/* Crypto using elliptic curves defined over the finite binary field GF(2^m) where m is prime. The curves used are the anomalous binary curves (ABC-curves) or also called Koblitz curves. This class of curves was chosen because it yields efficient implementation of operations. Curves available - their different NIST/SECG names and eqivalent symmetric security level: NIST SEC Group strength ------------------------------------ K-163 sect163k1 80 bit B-163 sect163r2 80 bit K-233 sect233k1 112 bit B-233 sect233r1 112 bit K-283 sect283k1 128 bit B-283 sect283r1 128 bit K-409 sect409k1 192 bit B-409 sect409r1 192 bit K-571 sect571k1 256 bit B-571 sect571r1 256 bit Curve parameters from: http://www.secg.org/sec2-v2.pdf http://csrc.nist.gov/publications/fips/fips186-3/fips_186-3.pdf Reference: https://www.ietf.org/rfc/rfc4492.txt */ #include #include "ecdh.h" /* margin for overhead needed in intermediate calculations */ #define BITVEC_MARGIN 3 #define BITVEC_NBITS (CURVE_DEGREE + BITVEC_MARGIN) #define BITVEC_NWORDS ((BITVEC_NBITS + 31) / 32) #define BITVEC_NBYTES (sizeof(uint32_t) * BITVEC_NWORDS) /* Disable assertions? */ #ifndef DISABLE_ASSERT #define DISABLE_ASSERT 0 #endif #if defined(DISABLE_ASSERT) && (DISABLE_ASSERT == 1) #define assert(...) #else #include #endif /* Default to a (somewhat) constant-time mode? NOTE: The library is _not_ capable of operating in constant-time and leaks information via timing. Even if all operations are written const-time-style, it requires the hardware is able to multiply in constant time. Multiplication on ARM Cortex-M processors takes a variable number of cycles depending on the operands... */ #ifndef CONST_TIME #define CONST_TIME 0 #endif /* Default to using ECC_CDH (cofactor multiplication-variation) ? */ #ifndef ECDH_COFACTOR_VARIANT #define ECDH_COFACTOR_VARIANT 0 #endif /******************************************************************************/ /* the following type will represent bit vectors of length (CURVE_DEGREE+MARGIN) */ typedef uint32_t bitvec_t[BITVEC_NWORDS]; typedef bitvec_t gf2elem_t; /* this type will represent field elements */ typedef bitvec_t scalar_t; /******************************************************************************/ /* Here the curve parameters are defined. */ #if defined (ECC_CURVE) && (ECC_CURVE != 0) #if (ECC_CURVE == NIST_K163) #define coeff_a 1 #define cofactor 2 /* NIST K-163 */ const gf2elem_t polynomial = { 0x000000c9, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000008 }; const gf2elem_t coeff_b = { 0x00000001, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000 }; const gf2elem_t base_x = { 0x5c94eee8, 0xde4e6d5e, 0xaa07d793, 0x7bbc11ac, 0xfe13c053, 0x00000002 }; const gf2elem_t base_y = { 0xccdaa3d9, 0x0536d538, 0x321f2e80, 0x5d38ff58, 0x89070fb0, 0x00000002 }; const scalar_t base_order = { 0x99f8a5ef, 0xa2e0cc0d, 0x00020108, 0x00000000, 0x00000000, 0x00000004 }; #endif #if (ECC_CURVE == NIST_B163) #define coeff_a 1 #define cofactor 2 /* NIST B-163 */ const gf2elem_t polynomial = { 0x000000c9, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000008 }; const gf2elem_t coeff_b = { 0x4a3205fd, 0x512f7874, 0x1481eb10, 0xb8c953ca, 0x0a601907, 0x00000002 }; const gf2elem_t base_x = { 0xe8343e36, 0xd4994637, 0xa0991168, 0x86a2d57e, 0xf0eba162, 0x00000003 }; const gf2elem_t base_y = { 0x797324f1, 0xb11c5c0c, 0xa2cdd545, 0x71a0094f, 0xd51fbc6c, 0x00000000 }; const scalar_t base_order = { 0xa4234c33, 0x77e70c12, 0x000292fe, 0x00000000, 0x00000000, 0x00000004 }; #endif #if (ECC_CURVE == NIST_K233) #define coeff_a 0 #define cofactor 4 /* NIST K-233 */ const gf2elem_t polynomial = { 0x00000001, 0x00000000, 0x00000400, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000200 }; const gf2elem_t coeff_b = { 0x00000001, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000 }; const gf2elem_t base_x = { 0xefad6126, 0x0a4c9d6e, 0x19c26bf5, 0x149563a4, 0x29f22ff4, 0x7e731af1, 0x32ba853a, 0x00000172 }; const gf2elem_t base_y = { 0x56fae6a3, 0x56e0c110, 0xf18aeb9b, 0x27a8cd9b, 0x555a67c4, 0x19b7f70f, 0x537dece8, 0x000001db }; const scalar_t base_order = { 0xf173abdf, 0x6efb1ad5, 0xb915bcd4, 0x00069d5b, 0x00000000, 0x00000000, 0x00000000, 0x00000080 }; #endif #if (ECC_CURVE == NIST_B233) #define coeff_a 1 #define cofactor 2 /* NIST B-233 */ const gf2elem_t polynomial = { 0x00000001, 0x00000000, 0x00000400, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000200 }; const gf2elem_t coeff_b = { 0x7d8f90ad, 0x81fe115f, 0x20e9ce42, 0x213b333b, 0x0923bb58, 0x332c7f8c, 0x647ede6c, 0x00000066 }; const gf2elem_t base_x = { 0x71fd558b, 0xf8f8eb73, 0x391f8b36, 0x5fef65bc, 0x39f1bb75, 0x8313bb21, 0xc9dfcbac, 0x000000fa }; const gf2elem_t base_y = { 0x01f81052, 0x36716f7e, 0xf867a7ca, 0xbf8a0bef, 0xe58528be, 0x03350678, 0x6a08a419, 0x00000100 }; const scalar_t base_order = { 0x03cfe0d7, 0x22031d26, 0xe72f8a69, 0x0013e974, 0x00000000, 0x00000000, 0x00000000, 0x00000100 }; #endif #if (ECC_CURVE == NIST_K283) #define coeff_a 0 #define cofactor 4 /* NIST K-283 */ const gf2elem_t polynomial = { 0x000010a1, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x08000000 }; const gf2elem_t coeff_b = { 0x00000001, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000 }; const gf2elem_t base_x = { 0x58492836, 0xb0c2ac24, 0x16876913, 0x23c1567a, 0x53cd265f, 0x62f188e5, 0x3f1a3b81, 0x78ca4488, 0x0503213f }; const gf2elem_t base_y = { 0x77dd2259, 0x4e341161, 0xe4596236, 0xe8184698, 0xe87e45c0, 0x07e5426f, 0x8d90f95d, 0x0f1c9e31, 0x01ccda38 }; const scalar_t base_order = { 0x1e163c61, 0x94451e06, 0x265dff7f, 0x2ed07577, 0xffffe9ae, 0xffffffff, 0xffffffff, 0xffffffff, 0x01ffffff }; #endif #if (ECC_CURVE == NIST_B283) #define coeff_a 1 #define cofactor 2 /* NIST B-283 */ const gf2elem_t polynomial = { 0x000010a1, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x08000000 }; const gf2elem_t coeff_b = { 0x3b79a2f5, 0xf6263e31, 0xa581485a, 0x45309fa2, 0xca97fd76, 0x19a0303f, 0xa5a4af8a, 0xc8b8596d, 0x027b680a }; const gf2elem_t base_x = { 0x86b12053, 0xf8cdbecd, 0x80e2e198, 0x557eac9c, 0x2eed25b8, 0x70b0dfec, 0xe1934f8c, 0x8db7dd90, 0x05f93925 }; const gf2elem_t base_y = { 0xbe8112f4, 0x13f0df45, 0x826779c8, 0x350eddb0, 0x516ff702, 0xb20d02b4, 0xb98fe6d4, 0xfe24141c, 0x03676854 }; const scalar_t base_order = { 0xefadb307, 0x5b042a7c, 0x938a9016, 0x399660fc, 0xffffef90, 0xffffffff, 0xffffffff, 0xffffffff, 0x03ffffff }; #endif #if (ECC_CURVE == NIST_K409) #define coeff_a 0 #define cofactor 4 /* NIST K-409 */ const gf2elem_t polynomial = { 0x00000001, 0x00000000, 0x00800000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x02000000 }; const gf2elem_t coeff_b = { 0x00000001, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000 }; const gf2elem_t base_x = { 0xe9023746, 0xb35540cf, 0xee222eb1, 0xb5aaaa62, 0xc460189e, 0xf9f67cc2, 0x27accfb8, 0xe307c84c, 0x0efd0987, 0x0f718421, 0xad3ab189, 0x658f49c1, 0x0060f05f }; const gf2elem_t base_y = { 0xd8e0286b, 0x5863ec48, 0xaa9ca27a, 0xe9c55215, 0xda5f6c42, 0xe9ea10e3, 0xe6325165, 0x918ea427, 0x3460782f, 0xbf04299c, 0xacba1dac, 0x0b7c4e42, 0x01e36905 }; const scalar_t base_order = { 0xe01e5fcf, 0x4b5c83b8, 0xe3e7ca5b, 0x557d5ed3, 0x20400ec4, 0x83b2d4ea, 0xfffffe5f, 0xffffffff, 0xffffffff, 0xffffffff, 0xffffffff, 0xffffffff, 0x007fffff }; #endif #if (ECC_CURVE == NIST_B409) #define coeff_a 1 #define cofactor 2 /* NIST B-409 */ const gf2elem_t polynomial = { 0x00000001, 0x00000000, 0x00800000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x02000000 }; const gf2elem_t coeff_b = { 0x7b13545f, 0x4f50ae31, 0xd57a55aa, 0x72822f6c, 0xa9a197b2, 0xd6ac27c8, 0x4761fa99, 0xf1f3dd67, 0x7fd6422e, 0x3b7b476b, 0x5c4b9a75, 0xc8ee9feb, 0x0021a5c2 }; const gf2elem_t base_x = { 0xbb7996a7, 0x60794e54, 0x5603aeab, 0x8a118051, 0xdc255a86, 0x34e59703, 0xb01ffe5b, 0xf1771d4d, 0x441cde4a, 0x64756260, 0x496b0c60, 0xd088ddb3, 0x015d4860 }; const gf2elem_t base_y = { 0x0273c706, 0x81c364ba, 0xd2181b36, 0xdf4b4f40, 0x38514f1f, 0x5488d08f, 0x0158aa4f, 0xa7bd198d, 0x7636b9c5, 0x24ed106a, 0x2bbfa783, 0xab6be5f3, 0x0061b1cf }; const scalar_t base_order = { 0xd9a21173, 0x8164cd37, 0x9e052f83, 0x5fa47c3c, 0xf33307be, 0xaad6a612, 0x000001e2, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x01000000 }; #endif #if (ECC_CURVE == NIST_K571) #define coeff_a 0 #define cofactor 4 /* NIST K-571 */ const gf2elem_t polynomial = { 0x00000425, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x08000000 }; const gf2elem_t coeff_b = { 0x00000001, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000 }; const gf2elem_t base_x = { 0xa01c8972, 0xe2945283, 0x4dca88c7, 0x988b4717, 0x494776fb, 0xbbd1ba39, 0xb4ceb08c, 0x47da304d, 0x93b205e6, 0x43709584, 0x01841ca4, 0x60248048, 0x0012d5d4, 0xac9ca297, 0xf8103fe4, 0x82189631, 0x59923fbc, 0x026eb7a8 }; const gf2elem_t base_y = { 0x3ef1c7a3, 0x01cd4c14, 0x591984f6, 0x320430c8, 0x7ba7af1b, 0xb620b01a, 0xf772aedc, 0x4fbebbb9, 0xac44aea7, 0x9d4979c0, 0x006d8a2c, 0xffc61efc, 0x9f307a54, 0x4dd58cec, 0x3bca9531, 0x4f4aeade, 0x7f4fbf37, 0x0349dc80 }; const scalar_t base_order = { 0x637c1001, 0x5cfe778f, 0x1e91deb4, 0xe5d63938, 0xb630d84b, 0x917f4138, 0xb391a8db, 0xf19a63e4, 0x131850e1, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x02000000 }; #endif #if (ECC_CURVE == NIST_B571) #define coeff_a 1 #define cofactor 2 /* NIST B-571 */ const gf2elem_t polynomial = { 0x00000425, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x00000000, 0x08000000 }; const gf2elem_t coeff_b = { 0x2955727a, 0x7ffeff7f, 0x39baca0c, 0x520e4de7, 0x78ff12aa, 0x4afd185a, 0x56a66e29, 0x2be7ad67, 0x8efa5933, 0x84ffabbd, 0x4a9a18ad, 0xcd6ba8ce, 0xcb8ceff1, 0x5c6a97ff, 0xb7f3d62f, 0xde297117, 0x2221f295, 0x02f40e7e }; const gf2elem_t base_x = { 0x8eec2d19, 0xe1e7769c, 0xc850d927, 0x4abfa3b4, 0x8614f139, 0x99ae6003, 0x5b67fb14, 0xcdd711a3, 0xf4c0d293, 0xbde53950, 0xdb7b2abd, 0xa5f40fc8, 0x955fa80a, 0x0a93d1d2, 0x0d3cd775, 0x6c16c0d4, 0x34b85629, 0x0303001d }; const gf2elem_t base_y = { 0x1b8ac15b, 0x1a4827af, 0x6e23dd3c, 0x16e2f151, 0x0485c19b, 0xb3531d2f, 0x461bb2a8, 0x6291af8f, 0xbab08a57, 0x84423e43, 0x3921e8a6, 0x1980f853, 0x009cbbca, 0x8c6c27a6, 0xb73d69d7, 0x6dccfffe, 0x42da639b, 0x037bf273 }; const scalar_t base_order = { 0x2fe84e47, 0x8382e9bb, 0x5174d66e, 0x161de93d, 0xc7dd9ca1, 0x6823851e, 0x08059b18, 0xff559873, 0xe661ce18, 0xffffffff, 0xffffffff, 0xffffffff, 0xffffffff, 0xffffffff, 0xffffffff, 0xffffffff, 0xffffffff, 0x03ffffff }; #endif #endif /*************************************************************************************************/ /* Private / static functions: */ /* some basic bit-manipulation routines that act on bit-vectors follow */ static int bitvec_get_bit(const bitvec_t x, const uint32_t idx) { return ((x[idx / 32U] >> (idx & 31U) & 1U)); } static void bitvec_clr_bit(bitvec_t x, const uint32_t idx) { x[idx / 32U] &= ~(1U << (idx & 31U)); } static void bitvec_copy(bitvec_t x, const bitvec_t y) { int i; for (i = 0; i < BITVEC_NWORDS; ++i) { x[i] = y[i]; } } static void bitvec_swap(bitvec_t x, bitvec_t y) { bitvec_t tmp; bitvec_copy(tmp, x); bitvec_copy(x, y); bitvec_copy(y, tmp); } #if defined(CONST_TIME) && (CONST_TIME == 0) /* fast version of equality test */ static int bitvec_equal(const bitvec_t x, const bitvec_t y) { int i; for (i = 0; i < BITVEC_NWORDS; ++i) { if (x[i] != y[i]) { return 0; } } return 1; } #else /* constant time version of equality test */ static int bitvec_equal(const bitvec_t x, const bitvec_t y) { int ret = 1; int i; for (i = 0; i < BITVEC_NWORDS; ++i) { ret &= (x[i] == y[i]); } return ret; } #endif static void bitvec_set_zero(bitvec_t x) { int i; for (i = 0; i < BITVEC_NWORDS; ++i) { x[i] = 0; } } #if defined(CONST_TIME) && (CONST_TIME == 0) /* fast implementation */ static int bitvec_is_zero(const bitvec_t x) { uint32_t i = 0; while (i < BITVEC_NWORDS) { if (x[i] != 0) { break; } i += 1; } return (i == BITVEC_NWORDS); } #else /* constant-time implementation */ static int bitvec_is_zero(const bitvec_t x) { int ret = 1; int i = 0; for (i = 0; i < BITVEC_NWORDS; ++i) { ret &= (x[i] == 0); } return ret; } #endif /* return the number of the highest one-bit + 1 */ static int bitvec_degree(const bitvec_t x) { int i = BITVEC_NWORDS * 32; /* Start at the back of the vector (MSB) */ x += BITVEC_NWORDS; /* Skip empty / zero words */ while ( (i > 0) && (*(--x)) == 0) { i -= 32; } /* Run through rest if count is not multiple of bitsize of DTYPE */ if (i != 0) { uint32_t u32mask = ((uint32_t)1 << 31); while (((*x) & u32mask) == 0) { u32mask >>= 1; i -= 1; } } return i; } /* left-shift by 'count' digits */ static void bitvec_lshift(bitvec_t x, const bitvec_t y, int nbits) { int nwords = (nbits / 32); /* Shift whole words first if nwords > 0 */ int i,j; for (i = 0; i < nwords; ++i) { /* Zero-initialize from least-significant word until offset reached */ x[i] = 0; } j = 0; /* Copy to x output */ while (i < BITVEC_NWORDS) { x[i] = y[j]; i += 1; j += 1; } /* Shift the rest if count was not multiple of bitsize of DTYPE */ nbits &= 31; if (nbits != 0) { /* Left shift rest */ int i; for (i = (BITVEC_NWORDS - 1); i > 0; --i) { x[i] = (x[i] << nbits) | (x[i - 1] >> (32 - nbits)); } x[0] <<= nbits; } } /*************************************************************************************************/ /* Code that does arithmetic on bit-vectors in the Galois Field GF(2^CURVE_DEGREE). */ /*************************************************************************************************/ static void gf2field_set_one(gf2elem_t x) { /* Set first word to one */ x[0] = 1; /* .. and the rest to zero */ int i; for (i = 1; i < BITVEC_NWORDS; ++i) { x[i] = 0; } } #if defined(CONST_TIME) && (CONST_TIME == 0) /* fastest check if x == 1 */ static int gf2field_is_one(const gf2elem_t x) { /* Check if first word == 1 */ if (x[0] != 1) { return 0; } /* ...and if rest of words == 0 */ int i; for (i = 1; i < BITVEC_NWORDS; ++i) { if (x[i] != 0) { break; } } return (i == BITVEC_NWORDS); } #else /* constant-time check */ static int gf2field_is_one(const gf2elem_t x) { int ret = 0; /* Check if first word == 1 */ if (x[0] == 1) { ret = 1; } /* ...and if rest of words == 0 */ int i; for (i = 1; i < BITVEC_NWORDS; ++i) { ret &= (x[i] == 0); } return ret; //(i == BITVEC_NWORDS); } #endif /* galois field(2^m) addition is modulo 2, so XOR is used instead - 'z := a + b' */ static void gf2field_add(gf2elem_t z, const gf2elem_t x, const gf2elem_t y) { int i; for (i = 0; i < BITVEC_NWORDS; ++i) { z[i] = (x[i] ^ y[i]); } } /* increment element */ static void gf2field_inc(gf2elem_t x) { x[0] ^= 1; } /* field multiplication 'z := (x * y)' */ static void gf2field_mul(gf2elem_t z, const gf2elem_t x, const gf2elem_t y) { int i; gf2elem_t tmp; #if defined(CONST_TIME) && (CONST_TIME == 1) gf2elem_t blind; bitvec_set_zero(blind); #endif assert(z != y); bitvec_copy(tmp, x); /* LSB set? Then start with x */ if (bitvec_get_bit(y, 0) != 0) { bitvec_copy(z, x); } else /* .. or else start with zero */ { bitvec_set_zero(z); } /* Then add 2^i * x for the rest */ for (i = 1; i < CURVE_DEGREE; ++i) { /* lshift 1 - doubling the value of tmp */ bitvec_lshift(tmp, tmp, 1); /* Modulo reduction polynomial if degree(tmp) > CURVE_DEGREE */ if (bitvec_get_bit(tmp, CURVE_DEGREE)) { gf2field_add(tmp, tmp, polynomial); } #if defined(CONST_TIME) && (CONST_TIME == 1) else /* blinding operation */ { gf2field_add(tmp, tmp, blind); } #endif /* Add 2^i * tmp if this factor in y is non-zero */ if (bitvec_get_bit(y, i)) { gf2field_add(z, z, tmp); } #if defined(CONST_TIME) && (CONST_TIME == 1) else /* blinding operation */ { gf2field_add(z, z, blind); } #endif } } /* field inversion 'z := 1/x' */ static void gf2field_inv(gf2elem_t z, const gf2elem_t x) { gf2elem_t u, v, g, h; int i; bitvec_copy(u, x); bitvec_copy(v, polynomial); bitvec_set_zero(g); gf2field_set_one(z); while (!gf2field_is_one(u)) { i = (bitvec_degree(u) - bitvec_degree(v)); if (i < 0) { bitvec_swap(u, v); bitvec_swap(g, z); i = -i; } #if defined(CONST_TIME) && (CONST_TIME == 1) else { bitvec_swap(u, v); bitvec_swap(v, u); } #endif bitvec_lshift(h, v, i); gf2field_add(u, u, h); bitvec_lshift(h, g, i); gf2field_add(z, z, h); } } /*************************************************************************************************/ /* The following code takes care of Galois-Field arithmetic. Elliptic curve points are represented by pairs (x,y) of bitvec_t. It is assumed that curve coefficient 'a' is {0,1} This is the case for all NIST binary curves. Coefficient 'b' is given in 'coeff_b'. '(base_x, base_y)' is a point that generates a large prime order group. */ /*************************************************************************************************/ static void gf2point_copy(gf2elem_t x1, gf2elem_t y1, const gf2elem_t x2, const gf2elem_t y2) { bitvec_copy(x1, x2); bitvec_copy(y1, y2); } static void gf2point_set_zero(gf2elem_t x, gf2elem_t y) { bitvec_set_zero(x); bitvec_set_zero(y); } static int gf2point_is_zero(const gf2elem_t x, const gf2elem_t y) { return ( bitvec_is_zero(x) && bitvec_is_zero(y)); } /* double the point (x,y) */ static void gf2point_double(gf2elem_t x, gf2elem_t y) { /* iff P = O (zero or infinity): 2 * P = P */ if (bitvec_is_zero(x)) { bitvec_set_zero(y); } else { gf2elem_t l; gf2field_inv(l, x); gf2field_mul(l, l, y); gf2field_add(l, l, x); gf2field_mul(y, x, x); gf2field_mul(x, l, l); #if (coeff_a == 1) gf2field_inc(l); #endif gf2field_add(x, x, l); gf2field_mul(l, l, x); gf2field_add(y, y, l); } } /* add two points together (x1, y1) := (x1, y1) + (x2, y2) */ static void gf2point_add(gf2elem_t x1, gf2elem_t y1, const gf2elem_t x2, const gf2elem_t y2) { if (!gf2point_is_zero(x2, y2)) { if (gf2point_is_zero(x1, y1)) { gf2point_copy(x1, y1, x2, y2); } else { if (bitvec_equal(x1, x2)) { if (bitvec_equal(y1, y2)) { gf2point_double(x1, y1); } else { gf2point_set_zero(x1, y1); } } else { /* Arithmetic with temporary variables */ gf2elem_t a, b, c, d; gf2field_add(a, y1, y2); gf2field_add(b, x1, x2); gf2field_inv(c, b); gf2field_mul(c, c, a); gf2field_mul(d, c, c); gf2field_add(d, d, c); gf2field_add(d, d, b); #if (coeff_a == 1) gf2field_inc(d); #endif gf2field_add(x1, x1, d); gf2field_mul(a, x1, c); gf2field_add(a, a, d); gf2field_add(y1, y1, a); bitvec_copy(x1, d); } } } } #if defined(CONST_TIME) && (CONST_TIME == 0) /* point multiplication via double-and-add algorithm */ static void gf2point_mul(gf2elem_t x, gf2elem_t y, const scalar_t exp) { gf2elem_t tmpx, tmpy; int i; int nbits = bitvec_degree(exp); gf2point_set_zero(tmpx, tmpy); for (i = (nbits - 1); i >= 0; --i) { gf2point_double(tmpx, tmpy); if (bitvec_get_bit(exp, i)) { gf2point_add(tmpx, tmpy, x, y); } } gf2point_copy(x, y, tmpx, tmpy); } #else /* point multiplication via double-and-add-always algorithm using scalar blinding */ static void gf2point_mul(gf2elem_t x, gf2elem_t y, const scalar_t exp) { gf2elem_t tmpx, tmpy; gf2elem_t dummyx, dummyy; int i; int nbits = bitvec_degree(exp); gf2point_set_zero(tmpx, tmpy); gf2point_set_zero(dummyx, dummyy); for (i = (nbits - 1); i >= 0; --i) { gf2point_double(tmpx, tmpy); /* Add point if bit(i) is set in exp */ if (bitvec_get_bit(exp, i)) { gf2point_add(tmpx, tmpy, x, y); } /* .. or add the neutral element to keep operation constant-time */ else { gf2point_add(tmpx, tmpy, dummyx, dummyy); } } gf2point_copy(x, y, tmpx, tmpy); } #endif /* check if y^2 + x*y = x^3 + a*x^2 + coeff_b holds */ static int gf2point_on_curve(const gf2elem_t x, const gf2elem_t y) { gf2elem_t a, b; if (gf2point_is_zero(x, y)) { return 1; } else { gf2field_mul(a, x, x); #if (coeff_a == 0) gf2field_mul(a, a, x); #else gf2field_mul(b, a, x); gf2field_add(a, a, b); #endif gf2field_add(a, a, coeff_b); gf2field_mul(b, y, y); gf2field_add(a, a, b); gf2field_mul(b, x, y); return bitvec_equal(a, b); } } /*************************************************************************************************/ /* Elliptic Curve Diffie-Hellman key exchange protocol. */ /*************************************************************************************************/ /* NOTE: private should contain random data a-priori! */ int ecdh_generate_keys(uint8_t* public_key, uint8_t* private_key) { /* Get copy of "base" point 'G' */ gf2point_copy((uint32_t*)public_key, (uint32_t*)(public_key + BITVEC_NBYTES), base_x, base_y); /* Abort key generation if random number is too small */ if (bitvec_degree((uint32_t*)private_key) < (CURVE_DEGREE / 2)) { return 0; } else { /* Clear bits > CURVE_DEGREE in highest word to satisfy constraint 1 <= exp < n. */ int nbits = bitvec_degree(base_order); int i; for (i = (nbits - 1); i < (BITVEC_NWORDS * 32); ++i) { bitvec_clr_bit((uint32_t*)private_key, i); } /* Multiply base-point with scalar (private-key) */ gf2point_mul((uint32_t*)public_key, (uint32_t*)(public_key + BITVEC_NBYTES), (uint32_t*)private_key); return 1; } } int ecdh_shared_secret(const uint8_t* private_key, const uint8_t* others_pub, uint8_t* output) { /* Do some basic validation of other party's public key */ if ( !gf2point_is_zero ((uint32_t*)others_pub, (uint32_t*)(others_pub + BITVEC_NBYTES)) && gf2point_on_curve((uint32_t*)others_pub, (uint32_t*)(others_pub + BITVEC_NBYTES)) ) { /* Copy other side's public key to output */ unsigned int i; for (i = 0; i < (BITVEC_NBYTES * 2); ++i) { output[i] = others_pub[i]; } /* Clear bits > CURVE_DEGREE in highest word to satisfy constraint 1 <= exp < n. */ int nbits = bitvec_degree(base_order); for (int i = (nbits - 1); i < (BITVEC_NWORDS * 32); ++i) { bitvec_clr_bit((uint32_t*)private_key, i); } /* Multiply other side's public key with own private key */ gf2point_mul((uint32_t*)output,(uint32_t*)(output + BITVEC_NBYTES), (const uint32_t*)private_key); /* Multiply outcome by cofactor if using ECC CDH-variant: */ #if defined(ECDH_COFACTOR_VARIANT) && (ECDH_COFACTOR_VARIANT == 1) #if (cofactor == 2) gf2point_double((uint32_t*)output, (uint32_t*)(output + BITVEC_NBYTES)); #elif (cofactor == 4) gf2point_double((uint32_t*)output, (uint32_t*)(output + BITVEC_NBYTES)); gf2point_double((uint32_t*)output, (uint32_t*)(output + BITVEC_NBYTES)); #endif #endif return 1; } else { return 0; } } /* ECDSA is broken :( ... */ int ecdsa_sign(const uint8_t* private_key, uint8_t* hash, uint8_t* random_k, uint8_t* signature) { /* 1) calculate e = HASH(m) 2) let z be the Ln leftmost bits of e, where Ln is the bit length of the group order n 3) Select a cryptographically secure random integer k from [1, n-1] 4) Calculate the curve point (x1, y1) = k * G 5) Calculate r = x1 mod n - if (r == 0) goto 3 6) Calculate s = inv(k) * (z + r * d) mod n - if (s == 0) goto 3 7) The signature is the pair (r, s) */ assert(private_key != 0); assert(hash != 0); assert(random_k != 0); assert(signature != 0); int success = 0; if ( (bitvec_degree((uint32_t*)private_key) >= (CURVE_DEGREE / 2)) && !bitvec_is_zero((uint32_t*)random_k) ) { gf2elem_t r, s, z, k; bitvec_set_zero(r); bitvec_set_zero(s); bitvec_copy(z, (uint32_t*)hash); /* 1 + 2 */ int nbits = bitvec_degree(base_order); int i; for (i = (nbits - 1); i < BITVEC_NBITS; ++i) { bitvec_clr_bit(z, i); } /* 3 */ bitvec_copy(k, (uint32_t*)random_k); /* 4 */ gf2point_copy(r, s, base_x, base_y); gf2point_mul(r, s, k); /* 5 */ if (!bitvec_is_zero(r)) { /* 6) s = inv(k) * (z + (r * d)) mod n ==> if (s == 0) goto 3 **/ gf2field_inv(s, k); /* s = inv(k) */ gf2field_mul(r, r, (uint32_t*)private_key); /* r = (r * d) */ gf2field_add(r, r, z); /* r = z + (r * d) */ nbits = bitvec_degree(r); /* r = r mod n */ for (i = (nbits - 1); i < BITVEC_NBITS; ++i) { printf("reduction r\n"); bitvec_clr_bit(r, i); } gf2field_mul(s, s, r); /* s = inv(k) * (z * (r * d)) */ nbits = bitvec_degree(s); /* s = s mod n */ for (i = (nbits - 1); i < BITVEC_NBITS; ++i) { printf("reduction s\n"); bitvec_clr_bit(s, i); } if (!bitvec_is_zero(s)) { bitvec_copy((uint32_t*)signature, r); bitvec_copy((uint32_t*)(signature + ECC_PRV_KEY_SIZE), s); success = 1; } } } return success; } int ecdsa_verify(const uint8_t* public_key, uint8_t* hash, const uint8_t* signature) { /* 1) Verify that (r,s) are in [1, n-1] 2) e = HASH(m) 3) z = Ln leftmost bits of e 4) w = inv(s) mod n 5) u1 = (z * w) mod n u2 = (r * w) mod n 6) (x,y) = (u1 * G) + (u2 * public) 7) Signature is valid if r == x mod n && (x,y) != (0,0) */ assert(public_key != 0); assert(hash != 0); assert(signature != 0); int success = 0; gf2elem_t r, s; bitvec_copy(r, (uint32_t*)(signature)); bitvec_copy(s, (uint32_t*)(signature + ECC_PRV_KEY_SIZE)); if ( !bitvec_is_zero(s) && !bitvec_is_zero(r)) { gf2elem_t x1, y1, u1, u2, w, z; /* 3) z = Ln leftmost bits of e */ bitvec_copy(z, (uint32_t*)hash); /* r,s,z are set */ uint32_t nbits = bitvec_degree(base_order); uint32_t i; for (i = (nbits - 1); i < BITVEC_NBITS; ++i) { bitvec_clr_bit(z, i); } /* 4) w = inv(s) mod n */ gf2field_inv(w, s); /* w = inv(s) */ /* Modulo reduction polynomial if degree(tmp) > CURVE_DEGREE */ if (bitvec_get_bit(w, CURVE_DEGREE)) { printf("reduction on w\n"); gf2field_add(w, w, polynomial); } /* 5) u1 = zw mod n, u2 = rw mod n*/ gf2field_mul(u1, z, w); /* u1 = z * w */ /* Modulo reduction polynomial if degree(tmp) > CURVE_DEGREE */ if (bitvec_get_bit(u1, CURVE_DEGREE)) { printf("reduction on u1\n"); gf2field_add(u1, u1, polynomial); } gf2field_mul(u2, r, w); /* u2 = r * w */ /* Modulo reduction polynomial if degree(tmp) > CURVE_DEGREE */ if (bitvec_get_bit(u2, CURVE_DEGREE)) { printf("reduction on u2\n"); gf2field_add(u2, u2, polynomial); } /* 6) (x,y) = (u1 * G) + (u2 * public) */ bitvec_copy(x1, base_x); bitvec_copy(y1, base_y); gf2field_mul(u1, x1, y1); /* u1 * G */ bitvec_copy(w, (uint32_t*)(public_key)); bitvec_copy(z, (uint32_t*)(public_key + ECC_PRV_KEY_SIZE)); gf2field_mul(u2, w, z); /* u2 * Q */ gf2point_add(x1, y1, w, z); if (bitvec_get_bit(x1, CURVE_DEGREE)) { printf("reduction on x1\n"); gf2field_add(x1, x1, polynomial); } success = bitvec_equal(r, x1); if (!success) { printf("x = '"); for (i = 0; i < BITVEC_NWORDS; ++i) { printf("%.08x", x1[i]); } printf("' [%u]\n", i); printf("r = '"); for (i = 0; i < BITVEC_NWORDS; ++i) { printf("%.08x", r[i]); } printf("' [%u]\n", i); } } else { printf("(s or r) == zero\n"); } return success; }