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linux-pinenote/lib/bch.c

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lib: add shared BCH ECC library This is a new software BCH encoding/decoding library, similar to the shared Reed-Solomon library. Binary BCH (Bose-Chaudhuri-Hocquenghem) codes are widely used to correct errors in NAND flash devices requiring more than 1-bit ecc correction; they are generally better suited for NAND flash than RS codes because NAND bit errors do not occur in bursts. Latest SLC NAND devices typically require at least 4-bit ecc protection per 512 bytes block. This library provides software encoding/decoding, but may also be used with ASIC/SoC hardware BCH engines to perform error correction. It is being currently used for this purpose on an OMAP3630 board (4bit/8bit HW BCH). It has also been used to decode raw dumps of NAND devices with on-die BCH ecc engines (e.g. Micron 4bit ecc SLC devices). Latest NAND devices (including SLC) can exhibit high error rates (typically a dozen or more bitflips per hour during stress tests); in order to minimize the performance impact of error correction, this library implements recently developed algorithms for fast polynomial root finding (see bch.c header for details) instead of the traditional exhaustive Chien root search; a few performance figures are provided below: Platform: arm926ejs @ 468 MHz, 32 KiB icache, 16 KiB dcache BCH ecc : 4-bit per 512 bytes Encoding average throughput: 250 Mbits/s Error correction time (compared with Chien search): average worst average (Chien) worst (Chien) ---------------------------------------------------------- 1 bit 8.5 µs 11 µs 200 µs 383 µs 2 bit 9.7 µs 12.5 µs 477 µs 728 µs 3 bit 18.1 µs 20.6 µs 758 µs 1010 µs 4 bit 19.5 µs 23 µs 1028 µs 1280 µs In the above figures, "worst" is meant in terms of error pattern, not in terms of cache miss / page faults effects (not taken into account here). The library has been extensively tested on the following platforms: x86, x86_64, arm926ejs, omap3630, qemu-ppc64, qemu-mips. Signed-off-by: Ivan Djelic <ivan.djelic@parrot.com> Signed-off-by: David Woodhouse <David.Woodhouse@intel.com>
2011-03-11 11:05:32 +01:00
/*
* Generic binary BCH encoding/decoding library
*
* This program is free software; you can redistribute it and/or modify it
* under the terms of the GNU General Public License version 2 as published by
* the Free Software Foundation.
*
* This program is distributed in the hope that it will be useful, but WITHOUT
* ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
* FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for
* more details.
*
* You should have received a copy of the GNU General Public License along with
* this program; if not, write to the Free Software Foundation, Inc., 51
* Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
*
* Copyright © 2011 Parrot S.A.
*
* Author: Ivan Djelic <ivan.djelic@parrot.com>
*
* Description:
*
* This library provides runtime configurable encoding/decoding of binary
* Bose-Chaudhuri-Hocquenghem (BCH) codes.
*
* Call init_bch to get a pointer to a newly allocated bch_control structure for
* the given m (Galois field order), t (error correction capability) and
* (optional) primitive polynomial parameters.
*
* Call encode_bch to compute and store ecc parity bytes to a given buffer.
* Call decode_bch to detect and locate errors in received data.
*
* On systems supporting hw BCH features, intermediate results may be provided
* to decode_bch in order to skip certain steps. See decode_bch() documentation
* for details.
*
* Option CONFIG_BCH_CONST_PARAMS can be used to force fixed values of
* parameters m and t; thus allowing extra compiler optimizations and providing
* better (up to 2x) encoding performance. Using this option makes sense when
* (m,t) are fixed and known in advance, e.g. when using BCH error correction
* on a particular NAND flash device.
*
* Algorithmic details:
*
* Encoding is performed by processing 32 input bits in parallel, using 4
* remainder lookup tables.
*
* The final stage of decoding involves the following internal steps:
* a. Syndrome computation
* b. Error locator polynomial computation using Berlekamp-Massey algorithm
* c. Error locator root finding (by far the most expensive step)
*
* In this implementation, step c is not performed using the usual Chien search.
* Instead, an alternative approach described in [1] is used. It consists in
* factoring the error locator polynomial using the Berlekamp Trace algorithm
* (BTA) down to a certain degree (4), after which ad hoc low-degree polynomial
* solving techniques [2] are used. The resulting algorithm, called BTZ, yields
* much better performance than Chien search for usual (m,t) values (typically
* m >= 13, t < 32, see [1]).
*
* [1] B. Biswas, V. Herbert. Efficient root finding of polynomials over fields
* of characteristic 2, in: Western European Workshop on Research in Cryptology
* - WEWoRC 2009, Graz, Austria, LNCS, Springer, July 2009, to appear.
* [2] [Zin96] V.A. Zinoviev. On the solution of equations of degree 10 over
* finite fields GF(2^q). In Rapport de recherche INRIA no 2829, 1996.
*/
#include <linux/kernel.h>
#include <linux/errno.h>
#include <linux/init.h>
#include <linux/module.h>
#include <linux/slab.h>
#include <linux/bitops.h>
#include <asm/byteorder.h>
#include <linux/bch.h>
#if defined(CONFIG_BCH_CONST_PARAMS)
#define GF_M(_p) (CONFIG_BCH_CONST_M)
#define GF_T(_p) (CONFIG_BCH_CONST_T)
#define GF_N(_p) ((1 << (CONFIG_BCH_CONST_M))-1)
#else
#define GF_M(_p) ((_p)->m)
#define GF_T(_p) ((_p)->t)
#define GF_N(_p) ((_p)->n)
#endif
#define BCH_ECC_WORDS(_p) DIV_ROUND_UP(GF_M(_p)*GF_T(_p), 32)
#define BCH_ECC_BYTES(_p) DIV_ROUND_UP(GF_M(_p)*GF_T(_p), 8)
#ifndef dbg
#define dbg(_fmt, args...) do {} while (0)
#endif
/*
* represent a polynomial over GF(2^m)
*/
struct gf_poly {
unsigned int deg; /* polynomial degree */
unsigned int c[0]; /* polynomial terms */
};
/* given its degree, compute a polynomial size in bytes */
#define GF_POLY_SZ(_d) (sizeof(struct gf_poly)+((_d)+1)*sizeof(unsigned int))
/* polynomial of degree 1 */
struct gf_poly_deg1 {
struct gf_poly poly;
unsigned int c[2];
};
/*
* same as encode_bch(), but process input data one byte at a time
*/
static void encode_bch_unaligned(struct bch_control *bch,
const unsigned char *data, unsigned int len,
uint32_t *ecc)
{
int i;
const uint32_t *p;
const int l = BCH_ECC_WORDS(bch)-1;
while (len--) {
p = bch->mod8_tab + (l+1)*(((ecc[0] >> 24)^(*data++)) & 0xff);
for (i = 0; i < l; i++)
ecc[i] = ((ecc[i] << 8)|(ecc[i+1] >> 24))^(*p++);
ecc[l] = (ecc[l] << 8)^(*p);
}
}
/*
* convert ecc bytes to aligned, zero-padded 32-bit ecc words
*/
static void load_ecc8(struct bch_control *bch, uint32_t *dst,
const uint8_t *src)
{
uint8_t pad[4] = {0, 0, 0, 0};
unsigned int i, nwords = BCH_ECC_WORDS(bch)-1;
for (i = 0; i < nwords; i++, src += 4)
dst[i] = (src[0] << 24)|(src[1] << 16)|(src[2] << 8)|src[3];
memcpy(pad, src, BCH_ECC_BYTES(bch)-4*nwords);
dst[nwords] = (pad[0] << 24)|(pad[1] << 16)|(pad[2] << 8)|pad[3];
}
/*
* convert 32-bit ecc words to ecc bytes
*/
static void store_ecc8(struct bch_control *bch, uint8_t *dst,
const uint32_t *src)
{
uint8_t pad[4];
unsigned int i, nwords = BCH_ECC_WORDS(bch)-1;
for (i = 0; i < nwords; i++) {
*dst++ = (src[i] >> 24);
*dst++ = (src[i] >> 16) & 0xff;
*dst++ = (src[i] >> 8) & 0xff;
*dst++ = (src[i] >> 0) & 0xff;
}
pad[0] = (src[nwords] >> 24);
pad[1] = (src[nwords] >> 16) & 0xff;
pad[2] = (src[nwords] >> 8) & 0xff;
pad[3] = (src[nwords] >> 0) & 0xff;
memcpy(dst, pad, BCH_ECC_BYTES(bch)-4*nwords);
}
/**
* encode_bch - calculate BCH ecc parity of data
* @bch: BCH control structure
* @data: data to encode
* @len: data length in bytes
* @ecc: ecc parity data, must be initialized by caller
*
* The @ecc parity array is used both as input and output parameter, in order to
* allow incremental computations. It should be of the size indicated by member
* @ecc_bytes of @bch, and should be initialized to 0 before the first call.
*
* The exact number of computed ecc parity bits is given by member @ecc_bits of
* @bch; it may be less than m*t for large values of t.
*/
void encode_bch(struct bch_control *bch, const uint8_t *data,
unsigned int len, uint8_t *ecc)
{
const unsigned int l = BCH_ECC_WORDS(bch)-1;
unsigned int i, mlen;
unsigned long m;
uint32_t w, r[l+1];
const uint32_t * const tab0 = bch->mod8_tab;
const uint32_t * const tab1 = tab0 + 256*(l+1);
const uint32_t * const tab2 = tab1 + 256*(l+1);
const uint32_t * const tab3 = tab2 + 256*(l+1);
const uint32_t *pdata, *p0, *p1, *p2, *p3;
if (ecc) {
/* load ecc parity bytes into internal 32-bit buffer */
load_ecc8(bch, bch->ecc_buf, ecc);
} else {
memset(bch->ecc_buf, 0, sizeof(r));
}
/* process first unaligned data bytes */
m = ((unsigned long)data) & 3;
if (m) {
mlen = (len < (4-m)) ? len : 4-m;
encode_bch_unaligned(bch, data, mlen, bch->ecc_buf);
data += mlen;
len -= mlen;
}
/* process 32-bit aligned data words */
pdata = (uint32_t *)data;
mlen = len/4;
data += 4*mlen;
len -= 4*mlen;
memcpy(r, bch->ecc_buf, sizeof(r));
/*
* split each 32-bit word into 4 polynomials of weight 8 as follows:
*
* 31 ...24 23 ...16 15 ... 8 7 ... 0
* xxxxxxxx yyyyyyyy zzzzzzzz tttttttt
* tttttttt mod g = r0 (precomputed)
* zzzzzzzz 00000000 mod g = r1 (precomputed)
* yyyyyyyy 00000000 00000000 mod g = r2 (precomputed)
* xxxxxxxx 00000000 00000000 00000000 mod g = r3 (precomputed)
* xxxxxxxx yyyyyyyy zzzzzzzz tttttttt mod g = r0^r1^r2^r3
*/
while (mlen--) {
/* input data is read in big-endian format */
w = r[0]^cpu_to_be32(*pdata++);
p0 = tab0 + (l+1)*((w >> 0) & 0xff);
p1 = tab1 + (l+1)*((w >> 8) & 0xff);
p2 = tab2 + (l+1)*((w >> 16) & 0xff);
p3 = tab3 + (l+1)*((w >> 24) & 0xff);
for (i = 0; i < l; i++)
r[i] = r[i+1]^p0[i]^p1[i]^p2[i]^p3[i];
r[l] = p0[l]^p1[l]^p2[l]^p3[l];
}
memcpy(bch->ecc_buf, r, sizeof(r));
/* process last unaligned bytes */
if (len)
encode_bch_unaligned(bch, data, len, bch->ecc_buf);
/* store ecc parity bytes into original parity buffer */
if (ecc)
store_ecc8(bch, ecc, bch->ecc_buf);
}
EXPORT_SYMBOL_GPL(encode_bch);
static inline int modulo(struct bch_control *bch, unsigned int v)
{
const unsigned int n = GF_N(bch);
while (v >= n) {
v -= n;
v = (v & n) + (v >> GF_M(bch));
}
return v;
}
/*
* shorter and faster modulo function, only works when v < 2N.
*/
static inline int mod_s(struct bch_control *bch, unsigned int v)
{
const unsigned int n = GF_N(bch);
return (v < n) ? v : v-n;
}
static inline int deg(unsigned int poly)
{
/* polynomial degree is the most-significant bit index */
return fls(poly)-1;
}
static inline int parity(unsigned int x)
{
/*
* public domain code snippet, lifted from
* http://www-graphics.stanford.edu/~seander/bithacks.html
*/
x ^= x >> 1;
x ^= x >> 2;
x = (x & 0x11111111U) * 0x11111111U;
return (x >> 28) & 1;
}
/* Galois field basic operations: multiply, divide, inverse, etc. */
static inline unsigned int gf_mul(struct bch_control *bch, unsigned int a,
unsigned int b)
{
return (a && b) ? bch->a_pow_tab[mod_s(bch, bch->a_log_tab[a]+
bch->a_log_tab[b])] : 0;
}
static inline unsigned int gf_sqr(struct bch_control *bch, unsigned int a)
{
return a ? bch->a_pow_tab[mod_s(bch, 2*bch->a_log_tab[a])] : 0;
}
static inline unsigned int gf_div(struct bch_control *bch, unsigned int a,
unsigned int b)
{
return a ? bch->a_pow_tab[mod_s(bch, bch->a_log_tab[a]+
GF_N(bch)-bch->a_log_tab[b])] : 0;
}
static inline unsigned int gf_inv(struct bch_control *bch, unsigned int a)
{
return bch->a_pow_tab[GF_N(bch)-bch->a_log_tab[a]];
}
static inline unsigned int a_pow(struct bch_control *bch, int i)
{
return bch->a_pow_tab[modulo(bch, i)];
}
static inline int a_log(struct bch_control *bch, unsigned int x)
{
return bch->a_log_tab[x];
}
static inline int a_ilog(struct bch_control *bch, unsigned int x)
{
return mod_s(bch, GF_N(bch)-bch->a_log_tab[x]);
}
/*
* compute 2t syndromes of ecc polynomial, i.e. ecc(a^j) for j=1..2t
*/
static void compute_syndromes(struct bch_control *bch, uint32_t *ecc,
unsigned int *syn)
{
int i, j, s;
unsigned int m;
uint32_t poly;
const int t = GF_T(bch);
s = bch->ecc_bits;
/* make sure extra bits in last ecc word are cleared */
m = ((unsigned int)s) & 31;
if (m)
ecc[s/32] &= ~((1u << (32-m))-1);
memset(syn, 0, 2*t*sizeof(*syn));
/* compute v(a^j) for j=1 .. 2t-1 */
do {
poly = *ecc++;
s -= 32;
while (poly) {
i = deg(poly);
for (j = 0; j < 2*t; j += 2)
syn[j] ^= a_pow(bch, (j+1)*(i+s));
poly ^= (1 << i);
}
} while (s > 0);
/* v(a^(2j)) = v(a^j)^2 */
for (j = 0; j < t; j++)
syn[2*j+1] = gf_sqr(bch, syn[j]);
}
static void gf_poly_copy(struct gf_poly *dst, struct gf_poly *src)
{
memcpy(dst, src, GF_POLY_SZ(src->deg));
}
static int compute_error_locator_polynomial(struct bch_control *bch,
const unsigned int *syn)
{
const unsigned int t = GF_T(bch);
const unsigned int n = GF_N(bch);
unsigned int i, j, tmp, l, pd = 1, d = syn[0];
struct gf_poly *elp = bch->elp;
struct gf_poly *pelp = bch->poly_2t[0];
struct gf_poly *elp_copy = bch->poly_2t[1];
int k, pp = -1;
memset(pelp, 0, GF_POLY_SZ(2*t));
memset(elp, 0, GF_POLY_SZ(2*t));
pelp->deg = 0;
pelp->c[0] = 1;
elp->deg = 0;
elp->c[0] = 1;
/* use simplified binary Berlekamp-Massey algorithm */
for (i = 0; (i < t) && (elp->deg <= t); i++) {
if (d) {
k = 2*i-pp;
gf_poly_copy(elp_copy, elp);
/* e[i+1](X) = e[i](X)+di*dp^-1*X^2(i-p)*e[p](X) */
tmp = a_log(bch, d)+n-a_log(bch, pd);
for (j = 0; j <= pelp->deg; j++) {
if (pelp->c[j]) {
l = a_log(bch, pelp->c[j]);
elp->c[j+k] ^= a_pow(bch, tmp+l);
}
}
/* compute l[i+1] = max(l[i]->c[l[p]+2*(i-p]) */
tmp = pelp->deg+k;
if (tmp > elp->deg) {
elp->deg = tmp;
gf_poly_copy(pelp, elp_copy);
pd = d;
pp = 2*i;
}
}
/* di+1 = S(2i+3)+elp[i+1].1*S(2i+2)+...+elp[i+1].lS(2i+3-l) */
if (i < t-1) {
d = syn[2*i+2];
for (j = 1; j <= elp->deg; j++)
d ^= gf_mul(bch, elp->c[j], syn[2*i+2-j]);
}
}
dbg("elp=%s\n", gf_poly_str(elp));
return (elp->deg > t) ? -1 : (int)elp->deg;
}
/*
* solve a m x m linear system in GF(2) with an expected number of solutions,
* and return the number of found solutions
*/
static int solve_linear_system(struct bch_control *bch, unsigned int *rows,
unsigned int *sol, int nsol)
{
const int m = GF_M(bch);
unsigned int tmp, mask;
int rem, c, r, p, k, param[m];
k = 0;
mask = 1 << m;
/* Gaussian elimination */
for (c = 0; c < m; c++) {
rem = 0;
p = c-k;
/* find suitable row for elimination */
for (r = p; r < m; r++) {
if (rows[r] & mask) {
if (r != p) {
tmp = rows[r];
rows[r] = rows[p];
rows[p] = tmp;
}
rem = r+1;
break;
}
}
if (rem) {
/* perform elimination on remaining rows */
tmp = rows[p];
for (r = rem; r < m; r++) {
if (rows[r] & mask)
rows[r] ^= tmp;
}
} else {
/* elimination not needed, store defective row index */
param[k++] = c;
}
mask >>= 1;
}
/* rewrite system, inserting fake parameter rows */
if (k > 0) {
p = k;
for (r = m-1; r >= 0; r--) {
if ((r > m-1-k) && rows[r])
/* system has no solution */
return 0;
rows[r] = (p && (r == param[p-1])) ?
p--, 1u << (m-r) : rows[r-p];
}
}
if (nsol != (1 << k))
/* unexpected number of solutions */
return 0;
for (p = 0; p < nsol; p++) {
/* set parameters for p-th solution */
for (c = 0; c < k; c++)
rows[param[c]] = (rows[param[c]] & ~1)|((p >> c) & 1);
/* compute unique solution */
tmp = 0;
for (r = m-1; r >= 0; r--) {
mask = rows[r] & (tmp|1);
tmp |= parity(mask) << (m-r);
}
sol[p] = tmp >> 1;
}
return nsol;
}
/*
* this function builds and solves a linear system for finding roots of a degree
* 4 affine monic polynomial X^4+aX^2+bX+c over GF(2^m).
*/
static int find_affine4_roots(struct bch_control *bch, unsigned int a,
unsigned int b, unsigned int c,
unsigned int *roots)
{
int i, j, k;
const int m = GF_M(bch);
unsigned int mask = 0xff, t, rows[16] = {0,};
j = a_log(bch, b);
k = a_log(bch, a);
rows[0] = c;
/* buid linear system to solve X^4+aX^2+bX+c = 0 */
for (i = 0; i < m; i++) {
rows[i+1] = bch->a_pow_tab[4*i]^
(a ? bch->a_pow_tab[mod_s(bch, k)] : 0)^
(b ? bch->a_pow_tab[mod_s(bch, j)] : 0);
j++;
k += 2;
}
/*
* transpose 16x16 matrix before passing it to linear solver
* warning: this code assumes m < 16
*/
for (j = 8; j != 0; j >>= 1, mask ^= (mask << j)) {
for (k = 0; k < 16; k = (k+j+1) & ~j) {
t = ((rows[k] >> j)^rows[k+j]) & mask;
rows[k] ^= (t << j);
rows[k+j] ^= t;
}
}
return solve_linear_system(bch, rows, roots, 4);
}
/*
* compute root r of a degree 1 polynomial over GF(2^m) (returned as log(1/r))
*/
static int find_poly_deg1_roots(struct bch_control *bch, struct gf_poly *poly,
unsigned int *roots)
{
int n = 0;
if (poly->c[0])
/* poly[X] = bX+c with c!=0, root=c/b */
roots[n++] = mod_s(bch, GF_N(bch)-bch->a_log_tab[poly->c[0]]+
bch->a_log_tab[poly->c[1]]);
return n;
}
/*
* compute roots of a degree 2 polynomial over GF(2^m)
*/
static int find_poly_deg2_roots(struct bch_control *bch, struct gf_poly *poly,
unsigned int *roots)
{
int n = 0, i, l0, l1, l2;
unsigned int u, v, r;
if (poly->c[0] && poly->c[1]) {
l0 = bch->a_log_tab[poly->c[0]];
l1 = bch->a_log_tab[poly->c[1]];
l2 = bch->a_log_tab[poly->c[2]];
/* using z=a/bX, transform aX^2+bX+c into z^2+z+u (u=ac/b^2) */
u = a_pow(bch, l0+l2+2*(GF_N(bch)-l1));
/*
* let u = sum(li.a^i) i=0..m-1; then compute r = sum(li.xi):
* r^2+r = sum(li.(xi^2+xi)) = sum(li.(a^i+Tr(a^i).a^k)) =
* u + sum(li.Tr(a^i).a^k) = u+a^k.Tr(sum(li.a^i)) = u+a^k.Tr(u)
* i.e. r and r+1 are roots iff Tr(u)=0
*/
r = 0;
v = u;
while (v) {
i = deg(v);
r ^= bch->xi_tab[i];
v ^= (1 << i);
}
/* verify root */
if ((gf_sqr(bch, r)^r) == u) {
/* reverse z=a/bX transformation and compute log(1/r) */
roots[n++] = modulo(bch, 2*GF_N(bch)-l1-
bch->a_log_tab[r]+l2);
roots[n++] = modulo(bch, 2*GF_N(bch)-l1-
bch->a_log_tab[r^1]+l2);
}
}
return n;
}
/*
* compute roots of a degree 3 polynomial over GF(2^m)
*/
static int find_poly_deg3_roots(struct bch_control *bch, struct gf_poly *poly,
unsigned int *roots)
{
int i, n = 0;
unsigned int a, b, c, a2, b2, c2, e3, tmp[4];
if (poly->c[0]) {
/* transform polynomial into monic X^3 + a2X^2 + b2X + c2 */
e3 = poly->c[3];
c2 = gf_div(bch, poly->c[0], e3);
b2 = gf_div(bch, poly->c[1], e3);
a2 = gf_div(bch, poly->c[2], e3);
/* (X+a2)(X^3+a2X^2+b2X+c2) = X^4+aX^2+bX+c (affine) */
c = gf_mul(bch, a2, c2); /* c = a2c2 */
b = gf_mul(bch, a2, b2)^c2; /* b = a2b2 + c2 */
a = gf_sqr(bch, a2)^b2; /* a = a2^2 + b2 */
/* find the 4 roots of this affine polynomial */
if (find_affine4_roots(bch, a, b, c, tmp) == 4) {
/* remove a2 from final list of roots */
for (i = 0; i < 4; i++) {
if (tmp[i] != a2)
roots[n++] = a_ilog(bch, tmp[i]);
}
}
}
return n;
}
/*
* compute roots of a degree 4 polynomial over GF(2^m)
*/
static int find_poly_deg4_roots(struct bch_control *bch, struct gf_poly *poly,
unsigned int *roots)
{
int i, l, n = 0;
unsigned int a, b, c, d, e = 0, f, a2, b2, c2, e4;
if (poly->c[0] == 0)
return 0;
/* transform polynomial into monic X^4 + aX^3 + bX^2 + cX + d */
e4 = poly->c[4];
d = gf_div(bch, poly->c[0], e4);
c = gf_div(bch, poly->c[1], e4);
b = gf_div(bch, poly->c[2], e4);
a = gf_div(bch, poly->c[3], e4);
/* use Y=1/X transformation to get an affine polynomial */
if (a) {
/* first, eliminate cX by using z=X+e with ae^2+c=0 */
if (c) {
/* compute e such that e^2 = c/a */
f = gf_div(bch, c, a);
l = a_log(bch, f);
l += (l & 1) ? GF_N(bch) : 0;
e = a_pow(bch, l/2);
/*
* use transformation z=X+e:
* z^4+e^4 + a(z^3+ez^2+e^2z+e^3) + b(z^2+e^2) +cz+ce+d
* z^4 + az^3 + (ae+b)z^2 + (ae^2+c)z+e^4+be^2+ae^3+ce+d
* z^4 + az^3 + (ae+b)z^2 + e^4+be^2+d
* z^4 + az^3 + b'z^2 + d'
*/
d = a_pow(bch, 2*l)^gf_mul(bch, b, f)^d;
b = gf_mul(bch, a, e)^b;
}
/* now, use Y=1/X to get Y^4 + b/dY^2 + a/dY + 1/d */
if (d == 0)
/* assume all roots have multiplicity 1 */
return 0;
c2 = gf_inv(bch, d);
b2 = gf_div(bch, a, d);
a2 = gf_div(bch, b, d);
} else {
/* polynomial is already affine */
c2 = d;
b2 = c;
a2 = b;
}
/* find the 4 roots of this affine polynomial */
if (find_affine4_roots(bch, a2, b2, c2, roots) == 4) {
for (i = 0; i < 4; i++) {
/* post-process roots (reverse transformations) */
f = a ? gf_inv(bch, roots[i]) : roots[i];
roots[i] = a_ilog(bch, f^e);
}
n = 4;
}
return n;
}
/*
* build monic, log-based representation of a polynomial
*/
static void gf_poly_logrep(struct bch_control *bch,
const struct gf_poly *a, int *rep)
{
int i, d = a->deg, l = GF_N(bch)-a_log(bch, a->c[a->deg]);
/* represent 0 values with -1; warning, rep[d] is not set to 1 */
for (i = 0; i < d; i++)
rep[i] = a->c[i] ? mod_s(bch, a_log(bch, a->c[i])+l) : -1;
}
/*
* compute polynomial Euclidean division remainder in GF(2^m)[X]
*/
static void gf_poly_mod(struct bch_control *bch, struct gf_poly *a,
const struct gf_poly *b, int *rep)
{
int la, p, m;
unsigned int i, j, *c = a->c;
const unsigned int d = b->deg;
if (a->deg < d)
return;
/* reuse or compute log representation of denominator */
if (!rep) {
rep = bch->cache;
gf_poly_logrep(bch, b, rep);
}
for (j = a->deg; j >= d; j--) {
if (c[j]) {
la = a_log(bch, c[j]);
p = j-d;
for (i = 0; i < d; i++, p++) {
m = rep[i];
if (m >= 0)
c[p] ^= bch->a_pow_tab[mod_s(bch,
m+la)];
}
}
}
a->deg = d-1;
while (!c[a->deg] && a->deg)
a->deg--;
}
/*
* compute polynomial Euclidean division quotient in GF(2^m)[X]
*/
static void gf_poly_div(struct bch_control *bch, struct gf_poly *a,
const struct gf_poly *b, struct gf_poly *q)
{
if (a->deg >= b->deg) {
q->deg = a->deg-b->deg;
/* compute a mod b (modifies a) */
gf_poly_mod(bch, a, b, NULL);
/* quotient is stored in upper part of polynomial a */
memcpy(q->c, &a->c[b->deg], (1+q->deg)*sizeof(unsigned int));
} else {
q->deg = 0;
q->c[0] = 0;
}
}
/*
* compute polynomial GCD (Greatest Common Divisor) in GF(2^m)[X]
*/
static struct gf_poly *gf_poly_gcd(struct bch_control *bch, struct gf_poly *a,
struct gf_poly *b)
{
struct gf_poly *tmp;
dbg("gcd(%s,%s)=", gf_poly_str(a), gf_poly_str(b));
if (a->deg < b->deg) {
tmp = b;
b = a;
a = tmp;
}
while (b->deg > 0) {
gf_poly_mod(bch, a, b, NULL);
tmp = b;
b = a;
a = tmp;
}
dbg("%s\n", gf_poly_str(a));
return a;
}
/*
* Given a polynomial f and an integer k, compute Tr(a^kX) mod f
* This is used in Berlekamp Trace algorithm for splitting polynomials
*/
static void compute_trace_bk_mod(struct bch_control *bch, int k,
const struct gf_poly *f, struct gf_poly *z,
struct gf_poly *out)
{
const int m = GF_M(bch);
int i, j;
/* z contains z^2j mod f */
z->deg = 1;
z->c[0] = 0;
z->c[1] = bch->a_pow_tab[k];
out->deg = 0;
memset(out, 0, GF_POLY_SZ(f->deg));
/* compute f log representation only once */
gf_poly_logrep(bch, f, bch->cache);
for (i = 0; i < m; i++) {
/* add a^(k*2^i)(z^(2^i) mod f) and compute (z^(2^i) mod f)^2 */
for (j = z->deg; j >= 0; j--) {
out->c[j] ^= z->c[j];
z->c[2*j] = gf_sqr(bch, z->c[j]);
z->c[2*j+1] = 0;
}
if (z->deg > out->deg)
out->deg = z->deg;
if (i < m-1) {
z->deg *= 2;
/* z^(2(i+1)) mod f = (z^(2^i) mod f)^2 mod f */
gf_poly_mod(bch, z, f, bch->cache);
}
}
while (!out->c[out->deg] && out->deg)
out->deg--;
dbg("Tr(a^%d.X) mod f = %s\n", k, gf_poly_str(out));
}
/*
* factor a polynomial using Berlekamp Trace algorithm (BTA)
*/
static void factor_polynomial(struct bch_control *bch, int k, struct gf_poly *f,
struct gf_poly **g, struct gf_poly **h)
{
struct gf_poly *f2 = bch->poly_2t[0];
struct gf_poly *q = bch->poly_2t[1];
struct gf_poly *tk = bch->poly_2t[2];
struct gf_poly *z = bch->poly_2t[3];
struct gf_poly *gcd;
dbg("factoring %s...\n", gf_poly_str(f));
*g = f;
*h = NULL;
/* tk = Tr(a^k.X) mod f */
compute_trace_bk_mod(bch, k, f, z, tk);
if (tk->deg > 0) {
/* compute g = gcd(f, tk) (destructive operation) */
gf_poly_copy(f2, f);
gcd = gf_poly_gcd(bch, f2, tk);
if (gcd->deg < f->deg) {
/* compute h=f/gcd(f,tk); this will modify f and q */
gf_poly_div(bch, f, gcd, q);
/* store g and h in-place (clobbering f) */
*h = &((struct gf_poly_deg1 *)f)[gcd->deg].poly;
gf_poly_copy(*g, gcd);
gf_poly_copy(*h, q);
}
}
}
/*
* find roots of a polynomial, using BTZ algorithm; see the beginning of this
* file for details
*/
static int find_poly_roots(struct bch_control *bch, unsigned int k,
struct gf_poly *poly, unsigned int *roots)
{
int cnt;
struct gf_poly *f1, *f2;
switch (poly->deg) {
/* handle low degree polynomials with ad hoc techniques */
case 1:
cnt = find_poly_deg1_roots(bch, poly, roots);
break;
case 2:
cnt = find_poly_deg2_roots(bch, poly, roots);
break;
case 3:
cnt = find_poly_deg3_roots(bch, poly, roots);
break;
case 4:
cnt = find_poly_deg4_roots(bch, poly, roots);
break;
default:
/* factor polynomial using Berlekamp Trace Algorithm (BTA) */
cnt = 0;
if (poly->deg && (k <= GF_M(bch))) {
factor_polynomial(bch, k, poly, &f1, &f2);
if (f1)
cnt += find_poly_roots(bch, k+1, f1, roots);
if (f2)
cnt += find_poly_roots(bch, k+1, f2, roots+cnt);
}
break;
}
return cnt;
}
#if defined(USE_CHIEN_SEARCH)
/*
* exhaustive root search (Chien) implementation - not used, included only for
* reference/comparison tests
*/
static int chien_search(struct bch_control *bch, unsigned int len,
struct gf_poly *p, unsigned int *roots)
{
int m;
unsigned int i, j, syn, syn0, count = 0;
const unsigned int k = 8*len+bch->ecc_bits;
/* use a log-based representation of polynomial */
gf_poly_logrep(bch, p, bch->cache);
bch->cache[p->deg] = 0;
syn0 = gf_div(bch, p->c[0], p->c[p->deg]);
for (i = GF_N(bch)-k+1; i <= GF_N(bch); i++) {
/* compute elp(a^i) */
for (j = 1, syn = syn0; j <= p->deg; j++) {
m = bch->cache[j];
if (m >= 0)
syn ^= a_pow(bch, m+j*i);
}
if (syn == 0) {
roots[count++] = GF_N(bch)-i;
if (count == p->deg)
break;
}
}
return (count == p->deg) ? count : 0;
}
#define find_poly_roots(_p, _k, _elp, _loc) chien_search(_p, len, _elp, _loc)
#endif /* USE_CHIEN_SEARCH */
/**
* decode_bch - decode received codeword and find bit error locations
* @bch: BCH control structure
* @data: received data, ignored if @calc_ecc is provided
* @len: data length in bytes, must always be provided
* @recv_ecc: received ecc, if NULL then assume it was XORed in @calc_ecc
* @calc_ecc: calculated ecc, if NULL then calc_ecc is computed from @data
* @syn: hw computed syndrome data (if NULL, syndrome is calculated)
* @errloc: output array of error locations
*
* Returns:
* The number of errors found, or -EBADMSG if decoding failed, or -EINVAL if
* invalid parameters were provided
*
* Depending on the available hw BCH support and the need to compute @calc_ecc
* separately (using encode_bch()), this function should be called with one of
* the following parameter configurations -
*
* by providing @data and @recv_ecc only:
* decode_bch(@bch, @data, @len, @recv_ecc, NULL, NULL, @errloc)
*
* by providing @recv_ecc and @calc_ecc:
* decode_bch(@bch, NULL, @len, @recv_ecc, @calc_ecc, NULL, @errloc)
*
* by providing ecc = recv_ecc XOR calc_ecc:
* decode_bch(@bch, NULL, @len, NULL, ecc, NULL, @errloc)
*
* by providing syndrome results @syn:
* decode_bch(@bch, NULL, @len, NULL, NULL, @syn, @errloc)
*
* Once decode_bch() has successfully returned with a positive value, error
* locations returned in array @errloc should be interpreted as follows -
*
* if (errloc[n] >= 8*len), then n-th error is located in ecc (no need for
* data correction)
*
* if (errloc[n] < 8*len), then n-th error is located in data and can be
* corrected with statement data[errloc[n]/8] ^= 1 << (errloc[n] % 8);
*
* Note that this function does not perform any data correction by itself, it
* merely indicates error locations.
*/
int decode_bch(struct bch_control *bch, const uint8_t *data, unsigned int len,
const uint8_t *recv_ecc, const uint8_t *calc_ecc,
const unsigned int *syn, unsigned int *errloc)
{
const unsigned int ecc_words = BCH_ECC_WORDS(bch);
unsigned int nbits;
int i, err, nroots;
uint32_t sum;
/* sanity check: make sure data length can be handled */
if (8*len > (bch->n-bch->ecc_bits))
return -EINVAL;
/* if caller does not provide syndromes, compute them */
if (!syn) {
if (!calc_ecc) {
/* compute received data ecc into an internal buffer */
if (!data || !recv_ecc)
return -EINVAL;
encode_bch(bch, data, len, NULL);
} else {
/* load provided calculated ecc */
load_ecc8(bch, bch->ecc_buf, calc_ecc);
}
/* load received ecc or assume it was XORed in calc_ecc */
if (recv_ecc) {
load_ecc8(bch, bch->ecc_buf2, recv_ecc);
/* XOR received and calculated ecc */
for (i = 0, sum = 0; i < (int)ecc_words; i++) {
bch->ecc_buf[i] ^= bch->ecc_buf2[i];
sum |= bch->ecc_buf[i];
}
if (!sum)
/* no error found */
return 0;
}
compute_syndromes(bch, bch->ecc_buf, bch->syn);
syn = bch->syn;
}
err = compute_error_locator_polynomial(bch, syn);
if (err > 0) {
nroots = find_poly_roots(bch, 1, bch->elp, errloc);
if (err != nroots)
err = -1;
}
if (err > 0) {
/* post-process raw error locations for easier correction */
nbits = (len*8)+bch->ecc_bits;
for (i = 0; i < err; i++) {
if (errloc[i] >= nbits) {
err = -1;
break;
}
errloc[i] = nbits-1-errloc[i];
errloc[i] = (errloc[i] & ~7)|(7-(errloc[i] & 7));
}
}
return (err >= 0) ? err : -EBADMSG;
}
EXPORT_SYMBOL_GPL(decode_bch);
/*
* generate Galois field lookup tables
*/
static int build_gf_tables(struct bch_control *bch, unsigned int poly)
{
unsigned int i, x = 1;
const unsigned int k = 1 << deg(poly);
/* primitive polynomial must be of degree m */
if (k != (1u << GF_M(bch)))
return -1;
for (i = 0; i < GF_N(bch); i++) {
bch->a_pow_tab[i] = x;
bch->a_log_tab[x] = i;
if (i && (x == 1))
/* polynomial is not primitive (a^i=1 with 0<i<2^m-1) */
return -1;
x <<= 1;
if (x & k)
x ^= poly;
}
bch->a_pow_tab[GF_N(bch)] = 1;
bch->a_log_tab[0] = 0;
return 0;
}
/*
* compute generator polynomial remainder tables for fast encoding
*/
static void build_mod8_tables(struct bch_control *bch, const uint32_t *g)
{
int i, j, b, d;
uint32_t data, hi, lo, *tab;
const int l = BCH_ECC_WORDS(bch);
const int plen = DIV_ROUND_UP(bch->ecc_bits+1, 32);
const int ecclen = DIV_ROUND_UP(bch->ecc_bits, 32);
memset(bch->mod8_tab, 0, 4*256*l*sizeof(*bch->mod8_tab));
for (i = 0; i < 256; i++) {
/* p(X)=i is a small polynomial of weight <= 8 */
for (b = 0; b < 4; b++) {
/* we want to compute (p(X).X^(8*b+deg(g))) mod g(X) */
tab = bch->mod8_tab + (b*256+i)*l;
data = i << (8*b);
while (data) {
d = deg(data);
/* subtract X^d.g(X) from p(X).X^(8*b+deg(g)) */
data ^= g[0] >> (31-d);
for (j = 0; j < ecclen; j++) {
hi = (d < 31) ? g[j] << (d+1) : 0;
lo = (j+1 < plen) ?
g[j+1] >> (31-d) : 0;
tab[j] ^= hi|lo;
}
}
}
}
}
/*
* build a base for factoring degree 2 polynomials
*/
static int build_deg2_base(struct bch_control *bch)
{
const int m = GF_M(bch);
int i, j, r;
unsigned int sum, x, y, remaining, ak = 0, xi[m];
/* find k s.t. Tr(a^k) = 1 and 0 <= k < m */
for (i = 0; i < m; i++) {
for (j = 0, sum = 0; j < m; j++)
sum ^= a_pow(bch, i*(1 << j));
if (sum) {
ak = bch->a_pow_tab[i];
break;
}
}
/* find xi, i=0..m-1 such that xi^2+xi = a^i+Tr(a^i).a^k */
remaining = m;
memset(xi, 0, sizeof(xi));
for (x = 0; (x <= GF_N(bch)) && remaining; x++) {
y = gf_sqr(bch, x)^x;
for (i = 0; i < 2; i++) {
r = a_log(bch, y);
if (y && (r < m) && !xi[r]) {
bch->xi_tab[r] = x;
xi[r] = 1;
remaining--;
dbg("x%d = %x\n", r, x);
break;
}
y ^= ak;
}
}
/* should not happen but check anyway */
return remaining ? -1 : 0;
}
static void *bch_alloc(size_t size, int *err)
{
void *ptr;
ptr = kmalloc(size, GFP_KERNEL);
if (ptr == NULL)
*err = 1;
return ptr;
}
/*
* compute generator polynomial for given (m,t) parameters.
*/
static uint32_t *compute_generator_polynomial(struct bch_control *bch)
{
const unsigned int m = GF_M(bch);
const unsigned int t = GF_T(bch);
int n, err = 0;
unsigned int i, j, nbits, r, word, *roots;
struct gf_poly *g;
uint32_t *genpoly;
g = bch_alloc(GF_POLY_SZ(m*t), &err);
roots = bch_alloc((bch->n+1)*sizeof(*roots), &err);
genpoly = bch_alloc(DIV_ROUND_UP(m*t+1, 32)*sizeof(*genpoly), &err);
if (err) {
kfree(genpoly);
genpoly = NULL;
goto finish;
}
/* enumerate all roots of g(X) */
memset(roots , 0, (bch->n+1)*sizeof(*roots));
for (i = 0; i < t; i++) {
for (j = 0, r = 2*i+1; j < m; j++) {
roots[r] = 1;
r = mod_s(bch, 2*r);
}
}
/* build generator polynomial g(X) */
g->deg = 0;
g->c[0] = 1;
for (i = 0; i < GF_N(bch); i++) {
if (roots[i]) {
/* multiply g(X) by (X+root) */
r = bch->a_pow_tab[i];
g->c[g->deg+1] = 1;
for (j = g->deg; j > 0; j--)
g->c[j] = gf_mul(bch, g->c[j], r)^g->c[j-1];
g->c[0] = gf_mul(bch, g->c[0], r);
g->deg++;
}
}
/* store left-justified binary representation of g(X) */
n = g->deg+1;
i = 0;
while (n > 0) {
nbits = (n > 32) ? 32 : n;
for (j = 0, word = 0; j < nbits; j++) {
if (g->c[n-1-j])
word |= 1u << (31-j);
}
genpoly[i++] = word;
n -= nbits;
}
bch->ecc_bits = g->deg;
finish:
kfree(g);
kfree(roots);
return genpoly;
}
/**
* init_bch - initialize a BCH encoder/decoder
* @m: Galois field order, should be in the range 5-15
* @t: maximum error correction capability, in bits
* @prim_poly: user-provided primitive polynomial (or 0 to use default)
*
* Returns:
* a newly allocated BCH control structure if successful, NULL otherwise
*
* This initialization can take some time, as lookup tables are built for fast
* encoding/decoding; make sure not to call this function from a time critical
* path. Usually, init_bch() should be called on module/driver init and
* free_bch() should be called to release memory on exit.
*
* You may provide your own primitive polynomial of degree @m in argument
* @prim_poly, or let init_bch() use its default polynomial.
*
* Once init_bch() has successfully returned a pointer to a newly allocated
* BCH control structure, ecc length in bytes is given by member @ecc_bytes of
* the structure.
*/
struct bch_control *init_bch(int m, int t, unsigned int prim_poly)
{
int err = 0;
unsigned int i, words;
uint32_t *genpoly;
struct bch_control *bch = NULL;
const int min_m = 5;
const int max_m = 15;
/* default primitive polynomials */
static const unsigned int prim_poly_tab[] = {
0x25, 0x43, 0x83, 0x11d, 0x211, 0x409, 0x805, 0x1053, 0x201b,
0x402b, 0x8003,
};
#if defined(CONFIG_BCH_CONST_PARAMS)
if ((m != (CONFIG_BCH_CONST_M)) || (t != (CONFIG_BCH_CONST_T))) {
printk(KERN_ERR "bch encoder/decoder was configured to support "
"parameters m=%d, t=%d only!\n",
CONFIG_BCH_CONST_M, CONFIG_BCH_CONST_T);
goto fail;
}
#endif
if ((m < min_m) || (m > max_m))
/*
* values of m greater than 15 are not currently supported;
* supporting m > 15 would require changing table base type
* (uint16_t) and a small patch in matrix transposition
*/
goto fail;
/* sanity checks */
if ((t < 1) || (m*t >= ((1 << m)-1)))
/* invalid t value */
goto fail;
/* select a primitive polynomial for generating GF(2^m) */
if (prim_poly == 0)
prim_poly = prim_poly_tab[m-min_m];
bch = kzalloc(sizeof(*bch), GFP_KERNEL);
if (bch == NULL)
goto fail;
bch->m = m;
bch->t = t;
bch->n = (1 << m)-1;
words = DIV_ROUND_UP(m*t, 32);
bch->ecc_bytes = DIV_ROUND_UP(m*t, 8);
bch->a_pow_tab = bch_alloc((1+bch->n)*sizeof(*bch->a_pow_tab), &err);
bch->a_log_tab = bch_alloc((1+bch->n)*sizeof(*bch->a_log_tab), &err);
bch->mod8_tab = bch_alloc(words*1024*sizeof(*bch->mod8_tab), &err);
bch->ecc_buf = bch_alloc(words*sizeof(*bch->ecc_buf), &err);
bch->ecc_buf2 = bch_alloc(words*sizeof(*bch->ecc_buf2), &err);
bch->xi_tab = bch_alloc(m*sizeof(*bch->xi_tab), &err);
bch->syn = bch_alloc(2*t*sizeof(*bch->syn), &err);
bch->cache = bch_alloc(2*t*sizeof(*bch->cache), &err);
bch->elp = bch_alloc((t+1)*sizeof(struct gf_poly_deg1), &err);
for (i = 0; i < ARRAY_SIZE(bch->poly_2t); i++)
bch->poly_2t[i] = bch_alloc(GF_POLY_SZ(2*t), &err);
if (err)
goto fail;
err = build_gf_tables(bch, prim_poly);
if (err)
goto fail;
/* use generator polynomial for computing encoding tables */
genpoly = compute_generator_polynomial(bch);
if (genpoly == NULL)
goto fail;
build_mod8_tables(bch, genpoly);
kfree(genpoly);
err = build_deg2_base(bch);
if (err)
goto fail;
return bch;
fail:
free_bch(bch);
return NULL;
}
EXPORT_SYMBOL_GPL(init_bch);
/**
* free_bch - free the BCH control structure
* @bch: BCH control structure to release
*/
void free_bch(struct bch_control *bch)
{
unsigned int i;
if (bch) {
kfree(bch->a_pow_tab);
kfree(bch->a_log_tab);
kfree(bch->mod8_tab);
kfree(bch->ecc_buf);
kfree(bch->ecc_buf2);
kfree(bch->xi_tab);
kfree(bch->syn);
kfree(bch->cache);
kfree(bch->elp);
for (i = 0; i < ARRAY_SIZE(bch->poly_2t); i++)
kfree(bch->poly_2t[i]);
kfree(bch);
}
}
EXPORT_SYMBOL_GPL(free_bch);
MODULE_LICENSE("GPL");
MODULE_AUTHOR("Ivan Djelic <ivan.djelic@parrot.com>");
MODULE_DESCRIPTION("Binary BCH encoder/decoder");
Reference in a new issue Copy permalink