]>
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2 * Generic binary BCH encoding/decoding library
4 * SPDX-License-Identifier: GPL-2.0
6 * Copyright © 2011 Parrot S.A.
8 * Author: Ivan Djelic <ivan.djelic@parrot.com>
12 * This library provides runtime configurable encoding/decoding of binary
13 * Bose-Chaudhuri-Hocquenghem (BCH) codes.
15 * Call init_bch to get a pointer to a newly allocated bch_control structure for
16 * the given m (Galois field order), t (error correction capability) and
17 * (optional) primitive polynomial parameters.
19 * Call encode_bch to compute and store ecc parity bytes to a given buffer.
20 * Call decode_bch to detect and locate errors in received data.
22 * On systems supporting hw BCH features, intermediate results may be provided
23 * to decode_bch in order to skip certain steps. See decode_bch() documentation
26 * Option CONFIG_BCH_CONST_PARAMS can be used to force fixed values of
27 * parameters m and t; thus allowing extra compiler optimizations and providing
28 * better (up to 2x) encoding performance. Using this option makes sense when
29 * (m,t) are fixed and known in advance, e.g. when using BCH error correction
30 * on a particular NAND flash device.
32 * Algorithmic details:
34 * Encoding is performed by processing 32 input bits in parallel, using 4
35 * remainder lookup tables.
37 * The final stage of decoding involves the following internal steps:
38 * a. Syndrome computation
39 * b. Error locator polynomial computation using Berlekamp-Massey algorithm
40 * c. Error locator root finding (by far the most expensive step)
42 * In this implementation, step c is not performed using the usual Chien search.
43 * Instead, an alternative approach described in [1] is used. It consists in
44 * factoring the error locator polynomial using the Berlekamp Trace algorithm
45 * (BTA) down to a certain degree (4), after which ad hoc low-degree polynomial
46 * solving techniques [2] are used. The resulting algorithm, called BTZ, yields
47 * much better performance than Chien search for usual (m,t) values (typically
48 * m >= 13, t < 32, see [1]).
50 * [1] B. Biswas, V. Herbert. Efficient root finding of polynomials over fields
51 * of characteristic 2, in: Western European Workshop on Research in Cryptology
52 * - WEWoRC 2009, Graz, Austria, LNCS, Springer, July 2009, to appear.
53 * [2] [Zin96] V.A. Zinoviev. On the solution of equations of degree 10 over
54 * finite fields GF(2^q). In Rapport de recherche INRIA no 2829, 1996.
59 #include <ubi_uboot.h>
61 #include <linux/bitops.h>
64 #if defined(__FreeBSD__)
65 #include <sys/endian.h>
74 #define cpu_to_be32 htobe32
75 #define DIV_ROUND_UP(n,d) (((n) + (d) - 1) / (d))
76 #define kmalloc(size, flags) malloc(size)
77 #define kzalloc(size, flags) calloc(1, size)
79 #define ARRAY_SIZE(arr) (sizeof(arr) / sizeof((arr)[0]))
82 #include <asm/byteorder.h>
83 #include <linux/bch.h>
85 #if defined(CONFIG_BCH_CONST_PARAMS)
86 #define GF_M(_p) (CONFIG_BCH_CONST_M)
87 #define GF_T(_p) (CONFIG_BCH_CONST_T)
88 #define GF_N(_p) ((1 << (CONFIG_BCH_CONST_M))-1)
90 #define GF_M(_p) ((_p)->m)
91 #define GF_T(_p) ((_p)->t)
92 #define GF_N(_p) ((_p)->n)
95 #define BCH_ECC_WORDS(_p) DIV_ROUND_UP(GF_M(_p)*GF_T(_p), 32)
96 #define BCH_ECC_BYTES(_p) DIV_ROUND_UP(GF_M(_p)*GF_T(_p), 8)
99 #define dbg(_fmt, args...) do {} while (0)
103 * represent a polynomial over GF(2^m)
106 unsigned int deg
; /* polynomial degree */
107 unsigned int c
[0]; /* polynomial terms */
110 /* given its degree, compute a polynomial size in bytes */
111 #define GF_POLY_SZ(_d) (sizeof(struct gf_poly)+((_d)+1)*sizeof(unsigned int))
113 /* polynomial of degree 1 */
114 struct gf_poly_deg1
{
120 #ifndef __BSD_VISIBLE
121 static int fls(int x
)
127 if (!(x
& 0xffff0000u
)) {
131 if (!(x
& 0xff000000u
)) {
135 if (!(x
& 0xf0000000u
)) {
139 if (!(x
& 0xc0000000u
)) {
143 if (!(x
& 0x80000000u
)) {
153 * same as encode_bch(), but process input data one byte at a time
155 static void encode_bch_unaligned(struct bch_control
*bch
,
156 const unsigned char *data
, unsigned int len
,
161 const int l
= BCH_ECC_WORDS(bch
)-1;
164 p
= bch
->mod8_tab
+ (l
+1)*(((ecc
[0] >> 24)^(*data
++)) & 0xff);
166 for (i
= 0; i
< l
; i
++)
167 ecc
[i
] = ((ecc
[i
] << 8)|(ecc
[i
+1] >> 24))^(*p
++);
169 ecc
[l
] = (ecc
[l
] << 8)^(*p
);
174 * convert ecc bytes to aligned, zero-padded 32-bit ecc words
176 static void load_ecc8(struct bch_control
*bch
, uint32_t *dst
,
179 uint8_t pad
[4] = {0, 0, 0, 0};
180 unsigned int i
, nwords
= BCH_ECC_WORDS(bch
)-1;
182 for (i
= 0; i
< nwords
; i
++, src
+= 4)
183 dst
[i
] = (src
[0] << 24)|(src
[1] << 16)|(src
[2] << 8)|src
[3];
185 memcpy(pad
, src
, BCH_ECC_BYTES(bch
)-4*nwords
);
186 dst
[nwords
] = (pad
[0] << 24)|(pad
[1] << 16)|(pad
[2] << 8)|pad
[3];
190 * convert 32-bit ecc words to ecc bytes
192 static void store_ecc8(struct bch_control
*bch
, uint8_t *dst
,
196 unsigned int i
, nwords
= BCH_ECC_WORDS(bch
)-1;
198 for (i
= 0; i
< nwords
; i
++) {
199 *dst
++ = (src
[i
] >> 24);
200 *dst
++ = (src
[i
] >> 16) & 0xff;
201 *dst
++ = (src
[i
] >> 8) & 0xff;
202 *dst
++ = (src
[i
] >> 0) & 0xff;
204 pad
[0] = (src
[nwords
] >> 24);
205 pad
[1] = (src
[nwords
] >> 16) & 0xff;
206 pad
[2] = (src
[nwords
] >> 8) & 0xff;
207 pad
[3] = (src
[nwords
] >> 0) & 0xff;
208 memcpy(dst
, pad
, BCH_ECC_BYTES(bch
)-4*nwords
);
212 * encode_bch - calculate BCH ecc parity of data
213 * @bch: BCH control structure
214 * @data: data to encode
215 * @len: data length in bytes
216 * @ecc: ecc parity data, must be initialized by caller
218 * The @ecc parity array is used both as input and output parameter, in order to
219 * allow incremental computations. It should be of the size indicated by member
220 * @ecc_bytes of @bch, and should be initialized to 0 before the first call.
222 * The exact number of computed ecc parity bits is given by member @ecc_bits of
223 * @bch; it may be less than m*t for large values of t.
225 void encode_bch(struct bch_control
*bch
, const uint8_t *data
,
226 unsigned int len
, uint8_t *ecc
)
228 const unsigned int l
= BCH_ECC_WORDS(bch
)-1;
229 unsigned int i
, mlen
;
232 const uint32_t * const tab0
= bch
->mod8_tab
;
233 const uint32_t * const tab1
= tab0
+ 256*(l
+1);
234 const uint32_t * const tab2
= tab1
+ 256*(l
+1);
235 const uint32_t * const tab3
= tab2
+ 256*(l
+1);
236 const uint32_t *pdata
, *p0
, *p1
, *p2
, *p3
;
239 /* load ecc parity bytes into internal 32-bit buffer */
240 load_ecc8(bch
, bch
->ecc_buf
, ecc
);
242 memset(bch
->ecc_buf
, 0, sizeof(r
));
245 /* process first unaligned data bytes */
246 m
= ((unsigned long)data
) & 3;
248 mlen
= (len
< (4-m
)) ? len
: 4-m
;
249 encode_bch_unaligned(bch
, data
, mlen
, bch
->ecc_buf
);
254 /* process 32-bit aligned data words */
255 pdata
= (uint32_t *)data
;
259 memcpy(r
, bch
->ecc_buf
, sizeof(r
));
262 * split each 32-bit word into 4 polynomials of weight 8 as follows:
264 * 31 ...24 23 ...16 15 ... 8 7 ... 0
265 * xxxxxxxx yyyyyyyy zzzzzzzz tttttttt
266 * tttttttt mod g = r0 (precomputed)
267 * zzzzzzzz 00000000 mod g = r1 (precomputed)
268 * yyyyyyyy 00000000 00000000 mod g = r2 (precomputed)
269 * xxxxxxxx 00000000 00000000 00000000 mod g = r3 (precomputed)
270 * xxxxxxxx yyyyyyyy zzzzzzzz tttttttt mod g = r0^r1^r2^r3
273 /* input data is read in big-endian format */
274 w
= r
[0]^cpu_to_be32(*pdata
++);
275 p0
= tab0
+ (l
+1)*((w
>> 0) & 0xff);
276 p1
= tab1
+ (l
+1)*((w
>> 8) & 0xff);
277 p2
= tab2
+ (l
+1)*((w
>> 16) & 0xff);
278 p3
= tab3
+ (l
+1)*((w
>> 24) & 0xff);
280 for (i
= 0; i
< l
; i
++)
281 r
[i
] = r
[i
+1]^p0
[i
]^p1
[i
]^p2
[i
]^p3
[i
];
283 r
[l
] = p0
[l
]^p1
[l
]^p2
[l
]^p3
[l
];
285 memcpy(bch
->ecc_buf
, r
, sizeof(r
));
287 /* process last unaligned bytes */
289 encode_bch_unaligned(bch
, data
, len
, bch
->ecc_buf
);
291 /* store ecc parity bytes into original parity buffer */
293 store_ecc8(bch
, ecc
, bch
->ecc_buf
);
296 static inline int modulo(struct bch_control
*bch
, unsigned int v
)
298 const unsigned int n
= GF_N(bch
);
301 v
= (v
& n
) + (v
>> GF_M(bch
));
307 * shorter and faster modulo function, only works when v < 2N.
309 static inline int mod_s(struct bch_control
*bch
, unsigned int v
)
311 const unsigned int n
= GF_N(bch
);
312 return (v
< n
) ? v
: v
-n
;
315 static inline int deg(unsigned int poly
)
317 /* polynomial degree is the most-significant bit index */
321 static inline int parity(unsigned int x
)
324 * public domain code snippet, lifted from
325 * http://www-graphics.stanford.edu/~seander/bithacks.html
329 x
= (x
& 0x11111111U
) * 0x11111111U
;
330 return (x
>> 28) & 1;
333 /* Galois field basic operations: multiply, divide, inverse, etc. */
335 static inline unsigned int gf_mul(struct bch_control
*bch
, unsigned int a
,
338 return (a
&& b
) ? bch
->a_pow_tab
[mod_s(bch
, bch
->a_log_tab
[a
]+
339 bch
->a_log_tab
[b
])] : 0;
342 static inline unsigned int gf_sqr(struct bch_control
*bch
, unsigned int a
)
344 return a
? bch
->a_pow_tab
[mod_s(bch
, 2*bch
->a_log_tab
[a
])] : 0;
347 static inline unsigned int gf_div(struct bch_control
*bch
, unsigned int a
,
350 return a
? bch
->a_pow_tab
[mod_s(bch
, bch
->a_log_tab
[a
]+
351 GF_N(bch
)-bch
->a_log_tab
[b
])] : 0;
354 static inline unsigned int gf_inv(struct bch_control
*bch
, unsigned int a
)
356 return bch
->a_pow_tab
[GF_N(bch
)-bch
->a_log_tab
[a
]];
359 static inline unsigned int a_pow(struct bch_control
*bch
, int i
)
361 return bch
->a_pow_tab
[modulo(bch
, i
)];
364 static inline int a_log(struct bch_control
*bch
, unsigned int x
)
366 return bch
->a_log_tab
[x
];
369 static inline int a_ilog(struct bch_control
*bch
, unsigned int x
)
371 return mod_s(bch
, GF_N(bch
)-bch
->a_log_tab
[x
]);
375 * compute 2t syndromes of ecc polynomial, i.e. ecc(a^j) for j=1..2t
377 static void compute_syndromes(struct bch_control
*bch
, uint32_t *ecc
,
383 const int t
= GF_T(bch
);
387 /* make sure extra bits in last ecc word are cleared */
388 m
= ((unsigned int)s
) & 31;
390 ecc
[s
/32] &= ~((1u << (32-m
))-1);
391 memset(syn
, 0, 2*t
*sizeof(*syn
));
393 /* compute v(a^j) for j=1 .. 2t-1 */
399 for (j
= 0; j
< 2*t
; j
+= 2)
400 syn
[j
] ^= a_pow(bch
, (j
+1)*(i
+s
));
406 /* v(a^(2j)) = v(a^j)^2 */
407 for (j
= 0; j
< t
; j
++)
408 syn
[2*j
+1] = gf_sqr(bch
, syn
[j
]);
411 static void gf_poly_copy(struct gf_poly
*dst
, struct gf_poly
*src
)
413 memcpy(dst
, src
, GF_POLY_SZ(src
->deg
));
416 static int compute_error_locator_polynomial(struct bch_control
*bch
,
417 const unsigned int *syn
)
419 const unsigned int t
= GF_T(bch
);
420 const unsigned int n
= GF_N(bch
);
421 unsigned int i
, j
, tmp
, l
, pd
= 1, d
= syn
[0];
422 struct gf_poly
*elp
= bch
->elp
;
423 struct gf_poly
*pelp
= bch
->poly_2t
[0];
424 struct gf_poly
*elp_copy
= bch
->poly_2t
[1];
427 memset(pelp
, 0, GF_POLY_SZ(2*t
));
428 memset(elp
, 0, GF_POLY_SZ(2*t
));
435 /* use simplified binary Berlekamp-Massey algorithm */
436 for (i
= 0; (i
< t
) && (elp
->deg
<= t
); i
++) {
439 gf_poly_copy(elp_copy
, elp
);
440 /* e[i+1](X) = e[i](X)+di*dp^-1*X^2(i-p)*e[p](X) */
441 tmp
= a_log(bch
, d
)+n
-a_log(bch
, pd
);
442 for (j
= 0; j
<= pelp
->deg
; j
++) {
444 l
= a_log(bch
, pelp
->c
[j
]);
445 elp
->c
[j
+k
] ^= a_pow(bch
, tmp
+l
);
448 /* compute l[i+1] = max(l[i]->c[l[p]+2*(i-p]) */
450 if (tmp
> elp
->deg
) {
452 gf_poly_copy(pelp
, elp_copy
);
457 /* di+1 = S(2i+3)+elp[i+1].1*S(2i+2)+...+elp[i+1].lS(2i+3-l) */
460 for (j
= 1; j
<= elp
->deg
; j
++)
461 d
^= gf_mul(bch
, elp
->c
[j
], syn
[2*i
+2-j
]);
464 dbg("elp=%s\n", gf_poly_str(elp
));
465 return (elp
->deg
> t
) ? -1 : (int)elp
->deg
;
469 * solve a m x m linear system in GF(2) with an expected number of solutions,
470 * and return the number of found solutions
472 static int solve_linear_system(struct bch_control
*bch
, unsigned int *rows
,
473 unsigned int *sol
, int nsol
)
475 const int m
= GF_M(bch
);
476 unsigned int tmp
, mask
;
477 int rem
, c
, r
, p
, k
, param
[m
];
482 /* Gaussian elimination */
483 for (c
= 0; c
< m
; c
++) {
486 /* find suitable row for elimination */
487 for (r
= p
; r
< m
; r
++) {
488 if (rows
[r
] & mask
) {
499 /* perform elimination on remaining rows */
501 for (r
= rem
; r
< m
; r
++) {
506 /* elimination not needed, store defective row index */
511 /* rewrite system, inserting fake parameter rows */
514 for (r
= m
-1; r
>= 0; r
--) {
515 if ((r
> m
-1-k
) && rows
[r
])
516 /* system has no solution */
519 rows
[r
] = (p
&& (r
== param
[p
-1])) ?
520 p
--, 1u << (m
-r
) : rows
[r
-p
];
524 if (nsol
!= (1 << k
))
525 /* unexpected number of solutions */
528 for (p
= 0; p
< nsol
; p
++) {
529 /* set parameters for p-th solution */
530 for (c
= 0; c
< k
; c
++)
531 rows
[param
[c
]] = (rows
[param
[c
]] & ~1)|((p
>> c
) & 1);
533 /* compute unique solution */
535 for (r
= m
-1; r
>= 0; r
--) {
536 mask
= rows
[r
] & (tmp
|1);
537 tmp
|= parity(mask
) << (m
-r
);
545 * this function builds and solves a linear system for finding roots of a degree
546 * 4 affine monic polynomial X^4+aX^2+bX+c over GF(2^m).
548 static int find_affine4_roots(struct bch_control
*bch
, unsigned int a
,
549 unsigned int b
, unsigned int c
,
553 const int m
= GF_M(bch
);
554 unsigned int mask
= 0xff, t
, rows
[16] = {0,};
560 /* buid linear system to solve X^4+aX^2+bX+c = 0 */
561 for (i
= 0; i
< m
; i
++) {
562 rows
[i
+1] = bch
->a_pow_tab
[4*i
]^
563 (a
? bch
->a_pow_tab
[mod_s(bch
, k
)] : 0)^
564 (b
? bch
->a_pow_tab
[mod_s(bch
, j
)] : 0);
569 * transpose 16x16 matrix before passing it to linear solver
570 * warning: this code assumes m < 16
572 for (j
= 8; j
!= 0; j
>>= 1, mask
^= (mask
<< j
)) {
573 for (k
= 0; k
< 16; k
= (k
+j
+1) & ~j
) {
574 t
= ((rows
[k
] >> j
)^rows
[k
+j
]) & mask
;
579 return solve_linear_system(bch
, rows
, roots
, 4);
583 * compute root r of a degree 1 polynomial over GF(2^m) (returned as log(1/r))
585 static int find_poly_deg1_roots(struct bch_control
*bch
, struct gf_poly
*poly
,
591 /* poly[X] = bX+c with c!=0, root=c/b */
592 roots
[n
++] = mod_s(bch
, GF_N(bch
)-bch
->a_log_tab
[poly
->c
[0]]+
593 bch
->a_log_tab
[poly
->c
[1]]);
598 * compute roots of a degree 2 polynomial over GF(2^m)
600 static int find_poly_deg2_roots(struct bch_control
*bch
, struct gf_poly
*poly
,
603 int n
= 0, i
, l0
, l1
, l2
;
604 unsigned int u
, v
, r
;
606 if (poly
->c
[0] && poly
->c
[1]) {
608 l0
= bch
->a_log_tab
[poly
->c
[0]];
609 l1
= bch
->a_log_tab
[poly
->c
[1]];
610 l2
= bch
->a_log_tab
[poly
->c
[2]];
612 /* using z=a/bX, transform aX^2+bX+c into z^2+z+u (u=ac/b^2) */
613 u
= a_pow(bch
, l0
+l2
+2*(GF_N(bch
)-l1
));
615 * let u = sum(li.a^i) i=0..m-1; then compute r = sum(li.xi):
616 * r^2+r = sum(li.(xi^2+xi)) = sum(li.(a^i+Tr(a^i).a^k)) =
617 * u + sum(li.Tr(a^i).a^k) = u+a^k.Tr(sum(li.a^i)) = u+a^k.Tr(u)
618 * i.e. r and r+1 are roots iff Tr(u)=0
628 if ((gf_sqr(bch
, r
)^r
) == u
) {
629 /* reverse z=a/bX transformation and compute log(1/r) */
630 roots
[n
++] = modulo(bch
, 2*GF_N(bch
)-l1
-
631 bch
->a_log_tab
[r
]+l2
);
632 roots
[n
++] = modulo(bch
, 2*GF_N(bch
)-l1
-
633 bch
->a_log_tab
[r
^1]+l2
);
640 * compute roots of a degree 3 polynomial over GF(2^m)
642 static int find_poly_deg3_roots(struct bch_control
*bch
, struct gf_poly
*poly
,
646 unsigned int a
, b
, c
, a2
, b2
, c2
, e3
, tmp
[4];
649 /* transform polynomial into monic X^3 + a2X^2 + b2X + c2 */
651 c2
= gf_div(bch
, poly
->c
[0], e3
);
652 b2
= gf_div(bch
, poly
->c
[1], e3
);
653 a2
= gf_div(bch
, poly
->c
[2], e3
);
655 /* (X+a2)(X^3+a2X^2+b2X+c2) = X^4+aX^2+bX+c (affine) */
656 c
= gf_mul(bch
, a2
, c2
); /* c = a2c2 */
657 b
= gf_mul(bch
, a2
, b2
)^c2
; /* b = a2b2 + c2 */
658 a
= gf_sqr(bch
, a2
)^b2
; /* a = a2^2 + b2 */
660 /* find the 4 roots of this affine polynomial */
661 if (find_affine4_roots(bch
, a
, b
, c
, tmp
) == 4) {
662 /* remove a2 from final list of roots */
663 for (i
= 0; i
< 4; i
++) {
665 roots
[n
++] = a_ilog(bch
, tmp
[i
]);
673 * compute roots of a degree 4 polynomial over GF(2^m)
675 static int find_poly_deg4_roots(struct bch_control
*bch
, struct gf_poly
*poly
,
679 unsigned int a
, b
, c
, d
, e
= 0, f
, a2
, b2
, c2
, e4
;
684 /* transform polynomial into monic X^4 + aX^3 + bX^2 + cX + d */
686 d
= gf_div(bch
, poly
->c
[0], e4
);
687 c
= gf_div(bch
, poly
->c
[1], e4
);
688 b
= gf_div(bch
, poly
->c
[2], e4
);
689 a
= gf_div(bch
, poly
->c
[3], e4
);
691 /* use Y=1/X transformation to get an affine polynomial */
693 /* first, eliminate cX by using z=X+e with ae^2+c=0 */
695 /* compute e such that e^2 = c/a */
696 f
= gf_div(bch
, c
, a
);
698 l
+= (l
& 1) ? GF_N(bch
) : 0;
701 * use transformation z=X+e:
702 * z^4+e^4 + a(z^3+ez^2+e^2z+e^3) + b(z^2+e^2) +cz+ce+d
703 * z^4 + az^3 + (ae+b)z^2 + (ae^2+c)z+e^4+be^2+ae^3+ce+d
704 * z^4 + az^3 + (ae+b)z^2 + e^4+be^2+d
705 * z^4 + az^3 + b'z^2 + d'
707 d
= a_pow(bch
, 2*l
)^gf_mul(bch
, b
, f
)^d
;
708 b
= gf_mul(bch
, a
, e
)^b
;
710 /* now, use Y=1/X to get Y^4 + b/dY^2 + a/dY + 1/d */
712 /* assume all roots have multiplicity 1 */
716 b2
= gf_div(bch
, a
, d
);
717 a2
= gf_div(bch
, b
, d
);
719 /* polynomial is already affine */
724 /* find the 4 roots of this affine polynomial */
725 if (find_affine4_roots(bch
, a2
, b2
, c2
, roots
) == 4) {
726 for (i
= 0; i
< 4; i
++) {
727 /* post-process roots (reverse transformations) */
728 f
= a
? gf_inv(bch
, roots
[i
]) : roots
[i
];
729 roots
[i
] = a_ilog(bch
, f
^e
);
737 * build monic, log-based representation of a polynomial
739 static void gf_poly_logrep(struct bch_control
*bch
,
740 const struct gf_poly
*a
, int *rep
)
742 int i
, d
= a
->deg
, l
= GF_N(bch
)-a_log(bch
, a
->c
[a
->deg
]);
744 /* represent 0 values with -1; warning, rep[d] is not set to 1 */
745 for (i
= 0; i
< d
; i
++)
746 rep
[i
] = a
->c
[i
] ? mod_s(bch
, a_log(bch
, a
->c
[i
])+l
) : -1;
750 * compute polynomial Euclidean division remainder in GF(2^m)[X]
752 static void gf_poly_mod(struct bch_control
*bch
, struct gf_poly
*a
,
753 const struct gf_poly
*b
, int *rep
)
756 unsigned int i
, j
, *c
= a
->c
;
757 const unsigned int d
= b
->deg
;
762 /* reuse or compute log representation of denominator */
765 gf_poly_logrep(bch
, b
, rep
);
768 for (j
= a
->deg
; j
>= d
; j
--) {
770 la
= a_log(bch
, c
[j
]);
772 for (i
= 0; i
< d
; i
++, p
++) {
775 c
[p
] ^= bch
->a_pow_tab
[mod_s(bch
,
781 while (!c
[a
->deg
] && a
->deg
)
786 * compute polynomial Euclidean division quotient in GF(2^m)[X]
788 static void gf_poly_div(struct bch_control
*bch
, struct gf_poly
*a
,
789 const struct gf_poly
*b
, struct gf_poly
*q
)
791 if (a
->deg
>= b
->deg
) {
792 q
->deg
= a
->deg
-b
->deg
;
793 /* compute a mod b (modifies a) */
794 gf_poly_mod(bch
, a
, b
, NULL
);
795 /* quotient is stored in upper part of polynomial a */
796 memcpy(q
->c
, &a
->c
[b
->deg
], (1+q
->deg
)*sizeof(unsigned int));
804 * compute polynomial GCD (Greatest Common Divisor) in GF(2^m)[X]
806 static struct gf_poly
*gf_poly_gcd(struct bch_control
*bch
, struct gf_poly
*a
,
811 dbg("gcd(%s,%s)=", gf_poly_str(a
), gf_poly_str(b
));
813 if (a
->deg
< b
->deg
) {
820 gf_poly_mod(bch
, a
, b
, NULL
);
826 dbg("%s\n", gf_poly_str(a
));
832 * Given a polynomial f and an integer k, compute Tr(a^kX) mod f
833 * This is used in Berlekamp Trace algorithm for splitting polynomials
835 static void compute_trace_bk_mod(struct bch_control
*bch
, int k
,
836 const struct gf_poly
*f
, struct gf_poly
*z
,
839 const int m
= GF_M(bch
);
842 /* z contains z^2j mod f */
845 z
->c
[1] = bch
->a_pow_tab
[k
];
848 memset(out
, 0, GF_POLY_SZ(f
->deg
));
850 /* compute f log representation only once */
851 gf_poly_logrep(bch
, f
, bch
->cache
);
853 for (i
= 0; i
< m
; i
++) {
854 /* add a^(k*2^i)(z^(2^i) mod f) and compute (z^(2^i) mod f)^2 */
855 for (j
= z
->deg
; j
>= 0; j
--) {
856 out
->c
[j
] ^= z
->c
[j
];
857 z
->c
[2*j
] = gf_sqr(bch
, z
->c
[j
]);
860 if (z
->deg
> out
->deg
)
865 /* z^(2(i+1)) mod f = (z^(2^i) mod f)^2 mod f */
866 gf_poly_mod(bch
, z
, f
, bch
->cache
);
869 while (!out
->c
[out
->deg
] && out
->deg
)
872 dbg("Tr(a^%d.X) mod f = %s\n", k
, gf_poly_str(out
));
876 * factor a polynomial using Berlekamp Trace algorithm (BTA)
878 static void factor_polynomial(struct bch_control
*bch
, int k
, struct gf_poly
*f
,
879 struct gf_poly
**g
, struct gf_poly
**h
)
881 struct gf_poly
*f2
= bch
->poly_2t
[0];
882 struct gf_poly
*q
= bch
->poly_2t
[1];
883 struct gf_poly
*tk
= bch
->poly_2t
[2];
884 struct gf_poly
*z
= bch
->poly_2t
[3];
887 dbg("factoring %s...\n", gf_poly_str(f
));
892 /* tk = Tr(a^k.X) mod f */
893 compute_trace_bk_mod(bch
, k
, f
, z
, tk
);
896 /* compute g = gcd(f, tk) (destructive operation) */
898 gcd
= gf_poly_gcd(bch
, f2
, tk
);
899 if (gcd
->deg
< f
->deg
) {
900 /* compute h=f/gcd(f,tk); this will modify f and q */
901 gf_poly_div(bch
, f
, gcd
, q
);
902 /* store g and h in-place (clobbering f) */
903 *h
= &((struct gf_poly_deg1
*)f
)[gcd
->deg
].poly
;
904 gf_poly_copy(*g
, gcd
);
911 * find roots of a polynomial, using BTZ algorithm; see the beginning of this
914 static int find_poly_roots(struct bch_control
*bch
, unsigned int k
,
915 struct gf_poly
*poly
, unsigned int *roots
)
918 struct gf_poly
*f1
, *f2
;
921 /* handle low degree polynomials with ad hoc techniques */
923 cnt
= find_poly_deg1_roots(bch
, poly
, roots
);
926 cnt
= find_poly_deg2_roots(bch
, poly
, roots
);
929 cnt
= find_poly_deg3_roots(bch
, poly
, roots
);
932 cnt
= find_poly_deg4_roots(bch
, poly
, roots
);
935 /* factor polynomial using Berlekamp Trace Algorithm (BTA) */
937 if (poly
->deg
&& (k
<= GF_M(bch
))) {
938 factor_polynomial(bch
, k
, poly
, &f1
, &f2
);
940 cnt
+= find_poly_roots(bch
, k
+1, f1
, roots
);
942 cnt
+= find_poly_roots(bch
, k
+1, f2
, roots
+cnt
);
949 #if defined(USE_CHIEN_SEARCH)
951 * exhaustive root search (Chien) implementation - not used, included only for
952 * reference/comparison tests
954 static int chien_search(struct bch_control
*bch
, unsigned int len
,
955 struct gf_poly
*p
, unsigned int *roots
)
958 unsigned int i
, j
, syn
, syn0
, count
= 0;
959 const unsigned int k
= 8*len
+bch
->ecc_bits
;
961 /* use a log-based representation of polynomial */
962 gf_poly_logrep(bch
, p
, bch
->cache
);
963 bch
->cache
[p
->deg
] = 0;
964 syn0
= gf_div(bch
, p
->c
[0], p
->c
[p
->deg
]);
966 for (i
= GF_N(bch
)-k
+1; i
<= GF_N(bch
); i
++) {
967 /* compute elp(a^i) */
968 for (j
= 1, syn
= syn0
; j
<= p
->deg
; j
++) {
971 syn
^= a_pow(bch
, m
+j
*i
);
974 roots
[count
++] = GF_N(bch
)-i
;
979 return (count
== p
->deg
) ? count
: 0;
981 #define find_poly_roots(_p, _k, _elp, _loc) chien_search(_p, len, _elp, _loc)
982 #endif /* USE_CHIEN_SEARCH */
985 * decode_bch - decode received codeword and find bit error locations
986 * @bch: BCH control structure
987 * @data: received data, ignored if @calc_ecc is provided
988 * @len: data length in bytes, must always be provided
989 * @recv_ecc: received ecc, if NULL then assume it was XORed in @calc_ecc
990 * @calc_ecc: calculated ecc, if NULL then calc_ecc is computed from @data
991 * @syn: hw computed syndrome data (if NULL, syndrome is calculated)
992 * @errloc: output array of error locations
995 * The number of errors found, or -EBADMSG if decoding failed, or -EINVAL if
996 * invalid parameters were provided
998 * Depending on the available hw BCH support and the need to compute @calc_ecc
999 * separately (using encode_bch()), this function should be called with one of
1000 * the following parameter configurations -
1002 * by providing @data and @recv_ecc only:
1003 * decode_bch(@bch, @data, @len, @recv_ecc, NULL, NULL, @errloc)
1005 * by providing @recv_ecc and @calc_ecc:
1006 * decode_bch(@bch, NULL, @len, @recv_ecc, @calc_ecc, NULL, @errloc)
1008 * by providing ecc = recv_ecc XOR calc_ecc:
1009 * decode_bch(@bch, NULL, @len, NULL, ecc, NULL, @errloc)
1011 * by providing syndrome results @syn:
1012 * decode_bch(@bch, NULL, @len, NULL, NULL, @syn, @errloc)
1014 * Once decode_bch() has successfully returned with a positive value, error
1015 * locations returned in array @errloc should be interpreted as follows -
1017 * if (errloc[n] >= 8*len), then n-th error is located in ecc (no need for
1020 * if (errloc[n] < 8*len), then n-th error is located in data and can be
1021 * corrected with statement data[errloc[n]/8] ^= 1 << (errloc[n] % 8);
1023 * Note that this function does not perform any data correction by itself, it
1024 * merely indicates error locations.
1026 int decode_bch(struct bch_control
*bch
, const uint8_t *data
, unsigned int len
,
1027 const uint8_t *recv_ecc
, const uint8_t *calc_ecc
,
1028 const unsigned int *syn
, unsigned int *errloc
)
1030 const unsigned int ecc_words
= BCH_ECC_WORDS(bch
);
1035 /* sanity check: make sure data length can be handled */
1036 if (8*len
> (bch
->n
-bch
->ecc_bits
))
1039 /* if caller does not provide syndromes, compute them */
1042 /* compute received data ecc into an internal buffer */
1043 if (!data
|| !recv_ecc
)
1045 encode_bch(bch
, data
, len
, NULL
);
1047 /* load provided calculated ecc */
1048 load_ecc8(bch
, bch
->ecc_buf
, calc_ecc
);
1050 /* load received ecc or assume it was XORed in calc_ecc */
1052 load_ecc8(bch
, bch
->ecc_buf2
, recv_ecc
);
1053 /* XOR received and calculated ecc */
1054 for (i
= 0, sum
= 0; i
< (int)ecc_words
; i
++) {
1055 bch
->ecc_buf
[i
] ^= bch
->ecc_buf2
[i
];
1056 sum
|= bch
->ecc_buf
[i
];
1059 /* no error found */
1062 compute_syndromes(bch
, bch
->ecc_buf
, bch
->syn
);
1066 err
= compute_error_locator_polynomial(bch
, syn
);
1068 nroots
= find_poly_roots(bch
, 1, bch
->elp
, errloc
);
1073 /* post-process raw error locations for easier correction */
1074 nbits
= (len
*8)+bch
->ecc_bits
;
1075 for (i
= 0; i
< err
; i
++) {
1076 if (errloc
[i
] >= nbits
) {
1080 errloc
[i
] = nbits
-1-errloc
[i
];
1081 errloc
[i
] = (errloc
[i
] & ~7)|(7-(errloc
[i
] & 7));
1084 return (err
>= 0) ? err
: -EBADMSG
;
1088 * generate Galois field lookup tables
1090 static int build_gf_tables(struct bch_control
*bch
, unsigned int poly
)
1092 unsigned int i
, x
= 1;
1093 const unsigned int k
= 1 << deg(poly
);
1095 /* primitive polynomial must be of degree m */
1096 if (k
!= (1u << GF_M(bch
)))
1099 for (i
= 0; i
< GF_N(bch
); i
++) {
1100 bch
->a_pow_tab
[i
] = x
;
1101 bch
->a_log_tab
[x
] = i
;
1103 /* polynomial is not primitive (a^i=1 with 0<i<2^m-1) */
1109 bch
->a_pow_tab
[GF_N(bch
)] = 1;
1110 bch
->a_log_tab
[0] = 0;
1116 * compute generator polynomial remainder tables for fast encoding
1118 static void build_mod8_tables(struct bch_control
*bch
, const uint32_t *g
)
1121 uint32_t data
, hi
, lo
, *tab
;
1122 const int l
= BCH_ECC_WORDS(bch
);
1123 const int plen
= DIV_ROUND_UP(bch
->ecc_bits
+1, 32);
1124 const int ecclen
= DIV_ROUND_UP(bch
->ecc_bits
, 32);
1126 memset(bch
->mod8_tab
, 0, 4*256*l
*sizeof(*bch
->mod8_tab
));
1128 for (i
= 0; i
< 256; i
++) {
1129 /* p(X)=i is a small polynomial of weight <= 8 */
1130 for (b
= 0; b
< 4; b
++) {
1131 /* we want to compute (p(X).X^(8*b+deg(g))) mod g(X) */
1132 tab
= bch
->mod8_tab
+ (b
*256+i
)*l
;
1136 /* subtract X^d.g(X) from p(X).X^(8*b+deg(g)) */
1137 data
^= g
[0] >> (31-d
);
1138 for (j
= 0; j
< ecclen
; j
++) {
1139 hi
= (d
< 31) ? g
[j
] << (d
+1) : 0;
1141 g
[j
+1] >> (31-d
) : 0;
1150 * build a base for factoring degree 2 polynomials
1152 static int build_deg2_base(struct bch_control
*bch
)
1154 const int m
= GF_M(bch
);
1156 unsigned int sum
, x
, y
, remaining
, ak
= 0, xi
[m
];
1158 /* find k s.t. Tr(a^k) = 1 and 0 <= k < m */
1159 for (i
= 0; i
< m
; i
++) {
1160 for (j
= 0, sum
= 0; j
< m
; j
++)
1161 sum
^= a_pow(bch
, i
*(1 << j
));
1164 ak
= bch
->a_pow_tab
[i
];
1168 /* find xi, i=0..m-1 such that xi^2+xi = a^i+Tr(a^i).a^k */
1170 memset(xi
, 0, sizeof(xi
));
1172 for (x
= 0; (x
<= GF_N(bch
)) && remaining
; x
++) {
1173 y
= gf_sqr(bch
, x
)^x
;
1174 for (i
= 0; i
< 2; i
++) {
1176 if (y
&& (r
< m
) && !xi
[r
]) {
1180 dbg("x%d = %x\n", r
, x
);
1186 /* should not happen but check anyway */
1187 return remaining
? -1 : 0;
1190 static void *bch_alloc(size_t size
, int *err
)
1194 ptr
= kmalloc(size
, GFP_KERNEL
);
1201 * compute generator polynomial for given (m,t) parameters.
1203 static uint32_t *compute_generator_polynomial(struct bch_control
*bch
)
1205 const unsigned int m
= GF_M(bch
);
1206 const unsigned int t
= GF_T(bch
);
1208 unsigned int i
, j
, nbits
, r
, word
, *roots
;
1212 g
= bch_alloc(GF_POLY_SZ(m
*t
), &err
);
1213 roots
= bch_alloc((bch
->n
+1)*sizeof(*roots
), &err
);
1214 genpoly
= bch_alloc(DIV_ROUND_UP(m
*t
+1, 32)*sizeof(*genpoly
), &err
);
1222 /* enumerate all roots of g(X) */
1223 memset(roots
, 0, (bch
->n
+1)*sizeof(*roots
));
1224 for (i
= 0; i
< t
; i
++) {
1225 for (j
= 0, r
= 2*i
+1; j
< m
; j
++) {
1227 r
= mod_s(bch
, 2*r
);
1230 /* build generator polynomial g(X) */
1233 for (i
= 0; i
< GF_N(bch
); i
++) {
1235 /* multiply g(X) by (X+root) */
1236 r
= bch
->a_pow_tab
[i
];
1238 for (j
= g
->deg
; j
> 0; j
--)
1239 g
->c
[j
] = gf_mul(bch
, g
->c
[j
], r
)^g
->c
[j
-1];
1241 g
->c
[0] = gf_mul(bch
, g
->c
[0], r
);
1245 /* store left-justified binary representation of g(X) */
1250 nbits
= (n
> 32) ? 32 : n
;
1251 for (j
= 0, word
= 0; j
< nbits
; j
++) {
1253 word
|= 1u << (31-j
);
1255 genpoly
[i
++] = word
;
1258 bch
->ecc_bits
= g
->deg
;
1268 * init_bch - initialize a BCH encoder/decoder
1269 * @m: Galois field order, should be in the range 5-15
1270 * @t: maximum error correction capability, in bits
1271 * @prim_poly: user-provided primitive polynomial (or 0 to use default)
1274 * a newly allocated BCH control structure if successful, NULL otherwise
1276 * This initialization can take some time, as lookup tables are built for fast
1277 * encoding/decoding; make sure not to call this function from a time critical
1278 * path. Usually, init_bch() should be called on module/driver init and
1279 * free_bch() should be called to release memory on exit.
1281 * You may provide your own primitive polynomial of degree @m in argument
1282 * @prim_poly, or let init_bch() use its default polynomial.
1284 * Once init_bch() has successfully returned a pointer to a newly allocated
1285 * BCH control structure, ecc length in bytes is given by member @ecc_bytes of
1288 struct bch_control
*init_bch(int m
, int t
, unsigned int prim_poly
)
1291 unsigned int i
, words
;
1293 struct bch_control
*bch
= NULL
;
1295 const int min_m
= 5;
1296 const int max_m
= 15;
1298 /* default primitive polynomials */
1299 static const unsigned int prim_poly_tab
[] = {
1300 0x25, 0x43, 0x83, 0x11d, 0x211, 0x409, 0x805, 0x1053, 0x201b,
1304 #if defined(CONFIG_BCH_CONST_PARAMS)
1305 if ((m
!= (CONFIG_BCH_CONST_M
)) || (t
!= (CONFIG_BCH_CONST_T
))) {
1306 printk(KERN_ERR
"bch encoder/decoder was configured to support "
1307 "parameters m=%d, t=%d only!\n",
1308 CONFIG_BCH_CONST_M
, CONFIG_BCH_CONST_T
);
1312 if ((m
< min_m
) || (m
> max_m
))
1314 * values of m greater than 15 are not currently supported;
1315 * supporting m > 15 would require changing table base type
1316 * (uint16_t) and a small patch in matrix transposition
1321 if ((t
< 1) || (m
*t
>= ((1 << m
)-1)))
1322 /* invalid t value */
1325 /* select a primitive polynomial for generating GF(2^m) */
1327 prim_poly
= prim_poly_tab
[m
-min_m
];
1329 bch
= kzalloc(sizeof(*bch
), GFP_KERNEL
);
1335 bch
->n
= (1 << m
)-1;
1336 words
= DIV_ROUND_UP(m
*t
, 32);
1337 bch
->ecc_bytes
= DIV_ROUND_UP(m
*t
, 8);
1338 bch
->a_pow_tab
= bch_alloc((1+bch
->n
)*sizeof(*bch
->a_pow_tab
), &err
);
1339 bch
->a_log_tab
= bch_alloc((1+bch
->n
)*sizeof(*bch
->a_log_tab
), &err
);
1340 bch
->mod8_tab
= bch_alloc(words
*1024*sizeof(*bch
->mod8_tab
), &err
);
1341 bch
->ecc_buf
= bch_alloc(words
*sizeof(*bch
->ecc_buf
), &err
);
1342 bch
->ecc_buf2
= bch_alloc(words
*sizeof(*bch
->ecc_buf2
), &err
);
1343 bch
->xi_tab
= bch_alloc(m
*sizeof(*bch
->xi_tab
), &err
);
1344 bch
->syn
= bch_alloc(2*t
*sizeof(*bch
->syn
), &err
);
1345 bch
->cache
= bch_alloc(2*t
*sizeof(*bch
->cache
), &err
);
1346 bch
->elp
= bch_alloc((t
+1)*sizeof(struct gf_poly_deg1
), &err
);
1348 for (i
= 0; i
< ARRAY_SIZE(bch
->poly_2t
); i
++)
1349 bch
->poly_2t
[i
] = bch_alloc(GF_POLY_SZ(2*t
), &err
);
1354 err
= build_gf_tables(bch
, prim_poly
);
1358 /* use generator polynomial for computing encoding tables */
1359 genpoly
= compute_generator_polynomial(bch
);
1360 if (genpoly
== NULL
)
1363 build_mod8_tables(bch
, genpoly
);
1366 err
= build_deg2_base(bch
);
1378 * free_bch - free the BCH control structure
1379 * @bch: BCH control structure to release
1381 void free_bch(struct bch_control
*bch
)
1386 kfree(bch
->a_pow_tab
);
1387 kfree(bch
->a_log_tab
);
1388 kfree(bch
->mod8_tab
);
1389 kfree(bch
->ecc_buf
);
1390 kfree(bch
->ecc_buf2
);
1396 for (i
= 0; i
< ARRAY_SIZE(bch
->poly_2t
); i
++)
1397 kfree(bch
->poly_2t
[i
]);