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
|
/*
* Reed-Solomon decoder, based on libfec
*
* Copyright (C) 2002, Phil Karn, KA9Q
* libcryptsetup modifications
* Copyright (C) 2017-2021 Red Hat, Inc. All rights reserved.
*
* This file is free software; you can redistribute it and/or
* modify it under the terms of the GNU Lesser General Public
* License as published by the Free Software Foundation; either
* version 2.1 of the License, or (at your option) any later version.
*
* This file 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
* Lesser General Public License for more details.
*
* You should have received a copy of the GNU Lesser General Public
* License along with this file; if not, write to the Free Software
* Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA.
*/
#include <string.h>
#include <stdlib.h>
#include "rs.h"
int decode_rs_char(struct rs* rs, data_t* data)
{
int deg_lambda, el, deg_omega, syn_error, count;
int i, j, r, k;
data_t q, tmp, num1, num2, den, discr_r;
/* FIXME: remove VLAs here */
data_t lambda[rs->nroots + 1], s[rs->nroots]; /* Err+Eras Locator poly and syndrome poly */
data_t b[rs->nroots + 1], t[rs->nroots + 1], omega[rs->nroots + 1];
data_t root[rs->nroots], reg[rs->nroots + 1], loc[rs->nroots];
memset(s, 0, rs->nroots * sizeof(data_t));
memset(b, 0, (rs->nroots + 1) * sizeof(data_t));
/* form the syndromes; i.e., evaluate data(x) at roots of g(x) */
for (i = 0; i < rs->nroots; i++)
s[i] = data[0];
for (j = 1; j < rs->nn - rs->pad; j++) {
for (i = 0; i < rs->nroots; i++) {
if (s[i] == 0) {
s[i] = data[j];
} else {
s[i] = data[j] ^ rs->alpha_to[modnn(rs, rs->index_of[s[i]] + (rs->fcr + i) * rs->prim)];
}
}
}
/* Convert syndromes to index form, checking for nonzero condition */
syn_error = 0;
for (i = 0; i < rs->nroots; i++) {
syn_error |= s[i];
s[i] = rs->index_of[s[i]];
}
/*
* if syndrome is zero, data[] is a codeword and there are no
* errors to correct. So return data[] unmodified
*/
if (!syn_error)
return 0;
memset(&lambda[1], 0, rs->nroots * sizeof(lambda[0]));
lambda[0] = 1;
for (i = 0; i < rs->nroots + 1; i++)
b[i] = rs->index_of[lambda[i]];
/*
* Begin Berlekamp-Massey algorithm to determine error+erasure
* locator polynomial
*/
r = 0;
el = 0;
while (++r <= rs->nroots) { /* r is the step number */
/* Compute discrepancy at the r-th step in poly-form */
discr_r = 0;
for (i = 0; i < r; i++) {
if ((lambda[i] != 0) && (s[r - i - 1] != A0)) {
discr_r ^= rs->alpha_to[modnn(rs, rs->index_of[lambda[i]] + s[r - i - 1])];
}
}
discr_r = rs->index_of[discr_r]; /* Index form */
if (discr_r == A0) {
/* 2 lines below: B(x) <-- x*B(x) */
memmove(&b[1], b, rs->nroots * sizeof(b[0]));
b[0] = A0;
} else {
/* 7 lines below: T(x) <-- lambda(x) - discr_r*x*b(x) */
t[0] = lambda[0];
for (i = 0; i < rs->nroots; i++) {
if (b[i] != A0)
t[i + 1] = lambda[i + 1] ^ rs->alpha_to[modnn(rs, discr_r + b[i])];
else
t[i + 1] = lambda[i + 1];
}
if (2 * el <= r - 1) {
el = r - el;
/*
* 2 lines below: B(x) <-- inv(discr_r) *
* lambda(x)
*/
for (i = 0; i <= rs->nroots; i++)
b[i] = (lambda[i] == 0) ? A0 : modnn(rs, rs->index_of[lambda[i]] - discr_r + rs->nn);
} else {
/* 2 lines below: B(x) <-- x*B(x) */
memmove(&b[1], b, rs->nroots * sizeof(b[0]));
b[0] = A0;
}
memcpy(lambda, t, (rs->nroots + 1) * sizeof(t[0]));
}
}
/* Convert lambda to index form and compute deg(lambda(x)) */
deg_lambda = 0;
for (i = 0; i < rs->nroots + 1; i++) {
lambda[i] = rs->index_of[lambda[i]];
if (lambda[i] != A0)
deg_lambda = i;
}
/* Find roots of the error+erasure locator polynomial by Chien search */
memcpy(®[1], &lambda[1], rs->nroots * sizeof(reg[0]));
count = 0; /* Number of roots of lambda(x) */
for (i = 1, k = rs->iprim - 1; i <= rs->nn; i++, k = modnn(rs, k + rs->iprim)) {
q = 1; /* lambda[0] is always 0 */
for (j = deg_lambda; j > 0; j--) {
if (reg[j] != A0) {
reg[j] = modnn(rs, reg[j] + j);
q ^= rs->alpha_to[reg[j]];
}
}
if (q != 0)
continue; /* Not a root */
/* store root (index-form) and error location number */
root[count] = i;
loc[count] = k;
/* If we've already found max possible roots, abort the search to save time */
if (++count == deg_lambda)
break;
}
/*
* deg(lambda) unequal to number of roots => uncorrectable
* error detected
*/
if (deg_lambda != count)
return -1;
/*
* Compute err+eras evaluator poly omega(x) = s(x)*lambda(x) (modulo
* x**rs->nroots). in index form. Also find deg(omega).
*/
deg_omega = deg_lambda - 1;
for (i = 0; i <= deg_omega; i++) {
tmp = 0;
for (j = i; j >= 0; j--) {
if ((s[i - j] != A0) && (lambda[j] != A0))
tmp ^= rs->alpha_to[modnn(rs, s[i - j] + lambda[j])];
}
omega[i] = rs->index_of[tmp];
}
/*
* Compute error values in poly-form. num1 = omega(inv(X(l))), num2 =
* inv(X(l))**(rs->fcr-1) and den = lambda_pr(inv(X(l))) all in poly-form
*/
for (j = count - 1; j >= 0; j--) {
num1 = 0;
for (i = deg_omega; i >= 0; i--) {
if (omega[i] != A0)
num1 ^= rs->alpha_to[modnn(rs, omega[i] + i * root[j])];
}
num2 = rs->alpha_to[modnn(rs, root[j] * (rs->fcr - 1) + rs->nn)];
den = 0;
/* lambda[i+1] for i even is the formal derivative lambda_pr of lambda[i] */
for (i = RS_MIN(deg_lambda, rs->nroots - 1) & ~1; i >= 0; i -= 2) {
if (lambda[i + 1] != A0)
den ^= rs->alpha_to[modnn(rs, lambda[i + 1] + i * root[j])];
}
/* Apply error to data */
if (num1 != 0 && loc[j] >= rs->pad) {
data[loc[j] - rs->pad] ^= rs->alpha_to[modnn(rs, rs->index_of[num1] +
rs->index_of[num2] + rs->nn - rs->index_of[den])];
}
}
return count;
}
|
