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>
70 #define cpu_to_be32 htobe32
71 #define DIV_ROUND_UP(n,d) (((n) + (d) - 1) / (d))
72 #define kmalloc(size, flags) malloc(size)
73 #define kzalloc(size, flags) calloc(1, size)
75 #define ARRAY_SIZE(arr) (sizeof(arr) / sizeof((arr)[0]))
78 #include <asm/byteorder.h>
79 #include <linux/bch.h>
81 #if defined(CONFIG_BCH_CONST_PARAMS)
82 #define GF_M(_p) (CONFIG_BCH_CONST_M)
83 #define GF_T(_p) (CONFIG_BCH_CONST_T)
84 #define GF_N(_p) ((1 << (CONFIG_BCH_CONST_M))-1)
86 #define GF_M(_p) ((_p)->m)
87 #define GF_T(_p) ((_p)->t)
88 #define GF_N(_p) ((_p)->n)
91 #define BCH_ECC_WORDS(_p) DIV_ROUND_UP(GF_M(_p)*GF_T(_p), 32)
92 #define BCH_ECC_BYTES(_p) DIV_ROUND_UP(GF_M(_p)*GF_T(_p), 8)
95 #define dbg(_fmt, args...) do {} while (0)
99 * represent a polynomial over GF(2^m)
102 unsigned int deg; /* polynomial degree */
103 unsigned int c[0]; /* polynomial terms */
106 /* given its degree, compute a polynomial size in bytes */
107 #define GF_POLY_SZ(_d) (sizeof(struct gf_poly)+((_d)+1)*sizeof(unsigned int))
109 /* polynomial of degree 1 */
110 struct gf_poly_deg1 {
116 static int fls(int x)
122 if (!(x & 0xffff0000u)) {
126 if (!(x & 0xff000000u)) {
130 if (!(x & 0xf0000000u)) {
134 if (!(x & 0xc0000000u)) {
138 if (!(x & 0x80000000u)) {
147 * same as encode_bch(), but process input data one byte at a time
149 static void encode_bch_unaligned(struct bch_control *bch,
150 const unsigned char *data, unsigned int len,
155 const int l = BCH_ECC_WORDS(bch)-1;
158 p = bch->mod8_tab + (l+1)*(((ecc[0] >> 24)^(*data++)) & 0xff);
160 for (i = 0; i < l; i++)
161 ecc[i] = ((ecc[i] << 8)|(ecc[i+1] >> 24))^(*p++);
163 ecc[l] = (ecc[l] << 8)^(*p);
168 * convert ecc bytes to aligned, zero-padded 32-bit ecc words
170 static void load_ecc8(struct bch_control *bch, uint32_t *dst,
173 uint8_t pad[4] = {0, 0, 0, 0};
174 unsigned int i, nwords = BCH_ECC_WORDS(bch)-1;
176 for (i = 0; i < nwords; i++, src += 4)
177 dst[i] = (src[0] << 24)|(src[1] << 16)|(src[2] << 8)|src[3];
179 memcpy(pad, src, BCH_ECC_BYTES(bch)-4*nwords);
180 dst[nwords] = (pad[0] << 24)|(pad[1] << 16)|(pad[2] << 8)|pad[3];
184 * convert 32-bit ecc words to ecc bytes
186 static void store_ecc8(struct bch_control *bch, uint8_t *dst,
190 unsigned int i, nwords = BCH_ECC_WORDS(bch)-1;
192 for (i = 0; i < nwords; i++) {
193 *dst++ = (src[i] >> 24);
194 *dst++ = (src[i] >> 16) & 0xff;
195 *dst++ = (src[i] >> 8) & 0xff;
196 *dst++ = (src[i] >> 0) & 0xff;
198 pad[0] = (src[nwords] >> 24);
199 pad[1] = (src[nwords] >> 16) & 0xff;
200 pad[2] = (src[nwords] >> 8) & 0xff;
201 pad[3] = (src[nwords] >> 0) & 0xff;
202 memcpy(dst, pad, BCH_ECC_BYTES(bch)-4*nwords);
206 * encode_bch - calculate BCH ecc parity of data
207 * @bch: BCH control structure
208 * @data: data to encode
209 * @len: data length in bytes
210 * @ecc: ecc parity data, must be initialized by caller
212 * The @ecc parity array is used both as input and output parameter, in order to
213 * allow incremental computations. It should be of the size indicated by member
214 * @ecc_bytes of @bch, and should be initialized to 0 before the first call.
216 * The exact number of computed ecc parity bits is given by member @ecc_bits of
217 * @bch; it may be less than m*t for large values of t.
219 void encode_bch(struct bch_control *bch, const uint8_t *data,
220 unsigned int len, uint8_t *ecc)
222 const unsigned int l = BCH_ECC_WORDS(bch)-1;
223 unsigned int i, mlen;
226 const uint32_t * const tab0 = bch->mod8_tab;
227 const uint32_t * const tab1 = tab0 + 256*(l+1);
228 const uint32_t * const tab2 = tab1 + 256*(l+1);
229 const uint32_t * const tab3 = tab2 + 256*(l+1);
230 const uint32_t *pdata, *p0, *p1, *p2, *p3;
233 /* load ecc parity bytes into internal 32-bit buffer */
234 load_ecc8(bch, bch->ecc_buf, ecc);
236 memset(bch->ecc_buf, 0, sizeof(r));
239 /* process first unaligned data bytes */
240 m = ((unsigned long)data) & 3;
242 mlen = (len < (4-m)) ? len : 4-m;
243 encode_bch_unaligned(bch, data, mlen, bch->ecc_buf);
248 /* process 32-bit aligned data words */
249 pdata = (uint32_t *)data;
253 memcpy(r, bch->ecc_buf, sizeof(r));
256 * split each 32-bit word into 4 polynomials of weight 8 as follows:
258 * 31 ...24 23 ...16 15 ... 8 7 ... 0
259 * xxxxxxxx yyyyyyyy zzzzzzzz tttttttt
260 * tttttttt mod g = r0 (precomputed)
261 * zzzzzzzz 00000000 mod g = r1 (precomputed)
262 * yyyyyyyy 00000000 00000000 mod g = r2 (precomputed)
263 * xxxxxxxx 00000000 00000000 00000000 mod g = r3 (precomputed)
264 * xxxxxxxx yyyyyyyy zzzzzzzz tttttttt mod g = r0^r1^r2^r3
267 /* input data is read in big-endian format */
268 w = r[0]^cpu_to_be32(*pdata++);
269 p0 = tab0 + (l+1)*((w >> 0) & 0xff);
270 p1 = tab1 + (l+1)*((w >> 8) & 0xff);
271 p2 = tab2 + (l+1)*((w >> 16) & 0xff);
272 p3 = tab3 + (l+1)*((w >> 24) & 0xff);
274 for (i = 0; i < l; i++)
275 r[i] = r[i+1]^p0[i]^p1[i]^p2[i]^p3[i];
277 r[l] = p0[l]^p1[l]^p2[l]^p3[l];
279 memcpy(bch->ecc_buf, r, sizeof(r));
281 /* process last unaligned bytes */
283 encode_bch_unaligned(bch, data, len, bch->ecc_buf);
285 /* store ecc parity bytes into original parity buffer */
287 store_ecc8(bch, ecc, bch->ecc_buf);
290 static inline int modulo(struct bch_control *bch, unsigned int v)
292 const unsigned int n = GF_N(bch);
295 v = (v & n) + (v >> GF_M(bch));
301 * shorter and faster modulo function, only works when v < 2N.
303 static inline int mod_s(struct bch_control *bch, unsigned int v)
305 const unsigned int n = GF_N(bch);
306 return (v < n) ? v : v-n;
309 static inline int deg(unsigned int poly)
311 /* polynomial degree is the most-significant bit index */
315 static inline int parity(unsigned int x)
318 * public domain code snippet, lifted from
319 * http://www-graphics.stanford.edu/~seander/bithacks.html
323 x = (x & 0x11111111U) * 0x11111111U;
324 return (x >> 28) & 1;
327 /* Galois field basic operations: multiply, divide, inverse, etc. */
329 static inline unsigned int gf_mul(struct bch_control *bch, unsigned int a,
332 return (a && b) ? bch->a_pow_tab[mod_s(bch, bch->a_log_tab[a]+
333 bch->a_log_tab[b])] : 0;
336 static inline unsigned int gf_sqr(struct bch_control *bch, unsigned int a)
338 return a ? bch->a_pow_tab[mod_s(bch, 2*bch->a_log_tab[a])] : 0;
341 static inline unsigned int gf_div(struct bch_control *bch, unsigned int a,
344 return a ? bch->a_pow_tab[mod_s(bch, bch->a_log_tab[a]+
345 GF_N(bch)-bch->a_log_tab[b])] : 0;
348 static inline unsigned int gf_inv(struct bch_control *bch, unsigned int a)
350 return bch->a_pow_tab[GF_N(bch)-bch->a_log_tab[a]];
353 static inline unsigned int a_pow(struct bch_control *bch, int i)
355 return bch->a_pow_tab[modulo(bch, i)];
358 static inline int a_log(struct bch_control *bch, unsigned int x)
360 return bch->a_log_tab[x];
363 static inline int a_ilog(struct bch_control *bch, unsigned int x)
365 return mod_s(bch, GF_N(bch)-bch->a_log_tab[x]);
369 * compute 2t syndromes of ecc polynomial, i.e. ecc(a^j) for j=1..2t
371 static void compute_syndromes(struct bch_control *bch, uint32_t *ecc,
377 const int t = GF_T(bch);
381 /* make sure extra bits in last ecc word are cleared */
382 m = ((unsigned int)s) & 31;
384 ecc[s/32] &= ~((1u << (32-m))-1);
385 memset(syn, 0, 2*t*sizeof(*syn));
387 /* compute v(a^j) for j=1 .. 2t-1 */
393 for (j = 0; j < 2*t; j += 2)
394 syn[j] ^= a_pow(bch, (j+1)*(i+s));
400 /* v(a^(2j)) = v(a^j)^2 */
401 for (j = 0; j < t; j++)
402 syn[2*j+1] = gf_sqr(bch, syn[j]);
405 static void gf_poly_copy(struct gf_poly *dst, struct gf_poly *src)
407 memcpy(dst, src, GF_POLY_SZ(src->deg));
410 static int compute_error_locator_polynomial(struct bch_control *bch,
411 const unsigned int *syn)
413 const unsigned int t = GF_T(bch);
414 const unsigned int n = GF_N(bch);
415 unsigned int i, j, tmp, l, pd = 1, d = syn[0];
416 struct gf_poly *elp = bch->elp;
417 struct gf_poly *pelp = bch->poly_2t[0];
418 struct gf_poly *elp_copy = bch->poly_2t[1];
421 memset(pelp, 0, GF_POLY_SZ(2*t));
422 memset(elp, 0, GF_POLY_SZ(2*t));
429 /* use simplified binary Berlekamp-Massey algorithm */
430 for (i = 0; (i < t) && (elp->deg <= t); i++) {
433 gf_poly_copy(elp_copy, elp);
434 /* e[i+1](X) = e[i](X)+di*dp^-1*X^2(i-p)*e[p](X) */
435 tmp = a_log(bch, d)+n-a_log(bch, pd);
436 for (j = 0; j <= pelp->deg; j++) {
438 l = a_log(bch, pelp->c[j]);
439 elp->c[j+k] ^= a_pow(bch, tmp+l);
442 /* compute l[i+1] = max(l[i]->c[l[p]+2*(i-p]) */
444 if (tmp > elp->deg) {
446 gf_poly_copy(pelp, elp_copy);
451 /* di+1 = S(2i+3)+elp[i+1].1*S(2i+2)+...+elp[i+1].lS(2i+3-l) */
454 for (j = 1; j <= elp->deg; j++)
455 d ^= gf_mul(bch, elp->c[j], syn[2*i+2-j]);
458 dbg("elp=%s\n", gf_poly_str(elp));
459 return (elp->deg > t) ? -1 : (int)elp->deg;
463 * solve a m x m linear system in GF(2) with an expected number of solutions,
464 * and return the number of found solutions
466 static int solve_linear_system(struct bch_control *bch, unsigned int *rows,
467 unsigned int *sol, int nsol)
469 const int m = GF_M(bch);
470 unsigned int tmp, mask;
471 int rem, c, r, p, k, param[m];
476 /* Gaussian elimination */
477 for (c = 0; c < m; c++) {
480 /* find suitable row for elimination */
481 for (r = p; r < m; r++) {
482 if (rows[r] & mask) {
493 /* perform elimination on remaining rows */
495 for (r = rem; r < m; r++) {
500 /* elimination not needed, store defective row index */
505 /* rewrite system, inserting fake parameter rows */
508 for (r = m-1; r >= 0; r--) {
509 if ((r > m-1-k) && rows[r])
510 /* system has no solution */
513 rows[r] = (p && (r == param[p-1])) ?
514 p--, 1u << (m-r) : rows[r-p];
518 if (nsol != (1 << k))
519 /* unexpected number of solutions */
522 for (p = 0; p < nsol; p++) {
523 /* set parameters for p-th solution */
524 for (c = 0; c < k; c++)
525 rows[param[c]] = (rows[param[c]] & ~1)|((p >> c) & 1);
527 /* compute unique solution */
529 for (r = m-1; r >= 0; r--) {
530 mask = rows[r] & (tmp|1);
531 tmp |= parity(mask) << (m-r);
539 * this function builds and solves a linear system for finding roots of a degree
540 * 4 affine monic polynomial X^4+aX^2+bX+c over GF(2^m).
542 static int find_affine4_roots(struct bch_control *bch, unsigned int a,
543 unsigned int b, unsigned int c,
547 const int m = GF_M(bch);
548 unsigned int mask = 0xff, t, rows[16] = {0,};
554 /* buid linear system to solve X^4+aX^2+bX+c = 0 */
555 for (i = 0; i < m; i++) {
556 rows[i+1] = bch->a_pow_tab[4*i]^
557 (a ? bch->a_pow_tab[mod_s(bch, k)] : 0)^
558 (b ? bch->a_pow_tab[mod_s(bch, j)] : 0);
563 * transpose 16x16 matrix before passing it to linear solver
564 * warning: this code assumes m < 16
566 for (j = 8; j != 0; j >>= 1, mask ^= (mask << j)) {
567 for (k = 0; k < 16; k = (k+j+1) & ~j) {
568 t = ((rows[k] >> j)^rows[k+j]) & mask;
573 return solve_linear_system(bch, rows, roots, 4);
577 * compute root r of a degree 1 polynomial over GF(2^m) (returned as log(1/r))
579 static int find_poly_deg1_roots(struct bch_control *bch, struct gf_poly *poly,
585 /* poly[X] = bX+c with c!=0, root=c/b */
586 roots[n++] = mod_s(bch, GF_N(bch)-bch->a_log_tab[poly->c[0]]+
587 bch->a_log_tab[poly->c[1]]);
592 * compute roots of a degree 2 polynomial over GF(2^m)
594 static int find_poly_deg2_roots(struct bch_control *bch, struct gf_poly *poly,
597 int n = 0, i, l0, l1, l2;
598 unsigned int u, v, r;
600 if (poly->c[0] && poly->c[1]) {
602 l0 = bch->a_log_tab[poly->c[0]];
603 l1 = bch->a_log_tab[poly->c[1]];
604 l2 = bch->a_log_tab[poly->c[2]];
606 /* using z=a/bX, transform aX^2+bX+c into z^2+z+u (u=ac/b^2) */
607 u = a_pow(bch, l0+l2+2*(GF_N(bch)-l1));
609 * let u = sum(li.a^i) i=0..m-1; then compute r = sum(li.xi):
610 * r^2+r = sum(li.(xi^2+xi)) = sum(li.(a^i+Tr(a^i).a^k)) =
611 * u + sum(li.Tr(a^i).a^k) = u+a^k.Tr(sum(li.a^i)) = u+a^k.Tr(u)
612 * i.e. r and r+1 are roots iff Tr(u)=0
622 if ((gf_sqr(bch, r)^r) == u) {
623 /* reverse z=a/bX transformation and compute log(1/r) */
624 roots[n++] = modulo(bch, 2*GF_N(bch)-l1-
625 bch->a_log_tab[r]+l2);
626 roots[n++] = modulo(bch, 2*GF_N(bch)-l1-
627 bch->a_log_tab[r^1]+l2);
634 * compute roots of a degree 3 polynomial over GF(2^m)
636 static int find_poly_deg3_roots(struct bch_control *bch, struct gf_poly *poly,
640 unsigned int a, b, c, a2, b2, c2, e3, tmp[4];
643 /* transform polynomial into monic X^3 + a2X^2 + b2X + c2 */
645 c2 = gf_div(bch, poly->c[0], e3);
646 b2 = gf_div(bch, poly->c[1], e3);
647 a2 = gf_div(bch, poly->c[2], e3);
649 /* (X+a2)(X^3+a2X^2+b2X+c2) = X^4+aX^2+bX+c (affine) */
650 c = gf_mul(bch, a2, c2); /* c = a2c2 */
651 b = gf_mul(bch, a2, b2)^c2; /* b = a2b2 + c2 */
652 a = gf_sqr(bch, a2)^b2; /* a = a2^2 + b2 */
654 /* find the 4 roots of this affine polynomial */
655 if (find_affine4_roots(bch, a, b, c, tmp) == 4) {
656 /* remove a2 from final list of roots */
657 for (i = 0; i < 4; i++) {
659 roots[n++] = a_ilog(bch, tmp[i]);
667 * compute roots of a degree 4 polynomial over GF(2^m)
669 static int find_poly_deg4_roots(struct bch_control *bch, struct gf_poly *poly,
673 unsigned int a, b, c, d, e = 0, f, a2, b2, c2, e4;
678 /* transform polynomial into monic X^4 + aX^3 + bX^2 + cX + d */
680 d = gf_div(bch, poly->c[0], e4);
681 c = gf_div(bch, poly->c[1], e4);
682 b = gf_div(bch, poly->c[2], e4);
683 a = gf_div(bch, poly->c[3], e4);
685 /* use Y=1/X transformation to get an affine polynomial */
687 /* first, eliminate cX by using z=X+e with ae^2+c=0 */
689 /* compute e such that e^2 = c/a */
690 f = gf_div(bch, c, a);
692 l += (l & 1) ? GF_N(bch) : 0;
695 * use transformation z=X+e:
696 * z^4+e^4 + a(z^3+ez^2+e^2z+e^3) + b(z^2+e^2) +cz+ce+d
697 * z^4 + az^3 + (ae+b)z^2 + (ae^2+c)z+e^4+be^2+ae^3+ce+d
698 * z^4 + az^3 + (ae+b)z^2 + e^4+be^2+d
699 * z^4 + az^3 + b'z^2 + d'
701 d = a_pow(bch, 2*l)^gf_mul(bch, b, f)^d;
702 b = gf_mul(bch, a, e)^b;
704 /* now, use Y=1/X to get Y^4 + b/dY^2 + a/dY + 1/d */
706 /* assume all roots have multiplicity 1 */
710 b2 = gf_div(bch, a, d);
711 a2 = gf_div(bch, b, d);
713 /* polynomial is already affine */
718 /* find the 4 roots of this affine polynomial */
719 if (find_affine4_roots(bch, a2, b2, c2, roots) == 4) {
720 for (i = 0; i < 4; i++) {
721 /* post-process roots (reverse transformations) */
722 f = a ? gf_inv(bch, roots[i]) : roots[i];
723 roots[i] = a_ilog(bch, f^e);
731 * build monic, log-based representation of a polynomial
733 static void gf_poly_logrep(struct bch_control *bch,
734 const struct gf_poly *a, int *rep)
736 int i, d = a->deg, l = GF_N(bch)-a_log(bch, a->c[a->deg]);
738 /* represent 0 values with -1; warning, rep[d] is not set to 1 */
739 for (i = 0; i < d; i++)
740 rep[i] = a->c[i] ? mod_s(bch, a_log(bch, a->c[i])+l) : -1;
744 * compute polynomial Euclidean division remainder in GF(2^m)[X]
746 static void gf_poly_mod(struct bch_control *bch, struct gf_poly *a,
747 const struct gf_poly *b, int *rep)
750 unsigned int i, j, *c = a->c;
751 const unsigned int d = b->deg;
756 /* reuse or compute log representation of denominator */
759 gf_poly_logrep(bch, b, rep);
762 for (j = a->deg; j >= d; j--) {
764 la = a_log(bch, c[j]);
766 for (i = 0; i < d; i++, p++) {
769 c[p] ^= bch->a_pow_tab[mod_s(bch,
775 while (!c[a->deg] && a->deg)
780 * compute polynomial Euclidean division quotient in GF(2^m)[X]
782 static void gf_poly_div(struct bch_control *bch, struct gf_poly *a,
783 const struct gf_poly *b, struct gf_poly *q)
785 if (a->deg >= b->deg) {
786 q->deg = a->deg-b->deg;
787 /* compute a mod b (modifies a) */
788 gf_poly_mod(bch, a, b, NULL);
789 /* quotient is stored in upper part of polynomial a */
790 memcpy(q->c, &a->c[b->deg], (1+q->deg)*sizeof(unsigned int));
798 * compute polynomial GCD (Greatest Common Divisor) in GF(2^m)[X]
800 static struct gf_poly *gf_poly_gcd(struct bch_control *bch, struct gf_poly *a,
805 dbg("gcd(%s,%s)=", gf_poly_str(a), gf_poly_str(b));
807 if (a->deg < b->deg) {
814 gf_poly_mod(bch, a, b, NULL);
820 dbg("%s\n", gf_poly_str(a));
826 * Given a polynomial f and an integer k, compute Tr(a^kX) mod f
827 * This is used in Berlekamp Trace algorithm for splitting polynomials
829 static void compute_trace_bk_mod(struct bch_control *bch, int k,
830 const struct gf_poly *f, struct gf_poly *z,
833 const int m = GF_M(bch);
836 /* z contains z^2j mod f */
839 z->c[1] = bch->a_pow_tab[k];
842 memset(out, 0, GF_POLY_SZ(f->deg));
844 /* compute f log representation only once */
845 gf_poly_logrep(bch, f, bch->cache);
847 for (i = 0; i < m; i++) {
848 /* add a^(k*2^i)(z^(2^i) mod f) and compute (z^(2^i) mod f)^2 */
849 for (j = z->deg; j >= 0; j--) {
850 out->c[j] ^= z->c[j];
851 z->c[2*j] = gf_sqr(bch, z->c[j]);
854 if (z->deg > out->deg)
859 /* z^(2(i+1)) mod f = (z^(2^i) mod f)^2 mod f */
860 gf_poly_mod(bch, z, f, bch->cache);
863 while (!out->c[out->deg] && out->deg)
866 dbg("Tr(a^%d.X) mod f = %s\n", k, gf_poly_str(out));
870 * factor a polynomial using Berlekamp Trace algorithm (BTA)
872 static void factor_polynomial(struct bch_control *bch, int k, struct gf_poly *f,
873 struct gf_poly **g, struct gf_poly **h)
875 struct gf_poly *f2 = bch->poly_2t[0];
876 struct gf_poly *q = bch->poly_2t[1];
877 struct gf_poly *tk = bch->poly_2t[2];
878 struct gf_poly *z = bch->poly_2t[3];
881 dbg("factoring %s...\n", gf_poly_str(f));
886 /* tk = Tr(a^k.X) mod f */
887 compute_trace_bk_mod(bch, k, f, z, tk);
890 /* compute g = gcd(f, tk) (destructive operation) */
892 gcd = gf_poly_gcd(bch, f2, tk);
893 if (gcd->deg < f->deg) {
894 /* compute h=f/gcd(f,tk); this will modify f and q */
895 gf_poly_div(bch, f, gcd, q);
896 /* store g and h in-place (clobbering f) */
897 *h = &((struct gf_poly_deg1 *)f)[gcd->deg].poly;
898 gf_poly_copy(*g, gcd);
905 * find roots of a polynomial, using BTZ algorithm; see the beginning of this
908 static int find_poly_roots(struct bch_control *bch, unsigned int k,
909 struct gf_poly *poly, unsigned int *roots)
912 struct gf_poly *f1, *f2;
915 /* handle low degree polynomials with ad hoc techniques */
917 cnt = find_poly_deg1_roots(bch, poly, roots);
920 cnt = find_poly_deg2_roots(bch, poly, roots);
923 cnt = find_poly_deg3_roots(bch, poly, roots);
926 cnt = find_poly_deg4_roots(bch, poly, roots);
929 /* factor polynomial using Berlekamp Trace Algorithm (BTA) */
931 if (poly->deg && (k <= GF_M(bch))) {
932 factor_polynomial(bch, k, poly, &f1, &f2);
934 cnt += find_poly_roots(bch, k+1, f1, roots);
936 cnt += find_poly_roots(bch, k+1, f2, roots+cnt);
943 #if defined(USE_CHIEN_SEARCH)
945 * exhaustive root search (Chien) implementation - not used, included only for
946 * reference/comparison tests
948 static int chien_search(struct bch_control *bch, unsigned int len,
949 struct gf_poly *p, unsigned int *roots)
952 unsigned int i, j, syn, syn0, count = 0;
953 const unsigned int k = 8*len+bch->ecc_bits;
955 /* use a log-based representation of polynomial */
956 gf_poly_logrep(bch, p, bch->cache);
957 bch->cache[p->deg] = 0;
958 syn0 = gf_div(bch, p->c[0], p->c[p->deg]);
960 for (i = GF_N(bch)-k+1; i <= GF_N(bch); i++) {
961 /* compute elp(a^i) */
962 for (j = 1, syn = syn0; j <= p->deg; j++) {
965 syn ^= a_pow(bch, m+j*i);
968 roots[count++] = GF_N(bch)-i;
973 return (count == p->deg) ? count : 0;
975 #define find_poly_roots(_p, _k, _elp, _loc) chien_search(_p, len, _elp, _loc)
976 #endif /* USE_CHIEN_SEARCH */
979 * decode_bch - decode received codeword and find bit error locations
980 * @bch: BCH control structure
981 * @data: received data, ignored if @calc_ecc is provided
982 * @len: data length in bytes, must always be provided
983 * @recv_ecc: received ecc, if NULL then assume it was XORed in @calc_ecc
984 * @calc_ecc: calculated ecc, if NULL then calc_ecc is computed from @data
985 * @syn: hw computed syndrome data (if NULL, syndrome is calculated)
986 * @errloc: output array of error locations
989 * The number of errors found, or -EBADMSG if decoding failed, or -EINVAL if
990 * invalid parameters were provided
992 * Depending on the available hw BCH support and the need to compute @calc_ecc
993 * separately (using encode_bch()), this function should be called with one of
994 * the following parameter configurations -
996 * by providing @data and @recv_ecc only:
997 * decode_bch(@bch, @data, @len, @recv_ecc, NULL, NULL, @errloc)
999 * by providing @recv_ecc and @calc_ecc:
1000 * decode_bch(@bch, NULL, @len, @recv_ecc, @calc_ecc, NULL, @errloc)
1002 * by providing ecc = recv_ecc XOR calc_ecc:
1003 * decode_bch(@bch, NULL, @len, NULL, ecc, NULL, @errloc)
1005 * by providing syndrome results @syn:
1006 * decode_bch(@bch, NULL, @len, NULL, NULL, @syn, @errloc)
1008 * Once decode_bch() has successfully returned with a positive value, error
1009 * locations returned in array @errloc should be interpreted as follows -
1011 * if (errloc[n] >= 8*len), then n-th error is located in ecc (no need for
1014 * if (errloc[n] < 8*len), then n-th error is located in data and can be
1015 * corrected with statement data[errloc[n]/8] ^= 1 << (errloc[n] % 8);
1017 * Note that this function does not perform any data correction by itself, it
1018 * merely indicates error locations.
1020 int decode_bch(struct bch_control *bch, const uint8_t *data, unsigned int len,
1021 const uint8_t *recv_ecc, const uint8_t *calc_ecc,
1022 const unsigned int *syn, unsigned int *errloc)
1024 const unsigned int ecc_words = BCH_ECC_WORDS(bch);
1029 /* sanity check: make sure data length can be handled */
1030 if (8*len > (bch->n-bch->ecc_bits))
1033 /* if caller does not provide syndromes, compute them */
1036 /* compute received data ecc into an internal buffer */
1037 if (!data || !recv_ecc)
1039 encode_bch(bch, data, len, NULL);
1041 /* load provided calculated ecc */
1042 load_ecc8(bch, bch->ecc_buf, calc_ecc);
1044 /* load received ecc or assume it was XORed in calc_ecc */
1046 load_ecc8(bch, bch->ecc_buf2, recv_ecc);
1047 /* XOR received and calculated ecc */
1048 for (i = 0, sum = 0; i < (int)ecc_words; i++) {
1049 bch->ecc_buf[i] ^= bch->ecc_buf2[i];
1050 sum |= bch->ecc_buf[i];
1053 /* no error found */
1056 compute_syndromes(bch, bch->ecc_buf, bch->syn);
1060 err = compute_error_locator_polynomial(bch, syn);
1062 nroots = find_poly_roots(bch, 1, bch->elp, errloc);
1067 /* post-process raw error locations for easier correction */
1068 nbits = (len*8)+bch->ecc_bits;
1069 for (i = 0; i < err; i++) {
1070 if (errloc[i] >= nbits) {
1074 errloc[i] = nbits-1-errloc[i];
1075 errloc[i] = (errloc[i] & ~7)|(7-(errloc[i] & 7));
1078 return (err >= 0) ? err : -EBADMSG;
1082 * generate Galois field lookup tables
1084 static int build_gf_tables(struct bch_control *bch, unsigned int poly)
1086 unsigned int i, x = 1;
1087 const unsigned int k = 1 << deg(poly);
1089 /* primitive polynomial must be of degree m */
1090 if (k != (1u << GF_M(bch)))
1093 for (i = 0; i < GF_N(bch); i++) {
1094 bch->a_pow_tab[i] = x;
1095 bch->a_log_tab[x] = i;
1097 /* polynomial is not primitive (a^i=1 with 0<i<2^m-1) */
1103 bch->a_pow_tab[GF_N(bch)] = 1;
1104 bch->a_log_tab[0] = 0;
1110 * compute generator polynomial remainder tables for fast encoding
1112 static void build_mod8_tables(struct bch_control *bch, const uint32_t *g)
1115 uint32_t data, hi, lo, *tab;
1116 const int l = BCH_ECC_WORDS(bch);
1117 const int plen = DIV_ROUND_UP(bch->ecc_bits+1, 32);
1118 const int ecclen = DIV_ROUND_UP(bch->ecc_bits, 32);
1120 memset(bch->mod8_tab, 0, 4*256*l*sizeof(*bch->mod8_tab));
1122 for (i = 0; i < 256; i++) {
1123 /* p(X)=i is a small polynomial of weight <= 8 */
1124 for (b = 0; b < 4; b++) {
1125 /* we want to compute (p(X).X^(8*b+deg(g))) mod g(X) */
1126 tab = bch->mod8_tab + (b*256+i)*l;
1130 /* subtract X^d.g(X) from p(X).X^(8*b+deg(g)) */
1131 data ^= g[0] >> (31-d);
1132 for (j = 0; j < ecclen; j++) {
1133 hi = (d < 31) ? g[j] << (d+1) : 0;
1135 g[j+1] >> (31-d) : 0;
1144 * build a base for factoring degree 2 polynomials
1146 static int build_deg2_base(struct bch_control *bch)
1148 const int m = GF_M(bch);
1150 unsigned int sum, x, y, remaining, ak = 0, xi[m];
1152 /* find k s.t. Tr(a^k) = 1 and 0 <= k < m */
1153 for (i = 0; i < m; i++) {
1154 for (j = 0, sum = 0; j < m; j++)
1155 sum ^= a_pow(bch, i*(1 << j));
1158 ak = bch->a_pow_tab[i];
1162 /* find xi, i=0..m-1 such that xi^2+xi = a^i+Tr(a^i).a^k */
1164 memset(xi, 0, sizeof(xi));
1166 for (x = 0; (x <= GF_N(bch)) && remaining; x++) {
1167 y = gf_sqr(bch, x)^x;
1168 for (i = 0; i < 2; i++) {
1170 if (y && (r < m) && !xi[r]) {
1174 dbg("x%d = %x\n", r, x);
1180 /* should not happen but check anyway */
1181 return remaining ? -1 : 0;
1184 static void *bch_alloc(size_t size, int *err)
1188 ptr = kmalloc(size, GFP_KERNEL);
1195 * compute generator polynomial for given (m,t) parameters.
1197 static uint32_t *compute_generator_polynomial(struct bch_control *bch)
1199 const unsigned int m = GF_M(bch);
1200 const unsigned int t = GF_T(bch);
1202 unsigned int i, j, nbits, r, word, *roots;
1206 g = bch_alloc(GF_POLY_SZ(m*t), &err);
1207 roots = bch_alloc((bch->n+1)*sizeof(*roots), &err);
1208 genpoly = bch_alloc(DIV_ROUND_UP(m*t+1, 32)*sizeof(*genpoly), &err);
1216 /* enumerate all roots of g(X) */
1217 memset(roots , 0, (bch->n+1)*sizeof(*roots));
1218 for (i = 0; i < t; i++) {
1219 for (j = 0, r = 2*i+1; j < m; j++) {
1221 r = mod_s(bch, 2*r);
1224 /* build generator polynomial g(X) */
1227 for (i = 0; i < GF_N(bch); i++) {
1229 /* multiply g(X) by (X+root) */
1230 r = bch->a_pow_tab[i];
1232 for (j = g->deg; j > 0; j--)
1233 g->c[j] = gf_mul(bch, g->c[j], r)^g->c[j-1];
1235 g->c[0] = gf_mul(bch, g->c[0], r);
1239 /* store left-justified binary representation of g(X) */
1244 nbits = (n > 32) ? 32 : n;
1245 for (j = 0, word = 0; j < nbits; j++) {
1247 word |= 1u << (31-j);
1249 genpoly[i++] = word;
1252 bch->ecc_bits = g->deg;
1262 * init_bch - initialize a BCH encoder/decoder
1263 * @m: Galois field order, should be in the range 5-15
1264 * @t: maximum error correction capability, in bits
1265 * @prim_poly: user-provided primitive polynomial (or 0 to use default)
1268 * a newly allocated BCH control structure if successful, NULL otherwise
1270 * This initialization can take some time, as lookup tables are built for fast
1271 * encoding/decoding; make sure not to call this function from a time critical
1272 * path. Usually, init_bch() should be called on module/driver init and
1273 * free_bch() should be called to release memory on exit.
1275 * You may provide your own primitive polynomial of degree @m in argument
1276 * @prim_poly, or let init_bch() use its default polynomial.
1278 * Once init_bch() has successfully returned a pointer to a newly allocated
1279 * BCH control structure, ecc length in bytes is given by member @ecc_bytes of
1282 struct bch_control *init_bch(int m, int t, unsigned int prim_poly)
1285 unsigned int i, words;
1287 struct bch_control *bch = NULL;
1289 const int min_m = 5;
1290 const int max_m = 15;
1292 /* default primitive polynomials */
1293 static const unsigned int prim_poly_tab[] = {
1294 0x25, 0x43, 0x83, 0x11d, 0x211, 0x409, 0x805, 0x1053, 0x201b,
1298 #if defined(CONFIG_BCH_CONST_PARAMS)
1299 if ((m != (CONFIG_BCH_CONST_M)) || (t != (CONFIG_BCH_CONST_T))) {
1300 printk(KERN_ERR "bch encoder/decoder was configured to support "
1301 "parameters m=%d, t=%d only!\n",
1302 CONFIG_BCH_CONST_M, CONFIG_BCH_CONST_T);
1306 if ((m < min_m) || (m > max_m))
1308 * values of m greater than 15 are not currently supported;
1309 * supporting m > 15 would require changing table base type
1310 * (uint16_t) and a small patch in matrix transposition
1315 if ((t < 1) || (m*t >= ((1 << m)-1)))
1316 /* invalid t value */
1319 /* select a primitive polynomial for generating GF(2^m) */
1321 prim_poly = prim_poly_tab[m-min_m];
1323 bch = kzalloc(sizeof(*bch), GFP_KERNEL);
1329 bch->n = (1 << m)-1;
1330 words = DIV_ROUND_UP(m*t, 32);
1331 bch->ecc_bytes = DIV_ROUND_UP(m*t, 8);
1332 bch->a_pow_tab = bch_alloc((1+bch->n)*sizeof(*bch->a_pow_tab), &err);
1333 bch->a_log_tab = bch_alloc((1+bch->n)*sizeof(*bch->a_log_tab), &err);
1334 bch->mod8_tab = bch_alloc(words*1024*sizeof(*bch->mod8_tab), &err);
1335 bch->ecc_buf = bch_alloc(words*sizeof(*bch->ecc_buf), &err);
1336 bch->ecc_buf2 = bch_alloc(words*sizeof(*bch->ecc_buf2), &err);
1337 bch->xi_tab = bch_alloc(m*sizeof(*bch->xi_tab), &err);
1338 bch->syn = bch_alloc(2*t*sizeof(*bch->syn), &err);
1339 bch->cache = bch_alloc(2*t*sizeof(*bch->cache), &err);
1340 bch->elp = bch_alloc((t+1)*sizeof(struct gf_poly_deg1), &err);
1342 for (i = 0; i < ARRAY_SIZE(bch->poly_2t); i++)
1343 bch->poly_2t[i] = bch_alloc(GF_POLY_SZ(2*t), &err);
1348 err = build_gf_tables(bch, prim_poly);
1352 /* use generator polynomial for computing encoding tables */
1353 genpoly = compute_generator_polynomial(bch);
1354 if (genpoly == NULL)
1357 build_mod8_tables(bch, genpoly);
1360 err = build_deg2_base(bch);
1372 * free_bch - free the BCH control structure
1373 * @bch: BCH control structure to release
1375 void free_bch(struct bch_control *bch)
1380 kfree(bch->a_pow_tab);
1381 kfree(bch->a_log_tab);
1382 kfree(bch->mod8_tab);
1383 kfree(bch->ecc_buf);
1384 kfree(bch->ecc_buf2);
1390 for (i = 0; i < ARRAY_SIZE(bch->poly_2t); i++)
1391 kfree(bch->poly_2t[i]);