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2 * Generic binary BCH encoding/decoding library
4 * This program is free software; you can redistribute it and/or modify it
5 * under the terms of the GNU General Public License version 2 as published by
6 * the Free Software Foundation.
8 * This program is distributed in the hope that it will be useful, but WITHOUT
9 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
10 * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for
13 * You should have received a copy of the GNU General Public License along with
14 * this program; if not, write to the Free Software Foundation, Inc., 51
15 * Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
17 * Copyright © 2011 Parrot S.A.
19 * Author: Ivan Djelic <ivan.djelic@parrot.com>
23 * This library provides runtime configurable encoding/decoding of binary
24 * Bose-Chaudhuri-Hocquenghem (BCH) codes.
26 * Call init_bch to get a pointer to a newly allocated bch_control structure for
27 * the given m (Galois field order), t (error correction capability) and
28 * (optional) primitive polynomial parameters.
30 * Call encode_bch to compute and store ecc parity bytes to a given buffer.
31 * Call decode_bch to detect and locate errors in received data.
33 * On systems supporting hw BCH features, intermediate results may be provided
34 * to decode_bch in order to skip certain steps. See decode_bch() documentation
37 * Option CONFIG_BCH_CONST_PARAMS can be used to force fixed values of
38 * parameters m and t; thus allowing extra compiler optimizations and providing
39 * better (up to 2x) encoding performance. Using this option makes sense when
40 * (m,t) are fixed and known in advance, e.g. when using BCH error correction
41 * on a particular NAND flash device.
43 * Algorithmic details:
45 * Encoding is performed by processing 32 input bits in parallel, using 4
46 * remainder lookup tables.
48 * The final stage of decoding involves the following internal steps:
49 * a. Syndrome computation
50 * b. Error locator polynomial computation using Berlekamp-Massey algorithm
51 * c. Error locator root finding (by far the most expensive step)
53 * In this implementation, step c is not performed using the usual Chien search.
54 * Instead, an alternative approach described in [1] is used. It consists in
55 * factoring the error locator polynomial using the Berlekamp Trace algorithm
56 * (BTA) down to a certain degree (4), after which ad hoc low-degree polynomial
57 * solving techniques [2] are used. The resulting algorithm, called BTZ, yields
58 * much better performance than Chien search for usual (m,t) values (typically
59 * m >= 13, t < 32, see [1]).
61 * [1] B. Biswas, V. Herbert. Efficient root finding of polynomials over fields
62 * of characteristic 2, in: Western European Workshop on Research in Cryptology
63 * - WEWoRC 2009, Graz, Austria, LNCS, Springer, July 2009, to appear.
64 * [2] [Zin96] V.A. Zinoviev. On the solution of equations of degree 10 over
65 * finite fields GF(2^q). In Rapport de recherche INRIA no 2829, 1996.
69 #include <ubi_uboot.h>
71 #include <linux/bitops.h>
72 #include <asm/byteorder.h>
73 #include <linux/bch.h>
75 #if defined(CONFIG_BCH_CONST_PARAMS)
76 #define GF_M(_p) (CONFIG_BCH_CONST_M)
77 #define GF_T(_p) (CONFIG_BCH_CONST_T)
78 #define GF_N(_p) ((1 << (CONFIG_BCH_CONST_M))-1)
80 #define GF_M(_p) ((_p)->m)
81 #define GF_T(_p) ((_p)->t)
82 #define GF_N(_p) ((_p)->n)
85 #define BCH_ECC_WORDS(_p) DIV_ROUND_UP(GF_M(_p)*GF_T(_p), 32)
86 #define BCH_ECC_BYTES(_p) DIV_ROUND_UP(GF_M(_p)*GF_T(_p), 8)
89 #define dbg(_fmt, args...) do {} while (0)
93 * represent a polynomial over GF(2^m)
96 unsigned int deg
; /* polynomial degree */
97 unsigned int c
[0]; /* polynomial terms */
100 /* given its degree, compute a polynomial size in bytes */
101 #define GF_POLY_SZ(_d) (sizeof(struct gf_poly)+((_d)+1)*sizeof(unsigned int))
103 /* polynomial of degree 1 */
104 struct gf_poly_deg1
{
110 * same as encode_bch(), but process input data one byte at a time
112 static void encode_bch_unaligned(struct bch_control
*bch
,
113 const unsigned char *data
, unsigned int len
,
118 const int l
= BCH_ECC_WORDS(bch
)-1;
121 p
= bch
->mod8_tab
+ (l
+1)*(((ecc
[0] >> 24)^(*data
++)) & 0xff);
123 for (i
= 0; i
< l
; i
++)
124 ecc
[i
] = ((ecc
[i
] << 8)|(ecc
[i
+1] >> 24))^(*p
++);
126 ecc
[l
] = (ecc
[l
] << 8)^(*p
);
131 * convert ecc bytes to aligned, zero-padded 32-bit ecc words
133 static void load_ecc8(struct bch_control
*bch
, uint32_t *dst
,
136 uint8_t pad
[4] = {0, 0, 0, 0};
137 unsigned int i
, nwords
= BCH_ECC_WORDS(bch
)-1;
139 for (i
= 0; i
< nwords
; i
++, src
+= 4)
140 dst
[i
] = (src
[0] << 24)|(src
[1] << 16)|(src
[2] << 8)|src
[3];
142 memcpy(pad
, src
, BCH_ECC_BYTES(bch
)-4*nwords
);
143 dst
[nwords
] = (pad
[0] << 24)|(pad
[1] << 16)|(pad
[2] << 8)|pad
[3];
147 * convert 32-bit ecc words to ecc bytes
149 static void store_ecc8(struct bch_control
*bch
, uint8_t *dst
,
153 unsigned int i
, nwords
= BCH_ECC_WORDS(bch
)-1;
155 for (i
= 0; i
< nwords
; i
++) {
156 *dst
++ = (src
[i
] >> 24);
157 *dst
++ = (src
[i
] >> 16) & 0xff;
158 *dst
++ = (src
[i
] >> 8) & 0xff;
159 *dst
++ = (src
[i
] >> 0) & 0xff;
161 pad
[0] = (src
[nwords
] >> 24);
162 pad
[1] = (src
[nwords
] >> 16) & 0xff;
163 pad
[2] = (src
[nwords
] >> 8) & 0xff;
164 pad
[3] = (src
[nwords
] >> 0) & 0xff;
165 memcpy(dst
, pad
, BCH_ECC_BYTES(bch
)-4*nwords
);
169 * encode_bch - calculate BCH ecc parity of data
170 * @bch: BCH control structure
171 * @data: data to encode
172 * @len: data length in bytes
173 * @ecc: ecc parity data, must be initialized by caller
175 * The @ecc parity array is used both as input and output parameter, in order to
176 * allow incremental computations. It should be of the size indicated by member
177 * @ecc_bytes of @bch, and should be initialized to 0 before the first call.
179 * The exact number of computed ecc parity bits is given by member @ecc_bits of
180 * @bch; it may be less than m*t for large values of t.
182 void encode_bch(struct bch_control
*bch
, const uint8_t *data
,
183 unsigned int len
, uint8_t *ecc
)
185 const unsigned int l
= BCH_ECC_WORDS(bch
)-1;
186 unsigned int i
, mlen
;
189 const uint32_t * const tab0
= bch
->mod8_tab
;
190 const uint32_t * const tab1
= tab0
+ 256*(l
+1);
191 const uint32_t * const tab2
= tab1
+ 256*(l
+1);
192 const uint32_t * const tab3
= tab2
+ 256*(l
+1);
193 const uint32_t *pdata
, *p0
, *p1
, *p2
, *p3
;
196 /* load ecc parity bytes into internal 32-bit buffer */
197 load_ecc8(bch
, bch
->ecc_buf
, ecc
);
199 memset(bch
->ecc_buf
, 0, sizeof(r
));
202 /* process first unaligned data bytes */
203 m
= ((unsigned long)data
) & 3;
205 mlen
= (len
< (4-m
)) ? len
: 4-m
;
206 encode_bch_unaligned(bch
, data
, mlen
, bch
->ecc_buf
);
211 /* process 32-bit aligned data words */
212 pdata
= (uint32_t *)data
;
216 memcpy(r
, bch
->ecc_buf
, sizeof(r
));
219 * split each 32-bit word into 4 polynomials of weight 8 as follows:
221 * 31 ...24 23 ...16 15 ... 8 7 ... 0
222 * xxxxxxxx yyyyyyyy zzzzzzzz tttttttt
223 * tttttttt mod g = r0 (precomputed)
224 * zzzzzzzz 00000000 mod g = r1 (precomputed)
225 * yyyyyyyy 00000000 00000000 mod g = r2 (precomputed)
226 * xxxxxxxx 00000000 00000000 00000000 mod g = r3 (precomputed)
227 * xxxxxxxx yyyyyyyy zzzzzzzz tttttttt mod g = r0^r1^r2^r3
230 /* input data is read in big-endian format */
231 w
= r
[0]^cpu_to_be32(*pdata
++);
232 p0
= tab0
+ (l
+1)*((w
>> 0) & 0xff);
233 p1
= tab1
+ (l
+1)*((w
>> 8) & 0xff);
234 p2
= tab2
+ (l
+1)*((w
>> 16) & 0xff);
235 p3
= tab3
+ (l
+1)*((w
>> 24) & 0xff);
237 for (i
= 0; i
< l
; i
++)
238 r
[i
] = r
[i
+1]^p0
[i
]^p1
[i
]^p2
[i
]^p3
[i
];
240 r
[l
] = p0
[l
]^p1
[l
]^p2
[l
]^p3
[l
];
242 memcpy(bch
->ecc_buf
, r
, sizeof(r
));
244 /* process last unaligned bytes */
246 encode_bch_unaligned(bch
, data
, len
, bch
->ecc_buf
);
248 /* store ecc parity bytes into original parity buffer */
250 store_ecc8(bch
, ecc
, bch
->ecc_buf
);
253 static inline int modulo(struct bch_control
*bch
, unsigned int v
)
255 const unsigned int n
= GF_N(bch
);
258 v
= (v
& n
) + (v
>> GF_M(bch
));
264 * shorter and faster modulo function, only works when v < 2N.
266 static inline int mod_s(struct bch_control
*bch
, unsigned int v
)
268 const unsigned int n
= GF_N(bch
);
269 return (v
< n
) ? v
: v
-n
;
272 static inline int deg(unsigned int poly
)
274 /* polynomial degree is the most-significant bit index */
278 static inline int parity(unsigned int x
)
281 * public domain code snippet, lifted from
282 * http://www-graphics.stanford.edu/~seander/bithacks.html
286 x
= (x
& 0x11111111U
) * 0x11111111U
;
287 return (x
>> 28) & 1;
290 /* Galois field basic operations: multiply, divide, inverse, etc. */
292 static inline unsigned int gf_mul(struct bch_control
*bch
, unsigned int a
,
295 return (a
&& b
) ? bch
->a_pow_tab
[mod_s(bch
, bch
->a_log_tab
[a
]+
296 bch
->a_log_tab
[b
])] : 0;
299 static inline unsigned int gf_sqr(struct bch_control
*bch
, unsigned int a
)
301 return a
? bch
->a_pow_tab
[mod_s(bch
, 2*bch
->a_log_tab
[a
])] : 0;
304 static inline unsigned int gf_div(struct bch_control
*bch
, unsigned int a
,
307 return a
? bch
->a_pow_tab
[mod_s(bch
, bch
->a_log_tab
[a
]+
308 GF_N(bch
)-bch
->a_log_tab
[b
])] : 0;
311 static inline unsigned int gf_inv(struct bch_control
*bch
, unsigned int a
)
313 return bch
->a_pow_tab
[GF_N(bch
)-bch
->a_log_tab
[a
]];
316 static inline unsigned int a_pow(struct bch_control
*bch
, int i
)
318 return bch
->a_pow_tab
[modulo(bch
, i
)];
321 static inline int a_log(struct bch_control
*bch
, unsigned int x
)
323 return bch
->a_log_tab
[x
];
326 static inline int a_ilog(struct bch_control
*bch
, unsigned int x
)
328 return mod_s(bch
, GF_N(bch
)-bch
->a_log_tab
[x
]);
332 * compute 2t syndromes of ecc polynomial, i.e. ecc(a^j) for j=1..2t
334 static void compute_syndromes(struct bch_control
*bch
, uint32_t *ecc
,
340 const int t
= GF_T(bch
);
344 /* make sure extra bits in last ecc word are cleared */
345 m
= ((unsigned int)s
) & 31;
347 ecc
[s
/32] &= ~((1u << (32-m
))-1);
348 memset(syn
, 0, 2*t
*sizeof(*syn
));
350 /* compute v(a^j) for j=1 .. 2t-1 */
356 for (j
= 0; j
< 2*t
; j
+= 2)
357 syn
[j
] ^= a_pow(bch
, (j
+1)*(i
+s
));
363 /* v(a^(2j)) = v(a^j)^2 */
364 for (j
= 0; j
< t
; j
++)
365 syn
[2*j
+1] = gf_sqr(bch
, syn
[j
]);
368 static void gf_poly_copy(struct gf_poly
*dst
, struct gf_poly
*src
)
370 memcpy(dst
, src
, GF_POLY_SZ(src
->deg
));
373 static int compute_error_locator_polynomial(struct bch_control
*bch
,
374 const unsigned int *syn
)
376 const unsigned int t
= GF_T(bch
);
377 const unsigned int n
= GF_N(bch
);
378 unsigned int i
, j
, tmp
, l
, pd
= 1, d
= syn
[0];
379 struct gf_poly
*elp
= bch
->elp
;
380 struct gf_poly
*pelp
= bch
->poly_2t
[0];
381 struct gf_poly
*elp_copy
= bch
->poly_2t
[1];
384 memset(pelp
, 0, GF_POLY_SZ(2*t
));
385 memset(elp
, 0, GF_POLY_SZ(2*t
));
392 /* use simplified binary Berlekamp-Massey algorithm */
393 for (i
= 0; (i
< t
) && (elp
->deg
<= t
); i
++) {
396 gf_poly_copy(elp_copy
, elp
);
397 /* e[i+1](X) = e[i](X)+di*dp^-1*X^2(i-p)*e[p](X) */
398 tmp
= a_log(bch
, d
)+n
-a_log(bch
, pd
);
399 for (j
= 0; j
<= pelp
->deg
; j
++) {
401 l
= a_log(bch
, pelp
->c
[j
]);
402 elp
->c
[j
+k
] ^= a_pow(bch
, tmp
+l
);
405 /* compute l[i+1] = max(l[i]->c[l[p]+2*(i-p]) */
407 if (tmp
> elp
->deg
) {
409 gf_poly_copy(pelp
, elp_copy
);
414 /* di+1 = S(2i+3)+elp[i+1].1*S(2i+2)+...+elp[i+1].lS(2i+3-l) */
417 for (j
= 1; j
<= elp
->deg
; j
++)
418 d
^= gf_mul(bch
, elp
->c
[j
], syn
[2*i
+2-j
]);
421 dbg("elp=%s\n", gf_poly_str(elp
));
422 return (elp
->deg
> t
) ? -1 : (int)elp
->deg
;
426 * solve a m x m linear system in GF(2) with an expected number of solutions,
427 * and return the number of found solutions
429 static int solve_linear_system(struct bch_control
*bch
, unsigned int *rows
,
430 unsigned int *sol
, int nsol
)
432 const int m
= GF_M(bch
);
433 unsigned int tmp
, mask
;
434 int rem
, c
, r
, p
, k
, param
[m
];
439 /* Gaussian elimination */
440 for (c
= 0; c
< m
; c
++) {
443 /* find suitable row for elimination */
444 for (r
= p
; r
< m
; r
++) {
445 if (rows
[r
] & mask
) {
456 /* perform elimination on remaining rows */
458 for (r
= rem
; r
< m
; r
++) {
463 /* elimination not needed, store defective row index */
468 /* rewrite system, inserting fake parameter rows */
471 for (r
= m
-1; r
>= 0; r
--) {
472 if ((r
> m
-1-k
) && rows
[r
])
473 /* system has no solution */
476 rows
[r
] = (p
&& (r
== param
[p
-1])) ?
477 p
--, 1u << (m
-r
) : rows
[r
-p
];
481 if (nsol
!= (1 << k
))
482 /* unexpected number of solutions */
485 for (p
= 0; p
< nsol
; p
++) {
486 /* set parameters for p-th solution */
487 for (c
= 0; c
< k
; c
++)
488 rows
[param
[c
]] = (rows
[param
[c
]] & ~1)|((p
>> c
) & 1);
490 /* compute unique solution */
492 for (r
= m
-1; r
>= 0; r
--) {
493 mask
= rows
[r
] & (tmp
|1);
494 tmp
|= parity(mask
) << (m
-r
);
502 * this function builds and solves a linear system for finding roots of a degree
503 * 4 affine monic polynomial X^4+aX^2+bX+c over GF(2^m).
505 static int find_affine4_roots(struct bch_control
*bch
, unsigned int a
,
506 unsigned int b
, unsigned int c
,
510 const int m
= GF_M(bch
);
511 unsigned int mask
= 0xff, t
, rows
[16] = {0,};
517 /* buid linear system to solve X^4+aX^2+bX+c = 0 */
518 for (i
= 0; i
< m
; i
++) {
519 rows
[i
+1] = bch
->a_pow_tab
[4*i
]^
520 (a
? bch
->a_pow_tab
[mod_s(bch
, k
)] : 0)^
521 (b
? bch
->a_pow_tab
[mod_s(bch
, j
)] : 0);
526 * transpose 16x16 matrix before passing it to linear solver
527 * warning: this code assumes m < 16
529 for (j
= 8; j
!= 0; j
>>= 1, mask
^= (mask
<< j
)) {
530 for (k
= 0; k
< 16; k
= (k
+j
+1) & ~j
) {
531 t
= ((rows
[k
] >> j
)^rows
[k
+j
]) & mask
;
536 return solve_linear_system(bch
, rows
, roots
, 4);
540 * compute root r of a degree 1 polynomial over GF(2^m) (returned as log(1/r))
542 static int find_poly_deg1_roots(struct bch_control
*bch
, struct gf_poly
*poly
,
548 /* poly[X] = bX+c with c!=0, root=c/b */
549 roots
[n
++] = mod_s(bch
, GF_N(bch
)-bch
->a_log_tab
[poly
->c
[0]]+
550 bch
->a_log_tab
[poly
->c
[1]]);
555 * compute roots of a degree 2 polynomial over GF(2^m)
557 static int find_poly_deg2_roots(struct bch_control
*bch
, struct gf_poly
*poly
,
560 int n
= 0, i
, l0
, l1
, l2
;
561 unsigned int u
, v
, r
;
563 if (poly
->c
[0] && poly
->c
[1]) {
565 l0
= bch
->a_log_tab
[poly
->c
[0]];
566 l1
= bch
->a_log_tab
[poly
->c
[1]];
567 l2
= bch
->a_log_tab
[poly
->c
[2]];
569 /* using z=a/bX, transform aX^2+bX+c into z^2+z+u (u=ac/b^2) */
570 u
= a_pow(bch
, l0
+l2
+2*(GF_N(bch
)-l1
));
572 * let u = sum(li.a^i) i=0..m-1; then compute r = sum(li.xi):
573 * r^2+r = sum(li.(xi^2+xi)) = sum(li.(a^i+Tr(a^i).a^k)) =
574 * u + sum(li.Tr(a^i).a^k) = u+a^k.Tr(sum(li.a^i)) = u+a^k.Tr(u)
575 * i.e. r and r+1 are roots iff Tr(u)=0
585 if ((gf_sqr(bch
, r
)^r
) == u
) {
586 /* reverse z=a/bX transformation and compute log(1/r) */
587 roots
[n
++] = modulo(bch
, 2*GF_N(bch
)-l1
-
588 bch
->a_log_tab
[r
]+l2
);
589 roots
[n
++] = modulo(bch
, 2*GF_N(bch
)-l1
-
590 bch
->a_log_tab
[r
^1]+l2
);
597 * compute roots of a degree 3 polynomial over GF(2^m)
599 static int find_poly_deg3_roots(struct bch_control
*bch
, struct gf_poly
*poly
,
603 unsigned int a
, b
, c
, a2
, b2
, c2
, e3
, tmp
[4];
606 /* transform polynomial into monic X^3 + a2X^2 + b2X + c2 */
608 c2
= gf_div(bch
, poly
->c
[0], e3
);
609 b2
= gf_div(bch
, poly
->c
[1], e3
);
610 a2
= gf_div(bch
, poly
->c
[2], e3
);
612 /* (X+a2)(X^3+a2X^2+b2X+c2) = X^4+aX^2+bX+c (affine) */
613 c
= gf_mul(bch
, a2
, c2
); /* c = a2c2 */
614 b
= gf_mul(bch
, a2
, b2
)^c2
; /* b = a2b2 + c2 */
615 a
= gf_sqr(bch
, a2
)^b2
; /* a = a2^2 + b2 */
617 /* find the 4 roots of this affine polynomial */
618 if (find_affine4_roots(bch
, a
, b
, c
, tmp
) == 4) {
619 /* remove a2 from final list of roots */
620 for (i
= 0; i
< 4; i
++) {
622 roots
[n
++] = a_ilog(bch
, tmp
[i
]);
630 * compute roots of a degree 4 polynomial over GF(2^m)
632 static int find_poly_deg4_roots(struct bch_control
*bch
, struct gf_poly
*poly
,
636 unsigned int a
, b
, c
, d
, e
= 0, f
, a2
, b2
, c2
, e4
;
641 /* transform polynomial into monic X^4 + aX^3 + bX^2 + cX + d */
643 d
= gf_div(bch
, poly
->c
[0], e4
);
644 c
= gf_div(bch
, poly
->c
[1], e4
);
645 b
= gf_div(bch
, poly
->c
[2], e4
);
646 a
= gf_div(bch
, poly
->c
[3], e4
);
648 /* use Y=1/X transformation to get an affine polynomial */
650 /* first, eliminate cX by using z=X+e with ae^2+c=0 */
652 /* compute e such that e^2 = c/a */
653 f
= gf_div(bch
, c
, a
);
655 l
+= (l
& 1) ? GF_N(bch
) : 0;
658 * use transformation z=X+e:
659 * z^4+e^4 + a(z^3+ez^2+e^2z+e^3) + b(z^2+e^2) +cz+ce+d
660 * z^4 + az^3 + (ae+b)z^2 + (ae^2+c)z+e^4+be^2+ae^3+ce+d
661 * z^4 + az^3 + (ae+b)z^2 + e^4+be^2+d
662 * z^4 + az^3 + b'z^2 + d'
664 d
= a_pow(bch
, 2*l
)^gf_mul(bch
, b
, f
)^d
;
665 b
= gf_mul(bch
, a
, e
)^b
;
667 /* now, use Y=1/X to get Y^4 + b/dY^2 + a/dY + 1/d */
669 /* assume all roots have multiplicity 1 */
673 b2
= gf_div(bch
, a
, d
);
674 a2
= gf_div(bch
, b
, d
);
676 /* polynomial is already affine */
681 /* find the 4 roots of this affine polynomial */
682 if (find_affine4_roots(bch
, a2
, b2
, c2
, roots
) == 4) {
683 for (i
= 0; i
< 4; i
++) {
684 /* post-process roots (reverse transformations) */
685 f
= a
? gf_inv(bch
, roots
[i
]) : roots
[i
];
686 roots
[i
] = a_ilog(bch
, f
^e
);
694 * build monic, log-based representation of a polynomial
696 static void gf_poly_logrep(struct bch_control
*bch
,
697 const struct gf_poly
*a
, int *rep
)
699 int i
, d
= a
->deg
, l
= GF_N(bch
)-a_log(bch
, a
->c
[a
->deg
]);
701 /* represent 0 values with -1; warning, rep[d] is not set to 1 */
702 for (i
= 0; i
< d
; i
++)
703 rep
[i
] = a
->c
[i
] ? mod_s(bch
, a_log(bch
, a
->c
[i
])+l
) : -1;
707 * compute polynomial Euclidean division remainder in GF(2^m)[X]
709 static void gf_poly_mod(struct bch_control
*bch
, struct gf_poly
*a
,
710 const struct gf_poly
*b
, int *rep
)
713 unsigned int i
, j
, *c
= a
->c
;
714 const unsigned int d
= b
->deg
;
719 /* reuse or compute log representation of denominator */
722 gf_poly_logrep(bch
, b
, rep
);
725 for (j
= a
->deg
; j
>= d
; j
--) {
727 la
= a_log(bch
, c
[j
]);
729 for (i
= 0; i
< d
; i
++, p
++) {
732 c
[p
] ^= bch
->a_pow_tab
[mod_s(bch
,
738 while (!c
[a
->deg
] && a
->deg
)
743 * compute polynomial Euclidean division quotient in GF(2^m)[X]
745 static void gf_poly_div(struct bch_control
*bch
, struct gf_poly
*a
,
746 const struct gf_poly
*b
, struct gf_poly
*q
)
748 if (a
->deg
>= b
->deg
) {
749 q
->deg
= a
->deg
-b
->deg
;
750 /* compute a mod b (modifies a) */
751 gf_poly_mod(bch
, a
, b
, NULL
);
752 /* quotient is stored in upper part of polynomial a */
753 memcpy(q
->c
, &a
->c
[b
->deg
], (1+q
->deg
)*sizeof(unsigned int));
761 * compute polynomial GCD (Greatest Common Divisor) in GF(2^m)[X]
763 static struct gf_poly
*gf_poly_gcd(struct bch_control
*bch
, struct gf_poly
*a
,
768 dbg("gcd(%s,%s)=", gf_poly_str(a
), gf_poly_str(b
));
770 if (a
->deg
< b
->deg
) {
777 gf_poly_mod(bch
, a
, b
, NULL
);
783 dbg("%s\n", gf_poly_str(a
));
789 * Given a polynomial f and an integer k, compute Tr(a^kX) mod f
790 * This is used in Berlekamp Trace algorithm for splitting polynomials
792 static void compute_trace_bk_mod(struct bch_control
*bch
, int k
,
793 const struct gf_poly
*f
, struct gf_poly
*z
,
796 const int m
= GF_M(bch
);
799 /* z contains z^2j mod f */
802 z
->c
[1] = bch
->a_pow_tab
[k
];
805 memset(out
, 0, GF_POLY_SZ(f
->deg
));
807 /* compute f log representation only once */
808 gf_poly_logrep(bch
, f
, bch
->cache
);
810 for (i
= 0; i
< m
; i
++) {
811 /* add a^(k*2^i)(z^(2^i) mod f) and compute (z^(2^i) mod f)^2 */
812 for (j
= z
->deg
; j
>= 0; j
--) {
813 out
->c
[j
] ^= z
->c
[j
];
814 z
->c
[2*j
] = gf_sqr(bch
, z
->c
[j
]);
817 if (z
->deg
> out
->deg
)
822 /* z^(2(i+1)) mod f = (z^(2^i) mod f)^2 mod f */
823 gf_poly_mod(bch
, z
, f
, bch
->cache
);
826 while (!out
->c
[out
->deg
] && out
->deg
)
829 dbg("Tr(a^%d.X) mod f = %s\n", k
, gf_poly_str(out
));
833 * factor a polynomial using Berlekamp Trace algorithm (BTA)
835 static void factor_polynomial(struct bch_control
*bch
, int k
, struct gf_poly
*f
,
836 struct gf_poly
**g
, struct gf_poly
**h
)
838 struct gf_poly
*f2
= bch
->poly_2t
[0];
839 struct gf_poly
*q
= bch
->poly_2t
[1];
840 struct gf_poly
*tk
= bch
->poly_2t
[2];
841 struct gf_poly
*z
= bch
->poly_2t
[3];
844 dbg("factoring %s...\n", gf_poly_str(f
));
849 /* tk = Tr(a^k.X) mod f */
850 compute_trace_bk_mod(bch
, k
, f
, z
, tk
);
853 /* compute g = gcd(f, tk) (destructive operation) */
855 gcd
= gf_poly_gcd(bch
, f2
, tk
);
856 if (gcd
->deg
< f
->deg
) {
857 /* compute h=f/gcd(f,tk); this will modify f and q */
858 gf_poly_div(bch
, f
, gcd
, q
);
859 /* store g and h in-place (clobbering f) */
860 *h
= &((struct gf_poly_deg1
*)f
)[gcd
->deg
].poly
;
861 gf_poly_copy(*g
, gcd
);
868 * find roots of a polynomial, using BTZ algorithm; see the beginning of this
871 static int find_poly_roots(struct bch_control
*bch
, unsigned int k
,
872 struct gf_poly
*poly
, unsigned int *roots
)
875 struct gf_poly
*f1
, *f2
;
878 /* handle low degree polynomials with ad hoc techniques */
880 cnt
= find_poly_deg1_roots(bch
, poly
, roots
);
883 cnt
= find_poly_deg2_roots(bch
, poly
, roots
);
886 cnt
= find_poly_deg3_roots(bch
, poly
, roots
);
889 cnt
= find_poly_deg4_roots(bch
, poly
, roots
);
892 /* factor polynomial using Berlekamp Trace Algorithm (BTA) */
894 if (poly
->deg
&& (k
<= GF_M(bch
))) {
895 factor_polynomial(bch
, k
, poly
, &f1
, &f2
);
897 cnt
+= find_poly_roots(bch
, k
+1, f1
, roots
);
899 cnt
+= find_poly_roots(bch
, k
+1, f2
, roots
+cnt
);
906 #if defined(USE_CHIEN_SEARCH)
908 * exhaustive root search (Chien) implementation - not used, included only for
909 * reference/comparison tests
911 static int chien_search(struct bch_control
*bch
, unsigned int len
,
912 struct gf_poly
*p
, unsigned int *roots
)
915 unsigned int i
, j
, syn
, syn0
, count
= 0;
916 const unsigned int k
= 8*len
+bch
->ecc_bits
;
918 /* use a log-based representation of polynomial */
919 gf_poly_logrep(bch
, p
, bch
->cache
);
920 bch
->cache
[p
->deg
] = 0;
921 syn0
= gf_div(bch
, p
->c
[0], p
->c
[p
->deg
]);
923 for (i
= GF_N(bch
)-k
+1; i
<= GF_N(bch
); i
++) {
924 /* compute elp(a^i) */
925 for (j
= 1, syn
= syn0
; j
<= p
->deg
; j
++) {
928 syn
^= a_pow(bch
, m
+j
*i
);
931 roots
[count
++] = GF_N(bch
)-i
;
936 return (count
== p
->deg
) ? count
: 0;
938 #define find_poly_roots(_p, _k, _elp, _loc) chien_search(_p, len, _elp, _loc)
939 #endif /* USE_CHIEN_SEARCH */
942 * decode_bch - decode received codeword and find bit error locations
943 * @bch: BCH control structure
944 * @data: received data, ignored if @calc_ecc is provided
945 * @len: data length in bytes, must always be provided
946 * @recv_ecc: received ecc, if NULL then assume it was XORed in @calc_ecc
947 * @calc_ecc: calculated ecc, if NULL then calc_ecc is computed from @data
948 * @syn: hw computed syndrome data (if NULL, syndrome is calculated)
949 * @errloc: output array of error locations
952 * The number of errors found, or -EBADMSG if decoding failed, or -EINVAL if
953 * invalid parameters were provided
955 * Depending on the available hw BCH support and the need to compute @calc_ecc
956 * separately (using encode_bch()), this function should be called with one of
957 * the following parameter configurations -
959 * by providing @data and @recv_ecc only:
960 * decode_bch(@bch, @data, @len, @recv_ecc, NULL, NULL, @errloc)
962 * by providing @recv_ecc and @calc_ecc:
963 * decode_bch(@bch, NULL, @len, @recv_ecc, @calc_ecc, NULL, @errloc)
965 * by providing ecc = recv_ecc XOR calc_ecc:
966 * decode_bch(@bch, NULL, @len, NULL, ecc, NULL, @errloc)
968 * by providing syndrome results @syn:
969 * decode_bch(@bch, NULL, @len, NULL, NULL, @syn, @errloc)
971 * Once decode_bch() has successfully returned with a positive value, error
972 * locations returned in array @errloc should be interpreted as follows -
974 * if (errloc[n] >= 8*len), then n-th error is located in ecc (no need for
977 * if (errloc[n] < 8*len), then n-th error is located in data and can be
978 * corrected with statement data[errloc[n]/8] ^= 1 << (errloc[n] % 8);
980 * Note that this function does not perform any data correction by itself, it
981 * merely indicates error locations.
983 int decode_bch(struct bch_control
*bch
, const uint8_t *data
, unsigned int len
,
984 const uint8_t *recv_ecc
, const uint8_t *calc_ecc
,
985 const unsigned int *syn
, unsigned int *errloc
)
987 const unsigned int ecc_words
= BCH_ECC_WORDS(bch
);
992 /* sanity check: make sure data length can be handled */
993 if (8*len
> (bch
->n
-bch
->ecc_bits
))
996 /* if caller does not provide syndromes, compute them */
999 /* compute received data ecc into an internal buffer */
1000 if (!data
|| !recv_ecc
)
1002 encode_bch(bch
, data
, len
, NULL
);
1004 /* load provided calculated ecc */
1005 load_ecc8(bch
, bch
->ecc_buf
, calc_ecc
);
1007 /* load received ecc or assume it was XORed in calc_ecc */
1009 load_ecc8(bch
, bch
->ecc_buf2
, recv_ecc
);
1010 /* XOR received and calculated ecc */
1011 for (i
= 0, sum
= 0; i
< (int)ecc_words
; i
++) {
1012 bch
->ecc_buf
[i
] ^= bch
->ecc_buf2
[i
];
1013 sum
|= bch
->ecc_buf
[i
];
1016 /* no error found */
1019 compute_syndromes(bch
, bch
->ecc_buf
, bch
->syn
);
1023 err
= compute_error_locator_polynomial(bch
, syn
);
1025 nroots
= find_poly_roots(bch
, 1, bch
->elp
, errloc
);
1030 /* post-process raw error locations for easier correction */
1031 nbits
= (len
*8)+bch
->ecc_bits
;
1032 for (i
= 0; i
< err
; i
++) {
1033 if (errloc
[i
] >= nbits
) {
1037 errloc
[i
] = nbits
-1-errloc
[i
];
1038 errloc
[i
] = (errloc
[i
] & ~7)|(7-(errloc
[i
] & 7));
1041 return (err
>= 0) ? err
: -EBADMSG
;
1045 * generate Galois field lookup tables
1047 static int build_gf_tables(struct bch_control
*bch
, unsigned int poly
)
1049 unsigned int i
, x
= 1;
1050 const unsigned int k
= 1 << deg(poly
);
1052 /* primitive polynomial must be of degree m */
1053 if (k
!= (1u << GF_M(bch
)))
1056 for (i
= 0; i
< GF_N(bch
); i
++) {
1057 bch
->a_pow_tab
[i
] = x
;
1058 bch
->a_log_tab
[x
] = i
;
1060 /* polynomial is not primitive (a^i=1 with 0<i<2^m-1) */
1066 bch
->a_pow_tab
[GF_N(bch
)] = 1;
1067 bch
->a_log_tab
[0] = 0;
1073 * compute generator polynomial remainder tables for fast encoding
1075 static void build_mod8_tables(struct bch_control
*bch
, const uint32_t *g
)
1078 uint32_t data
, hi
, lo
, *tab
;
1079 const int l
= BCH_ECC_WORDS(bch
);
1080 const int plen
= DIV_ROUND_UP(bch
->ecc_bits
+1, 32);
1081 const int ecclen
= DIV_ROUND_UP(bch
->ecc_bits
, 32);
1083 memset(bch
->mod8_tab
, 0, 4*256*l
*sizeof(*bch
->mod8_tab
));
1085 for (i
= 0; i
< 256; i
++) {
1086 /* p(X)=i is a small polynomial of weight <= 8 */
1087 for (b
= 0; b
< 4; b
++) {
1088 /* we want to compute (p(X).X^(8*b+deg(g))) mod g(X) */
1089 tab
= bch
->mod8_tab
+ (b
*256+i
)*l
;
1093 /* subtract X^d.g(X) from p(X).X^(8*b+deg(g)) */
1094 data
^= g
[0] >> (31-d
);
1095 for (j
= 0; j
< ecclen
; j
++) {
1096 hi
= (d
< 31) ? g
[j
] << (d
+1) : 0;
1098 g
[j
+1] >> (31-d
) : 0;
1107 * build a base for factoring degree 2 polynomials
1109 static int build_deg2_base(struct bch_control
*bch
)
1111 const int m
= GF_M(bch
);
1113 unsigned int sum
, x
, y
, remaining
, ak
= 0, xi
[m
];
1115 /* find k s.t. Tr(a^k) = 1 and 0 <= k < m */
1116 for (i
= 0; i
< m
; i
++) {
1117 for (j
= 0, sum
= 0; j
< m
; j
++)
1118 sum
^= a_pow(bch
, i
*(1 << j
));
1121 ak
= bch
->a_pow_tab
[i
];
1125 /* find xi, i=0..m-1 such that xi^2+xi = a^i+Tr(a^i).a^k */
1127 memset(xi
, 0, sizeof(xi
));
1129 for (x
= 0; (x
<= GF_N(bch
)) && remaining
; x
++) {
1130 y
= gf_sqr(bch
, x
)^x
;
1131 for (i
= 0; i
< 2; i
++) {
1133 if (y
&& (r
< m
) && !xi
[r
]) {
1137 dbg("x%d = %x\n", r
, x
);
1143 /* should not happen but check anyway */
1144 return remaining
? -1 : 0;
1147 static void *bch_alloc(size_t size
, int *err
)
1151 ptr
= kmalloc(size
, GFP_KERNEL
);
1158 * compute generator polynomial for given (m,t) parameters.
1160 static uint32_t *compute_generator_polynomial(struct bch_control
*bch
)
1162 const unsigned int m
= GF_M(bch
);
1163 const unsigned int t
= GF_T(bch
);
1165 unsigned int i
, j
, nbits
, r
, word
, *roots
;
1169 g
= bch_alloc(GF_POLY_SZ(m
*t
), &err
);
1170 roots
= bch_alloc((bch
->n
+1)*sizeof(*roots
), &err
);
1171 genpoly
= bch_alloc(DIV_ROUND_UP(m
*t
+1, 32)*sizeof(*genpoly
), &err
);
1179 /* enumerate all roots of g(X) */
1180 memset(roots
, 0, (bch
->n
+1)*sizeof(*roots
));
1181 for (i
= 0; i
< t
; i
++) {
1182 for (j
= 0, r
= 2*i
+1; j
< m
; j
++) {
1184 r
= mod_s(bch
, 2*r
);
1187 /* build generator polynomial g(X) */
1190 for (i
= 0; i
< GF_N(bch
); i
++) {
1192 /* multiply g(X) by (X+root) */
1193 r
= bch
->a_pow_tab
[i
];
1195 for (j
= g
->deg
; j
> 0; j
--)
1196 g
->c
[j
] = gf_mul(bch
, g
->c
[j
], r
)^g
->c
[j
-1];
1198 g
->c
[0] = gf_mul(bch
, g
->c
[0], r
);
1202 /* store left-justified binary representation of g(X) */
1207 nbits
= (n
> 32) ? 32 : n
;
1208 for (j
= 0, word
= 0; j
< nbits
; j
++) {
1210 word
|= 1u << (31-j
);
1212 genpoly
[i
++] = word
;
1215 bch
->ecc_bits
= g
->deg
;
1225 * init_bch - initialize a BCH encoder/decoder
1226 * @m: Galois field order, should be in the range 5-15
1227 * @t: maximum error correction capability, in bits
1228 * @prim_poly: user-provided primitive polynomial (or 0 to use default)
1231 * a newly allocated BCH control structure if successful, NULL otherwise
1233 * This initialization can take some time, as lookup tables are built for fast
1234 * encoding/decoding; make sure not to call this function from a time critical
1235 * path. Usually, init_bch() should be called on module/driver init and
1236 * free_bch() should be called to release memory on exit.
1238 * You may provide your own primitive polynomial of degree @m in argument
1239 * @prim_poly, or let init_bch() use its default polynomial.
1241 * Once init_bch() has successfully returned a pointer to a newly allocated
1242 * BCH control structure, ecc length in bytes is given by member @ecc_bytes of
1245 struct bch_control
*init_bch(int m
, int t
, unsigned int prim_poly
)
1248 unsigned int i
, words
;
1250 struct bch_control
*bch
= NULL
;
1252 const int min_m
= 5;
1253 const int max_m
= 15;
1255 /* default primitive polynomials */
1256 static const unsigned int prim_poly_tab
[] = {
1257 0x25, 0x43, 0x83, 0x11d, 0x211, 0x409, 0x805, 0x1053, 0x201b,
1261 #if defined(CONFIG_BCH_CONST_PARAMS)
1262 if ((m
!= (CONFIG_BCH_CONST_M
)) || (t
!= (CONFIG_BCH_CONST_T
))) {
1263 printk(KERN_ERR
"bch encoder/decoder was configured to support "
1264 "parameters m=%d, t=%d only!\n",
1265 CONFIG_BCH_CONST_M
, CONFIG_BCH_CONST_T
);
1269 if ((m
< min_m
) || (m
> max_m
))
1271 * values of m greater than 15 are not currently supported;
1272 * supporting m > 15 would require changing table base type
1273 * (uint16_t) and a small patch in matrix transposition
1278 if ((t
< 1) || (m
*t
>= ((1 << m
)-1)))
1279 /* invalid t value */
1282 /* select a primitive polynomial for generating GF(2^m) */
1284 prim_poly
= prim_poly_tab
[m
-min_m
];
1286 bch
= kzalloc(sizeof(*bch
), GFP_KERNEL
);
1292 bch
->n
= (1 << m
)-1;
1293 words
= DIV_ROUND_UP(m
*t
, 32);
1294 bch
->ecc_bytes
= DIV_ROUND_UP(m
*t
, 8);
1295 bch
->a_pow_tab
= bch_alloc((1+bch
->n
)*sizeof(*bch
->a_pow_tab
), &err
);
1296 bch
->a_log_tab
= bch_alloc((1+bch
->n
)*sizeof(*bch
->a_log_tab
), &err
);
1297 bch
->mod8_tab
= bch_alloc(words
*1024*sizeof(*bch
->mod8_tab
), &err
);
1298 bch
->ecc_buf
= bch_alloc(words
*sizeof(*bch
->ecc_buf
), &err
);
1299 bch
->ecc_buf2
= bch_alloc(words
*sizeof(*bch
->ecc_buf2
), &err
);
1300 bch
->xi_tab
= bch_alloc(m
*sizeof(*bch
->xi_tab
), &err
);
1301 bch
->syn
= bch_alloc(2*t
*sizeof(*bch
->syn
), &err
);
1302 bch
->cache
= bch_alloc(2*t
*sizeof(*bch
->cache
), &err
);
1303 bch
->elp
= bch_alloc((t
+1)*sizeof(struct gf_poly_deg1
), &err
);
1305 for (i
= 0; i
< ARRAY_SIZE(bch
->poly_2t
); i
++)
1306 bch
->poly_2t
[i
] = bch_alloc(GF_POLY_SZ(2*t
), &err
);
1311 err
= build_gf_tables(bch
, prim_poly
);
1315 /* use generator polynomial for computing encoding tables */
1316 genpoly
= compute_generator_polynomial(bch
);
1317 if (genpoly
== NULL
)
1320 build_mod8_tables(bch
, genpoly
);
1323 err
= build_deg2_base(bch
);
1335 * free_bch - free the BCH control structure
1336 * @bch: BCH control structure to release
1338 void free_bch(struct bch_control
*bch
)
1343 kfree(bch
->a_pow_tab
);
1344 kfree(bch
->a_log_tab
);
1345 kfree(bch
->mod8_tab
);
1346 kfree(bch
->ecc_buf
);
1347 kfree(bch
->ecc_buf2
);
1353 for (i
= 0; i
< ARRAY_SIZE(bch
->poly_2t
); i
++)
1354 kfree(bch
->poly_2t
[i
]);