| #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<ECC_BYTES; ++i) |
| { |
| p_vli[i] = 0; |
| } |
| } |
| |
| /* Returns 1 if p_vli == 0, 0 otherwise. */ |
| static uint8_t vli_isZero(const uint8_t *p_vli) |
| { |
| uint i; |
| for(i = 0; i < ECC_BYTES; ++i) |
| { |
| if(p_vli[i]) |
| { |
| return 0; |
| } |
| } |
| return 1; |
| } |
| |
| /* Returns nonzero if bit p_bit of p_vli is set. */ |
| static uint8_t vli_testBit(const uint8_t *p_vli, uint p_bit) |
| { |
| return (p_vli[p_bit/8] & ((uint8_t)1 << (p_bit % 8))); |
| } |
| |
| /* Counts the number of 8-bit "digits" in p_vli. */ |
| static uint vli_numDigits(const uint8_t *p_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 = ECC_BYTES - 1; 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<ECC_BYTES; ++i) |
| { |
| p_dest[i] = p_src[i]; |
| } |
| } |
| |
| /* Returns sign of p_left - p_right. */ |
| static int vli_cmp(uint8_t *p_left, uint8_t *p_right) |
| { |
| int i; |
| for(i = ECC_BYTES-1; 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<ECC_BYTES; ++i) |
| { |
| uint16_t l_sum = (uint16_t)p_left[i] + p_right[i] + l_carry; |
| p_result[i] = (uint8_t)l_sum; |
| l_carry = l_sum >> 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<ECC_BYTES; ++i) |
| { |
| uint16_t l_diff = (uint16_t)p_left[i] - p_right[i] - l_borrow; |
| p_result[i] = (uint8_t)l_diff; |
| l_borrow = (l_diff >> 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<ECC_BYTES; ++i) |
| { |
| uint16_t l_product = (uint16_t)p_left[i] * p_right[k-i]; |
| 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; |
| } |
| |
| #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<ECC_BYTES+4; ++i) |
| { |
| uint16_t l_sum = (uint16_t)p_product[i] + l_tmp[i] + l_carry; |
| p_product[i] = (uint8_t)l_sum; |
| l_carry = l_sum >> 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); |
| } |
