/* * Copyright (c) 2013, 2014 Kenneth MacKay. All rights reserved. * Copyright (c) 2019 Vitaly Chikunov * * 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. */ #include #include #include #include #include #include #include #include #include #include "ecc.h" #include "ecc_curve_defs.h" typedef struct { u64 m_low; u64 m_high; } uint128_t; static inline const struct ecc_curve *ecc_get_curve(unsigned int curve_id) { switch (curve_id) { /* In FIPS mode only allow P256 and higher */ case ECC_CURVE_NIST_P192: return fips_enabled ? NULL : &nist_p192; case ECC_CURVE_NIST_P256: return &nist_p256; default: return NULL; } } static u64 *ecc_alloc_digits_space(unsigned int ndigits) { size_t len = ndigits * sizeof(u64); if (!len) return NULL; return kmalloc(len, GFP_KERNEL); } static void ecc_free_digits_space(u64 *space) { kzfree(space); } static struct ecc_point *ecc_alloc_point(unsigned int ndigits) { struct ecc_point *p = kmalloc(sizeof(*p), GFP_KERNEL); if (!p) return NULL; p->x = ecc_alloc_digits_space(ndigits); if (!p->x) goto err_alloc_x; p->y = ecc_alloc_digits_space(ndigits); if (!p->y) goto err_alloc_y; p->ndigits = ndigits; return p; err_alloc_y: ecc_free_digits_space(p->x); err_alloc_x: kfree(p); return NULL; } static void ecc_free_point(struct ecc_point *p) { if (!p) return; kzfree(p->x); kzfree(p->y); kzfree(p); } static void vli_clear(u64 *vli, unsigned int ndigits) { int i; for (i = 0; i < ndigits; i++) vli[i] = 0; } /* Returns true if vli == 0, false otherwise. */ bool vli_is_zero(const u64 *vli, unsigned int ndigits) { int i; for (i = 0; i < ndigits; i++) { if (vli[i]) return false; } return true; } EXPORT_SYMBOL(vli_is_zero); /* Returns nonzero if bit bit of vli is set. */ static u64 vli_test_bit(const u64 *vli, unsigned int bit) { return (vli[bit / 64] & ((u64)1 << (bit % 64))); } static bool vli_is_negative(const u64 *vli, unsigned int ndigits) { return vli_test_bit(vli, ndigits * 64 - 1); } /* Counts the number of 64-bit "digits" in vli. */ static unsigned int vli_num_digits(const u64 *vli, unsigned int ndigits) { 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 = ndigits - 1; i >= 0 && vli[i] == 0; i--); return (i + 1); } /* Counts the number of bits required for vli. */ static unsigned int vli_num_bits(const u64 *vli, unsigned int ndigits) { unsigned int i, num_digits; u64 digit; num_digits = vli_num_digits(vli, ndigits); if (num_digits == 0) return 0; digit = vli[num_digits - 1]; for (i = 0; digit; i++) digit >>= 1; return ((num_digits - 1) * 64 + i); } /* Set dest from unaligned bit string src. */ void vli_from_be64(u64 *dest, const void *src, unsigned int ndigits) { int i; const u64 *from = src; for (i = 0; i < ndigits; i++) dest[i] = get_unaligned_be64(&from[ndigits - 1 - i]); } EXPORT_SYMBOL(vli_from_be64); void vli_from_le64(u64 *dest, const void *src, unsigned int ndigits) { int i; const u64 *from = src; for (i = 0; i < ndigits; i++) dest[i] = get_unaligned_le64(&from[i]); } EXPORT_SYMBOL(vli_from_le64); /* Sets dest = src. */ static void vli_set(u64 *dest, const u64 *src, unsigned int ndigits) { int i; for (i = 0; i < ndigits; i++) dest[i] = src[i]; } /* Returns sign of left - right. */ int vli_cmp(const u64 *left, const u64 *right, unsigned int ndigits) { int i; for (i = ndigits - 1; i >= 0; i--) { if (left[i] > right[i]) return 1; else if (left[i] < right[i]) return -1; } return 0; } EXPORT_SYMBOL(vli_cmp); /* Computes result = in << c, returning carry. Can modify in place * (if result == in). 0 < shift < 64. */ static u64 vli_lshift(u64 *result, const u64 *in, unsigned int shift, unsigned int ndigits) { u64 carry = 0; int i; for (i = 0; i < ndigits; i++) { u64 temp = in[i]; result[i] = (temp << shift) | carry; carry = temp >> (64 - shift); } return carry; } /* Computes vli = vli >> 1. */ static void vli_rshift1(u64 *vli, unsigned int ndigits) { u64 *end = vli; u64 carry = 0; vli += ndigits; while (vli-- > end) { u64 temp = *vli; *vli = (temp >> 1) | carry; carry = temp << 63; } } /* Computes result = left + right, returning carry. Can modify in place. */ static u64 vli_add(u64 *result, const u64 *left, const u64 *right, unsigned int ndigits) { u64 carry = 0; int i; for (i = 0; i < ndigits; i++) { u64 sum; sum = left[i] + right[i] + carry; if (sum != left[i]) carry = (sum < left[i]); result[i] = sum; } return carry; } /* Computes result = left + right, returning carry. Can modify in place. */ static u64 vli_uadd(u64 *result, const u64 *left, u64 right, unsigned int ndigits) { u64 carry = right; int i; for (i = 0; i < ndigits; i++) { u64 sum; sum = left[i] + carry; if (sum != left[i]) carry = (sum < left[i]); else carry = !!carry; result[i] = sum; } return carry; } /* Computes result = left - right, returning borrow. Can modify in place. */ u64 vli_sub(u64 *result, const u64 *left, const u64 *right, unsigned int ndigits) { u64 borrow = 0; int i; for (i = 0; i < ndigits; i++) { u64 diff; diff = left[i] - right[i] - borrow; if (diff != left[i]) borrow = (diff > left[i]); result[i] = diff; } return borrow; } EXPORT_SYMBOL(vli_sub); /* Computes result = left - right, returning borrow. Can modify in place. */ static u64 vli_usub(u64 *result, const u64 *left, u64 right, unsigned int ndigits) { u64 borrow = right; int i; for (i = 0; i < ndigits; i++) { u64 diff; diff = left[i] - borrow; if (diff != left[i]) borrow = (diff > left[i]); result[i] = diff; } return borrow; } static uint128_t mul_64_64(u64 left, u64 right) { uint128_t result; #if defined(CONFIG_ARCH_SUPPORTS_INT128) && defined(__SIZEOF_INT128__) unsigned __int128 m = (unsigned __int128)left * right; result.m_low = m; result.m_high = m >> 64; #else u64 a0 = left & 0xffffffffull; u64 a1 = left >> 32; u64 b0 = right & 0xffffffffull; u64 b1 = right >> 32; u64 m0 = a0 * b0; u64 m1 = a0 * b1; u64 m2 = a1 * b0; u64 m3 = a1 * b1; m2 += (m0 >> 32); m2 += m1; /* Overflow */ if (m2 < m1) m3 += 0x100000000ull; result.m_low = (m0 & 0xffffffffull) | (m2 << 32); result.m_high = m3 + (m2 >> 32); #endif return result; } static uint128_t add_128_128(uint128_t a, uint128_t b) { uint128_t result; result.m_low = a.m_low + b.m_low; result.m_high = a.m_high + b.m_high + (result.m_low < a.m_low); return result; } static void vli_mult(u64 *result, const u64 *left, const u64 *right, unsigned int ndigits) { uint128_t r01 = { 0, 0 }; u64 r2 = 0; unsigned int i, k; /* Compute each digit of result in sequence, maintaining the * carries. */ for (k = 0; k < ndigits * 2 - 1; k++) { unsigned int min; if (k < ndigits) min = 0; else min = (k + 1) - ndigits; for (i = min; i <= k && i < ndigits; i++) { uint128_t product; product = mul_64_64(left[i], right[k - i]); r01 = add_128_128(r01, product); r2 += (r01.m_high < product.m_high); } result[k] = r01.m_low; r01.m_low = r01.m_high; r01.m_high = r2; r2 = 0; } result[ndigits * 2 - 1] = r01.m_low; } /* Compute product = left * right, for a small right value. */ static void vli_umult(u64 *result, const u64 *left, u32 right, unsigned int ndigits) { uint128_t r01 = { 0 }; unsigned int k; for (k = 0; k < ndigits; k++) { uint128_t product; product = mul_64_64(left[k], right); r01 = add_128_128(r01, product); /* no carry */ result[k] = r01.m_low; r01.m_low = r01.m_high; r01.m_high = 0; } result[k] = r01.m_low; for (++k; k < ndigits * 2; k++) result[k] = 0; } static void vli_square(u64 *result, const u64 *left, unsigned int ndigits) { uint128_t r01 = { 0, 0 }; u64 r2 = 0; int i, k; for (k = 0; k < ndigits * 2 - 1; k++) { unsigned int min; if (k < ndigits) min = 0; else min = (k + 1) - ndigits; for (i = min; i <= k && i <= k - i; i++) { uint128_t product; product = mul_64_64(left[i], left[k - i]); if (i < k - i) { r2 += product.m_high >> 63; product.m_high = (product.m_high << 1) | (product.m_low >> 63); product.m_low <<= 1; } r01 = add_128_128(r01, product); r2 += (r01.m_high < product.m_high); } result[k] = r01.m_low; r01.m_low = r01.m_high; r01.m_high = r2; r2 = 0; } result[ndigits * 2 - 1] = r01.m_low; } /* Computes result = (left + right) % mod. * Assumes that left < mod and right < mod, result != mod. */ static void vli_mod_add(u64 *result, const u64 *left, const u64 *right, const u64 *mod, unsigned int ndigits) { u64 carry; carry = vli_add(result, left, right, ndigits); /* result > mod (result = mod + remainder), so subtract mod to * get remainder. */ if (carry || vli_cmp(result, mod, ndigits) >= 0) vli_sub(result, result, mod, ndigits); } /* Computes result = (left - right) % mod. * Assumes that left < mod and right < mod, result != mod. */ static void vli_mod_sub(u64 *result, const u64 *left, const u64 *right, const u64 *mod, unsigned int ndigits) { u64 borrow = vli_sub(result, left, right, ndigits); /* In this case, p_result == -diff == (max int) - diff. * Since -x % d == d - x, we can get the correct result from * result + mod (with overflow). */ if (borrow) vli_add(result, result, mod, ndigits); } /* * Computes result = product % mod * for special form moduli: p = 2^k-c, for small c (note the minus sign) * * References: * R. Crandall, C. Pomerance. Prime Numbers: A Computational Perspective. * 9 Fast Algorithms for Large-Integer Arithmetic. 9.2.3 Moduli of special form * Algorithm 9.2.13 (Fast mod operation for special-form moduli). */ static void vli_mmod_special(u64 *result, const u64 *product, const u64 *mod, unsigned int ndigits) { u64 c = -mod[0]; u64 t[ECC_MAX_DIGITS * 2]; u64 r[ECC_MAX_DIGITS * 2]; vli_set(r, product, ndigits * 2); while (!vli_is_zero(r + ndigits, ndigits)) { vli_umult(t, r + ndigits, c, ndigits); vli_clear(r + ndigits, ndigits); vli_add(r, r, t, ndigits * 2); } vli_set(t, mod, ndigits); vli_clear(t + ndigits, ndigits); while (vli_cmp(r, t, ndigits * 2) >= 0) vli_sub(r, r, t, ndigits * 2); vli_set(result, r, ndigits); } /* * Computes result = product % mod * for special form moduli: p = 2^{k-1}+c, for small c (note the plus sign) * where k-1 does not fit into qword boundary by -1 bit (such as 255). * References (loosely based on): * A. Menezes, P. van Oorschot, S. Vanstone. Handbook of Applied Cryptography. * 14.3.4 Reduction methods for moduli of special form. Algorithm 14.47. * URL: http://cacr.uwaterloo.ca/hac/about/chap14.pdf * * H. Cohen, G. Frey, R. Avanzi, C. Doche, T. Lange, K. Nguyen, F. Vercauteren. * Handbook of Elliptic and Hyperelliptic Curve Cryptography. * Algorithm 10.25 Fast reduction for special form moduli */ static void vli_mmod_special2(u64 *result, const u64 *product, const u64 *mod, unsigned int ndigits) { u64 c2 = mod[0] * 2; u64 q[ECC_MAX_DIGITS]; u64 r[ECC_MAX_DIGITS * 2]; u64 m[ECC_MAX_DIGITS * 2]; /* expanded mod */ int carry; /* last bit that doesn't fit into q */ int i; vli_set(m, mod, ndigits); vli_clear(m + ndigits, ndigits); vli_set(r, product, ndigits); /* q and carry are top bits */ vli_set(q, product + ndigits, ndigits); vli_clear(r + ndigits, ndigits); carry = vli_is_negative(r, ndigits); if (carry) r[ndigits - 1] &= (1ull << 63) - 1; for (i = 1; carry || !vli_is_zero(q, ndigits); i++) { u64 qc[ECC_MAX_DIGITS * 2]; vli_umult(qc, q, c2, ndigits); if (carry) vli_uadd(qc, qc, mod[0], ndigits * 2); vli_set(q, qc + ndigits, ndigits); vli_clear(qc + ndigits, ndigits); carry = vli_is_negative(qc, ndigits); if (carry) qc[ndigits - 1] &= (1ull << 63) - 1; if (i & 1) vli_sub(r, r, qc, ndigits * 2); else vli_add(r, r, qc, ndigits * 2); } while (vli_is_negative(r, ndigits * 2)) vli_add(r, r, m, ndigits * 2); while (vli_cmp(r, m, ndigits * 2) >= 0) vli_sub(r, r, m, ndigits * 2); vli_set(result, r, ndigits); } /* * Computes result = product % mod, where product is 2N words long. * Reference: Ken MacKay's micro-ecc. * Currently only designed to work for curve_p or curve_n. */ static void vli_mmod_slow(u64 *result, u64 *product, const u64 *mod, unsigned int ndigits) { u64 mod_m[2 * ECC_MAX_DIGITS]; u64 tmp[2 * ECC_MAX_DIGITS]; u64 *v[2] = { tmp, product }; u64 carry = 0; unsigned int i; /* Shift mod so its highest set bit is at the maximum position. */ int shift = (ndigits * 2 * 64) - vli_num_bits(mod, ndigits); int word_shift = shift / 64; int bit_shift = shift % 64; vli_clear(mod_m, word_shift); if (bit_shift > 0) { for (i = 0; i < ndigits; ++i) { mod_m[word_shift + i] = (mod[i] << bit_shift) | carry; carry = mod[i] >> (64 - bit_shift); } } else vli_set(mod_m + word_shift, mod, ndigits); for (i = 1; shift >= 0; --shift) { u64 borrow = 0; unsigned int j; for (j = 0; j < ndigits * 2; ++j) { u64 diff = v[i][j] - mod_m[j] - borrow; if (diff != v[i][j]) borrow = (diff > v[i][j]); v[1 - i][j] = diff; } i = !(i ^ borrow); /* Swap the index if there was no borrow */ vli_rshift1(mod_m, ndigits); mod_m[ndigits - 1] |= mod_m[ndigits] << (64 - 1); vli_rshift1(mod_m + ndigits, ndigits); } vli_set(result, v[i], ndigits); } /* Computes result = product % mod using Barrett's reduction with precomputed * value mu appended to the mod after ndigits, mu = (2^{2w} / mod) and have * length ndigits + 1, where mu * (2^w - 1) should not overflow ndigits * boundary. * * Reference: * R. Brent, P. Zimmermann. Modern Computer Arithmetic. 2010. * 2.4.1 Barrett's algorithm. Algorithm 2.5. */ static void vli_mmod_barrett(u64 *result, u64 *product, const u64 *mod, unsigned int ndigits) { u64 q[ECC_MAX_DIGITS * 2]; u64 r[ECC_MAX_DIGITS * 2]; const u64 *mu = mod + ndigits; vli_mult(q, product + ndigits, mu, ndigits); if (mu[ndigits]) vli_add(q + ndigits, q + ndigits, product + ndigits, ndigits); vli_mult(r, mod, q + ndigits, ndigits); vli_sub(r, product, r, ndigits * 2); while (!vli_is_zero(r + ndigits, ndigits) || vli_cmp(r, mod, ndigits) != -1) { u64 carry; carry = vli_sub(r, r, mod, ndigits); vli_usub(r + ndigits, r + ndigits, carry, ndigits); } vli_set(result, r, ndigits); } /* Computes p_result = p_product % 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_192(u64 *result, const u64 *product, const u64 *curve_prime, u64 *tmp) { const unsigned int ndigits = 3; int carry; vli_set(result, product, ndigits); vli_set(tmp, &product[3], ndigits); carry = vli_add(result, result, tmp, ndigits); tmp[0] = 0; tmp[1] = product[3]; tmp[2] = product[4]; carry += vli_add(result, result, tmp, ndigits); tmp[0] = tmp[1] = product[5]; tmp[2] = 0; carry += vli_add(result, result, tmp, ndigits); while (carry || vli_cmp(curve_prime, result, ndigits) != 1) carry -= vli_sub(result, result, curve_prime, ndigits); } /* Computes result = product % curve_prime * from http://www.nsa.gov/ia/_files/nist-routines.pdf */ static void vli_mmod_fast_256(u64 *result, const u64 *product, const u64 *curve_prime, u64 *tmp) { int carry; const unsigned int ndigits = 4; /* t */ vli_set(result, product, ndigits); /* s1 */ tmp[0] = 0; tmp[1] = product[5] & 0xffffffff00000000ull; tmp[2] = product[6]; tmp[3] = product[7]; carry = vli_lshift(tmp, tmp, 1, ndigits); carry += vli_add(result, result, tmp, ndigits); /* s2 */ tmp[1] = product[6] << 32; tmp[2] = (product[6] >> 32) | (product[7] << 32); tmp[3] = product[7] >> 32; carry += vli_lshift(tmp, tmp, 1, ndigits); carry += vli_add(result, result, tmp, ndigits); /* s3 */ tmp[0] = product[4]; tmp[1] = product[5] & 0xffffffff; tmp[2] = 0; tmp[3] = product[7]; carry += vli_add(result, result, tmp, ndigits); /* s4 */ tmp[0] = (product[4] >> 32) | (product[5] << 32); tmp[1] = (product[5] >> 32) | (product[6] & 0xffffffff00000000ull); tmp[2] = product[7]; tmp[3] = (product[6] >> 32) | (product[4] << 32); carry += vli_add(result, result, tmp, ndigits); /* d1 */ tmp[0] = (product[5] >> 32) | (product[6] << 32); tmp[1] = (product[6] >> 32); tmp[2] = 0; tmp[3] = (product[4] & 0xffffffff) | (product[5] << 32); carry -= vli_sub(result, result, tmp, ndigits); /* d2 */ tmp[0] = product[6]; tmp[1] = product[7]; tmp[2] = 0; tmp[3] = (product[4] >> 32) | (product[5] & 0xffffffff00000000ull); carry -= vli_sub(result, result, tmp, ndigits); /* d3 */ tmp[0] = (product[6] >> 32) | (product[7] << 32); tmp[1] = (product[7] >> 32) | (product[4] << 32); tmp[2] = (product[4] >> 32) | (product[5] << 32); tmp[3] = (product[6] << 32); carry -= vli_sub(result, result, tmp, ndigits); /* d4 */ tmp[0] = product[7]; tmp[1] = product[4] & 0xffffffff00000000ull; tmp[2] = product[5]; tmp[3] = product[6] & 0xffffffff00000000ull; carry -= vli_sub(result, result, tmp, ndigits); if (carry < 0) { do { carry += vli_add(result, result, curve_prime, ndigits); } while (carry < 0); } else { while (carry || vli_cmp(curve_prime, result, ndigits) != 1) carry -= vli_sub(result, result, curve_prime, ndigits); } } /* Computes result = product % curve_prime for different curve_primes. * * Note that curve_primes are distinguished just by heuristic check and * not by complete conformance check. */ static bool vli_mmod_fast(u64 *result, u64 *product, const u64 *curve_prime, unsigned int ndigits) { u64 tmp[2 * ECC_MAX_DIGITS]; /* Currently, both NIST primes have -1 in lowest qword. */ if (curve_prime[0] != -1ull) { /* Try to handle Pseudo-Marsenne primes. */ if (curve_prime[ndigits - 1] == -1ull) { vli_mmod_special(result, product, curve_prime, ndigits); return true; } else if (curve_prime[ndigits - 1] == 1ull << 63 && curve_prime[ndigits - 2] == 0) { vli_mmod_special2(result, product, curve_prime, ndigits); return true; } vli_mmod_barrett(result, product, curve_prime, ndigits); return true; } switch (ndigits) { case 3: vli_mmod_fast_192(result, product, curve_prime, tmp); break; case 4: vli_mmod_fast_256(result, product, curve_prime, tmp); break; default: pr_err_ratelimited("ecc: unsupported digits size!\n"); return false; } return true; } /* Computes result = (left * right) % mod. * Assumes that mod is big enough curve order. */ void vli_mod_mult_slow(u64 *result, const u64 *left, const u64 *right, const u64 *mod, unsigned int ndigits) { u64 product[ECC_MAX_DIGITS * 2]; vli_mult(product, left, right, ndigits); vli_mmod_slow(result, product, mod, ndigits); } EXPORT_SYMBOL(vli_mod_mult_slow); /* Computes result = (left * right) % curve_prime. */ static void vli_mod_mult_fast(u64 *result, const u64 *left, const u64 *right, const u64 *curve_prime, unsigned int ndigits) { u64 product[2 * ECC_MAX_DIGITS]; vli_mult(product, left, right, ndigits); vli_mmod_fast(result, product, curve_prime, ndigits); } /* Computes result = left^2 % curve_prime. */ static void vli_mod_square_fast(u64 *result, const u64 *left, const u64 *curve_prime, unsigned int ndigits) { u64 product[2 * ECC_MAX_DIGITS]; vli_square(product, left, ndigits); vli_mmod_fast(result, product, curve_prime, ndigits); } #define EVEN(vli) (!(vli[0] & 1)) /* Computes result = (1 / p_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 */ void vli_mod_inv(u64 *result, const u64 *input, const u64 *mod, unsigned int ndigits) { u64 a[ECC_MAX_DIGITS], b[ECC_MAX_DIGITS]; u64 u[ECC_MAX_DIGITS], v[ECC_MAX_DIGITS]; u64 carry; int cmp_result; if (vli_is_zero(input, ndigits)) { vli_clear(result, ndigits); return; } vli_set(a, input, ndigits); vli_set(b, mod, ndigits); vli_clear(u, ndigits); u[0] = 1; vli_clear(v, ndigits); while ((cmp_result = vli_cmp(a, b, ndigits)) != 0) { carry = 0; if (EVEN(a)) { vli_rshift1(a, ndigits); if (!EVEN(u)) carry = vli_add(u, u, mod, ndigits); vli_rshift1(u, ndigits); if (carry) u[ndigits - 1] |= 0x8000000000000000ull; } else if (EVEN(b)) { vli_rshift1(b, ndigits); if (!EVEN(v)) carry = vli_add(v, v, mod, ndigits); vli_rshift1(v, ndigits); if (carry) v[ndigits - 1] |= 0x8000000000000000ull; } else if (cmp_result > 0) { vli_sub(a, a, b, ndigits); vli_rshift1(a, ndigits); if (vli_cmp(u, v, ndigits) < 0) vli_add(u, u, mod, ndigits); vli_sub(u, u, v, ndigits); if (!EVEN(u)) carry = vli_add(u, u, mod, ndigits); vli_rshift1(u, ndigits); if (carry) u[ndigits - 1] |= 0x8000000000000000ull; } else { vli_sub(b, b, a, ndigits); vli_rshift1(b, ndigits); if (vli_cmp(v, u, ndigits) < 0) vli_add(v, v, mod, ndigits); vli_sub(v, v, u, ndigits); if (!EVEN(v)) carry = vli_add(v, v, mod, ndigits); vli_rshift1(v, ndigits); if (carry) v[ndigits - 1] |= 0x8000000000000000ull; } } vli_set(result, u, ndigits); } EXPORT_SYMBOL(vli_mod_inv); /* ------ Point operations ------ */ /* Returns true if p_point is the point at infinity, false otherwise. */ static bool ecc_point_is_zero(const struct ecc_point *point) { return (vli_is_zero(point->x, point->ndigits) && vli_is_zero(point->y, point->ndigits)); } /* Point multiplication algorithm using Montgomery's ladder with co-Z * coordinates. From http://eprint.iacr.org/2011/338.pdf */ /* Double in place */ static void ecc_point_double_jacobian(u64 *x1, u64 *y1, u64 *z1, u64 *curve_prime, unsigned int ndigits) { /* t1 = x, t2 = y, t3 = z */ u64 t4[ECC_MAX_DIGITS]; u64 t5[ECC_MAX_DIGITS]; if (vli_is_zero(z1, ndigits)) return; /* t4 = y1^2 */ vli_mod_square_fast(t4, y1, curve_prime, ndigits); /* t5 = x1*y1^2 = A */ vli_mod_mult_fast(t5, x1, t4, curve_prime, ndigits); /* t4 = y1^4 */ vli_mod_square_fast(t4, t4, curve_prime, ndigits); /* t2 = y1*z1 = z3 */ vli_mod_mult_fast(y1, y1, z1, curve_prime, ndigits); /* t3 = z1^2 */ vli_mod_square_fast(z1, z1, curve_prime, ndigits); /* t1 = x1 + z1^2 */ vli_mod_add(x1, x1, z1, curve_prime, ndigits); /* t3 = 2*z1^2 */ vli_mod_add(z1, z1, z1, curve_prime, ndigits); /* t3 = x1 - z1^2 */ vli_mod_sub(z1, x1, z1, curve_prime, ndigits); /* t1 = x1^2 - z1^4 */ vli_mod_mult_fast(x1, x1, z1, curve_prime, ndigits); /* t3 = 2*(x1^2 - z1^4) */ vli_mod_add(z1, x1, x1, curve_prime, ndigits); /* t1 = 3*(x1^2 - z1^4) */ vli_mod_add(x1, x1, z1, curve_prime, ndigits); if (vli_test_bit(x1, 0)) { u64 carry = vli_add(x1, x1, curve_prime, ndigits); vli_rshift1(x1, ndigits); x1[ndigits - 1] |= carry << 63; } else { vli_rshift1(x1, ndigits); } /* t1 = 3/2*(x1^2 - z1^4) = B */ /* t3 = B^2 */ vli_mod_square_fast(z1, x1, curve_prime, ndigits); /* t3 = B^2 - A */ vli_mod_sub(z1, z1, t5, curve_prime, ndigits); /* t3 = B^2 - 2A = x3 */ vli_mod_sub(z1, z1, t5, curve_prime, ndigits); /* t5 = A - x3 */ vli_mod_sub(t5, t5, z1, curve_prime, ndigits); /* t1 = B * (A - x3) */ vli_mod_mult_fast(x1, x1, t5, curve_prime, ndigits); /* t4 = B * (A - x3) - y1^4 = y3 */ vli_mod_sub(t4, x1, t4, curve_prime, ndigits); vli_set(x1, z1, ndigits); vli_set(z1, y1, ndigits); vli_set(y1, t4, ndigits); } /* Modify (x1, y1) => (x1 * z^2, y1 * z^3) */ static void apply_z(u64 *x1, u64 *y1, u64 *z, u64 *curve_prime, unsigned int ndigits) { u64 t1[ECC_MAX_DIGITS]; vli_mod_square_fast(t1, z, curve_prime, ndigits); /* z^2 */ vli_mod_mult_fast(x1, x1, t1, curve_prime, ndigits); /* x1 * z^2 */ vli_mod_mult_fast(t1, t1, z, curve_prime, ndigits); /* z^3 */ vli_mod_mult_fast(y1, y1, t1, curve_prime, ndigits); /* y1 * z^3 */ } /* P = (x1, y1) => 2P, (x2, y2) => P' */ static void xycz_initial_double(u64 *x1, u64 *y1, u64 *x2, u64 *y2, u64 *p_initial_z, u64 *curve_prime, unsigned int ndigits) { u64 z[ECC_MAX_DIGITS]; vli_set(x2, x1, ndigits); vli_set(y2, y1, ndigits); vli_clear(z, ndigits); z[0] = 1; if (p_initial_z) vli_set(z, p_initial_z, ndigits); apply_z(x1, y1, z, curve_prime, ndigits); ecc_point_double_jacobian(x1, y1, z, curve_prime, ndigits); apply_z(x2, y2, z, curve_prime, ndigits); } /* 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(u64 *x1, u64 *y1, u64 *x2, u64 *y2, u64 *curve_prime, unsigned int ndigits) { /* t1 = X1, t2 = Y1, t3 = X2, t4 = Y2 */ u64 t5[ECC_MAX_DIGITS]; /* t5 = x2 - x1 */ vli_mod_sub(t5, x2, x1, curve_prime, ndigits); /* t5 = (x2 - x1)^2 = A */ vli_mod_square_fast(t5, t5, curve_prime, ndigits); /* t1 = x1*A = B */ vli_mod_mult_fast(x1, x1, t5, curve_prime, ndigits); /* t3 = x2*A = C */ vli_mod_mult_fast(x2, x2, t5, curve_prime, ndigits); /* t4 = y2 - y1 */ vli_mod_sub(y2, y2, y1, curve_prime, ndigits); /* t5 = (y2 - y1)^2 = D */ vli_mod_square_fast(t5, y2, curve_prime, ndigits); /* t5 = D - B */ vli_mod_sub(t5, t5, x1, curve_prime, ndigits); /* t5 = D - B - C = x3 */ vli_mod_sub(t5, t5, x2, curve_prime, ndigits); /* t3 = C - B */ vli_mod_sub(x2, x2, x1, curve_prime, ndigits); /* t2 = y1*(C - B) */ vli_mod_mult_fast(y1, y1, x2, curve_prime, ndigits); /* t3 = B - x3 */ vli_mod_sub(x2, x1, t5, curve_prime, ndigits); /* t4 = (y2 - y1)*(B - x3) */ vli_mod_mult_fast(y2, y2, x2, curve_prime, ndigits); /* t4 = y3 */ vli_mod_sub(y2, y2, y1, curve_prime, ndigits); vli_set(x2, t5, ndigits); } /* 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_add_c(u64 *x1, u64 *y1, u64 *x2, u64 *y2, u64 *curve_prime, unsigned int ndigits) { /* t1 = X1, t2 = Y1, t3 = X2, t4 = Y2 */ u64 t5[ECC_MAX_DIGITS]; u64 t6[ECC_MAX_DIGITS]; u64 t7[ECC_MAX_DIGITS]; /* t5 = x2 - x1 */ vli_mod_sub(t5, x2, x1, curve_prime, ndigits); /* t5 = (x2 - x1)^2 = A */ vli_mod_square_fast(t5, t5, curve_prime, ndigits); /* t1 = x1*A = B */ vli_mod_mult_fast(x1, x1, t5, curve_prime, ndigits); /* t3 = x2*A = C */ vli_mod_mult_fast(x2, x2, t5, curve_prime, ndigits); /* t4 = y2 + y1 */ vli_mod_add(t5, y2, y1, curve_prime, ndigits); /* t4 = y2 - y1 */ vli_mod_sub(y2, y2, y1, curve_prime, ndigits); /* t6 = C - B */ vli_mod_sub(t6, x2, x1, curve_prime, ndigits); /* t2 = y1 * (C - B) */ vli_mod_mult_fast(y1, y1, t6, curve_prime, ndigits); /* t6 = B + C */ vli_mod_add(t6, x1, x2, curve_prime, ndigits); /* t3 = (y2 - y1)^2 */ vli_mod_square_fast(x2, y2, curve_prime, ndigits); /* t3 = x3 */ vli_mod_sub(x2, x2, t6, curve_prime, ndigits); /* t7 = B - x3 */ vli_mod_sub(t7, x1, x2, curve_prime, ndigits); /* t4 = (y2 - y1)*(B - x3) */ vli_mod_mult_fast(y2, y2, t7, curve_prime, ndigits); /* t4 = y3 */ vli_mod_sub(y2, y2, y1, curve_prime, ndigits); /* t7 = (y2 + y1)^2 = F */ vli_mod_square_fast(t7, t5, curve_prime, ndigits); /* t7 = x3' */ vli_mod_sub(t7, t7, t6, curve_prime, ndigits); /* t6 = x3' - B */ vli_mod_sub(t6, t7, x1, curve_prime, ndigits); /* t6 = (y2 + y1)*(x3' - B) */ vli_mod_mult_fast(t6, t6, t5, curve_prime, ndigits); /* t2 = y3' */ vli_mod_sub(y1, t6, y1, curve_prime, ndigits); vli_set(x1, t7, ndigits); } static void ecc_point_mult(struct ecc_point *result, const struct ecc_point *point, const u64 *scalar, u64 *initial_z, const struct ecc_curve *curve, unsigned int ndigits) { /* R0 and R1 */ u64 rx[2][ECC_MAX_DIGITS]; u64 ry[2][ECC_MAX_DIGITS]; u64 z[ECC_MAX_DIGITS]; u64 sk[2][ECC_MAX_DIGITS]; u64 *curve_prime = curve->p; int i, nb; int num_bits; int carry; carry = vli_add(sk[0], scalar, curve->n, ndigits); vli_add(sk[1], sk[0], curve->n, ndigits); scalar = sk[!carry]; num_bits = sizeof(u64) * ndigits * 8 + 1; vli_set(rx[1], point->x, ndigits); vli_set(ry[1], point->y, ndigits); xycz_initial_double(rx[1], ry[1], rx[0], ry[0], initial_z, curve_prime, ndigits); for (i = num_bits - 2; i > 0; i--) { nb = !vli_test_bit(scalar, i); xycz_add_c(rx[1 - nb], ry[1 - nb], rx[nb], ry[nb], curve_prime, ndigits); xycz_add(rx[nb], ry[nb], rx[1 - nb], ry[1 - nb], curve_prime, ndigits); } nb = !vli_test_bit(scalar, 0); xycz_add_c(rx[1 - nb], ry[1 - nb], rx[nb], ry[nb], curve_prime, ndigits); /* Find final 1/Z value. */ /* X1 - X0 */ vli_mod_sub(z, rx[1], rx[0], curve_prime, ndigits); /* Yb * (X1 - X0) */ vli_mod_mult_fast(z, z, ry[1 - nb], curve_prime, ndigits); /* xP * Yb * (X1 - X0) */ vli_mod_mult_fast(z, z, point->x, curve_prime, ndigits); /* 1 / (xP * Yb * (X1 - X0)) */ vli_mod_inv(z, z, curve_prime, point->ndigits); /* yP / (xP * Yb * (X1 - X0)) */ vli_mod_mult_fast(z, z, point->y, curve_prime, ndigits); /* Xb * yP / (xP * Yb * (X1 - X0)) */ vli_mod_mult_fast(z, z, rx[1 - nb], curve_prime, ndigits); /* End 1/Z calculation */ xycz_add(rx[nb], ry[nb], rx[1 - nb], ry[1 - nb], curve_prime, ndigits); apply_z(rx[0], ry[0], z, curve_prime, ndigits); vli_set(result->x, rx[0], ndigits); vli_set(result->y, ry[0], ndigits); } /* Computes R = P + Q mod p */ static void ecc_point_add(const struct ecc_point *result, const struct ecc_point *p, const struct ecc_point *q, const struct ecc_curve *curve) { u64 z[ECC_MAX_DIGITS]; u64 px[ECC_MAX_DIGITS]; u64 py[ECC_MAX_DIGITS]; unsigned int ndigits = curve->g.ndigits; vli_set(result->x, q->x, ndigits); vli_set(result->y, q->y, ndigits); vli_mod_sub(z, result->x, p->x, curve->p, ndigits); vli_set(px, p->x, ndigits); vli_set(py, p->y, ndigits); xycz_add(px, py, result->x, result->y, curve->p, ndigits); vli_mod_inv(z, z, curve->p, ndigits); apply_z(result->x, result->y, z, curve->p, ndigits); } /* Computes R = u1P + u2Q mod p using Shamir's trick. * Based on: Kenneth MacKay's micro-ecc (2014). */ void ecc_point_mult_shamir(const struct ecc_point *result, const u64 *u1, const struct ecc_point *p, const u64 *u2, const struct ecc_point *q, const struct ecc_curve *curve) { u64 z[ECC_MAX_DIGITS]; u64 sump[2][ECC_MAX_DIGITS]; u64 *rx = result->x; u64 *ry = result->y; unsigned int ndigits = curve->g.ndigits; unsigned int num_bits; struct ecc_point sum = ECC_POINT_INIT(sump[0], sump[1], ndigits); const struct ecc_point *points[4]; const struct ecc_point *point; unsigned int idx; int i; ecc_point_add(&sum, p, q, curve); points[0] = NULL; points[1] = p; points[2] = q; points[3] = ∑ num_bits = max(vli_num_bits(u1, ndigits), vli_num_bits(u2, ndigits)); i = num_bits - 1; idx = (!!vli_test_bit(u1, i)) | ((!!vli_test_bit(u2, i)) << 1); point = points[idx]; vli_set(rx, point->x, ndigits); vli_set(ry, point->y, ndigits); vli_clear(z + 1, ndigits - 1); z[0] = 1; for (--i; i >= 0; i--) { ecc_point_double_jacobian(rx, ry, z, curve->p, ndigits); idx = (!!vli_test_bit(u1, i)) | ((!!vli_test_bit(u2, i)) << 1); point = points[idx]; if (point) { u64 tx[ECC_MAX_DIGITS]; u64 ty[ECC_MAX_DIGITS]; u64 tz[ECC_MAX_DIGITS]; vli_set(tx, point->x, ndigits); vli_set(ty, point->y, ndigits); apply_z(tx, ty, z, curve->p, ndigits); vli_mod_sub(tz, rx, tx, curve->p, ndigits); xycz_add(tx, ty, rx, ry, curve->p, ndigits); vli_mod_mult_fast(z, z, tz, curve->p, ndigits); } } vli_mod_inv(z, z, curve->p, ndigits); apply_z(rx, ry, z, curve->p, ndigits); } EXPORT_SYMBOL(ecc_point_mult_shamir); static inline void ecc_swap_digits(const u64 *in, u64 *out, unsigned int ndigits) { int i; for (i = 0; i < ndigits; i++) out[i] = __swab64(in[ndigits - 1 - i]); } static int __ecc_is_key_valid(const struct ecc_curve *curve, const u64 *private_key, unsigned int ndigits) { u64 one[ECC_MAX_DIGITS] = { 1, }; u64 res[ECC_MAX_DIGITS]; if (!private_key) return -EINVAL; if (curve->g.ndigits != ndigits) return -EINVAL; /* Make sure the private key is in the range [2, n-3]. */ if (vli_cmp(one, private_key, ndigits) != -1) return -EINVAL; vli_sub(res, curve->n, one, ndigits); vli_sub(res, res, one, ndigits); if (vli_cmp(res, private_key, ndigits) != 1) return -EINVAL; return 0; } int ecc_is_key_valid(unsigned int curve_id, unsigned int ndigits, const u64 *private_key, unsigned int private_key_len) { int nbytes; const struct ecc_curve *curve = ecc_get_curve(curve_id); nbytes = ndigits << ECC_DIGITS_TO_BYTES_SHIFT; if (private_key_len != nbytes) return -EINVAL; return __ecc_is_key_valid(curve, private_key, ndigits); } EXPORT_SYMBOL(ecc_is_key_valid); /* * ECC private keys are generated using the method of extra random bits, * equivalent to that described in FIPS 186-4, Appendix B.4.1. * * d = (c mod(n–1)) + 1 where c is a string of random bits, 64 bits longer * than requested * 0 <= c mod(n-1) <= n-2 and implies that * 1 <= d <= n-1 * * This method generates a private key uniformly distributed in the range * [1, n-1]. */ int ecc_gen_privkey(unsigned int curve_id, unsigned int ndigits, u64 *privkey) { const struct ecc_curve *curve = ecc_get_curve(curve_id); u64 priv[ECC_MAX_DIGITS]; unsigned int nbytes = ndigits << ECC_DIGITS_TO_BYTES_SHIFT; unsigned int nbits = vli_num_bits(curve->n, ndigits); int err; /* Check that N is included in Table 1 of FIPS 186-4, section 6.1.1 */ if (nbits < 160 || ndigits > ARRAY_SIZE(priv)) return -EINVAL; /* * FIPS 186-4 recommends that the private key should be obtained from a * RBG with a security strength equal to or greater than the security * strength associated with N. * * The maximum security strength identified by NIST SP800-57pt1r4 for * ECC is 256 (N >= 512). * * This condition is met by the default RNG because it selects a favored * DRBG with a security strength of 256. */ if (crypto_get_default_rng()) return -EFAULT; err = crypto_rng_get_bytes(crypto_default_rng, (u8 *)priv, nbytes); crypto_put_default_rng(); if (err) return err; /* Make sure the private key is in the valid range. */ if (__ecc_is_key_valid(curve, priv, ndigits)) return -EINVAL; ecc_swap_digits(priv, privkey, ndigits); return 0; } EXPORT_SYMBOL(ecc_gen_privkey); int ecc_make_pub_key(unsigned int curve_id, unsigned int ndigits, const u64 *private_key, u64 *public_key) { int ret = 0; struct ecc_point *pk; u64 priv[ECC_MAX_DIGITS]; const struct ecc_curve *curve = ecc_get_curve(curve_id); if (!private_key || !curve || ndigits > ARRAY_SIZE(priv)) { ret = -EINVAL; goto out; } ecc_swap_digits(private_key, priv, ndigits); pk = ecc_alloc_point(ndigits); if (!pk) { ret = -ENOMEM; goto out; } ecc_point_mult(pk, &curve->g, priv, NULL, curve, ndigits); if (ecc_point_is_zero(pk)) { ret = -EAGAIN; goto err_free_point; } ecc_swap_digits(pk->x, public_key, ndigits); ecc_swap_digits(pk->y, &public_key[ndigits], ndigits); err_free_point: ecc_free_point(pk); out: return ret; } EXPORT_SYMBOL(ecc_make_pub_key); /* SP800-56A section 5.6.2.3.4 partial verification: ephemeral keys only */ int ecc_is_pubkey_valid_partial(const struct ecc_curve *curve, struct ecc_point *pk) { u64 yy[ECC_MAX_DIGITS], xxx[ECC_MAX_DIGITS], w[ECC_MAX_DIGITS]; if (WARN_ON(pk->ndigits != curve->g.ndigits)) return -EINVAL; /* Check 1: Verify key is not the zero point. */ if (ecc_point_is_zero(pk)) return -EINVAL; /* Check 2: Verify key is in the range [1, p-1]. */ if (vli_cmp(curve->p, pk->x, pk->ndigits) != 1) return -EINVAL; if (vli_cmp(curve->p, pk->y, pk->ndigits) != 1) return -EINVAL; /* Check 3: Verify that y^2 == (x^3 + a·x + b) mod p */ vli_mod_square_fast(yy, pk->y, curve->p, pk->ndigits); /* y^2 */ vli_mod_square_fast(xxx, pk->x, curve->p, pk->ndigits); /* x^2 */ vli_mod_mult_fast(xxx, xxx, pk->x, curve->p, pk->ndigits); /* x^3 */ vli_mod_mult_fast(w, curve->a, pk->x, curve->p, pk->ndigits); /* a·x */ vli_mod_add(w, w, curve->b, curve->p, pk->ndigits); /* a·x + b */ vli_mod_add(w, w, xxx, curve->p, pk->ndigits); /* x^3 + a·x + b */ if (vli_cmp(yy, w, pk->ndigits) != 0) /* Equation */ return -EINVAL; return 0; } EXPORT_SYMBOL(ecc_is_pubkey_valid_partial); int crypto_ecdh_shared_secret(unsigned int curve_id, unsigned int ndigits, const u64 *private_key, const u64 *public_key, u64 *secret) { int ret = 0; struct ecc_point *product, *pk; u64 priv[ECC_MAX_DIGITS]; u64 rand_z[ECC_MAX_DIGITS]; unsigned int nbytes; const struct ecc_curve *curve = ecc_get_curve(curve_id); if (!private_key || !public_key || !curve || ndigits > ARRAY_SIZE(priv) || ndigits > ARRAY_SIZE(rand_z)) { ret = -EINVAL; goto out; } nbytes = ndigits << ECC_DIGITS_TO_BYTES_SHIFT; get_random_bytes(rand_z, nbytes); pk = ecc_alloc_point(ndigits); if (!pk) { ret = -ENOMEM; goto out; } ecc_swap_digits(public_key, pk->x, ndigits); ecc_swap_digits(&public_key[ndigits], pk->y, ndigits); ret = ecc_is_pubkey_valid_partial(curve, pk); if (ret) goto err_alloc_product; ecc_swap_digits(private_key, priv, ndigits); product = ecc_alloc_point(ndigits); if (!product) { ret = -ENOMEM; goto err_alloc_product; } ecc_point_mult(product, pk, priv, rand_z, curve, ndigits); ecc_swap_digits(product->x, secret, ndigits); if (ecc_point_is_zero(product)) ret = -EFAULT; ecc_free_point(product); err_alloc_product: ecc_free_point(pk); out: return ret; } EXPORT_SYMBOL(crypto_ecdh_shared_secret); MODULE_LICENSE("Dual BSD/GPL");