ftape-ecc.c

h内核

/*
 *
 *      Copyright (c) 1993 Ning and David Mosberger.
 
 This is based on code originally written by Bas Laarhoven ([e-mail])
 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 <linux/ftape.h>

#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., L0<L1<L2.
 *
 * The linear system that needs to be solved for the error magnitudes
 * is A * b = s, where s is the known vector of syndromes, b is the
 * vector of error magnitudes and A in the ORDER=3 case:
 *
 *    A_3 = {{1/r^L[0], 1/r^L[1], 1/r^L[2]},
 *          {        1,        1,        1},
 *          { r^L[0], r^L[1], r^L[2]}} 
 */
static inline int gfinv3(__u8 l0,
			 __u8 l1, 
			 __u8 l2, 
			 Matrix Ainv)
{
	__u8 det;
	__u8 t20, t10, t21, t12, t01, t02;
	int log_det;

	/* compute some intermediate results: */
	t20 = gfpow[l2 - l0];	        /* t20 = r^l2/r^l0 */
	t10 = gfpow[l1 - l0];	        /* t10 = r^l1/r^l0 */
	t21 = gfpow[l2 - l1];	        /* t21 = r^l2/r^l1 */
	t12 = gfpow[l1 - l2 + 255];	/* t12 = r^l1/r^l2 */
	t01 = gfpow[l0 - l1 + 255];	/* t01 = r^l0/r^l1 */
	t02 = gfpow[l0 - l2 + 255];	/* t02 = r^l0/r^l2 */
	/* Calculate the determinant of matrix A_3^-1 (sometimes
	 * called the Vandermonde determinant):
	 */
	det = gfadd(t20, gfadd(t10, gfadd(t21, gfadd(t12, gfadd(t01, t02)))));
	if (!det) {
		TRACE_FUN(ft_t_any);
		TRACE_ABORT(0, ft_t_err,
			   "Inversion failed (3 CRC errors, >0 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;