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bch.c
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bch.c
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/*
* 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 bch_init 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 bch_encode to compute and store ecc parity bytes to a given buffer.
* Call bch_decode to detect and locate errors in received data.
*
* On systems supporting hw BCH features, intermediate results may be provided
* to bch_decode in order to skip certain steps. See bch_decode() 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)
#define BCH_MAX_M (CONFIG_BCH_CONST_M)
#define BCH_MAX_T (CONFIG_BCH_CONST_T)
#else
#define GF_M(_p) ((_p)->m)
#define GF_T(_p) ((_p)->t)
#define GF_N(_p) ((_p)->n)
#define BCH_MAX_M 15 /* 2KB */
#define BCH_MAX_T 64 /* 64 bit correction */
#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)
#define BCH_ECC_MAX_WORDS DIV_ROUND_UP(BCH_MAX_M * BCH_MAX_T, 32)
#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[]; /* 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];
};
static u8 swap_bits_table[] = {
0x00, 0x80, 0x40, 0xc0, 0x20, 0xa0, 0x60, 0xe0,
0x10, 0x90, 0x50, 0xd0, 0x30, 0xb0, 0x70, 0xf0,
0x08, 0x88, 0x48, 0xc8, 0x28, 0xa8, 0x68, 0xe8,
0x18, 0x98, 0x58, 0xd8, 0x38, 0xb8, 0x78, 0xf8,
0x04, 0x84, 0x44, 0xc4, 0x24, 0xa4, 0x64, 0xe4,
0x14, 0x94, 0x54, 0xd4, 0x34, 0xb4, 0x74, 0xf4,
0x0c, 0x8c, 0x4c, 0xcc, 0x2c, 0xac, 0x6c, 0xec,
0x1c, 0x9c, 0x5c, 0xdc, 0x3c, 0xbc, 0x7c, 0xfc,
0x02, 0x82, 0x42, 0xc2, 0x22, 0xa2, 0x62, 0xe2,
0x12, 0x92, 0x52, 0xd2, 0x32, 0xb2, 0x72, 0xf2,
0x0a, 0x8a, 0x4a, 0xca, 0x2a, 0xaa, 0x6a, 0xea,
0x1a, 0x9a, 0x5a, 0xda, 0x3a, 0xba, 0x7a, 0xfa,
0x06, 0x86, 0x46, 0xc6, 0x26, 0xa6, 0x66, 0xe6,
0x16, 0x96, 0x56, 0xd6, 0x36, 0xb6, 0x76, 0xf6,
0x0e, 0x8e, 0x4e, 0xce, 0x2e, 0xae, 0x6e, 0xee,
0x1e, 0x9e, 0x5e, 0xde, 0x3e, 0xbe, 0x7e, 0xfe,
0x01, 0x81, 0x41, 0xc1, 0x21, 0xa1, 0x61, 0xe1,
0x11, 0x91, 0x51, 0xd1, 0x31, 0xb1, 0x71, 0xf1,
0x09, 0x89, 0x49, 0xc9, 0x29, 0xa9, 0x69, 0xe9,
0x19, 0x99, 0x59, 0xd9, 0x39, 0xb9, 0x79, 0xf9,
0x05, 0x85, 0x45, 0xc5, 0x25, 0xa5, 0x65, 0xe5,
0x15, 0x95, 0x55, 0xd5, 0x35, 0xb5, 0x75, 0xf5,
0x0d, 0x8d, 0x4d, 0xcd, 0x2d, 0xad, 0x6d, 0xed,
0x1d, 0x9d, 0x5d, 0xdd, 0x3d, 0xbd, 0x7d, 0xfd,
0x03, 0x83, 0x43, 0xc3, 0x23, 0xa3, 0x63, 0xe3,
0x13, 0x93, 0x53, 0xd3, 0x33, 0xb3, 0x73, 0xf3,
0x0b, 0x8b, 0x4b, 0xcb, 0x2b, 0xab, 0x6b, 0xeb,
0x1b, 0x9b, 0x5b, 0xdb, 0x3b, 0xbb, 0x7b, 0xfb,
0x07, 0x87, 0x47, 0xc7, 0x27, 0xa7, 0x67, 0xe7,
0x17, 0x97, 0x57, 0xd7, 0x37, 0xb7, 0x77, 0xf7,
0x0f, 0x8f, 0x4f, 0xcf, 0x2f, 0xaf, 0x6f, 0xef,
0x1f, 0x9f, 0x5f, 0xdf, 0x3f, 0xbf, 0x7f, 0xff,
};
static u8 swap_bits(struct bch_control *bch, u8 in)
{
if (!bch->swap_bits)
return in;
return swap_bits_table[in];
}
/*
* same as bch_encode(), but process input data one byte at a time
*/
static void bch_encode_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--) {
u8 tmp = swap_bits(bch, *data++);
p = bch->mod8_tab + (l+1)*(((ecc[0] >> 24)^(tmp)) & 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] = ((u32)swap_bits(bch, src[0]) << 24) |
((u32)swap_bits(bch, src[1]) << 16) |
((u32)swap_bits(bch, src[2]) << 8) |
swap_bits(bch, src[3]);
memcpy(pad, src, BCH_ECC_BYTES(bch)-4*nwords);
dst[nwords] = ((u32)swap_bits(bch, pad[0]) << 24) |
((u32)swap_bits(bch, pad[1]) << 16) |
((u32)swap_bits(bch, pad[2]) << 8) |
swap_bits(bch, 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++ = swap_bits(bch, src[i] >> 24);
*dst++ = swap_bits(bch, src[i] >> 16);
*dst++ = swap_bits(bch, src[i] >> 8);
*dst++ = swap_bits(bch, src[i]);
}
pad[0] = swap_bits(bch, src[nwords] >> 24);
pad[1] = swap_bits(bch, src[nwords] >> 16);
pad[2] = swap_bits(bch, src[nwords] >> 8);
pad[3] = swap_bits(bch, src[nwords]);
memcpy(dst, pad, BCH_ECC_BYTES(bch)-4*nwords);
}
/**
* bch_encode - 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 bch_encode(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[BCH_ECC_MAX_WORDS];
const size_t r_bytes = BCH_ECC_WORDS(bch) * sizeof(*r);
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 (WARN_ON(r_bytes > sizeof(r)))
return;
if (ecc) {
/* load ecc parity bytes into internal 32-bit buffer */
load_ecc8(bch, bch->ecc_buf, ecc);
} else {
memset(bch->ecc_buf, 0, r_bytes);
}
/* process first unaligned data bytes */
m = ((unsigned long)data) & 3;
if (m) {
mlen = (len < (4-m)) ? len : 4-m;
bch_encode_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, r_bytes);
/*
* 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 = cpu_to_be32(*pdata++);
if (bch->swap_bits)
w = (u32)swap_bits(bch, w) |
((u32)swap_bits(bch, w >> 8) << 8) |
((u32)swap_bits(bch, w >> 16) << 16) |
((u32)swap_bits(bch, w >> 24) << 24);
w ^= r[0];
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, r_bytes);
/* process last unaligned bytes */
if (len)
bch_encode_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(bch_encode);
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[BCH_MAX_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;
/* build 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) {