/* * BCH / Reed-Solomon * encoder() * decoder() (Euklid. Alg.) * * * author: zilog80 * Vaisala RS92, RS41: RS(255, 231), t=12 f=X^8+X^4+X^3+X^2+1, b=0 g(X) = (X-alpha^0)...(X-alpha^(2t-1)) LMS6: RS(255, 223), t=16 (CCSDS) f=X^8+X^7+X^2+X+1, b=112 g(X) = (X-(alpha^11)^112)...(X-(alpha^11)^(112+2t-1)) Meisei: bin.BCH(63, 51), t=2 g(X) = (X^6+X+1)(X^6+X^4+X^2+X+1) g(a) = 0 fuer a = alpha^1,...,alpha^4 Es koennen 2 Fehler korrigiert werden; diese koennen auch direkt mit L(x) = 1 + L1 x + L2 x^2, L1=L1(S1), L2=L2(S1,S3) gefunden werden. Problem: 3 Fehler und mehr erkennen. Auch bei 3 Fehlern ist deg(Lambda)=2 und Lambda hat auch 2 Loesungen. Meisei-Bloecke sind auf 46 bit gekuerzt und enthalten 2 parity bits. -> Wenn decodierte Bloecke bits in Position 46-63 schalten oder einer der parity-checks fehlschlaegt, dann Block nicht korrigierbar. Es werden - 54% der 3-Fehler-Bloecke erkannt - 39% der 3-Fehler-Bloecke werden durch Position/Parity erkannt - 7% der 3-Fehler-Bloecke werden falsch korrigiert * */ /* // // --- bch_ecc.h --- // typedef unsigned char ui8_t; typedef unsigned int ui32_t; */ int rs_init_RS255(void); int rs_init_BCH64(void); int rs_encode(ui8_t cw[]); int rs_decode(ui8_t cw[], ui8_t *err_pos, ui8_t *err_val); int rs_decode_ErrEra(ui8_t cw[], int nera, ui8_t era_pos[], ui8_t *err_pos, ui8_t *err_val); int rs_decode_bch_gf2t2(ui8_t cw[], ui8_t *err_pos, ui8_t *err_val); // --- #define MAX_DEG 254 // max N-1 typedef struct { ui32_t f; ui32_t ord; ui8_t alpha; } GF_t; static GF_t GF256RS = { 0x11D, // RS-GF(2^8): X^8 + X^4 + X^3 + X^2 + 1 : 0x11D 256, // 2^8 0x02 }; // generator: alpha = X static GF_t GF256RSccsds = { 0x187, // RS-GF(2^8): X^8 + X^7 + X^2 + X + 1 : 0x187 256, // 2^8 0x02 }; // generator: alpha = X static GF_t GF64BCH = { 0x43, // BCH-GF(2^6): X^6 + X + 1 : 0x43 64, // 2^6 0x02 }; // generator: alpha = X static GF_t GF16RS = { 0x13, // RS-GF(2^4): X^4 + X + 1 : 0x13 16, // 2^4 0x02 }; // generator: alpha = X static GF_t GF256AES = { 0x11B, // AES-GF(2^8): X^8 + X^4 + X^3 + X + 1 : 0x11B 256, // 2^8 0x03 }; // generator: alpha = X+1 typedef struct { ui8_t N; ui8_t t; ui8_t R; // RS: R=2t, BCH: R<=mt ui8_t K; // K=N-R ui8_t b; ui8_t p; ui8_t ip; // p*ip = 1 mod N ui8_t g[MAX_DEG+1]; // ohne g[] eventuell als init_return } RS_t; static RS_t RS256 = { 255, 12, 24, 231, 0, 1, 1, {0}}; static RS_t RS256ccsds = { 255, 16, 32, 223, 112, 11, 116, {0}}; static RS_t BCH64 = { 63, 2, 12, 51, 1, 1, 1, {0}}; // static RS_t RS16_0 = { 15, 3, 6, 9, 0, 1, 1, {0}}; static RS_t RS16ccsds = { 15, 2, 4, 11, 6, 1, 1, {0}}; static GF_t GF; static RS_t RS; static ui8_t exp_a[256], log_a[256]; /* --------------------------------------------------------------------------------------------- */ static int GF_deg(ui32_t p) { ui32_t d = 31; if (p == 0) return -1; /* deg(0) = -infty */ else { while (d && !(p & (1<>= 1; if (b & 1) ab ^= (ui8_t)aa; /* b_{i+1} > 0 ? */ } return ab; } static int GF_genTab(GF_t gf, ui8_t expa[], ui8_t loga[]) { int i, j; ui8_t b; // GF.f = f; // GF.ord = 1 << GF_deg(GF.f); b = 0x01; for (i = 0; i < gf.ord; i++) { expa[i] = b; // b_i = a^i b = GF2m_mul(gf.alpha, b); } loga[0] = -00; // log(0) = -inf for (i = 1; i < gf.ord; i++) { b = 0x01; j = 0; while (b != i) { b = GF2m_mul(gf.alpha, b); j++; if (j > gf.ord-1) { return -1; // a not primitive } } // j = log_a(i) loga[i] = j; } return 0; } /* --------------------------------------------------------------------------------------------- */ /* static ui32_t f256 = 0x11D; static ui8_t f256FF = 0x1D; static int gf256_deg(ui8_t p) { int d = 7; // sizeof(p)*8 - 1 = 7 fuer ui8_t if (p == 0) return -0xFF; // deg(0) = -infty else { while (d && !(p & (1<>= 1; if (q & 1) h ^= p; // q_{i+1} > 0 ? } return h; } static int gf256_divmod(ui8_t p, ui8_t q, ui8_t *s, ui8_t *r) { int deg_p, deg_q = gf256_deg(q); // p = s*q + r *s = 0; if (q == 0) { *s = -1; *r = -1; return -1;} // DIV_BY_ZERO if (q == 1) { *s = p; *r = 0; } else { deg_p = gf256_deg(p); if (p == 0) { p = f256FF; // (ui8_t) f256 = f256 & 0xFF = f256FF deg_p = 8; // deg(f256) = 8 } while (deg_p >= deg_q) { *s |= 1 << (deg_p-deg_q); p ^= q << (deg_p-deg_q); deg_p = gf256_deg(p); } *r = p; } return 0; } static ui8_t gf256_inv(ui8_t a) { // 1 = x*a + y*f , ggT(a, f) = 1 ui8_t rem, rem1, rem2, aux, aux1, aux2, quo; if (a == 0) return 0; // nicht definiert; DIV_BY_ZERO if (a == 1) return 1; rem1 = a; rem2 = 0; // = f256 aux1 = 0x1; aux2 = 0x0; rem = rem1; aux = aux1; while (rem > 0x1) { gf256_divmod(rem2, rem1, &quo, &rem); aux = gf256_mul(quo, aux1) ^ aux2; // aux = aux2 - quo*aux1 rem2 = rem1; rem1 = rem; aux2 = aux1; aux1 = aux; } return aux; } */ /* --------------------------------------------------------------------------------------------- */ /* // F2[X] mod X^8 + X^4 + X^3 + X + 1 static ui8_t exp_11B[256] = { // 0x11B: a^n , a = 0x03 = X+1 0x01, 0x03, 0x05, 0x0F, 0x11, 0x33, 0x55, 0xFF, 0x1A, 0x2E, 0x72, 0x96, 0xA1, 0xF8, 0x13, 0x35, 0x5F, 0xE1, 0x38, 0x48, 0xD8, 0x73, 0x95, 0xA4, 0xF7, 0x02, 0x06, 0x0A, 0x1E, 0x22, 0x66, 0xAA, 0xE5, 0x34, 0x5C, 0xE4, 0x37, 0x59, 0xEB, 0x26, 0x6A, 0xBE, 0xD9, 0x70, 0x90, 0xAB, 0xE6, 0x31, 0x53, 0xF5, 0x04, 0x0C, 0x14, 0x3C, 0x44, 0xCC, 0x4F, 0xD1, 0x68, 0xB8, 0xD3, 0x6E, 0xB2, 0xCD, 0x4C, 0xD4, 0x67, 0xA9, 0xE0, 0x3B, 0x4D, 0xD7, 0x62, 0xA6, 0xF1, 0x08, 0x18, 0x28, 0x78, 0x88, 0x83, 0x9E, 0xB9, 0xD0, 0x6B, 0xBD, 0xDC, 0x7F, 0x81, 0x98, 0xB3, 0xCE, 0x49, 0xDB, 0x76, 0x9A, 0xB5, 0xC4, 0x57, 0xF9, 0x10, 0x30, 0x50, 0xF0, 0x0B, 0x1D, 0x27, 0x69, 0xBB, 0xD6, 0x61, 0xA3, 0xFE, 0x19, 0x2B, 0x7D, 0x87, 0x92, 0xAD, 0xEC, 0x2F, 0x71, 0x93, 0xAE, 0xE9, 0x20, 0x60, 0xA0, 0xFB, 0x16, 0x3A, 0x4E, 0xD2, 0x6D, 0xB7, 0xC2, 0x5D, 0xE7, 0x32, 0x56, 0xFA, 0x15, 0x3F, 0x41, 0xC3, 0x5E, 0xE2, 0x3D, 0x47, 0xC9, 0x40, 0xC0, 0x5B, 0xED, 0x2C, 0x74, 0x9C, 0xBF, 0xDA, 0x75, 0x9F, 0xBA, 0xD5, 0x64, 0xAC, 0xEF, 0x2A, 0x7E, 0x82, 0x9D, 0xBC, 0xDF, 0x7A, 0x8E, 0x89, 0x80, 0x9B, 0xB6, 0xC1, 0x58, 0xE8, 0x23, 0x65, 0xAF, 0xEA, 0x25, 0x6F, 0xB1, 0xC8, 0x43, 0xC5, 0x54, 0xFC, 0x1F, 0x21, 0x63, 0xA5, 0xF4, 0x07, 0x09, 0x1B, 0x2D, 0x77, 0x99, 0xB0, 0xCB, 0x46, 0xCA, 0x45, 0xCF, 0x4A, 0xDE, 0x79, 0x8B, 0x86, 0x91, 0xA8, 0xE3, 0x3E, 0x42, 0xC6, 0x51, 0xF3, 0x0E, 0x12, 0x36, 0x5A, 0xEE, 0x29, 0x7B, 0x8D, 0x8C, 0x8F, 0x8A, 0x85, 0x94, 0xA7, 0xF2, 0x0D, 0x17, 0x39, 0x4B, 0xDD, 0x7C, 0x84, 0x97, 0xA2, 0xFD, 0x1C, 0x24, 0x6C, 0xB4, 0xC7, 0x52, 0xF6, 0x01}; static ui8_t log_11B[256] = { // 0x11B: log_a , a = 0x03 = X+1 -00 , 0x00, 0x19, 0x01, 0x32, 0x02, 0x1A, 0xC6, 0x4B, 0xC7, 0x1B, 0x68, 0x33, 0xEE, 0xDF, 0x03, 0x64, 0x04, 0xE0, 0x0E, 0x34, 0x8D, 0x81, 0xEF, 0x4C, 0x71, 0x08, 0xC8, 0xF8, 0x69, 0x1C, 0xC1, 0x7D, 0xC2, 0x1D, 0xB5, 0xF9, 0xB9, 0x27, 0x6A, 0x4D, 0xE4, 0xA6, 0x72, 0x9A, 0xC9, 0x09, 0x78, 0x65, 0x2F, 0x8A, 0x05, 0x21, 0x0F, 0xE1, 0x24, 0x12, 0xF0, 0x82, 0x45, 0x35, 0x93, 0xDA, 0x8E, 0x96, 0x8F, 0xDB, 0xBD, 0x36, 0xD0, 0xCE, 0x94, 0x13, 0x5C, 0xD2, 0xF1, 0x40, 0x46, 0x83, 0x38, 0x66, 0xDD, 0xFD, 0x30, 0xBF, 0x06, 0x8B, 0x62, 0xB3, 0x25, 0xE2, 0x98, 0x22, 0x88, 0x91, 0x10, 0x7E, 0x6E, 0x48, 0xC3, 0xA3, 0xB6, 0x1E, 0x42, 0x3A, 0x6B, 0x28, 0x54, 0xFA, 0x85, 0x3D, 0xBA, 0x2B, 0x79, 0x0A, 0x15, 0x9B, 0x9F, 0x5E, 0xCA, 0x4E, 0xD4, 0xAC, 0xE5, 0xF3, 0x73, 0xA7, 0x57, 0xAF, 0x58, 0xA8, 0x50, 0xF4, 0xEA, 0xD6, 0x74, 0x4F, 0xAE, 0xE9, 0xD5, 0xE7, 0xE6, 0xAD, 0xE8, 0x2C, 0xD7, 0x75, 0x7A, 0xEB, 0x16, 0x0B, 0xF5, 0x59, 0xCB, 0x5F, 0xB0, 0x9C, 0xA9, 0x51, 0xA0, 0x7F, 0x0C, 0xF6, 0x6F, 0x17, 0xC4, 0x49, 0xEC, 0xD8, 0x43, 0x1F, 0x2D, 0xA4, 0x76, 0x7B, 0xB7, 0xCC, 0xBB, 0x3E, 0x5A, 0xFB, 0x60, 0xB1, 0x86, 0x3B, 0x52, 0xA1, 0x6C, 0xAA, 0x55, 0x29, 0x9D, 0x97, 0xB2, 0x87, 0x90, 0x61, 0xBE, 0xDC, 0xFC, 0xBC, 0x95, 0xCF, 0xCD, 0x37, 0x3F, 0x5B, 0xD1, 0x53, 0x39, 0x84, 0x3C, 0x41, 0xA2, 0x6D, 0x47, 0x14, 0x2A, 0x9E, 0x5D, 0x56, 0xF2, 0xD3, 0xAB, 0x44, 0x11, 0x92, 0xD9, 0x23, 0x20, 0x2E, 0x89, 0xB4, 0x7C, 0xB8, 0x26, 0x77, 0x99, 0xE3, 0xA5, 0x67, 0x4A, 0xED, 0xDE, 0xC5, 0x31, 0xFE, 0x18, 0x0D, 0x63, 0x8C, 0x80, 0xC0, 0xF7, 0x70, 0x07}; // ------------------------------------------------------------------------------------------------ // F2[X] mod X^8 + X^4 + X^3 + X^2 + 1 static ui8_t exp_11D[256] = { // 0x11D: a^n , a = 0x02 = X 0x01, 0x02, 0x04, 0x08, 0x10, 0x20, 0x40, 0x80, 0x1D, 0x3A, 0x74, 0xE8, 0xCD, 0x87, 0x13, 0x26, 0x4C, 0x98, 0x2D, 0x5A, 0xB4, 0x75, 0xEA, 0xC9, 0x8F, 0x03, 0x06, 0x0C, 0x18, 0x30, 0x60, 0xC0, 0x9D, 0x27, 0x4E, 0x9C, 0x25, 0x4A, 0x94, 0x35, 0x6A, 0xD4, 0xB5, 0x77, 0xEE, 0xC1, 0x9F, 0x23, 0x46, 0x8C, 0x05, 0x0A, 0x14, 0x28, 0x50, 0xA0, 0x5D, 0xBA, 0x69, 0xD2, 0xB9, 0x6F, 0xDE, 0xA1, 0x5F, 0xBE, 0x61, 0xC2, 0x99, 0x2F, 0x5E, 0xBC, 0x65, 0xCA, 0x89, 0x0F, 0x1E, 0x3C, 0x78, 0xF0, 0xFD, 0xE7, 0xD3, 0xBB, 0x6B, 0xD6, 0xB1, 0x7F, 0xFE, 0xE1, 0xDF, 0xA3, 0x5B, 0xB6, 0x71, 0xE2, 0xD9, 0xAF, 0x43, 0x86, 0x11, 0x22, 0x44, 0x88, 0x0D, 0x1A, 0x34, 0x68, 0xD0, 0xBD, 0x67, 0xCE, 0x81, 0x1F, 0x3E, 0x7C, 0xF8, 0xED, 0xC7, 0x93, 0x3B, 0x76, 0xEC, 0xC5, 0x97, 0x33, 0x66, 0xCC, 0x85, 0x17, 0x2E, 0x5C, 0xB8, 0x6D, 0xDA, 0xA9, 0x4F, 0x9E, 0x21, 0x42, 0x84, 0x15, 0x2A, 0x54, 0xA8, 0x4D, 0x9A, 0x29, 0x52, 0xA4, 0x55, 0xAA, 0x49, 0x92, 0x39, 0x72, 0xE4, 0xD5, 0xB7, 0x73, 0xE6, 0xD1, 0xBF, 0x63, 0xC6, 0x91, 0x3F, 0x7E, 0xFC, 0xE5, 0xD7, 0xB3, 0x7B, 0xF6, 0xF1, 0xFF, 0xE3, 0xDB, 0xAB, 0x4B, 0x96, 0x31, 0x62, 0xC4, 0x95, 0x37, 0x6E, 0xDC, 0xA5, 0x57, 0xAE, 0x41, 0x82, 0x19, 0x32, 0x64, 0xC8, 0x8D, 0x07, 0x0E, 0x1C, 0x38, 0x70, 0xE0, 0xDD, 0xA7, 0x53, 0xA6, 0x51, 0xA2, 0x59, 0xB2, 0x79, 0xF2, 0xF9, 0xEF, 0xC3, 0x9B, 0x2B, 0x56, 0xAC, 0x45, 0x8A, 0x09, 0x12, 0x24, 0x48, 0x90, 0x3D, 0x7A, 0xF4, 0xF5, 0xF7, 0xF3, 0xFB, 0xEB, 0xCB, 0x8B, 0x0B, 0x16, 0x2C, 0x58, 0xB0, 0x7D, 0xFA, 0xE9, 0xCF, 0x83, 0x1B, 0x36, 0x6C, 0xD8, 0xAD, 0x47, 0x8E, 0x01}; static ui8_t log_11D[256] = { // 0x11D: log_a , a = 0x02 = X -00 , 0x00, 0x01, 0x19, 0x02, 0x32, 0x1A, 0xC6, 0x03, 0xDF, 0x33, 0xEE, 0x1B, 0x68, 0xC7, 0x4B, 0x04, 0x64, 0xE0, 0x0E, 0x34, 0x8D, 0xEF, 0x81, 0x1C, 0xC1, 0x69, 0xF8, 0xC8, 0x08, 0x4C, 0x71, 0x05, 0x8A, 0x65, 0x2F, 0xE1, 0x24, 0x0F, 0x21, 0x35, 0x93, 0x8E, 0xDA, 0xF0, 0x12, 0x82, 0x45, 0x1D, 0xB5, 0xC2, 0x7D, 0x6A, 0x27, 0xF9, 0xB9, 0xC9, 0x9A, 0x09, 0x78, 0x4D, 0xE4, 0x72, 0xA6, 0x06, 0xBF, 0x8B, 0x62, 0x66, 0xDD, 0x30, 0xFD, 0xE2, 0x98, 0x25, 0xB3, 0x10, 0x91, 0x22, 0x88, 0x36, 0xD0, 0x94, 0xCE, 0x8F, 0x96, 0xDB, 0xBD, 0xF1, 0xD2, 0x13, 0x5C, 0x83, 0x38, 0x46, 0x40, 0x1E, 0x42, 0xB6, 0xA3, 0xC3, 0x48, 0x7E, 0x6E, 0x6B, 0x3A, 0x28, 0x54, 0xFA, 0x85, 0xBA, 0x3D, 0xCA, 0x5E, 0x9B, 0x9F, 0x0A, 0x15, 0x79, 0x2B, 0x4E, 0xD4, 0xE5, 0xAC, 0x73, 0xF3, 0xA7, 0x57, 0x07, 0x70, 0xC0, 0xF7, 0x8C, 0x80, 0x63, 0x0D, 0x67, 0x4A, 0xDE, 0xED, 0x31, 0xC5, 0xFE, 0x18, 0xE3, 0xA5, 0x99, 0x77, 0x26, 0xB8, 0xB4, 0x7C, 0x11, 0x44, 0x92, 0xD9, 0x23, 0x20, 0x89, 0x2E, 0x37, 0x3F, 0xD1, 0x5B, 0x95, 0xBC, 0xCF, 0xCD, 0x90, 0x87, 0x97, 0xB2, 0xDC, 0xFC, 0xBE, 0x61, 0xF2, 0x56, 0xD3, 0xAB, 0x14, 0x2A, 0x5D, 0x9E, 0x84, 0x3C, 0x39, 0x53, 0x47, 0x6D, 0x41, 0xA2, 0x1F, 0x2D, 0x43, 0xD8, 0xB7, 0x7B, 0xA4, 0x76, 0xC4, 0x17, 0x49, 0xEC, 0x7F, 0x0C, 0x6F, 0xF6, 0x6C, 0xA1, 0x3B, 0x52, 0x29, 0x9D, 0x55, 0xAA, 0xFB, 0x60, 0x86, 0xB1, 0xBB, 0xCC, 0x3E, 0x5A, 0xCB, 0x59, 0x5F, 0xB0, 0x9C, 0xA9, 0xA0, 0x51, 0x0B, 0xF5, 0x16, 0xEB, 0x7A, 0x75, 0x2C, 0xD7, 0x4F, 0xAE, 0xD5, 0xE9, 0xE6, 0xE7, 0xAD, 0xE8, 0x74, 0xD6, 0xF4, 0xEA, 0xA8, 0x50, 0x58, 0xAF}; // ------------------------------------------------------------------------------------------------ // F2[X] mod X^6 + X + 1 : 0x43 static ui8_t exp64[64] = { // 0x43: a^n , a = 0x02 = X 0x01, 0x02, 0x04, 0x08, 0x10, 0x20, 0x03, 0x06, 0x0C, 0x18, 0x30, 0x23, 0x05, 0x0A, 0x14, 0x28, 0x13, 0x26, 0x0F, 0x1E, 0x3C, 0x3B, 0x35, 0x29, 0x11, 0x22, 0x07, 0x0E, 0x1C, 0x38, 0x33, 0x25, 0x09, 0x12, 0x24, 0x0B, 0x16, 0x2C, 0x1B, 0x36, 0x2F, 0x1D, 0x3A, 0x37, 0x2D, 0x19, 0x32, 0x27, 0x0D, 0x1A, 0x34, 0x2B, 0x15, 0x2A, 0x17, 0x2E, 0x1F, 0x3E, 0x3F, 0x3D, 0x39, 0x31, 0x21, 0x01}; static ui8_t log64[64] = { -00 , 0x00, 0x01, 0x06, 0x02, 0x0C, 0x07, 0x1A, 0x03, 0x20, 0x0D, 0x23, 0x08, 0x30, 0x1B, 0x12, 0x04, 0x18, 0x21, 0x10, 0x0E, 0x34, 0x24, 0x36, 0x09, 0x2D, 0x31, 0x26, 0x1C, 0x29, 0x13, 0x38, 0x05, 0x3E, 0x19, 0x0B, 0x22, 0x1F, 0x11, 0x2F, 0x0F, 0x17, 0x35, 0x33, 0x25, 0x2C, 0x37, 0x28, 0x0A, 0x3D, 0x2E, 0x1E, 0x32, 0x16, 0x27, 0x2B, 0x1D, 0x3C, 0x2A, 0x15, 0x14, 0x3B, 0x39, 0x3A}; // ------------------------------------------------------------------------------------------------ // F2[X] mod X^4 + X + 1 : 0x13 static ui8_t exp16[16] = { // 0x43: a^n , a = 0x02 = X 0x1, 0x2, 0x4, 0x8, 0x3, 0x6, 0xC, 0xB, 0x5, 0xA, 0x7, 0xE, 0xF, 0xD, 0x9, 0x1}; static ui8_t log16[16] = { -00, 0x0, 0x1, 0x4, 0x2, 0x8, 0x5, 0xA, 0x3, 0xE, 0x9, 0x7, 0x6, 0xD, 0xB, 0xC}; */ /* --------------------------------------------------------------------------------------------- */ static ui8_t GF_mul(ui8_t p, ui8_t q) { ui32_t x; if ((p == 0) || (q == 0)) return 0; x = (ui32_t)log_a[p] + log_a[q]; return exp_a[x % (GF.ord-1)]; // a^(ord-1) = 1 } static ui8_t GF_inv(ui8_t p) { if (p == 0) return 0; // DIV_BY_ZERO return exp_a[GF.ord-1-log_a[p]]; // a^(ord-1) = 1 } /* --------------------------------------------------------------------------------------------- */ /* * p(x) = p[0] + p[1]x + ... + p[N-1]x^(N-1) */ static ui8_t poly_eval(ui8_t poly[], ui8_t x) { int n; ui8_t xn, y; y = poly[0]; if (x != 0) { for (n = 1; n < GF.ord-1; n++) { xn = exp_a[(n*log_a[x]) % (GF.ord-1)]; y ^= GF_mul(poly[n], xn); } } return y; } static ui8_t poly_evalH(ui8_t poly[], ui8_t x) { int n; ui8_t y; y = poly[GF.ord-1]; for (n = GF.ord-2; n >= 0; n--) { y = GF_mul(y, x) ^ poly[n]; } return y; } static int poly_deg(ui8_t p[]) { int n = MAX_DEG; while (p[n] == 0 && n > 0) n--; if (p[n] == 0) n--; // deg(0) = -inf return n; } static int poly_divmod(ui8_t p[], ui8_t q[], ui8_t *d, ui8_t *r) { int deg_p, deg_q; // p(x) = q(x)d(x) + r(x) int i; // deg(r) < deg(q) ui8_t c; deg_p = poly_deg(p); deg_q = poly_deg(q); if (deg_q < 0) return -1; // q=0: DIV_BY_ZERO for (i = 0; i <= MAX_DEG; i++) d[i] = 0; for (i = 0; i <= MAX_DEG; i++) r[i] = 0; c = GF_mul( p[deg_p], GF_inv(q[deg_q])); if (deg_q == 0) { for (i = 0; i <= deg_p; i++) d[i] = GF_mul(p[i], c); for (i = 0; i <= MAX_DEG; i++) r[i] = 0; } else if (deg_p < 0) { // p=0 for (i = 0; i <= MAX_DEG; i++) { d[i] = 0; r[i] = 0; } } else if (deg_p < deg_q) { for (i = 0; i <= MAX_DEG; i++) d[i] = 0; for (i = 0; i <= deg_p; i++) r[i] = p[i]; // r(x)=p(x), deg(r)= deg_q) { d[deg_p-deg_q] = c; for (i = 0; i <= deg_q; i++) { r[deg_p-i] ^= GF_mul( q[deg_q-i], c); } while (r[deg_p] == 0 && deg_p > 0) deg_p--; if (r[deg_p] == 0) deg_p--; if (deg_p >= 0) c = GF_mul( r[deg_p], GF_inv(q[deg_q])); } } return 0; } static int poly_add(ui8_t a[], ui8_t b[], ui8_t *sum) { int i; ui8_t c[MAX_DEG+1]; for (i = 0; i <= MAX_DEG; i++) { c[i] = a[i] ^ b[i]; } for (i = 0; i <= MAX_DEG; i++) { sum[i] = c[i]; } return 0; } static int poly_mul(ui8_t a[], ui8_t b[], ui8_t *ab) { int i, j; ui8_t c[MAX_DEG+1]; if (poly_deg(a)+poly_deg(b) > MAX_DEG) { return -1; } for (i = 0; i <= MAX_DEG; i++) { c[i] = 0; } for (i = 0; i <= poly_deg(a); i++) { for (j = 0; j <= poly_deg(b); j++) { c[i+j] ^= GF_mul(a[i], b[j]); } } for (i = 0; i <= MAX_DEG; i++) { ab[i] = c[i]; } return 0; } static int polyGF_eggT(int deg, ui8_t a[], ui8_t b[], // in ui8_t *u, ui8_t *v, ui8_t *ggt // out ) { // deg = 0: // a(x)u(x)+b(x)v(x) = ggt(x) // RS: // deg=t, a(x)=S(x), b(x)=x^2t int i; ui8_t r0[MAX_DEG+1], r1[MAX_DEG+1], r2[MAX_DEG+1], s0[MAX_DEG+1], s1[MAX_DEG+1], s2[MAX_DEG+1], t0[MAX_DEG+1], t1[MAX_DEG+1], t2[MAX_DEG+1], quo[MAX_DEG+1]; for (i = 0; i <= MAX_DEG; i++) { u[i] = 0; } for (i = 0; i <= MAX_DEG; i++) { v[i] = 0; } for (i = 0; i <= MAX_DEG; i++) { ggt[i] = 0; } for (i = 0; i <= MAX_DEG; i++) { r0[i] = a[i]; } for (i = 0; i <= MAX_DEG; i++) { r1[i] = b[i]; } s0[0] = 1; for (i = 1; i <= MAX_DEG; i++) { s0[i] = 0; } // s0 = 1 s1[0] = 0; for (i = 1; i <= MAX_DEG; i++) { s1[i] = 0; } // s1 = 0 t0[0] = 0; for (i = 1; i <= MAX_DEG; i++) { t0[i] = 0; } // t0 = 0 t1[0] = 1; for (i = 1; i <= MAX_DEG; i++) { t1[i] = 0; } // t1 = 1 for (i = 0; i <= MAX_DEG; i++) { r2[i] = 0; } for (i = 0; i <= MAX_DEG; i++) { s2[i] = 0; } for (i = 0; i <= MAX_DEG; i++) { t2[i] = 0; } while ( poly_deg(r1) >= deg ) { poly_divmod(r0, r1, quo, r2); for (i = 0; i <= MAX_DEG; i++) { r0[i] = r1[i]; } for (i = 0; i <= MAX_DEG; i++) { r1[i] = r2[i]; } poly_mul(quo, s1, s2); poly_add(s0, s2, s2); for (i = 0; i <= MAX_DEG; i++) { s0[i] = s1[i]; } for (i = 0; i <= MAX_DEG; i++) { s1[i] = s2[i]; } poly_mul(quo, t1, t2); poly_add(t0, t2, t2); for (i = 0; i <= MAX_DEG; i++) { t0[i] = t1[i]; } for (i = 0; i <= MAX_DEG; i++) { t1[i] = t2[i]; } } if (deg > 0) { for (i = 0; i <= MAX_DEG; i++) { ggt[i] = r1[i]; } // deg=0: r0 for (i = 0; i <= MAX_DEG; i++) { u[i] = s1[i]; } // deg=0: s0 for (i = 0; i <= MAX_DEG; i++) { v[i] = t1[i]; } } else { for (i = 0; i <= MAX_DEG; i++) { ggt[i] = r0[i]; } for (i = 0; i <= MAX_DEG; i++) { u[i] = s0[i]; } for (i = 0; i <= MAX_DEG; i++) { v[i] = t0[i]; } } return 0; } static int polyGF_lfsr(int deg, int x2t, ui8_t S[], ui8_t *Lambda, ui8_t *Omega ) { // BCH/RS/LFSR: deg=t+e/2, e=#erasures // S(x)Lambda(x) = Omega(x) mod x^2t int i; ui8_t r0[MAX_DEG+1], r1[MAX_DEG+1], r2[MAX_DEG+1], s0[MAX_DEG+1], s1[MAX_DEG+1], s2[MAX_DEG+1], quo[MAX_DEG+1]; for (i = 0; i <= MAX_DEG; i++) { Lambda[i] = 0; } for (i = 0; i <= MAX_DEG; i++) { Omega[i] = 0; } for (i = 0; i <= MAX_DEG; i++) { r0[i] = S[i]; } for (i = 0; i <= MAX_DEG; i++) { r1[i] = 0; } r1[x2t] = 1; //x^2t s0[0] = 1; for (i = 1; i <= MAX_DEG; i++) { s0[i] = 0; } s1[0] = 0; for (i = 1; i <= MAX_DEG; i++) { s1[i] = 0; } for (i = 0; i <= MAX_DEG; i++) { r2[i] = 0; } for (i = 0; i <= MAX_DEG; i++) { s2[i] = 0; } while ( poly_deg(r1) >= deg ) { // deg=t+e/2 poly_divmod(r0, r1, quo, r2); for (i = 0; i <= MAX_DEG; i++) { r0[i] = r1[i]; } for (i = 0; i <= MAX_DEG; i++) { r1[i] = r2[i]; } poly_mul(quo, s1, s2); poly_add(s0, s2, s2); for (i = 0; i <= MAX_DEG; i++) { s0[i] = s1[i]; } for (i = 0; i <= MAX_DEG; i++) { s1[i] = s2[i]; } } // deg > 0: for (i = 0; i <= MAX_DEG; i++) { Omega[i] = r1[i]; } for (i = 0; i <= MAX_DEG; i++) { Lambda[i] = s1[i]; } return 0; } static int poly_D(ui8_t a[], ui8_t *Da) { int i; for (i = 0; i <= MAX_DEG; i++) { Da[i] = 0; } // unten werden nicht immer // alle Koeffizienten gesetzt for (i = 1; i <= poly_deg(a); i++) { if (i % 2) Da[i-1] = a[i]; // GF(2^n): b+b=0 } return 0; } static ui8_t forney(ui8_t x, ui8_t Omega[], ui8_t Lambda[]) { ui8_t DLam[MAX_DEG+1]; ui8_t w, z, Y; // x=X^(-1), Y = x^(b-1) * Omega(x)/Lambda'(x) // Y = X^(1-b) * Omega(X^(-1))/Lambda'(X^(-1)) poly_D(Lambda, DLam); w = poly_eval(Omega, x); z = poly_eval(DLam, x); if (z == 0) { return -00; } Y = GF_mul(w, GF_inv(z)); if (RS.b == 0) Y = GF_mul(GF_inv(x), Y); else if (RS.b > 1) { ui8_t xb1 = exp_a[((RS.b-1)*log_a[x]) % (GF.ord-1)]; Y = GF_mul(xb1, Y); } return Y; } static int era_sigma(int n, ui8_t era_pos[], ui8_t *sigma) { int i; ui8_t Xa[MAX_DEG+1], sig[MAX_DEG+1]; ui8_t a_i; for (i = 0; i <= MAX_DEG; i++) sig[i] = 0; for (i = 0; i <= MAX_DEG; i++) Xa[i] = 0; // sigma(X)=(1 - alpha^j1 X)...(1 - alpha^jn X) // j_{i+1} = era_pos[i] sig[0] = 1; Xa[0] = 1; for (i = 0; i < n; i++) { // n <= 2*RS.t a_i = exp_a[(RS.p*era_pos[i]) % (GF.ord-1)]; Xa[1] = a_i; // Xalp[0..1]: (1-alpha^(j_)X) poly_mul(sig, Xa, sig); } for (i = 0; i <= MAX_DEG; i++) sigma[i] = sig[i]; return 0; } static int syndromes(ui8_t cw[], ui8_t *S) { int i, errors = 0; ui8_t a_i; // syndromes: e_j=S((alpha^p)^(b+i)) (wie in g(X)) for (i = 0; i < 2*RS.t; i++) { a_i = exp_a[(RS.p*(RS.b+i)) % (GF.ord-1)]; // (alpha^p)^(b+i) S[i] = poly_eval(cw, a_i); if (S[i]) errors = 1; } return errors; } static int prn_GFpoly(ui32_t p) { int i, s; s = 0; if (p == 0) printf("0"); else { if (p != (p & 0x1FF)) printf(" (& 0x1FF) "); for (i=8; i>1; i--) { if ( (p>>i) & 1 ) { if (s) printf(" + "); printf("X^%d", i); s = 1; } } if ( (p>>1) & 1 ) { if (s) printf(" + "); printf("X"); s = 1; } if ( p & 1 ) { if (s) printf(" + "); printf("1"); } } return 0; } static void prn_table(void) { int i; printf("F2[X] mod "); prn_GFpoly(GF.f); printf(" : 0x%X\n", GF.f); printf("\n"); printf("a^n[%d] = {\n", GF.ord); printf(" "); for (i=0; i ip=116 for (i = 1; i < GF.ord-1; i++) { if ( (RS.p * i) % (GF.ord-1) == 1 ) { RS.ip = i; break; } } // g(X)=(X-(alpha^p)^b)...(X-(alpha^p)^(b+2t-1)), b=112 RS.g[0] = 0x01; Xalp[1] = 0x01; // X for (i = 0; i < 2*RS.t; i++) { Xalp[0] = exp_a[(RS.p*(RS.b+i)) % (GF.ord-1)]; // Xalp[0..1]: X - (alpha^p)^(b+i) poly_mul(RS.g, Xalp, RS.g); } /* RS.g[ 0] = RS.g[32] = exp_a[0]; RS.g[ 1] = RS.g[31] = exp_a[249]; RS.g[ 2] = RS.g[30] = exp_a[59]; RS.g[ 3] = RS.g[29] = exp_a[66]; RS.g[ 4] = RS.g[28] = exp_a[4]; RS.g[ 5] = RS.g[27] = exp_a[43]; RS.g[ 6] = RS.g[26] = exp_a[126]; RS.g[ 7] = RS.g[25] = exp_a[251]; RS.g[ 8] = RS.g[24] = exp_a[97]; RS.g[ 9] = RS.g[23] = exp_a[30]; RS.g[10] = RS.g[22] = exp_a[3]; RS.g[11] = RS.g[21] = exp_a[213]; RS.g[12] = RS.g[20] = exp_a[50]; RS.g[13] = RS.g[19] = exp_a[66]; RS.g[14] = RS.g[18] = exp_a[170]; RS.g[15] = RS.g[17] = exp_a[5]; RS.g[16] = exp_a[24]; */ return check_gen; } int rs_init_BCH64() { int i, check_gen; GF = GF64BCH; check_gen = GF_genTab( GF, exp_a, log_a); RS = BCH64; // N=63, t=2, b=1 for (i = 0; i <= MAX_DEG; i++) RS.g[i] = 0; // g(X)=X^12+X^10+X^8+X^5+X^4+X^3+1 // =(X^6+X+1)(X^6+X^4+X^2+X+1) RS.g[0] = RS.g[3] = RS.g[4] = RS.g[5] = RS.g[8] = RS.g[10] = RS.g[12] = 1; return check_gen; } int rs_init_RS15ccsds() { int i, check_gen; ui8_t Xalp[MAX_DEG+1]; GF = GF16RS; check_gen = GF_genTab( GF, exp_a, log_a); //RS = RS16_0; // N=15, t=3, b=0, p=1 RS = RS16ccsds; // N=15, t=2, b=6, p=1 for (i = 0; i <= MAX_DEG; i++) RS.g[i] = 0; for (i = 0; i <= MAX_DEG; i++) Xalp[i] = 0; // g(X)=(X-alpha^b)...(X-alpha^(b+2t-1)) RS.g[0] = 0x01; Xalp[1] = 0x01; // X for (i = 0; i < 2*RS.t; i++) { Xalp[0] = exp_a[(RS.b+i) % (GF.ord-1)]; // Xalp[0..1]: X - alpha^(b+i) poly_mul(RS.g, Xalp, RS.g); } return check_gen; } int rs_encode(ui8_t cw[]) { int j; ui8_t __cw[MAX_DEG+1], parity[MAX_DEG+1], d[MAX_DEG+1]; for (j = 0; j <= MAX_DEG; j++) parity[j] = 0; for (j = 0; j <= MAX_DEG; j++) __cw[j] = 0; for (j = RS.R; j < RS.N; j++) __cw[j] = cw[j]; poly_divmod(__cw, RS.g, d, parity); //if (poly_deg(parity) >= RS.R) return -1; for (j = 0; j < RS.R; j++) cw[j] = parity[j]; return 0; } // 2*Errors + Erasure <= 2*t int rs_decode_ErrEra(ui8_t cw[], int nera, ui8_t era_pos[], ui8_t *err_pos, ui8_t *err_val) { ui8_t x, gamma; ui8_t S[MAX_DEG+1], Lambda[MAX_DEG+1], Omega[MAX_DEG+1], sigma[MAX_DEG+1], sigLam[MAX_DEG+1]; int deg_sigLam, deg_Lambda, deg_Omega; int i, nerr, errera = 0; if (nera > 2*RS.t) { return -4; } for (i = 0; i < 2*RS.t; i++) { err_pos[i] = 0; } for (i = 0; i < 2*RS.t; i++) { err_val[i] = 0; } // IF: erasures set 0 // for (i = 0; i < nera; i++) cw[era_pos[i]] = 0x00; // erasures // THEN: restore cw[era_pos[i]], if errera < 0 for (i = 0; i <= MAX_DEG; i++) { S[i] = 0; } errera = syndromes(cw, S); // wenn S(x)=0 , dann poly_divmod(cw, RS.g, d, rem): rem=0 for (i = 0; i <= MAX_DEG; i++) { sigma[i] = 0; } sigma[0] = 1; if (nera > 0) { era_sigma(nera, era_pos, sigma); poly_mul(sigma, S, S); for (i = 2*RS.t; i <= MAX_DEG; i++) S[i] = 0; // S = sig*S mod x^2t } if (errera) { polyGF_lfsr(RS.t+nera/2, 2*RS.t, S, Lambda, Omega); deg_Lambda = poly_deg(Lambda); deg_Omega = poly_deg(Omega); if (deg_Omega >= deg_Lambda + nera) { errera = -3; return errera; } gamma = Lambda[0]; if (gamma) { for (i = deg_Lambda; i >= 0; i--) Lambda[i] = GF_mul(Lambda[i], GF_inv(gamma)); for (i = deg_Omega ; i >= 0; i--) Omega[i] = GF_mul( Omega[i], GF_inv(gamma)); poly_mul(sigma, Lambda, sigLam); deg_sigLam = poly_deg(sigLam); } else { errera = -2; return errera; } nerr = 0; // Errors + Erasures (erasure-pos bereits bekannt) for (i = 1; i < GF.ord ; i++) { // Lambda(0)=1 x = (ui8_t)i; // roll-over if (poly_eval(sigLam, x) == 0) { // Lambda(x)=0 fuer x in erasures[] moeglich // error location index ui8_t x1 = GF_inv(x); err_pos[nerr] = (log_a[x1]*RS.ip) % (GF.ord-1); // error value; bin-BCH: err_val=1 err_val[nerr] = forney(x, Omega, sigLam); //err_val[nerr] == 0, wenn era_val[pos]=0, d.h. cw[pos] schon korrekt nerr++; } if (nerr >= deg_sigLam) break; } // 2*Errors + Erasure <= 2*t if (nerr < deg_sigLam) errera = -1; // uncorrectable errors else { errera = nerr; for (i = 0; i < errera; i++) cw[err_pos[i]] ^= err_val[i]; } } return errera; } // Errors <= t int rs_decode(ui8_t cw[], ui8_t *err_pos, ui8_t *err_val) { ui8_t tmp[1] = {0}; return rs_decode_ErrEra(cw, 0, tmp, err_pos, err_val); } int rs_decode_bch_gf2t2(ui8_t cw[], ui8_t *err_pos, ui8_t *err_val) { // binary 2-error correcting BCH ui8_t x, gamma, S[MAX_DEG+1], L[MAX_DEG+1], L2, Lambda[MAX_DEG+1], Omega[MAX_DEG+1]; int i, n, errors = 0; for (i = 0; i < RS.t; i++) { err_pos[i] = 0; } for (i = 0; i < RS.t; i++) { err_val[i] = 0; } for (i = 0; i <= MAX_DEG; i++) { S[i] = 0; } errors = syndromes(cw, S); // wenn S(x)=0 , dann poly_divmod(cw, RS.g, d, rem): rem=0 if (errors) { polyGF_lfsr(RS.t, 2*RS.t, S, Lambda, Omega); gamma = Lambda[0]; if (gamma) { for (i = poly_deg(Lambda); i >= 0; i--) Lambda[i] = GF_mul(Lambda[i], GF_inv(gamma)); for (i = poly_deg(Omega) ; i >= 0; i--) Omega[i] = GF_mul( Omega[i], GF_inv(gamma)); } else { errors = -2; return errors; } // GF(2)-BCH, t=2: // S1 = S[0] // S1^2 = S2 , S2^2 = S4 // L(x) = 1 + L1 x + L2 x^2 (1-2 errors) // L1 = S1 , L2 = (S3 + S1^3)/S1 if ( RS.t == 2 ) { for (i = 0; i <= MAX_DEG; i++) { L[i] = 0; } L[0] = 1; L[1] = S[0]; L2 = GF_mul(S[0], S[0]); L2 = GF_mul(L2, S[0]); L2 ^= S[2]; L2 = GF_mul(L2, GF_inv(S[0])); L[2] = L2; if (S[1] != GF_mul(S[0],S[0]) || S[3] != GF_mul(S[1],S[1])) { errors = -2; return errors; } if (L[1] != Lambda[1] || L[2] != Lambda[2] ) { errors = -2; return errors; } } n = 0; for (i = 1; i < GF.ord ; i++) { // Lambda(0)=1 x = (ui8_t)i; // roll-over if (poly_eval(Lambda, x) == 0) { // error location index err_pos[n] = log_a[GF_inv(x)]; // error value; bin-BCH: err_val=1 err_val[n] = 1; // = forney(x, Omega, Lambda); n++; } if (n >= poly_deg(Lambda)) break; } if (n < poly_deg(Lambda)) errors = -1; // uncorrectable errors else { errors = n; for (i = 0; i < errors; i++) cw[err_pos[i]] ^= err_val[i]; } } return errors; }