////////////////////////////////////////////////////////////////////////////// // This is EASY-ECC by Kenneth MacKay // https://github.com/esxgx/easy-ecc // This code is under the BSD 2-clause license, not ZeroTier's license ////////////////////////////////////////////////////////////////////////////// #include #include #include #include #include "Constants.hpp" #include "ECC384.hpp" #include "Utils.hpp" namespace ZeroTier { namespace { #define secp384r1 48 #define ECC_CURVE secp384r1 #define ECC_BYTES ECC_CURVE #define NUM_ECC_DIGITS (ECC_BYTES/8) #define MAX_TRIES 1024 typedef unsigned int uint; #if defined(__SIZEOF_INT128__) || ((__clang_major__ * 100 + __clang_minor__) >= 302) #define SUPPORTS_INT128 1 #else #define SUPPORTS_INT128 0 #endif #if SUPPORTS_INT128 typedef unsigned __int128 uint128_t; #else typedef struct { uint64_t m_low; uint64_t m_high; } uint128_t; #endif typedef struct EccPoint { uint64_t x[NUM_ECC_DIGITS]; uint64_t y[NUM_ECC_DIGITS]; } EccPoint; #define CONCAT1(a, b) a##b #define CONCAT(a, b) CONCAT1(a, b) #define Curve_P_48 {0x00000000FFFFFFFF, 0xFFFFFFFF00000000, 0xFFFFFFFFFFFFFFFE, 0xFFFFFFFFFFFFFFFF, 0xFFFFFFFFFFFFFFFF, 0xFFFFFFFFFFFFFFFF} #define Curve_B_48 {0x2A85C8EDD3EC2AEF, 0xC656398D8A2ED19D, 0x0314088F5013875A, 0x181D9C6EFE814112, 0x988E056BE3F82D19, 0xB3312FA7E23EE7E4} #define Curve_G_48 {{0x3A545E3872760AB7, 0x5502F25DBF55296C, 0x59F741E082542A38, 0x6E1D3B628BA79B98, 0x8EB1C71EF320AD74, 0xAA87CA22BE8B0537}, {0x7A431D7C90EA0E5F, 0x0A60B1CE1D7E819D, 0xE9DA3113B5F0B8C0, 0xF8F41DBD289A147C, 0x5D9E98BF9292DC29, 0x3617DE4A96262C6F}} #define Curve_N_48 {0xECEC196ACCC52973, 0x581A0DB248B0A77A, 0xC7634D81F4372DDF, 0xFFFFFFFFFFFFFFFF, 0xFFFFFFFFFFFFFFFF, 0xFFFFFFFFFFFFFFFF} static uint64_t curve_p[NUM_ECC_DIGITS] = CONCAT(Curve_P_, ECC_CURVE); static uint64_t curve_b[NUM_ECC_DIGITS] = CONCAT(Curve_B_, ECC_CURVE); static EccPoint curve_G = CONCAT(Curve_G_, ECC_CURVE); static uint64_t curve_n[NUM_ECC_DIGITS] = CONCAT(Curve_N_, ECC_CURVE); // Use ZeroTier's secure PRNG static ZT_ALWAYS_INLINE int getRandomNumber(uint64_t *p_vli) { Utils::getSecureRandom(p_vli,ECC_BYTES); return 1; } static ZT_ALWAYS_INLINE void vli_clear(uint64_t *p_vli) { uint i; for(i=0; i= 0 && p_vli[i] == 0; --i) { } return (i + 1); } /* Counts the number of bits required for p_vli. */ static ZT_ALWAYS_INLINE uint vli_numBits(uint64_t *p_vli) { uint i; uint64_t l_digit; uint l_numDigits = vli_numDigits(p_vli); if(l_numDigits == 0) { return 0; } l_digit = p_vli[l_numDigits - 1]; for(i=0; l_digit; ++i) { l_digit >>= 1; } return ((l_numDigits - 1) * 64 + i); } /* Sets p_dest = p_src. */ static ZT_ALWAYS_INLINE void vli_set(uint64_t *p_dest, uint64_t *p_src) { uint i; for(i=0; i= 0; --i) { if(p_left[i] > p_right[i]) { return 1; } else if(p_left[i] < p_right[i]) { return -1; } } return 0; } /* Computes p_result = p_in << c, returning carry. Can modify in place (if p_result == p_in). 0 < p_shift < 64. */ static inline uint64_t vli_lshift(uint64_t *p_result, uint64_t *p_in, uint p_shift) { uint64_t l_carry = 0; uint i; for(i = 0; i < NUM_ECC_DIGITS; ++i) { uint64_t l_temp = p_in[i]; p_result[i] = (l_temp << p_shift) | l_carry; l_carry = l_temp >> (64 - p_shift); } return l_carry; } /* Computes p_vli = p_vli >> 1. */ static inline void vli_rshift1(uint64_t *p_vli) { uint64_t *l_end = p_vli; uint64_t l_carry = 0; p_vli += NUM_ECC_DIGITS; while(p_vli-- > l_end) { uint64_t l_temp = *p_vli; *p_vli = (l_temp >> 1) | l_carry; l_carry = l_temp << 63; } } /* Computes p_result = p_left + p_right, returning carry. Can modify in place. */ static inline uint64_t vli_add(uint64_t *p_result, uint64_t *p_left, uint64_t *p_right) { uint64_t l_carry = 0; uint i; for(i=0; i p_left[i]); } p_result[i] = l_diff; } return l_borrow; } #if SUPPORTS_INT128 /* Computes p_result = p_left * p_right. */ static inline void vli_mult(uint64_t *p_result, uint64_t *p_left, uint64_t *p_right) { uint128_t r01 = 0; uint64_t r2 = 0; uint i, k; /* Compute each digit of p_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> 64) | (((uint128_t)r2) << 64); r2 = 0; } p_result[NUM_ECC_DIGITS*2 - 1] = (uint64_t)r01; } /* Computes p_result = p_left^2. */ static inline void vli_square(uint64_t *p_result, uint64_t *p_left) { uint128_t r01 = 0; uint64_t r2 = 0; uint i, k; 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<=k-i; ++i) { uint128_t l_product = (uint128_t)p_left[i] * p_left[k-i]; if(i < k-i) { r2 += l_product >> 127; l_product *= 2; } r01 += l_product; r2 += (r01 < l_product); } p_result[k] = (uint64_t)r01; r01 = (r01 >> 64) | (((uint128_t)r2) << 64); r2 = 0; } p_result[NUM_ECC_DIGITS*2 - 1] = (uint64_t)r01; } #else /* #if SUPPORTS_INT128 */ static inline uint128_t mul_64_64(uint64_t p_left, uint64_t p_right) { uint128_t l_result; uint64_t a0 = p_left & 0xffffffffull; uint64_t a1 = p_left >> 32; uint64_t b0 = p_right & 0xffffffffull; uint64_t b1 = p_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) { // overflow m3 += 0x100000000ull; } l_result.m_low = (m0 & 0xffffffffull) | (m2 << 32); l_result.m_high = m3 + (m2 >> 32); return l_result; } static inline 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 inline void vli_mult(uint64_t *p_result, uint64_t *p_left, uint64_t *p_right) { uint128_t r01 = {0, 0}; uint64_t r2 = 0; uint i, k; /* Compute each digit of p_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> 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); } p_result[k] = r01.m_low; r01.m_low = r01.m_high; r01.m_high = r2; r2 = 0; } p_result[NUM_ECC_DIGITS*2 - 1] = r01.m_low; } #endif /* SUPPORTS_INT128 */ /* Computes p_result = (p_left + p_right) % p_mod. Assumes that p_left < p_mod and p_right < p_mod, p_result != p_mod. */ static ZT_ALWAYS_INLINE void vli_modAdd(uint64_t *p_result, uint64_t *p_left, uint64_t *p_right, uint64_t *p_mod) { uint64_t l_carry = vli_add(p_result, p_left, p_right); if(l_carry || vli_cmp(p_result, p_mod) >= 0) { /* p_result > p_mod (p_result = p_mod + remainder), so subtract p_mod to get remainder. */ vli_sub(p_result, p_result, p_mod); } } /* Computes p_result = (p_left - p_right) % p_mod. Assumes that p_left < p_mod and p_right < p_mod, p_result != p_mod. */ static ZT_ALWAYS_INLINE void vli_modSub(uint64_t *p_result, uint64_t *p_left, uint64_t *p_right, uint64_t *p_mod) { uint64_t l_borrow = vli_sub(p_result, p_left, p_right); if(l_borrow) { /* In this case, p_result == -diff == (max int) - diff. Since -x % d == d - x, we can get the correct result from p_result + p_mod (with overflow). */ vli_add(p_result, p_result, p_mod); } } //#elif ECC_CURVE == secp384r1 static inline void omega_mult(uint64_t *p_result, uint64_t *p_right) { uint64_t l_tmp[NUM_ECC_DIGITS]; uint64_t l_carry, l_diff; /* Multiply by (2^128 + 2^96 - 2^32 + 1). */ vli_set(p_result, p_right); /* 1 */ l_carry = vli_lshift(l_tmp, p_right, 32); p_result[1 + NUM_ECC_DIGITS] = l_carry + vli_add(p_result + 1, p_result + 1, l_tmp); /* 2^96 + 1 */ p_result[2 + NUM_ECC_DIGITS] = vli_add(p_result + 2, p_result + 2, p_right); /* 2^128 + 2^96 + 1 */ l_carry += vli_sub(p_result, p_result, l_tmp); /* 2^128 + 2^96 - 2^32 + 1 */ l_diff = p_result[NUM_ECC_DIGITS] - l_carry; if(l_diff > p_result[NUM_ECC_DIGITS]) { /* Propagate borrow if necessary. */ uint i; for(i = 1 + NUM_ECC_DIGITS; ; ++i) { --p_result[i]; if(p_result[i] != (uint64_t)-1) { break; } } } p_result[NUM_ECC_DIGITS] = l_diff; } /* Computes p_result = p_product % curve_p see PDF "Comparing Elliptic Curve Cryptography and RSA on 8-bit CPUs" section "Curve-Specific Optimizations" */ static inline void vli_mmod_fast(uint64_t *p_result, uint64_t *p_product) { uint64_t l_tmp[2*NUM_ECC_DIGITS]; while(!vli_isZero(p_product + NUM_ECC_DIGITS)) /* While c1 != 0 */ { uint64_t l_carry = 0; uint i; vli_clear(l_tmp); vli_clear(l_tmp + NUM_ECC_DIGITS); omega_mult(l_tmp, p_product + NUM_ECC_DIGITS); /* tmp = w * c1 */ vli_clear(p_product + NUM_ECC_DIGITS); /* p = c0 */ /* (c1, c0) = c0 + w * c1 */ for(i=0; i 0) { vli_sub(p_product, p_product, curve_p); } vli_set(p_result, p_product); } //#endif /* Computes p_result = (p_left * p_right) % curve_p. */ static ZT_ALWAYS_INLINE void vli_modMult_fast(uint64_t *p_result, uint64_t *p_left, uint64_t *p_right) { uint64_t l_product[2 * NUM_ECC_DIGITS]; vli_mult(l_product, p_left, p_right); vli_mmod_fast(p_result, l_product); } /* Computes p_result = p_left^2 % curve_p. */ static ZT_ALWAYS_INLINE void vli_modSquare_fast(uint64_t *p_result, uint64_t *p_left) { uint64_t l_product[2 * NUM_ECC_DIGITS]; vli_square(l_product, p_left); vli_mmod_fast(p_result, l_product); } #define EVEN(vli) (!(vli[0] & 1)) /* Computes p_result = (1 / p_input) % p_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 inline void vli_modInv(uint64_t *p_result, uint64_t *p_input, uint64_t *p_mod) { uint64_t a[NUM_ECC_DIGITS], b[NUM_ECC_DIGITS], u[NUM_ECC_DIGITS], v[NUM_ECC_DIGITS]; uint64_t l_carry; int l_cmpResult; if(vli_isZero(p_input)) { vli_clear(p_result); return; } vli_set(a, p_input); vli_set(b, p_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, p_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, p_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, p_mod); } vli_sub(u, u, v); if(!EVEN(u)) { l_carry = vli_add(u, u, p_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, p_mod); } vli_sub(v, v, u); if(!EVEN(v)) { l_carry = vli_add(v, v, p_mod); } vli_rshift1(v); if(l_carry) { v[NUM_ECC_DIGITS-1] |= 0x8000000000000000ull; } } } vli_set(p_result, u); } /* ------ Point operations ------ */ /* Returns 1 if p_point is the point at infinity, 0 otherwise. */ static ZT_ALWAYS_INLINE int EccPoint_isZero(EccPoint *p_point) { return (vli_isZero(p_point->x) && vli_isZero(p_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 inline void EccPoint_double_jacobian(uint64_t *X1, uint64_t *Y1, uint64_t *Z1) { /* t1 = X, t2 = Y, t3 = Z */ uint64_t t4[NUM_ECC_DIGITS]; uint64_t t5[NUM_ECC_DIGITS]; 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, curve_p); /* t1 = x1 + z1^2 */ vli_modAdd(Z1, Z1, Z1, curve_p); /* t3 = 2*z1^2 */ vli_modSub(Z1, X1, Z1, curve_p); /* t3 = x1 - z1^2 */ vli_modMult_fast(X1, X1, Z1); /* t1 = x1^2 - z1^4 */ vli_modAdd(Z1, X1, X1, curve_p); /* t3 = 2*(x1^2 - z1^4) */ vli_modAdd(X1, X1, Z1, curve_p); /* t1 = 3*(x1^2 - z1^4) */ if(vli_testBit(X1, 0)) { uint64_t l_carry = vli_add(X1, X1, 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 */ vli_modSquare_fast(Z1, X1); /* t3 = B^2 */ vli_modSub(Z1, Z1, t5, curve_p); /* t3 = B^2 - A */ vli_modSub(Z1, Z1, t5, curve_p); /* t3 = B^2 - 2A = x3 */ vli_modSub(t5, t5, Z1, curve_p); /* t5 = A - x3 */ vli_modMult_fast(X1, X1, t5); /* t1 = B * (A - x3) */ vli_modSub(t4, X1, t4, curve_p); /* t4 = B * (A - x3) - y1^4 = y3 */ vli_set(X1, Z1); vli_set(Z1, Y1); vli_set(Y1, t4); } /* Modify (x1, y1) => (x1 * z^2, y1 * z^3) */ static ZT_ALWAYS_INLINE void apply_z(uint64_t *X1, uint64_t *Y1, uint64_t *Z) { uint64_t t1[NUM_ECC_DIGITS]; vli_modSquare_fast(t1, Z); /* z^2 */ vli_modMult_fast(X1, X1, t1); /* x1 * z^2 */ vli_modMult_fast(t1, t1, Z); /* z^3 */ vli_modMult_fast(Y1, Y1, t1); /* y1 * z^3 */ } /* P = (x1, y1) => 2P, (x2, y2) => P' */ static inline void XYcZ_initial_double(uint64_t *X1, uint64_t *Y1, uint64_t *X2, uint64_t *Y2, uint64_t *p_initialZ) { uint64_t z[NUM_ECC_DIGITS]; vli_set(X2, X1); vli_set(Y2, Y1); vli_clear(z); z[0] = 1; if(p_initialZ) { vli_set(z, p_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 inline void XYcZ_add(uint64_t *X1, uint64_t *Y1, uint64_t *X2, uint64_t *Y2) { /* t1 = X1, t2 = Y1, t3 = X2, t4 = Y2 */ uint64_t t5[NUM_ECC_DIGITS]; vli_modSub(t5, X2, X1, curve_p); /* t5 = x2 - x1 */ vli_modSquare_fast(t5, t5); /* t5 = (x2 - x1)^2 = A */ vli_modMult_fast(X1, X1, t5); /* t1 = x1*A = B */ vli_modMult_fast(X2, X2, t5); /* t3 = x2*A = C */ vli_modSub(Y2, Y2, Y1, curve_p); /* t4 = y2 - y1 */ vli_modSquare_fast(t5, Y2); /* t5 = (y2 - y1)^2 = D */ vli_modSub(t5, t5, X1, curve_p); /* t5 = D - B */ vli_modSub(t5, t5, X2, curve_p); /* t5 = D - B - C = x3 */ vli_modSub(X2, X2, X1, curve_p); /* t3 = C - B */ vli_modMult_fast(Y1, Y1, X2); /* t2 = y1*(C - B) */ vli_modSub(X2, X1, t5, curve_p); /* t3 = B - x3 */ vli_modMult_fast(Y2, Y2, X2); /* t4 = (y2 - y1)*(B - x3) */ vli_modSub(Y2, Y2, Y1, curve_p); /* t4 = y3 */ 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 inline void XYcZ_addC(uint64_t *X1, uint64_t *Y1, uint64_t *X2, 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]; vli_modSub(t5, X2, X1, curve_p); /* t5 = x2 - x1 */ vli_modSquare_fast(t5, t5); /* t5 = (x2 - x1)^2 = A */ vli_modMult_fast(X1, X1, t5); /* t1 = x1*A = B */ vli_modMult_fast(X2, X2, t5); /* t3 = x2*A = C */ vli_modAdd(t5, Y2, Y1, curve_p); /* t4 = y2 + y1 */ vli_modSub(Y2, Y2, Y1, curve_p); /* t4 = y2 - y1 */ vli_modSub(t6, X2, X1, curve_p); /* t6 = C - B */ vli_modMult_fast(Y1, Y1, t6); /* t2 = y1 * (C - B) */ vli_modAdd(t6, X1, X2, curve_p); /* t6 = B + C */ vli_modSquare_fast(X2, Y2); /* t3 = (y2 - y1)^2 */ vli_modSub(X2, X2, t6, curve_p); /* t3 = x3 */ vli_modSub(t7, X1, X2, curve_p); /* t7 = B - x3 */ vli_modMult_fast(Y2, Y2, t7); /* t4 = (y2 - y1)*(B - x3) */ vli_modSub(Y2, Y2, Y1, curve_p); /* t4 = y3 */ vli_modSquare_fast(t7, t5); /* t7 = (y2 + y1)^2 = F */ vli_modSub(t7, t7, t6, curve_p); /* t7 = x3' */ vli_modSub(t6, t7, X1, curve_p); /* t6 = x3' - B */ vli_modMult_fast(t6, t6, t5); /* t6 = (y2 + y1)*(x3' - B) */ vli_modSub(Y1, t6, Y1, curve_p); /* t2 = y3' */ vli_set(X1, t7); } static inline void EccPoint_mult(EccPoint *p_result, EccPoint *p_point, uint64_t *p_scalar, uint64_t *p_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 i, nb; vli_set(Rx[1], p_point->x); vli_set(Ry[1], p_point->y); XYcZ_initial_double(Rx[1], Ry[1], Rx[0], Ry[0], p_initialZ); for(i = vli_numBits(p_scalar) - 2; i > 0; --i) { nb = !vli_testBit(p_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(p_scalar, 0); XYcZ_addC(Rx[1-nb], Ry[1-nb], Rx[nb], Ry[nb]); /* Find final 1/Z value. */ vli_modSub(z, Rx[1], Rx[0], curve_p); /* X1 - X0 */ vli_modMult_fast(z, z, Ry[1-nb]); /* Yb * (X1 - X0) */ vli_modMult_fast(z, z, p_point->x); /* xP * Yb * (X1 - X0) */ vli_modInv(z, z, curve_p); /* 1 / (xP * Yb * (X1 - X0)) */ vli_modMult_fast(z, z, p_point->y); /* yP / (xP * Yb * (X1 - X0)) */ vli_modMult_fast(z, z, Rx[1-nb]); /* Xb * yP / (xP * Yb * (X1 - X0)) */ /* 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(p_result->x, Rx[0]); vli_set(p_result->y, Ry[0]); } static ZT_ALWAYS_INLINE void ecc_bytes2native(uint64_t p_native[NUM_ECC_DIGITS], const uint8_t p_bytes[ECC_BYTES]) { unsigned i; for(i=0; i> 56; p_digit[1] = p_native[i] >> 48; p_digit[2] = p_native[i] >> 40; p_digit[3] = p_native[i] >> 32; p_digit[4] = p_native[i] >> 24; p_digit[5] = p_native[i] >> 16; p_digit[6] = p_native[i] >> 8; p_digit[7] = p_native[i]; } } /* Compute a = sqrt(a) (mod curve_p). */ static inline void mod_sqrt(uint64_t a[NUM_ECC_DIGITS]) { unsigned i; uint64_t p1[NUM_ECC_DIGITS] = {1}; uint64_t l_result[NUM_ECC_DIGITS] = {1}; /* Since curve_p == 3 (mod 4) for all supported curves, we can compute sqrt(a) = a^((curve_p + 1) / 4) (mod curve_p). */ vli_add(p1, curve_p, p1); /* p1 = 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 inline void ecc_point_decompress(EccPoint *p_point, const uint8_t p_compressed[ECC_BYTES+1]) { uint64_t _3[NUM_ECC_DIGITS] = {3}; /* -a = 3 */ ecc_bytes2native(p_point->x, p_compressed+1); vli_modSquare_fast(p_point->y, p_point->x); /* y = x^2 */ vli_modSub(p_point->y, p_point->y, _3, curve_p); /* y = x^2 - 3 */ vli_modMult_fast(p_point->y, p_point->y, p_point->x); /* y = x^3 - 3x */ vli_modAdd(p_point->y, p_point->y, curve_b, curve_p); /* y = x^3 - 3x + b */ mod_sqrt(p_point->y); if((p_point->y[0] & 0x01) != (p_compressed[0] & 0x01)) { vli_sub(p_point->y, curve_p, p_point->y); } } static inline int ecc_make_key(uint8_t p_publicKey[ECC_BYTES+1], uint8_t p_privateKey[ECC_BYTES]) { uint64_t l_private[NUM_ECC_DIGITS]; EccPoint l_public; unsigned l_tries = 0; do { if(!getRandomNumber(l_private) || (l_tries++ >= MAX_TRIES)) { return 0; } 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(curve_n, l_private) != 1) { vli_sub(l_private, l_private, curve_n); } EccPoint_mult(&l_public, &curve_G, l_private, NULL); } while(EccPoint_isZero(&l_public)); ecc_native2bytes(p_privateKey, l_private); ecc_native2bytes(p_publicKey + 1, l_public.x); p_publicKey[0] = 2 + (l_public.y[0] & 0x01); return 1; } static inline int ecdh_shared_secret(const uint8_t p_publicKey[ECC_BYTES+1], const uint8_t p_privateKey[ECC_BYTES], uint8_t p_secret[ECC_BYTES]) { EccPoint l_public; uint64_t l_private[NUM_ECC_DIGITS]; uint64_t l_random[NUM_ECC_DIGITS]; if(!getRandomNumber(l_random)) { return 0; } ecc_point_decompress(&l_public, p_publicKey); ecc_bytes2native(l_private, p_privateKey); EccPoint l_product; EccPoint_mult(&l_product, &l_public, l_private, l_random); ecc_native2bytes(p_secret, l_product.x); return !EccPoint_isZero(&l_product); } /* -------- ECDSA code -------- */ /* Computes p_result = (p_left * p_right) % p_mod. */ static inline void vli_modMult(uint64_t *p_result, uint64_t *p_left, uint64_t *p_right, uint64_t *p_mod) { uint64_t l_product[2 * NUM_ECC_DIGITS]; uint64_t l_modMultiple[2 * NUM_ECC_DIGITS]; uint l_digitShift, l_bitShift; uint l_productBits; uint l_modBits = vli_numBits(p_mod); vli_mult(l_product, p_left, p_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 < p_mod. */ vli_set(p_result, l_product); return; } /* Shift p_mod by (l_leftBits - l_modBits). This multiplies p_mod by the largest power of two possible while still resulting in a number less than p_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, p_mod, l_bitShift); } else { vli_set(l_modMultiple + l_digitShift, p_mod); } /* Subtract all multiples of p_mod to get the remainder. */ vli_clear(p_result); p_result[0] = 1; /* Use p_result as a temp var to store 1 (for subtraction) */ while(l_productBits > NUM_ECC_DIGITS * 64 || vli_cmp(l_modMultiple, p_mod) >= 0) { int 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, p_result); } vli_sub(l_product + NUM_ECC_DIGITS, l_product + NUM_ECC_DIGITS, l_modMultiple + NUM_ECC_DIGITS); } uint64_t 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(p_result, l_product); } static ZT_ALWAYS_INLINE uint umax(uint a, uint b) { return (a > b ? a : b); } static inline int ecdsa_sign(const uint8_t p_privateKey[ECC_BYTES], const uint8_t p_hash[ECC_BYTES], uint8_t p_signature[ECC_BYTES*2]) { uint64_t k[NUM_ECC_DIGITS]; uint64_t l_tmp[NUM_ECC_DIGITS]; uint64_t l_s[NUM_ECC_DIGITS]; EccPoint p; unsigned l_tries = 0; do { if(!getRandomNumber(k) || (l_tries++ >= MAX_TRIES)) { return 0; } if(vli_isZero(k)) { continue; } if(vli_cmp(curve_n, k) != 1) { vli_sub(k, k, curve_n); } /* tmp = k * G */ EccPoint_mult(&p, &curve_G, k, NULL); /* r = x1 (mod n) */ if(vli_cmp(curve_n, p.x) != 1) { vli_sub(p.x, p.x, curve_n); } } while(vli_isZero(p.x)); ecc_native2bytes(p_signature, p.x); ecc_bytes2native(l_tmp, p_privateKey); vli_modMult(l_s, p.x, l_tmp, curve_n); /* s = r*d */ ecc_bytes2native(l_tmp, p_hash); vli_modAdd(l_s, l_tmp, l_s, curve_n); /* s = e + r*d */ vli_modInv(k, k, curve_n); /* k = 1 / k */ vli_modMult(l_s, l_s, k, curve_n); /* s = (e + r*d) / k */ ecc_native2bytes(p_signature + ECC_BYTES, l_s); return 1; } static inline int ecdsa_verify(const uint8_t p_publicKey[ECC_BYTES+1], const uint8_t p_hash[ECC_BYTES], const uint8_t p_signature[ECC_BYTES*2]) { uint64_t u1[NUM_ECC_DIGITS], u2[NUM_ECC_DIGITS]; uint64_t z[NUM_ECC_DIGITS]; EccPoint l_public, l_sum; 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], l_s[NUM_ECC_DIGITS]; ecc_point_decompress(&l_public, p_publicKey); ecc_bytes2native(l_r, p_signature); ecc_bytes2native(l_s, p_signature + ECC_BYTES); if(vli_isZero(l_r) || vli_isZero(l_s)) { /* r, s must not be 0. */ return 0; } if(vli_cmp(curve_n, l_r) != 1 || vli_cmp(curve_n, l_s) != 1) { /* r, s must be < n. */ return 0; } /* Calculate u1 and u2. */ vli_modInv(z, l_s, curve_n); /* Z = s^-1 */ ecc_bytes2native(u1, p_hash); vli_modMult(u1, u1, z, curve_n); /* u1 = e/s */ vli_modMult(u2, l_r, z, curve_n); /* u2 = r/s */ /* Calculate l_sum = G + Q. */ vli_set(l_sum.x, l_public.x); vli_set(l_sum.y, l_public.y); vli_set(tx, curve_G.x); vli_set(ty, curve_G.y); vli_modSub(z, l_sum.x, tx, curve_p); /* Z = x2 - x1 */ XYcZ_add(tx, ty, l_sum.x, l_sum.y); vli_modInv(z, z, curve_p); /* Z = 1/Z */ apply_z(l_sum.x, l_sum.y, z); /* Use Shamir's trick to calculate u1*G + u2*Q */ EccPoint *l_points[4] = {NULL, &curve_G, &l_public, &l_sum}; uint l_numBits = umax(vli_numBits(u1), vli_numBits(u2)); EccPoint *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; int i; for(i = l_numBits - 2; i >= 0; --i) { EccPoint_double_jacobian(rx, ry, z); int l_index = (!!vli_testBit(u1, i)) | ((!!vli_testBit(u2, i)) << 1); EccPoint *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, curve_p); /* Z = x2 - x1 */ XYcZ_add(tx, ty, rx, ry); vli_modMult_fast(z, z, tz); } } vli_modInv(z, z, curve_p); /* Z = 1/Z */ apply_z(rx, ry, z); /* v = x1 (mod n) */ if(vli_cmp(curve_n, rx) != 1) { vli_sub(rx, rx, curve_n); } /* Accept only if v == r. */ return (vli_cmp(rx, l_r) == 0); } ////////////////////////////////////////////////////////////////////////////// ////////////////////////////////////////////////////////////////////////////// ////////////////////////////////////////////////////////////////////////////// } // anonymous namespace void ECC384GenerateKey(uint8_t pub[ZT_ECC384_PUBLIC_KEY_SIZE],uint8_t priv[ZT_ECC384_PRIVATE_KEY_SIZE]) { if (!ecc_make_key(pub,priv)) { fprintf(stderr,"FATAL: ecdsa_make_key() failed!" ZT_EOL_S); abort(); } } void ECC384ECDSASign(const uint8_t priv[ZT_ECC384_PRIVATE_KEY_SIZE],const uint8_t hash[ZT_ECC384_SIGNATURE_HASH_SIZE],uint8_t sig[ZT_ECC384_SIGNATURE_SIZE]) { if (!ecdsa_sign(priv,hash,sig)) { fprintf(stderr,"FATAL: ecdsa_sign() failed!" ZT_EOL_S); abort(); } } bool ECC384ECDSAVerify(const uint8_t pub[ZT_ECC384_PUBLIC_KEY_SIZE],const uint8_t hash[ZT_ECC384_SIGNATURE_HASH_SIZE],const uint8_t sig[ZT_ECC384_SIGNATURE_SIZE]) { return (ecdsa_verify(pub,hash,sig) != 0); } bool ECC384ECDH(const uint8_t theirPub[ZT_ECC384_PUBLIC_KEY_SIZE],const uint8_t ourPriv[ZT_ECC384_PRIVATE_KEY_SIZE],uint8_t secret[ZT_ECC384_SHARED_SECRET_SIZE]) { return (ecdh_shared_secret(theirPub,ourPriv,secret) != 0); } } // namespace ZeroTier