aboutsummaryrefslogtreecommitdiffstats
path: root/drivers/char/ftape/lowlevel/ftape-ecc.c
blob: e5632f674bc8b7e11612fa2e45009065b662d572 (plain) (blame)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
/*
 *
 *      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;
	}
}



/* 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;
}