n); /* Z = s^-1 */ ecc_bytes2native(u1, hash); vli_modmult(u1, u1, z, g_curve_n); /* u1 = e/s */ /* u2 = r/s */ vli_modmult(u2, l_r, z, g_curve_n); /* Calculate l_sum = G + Q. */ vli_set(l_sum.x, l_public.x); vli_set(l_sum.y, l_public.y); vli_set(tx, g_curve_g.x); vli_set(ty, g_curve_g.y); vli_modsub(z, l_sum.x, tx, g_curve_p); /* Z = x2 - x1 */ xycz_add(tx, ty, l_sum.x, l_sum.y); vli_modinv(z, z, g_curve_p); /* Z = 1/Z */ apply_z(l_sum.x, l_sum.y, z); l_numbits = umax(vli_numbits(u1), vli_numbits(u2)); l_point = l_points[(!!vli_testbit(u1, l_numbits - 1)) | ((!!vli_testbit(u2, l_numbits - 1)) << 1)]; vli_set(rx, l_point->x); vli_set(ry, l_point->y); vli_clear(z); z[0] = 1; for (i = l_numbits - 2; i >= 0; --i) { eccpoint_double_jacobian(rx, ry, z); l_index = (!!vli_testbit(u1, i)) | ((!!vli_testbit(u2, i)) << 1); l_point = l_points[l_index]; if (l_point) { vli_set(tx, l_point->x); vli_set(ty, l_point->y); apply_z(tx, ty, z); vli_modsub(tz, rx, tx, g_curve_p); /* Z = x2 - x1 */ xycz_add(tx, ty, rx, ry); vli_modmult_fast(z, z, tz); } } vli_modinv(z, z, g_curve_p); /* Z = 1/Z */ apply_z(rx, ry, z); /* v = x1 (mod n) */ if (vli_cmp(g_curve_n, rx) != 1) { vli_sub(rx, rx, g_curve_n); } /* Accept only if v == r. */ return vli_cmp(rx, l_r) == 0; }