1 // SPDX-License-Identifier: GPL-2.0
3 * Generic binary BCH encoding/decoding library
5 * Copyright © 2011 Parrot S.A.
7 * Author: Ivan Djelic <ivan.djelic@parrot.com>
11 * This library provides runtime configurable encoding/decoding of binary
12 * Bose-Chaudhuri-Hocquenghem (BCH) codes.
14 * Call init_bch to get a pointer to a newly allocated bch_control structure for
15 * the given m (Galois field order), t (error correction capability) and
16 * (optional) primitive polynomial parameters.
18 * Call encode_bch to compute and store ecc parity bytes to a given buffer.
19 * Call decode_bch to detect and locate errors in received data.
21 * On systems supporting hw BCH features, intermediate results may be provided
22 * to decode_bch in order to skip certain steps. See decode_bch() documentation
25 * Option CONFIG_BCH_CONST_PARAMS can be used to force fixed values of
26 * parameters m and t; thus allowing extra compiler optimizations and providing
27 * better (up to 2x) encoding performance. Using this option makes sense when
28 * (m,t) are fixed and known in advance, e.g. when using BCH error correction
29 * on a particular NAND flash device.
31 * Algorithmic details:
33 * Encoding is performed by processing 32 input bits in parallel, using 4
34 * remainder lookup tables.
36 * The final stage of decoding involves the following internal steps:
37 * a. Syndrome computation
38 * b. Error locator polynomial computation using Berlekamp-Massey algorithm
39 * c. Error locator root finding (by far the most expensive step)
41 * In this implementation, step c is not performed using the usual Chien search.
42 * Instead, an alternative approach described in [1] is used. It consists in
43 * factoring the error locator polynomial using the Berlekamp Trace algorithm
44 * (BTA) down to a certain degree (4), after which ad hoc low-degree polynomial
45 * solving techniques [2] are used. The resulting algorithm, called BTZ, yields
46 * much better performance than Chien search for usual (m,t) values (typically
47 * m >= 13, t < 32, see [1]).
49 * [1] B. Biswas, V. Herbert. Efficient root finding of polynomials over fields
50 * of characteristic 2, in: Western European Workshop on Research in Cryptology
51 * - WEWoRC 2009, Graz, Austria, LNCS, Springer, July 2009, to appear.
52 * [2] [Zin96] V.A. Zinoviev. On the solution of equations of degree 10 over
53 * finite fields GF(2^q). In Rapport de recherche INRIA no 2829, 1996.
58 #include <ubi_uboot.h>
60 #include <linux/bitops.h>
63 #if defined(__FreeBSD__)
64 #include <sys/endian.h>
65 #elif defined(__APPLE__)
66 #include <machine/endian.h>
67 #include <libkern/OSByteOrder.h>
76 #if defined(__APPLE__)
77 #define cpu_to_be32 OSSwapHostToBigInt32
79 #define cpu_to_be32 htobe32
81 #define DIV_ROUND_UP(n,d) (((n) + (d) - 1) / (d))
82 #define kmalloc(size, flags) malloc(size)
83 #define kzalloc(size, flags) calloc(1, size)
85 #define ARRAY_SIZE(arr) (sizeof(arr) / sizeof((arr)[0]))
88 #include <asm/byteorder.h>
89 #include <linux/bch.h>
91 #if defined(CONFIG_BCH_CONST_PARAMS)
92 #define GF_M(_p) (CONFIG_BCH_CONST_M)
93 #define GF_T(_p) (CONFIG_BCH_CONST_T)
94 #define GF_N(_p) ((1 << (CONFIG_BCH_CONST_M))-1)
96 #define GF_M(_p) ((_p)->m)
97 #define GF_T(_p) ((_p)->t)
98 #define GF_N(_p) ((_p)->n)
101 #define BCH_ECC_WORDS(_p) DIV_ROUND_UP(GF_M(_p)*GF_T(_p), 32)
102 #define BCH_ECC_BYTES(_p) DIV_ROUND_UP(GF_M(_p)*GF_T(_p), 8)
105 #define dbg(_fmt, args...) do {} while (0)
109 * represent a polynomial over GF(2^m)
112 unsigned int deg
; /* polynomial degree */
113 unsigned int c
[0]; /* polynomial terms */
116 /* given its degree, compute a polynomial size in bytes */
117 #define GF_POLY_SZ(_d) (sizeof(struct gf_poly)+((_d)+1)*sizeof(unsigned int))
119 /* polynomial of degree 1 */
120 struct gf_poly_deg1
{
126 #if !defined(__DragonFly__) && !defined(__FreeBSD__) && !defined(__APPLE__)
127 static int fls(int x
)
133 if (!(x
& 0xffff0000u
)) {
137 if (!(x
& 0xff000000u
)) {
141 if (!(x
& 0xf0000000u
)) {
145 if (!(x
& 0xc0000000u
)) {
149 if (!(x
& 0x80000000u
)) {
159 * same as encode_bch(), but process input data one byte at a time
161 static void encode_bch_unaligned(struct bch_control
*bch
,
162 const unsigned char *data
, unsigned int len
,
167 const int l
= BCH_ECC_WORDS(bch
)-1;
170 p
= bch
->mod8_tab
+ (l
+1)*(((ecc
[0] >> 24)^(*data
++)) & 0xff);
172 for (i
= 0; i
< l
; i
++)
173 ecc
[i
] = ((ecc
[i
] << 8)|(ecc
[i
+1] >> 24))^(*p
++);
175 ecc
[l
] = (ecc
[l
] << 8)^(*p
);
180 * convert ecc bytes to aligned, zero-padded 32-bit ecc words
182 static void load_ecc8(struct bch_control
*bch
, uint32_t *dst
,
185 uint8_t pad
[4] = {0, 0, 0, 0};
186 unsigned int i
, nwords
= BCH_ECC_WORDS(bch
)-1;
188 for (i
= 0; i
< nwords
; i
++, src
+= 4)
189 dst
[i
] = (src
[0] << 24)|(src
[1] << 16)|(src
[2] << 8)|src
[3];
191 memcpy(pad
, src
, BCH_ECC_BYTES(bch
)-4*nwords
);
192 dst
[nwords
] = (pad
[0] << 24)|(pad
[1] << 16)|(pad
[2] << 8)|pad
[3];
196 * convert 32-bit ecc words to ecc bytes
198 static void store_ecc8(struct bch_control
*bch
, uint8_t *dst
,
202 unsigned int i
, nwords
= BCH_ECC_WORDS(bch
)-1;
204 for (i
= 0; i
< nwords
; i
++) {
205 *dst
++ = (src
[i
] >> 24);
206 *dst
++ = (src
[i
] >> 16) & 0xff;
207 *dst
++ = (src
[i
] >> 8) & 0xff;
208 *dst
++ = (src
[i
] >> 0) & 0xff;
210 pad
[0] = (src
[nwords
] >> 24);
211 pad
[1] = (src
[nwords
] >> 16) & 0xff;
212 pad
[2] = (src
[nwords
] >> 8) & 0xff;
213 pad
[3] = (src
[nwords
] >> 0) & 0xff;
214 memcpy(dst
, pad
, BCH_ECC_BYTES(bch
)-4*nwords
);
218 * encode_bch - calculate BCH ecc parity of data
219 * @bch: BCH control structure
220 * @data: data to encode
221 * @len: data length in bytes
222 * @ecc: ecc parity data, must be initialized by caller
224 * The @ecc parity array is used both as input and output parameter, in order to
225 * allow incremental computations. It should be of the size indicated by member
226 * @ecc_bytes of @bch, and should be initialized to 0 before the first call.
228 * The exact number of computed ecc parity bits is given by member @ecc_bits of
229 * @bch; it may be less than m*t for large values of t.
231 void encode_bch(struct bch_control
*bch
, const uint8_t *data
,
232 unsigned int len
, uint8_t *ecc
)
234 const unsigned int l
= BCH_ECC_WORDS(bch
)-1;
235 unsigned int i
, mlen
;
238 const uint32_t * const tab0
= bch
->mod8_tab
;
239 const uint32_t * const tab1
= tab0
+ 256*(l
+1);
240 const uint32_t * const tab2
= tab1
+ 256*(l
+1);
241 const uint32_t * const tab3
= tab2
+ 256*(l
+1);
242 const uint32_t *pdata
, *p0
, *p1
, *p2
, *p3
;
245 /* load ecc parity bytes into internal 32-bit buffer */
246 load_ecc8(bch
, bch
->ecc_buf
, ecc
);
248 memset(bch
->ecc_buf
, 0, sizeof(r
));
251 /* process first unaligned data bytes */
252 m
= ((unsigned long)data
) & 3;
254 mlen
= (len
< (4-m
)) ? len
: 4-m
;
255 encode_bch_unaligned(bch
, data
, mlen
, bch
->ecc_buf
);
260 /* process 32-bit aligned data words */
261 pdata
= (uint32_t *)data
;
265 memcpy(r
, bch
->ecc_buf
, sizeof(r
));
268 * split each 32-bit word into 4 polynomials of weight 8 as follows:
270 * 31 ...24 23 ...16 15 ... 8 7 ... 0
271 * xxxxxxxx yyyyyyyy zzzzzzzz tttttttt
272 * tttttttt mod g = r0 (precomputed)
273 * zzzzzzzz 00000000 mod g = r1 (precomputed)
274 * yyyyyyyy 00000000 00000000 mod g = r2 (precomputed)
275 * xxxxxxxx 00000000 00000000 00000000 mod g = r3 (precomputed)
276 * xxxxxxxx yyyyyyyy zzzzzzzz tttttttt mod g = r0^r1^r2^r3
279 /* input data is read in big-endian format */
280 w
= r
[0]^cpu_to_be32(*pdata
++);
281 p0
= tab0
+ (l
+1)*((w
>> 0) & 0xff);
282 p1
= tab1
+ (l
+1)*((w
>> 8) & 0xff);
283 p2
= tab2
+ (l
+1)*((w
>> 16) & 0xff);
284 p3
= tab3
+ (l
+1)*((w
>> 24) & 0xff);
286 for (i
= 0; i
< l
; i
++)
287 r
[i
] = r
[i
+1]^p0
[i
]^p1
[i
]^p2
[i
]^p3
[i
];
289 r
[l
] = p0
[l
]^p1
[l
]^p2
[l
]^p3
[l
];
291 memcpy(bch
->ecc_buf
, r
, sizeof(r
));
293 /* process last unaligned bytes */
295 encode_bch_unaligned(bch
, data
, len
, bch
->ecc_buf
);
297 /* store ecc parity bytes into original parity buffer */
299 store_ecc8(bch
, ecc
, bch
->ecc_buf
);
302 static inline int modulo(struct bch_control
*bch
, unsigned int v
)
304 const unsigned int n
= GF_N(bch
);
307 v
= (v
& n
) + (v
>> GF_M(bch
));
313 * shorter and faster modulo function, only works when v < 2N.
315 static inline int mod_s(struct bch_control
*bch
, unsigned int v
)
317 const unsigned int n
= GF_N(bch
);
318 return (v
< n
) ? v
: v
-n
;
321 static inline int deg(unsigned int poly
)
323 /* polynomial degree is the most-significant bit index */
327 static inline int parity(unsigned int x
)
330 * public domain code snippet, lifted from
331 * http://www-graphics.stanford.edu/~seander/bithacks.html
335 x
= (x
& 0x11111111U
) * 0x11111111U
;
336 return (x
>> 28) & 1;
339 /* Galois field basic operations: multiply, divide, inverse, etc. */
341 static inline unsigned int gf_mul(struct bch_control
*bch
, unsigned int a
,
344 return (a
&& b
) ? bch
->a_pow_tab
[mod_s(bch
, bch
->a_log_tab
[a
]+
345 bch
->a_log_tab
[b
])] : 0;
348 static inline unsigned int gf_sqr(struct bch_control
*bch
, unsigned int a
)
350 return a
? bch
->a_pow_tab
[mod_s(bch
, 2*bch
->a_log_tab
[a
])] : 0;
353 static inline unsigned int gf_div(struct bch_control
*bch
, unsigned int a
,
356 return a
? bch
->a_pow_tab
[mod_s(bch
, bch
->a_log_tab
[a
]+
357 GF_N(bch
)-bch
->a_log_tab
[b
])] : 0;
360 static inline unsigned int gf_inv(struct bch_control
*bch
, unsigned int a
)
362 return bch
->a_pow_tab
[GF_N(bch
)-bch
->a_log_tab
[a
]];
365 static inline unsigned int a_pow(struct bch_control
*bch
, int i
)
367 return bch
->a_pow_tab
[modulo(bch
, i
)];
370 static inline int a_log(struct bch_control
*bch
, unsigned int x
)
372 return bch
->a_log_tab
[x
];
375 static inline int a_ilog(struct bch_control
*bch
, unsigned int x
)
377 return mod_s(bch
, GF_N(bch
)-bch
->a_log_tab
[x
]);
381 * compute 2t syndromes of ecc polynomial, i.e. ecc(a^j) for j=1..2t
383 static void compute_syndromes(struct bch_control
*bch
, uint32_t *ecc
,
389 const int t
= GF_T(bch
);
393 /* make sure extra bits in last ecc word are cleared */
394 m
= ((unsigned int)s
) & 31;
396 ecc
[s
/32] &= ~((1u << (32-m
))-1);
397 memset(syn
, 0, 2*t
*sizeof(*syn
));
399 /* compute v(a^j) for j=1 .. 2t-1 */
405 for (j
= 0; j
< 2*t
; j
+= 2)
406 syn
[j
] ^= a_pow(bch
, (j
+1)*(i
+s
));
412 /* v(a^(2j)) = v(a^j)^2 */
413 for (j
= 0; j
< t
; j
++)
414 syn
[2*j
+1] = gf_sqr(bch
, syn
[j
]);
417 static void gf_poly_copy(struct gf_poly
*dst
, struct gf_poly
*src
)
419 memcpy(dst
, src
, GF_POLY_SZ(src
->deg
));
422 static int compute_error_locator_polynomial(struct bch_control
*bch
,
423 const unsigned int *syn
)
425 const unsigned int t
= GF_T(bch
);
426 const unsigned int n
= GF_N(bch
);
427 unsigned int i
, j
, tmp
, l
, pd
= 1, d
= syn
[0];
428 struct gf_poly
*elp
= bch
->elp
;
429 struct gf_poly
*pelp
= bch
->poly_2t
[0];
430 struct gf_poly
*elp_copy
= bch
->poly_2t
[1];
433 memset(pelp
, 0, GF_POLY_SZ(2*t
));
434 memset(elp
, 0, GF_POLY_SZ(2*t
));
441 /* use simplified binary Berlekamp-Massey algorithm */
442 for (i
= 0; (i
< t
) && (elp
->deg
<= t
); i
++) {
445 gf_poly_copy(elp_copy
, elp
);
446 /* e[i+1](X) = e[i](X)+di*dp^-1*X^2(i-p)*e[p](X) */
447 tmp
= a_log(bch
, d
)+n
-a_log(bch
, pd
);
448 for (j
= 0; j
<= pelp
->deg
; j
++) {
450 l
= a_log(bch
, pelp
->c
[j
]);
451 elp
->c
[j
+k
] ^= a_pow(bch
, tmp
+l
);
454 /* compute l[i+1] = max(l[i]->c[l[p]+2*(i-p]) */
456 if (tmp
> elp
->deg
) {
458 gf_poly_copy(pelp
, elp_copy
);
463 /* di+1 = S(2i+3)+elp[i+1].1*S(2i+2)+...+elp[i+1].lS(2i+3-l) */
466 for (j
= 1; j
<= elp
->deg
; j
++)
467 d
^= gf_mul(bch
, elp
->c
[j
], syn
[2*i
+2-j
]);
470 dbg("elp=%s\n", gf_poly_str(elp
));
471 return (elp
->deg
> t
) ? -1 : (int)elp
->deg
;
475 * solve a m x m linear system in GF(2) with an expected number of solutions,
476 * and return the number of found solutions
478 static int solve_linear_system(struct bch_control
*bch
, unsigned int *rows
,
479 unsigned int *sol
, int nsol
)
481 const int m
= GF_M(bch
);
482 unsigned int tmp
, mask
;
483 int rem
, c
, r
, p
, k
, param
[m
];
488 /* Gaussian elimination */
489 for (c
= 0; c
< m
; c
++) {
492 /* find suitable row for elimination */
493 for (r
= p
; r
< m
; r
++) {
494 if (rows
[r
] & mask
) {
505 /* perform elimination on remaining rows */
507 for (r
= rem
; r
< m
; r
++) {
512 /* elimination not needed, store defective row index */
517 /* rewrite system, inserting fake parameter rows */
520 for (r
= m
-1; r
>= 0; r
--) {
521 if ((r
> m
-1-k
) && rows
[r
])
522 /* system has no solution */
525 rows
[r
] = (p
&& (r
== param
[p
-1])) ?
526 p
--, 1u << (m
-r
) : rows
[r
-p
];
530 if (nsol
!= (1 << k
))
531 /* unexpected number of solutions */
534 for (p
= 0; p
< nsol
; p
++) {
535 /* set parameters for p-th solution */
536 for (c
= 0; c
< k
; c
++)
537 rows
[param
[c
]] = (rows
[param
[c
]] & ~1)|((p
>> c
) & 1);
539 /* compute unique solution */
541 for (r
= m
-1; r
>= 0; r
--) {
542 mask
= rows
[r
] & (tmp
|1);
543 tmp
|= parity(mask
) << (m
-r
);
551 * this function builds and solves a linear system for finding roots of a degree
552 * 4 affine monic polynomial X^4+aX^2+bX+c over GF(2^m).
554 static int find_affine4_roots(struct bch_control
*bch
, unsigned int a
,
555 unsigned int b
, unsigned int c
,
559 const int m
= GF_M(bch
);
560 unsigned int mask
= 0xff, t
, rows
[16] = {0,};
566 /* buid linear system to solve X^4+aX^2+bX+c = 0 */
567 for (i
= 0; i
< m
; i
++) {
568 rows
[i
+1] = bch
->a_pow_tab
[4*i
]^
569 (a
? bch
->a_pow_tab
[mod_s(bch
, k
)] : 0)^
570 (b
? bch
->a_pow_tab
[mod_s(bch
, j
)] : 0);
575 * transpose 16x16 matrix before passing it to linear solver
576 * warning: this code assumes m < 16
578 for (j
= 8; j
!= 0; j
>>= 1, mask
^= (mask
<< j
)) {
579 for (k
= 0; k
< 16; k
= (k
+j
+1) & ~j
) {
580 t
= ((rows
[k
] >> j
)^rows
[k
+j
]) & mask
;
585 return solve_linear_system(bch
, rows
, roots
, 4);
589 * compute root r of a degree 1 polynomial over GF(2^m) (returned as log(1/r))
591 static int find_poly_deg1_roots(struct bch_control
*bch
, struct gf_poly
*poly
,
597 /* poly[X] = bX+c with c!=0, root=c/b */
598 roots
[n
++] = mod_s(bch
, GF_N(bch
)-bch
->a_log_tab
[poly
->c
[0]]+
599 bch
->a_log_tab
[poly
->c
[1]]);
604 * compute roots of a degree 2 polynomial over GF(2^m)
606 static int find_poly_deg2_roots(struct bch_control
*bch
, struct gf_poly
*poly
,
609 int n
= 0, i
, l0
, l1
, l2
;
610 unsigned int u
, v
, r
;
612 if (poly
->c
[0] && poly
->c
[1]) {
614 l0
= bch
->a_log_tab
[poly
->c
[0]];
615 l1
= bch
->a_log_tab
[poly
->c
[1]];
616 l2
= bch
->a_log_tab
[poly
->c
[2]];
618 /* using z=a/bX, transform aX^2+bX+c into z^2+z+u (u=ac/b^2) */
619 u
= a_pow(bch
, l0
+l2
+2*(GF_N(bch
)-l1
));
621 * let u = sum(li.a^i) i=0..m-1; then compute r = sum(li.xi):
622 * r^2+r = sum(li.(xi^2+xi)) = sum(li.(a^i+Tr(a^i).a^k)) =
623 * u + sum(li.Tr(a^i).a^k) = u+a^k.Tr(sum(li.a^i)) = u+a^k.Tr(u)
624 * i.e. r and r+1 are roots iff Tr(u)=0
634 if ((gf_sqr(bch
, r
)^r
) == u
) {
635 /* reverse z=a/bX transformation and compute log(1/r) */
636 roots
[n
++] = modulo(bch
, 2*GF_N(bch
)-l1
-
637 bch
->a_log_tab
[r
]+l2
);
638 roots
[n
++] = modulo(bch
, 2*GF_N(bch
)-l1
-
639 bch
->a_log_tab
[r
^1]+l2
);
646 * compute roots of a degree 3 polynomial over GF(2^m)
648 static int find_poly_deg3_roots(struct bch_control
*bch
, struct gf_poly
*poly
,
652 unsigned int a
, b
, c
, a2
, b2
, c2
, e3
, tmp
[4];
655 /* transform polynomial into monic X^3 + a2X^2 + b2X + c2 */
657 c2
= gf_div(bch
, poly
->c
[0], e3
);
658 b2
= gf_div(bch
, poly
->c
[1], e3
);
659 a2
= gf_div(bch
, poly
->c
[2], e3
);
661 /* (X+a2)(X^3+a2X^2+b2X+c2) = X^4+aX^2+bX+c (affine) */
662 c
= gf_mul(bch
, a2
, c2
); /* c = a2c2 */
663 b
= gf_mul(bch
, a2
, b2
)^c2
; /* b = a2b2 + c2 */
664 a
= gf_sqr(bch
, a2
)^b2
; /* a = a2^2 + b2 */
666 /* find the 4 roots of this affine polynomial */
667 if (find_affine4_roots(bch
, a
, b
, c
, tmp
) == 4) {
668 /* remove a2 from final list of roots */
669 for (i
= 0; i
< 4; i
++) {
671 roots
[n
++] = a_ilog(bch
, tmp
[i
]);
679 * compute roots of a degree 4 polynomial over GF(2^m)
681 static int find_poly_deg4_roots(struct bch_control
*bch
, struct gf_poly
*poly
,
685 unsigned int a
, b
, c
, d
, e
= 0, f
, a2
, b2
, c2
, e4
;
690 /* transform polynomial into monic X^4 + aX^3 + bX^2 + cX + d */
692 d
= gf_div(bch
, poly
->c
[0], e4
);
693 c
= gf_div(bch
, poly
->c
[1], e4
);
694 b
= gf_div(bch
, poly
->c
[2], e4
);
695 a
= gf_div(bch
, poly
->c
[3], e4
);
697 /* use Y=1/X transformation to get an affine polynomial */
699 /* first, eliminate cX by using z=X+e with ae^2+c=0 */
701 /* compute e such that e^2 = c/a */
702 f
= gf_div(bch
, c
, a
);
704 l
+= (l
& 1) ? GF_N(bch
) : 0;
707 * use transformation z=X+e:
708 * z^4+e^4 + a(z^3+ez^2+e^2z+e^3) + b(z^2+e^2) +cz+ce+d
709 * z^4 + az^3 + (ae+b)z^2 + (ae^2+c)z+e^4+be^2+ae^3+ce+d
710 * z^4 + az^3 + (ae+b)z^2 + e^4+be^2+d
711 * z^4 + az^3 + b'z^2 + d'
713 d
= a_pow(bch
, 2*l
)^gf_mul(bch
, b
, f
)^d
;
714 b
= gf_mul(bch
, a
, e
)^b
;
716 /* now, use Y=1/X to get Y^4 + b/dY^2 + a/dY + 1/d */
718 /* assume all roots have multiplicity 1 */
722 b2
= gf_div(bch
, a
, d
);
723 a2
= gf_div(bch
, b
, d
);
725 /* polynomial is already affine */
730 /* find the 4 roots of this affine polynomial */
731 if (find_affine4_roots(bch
, a2
, b2
, c2
, roots
) == 4) {
732 for (i
= 0; i
< 4; i
++) {
733 /* post-process roots (reverse transformations) */
734 f
= a
? gf_inv(bch
, roots
[i
]) : roots
[i
];
735 roots
[i
] = a_ilog(bch
, f
^e
);
743 * build monic, log-based representation of a polynomial
745 static void gf_poly_logrep(struct bch_control
*bch
,
746 const struct gf_poly
*a
, int *rep
)
748 int i
, d
= a
->deg
, l
= GF_N(bch
)-a_log(bch
, a
->c
[a
->deg
]);
750 /* represent 0 values with -1; warning, rep[d] is not set to 1 */
751 for (i
= 0; i
< d
; i
++)
752 rep
[i
] = a
->c
[i
] ? mod_s(bch
, a_log(bch
, a
->c
[i
])+l
) : -1;
756 * compute polynomial Euclidean division remainder in GF(2^m)[X]
758 static void gf_poly_mod(struct bch_control
*bch
, struct gf_poly
*a
,
759 const struct gf_poly
*b
, int *rep
)
762 unsigned int i
, j
, *c
= a
->c
;
763 const unsigned int d
= b
->deg
;
768 /* reuse or compute log representation of denominator */
771 gf_poly_logrep(bch
, b
, rep
);
774 for (j
= a
->deg
; j
>= d
; j
--) {
776 la
= a_log(bch
, c
[j
]);
778 for (i
= 0; i
< d
; i
++, p
++) {
781 c
[p
] ^= bch
->a_pow_tab
[mod_s(bch
,
787 while (!c
[a
->deg
] && a
->deg
)
792 * compute polynomial Euclidean division quotient in GF(2^m)[X]
794 static void gf_poly_div(struct bch_control
*bch
, struct gf_poly
*a
,
795 const struct gf_poly
*b
, struct gf_poly
*q
)
797 if (a
->deg
>= b
->deg
) {
798 q
->deg
= a
->deg
-b
->deg
;
799 /* compute a mod b (modifies a) */
800 gf_poly_mod(bch
, a
, b
, NULL
);
801 /* quotient is stored in upper part of polynomial a */
802 memcpy(q
->c
, &a
->c
[b
->deg
], (1+q
->deg
)*sizeof(unsigned int));
810 * compute polynomial GCD (Greatest Common Divisor) in GF(2^m)[X]
812 static struct gf_poly
*gf_poly_gcd(struct bch_control
*bch
, struct gf_poly
*a
,
817 dbg("gcd(%s,%s)=", gf_poly_str(a
), gf_poly_str(b
));
819 if (a
->deg
< b
->deg
) {
826 gf_poly_mod(bch
, a
, b
, NULL
);
832 dbg("%s\n", gf_poly_str(a
));
838 * Given a polynomial f and an integer k, compute Tr(a^kX) mod f
839 * This is used in Berlekamp Trace algorithm for splitting polynomials
841 static void compute_trace_bk_mod(struct bch_control
*bch
, int k
,
842 const struct gf_poly
*f
, struct gf_poly
*z
,
845 const int m
= GF_M(bch
);
848 /* z contains z^2j mod f */
851 z
->c
[1] = bch
->a_pow_tab
[k
];
854 memset(out
, 0, GF_POLY_SZ(f
->deg
));
856 /* compute f log representation only once */
857 gf_poly_logrep(bch
, f
, bch
->cache
);
859 for (i
= 0; i
< m
; i
++) {
860 /* add a^(k*2^i)(z^(2^i) mod f) and compute (z^(2^i) mod f)^2 */
861 for (j
= z
->deg
; j
>= 0; j
--) {
862 out
->c
[j
] ^= z
->c
[j
];
863 z
->c
[2*j
] = gf_sqr(bch
, z
->c
[j
]);
866 if (z
->deg
> out
->deg
)
871 /* z^(2(i+1)) mod f = (z^(2^i) mod f)^2 mod f */
872 gf_poly_mod(bch
, z
, f
, bch
->cache
);
875 while (!out
->c
[out
->deg
] && out
->deg
)
878 dbg("Tr(a^%d.X) mod f = %s\n", k
, gf_poly_str(out
));
882 * factor a polynomial using Berlekamp Trace algorithm (BTA)
884 static void factor_polynomial(struct bch_control
*bch
, int k
, struct gf_poly
*f
,
885 struct gf_poly
**g
, struct gf_poly
**h
)
887 struct gf_poly
*f2
= bch
->poly_2t
[0];
888 struct gf_poly
*q
= bch
->poly_2t
[1];
889 struct gf_poly
*tk
= bch
->poly_2t
[2];
890 struct gf_poly
*z
= bch
->poly_2t
[3];
893 dbg("factoring %s...\n", gf_poly_str(f
));
898 /* tk = Tr(a^k.X) mod f */
899 compute_trace_bk_mod(bch
, k
, f
, z
, tk
);
902 /* compute g = gcd(f, tk) (destructive operation) */
904 gcd
= gf_poly_gcd(bch
, f2
, tk
);
905 if (gcd
->deg
< f
->deg
) {
906 /* compute h=f/gcd(f,tk); this will modify f and q */
907 gf_poly_div(bch
, f
, gcd
, q
);
908 /* store g and h in-place (clobbering f) */
909 *h
= &((struct gf_poly_deg1
*)f
)[gcd
->deg
].poly
;
910 gf_poly_copy(*g
, gcd
);
917 * find roots of a polynomial, using BTZ algorithm; see the beginning of this
920 static int find_poly_roots(struct bch_control
*bch
, unsigned int k
,
921 struct gf_poly
*poly
, unsigned int *roots
)
924 struct gf_poly
*f1
, *f2
;
927 /* handle low degree polynomials with ad hoc techniques */
929 cnt
= find_poly_deg1_roots(bch
, poly
, roots
);
932 cnt
= find_poly_deg2_roots(bch
, poly
, roots
);
935 cnt
= find_poly_deg3_roots(bch
, poly
, roots
);
938 cnt
= find_poly_deg4_roots(bch
, poly
, roots
);
941 /* factor polynomial using Berlekamp Trace Algorithm (BTA) */
943 if (poly
->deg
&& (k
<= GF_M(bch
))) {
944 factor_polynomial(bch
, k
, poly
, &f1
, &f2
);
946 cnt
+= find_poly_roots(bch
, k
+1, f1
, roots
);
948 cnt
+= find_poly_roots(bch
, k
+1, f2
, roots
+cnt
);
955 #if defined(USE_CHIEN_SEARCH)
957 * exhaustive root search (Chien) implementation - not used, included only for
958 * reference/comparison tests
960 static int chien_search(struct bch_control
*bch
, unsigned int len
,
961 struct gf_poly
*p
, unsigned int *roots
)
964 unsigned int i
, j
, syn
, syn0
, count
= 0;
965 const unsigned int k
= 8*len
+bch
->ecc_bits
;
967 /* use a log-based representation of polynomial */
968 gf_poly_logrep(bch
, p
, bch
->cache
);
969 bch
->cache
[p
->deg
] = 0;
970 syn0
= gf_div(bch
, p
->c
[0], p
->c
[p
->deg
]);
972 for (i
= GF_N(bch
)-k
+1; i
<= GF_N(bch
); i
++) {
973 /* compute elp(a^i) */
974 for (j
= 1, syn
= syn0
; j
<= p
->deg
; j
++) {
977 syn
^= a_pow(bch
, m
+j
*i
);
980 roots
[count
++] = GF_N(bch
)-i
;
985 return (count
== p
->deg
) ? count
: 0;
987 #define find_poly_roots(_p, _k, _elp, _loc) chien_search(_p, len, _elp, _loc)
988 #endif /* USE_CHIEN_SEARCH */
991 * decode_bch - decode received codeword and find bit error locations
992 * @bch: BCH control structure
993 * @data: received data, ignored if @calc_ecc is provided
994 * @len: data length in bytes, must always be provided
995 * @recv_ecc: received ecc, if NULL then assume it was XORed in @calc_ecc
996 * @calc_ecc: calculated ecc, if NULL then calc_ecc is computed from @data
997 * @syn: hw computed syndrome data (if NULL, syndrome is calculated)
998 * @errloc: output array of error locations
1001 * The number of errors found, or -EBADMSG if decoding failed, or -EINVAL if
1002 * invalid parameters were provided
1004 * Depending on the available hw BCH support and the need to compute @calc_ecc
1005 * separately (using encode_bch()), this function should be called with one of
1006 * the following parameter configurations -
1008 * by providing @data and @recv_ecc only:
1009 * decode_bch(@bch, @data, @len, @recv_ecc, NULL, NULL, @errloc)
1011 * by providing @recv_ecc and @calc_ecc:
1012 * decode_bch(@bch, NULL, @len, @recv_ecc, @calc_ecc, NULL, @errloc)
1014 * by providing ecc = recv_ecc XOR calc_ecc:
1015 * decode_bch(@bch, NULL, @len, NULL, ecc, NULL, @errloc)
1017 * by providing syndrome results @syn:
1018 * decode_bch(@bch, NULL, @len, NULL, NULL, @syn, @errloc)
1020 * Once decode_bch() has successfully returned with a positive value, error
1021 * locations returned in array @errloc should be interpreted as follows -
1023 * if (errloc[n] >= 8*len), then n-th error is located in ecc (no need for
1026 * if (errloc[n] < 8*len), then n-th error is located in data and can be
1027 * corrected with statement data[errloc[n]/8] ^= 1 << (errloc[n] % 8);
1029 * Note that this function does not perform any data correction by itself, it
1030 * merely indicates error locations.
1032 int decode_bch(struct bch_control
*bch
, const uint8_t *data
, unsigned int len
,
1033 const uint8_t *recv_ecc
, const uint8_t *calc_ecc
,
1034 const unsigned int *syn
, unsigned int *errloc
)
1036 const unsigned int ecc_words
= BCH_ECC_WORDS(bch
);
1041 /* sanity check: make sure data length can be handled */
1042 if (8*len
> (bch
->n
-bch
->ecc_bits
))
1045 /* if caller does not provide syndromes, compute them */
1048 /* compute received data ecc into an internal buffer */
1049 if (!data
|| !recv_ecc
)
1051 encode_bch(bch
, data
, len
, NULL
);
1053 /* load provided calculated ecc */
1054 load_ecc8(bch
, bch
->ecc_buf
, calc_ecc
);
1056 /* load received ecc or assume it was XORed in calc_ecc */
1058 load_ecc8(bch
, bch
->ecc_buf2
, recv_ecc
);
1059 /* XOR received and calculated ecc */
1060 for (i
= 0, sum
= 0; i
< (int)ecc_words
; i
++) {
1061 bch
->ecc_buf
[i
] ^= bch
->ecc_buf2
[i
];
1062 sum
|= bch
->ecc_buf
[i
];
1065 /* no error found */
1068 compute_syndromes(bch
, bch
->ecc_buf
, bch
->syn
);
1072 err
= compute_error_locator_polynomial(bch
, syn
);
1074 nroots
= find_poly_roots(bch
, 1, bch
->elp
, errloc
);
1079 /* post-process raw error locations for easier correction */
1080 nbits
= (len
*8)+bch
->ecc_bits
;
1081 for (i
= 0; i
< err
; i
++) {
1082 if (errloc
[i
] >= nbits
) {
1086 errloc
[i
] = nbits
-1-errloc
[i
];
1087 errloc
[i
] = (errloc
[i
] & ~7)|(7-(errloc
[i
] & 7));
1090 return (err
>= 0) ? err
: -EBADMSG
;
1094 * generate Galois field lookup tables
1096 static int build_gf_tables(struct bch_control
*bch
, unsigned int poly
)
1098 unsigned int i
, x
= 1;
1099 const unsigned int k
= 1 << deg(poly
);
1101 /* primitive polynomial must be of degree m */
1102 if (k
!= (1u << GF_M(bch
)))
1105 for (i
= 0; i
< GF_N(bch
); i
++) {
1106 bch
->a_pow_tab
[i
] = x
;
1107 bch
->a_log_tab
[x
] = i
;
1109 /* polynomial is not primitive (a^i=1 with 0<i<2^m-1) */
1115 bch
->a_pow_tab
[GF_N(bch
)] = 1;
1116 bch
->a_log_tab
[0] = 0;
1122 * compute generator polynomial remainder tables for fast encoding
1124 static void build_mod8_tables(struct bch_control
*bch
, const uint32_t *g
)
1127 uint32_t data
, hi
, lo
, *tab
;
1128 const int l
= BCH_ECC_WORDS(bch
);
1129 const int plen
= DIV_ROUND_UP(bch
->ecc_bits
+1, 32);
1130 const int ecclen
= DIV_ROUND_UP(bch
->ecc_bits
, 32);
1132 memset(bch
->mod8_tab
, 0, 4*256*l
*sizeof(*bch
->mod8_tab
));
1134 for (i
= 0; i
< 256; i
++) {
1135 /* p(X)=i is a small polynomial of weight <= 8 */
1136 for (b
= 0; b
< 4; b
++) {
1137 /* we want to compute (p(X).X^(8*b+deg(g))) mod g(X) */
1138 tab
= bch
->mod8_tab
+ (b
*256+i
)*l
;
1142 /* subtract X^d.g(X) from p(X).X^(8*b+deg(g)) */
1143 data
^= g
[0] >> (31-d
);
1144 for (j
= 0; j
< ecclen
; j
++) {
1145 hi
= (d
< 31) ? g
[j
] << (d
+1) : 0;
1147 g
[j
+1] >> (31-d
) : 0;
1156 * build a base for factoring degree 2 polynomials
1158 static int build_deg2_base(struct bch_control
*bch
)
1160 const int m
= GF_M(bch
);
1162 unsigned int sum
, x
, y
, remaining
, ak
= 0, xi
[m
];
1164 /* find k s.t. Tr(a^k) = 1 and 0 <= k < m */
1165 for (i
= 0; i
< m
; i
++) {
1166 for (j
= 0, sum
= 0; j
< m
; j
++)
1167 sum
^= a_pow(bch
, i
*(1 << j
));
1170 ak
= bch
->a_pow_tab
[i
];
1174 /* find xi, i=0..m-1 such that xi^2+xi = a^i+Tr(a^i).a^k */
1176 memset(xi
, 0, sizeof(xi
));
1178 for (x
= 0; (x
<= GF_N(bch
)) && remaining
; x
++) {
1179 y
= gf_sqr(bch
, x
)^x
;
1180 for (i
= 0; i
< 2; i
++) {
1182 if (y
&& (r
< m
) && !xi
[r
]) {
1186 dbg("x%d = %x\n", r
, x
);
1192 /* should not happen but check anyway */
1193 return remaining
? -1 : 0;
1196 static void *bch_alloc(size_t size
, int *err
)
1200 ptr
= kmalloc(size
, GFP_KERNEL
);
1207 * compute generator polynomial for given (m,t) parameters.
1209 static uint32_t *compute_generator_polynomial(struct bch_control
*bch
)
1211 const unsigned int m
= GF_M(bch
);
1212 const unsigned int t
= GF_T(bch
);
1214 unsigned int i
, j
, nbits
, r
, word
, *roots
;
1218 g
= bch_alloc(GF_POLY_SZ(m
*t
), &err
);
1219 roots
= bch_alloc((bch
->n
+1)*sizeof(*roots
), &err
);
1220 genpoly
= bch_alloc(DIV_ROUND_UP(m
*t
+1, 32)*sizeof(*genpoly
), &err
);
1228 /* enumerate all roots of g(X) */
1229 memset(roots
, 0, (bch
->n
+1)*sizeof(*roots
));
1230 for (i
= 0; i
< t
; i
++) {
1231 for (j
= 0, r
= 2*i
+1; j
< m
; j
++) {
1233 r
= mod_s(bch
, 2*r
);
1236 /* build generator polynomial g(X) */
1239 for (i
= 0; i
< GF_N(bch
); i
++) {
1241 /* multiply g(X) by (X+root) */
1242 r
= bch
->a_pow_tab
[i
];
1244 for (j
= g
->deg
; j
> 0; j
--)
1245 g
->c
[j
] = gf_mul(bch
, g
->c
[j
], r
)^g
->c
[j
-1];
1247 g
->c
[0] = gf_mul(bch
, g
->c
[0], r
);
1251 /* store left-justified binary representation of g(X) */
1256 nbits
= (n
> 32) ? 32 : n
;
1257 for (j
= 0, word
= 0; j
< nbits
; j
++) {
1259 word
|= 1u << (31-j
);
1261 genpoly
[i
++] = word
;
1264 bch
->ecc_bits
= g
->deg
;
1274 * init_bch - initialize a BCH encoder/decoder
1275 * @m: Galois field order, should be in the range 5-15
1276 * @t: maximum error correction capability, in bits
1277 * @prim_poly: user-provided primitive polynomial (or 0 to use default)
1280 * a newly allocated BCH control structure if successful, NULL otherwise
1282 * This initialization can take some time, as lookup tables are built for fast
1283 * encoding/decoding; make sure not to call this function from a time critical
1284 * path. Usually, init_bch() should be called on module/driver init and
1285 * free_bch() should be called to release memory on exit.
1287 * You may provide your own primitive polynomial of degree @m in argument
1288 * @prim_poly, or let init_bch() use its default polynomial.
1290 * Once init_bch() has successfully returned a pointer to a newly allocated
1291 * BCH control structure, ecc length in bytes is given by member @ecc_bytes of
1294 struct bch_control
*init_bch(int m
, int t
, unsigned int prim_poly
)
1297 unsigned int i
, words
;
1299 struct bch_control
*bch
= NULL
;
1301 const int min_m
= 5;
1302 const int max_m
= 15;
1304 /* default primitive polynomials */
1305 static const unsigned int prim_poly_tab
[] = {
1306 0x25, 0x43, 0x83, 0x11d, 0x211, 0x409, 0x805, 0x1053, 0x201b,
1310 #if defined(CONFIG_BCH_CONST_PARAMS)
1311 if ((m
!= (CONFIG_BCH_CONST_M
)) || (t
!= (CONFIG_BCH_CONST_T
))) {
1312 printk(KERN_ERR
"bch encoder/decoder was configured to support "
1313 "parameters m=%d, t=%d only!\n",
1314 CONFIG_BCH_CONST_M
, CONFIG_BCH_CONST_T
);
1318 if ((m
< min_m
) || (m
> max_m
))
1320 * values of m greater than 15 are not currently supported;
1321 * supporting m > 15 would require changing table base type
1322 * (uint16_t) and a small patch in matrix transposition
1327 if ((t
< 1) || (m
*t
>= ((1 << m
)-1)))
1328 /* invalid t value */
1331 /* select a primitive polynomial for generating GF(2^m) */
1333 prim_poly
= prim_poly_tab
[m
-min_m
];
1335 bch
= kzalloc(sizeof(*bch
), GFP_KERNEL
);
1341 bch
->n
= (1 << m
)-1;
1342 words
= DIV_ROUND_UP(m
*t
, 32);
1343 bch
->ecc_bytes
= DIV_ROUND_UP(m
*t
, 8);
1344 bch
->a_pow_tab
= bch_alloc((1+bch
->n
)*sizeof(*bch
->a_pow_tab
), &err
);
1345 bch
->a_log_tab
= bch_alloc((1+bch
->n
)*sizeof(*bch
->a_log_tab
), &err
);
1346 bch
->mod8_tab
= bch_alloc(words
*1024*sizeof(*bch
->mod8_tab
), &err
);
1347 bch
->ecc_buf
= bch_alloc(words
*sizeof(*bch
->ecc_buf
), &err
);
1348 bch
->ecc_buf2
= bch_alloc(words
*sizeof(*bch
->ecc_buf2
), &err
);
1349 bch
->xi_tab
= bch_alloc(m
*sizeof(*bch
->xi_tab
), &err
);
1350 bch
->syn
= bch_alloc(2*t
*sizeof(*bch
->syn
), &err
);
1351 bch
->cache
= bch_alloc(2*t
*sizeof(*bch
->cache
), &err
);
1352 bch
->elp
= bch_alloc((t
+1)*sizeof(struct gf_poly_deg1
), &err
);
1354 for (i
= 0; i
< ARRAY_SIZE(bch
->poly_2t
); i
++)
1355 bch
->poly_2t
[i
] = bch_alloc(GF_POLY_SZ(2*t
), &err
);
1360 err
= build_gf_tables(bch
, prim_poly
);
1364 /* use generator polynomial for computing encoding tables */
1365 genpoly
= compute_generator_polynomial(bch
);
1366 if (genpoly
== NULL
)
1369 build_mod8_tables(bch
, genpoly
);
1372 err
= build_deg2_base(bch
);
1384 * free_bch - free the BCH control structure
1385 * @bch: BCH control structure to release
1387 void free_bch(struct bch_control
*bch
)
1392 kfree(bch
->a_pow_tab
);
1393 kfree(bch
->a_log_tab
);
1394 kfree(bch
->mod8_tab
);
1395 kfree(bch
->ecc_buf
);
1396 kfree(bch
->ecc_buf2
);
1402 for (i
= 0; i
< ARRAY_SIZE(bch
->poly_2t
); i
++)
1403 kfree(bch
->poly_2t
[i
]);