diff --git a/ecc.c b/ecc.c new file mode 100644 index 0000000..ead3c24 --- /dev/null +++ b/ecc.c @@ -0,0 +1,831 @@ +#include "ecc.h" + +typedef unsigned int uint; + +typedef struct EccPoint +{ + uint8_t x[ECC_BYTES]; + uint8_t y[ECC_BYTES]; +} EccPoint; + +#define MAX_TRIES 16 + +#define Curve_P_1 {0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, \ + 0xFF, 0xFF, 0xFF, 0xFF, 0xFD, 0xFF, 0xFF, 0xFF} +#define Curve_P_2 {0xFF, 0xFF, 0xFF, 0x7F, 0xFF, 0xFF, 0xFF, 0xFF, \ + 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, \ + 0xFF, 0xFF, 0xFF, 0xFF} +#define Curve_P_3 {0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, \ + 0xFE, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, \ + 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF} +#define Curve_P_4 {0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, \ + 0xFF, 0xFF, 0xFF, 0xFF, 0x00, 0x00, 0x00, 0x00, \ + 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, 0x00, \ + 0x01, 0x00, 0x00, 0x00, 0xFF, 0xFF, 0xFF, 0xFF} + +#define Curve_B_1 {0xD3, 0x5E, 0xEE, 0x2C, 0x3C, 0x99, 0x24, 0xD8, \ + 0x3D, 0xF4, 0x79, 0x10, 0xC1, 0x79, 0x75, 0xE8} +#define Curve_B_2 {0x45, 0xFA, 0x65, 0xC5, 0xAD, 0xD4, 0xD4, 0x81, \ + 0x9F, 0xF8, 0xAC, 0x65, 0x8B, 0x7A, 0xBD, 0x54, \ + 0xFC, 0xBE, 0x97, 0x1C} +#define Curve_B_3 {0xB1, 0xB9, 0x46, 0xC1, 0xEC, 0xDE, 0xB8, 0xFE, \ + 0x49, 0x30, 0x24, 0x72, 0xAB, 0xE9, 0xA7, 0x0F, \ + 0xE7, 0x80, 0x9C, 0xE5, 0x19, 0x05, 0x21, 0x64} +#define Curve_B_4 {0x4B, 0x60, 0xD2, 0x27, 0x3E, 0x3C, 0xCE, 0x3B, \ + 0xF6, 0xB0, 0x53, 0xCC, 0xB0, 0x06, 0x1D, 0x65, \ + 0xBC, 0x86, 0x98, 0x76, 0x55, 0xBD, 0xEB, 0xB3, \ + 0xE7, 0x93, 0x3A, 0xAA, 0xD8, 0x35, 0xC6, 0x5A} + +#define Curve_G_1 { \ + {0x86, 0x5B, 0x2C, 0xA5, 0x7C, 0x60, 0x28, 0x0C, \ + 0x2D, 0x9B, 0x89, 0x8B, 0x52, 0xF7, 0x1F, 0x16}, \ + {0x83, 0x7A, 0xED, 0xDD, 0x92, 0xA2, 0x2D, 0xC0, \ + 0x13, 0xEB, 0xAF, 0x5B, 0x39, 0xC8, 0x5A, 0xCF}} + +#define Curve_G_2 { \ + {0x82, 0xFC, 0xCB, 0x13, 0xB9, 0x8B, 0xC3, 0x68, \ + 0x89, 0x69, 0x64, 0x46, 0x28, 0x73, 0xF5, 0x8E, \ + 0x68, 0xB5, 0x96, 0x4A}, \ + {0x32, 0xFB, 0xC5, 0x7A, 0x37, 0x51, 0x23, 0x04, \ + 0x12, 0xC9, 0xDC, 0x59, 0x7D, 0x94, 0x68, 0x31, \ + 0x55, 0x28, 0xA6, 0x23}} + +#define Curve_G_3 { \ + {0x12, 0x10, 0xFF, 0x82, 0xFD, 0x0A, 0xFF, 0xF4, \ + 0x00, 0x88, 0xA1, 0x43, 0xEB, 0x20, 0xBF, 0x7C, \ + 0xF6, 0x90, 0x30, 0xB0, 0x0E, 0xA8, 0x8D, 0x18}, \ + {0x11, 0x48, 0x79, 0x1E, 0xA1, 0x77, 0xF9, 0x73, \ + 0xD5, 0xCD, 0x24, 0x6B, 0xED, 0x11, 0x10, 0x63, \ + 0x78, 0xDA, 0xC8, 0xFF, 0x95, 0x2B, 0x19, 0x07}} + +#define Curve_G_4 { \ + {0x96, 0xC2, 0x98, 0xD8, 0x45, 0x39, 0xA1, 0xF4, \ + 0xA0, 0x33, 0xEB, 0x2D, 0x81, 0x7D, 0x03, 0x77, \ + 0xF2, 0x40, 0xA4, 0x63, 0xE5, 0xE6, 0xBC, 0xF8, \ + 0x47, 0x42, 0x2C, 0xE1, 0xF2, 0xD1, 0x17, 0x6B}, \ + {0xF5, 0x51, 0xBF, 0x37, 0x68, 0x40, 0xB6, 0xCB, \ + 0xCE, 0x5E, 0x31, 0x6B, 0x57, 0x33, 0xCE, 0x2B, \ + 0x16, 0x9E, 0x0F, 0x7C, 0x4A, 0xEB, 0xE7, 0x8E, \ + 0x9B, 0x7F, 0x1A, 0xFE, 0xE2, 0x42, 0xE3, 0x4F}} + +#define Curve_N_1 {0x15, 0xA1, 0x38, 0x90, 0x1B, 0x0D, 0xA3, 0x75, \ + 0x00, 0x00, 0x00, 0x00, 0xFE, 0xFF, 0xFF, 0xFF} +#define Curve_N_2 {0x57, 0x22, 0x75, 0xCA, 0xD3, 0xAE, 0x27, 0xF9, \ + 0xC8, 0xF4, 0x01, 0x00, 0x00, 0x00, 0x00, 0x00, \ + 0x00, 0x00, 0x00, 0x00} /* 01 */ +#define Curve_N_3 {0x31, 0x28, 0xD2, 0xB4, 0xB1, 0xC9, 0x6B, 0x14, \ + 0x36, 0xF8, 0xDE, 0x99, 0xFF, 0xFF, 0xFF, 0xFF, \ + 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF} +#define Curve_N_4 {0x51, 0x25, 0x63, 0xFC, 0xC2, 0xCA, 0xB9, 0xF3, \ + 0x84, 0x9E, 0x17, 0xA7, 0xAD, 0xFA, 0xE6, 0xBC, \ + 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, 0xFF, \ + 0x00, 0x00, 0x00, 0x00, 0xFF, 0xFF, 0xFF, 0xFF} + +static uint8_t curve_p[ECC_BYTES] = ECC_CONCAT(Curve_P_, ECC_CURVE); +static uint8_t curve_b[ECC_BYTES] = ECC_CONCAT(Curve_B_, ECC_CURVE); +static EccPoint curve_G = ECC_CONCAT(Curve_G_, ECC_CURVE); +static uint8_t curve_n[ECC_BYTES] = ECC_CONCAT(Curve_N_, ECC_CURVE); + +static int fake_RNG(uint8_t *p_dest, unsigned p_size) +{ + return 0; +} + +static RNG_Function g_rng = &fake_RNG; + +void ecc_set_rng(RNG_Function p_rng) +{ + g_rng = p_rng; +} + +static void vli_clear(uint8_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 uint vli_numBits(const uint8_t *p_vli) +{ + uint i; + uint8_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) * 8 + i); +} + +/* Sets p_dest = p_src. */ +static void vli_set(uint8_t *p_dest, const uint8_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 < 8. */ +static uint8_t vli_lshift(uint8_t *p_result, uint8_t *p_in, uint8_t p_shift) +{ + uint64_t l_carry = 0; + uint i; + for(i = 0; i < ECC_BYTES; ++i) + { + uint8_t l_temp = p_in[i]; + p_result[i] = (l_temp << p_shift) | l_carry; + l_carry = l_temp >> (8 - p_shift); + } + + return l_carry; +} + +/* Computes p_vli = p_vli >> 1. */ +static void vli_rshift1(uint8_t *p_vli) +{ + uint8_t *l_end = p_vli; + uint8_t l_carry = 0; + + p_vli += ECC_BYTES; + while(p_vli-- > l_end) + { + uint8_t l_temp = *p_vli; + *p_vli = (l_temp >> 1) | l_carry; + l_carry = l_temp << 7; + } +} + +/* Computes p_result = p_left + p_right, returning carry. Can modify in place. */ +static uint8_t vli_add(uint8_t *p_result, uint8_t *p_left, uint8_t *p_right) +{ + uint8_t l_carry = 0; + uint i; + for(i=0; i> 8; + } + return l_carry; +} + +/* Computes p_result = p_left - p_right, returning borrow. Can modify in place. */ +static uint8_t vli_sub(uint8_t *p_result, uint8_t *p_left, uint8_t *p_right) +{ + uint8_t l_borrow = 0; + uint i; + for(i=0; i> 8) & 0x01; + } + return l_borrow; +} + +static void vli_mult(uint8_t *p_result, uint8_t *p_left, uint8_t *p_right) +{ + uint16_t r01 = 0; + uint8_t r2 = 0; + + uint8_t i, k; + + /* Compute each digit of p_result in sequence, maintaining the carries. */ + for(k=0; k < ECC_BYTES*2 - 1; ++k) + { + uint8_t l_min = (k < ECC_BYTES ? 0 : (k + 1) - ECC_BYTES); + for(i=l_min; i<=k && i> 8) | (((uint16_t)r2) << 8); + r2 = 0; + } + + p_result[ECC_BYTES*2 - 1] = (uint8_t)r01; +} + +#if ECC_SQUARE_FUNC + +static void vli_square(uint8_t *p_result, uint8_t *p_left) +{ + uint16_t r01 = 0; + uint8_t r2 = 0; + + uint8_t i, k; + for(k=0; k < ECC_BYTES*2 - 1; ++k) + { + uint8_t l_min = (k < ECC_BYTES ? 0 : (k + 1) - ECC_BYTES); + for(i=l_min; i<=k && i<=k-i; ++i) + { + uint16_t l_product = (uint16_t)p_left[i] * p_left[k-i]; + if(i < k-i) + { + r2 += l_product >> 15; + l_product *= 2; + } + r01 += l_product; + r2 += (r01 < l_product); + } + p_result[k] = (uint8_t)r01; + r01 = (r01 >> 8) | (((uint16_t)r2) << 8); + r2 = 0; + } + + p_result[ECC_BYTES*2 - 1] = (uint8_t)r01; +} + +#else /* ECC_SQUARE_FUNC */ + +#define vli_square(result, left, size) vli_mult((result), (left), (left), (size)) + +#endif /* ECC_SQUARE_FUNC */ + + +/* 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 void vli_modAdd(uint8_t *p_result, uint8_t *p_left, uint8_t *p_right, uint8_t *p_mod) +{ + uint8_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 void vli_modSub(uint8_t *p_result, uint8_t *p_left, uint8_t *p_right, uint8_t *p_mod) +{ + uint8_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); + } +} + +#if ECC_CURVE == secp128r1 + +/* 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(uint8_t *p_result, uint8_t *p_product) +{ + /* TODO */ +} + +#elif ECC_CURVE == secp160r1 + +static void omega_mult(uint8_t * restrict p_result, uint8_t * restrict p_right) +{ + uint8_t l_carry; + uint8_t i; + + /* Multiply by (2^31 + 1). */ + vli_set(p_result + 4, p_right); /* 2^32 */ + vli_rshift1(p_result + 4); /* 2^31 */ + p_result[3] = p_right[0] << 7; /* get last bit from shift */ + + l_carry = vli_add(p_result, p_result, p_right); /* 2^31 + 1 */ + for(i = ECC_BYTES; l_carry; ++i) + { + uint16_t l_sum = (uint16_t)p_result[i] + l_carry; + p_result[i] = (uint8_t)l_sum; + l_carry = l_sum >> 8; + } +} + +/* Computes p_result = p_product % curve_p + see PDF "Comparing Elliptic Curve Cryptography and RSA on 8-bit CPUs" + section "Curve-Specific Optimizations" */ +static void vli_mmod_fast(uint8_t * restrict p_result, uint8_t * restrict p_product) +{ + uint8_t l_tmp[2*ECC_BYTES]; + + while(!vli_isZero(p_product + ECC_BYTES)) /* While c1 != 0 */ + { + uint8_t l_carry = 0; + uint8_t i; + + vli_clear(l_tmp); + vli_clear(l_tmp + ECC_BYTES); + omega_mult(l_tmp, p_product + ECC_BYTES); /* tmp = w * c1 */ + vli_clear(p_product + ECC_BYTES); /* p = c0 */ + + /* (c1, c0) = c0 + w * c1 */ + for(i=0; i> 8; + } + p_product[ECC_BYTES+4] = l_carry; + } + + while(vli_cmp(p_product, curve_p) > 0) + { + vli_sub(p_product, p_product, curve_p); + } + vli_set(p_result, p_product); +} + +#elif ECC_CURVE == secp192r1 + +/* 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(uint64_t *p_result, uint64_t *p_product) +{ + /* TODO */ +} + +#elif ECC_CURVE == secp256r1 + +/* Computes p_result = p_product % curve_p + from http://www.nsa.gov/ia/_files/nist-routines.pdf */ +static void vli_mmod_fast(uint64_t *p_result, uint64_t *p_product) +{ + /* TODO */ +} + +#endif + + +/* Computes p_result = (p_left * p_right) % curve_p. */ +static void vli_modMult_fast(uint8_t *p_result, uint8_t *p_left, uint8_t *p_right) +{ + uint8_t l_product[2 * ECC_BYTES]; + vli_mult(l_product, p_left, p_right); + vli_mmod_fast(p_result, l_product); +} + +#if ECC_SQUARE_FUNC + +/* Computes p_result = p_left^2 % curve_p. */ +static void vli_modSquare_fast(uint8_t *p_result, uint8_t *p_left) +{ + uint8_t l_product[2 * ECC_BYTES]; + vli_square(l_product, p_left); + vli_mmod_fast(p_result, l_product); +} + +#else /* ECC_SQUARE_FUNC */ + +#define vli_modSquare_fast(result, left) vli_modMult_fast((result), (left), (left)) + +#endif /* ECC_SQUARE_FUNC */ + + +#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 void vli_modInv(uint8_t *p_result, uint8_t *p_input, uint8_t *p_mod) +{ + uint8_t a[ECC_BYTES], b[ECC_BYTES], u[ECC_BYTES], v[ECC_BYTES]; + uint8_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[ECC_BYTES-1] |= 0x80; + } + } + 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[ECC_BYTES-1] |= 0x80; + } + } + 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[ECC_BYTES-1] |= 0x80; + } + } + 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[ECC_BYTES-1] |= 0x80; + } + } + } + + vli_set(p_result, u); +} + +/* ------ Point operations ------ */ + +/* Returns 1 if p_point is the point at infinity, 0 otherwise. */ +static 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 void EccPoint_double_jacobian(uint8_t * restrict X1, uint8_t * restrict Y1, uint8_t * restrict Z1) +{ + /* t1 = X, t2 = Y, t3 = Z */ + uint8_t t4[ECC_BYTES]; + uint8_t t5[ECC_BYTES]; + + 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)) + { + uint8_t l_carry = vli_add(X1, X1, curve_p); + vli_rshift1(X1); + X1[ECC_BYTES-1] |= l_carry << 7; + } + 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 void apply_z(uint8_t * restrict X1, uint8_t * restrict Y1, uint8_t * restrict Z) +{ + uint8_t t1[ECC_BYTES]; + + 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 void XYcZ_initial_double(uint8_t * restrict X1, uint8_t * restrict Y1, + uint8_t * restrict X2, uint8_t * restrict Y2, const uint8_t * restrict p_initialZ) +{ + uint8_t z[ECC_BYTES]; + + 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 void XYcZ_add(uint8_t * restrict X1, uint8_t * restrict Y1, uint8_t * restrict X2, uint8_t * restrict Y2) +{ + /* t1 = X1, t2 = Y1, t3 = X2, t4 = Y2 */ + uint8_t t5[ECC_BYTES]; + + 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 void XYcZ_addC(uint8_t * restrict X1, uint8_t * restrict Y1, uint8_t * restrict X2, uint8_t * restrict Y2) +{ + /* t1 = X1, t2 = Y1, t3 = X2, t4 = Y2 */ + uint8_t t5[ECC_BYTES]; + uint8_t t6[ECC_BYTES]; + uint8_t t7[ECC_BYTES]; + + 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 void EccPoint_mult(EccPoint * restrict p_result, EccPoint * restrict p_point, + const uint8_t * restrict p_scalar, const uint8_t * restrict p_initialZ) +{ + /* R0 and R1 */ + uint8_t Rx[2][ECC_BYTES]; + uint8_t Ry[2][ECC_BYTES]; + uint8_t z[ECC_BYTES]; + + int i; + uint8_t 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]); +} + +/* Compute a = sqrt(a) (mod curve_p). */ +static void mod_sqrt(uint8_t a[ECC_BYTES]) +{ + uint i; + uint8_t p1[ECC_BYTES] = {1}; + uint8_t l_result[ECC_BYTES] = {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 void ecc_point_decompress(EccPoint *p_point, const uint8_t p_compressed[ECC_BYTES+1]) +{ + uint8_t _3[ECC_BYTES] = {3}; /* -a = 3 */ + vli_set(p_point->x, p_compressed); + + 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[ECC_BYTES] & 0x01)) + { + vli_sub(p_point->y, curve_p, p_point->y); + } +} + +int ecc_make_key(uint8_t p_publicKey[ECC_BYTES+1], uint8_t p_privateKey[ECC_BYTES]) +{ + EccPoint l_public; + uint l_tries = 0; + + do + { + if(!g_rng(p_privateKey, ECC_BYTES) || (l_tries++ >= MAX_TRIES)) + { + return 0; + } + if(vli_isZero(p_privateKey)) + { + 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 ECC_CURVE != secp160r1 + if(vli_cmp(curve_n, p_privateKey) != 1) + { + vli_sub(p_privateKey, p_privateKey, curve_n); + } + #endif + + EccPoint_mult(&l_public, &curve_G, p_privateKey, 0); + } while(EccPoint_isZero(&l_public)); + + vli_set(p_publicKey, l_public.x); + p_publicKey[ECC_BYTES] = 2 + (l_public.y[0] & 0x01); + return 1; +} + +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; + uint8_t l_random[ECC_BYTES]; + + if(!g_rng(l_random, ECC_BYTES)) + { + return 0; + } + + ecc_point_decompress(&l_public, p_publicKey); + + EccPoint l_product; + EccPoint_mult(&l_product, &l_public, p_privateKey, l_random); + + vli_set(p_secret, l_product.x); + + return !EccPoint_isZero(&l_product); +} diff --git a/ecc.h b/ecc.h new file mode 100644 index 0000000..fd70564 --- /dev/null +++ b/ecc.h @@ -0,0 +1,93 @@ +#ifndef _AVR_ECC_H_ +#define _AVR_ECC_H_ + +#include + +/* Optimization settings. Define as 1 to enable an optimization, 0 to disable it. +ECC_SQUARE_FUNC - If enabled, this will cause a specific function to be used for (scalar) squaring instead of the generic + multiplication function. +*/ +#define ECC_SQUARE_FUNC 1 + +/* Inline assembly options. +Inline assembly (gcc format) is provided for selected operations for AVR (requires MUL support). +*/ +#define ecc_asm_none 0 +#define ecc_asm_avr 1 +#ifndef ECC_ASM + #define ECC_ASM ecc_asm_none +#endif + +#define ECC_CONCAT1(a, b) a##b +#define ECC_CONCAT(a, b) ECC_CONCAT1(a, b) + +/* Curve selection options. */ +#define secp128r1 1 +#define secp160r1 2 +#define secp192r1 3 +#define secp256r1 4 + +#ifndef ECC_CURVE + #define ECC_CURVE secp160r1 +#endif + +#define ecc_size_1 16 +#define ecc_size_2 20 +#define ecc_size_3 24 +#define ecc_size_4 32 + +#define ECC_BYTES ECC_CONCAT(ecc_size_, ECC_CURVE) + +#ifdef __cplusplus +extern "C" +{ +#endif + +/* RNG_Function type +The RNG function should fill p_size random bytes into p_dest. It should return 1 if +p_dest was filled with random data, or 0 if the random data could not be generated. +*/ +typedef int (*RNG_Function)(uint8_t *p_dest, unsigned p_size); + +/* ecc_set_rng() function. +Set the function that will be used to generate random bytes. The RNG function should +return 1 if the random data was generated, or 0 if the random data could not be generated. + +This must be called before ecc_make_key(), ecdh_shared_secret(), or ecdsa_sign() are used. + +Inputs: + p_rng - The function that will be used to generate random bytes. +*/ +void ecc_set_rng(RNG_Function p_rng); + +/* ecc_make_key() function. +Create a public/private key pair. + +Outputs: + p_publicKey - Will be filled in with the public key. + p_privateKey - Will be filled in with the private key. + +Returns 1 if the key pair was generated successfully, 0 if an error occurred. +*/ +int ecc_make_key(uint8_t p_publicKey[ECC_BYTES+1], uint8_t p_privateKey[ECC_BYTES]); + +/* ecdh_shared_secret() function. +Compute a shared secret given your secret key and someone else's public key. +Note: It is recommended that you hash the result of ecdh_shared_secret before using it for symmetric encryption or HMAC. + +Inputs: + p_publicKey - The public key of the remote party. + p_privateKey - Your private key. + +Outputs: + p_secret - Will be filled in with the shared secret value. + +Returns 1 if the shared secret was generated successfully, 0 if an error occurred. +*/ +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]); + +#ifdef __cplusplus +} /* end of extern "C" */ +#endif + +#endif /* _EASY_ECC_H_ */ diff --git a/emk_rules.py b/emk_rules.py new file mode 100644 index 0000000..f1b9fb6 --- /dev/null +++ b/emk_rules.py @@ -0,0 +1,2 @@ +c, link = emk.module("c", "link") +c.flags += ["-O3", "-Wall", "-Wextra", "-Werror", "-Wno-unused-parameter", "-Wno-unused-variable", "-Wno-unused-function"] diff --git a/verify_ecdh.c b/verify_ecdh.c new file mode 100644 index 0000000..046aeb0 --- /dev/null +++ b/verify_ecdh.c @@ -0,0 +1,93 @@ +#include "ecc.h" + +#include +#include +#include +#include + +void vli_print(uint8_t *p_vli, unsigned int p_size) +{ + while(p_size) + { + printf("%02X ", (unsigned)p_vli[p_size - 1]); + --p_size; + } +} + +int randfd; + +int RNG(uint8_t *p_dest, unsigned p_size) +{ + if(read(randfd, p_dest, p_size) != (int)p_size) + { + printf("Failed to get random bytes.\n"); + return 0; + } + return 1; +} + +int main() +{ + int i; + + uint8_t l_private1[ECC_BYTES]; + uint8_t l_private2[ECC_BYTES]; + + uint8_t l_public1[ECC_BYTES + 1]; + uint8_t l_public2[ECC_BYTES + 1]; + + uint8_t l_secret1[ECC_BYTES]; + uint8_t l_secret2[ECC_BYTES]; + + randfd = open("/dev/urandom", O_RDONLY); + if(randfd == -1) + { + printf("No access to urandom\n"); + return -1; + } + + ecc_set_rng(&RNG); + + printf("Testing 256 random private key pairs\n"); + + for(i=0; i<256; ++i) + { + printf("."); + fflush(stdout); + + ecc_make_key(l_public1, l_private1); + ecc_make_key(l_public2, l_private2); + + if(!ecdh_shared_secret(l_public2, l_private1, l_secret1)) + { + printf("shared_secret() failed (1)\n"); + return 1; + } + + if(!ecdh_shared_secret(l_public1, l_private2, l_secret2)) + { + printf("shared_secret() failed (2)\n"); + return 1; + } + + if(memcmp(l_secret1, l_secret2, sizeof(l_secret1)) != 0) + { + printf("Shared secrets are not identical!\n"); + printf("Shared secret 1 = "); + vli_print(l_secret1, ECC_BYTES); + printf("\n"); + printf("Shared secret 2 = "); + vli_print(l_secret2, ECC_BYTES); + printf("\n"); + printf("Private key 1 = "); + vli_print(l_private1, ECC_BYTES); + printf("\n"); + printf("Private key 2 = "); + vli_print(l_private2, ECC_BYTES); + printf("\n"); + } + } + printf("\n"); + + return 0; +}