/* * * Copyright (c) 1993 Ning and David Mosberger. This is based on code originally written by Bas Laarhoven (bas@vimec.nl) and David L. Brown, Jr., and incorporates improvements suggested by Kai Harrekilde-Petersen. This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this program; see the file COPYING. If not, write to the Free Software Foundation, 675 Mass Ave, Cambridge, MA 02139, USA. * * $Source: /homes/cvs/ftape-stacked/ftape/lowlevel/ftape-ecc.c,v $ * $Revision: 1.3 $ * $Date: 1997/10/05 19:18:10 $ * * This file contains the Reed-Solomon error correction code * for the QIC-40/80 floppy-tape driver for Linux. */ #include #include "../lowlevel/ftape-tracing.h" #include "../lowlevel/ftape-ecc.h" /* Machines that are big-endian should define macro BIG_ENDIAN. * Unfortunately, there doesn't appear to be a standard include file * that works for all OSs. */ #if defined(__sparc__) || defined(__hppa) #define BIG_ENDIAN #endif /* __sparc__ || __hppa */ #if defined(__mips__) #error Find a smart way to determine the Endianness of the MIPS CPU #endif /* Notice: to minimize the potential for confusion, we use r to * denote the independent variable of the polynomials in the * Galois Field GF(2^8). We reserve x for polynomials that * that have coefficients in GF(2^8). * * The Galois Field in which coefficient arithmetic is performed are * the polynomials over Z_2 (i.e., 0 and 1) modulo the irreducible * polynomial f(r), where f(r)=r^8 + r^7 + r^2 + r + 1. A polynomial * is represented as a byte with the MSB as the coefficient of r^7 and * the LSB as the coefficient of r^0. For example, the binary * representation of f(x) is 0x187 (of course, this doesn't fit into 8 * bits). In this field, the polynomial r is a primitive element. * That is, r^i with i in 0,...,255 enumerates all elements in the * field. * * The generator polynomial for the QIC-80 ECC is * * g(x) = x^3 + r^105*x^2 + r^105*x + 1 * * which can be factored into: * * g(x) = (x-r^-1)(x-r^0)(x-r^1) * * the byte representation of the coefficients are: * * r^105 = 0xc0 * r^-1 = 0xc3 * r^0 = 0x01 * r^1 = 0x02 * * Notice that r^-1 = r^254 as exponent arithmetic is performed * modulo 2^8-1 = 255. * * For more information on Galois Fields and Reed-Solomon codes, refer * to any good book. I found _An Introduction to Error Correcting * Codes with Applications_ by S. A. Vanstone and P. C. van Oorschot * to be a good introduction into the former. _CODING THEORY: The * Essentials_ I found very useful for its concise description of * Reed-Solomon encoding/decoding. * */ typedef __u8 Matrix[3][3]; /* * gfpow[] is defined such that gfpow[i] returns r^i if * i is in the range [0..255]. */ static const __u8 gfpow[] = { 0x01, 0x02, 0x04, 0x08, 0x10, 0x20, 0x40, 0x80, 0x87, 0x89, 0x95, 0xad, 0xdd, 0x3d, 0x7a, 0xf4, 0x6f, 0xde, 0x3b, 0x76, 0xec, 0x5f, 0xbe, 0xfb, 0x71, 0xe2, 0x43, 0x86, 0x8b, 0x91, 0xa5, 0xcd, 0x1d, 0x3a, 0x74, 0xe8, 0x57, 0xae, 0xdb, 0x31, 0x62, 0xc4, 0x0f, 0x1e, 0x3c, 0x78, 0xf0, 0x67, 0xce, 0x1b, 0x36, 0x6c, 0xd8, 0x37, 0x6e, 0xdc, 0x3f, 0x7e, 0xfc, 0x7f, 0xfe, 0x7b, 0xf6, 0x6b, 0xd6, 0x2b, 0x56, 0xac, 0xdf, 0x39, 0x72, 0xe4, 0x4f, 0x9e, 0xbb, 0xf1, 0x65, 0xca, 0x13, 0x26, 0x4c, 0x98, 0xb7, 0xe9, 0x55, 0xaa, 0xd3, 0x21, 0x42, 0x84, 0x8f, 0x99, 0xb5, 0xed, 0x5d, 0xba, 0xf3, 0x61, 0xc2, 0x03, 0x06, 0x0c, 0x18, 0x30, 0x60, 0xc0, 0x07, 0x0e, 0x1c, 0x38, 0x70, 0xe0, 0x47, 0x8e, 0x9b, 0xb1, 0xe5, 0x4d, 0x9a, 0xb3, 0xe1, 0x45, 0x8a, 0x93, 0xa1, 0xc5, 0x0d, 0x1a, 0x34, 0x68, 0xd0, 0x27, 0x4e, 0x9c, 0xbf, 0xf9, 0x75, 0xea, 0x53, 0xa6, 0xcb, 0x11, 0x22, 0x44, 0x88, 0x97, 0xa9, 0xd5, 0x2d, 0x5a, 0xb4, 0xef, 0x59, 0xb2, 0xe3, 0x41, 0x82, 0x83, 0x81, 0x85, 0x8d, 0x9d, 0xbd, 0xfd, 0x7d, 0xfa, 0x73, 0xe6, 0x4b, 0x96, 0xab, 0xd1, 0x25, 0x4a, 0x94, 0xaf, 0xd9, 0x35, 0x6a, 0xd4, 0x2f, 0x5e, 0xbc, 0xff, 0x79, 0xf2, 0x63, 0xc6, 0x0b, 0x16, 0x2c, 0x58, 0xb0, 0xe7, 0x49, 0x92, 0xa3, 0xc1, 0x05, 0x0a, 0x14, 0x28, 0x50, 0xa0, 0xc7, 0x09, 0x12, 0x24, 0x48, 0x90, 0xa7, 0xc9, 0x15, 0x2a, 0x54, 0xa8, 0xd7, 0x29, 0x52, 0xa4, 0xcf, 0x19, 0x32, 0x64, 0xc8, 0x17, 0x2e, 0x5c, 0xb8, 0xf7, 0x69, 0xd2, 0x23, 0x46, 0x8c, 0x9f, 0xb9, 0xf5, 0x6d, 0xda, 0x33, 0x66, 0xcc, 0x1f, 0x3e, 0x7c, 0xf8, 0x77, 0xee, 0x5b, 0xb6, 0xeb, 0x51, 0xa2, 0xc3, 0x01 }; /* * This is a log table. That is, gflog[r^i] returns i (modulo f(r)). * gflog[0] is undefined and the first element is therefore not valid. */ static const __u8 gflog[256] = { 0xff, 0x00, 0x01, 0x63, 0x02, 0xc6, 0x64, 0x6a, 0x03, 0xcd, 0xc7, 0xbc, 0x65, 0x7e, 0x6b, 0x2a, 0x04, 0x8d, 0xce, 0x4e, 0xc8, 0xd4, 0xbd, 0xe1, 0x66, 0xdd, 0x7f, 0x31, 0x6c, 0x20, 0x2b, 0xf3, 0x05, 0x57, 0x8e, 0xe8, 0xcf, 0xac, 0x4f, 0x83, 0xc9, 0xd9, 0xd5, 0x41, 0xbe, 0x94, 0xe2, 0xb4, 0x67, 0x27, 0xde, 0xf0, 0x80, 0xb1, 0x32, 0x35, 0x6d, 0x45, 0x21, 0x12, 0x2c, 0x0d, 0xf4, 0x38, 0x06, 0x9b, 0x58, 0x1a, 0x8f, 0x79, 0xe9, 0x70, 0xd0, 0xc2, 0xad, 0xa8, 0x50, 0x75, 0x84, 0x48, 0xca, 0xfc, 0xda, 0x8a, 0xd6, 0x54, 0x42, 0x24, 0xbf, 0x98, 0x95, 0xf9, 0xe3, 0x5e, 0xb5, 0x15, 0x68, 0x61, 0x28, 0xba, 0xdf, 0x4c, 0xf1, 0x2f, 0x81, 0xe6, 0xb2, 0x3f, 0x33, 0xee, 0x36, 0x10, 0x6e, 0x18, 0x46, 0xa6, 0x22, 0x88, 0x13, 0xf7, 0x2d, 0xb8, 0x0e, 0x3d, 0xf5, 0xa4, 0x39, 0x3b, 0x07, 0x9e, 0x9c, 0x9d, 0x59, 0x9f, 0x1b, 0x08, 0x90, 0x09, 0x7a, 0x1c, 0xea, 0xa0, 0x71, 0x5a, 0xd1, 0x1d, 0xc3, 0x7b, 0xae, 0x0a, 0xa9, 0x91, 0x51, 0x5b, 0x76, 0x72, 0x85, 0xa1, 0x49, 0xeb, 0xcb, 0x7c, 0xfd, 0xc4, 0xdb, 0x1e, 0x8b, 0xd2, 0xd7, 0x92, 0x55, 0xaa, 0x43, 0x0b, 0x25, 0xaf, 0xc0, 0x73, 0x99, 0x77, 0x96, 0x5c, 0xfa, 0x52, 0xe4, 0xec, 0x5f, 0x4a, 0xb6, 0xa2, 0x16, 0x86, 0x69, 0xc5, 0x62, 0xfe, 0x29, 0x7d, 0xbb, 0xcc, 0xe0, 0xd3, 0x4d, 0x8c, 0xf2, 0x1f, 0x30, 0xdc, 0x82, 0xab, 0xe7, 0x56, 0xb3, 0x93, 0x40, 0xd8, 0x34, 0xb0, 0xef, 0x26, 0x37, 0x0c, 0x11, 0x44, 0x6f, 0x78, 0x19, 0x9a, 0x47, 0x74, 0xa7, 0xc1, 0x23, 0x53, 0x89, 0xfb, 0x14, 0x5d, 0xf8, 0x97, 0x2e, 0x4b, 0xb9, 0x60, 0x0f, 0xed, 0x3e, 0xe5, 0xf6, 0x87, 0xa5, 0x17, 0x3a, 0xa3, 0x3c, 0xb7 }; /* This is a multiplication table for the factor 0xc0 (i.e., r^105 (mod f(r)). * gfmul_c0[f] returns r^105 * f(r) (modulo f(r)). */ static const __u8 gfmul_c0[256] = { 0x00, 0xc0, 0x07, 0xc7, 0x0e, 0xce, 0x09, 0xc9, 0x1c, 0xdc, 0x1b, 0xdb, 0x12, 0xd2, 0x15, 0xd5, 0x38, 0xf8, 0x3f, 0xff, 0x36, 0xf6, 0x31, 0xf1, 0x24, 0xe4, 0x23, 0xe3, 0x2a, 0xea, 0x2d, 0xed, 0x70, 0xb0, 0x77, 0xb7, 0x7e, 0xbe, 0x79, 0xb9, 0x6c, 0xac, 0x6b, 0xab, 0x62, 0xa2, 0x65, 0xa5, 0x48, 0x88, 0x4f, 0x8f, 0x46, 0x86, 0x41, 0x81, 0x54, 0x94, 0x53, 0x93, 0x5a, 0x9a, 0x5d, 0x9d, 0xe0, 0x20, 0xe7, 0x27, 0xee, 0x2e, 0xe9, 0x29, 0xfc, 0x3c, 0xfb, 0x3b, 0xf2, 0x32, 0xf5, 0x35, 0xd8, 0x18, 0xdf, 0x1f, 0xd6, 0x16, 0xd1, 0x11, 0xc4, 0x04, 0xc3, 0x03, 0xca, 0x0a, 0xcd, 0x0d, 0x90, 0x50, 0x97, 0x57, 0x9e, 0x5e, 0x99, 0x59, 0x8c, 0x4c, 0x8b, 0x4b, 0x82, 0x42, 0x85, 0x45, 0xa8, 0x68, 0xaf, 0x6f, 0xa6, 0x66, 0xa1, 0x61, 0xb4, 0x74, 0xb3, 0x73, 0xba, 0x7a, 0xbd, 0x7d, 0x47, 0x87, 0x40, 0x80, 0x49, 0x89, 0x4e, 0x8e, 0x5b, 0x9b, 0x5c, 0x9c, 0x55, 0x95, 0x52, 0x92, 0x7f, 0xbf, 0x78, 0xb8, 0x71, 0xb1, 0x76, 0xb6, 0x63, 0xa3, 0x64, 0xa4, 0x6d, 0xad, 0x6a, 0xaa, 0x37, 0xf7, 0x30, 0xf0, 0x39, 0xf9, 0x3e, 0xfe, 0x2b, 0xeb, 0x2c, 0xec, 0x25, 0xe5, 0x22, 0xe2, 0x0f, 0xcf, 0x08, 0xc8, 0x01, 0xc1, 0x06, 0xc6, 0x13, 0xd3, 0x14, 0xd4, 0x1d, 0xdd, 0x1a, 0xda, 0xa7, 0x67, 0xa0, 0x60, 0xa9, 0x69, 0xae, 0x6e, 0xbb, 0x7b, 0xbc, 0x7c, 0xb5, 0x75, 0xb2, 0x72, 0x9f, 0x5f, 0x98, 0x58, 0x91, 0x51, 0x96, 0x56, 0x83, 0x43, 0x84, 0x44, 0x8d, 0x4d, 0x8a, 0x4a, 0xd7, 0x17, 0xd0, 0x10, 0xd9, 0x19, 0xde, 0x1e, 0xcb, 0x0b, 0xcc, 0x0c, 0xc5, 0x05, 0xc2, 0x02, 0xef, 0x2f, 0xe8, 0x28, 0xe1, 0x21, 0xe6, 0x26, 0xf3, 0x33, 0xf4, 0x34, 0xfd, 0x3d, 0xfa, 0x3a }; /* Returns V modulo 255 provided V is in the range -255,-254,...,509. */ static inline __u8 mod255(int v) { if (v > 0) { if (v < 255) { return v; } else { return v - 255; } } else { return v + 255; } } /* Add two numbers in the field. Addition in this field is equivalent * to a bit-wise exclusive OR operation---subtraction is therefore * identical to addition. */ static inline __u8 gfadd(__u8 a, __u8 b) { return a ^ b; } /* Add two vectors of numbers in the field. Each byte in A and B gets * added individually. */ static inline unsigned long gfadd_long(unsigned long a, unsigned long b) { return a ^ b; } /* Multiply two numbers in the field: */ static inline __u8 gfmul(__u8 a, __u8 b) { if (a && b) { return gfpow[mod255(gflog[a] + gflog[b])]; } else { return 0; } } /* Just like gfmul, except we have already looked up the log of the * second number. */ static inline __u8 gfmul_exp(__u8 a, int b) { if (a) { return gfpow[mod255(gflog[a] + b)]; } else { return 0; } } /* Just like gfmul_exp, except that A is a vector of numbers. That * is, each byte in A gets multiplied by gfpow[mod255(B)]. */ static inline unsigned long gfmul_exp_long(unsigned long a, int b) { __u8 t; if (sizeof(long) == 4) { return ( ((t = (__u32)a >> 24 & 0xff) ? (((__u32) gfpow[mod255(gflog[t] + b)]) << 24) : 0) | ((t = (__u32)a >> 16 & 0xff) ? (((__u32) gfpow[mod255(gflog[t] + b)]) << 16) : 0) | ((t = (__u32)a >> 8 & 0xff) ? (((__u32) gfpow[mod255(gflog[t] + b)]) << 8) : 0) | ((t = (__u32)a >> 0 & 0xff) ? (((__u32) gfpow[mod255(gflog[t] + b)]) << 0) : 0)); } else if (sizeof(long) == 8) { return ( ((t = (__u64)a >> 56 & 0xff) ? (((__u64) gfpow[mod255(gflog[t] + b)]) << 56) : 0) | ((t = (__u64)a >> 48 & 0xff) ? (((__u64) gfpow[mod255(gflog[t] + b)]) << 48) : 0) | ((t = (__u64)a >> 40 & 0xff) ? (((__u64) gfpow[mod255(gflog[t] + b)]) << 40) : 0) | ((t = (__u64)a >> 32 & 0xff) ? (((__u64) gfpow[mod255(gflog[t] + b)]) << 32) : 0) | ((t = (__u64)a >> 24 & 0xff) ? (((__u64) gfpow[mod255(gflog[t] + b)]) << 24) : 0) | ((t = (__u64)a >> 16 & 0xff) ? (((__u64) gfpow[mod255(gflog[t] + b)]) << 16) : 0) | ((t = (__u64)a >> 8 & 0xff) ? (((__u64) gfpow[mod255(gflog[t] + b)]) << 8) : 0) | ((t = (__u64)a >> 0 & 0xff) ? (((__u64) gfpow[mod255(gflog[t] + b)]) << 0) : 0)); } else { TRACE_FUN(ft_t_any); TRACE_ABORT(-1, ft_t_err, "Error: size of long is %d bytes", (int)sizeof(long)); } } /* Divide two numbers in the field. Returns a/b (modulo f(x)). */ static inline __u8 gfdiv(__u8 a, __u8 b) { if (!b) { TRACE_FUN(ft_t_any); TRACE_ABORT(0xff, ft_t_bug, "Error: division by zero"); } else if (a == 0) { return 0; } else { return gfpow[mod255(gflog[a] - gflog[b])]; } } /* The following functions return the inverse of the matrix of the * linear system that needs to be solved to determine the error * magnitudes. The first deals with matrices of rank 3, while the * second deals with matrices of rank 2. The error indices are passed * in arguments L0,..,L2 (0=first sector, 31=last sector). The error * indices must be sorted in ascending order, i.e., L00 CRC failures)"); } log_det = 255 - gflog[det]; /* Now, calculate all of the coefficients: */ Ainv[0][0]= gfmul_exp(gfadd(gfpow[l1], gfpow[l2]), log_det); Ainv[0][1]= gfmul_exp(gfadd(t21, t12), log_det); Ainv[0][2]= gfmul_exp(gfadd(gfpow[255 - l1], gfpow[255 - l2]),log_det); Ainv[1][0]= gfmul_exp(gfadd(gfpow[l0], gfpow[l2]), log_det); Ainv[1][1]= gfmul_exp(gfadd(t20, t02), log_det); Ainv[1][2]= gfmul_exp(gfadd(gfpow[255 - l0], gfpow[255 - l2]),log_det); Ainv[2][0]= gfmul_exp(gfadd(gfpow[l0], gfpow[l1]), log_det); Ainv[2][1]= gfmul_exp(gfadd(t10, t01), log_det); Ainv[2][2]= gfmul_exp(gfadd(gfpow[255 - l0], gfpow[255 - l1]),log_det); return 1; } static inline int gfinv2(__u8 l0, __u8 l1, Matrix Ainv) { __u8 det; __u8 t1, t2; int log_det; t1 = gfpow[255 - l0]; t2 = gfpow[255 - l1]; det = gfadd(t1, t2); if (!det) { TRACE_FUN(ft_t_any); TRACE_ABORT(0, ft_t_err, "Inversion failed (2 CRC errors, >0 CRC failures)"); } log_det = 255 - gflog[det]; /* Now, calculate all of the coefficients: */ Ainv[0][0] = Ainv[1][0] = gfpow[log_det]; Ainv[0][1] = gfmul_exp(t2, log_det); Ainv[1][1] = gfmul_exp(t1, log_det); return 1; } /* Multiply matrix A by vector S and return result in vector B. M is * assumed to be of order NxN, S and B of order Nx1. */ static inline void gfmat_mul(int n, Matrix A, __u8 *s, __u8 *b) { int i, j; __u8 dot_prod; for (i = 0; i < n; ++i) { dot_prod = 0; for (j = 0; j < n; ++j) { dot_prod = gfadd(dot_prod, gfmul(A[i][j], s[j])); } b[i] = dot_prod; } } /* The Reed Solomon ECC codes are computed over the N-th byte of each * block, where N=SECTOR_SIZE. There are up to 29 blocks of data, and * 3 blocks of ECC. The blocks are stored contiguously in memory. A * segment, consequently, is assumed to have at least 4 blocks: one or * more data blocks plus three ECC blocks. * * Notice: In QIC-80 speak, a CRC error is a sector with an incorrect * CRC. A CRC failure is a sector with incorrect data, but * a valid CRC. In the error control literature, the former * is usually called "erasure", the latter "error." */ /* Compute the parity bytes for C columns of data, where C is the * number of bytes that fit into a long integer. We use a linear * feed-back register to do this. The parity bytes P[0], P[STRIDE], * P[2*STRIDE] are computed such that: * * x^k * p(x) + m(x) = 0 (modulo g(x)) * * where k = NBLOCKS, * p(x) = P[0] + P[STRIDE]*x + P[2*STRIDE]*x^2, and * m(x) = sum_{i=0}^k m_i*x^i. * m_i = DATA[i*SECTOR_SIZE] */ static inline void set_parity(unsigned long *data, int nblocks, unsigned long *p, int stride) { unsigned long p0, p1, p2, t1, t2, *end; end = data + nblocks * (FT_SECTOR_SIZE / sizeof(long)); p0 = p1 = p2 = 0; while (data < end) { /* The new parity bytes p0_i, p1_i, p2_i are computed * from the old values p0_{i-1}, p1_{i-1}, p2_{i-1} * recursively as: * * p0_i = p1_{i-1} + r^105 * (m_{i-1} - p0_{i-1}) * p1_i = p2_{i-1} + r^105 * (m_{i-1} - p0_{i-1}) * p2_i = (m_{i-1} - p0_{i-1}) * * With the initial condition: p0_0 = p1_0 = p2_0 = 0. */ t1 = gfadd_long(*data, p0); /* * Multiply each byte in t1 by 0xc0: */ if (sizeof(long) == 4) { t2= (((__u32) gfmul_c0[(__u32)t1 >> 24 & 0xff]) << 24 | ((__u32) gfmul_c0[(__u32)t1 >> 16 & 0xff]) << 16 | ((__u32) gfmul_c0[(__u32)t1 >> 8 & 0xff]) << 8 | ((__u32) gfmul_c0[(__u32)t1 >> 0 & 0xff]) << 0); } else if (sizeof(long) == 8) { t2= (((__u64) gfmul_c0[(__u64)t1 >> 56 & 0xff]) << 56 | ((__u64) gfmul_c0[(__u64)t1 >> 48 & 0xff]) << 48 | ((__u64) gfmul_c0[(__u64)t1 >> 40 & 0xff]) << 40 | ((__u64) gfmul_c0[(__u64)t1 >> 32 & 0xff]) << 32 | ((__u64) gfmul_c0[(__u64)t1 >> 24 & 0xff]) << 24 | ((__u64) gfmul_c0[(__u64)t1 >> 16 & 0xff]) << 16 | ((__u64) gfmul_c0[(__u64)t1 >> 8 & 0xff]) << 8 | ((__u64) gfmul_c0[(__u64)t1 >> 0 & 0xff]) << 0); } else { TRACE_FUN(ft_t_any); TRACE(ft_t_err, "Error: long is of size %d", (int) sizeof(long)); TRACE_EXIT; } p0 = gfadd_long(t2, p1); p1 = gfadd_long(t2, p2); p2 = t1; data += FT_SECTOR_SIZE / sizeof(long); } *p = p0; p += stride; *p = p1; p += stride; *p = p2; return; } /* Compute the 3 syndrome values. DATA should point to the first byte * of the column for which the syndromes are desired. The syndromes * are computed over the first NBLOCKS of rows. The three bytes will * be placed in S[0], S[1], and S[2]. * * S[i] is the value of the "message" polynomial m(x) evaluated at the * i-th root of the generator polynomial g(x). * * As g(x)=(x-r^-1)(x-1)(x-r^1) we evaluate the message polynomial at * x=r^-1 to get S[0], at x=r^0=1 to get S[1], and at x=r to get S[2]. * This could be done directly and efficiently via the Horner scheme. * However, it would require multiplication tables for the factors * r^-1 (0xc3) and r (0x02). The following scheme does not require * any multiplication tables beyond what's needed for set_parity() * anyway and is slightly faster if there are no errors and slightly * slower if there are errors. The latter is hopefully the infrequent * case. * * To understand the alternative algorithm, notice that set_parity(m, * k, p) computes parity bytes such that: * * x^k * p(x) = m(x) (modulo g(x)). * * That is, to evaluate m(r^m), where r^m is a root of g(x), we can * simply evaluate (r^m)^k*p(r^m). Also, notice that p is 0 if and * only if s is zero. That is, if all parity bytes are 0, we know * there is no error in the data and consequently there is no need to * compute s(x) at all! In all other cases, we compute s(x) from p(x) * by evaluating (r^m)^k*p(r^m) for m=-1, m=0, and m=1. The p(x) * polynomial is evaluated via the Horner scheme. */ static int compute_syndromes(unsigned long *data, int nblocks, unsigned long *s) { unsigned long p[3]; set_parity(data, nblocks, p, 1); if (p[0] | p[1] | p[2]) { /* Some of the checked columns do not have a zero * syndrome. For simplicity, we compute the syndromes * for all columns that we have computed the * remainders for. */ s[0] = gfmul_exp_long( gfadd_long(p[0], gfmul_exp_long( gfadd_long(p[1], gfmul_exp_long(p[2], -1)), -1)), -nblocks); s[1] = gfadd_long(gfadd_long(p[2], p[1]), p[0]); s[2] = gfmul_exp_long( gfadd_long(p[0], gfmul_exp_long( gfadd_long(p[1], gfmul_exp_long(p[2], 1)), 1)), nblocks); return 0; } else { return 1; } } /* Correct the block in the column pointed to by DATA. There are NBAD * CRC errors and their indices are in BAD_LOC[0], up to * BAD_LOC[NBAD-1]. If NBAD>1, Ainv holds the inverse of the matrix * of the linear system that needs to be solved to determine the error * magnitudes. S[0], S[1], and S[2] are the syndrome values. If row * j gets corrected, then bit j will be set in CORRECTION_MAP. */ static inline int correct_block(__u8 *data, int nblocks, int nbad, int *bad_loc, Matrix Ainv, __u8 *s, SectorMap * correction_map) { int ncorrected = 0; int i; __u8 t1, t2; __u8 c0, c1, c2; /* check bytes */ __u8 error_mag[3], log_error_mag; __u8 *dp, l, e; TRACE_FUN(ft_t_any); switch (nbad) { case 0: /* might have a CRC failure: */ if (s[0] == 0) { /* more than one error */ TRACE_ABORT(-1, ft_t_err, "ECC failed (0 CRC errors, >1 CRC failures)"); } t1 = gfdiv(s[1], s[0]); if ((bad_loc[nbad++] = gflog[t1]) >= nblocks) { TRACE(ft_t_err, "ECC failed (0 CRC errors, >1 CRC failures)"); TRACE_ABORT(-1, ft_t_err, "attempt to correct data at %d", bad_loc[0]); } error_mag[0] = s[1]; break; case 1: t1 = gfadd(gfmul_exp(s[1], bad_loc[0]), s[2]); t2 = gfadd(gfmul_exp(s[0], bad_loc[0]), s[1]); if (t1 == 0 && t2 == 0) { /* one erasure, no error: */ Ainv[0][0] = gfpow[bad_loc[0]]; } else if (t1 == 0 || t2 == 0) { /* one erasure and more than one error: */ TRACE_ABORT(-1, ft_t_err, "ECC failed (1 erasure, >1 error)"); } else { /* one erasure, one error: */ if ((bad_loc[nbad++] = gflog[gfdiv(t1, t2)]) >= nblocks) { TRACE(ft_t_err, "ECC failed " "(1 CRC errors, >1 CRC failures)"); TRACE_ABORT(-1, ft_t_err, "attempt to correct data at %d", bad_loc[1]); } if (!gfinv2(bad_loc[0], bad_loc[1], Ainv)) { /* inversion failed---must have more * than one error */ TRACE_EXIT -1; } } /* FALL THROUGH TO ERROR MAGNITUDE COMPUTATION: */ case 2: case 3: /* compute error magnitudes: */ gfmat_mul(nbad, Ainv, s, error_mag); break; default: TRACE_ABORT(-1, ft_t_err, "Internal Error: number of CRC errors > 3"); } /* Perform correction by adding ERROR_MAG[i] to the byte at * offset BAD_LOC[i]. Also add the value of the computed * error polynomial to the syndrome values. If the correction * was successful, the resulting check bytes should be zero * (i.e., the corrected data is a valid code word). */ c0 = s[0]; c1 = s[1]; c2 = s[2]; for (i = 0; i < nbad; ++i) { e = error_mag[i]; if (e) { /* correct the byte at offset L by magnitude E: */ l = bad_loc[i]; dp = &data[l * FT_SECTOR_SIZE]; *dp = gfadd(*dp, e); *correction_map |= 1 << l; ++ncorrected; log_error_mag = gflog[e]; c0 = gfadd(c0, gfpow[mod255(log_error_mag - l)]); c1 = gfadd(c1, e); c2 = gfadd(c2, gfpow[mod255(log_error_mag + l)]); } } if (c0 || c1 || c2) { TRACE_ABORT(-1, ft_t_err, "ECC self-check failed, too many errors"); } TRACE_EXIT ncorrected; } #if defined(ECC_SANITY_CHECK) || defined(ECC_PARANOID) /* Perform a sanity check on the computed parity bytes: */ static int sanity_check(unsigned long *data, int nblocks) { TRACE_FUN(ft_t_any); unsigned long s[3]; if (!compute_syndromes(data, nblocks, s)) { TRACE_ABORT(0, ft_bug, "Internal Error: syndrome self-check failed"); } TRACE_EXIT 1; } #endif /* defined(ECC_SANITY_CHECK) || defined(ECC_PARANOID) */ /* Compute the parity for an entire segment of data. */ int ftape_ecc_set_segment_parity(struct memory_segment *mseg) { int i; __u8 *parity_bytes; parity_bytes = &mseg->data[(mseg->blocks - 3) * FT_SECTOR_SIZE]; for (i = 0; i < FT_SECTOR_SIZE; i += sizeof(long)) { set_parity((unsigned long *) &mseg->data[i], mseg->blocks - 3, (unsigned long *) &parity_bytes[i], FT_SECTOR_SIZE / sizeof(long)); #ifdef ECC_PARANOID if (!sanity_check((unsigned long *) &mseg->data[i], mseg->blocks)) { return -1; } #endif /* ECC_PARANOID */ } return 0; } /* Checks and corrects (if possible) the segment MSEG. Returns one of * ECC_OK, ECC_CORRECTED, and ECC_FAILED. */ int ftape_ecc_correct_data(struct memory_segment *mseg) { int col, i, result; int ncorrected = 0; int nerasures = 0; /* # of erasures (CRC errors) */ int erasure_loc[3]; /* erasure locations */ unsigned long ss[3]; __u8 s[3]; Matrix Ainv; TRACE_FUN(ft_t_flow); mseg->corrected = 0; /* find first column that has non-zero syndromes: */ for (col = 0; col < FT_SECTOR_SIZE; col += sizeof(long)) { if (!compute_syndromes((unsigned long *) &mseg->data[col], mseg->blocks, ss)) { /* something is wrong---have to fix things */ break; } } if (col >= FT_SECTOR_SIZE) { /* all syndromes are ok, therefore nothing to correct */ TRACE_EXIT ECC_OK; } /* count the number of CRC errors if there were any: */ if (mseg->read_bad) { for (i = 0; i < mseg->blocks; i++) { if (BAD_CHECK(mseg->read_bad, i)) { if (nerasures >= 3) { /* this is too much for ECC */ TRACE_ABORT(ECC_FAILED, ft_t_err, "ECC failed (>3 CRC errors)"); } /* if */ erasure_loc[nerasures++] = i; } } } /* * If there are at least 2 CRC errors, determine inverse of matrix * of linear system to be solved: */ switch (nerasures) { case 2: if (!gfinv2(erasure_loc[0], erasure_loc[1], Ainv)) { TRACE_EXIT ECC_FAILED; } break; case 3: if (!gfinv3(erasure_loc[0], erasure_loc[1], erasure_loc[2], Ainv)) { TRACE_EXIT ECC_FAILED; } break; default: /* this is not an error condition... */ break; } do { for (i = 0; i < sizeof(long); ++i) { s[0] = ss[0]; s[1] = ss[1]; s[2] = ss[2]; if (s[0] | s[1] | s[2]) { #ifdef BIG_ENDIAN result = correct_block( &mseg->data[col + sizeof(long) - 1 - i], mseg->blocks, nerasures, erasure_loc, Ainv, s, &mseg->corrected); #else result = correct_block(&mseg->data[col + i], mseg->blocks, nerasures, erasure_loc, Ainv, s, &mseg->corrected); #endif if (result < 0) { TRACE_EXIT ECC_FAILED; } ncorrected += result; } ss[0] >>= 8; ss[1] >>= 8; ss[2] >>= 8; } #ifdef ECC_SANITY_CHECK if (!sanity_check((unsigned long *) &mseg->data[col], mseg->blocks)) { TRACE_EXIT ECC_FAILED; } #endif /* ECC_SANITY_CHECK */ /* find next column with non-zero syndromes: */ while ((col += sizeof(long)) < FT_SECTOR_SIZE) { if (!compute_syndromes((unsigned long *) &mseg->data[col], mseg->blocks, ss)) { /* something is wrong---have to fix things */ break; } } } while (col < FT_SECTOR_SIZE); if (ncorrected && nerasures == 0) { TRACE(ft_t_warn, "block contained error not caught by CRC"); } TRACE((ncorrected > 0) ? ft_t_noise : ft_t_any, "number of corrections: %d", ncorrected); TRACE_EXIT ncorrected ? ECC_CORRECTED : ECC_OK; }