Linux-2.6.12-rc2

[linux-2.6/next.git] / drivers / char / ftape / lowlevel / ftape-ecc.c

/*
 *
 *      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 <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;

        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;
        }
}
\f


/* 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;
}