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* Generic binary BCH encoding/decoding library
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* This program is free software; you can redistribute it and/or modify it
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* under the terms of the GNU General Public License version 2 as published by
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* the Free Software Foundation.
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* This program is distributed in the hope that it will be useful, but WITHOUT
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* ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
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* FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for
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* You should have received a copy of the GNU General Public License along with
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* this program; if not, write to the Free Software Foundation, Inc., 51
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* Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
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* Copyright Ā© 2011 Parrot S.A.
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* Author: Ivan Djelic <ivan.djelic@parrot.com>
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* This library provides runtime configurable encoding/decoding of binary
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* Bose-Chaudhuri-Hocquenghem (BCH) codes.
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* Call init_bch to get a pointer to a newly allocated bch_control structure for
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* the given m (Galois field order), t (error correction capability) and
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* (optional) primitive polynomial parameters.
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* Call encode_bch to compute and store ecc parity bytes to a given buffer.
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* Call decode_bch to detect and locate errors in received data.
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* On systems supporting hw BCH features, intermediate results may be provided
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* to decode_bch in order to skip certain steps. See decode_bch() documentation
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* Option CONFIG_BCH_CONST_PARAMS can be used to force fixed values of
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* parameters m and t; thus allowing extra compiler optimizations and providing
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* better (up to 2x) encoding performance. Using this option makes sense when
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* (m,t) are fixed and known in advance, e.g. when using BCH error correction
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* on a particular NAND flash device.
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* Algorithmic details:
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* Encoding is performed by processing 32 input bits in parallel, using 4
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* remainder lookup tables.
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* The final stage of decoding involves the following internal steps:
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* a. Syndrome computation
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* b. Error locator polynomial computation using Berlekamp-Massey algorithm
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* c. Error locator root finding (by far the most expensive step)
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* In this implementation, step c is not performed using the usual Chien search.
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* Instead, an alternative approach described in [1] is used. It consists in
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* factoring the error locator polynomial using the Berlekamp Trace algorithm
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* (BTA) down to a certain degree (4), after which ad hoc low-degree polynomial
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* solving techniques [2] are used. The resulting algorithm, called BTZ, yields
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* much better performance than Chien search for usual (m,t) values (typically
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* m >= 13, t < 32, see [1]).
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* [1] B. Biswas, V. Herbert. Efficient root finding of polynomials over fields
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* of characteristic 2, in: Western European Workshop on Research in Cryptology
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* - WEWoRC 2009, Graz, Austria, LNCS, Springer, July 2009, to appear.
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* [2] [Zin96] V.A. Zinoviev. On the solution of equations of degree 10 over
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* finite fields GF(2^q). In Rapport de recherche INRIA no 2829, 1996.
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#include <linux/kernel.h>
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#include <linux/errno.h>
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#include <linux/init.h>
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#include <linux/module.h>
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#include <linux/slab.h>
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#include <linux/bitops.h>
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#include <asm/byteorder.h>
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#include <linux/bch.h>
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#if defined(CONFIG_BCH_CONST_PARAMS)
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#define GF_M(_p) (CONFIG_BCH_CONST_M)
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#define GF_T(_p) (CONFIG_BCH_CONST_T)
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#define GF_N(_p) ((1 << (CONFIG_BCH_CONST_M))-1)
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#define GF_M(_p) ((_p)->m)
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#define GF_T(_p) ((_p)->t)
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#define GF_N(_p) ((_p)->n)
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#define BCH_ECC_WORDS(_p) DIV_ROUND_UP(GF_M(_p)*GF_T(_p), 32)
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#define BCH_ECC_BYTES(_p) DIV_ROUND_UP(GF_M(_p)*GF_T(_p), 8)
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#define dbg(_fmt, args...) do {} while (0)
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* represent a polynomial over GF(2^m)
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unsigned int deg; /* polynomial degree */
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unsigned int c[0]; /* polynomial terms */
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/* given its degree, compute a polynomial size in bytes */
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#define GF_POLY_SZ(_d) (sizeof(struct gf_poly)+((_d)+1)*sizeof(unsigned int))
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/* polynomial of degree 1 */
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struct gf_poly_deg1 {
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* same as encode_bch(), but process input data one byte at a time
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static void encode_bch_unaligned(struct bch_control *bch,
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const unsigned char *data, unsigned int len,
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const int l = BCH_ECC_WORDS(bch)-1;
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p = bch->mod8_tab + (l+1)*(((ecc[0] >> 24)^(*data++)) & 0xff);
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for (i = 0; i < l; i++)
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ecc[i] = ((ecc[i] << 8)|(ecc[i+1] >> 24))^(*p++);
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ecc[l] = (ecc[l] << 8)^(*p);
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* convert ecc bytes to aligned, zero-padded 32-bit ecc words
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static void load_ecc8(struct bch_control *bch, uint32_t *dst,
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uint8_t pad[4] = {0, 0, 0, 0};
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unsigned int i, nwords = BCH_ECC_WORDS(bch)-1;
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for (i = 0; i < nwords; i++, src += 4)
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dst[i] = (src[0] << 24)|(src[1] << 16)|(src[2] << 8)|src[3];
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memcpy(pad, src, BCH_ECC_BYTES(bch)-4*nwords);
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dst[nwords] = (pad[0] << 24)|(pad[1] << 16)|(pad[2] << 8)|pad[3];
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* convert 32-bit ecc words to ecc bytes
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static void store_ecc8(struct bch_control *bch, uint8_t *dst,
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unsigned int i, nwords = BCH_ECC_WORDS(bch)-1;
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for (i = 0; i < nwords; i++) {
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*dst++ = (src[i] >> 24);
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*dst++ = (src[i] >> 16) & 0xff;
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*dst++ = (src[i] >> 8) & 0xff;
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*dst++ = (src[i] >> 0) & 0xff;
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pad[0] = (src[nwords] >> 24);
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pad[1] = (src[nwords] >> 16) & 0xff;
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pad[2] = (src[nwords] >> 8) & 0xff;
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pad[3] = (src[nwords] >> 0) & 0xff;
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memcpy(dst, pad, BCH_ECC_BYTES(bch)-4*nwords);
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* encode_bch - calculate BCH ecc parity of data
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* @bch: BCH control structure
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* @data: data to encode
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* @len: data length in bytes
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* @ecc: ecc parity data, must be initialized by caller
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* The @ecc parity array is used both as input and output parameter, in order to
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* allow incremental computations. It should be of the size indicated by member
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* @ecc_bytes of @bch, and should be initialized to 0 before the first call.
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* The exact number of computed ecc parity bits is given by member @ecc_bits of
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* @bch; it may be less than m*t for large values of t.
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void encode_bch(struct bch_control *bch, const uint8_t *data,
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unsigned int len, uint8_t *ecc)
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const unsigned int l = BCH_ECC_WORDS(bch)-1;
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unsigned int i, mlen;
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const uint32_t * const tab0 = bch->mod8_tab;
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const uint32_t * const tab1 = tab0 + 256*(l+1);
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const uint32_t * const tab2 = tab1 + 256*(l+1);
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const uint32_t * const tab3 = tab2 + 256*(l+1);
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const uint32_t *pdata, *p0, *p1, *p2, *p3;
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/* load ecc parity bytes into internal 32-bit buffer */
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load_ecc8(bch, bch->ecc_buf, ecc);
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memset(bch->ecc_buf, 0, sizeof(r));
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/* process first unaligned data bytes */
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m = ((unsigned long)data) & 3;
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mlen = (len < (4-m)) ? len : 4-m;
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encode_bch_unaligned(bch, data, mlen, bch->ecc_buf);
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/* process 32-bit aligned data words */
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pdata = (uint32_t *)data;
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memcpy(r, bch->ecc_buf, sizeof(r));
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* split each 32-bit word into 4 polynomials of weight 8 as follows:
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* 31 ...24 23 ...16 15 ... 8 7 ... 0
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* xxxxxxxx yyyyyyyy zzzzzzzz tttttttt
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* tttttttt mod g = r0 (precomputed)
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* zzzzzzzz 00000000 mod g = r1 (precomputed)
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* yyyyyyyy 00000000 00000000 mod g = r2 (precomputed)
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* xxxxxxxx 00000000 00000000 00000000 mod g = r3 (precomputed)
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* xxxxxxxx yyyyyyyy zzzzzzzz tttttttt mod g = r0^r1^r2^r3
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/* input data is read in big-endian format */
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w = r[0]^cpu_to_be32(*pdata++);
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p0 = tab0 + (l+1)*((w >> 0) & 0xff);
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p1 = tab1 + (l+1)*((w >> 8) & 0xff);
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p2 = tab2 + (l+1)*((w >> 16) & 0xff);
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p3 = tab3 + (l+1)*((w >> 24) & 0xff);
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for (i = 0; i < l; i++)
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r[i] = r[i+1]^p0[i]^p1[i]^p2[i]^p3[i];
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r[l] = p0[l]^p1[l]^p2[l]^p3[l];
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memcpy(bch->ecc_buf, r, sizeof(r));
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/* process last unaligned bytes */
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encode_bch_unaligned(bch, data, len, bch->ecc_buf);
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/* store ecc parity bytes into original parity buffer */
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store_ecc8(bch, ecc, bch->ecc_buf);
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EXPORT_SYMBOL_GPL(encode_bch);
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static inline int modulo(struct bch_control *bch, unsigned int v)
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const unsigned int n = GF_N(bch);
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v = (v & n) + (v >> GF_M(bch));
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* shorter and faster modulo function, only works when v < 2N.
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static inline int mod_s(struct bch_control *bch, unsigned int v)
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const unsigned int n = GF_N(bch);
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return (v < n) ? v : v-n;
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static inline int deg(unsigned int poly)
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/* polynomial degree is the most-significant bit index */
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static inline int parity(unsigned int x)
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* public domain code snippet, lifted from
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* http://www-graphics.stanford.edu/~seander/bithacks.html
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x = (x & 0x11111111U) * 0x11111111U;
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return (x >> 28) & 1;
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/* Galois field basic operations: multiply, divide, inverse, etc. */
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static inline unsigned int gf_mul(struct bch_control *bch, unsigned int a,
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return (a && b) ? bch->a_pow_tab[mod_s(bch, bch->a_log_tab[a]+
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bch->a_log_tab[b])] : 0;
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static inline unsigned int gf_sqr(struct bch_control *bch, unsigned int a)
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return a ? bch->a_pow_tab[mod_s(bch, 2*bch->a_log_tab[a])] : 0;
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static inline unsigned int gf_div(struct bch_control *bch, unsigned int a,
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return a ? bch->a_pow_tab[mod_s(bch, bch->a_log_tab[a]+
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GF_N(bch)-bch->a_log_tab[b])] : 0;
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static inline unsigned int gf_inv(struct bch_control *bch, unsigned int a)
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return bch->a_pow_tab[GF_N(bch)-bch->a_log_tab[a]];
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static inline unsigned int a_pow(struct bch_control *bch, int i)
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return bch->a_pow_tab[modulo(bch, i)];
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static inline int a_log(struct bch_control *bch, unsigned int x)
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return bch->a_log_tab[x];
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static inline int a_ilog(struct bch_control *bch, unsigned int x)
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return mod_s(bch, GF_N(bch)-bch->a_log_tab[x]);
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* compute 2t syndromes of ecc polynomial, i.e. ecc(a^j) for j=1..2t
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static void compute_syndromes(struct bch_control *bch, uint32_t *ecc,
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const int t = GF_T(bch);
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/* make sure extra bits in last ecc word are cleared */
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m = ((unsigned int)s) & 31;
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ecc[s/32] &= ~((1u << (32-m))-1);
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memset(syn, 0, 2*t*sizeof(*syn));
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/* compute v(a^j) for j=1 .. 2t-1 */
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for (j = 0; j < 2*t; j += 2)
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syn[j] ^= a_pow(bch, (j+1)*(i+s));
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/* v(a^(2j)) = v(a^j)^2 */
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for (j = 0; j < t; j++)
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syn[2*j+1] = gf_sqr(bch, syn[j]);
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static void gf_poly_copy(struct gf_poly *dst, struct gf_poly *src)
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memcpy(dst, src, GF_POLY_SZ(src->deg));
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static int compute_error_locator_polynomial(struct bch_control *bch,
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const unsigned int *syn)
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const unsigned int t = GF_T(bch);
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const unsigned int n = GF_N(bch);
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unsigned int i, j, tmp, l, pd = 1, d = syn[0];
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struct gf_poly *elp = bch->elp;
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struct gf_poly *pelp = bch->poly_2t[0];
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struct gf_poly *elp_copy = bch->poly_2t[1];
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memset(pelp, 0, GF_POLY_SZ(2*t));
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memset(elp, 0, GF_POLY_SZ(2*t));
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/* use simplified binary Berlekamp-Massey algorithm */
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for (i = 0; (i < t) && (elp->deg <= t); i++) {
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gf_poly_copy(elp_copy, elp);
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/* e[i+1](X) = e[i](X)+di*dp^-1*X^2(i-p)*e[p](X) */
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tmp = a_log(bch, d)+n-a_log(bch, pd);
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for (j = 0; j <= pelp->deg; j++) {
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l = a_log(bch, pelp->c[j]);
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elp->c[j+k] ^= a_pow(bch, tmp+l);
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/* compute l[i+1] = max(l[i]->c[l[p]+2*(i-p]) */
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if (tmp > elp->deg) {
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gf_poly_copy(pelp, elp_copy);
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/* di+1 = S(2i+3)+elp[i+1].1*S(2i+2)+...+elp[i+1].lS(2i+3-l) */
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for (j = 1; j <= elp->deg; j++)
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d ^= gf_mul(bch, elp->c[j], syn[2*i+2-j]);
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dbg("elp=%s\n", gf_poly_str(elp));
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return (elp->deg > t) ? -1 : (int)elp->deg;
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* solve a m x m linear system in GF(2) with an expected number of solutions,
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* and return the number of found solutions
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static int solve_linear_system(struct bch_control *bch, unsigned int *rows,
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unsigned int *sol, int nsol)
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const int m = GF_M(bch);
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unsigned int tmp, mask;
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int rem, c, r, p, k, param[m];
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/* Gaussian elimination */
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for (c = 0; c < m; c++) {
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/* find suitable row for elimination */
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for (r = p; r < m; r++) {
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if (rows[r] & mask) {
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/* perform elimination on remaining rows */
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for (r = rem; r < m; r++) {
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/* elimination not needed, store defective row index */
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/* rewrite system, inserting fake parameter rows */
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for (r = m-1; r >= 0; r--) {
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if ((r > m-1-k) && rows[r])
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/* system has no solution */
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rows[r] = (p && (r == param[p-1])) ?
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p--, 1u << (m-r) : rows[r-p];
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if (nsol != (1 << k))
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/* unexpected number of solutions */
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for (p = 0; p < nsol; p++) {
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/* set parameters for p-th solution */
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for (c = 0; c < k; c++)
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rows[param[c]] = (rows[param[c]] & ~1)|((p >> c) & 1);
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/* compute unique solution */
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for (r = m-1; r >= 0; r--) {
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mask = rows[r] & (tmp|1);
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tmp |= parity(mask) << (m-r);
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* this function builds and solves a linear system for finding roots of a degree
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* 4 affine monic polynomial X^4+aX^2+bX+c over GF(2^m).
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static int find_affine4_roots(struct bch_control *bch, unsigned int a,
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unsigned int b, unsigned int c,
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const int m = GF_M(bch);
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unsigned int mask = 0xff, t, rows[16] = {0,};
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/* buid linear system to solve X^4+aX^2+bX+c = 0 */
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for (i = 0; i < m; i++) {
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rows[i+1] = bch->a_pow_tab[4*i]^
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(a ? bch->a_pow_tab[mod_s(bch, k)] : 0)^
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(b ? bch->a_pow_tab[mod_s(bch, j)] : 0);
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* transpose 16x16 matrix before passing it to linear solver
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* warning: this code assumes m < 16
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for (j = 8; j != 0; j >>= 1, mask ^= (mask << j)) {
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for (k = 0; k < 16; k = (k+j+1) & ~j) {
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t = ((rows[k] >> j)^rows[k+j]) & mask;
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return solve_linear_system(bch, rows, roots, 4);
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* compute root r of a degree 1 polynomial over GF(2^m) (returned as log(1/r))
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static int find_poly_deg1_roots(struct bch_control *bch, struct gf_poly *poly,
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/* poly[X] = bX+c with c!=0, root=c/b */
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roots[n++] = mod_s(bch, GF_N(bch)-bch->a_log_tab[poly->c[0]]+
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bch->a_log_tab[poly->c[1]]);
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* compute roots of a degree 2 polynomial over GF(2^m)
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static int find_poly_deg2_roots(struct bch_control *bch, struct gf_poly *poly,
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int n = 0, i, l0, l1, l2;
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unsigned int u, v, r;
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if (poly->c[0] && poly->c[1]) {
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l0 = bch->a_log_tab[poly->c[0]];
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l1 = bch->a_log_tab[poly->c[1]];
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l2 = bch->a_log_tab[poly->c[2]];
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/* using z=a/bX, transform aX^2+bX+c into z^2+z+u (u=ac/b^2) */
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u = a_pow(bch, l0+l2+2*(GF_N(bch)-l1));
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* let u = sum(li.a^i) i=0..m-1; then compute r = sum(li.xi):
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* r^2+r = sum(li.(xi^2+xi)) = sum(li.(a^i+Tr(a^i).a^k)) =
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* u + sum(li.Tr(a^i).a^k) = u+a^k.Tr(sum(li.a^i)) = u+a^k.Tr(u)
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* i.e. r and r+1 are roots iff Tr(u)=0
588
if ((gf_sqr(bch, r)^r) == u) {
589
/* reverse z=a/bX transformation and compute log(1/r) */
590
roots[n++] = modulo(bch, 2*GF_N(bch)-l1-
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bch->a_log_tab[r]+l2);
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roots[n++] = modulo(bch, 2*GF_N(bch)-l1-
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bch->a_log_tab[r^1]+l2);
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* compute roots of a degree 3 polynomial over GF(2^m)
602
static int find_poly_deg3_roots(struct bch_control *bch, struct gf_poly *poly,
606
unsigned int a, b, c, a2, b2, c2, e3, tmp[4];
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/* transform polynomial into monic X^3 + a2X^2 + b2X + c2 */
611
c2 = gf_div(bch, poly->c[0], e3);
612
b2 = gf_div(bch, poly->c[1], e3);
613
a2 = gf_div(bch, poly->c[2], e3);
615
/* (X+a2)(X^3+a2X^2+b2X+c2) = X^4+aX^2+bX+c (affine) */
616
c = gf_mul(bch, a2, c2); /* c = a2c2 */
617
b = gf_mul(bch, a2, b2)^c2; /* b = a2b2 + c2 */
618
a = gf_sqr(bch, a2)^b2; /* a = a2^2 + b2 */
620
/* find the 4 roots of this affine polynomial */
621
if (find_affine4_roots(bch, a, b, c, tmp) == 4) {
622
/* remove a2 from final list of roots */
623
for (i = 0; i < 4; i++) {
625
roots[n++] = a_ilog(bch, tmp[i]);
633
* compute roots of a degree 4 polynomial over GF(2^m)
635
static int find_poly_deg4_roots(struct bch_control *bch, struct gf_poly *poly,
639
unsigned int a, b, c, d, e = 0, f, a2, b2, c2, e4;
644
/* transform polynomial into monic X^4 + aX^3 + bX^2 + cX + d */
646
d = gf_div(bch, poly->c[0], e4);
647
c = gf_div(bch, poly->c[1], e4);
648
b = gf_div(bch, poly->c[2], e4);
649
a = gf_div(bch, poly->c[3], e4);
651
/* use Y=1/X transformation to get an affine polynomial */
653
/* first, eliminate cX by using z=X+e with ae^2+c=0 */
655
/* compute e such that e^2 = c/a */
656
f = gf_div(bch, c, a);
658
l += (l & 1) ? GF_N(bch) : 0;
661
* use transformation z=X+e:
662
* z^4+e^4 + a(z^3+ez^2+e^2z+e^3) + b(z^2+e^2) +cz+ce+d
663
* z^4 + az^3 + (ae+b)z^2 + (ae^2+c)z+e^4+be^2+ae^3+ce+d
664
* z^4 + az^3 + (ae+b)z^2 + e^4+be^2+d
665
* z^4 + az^3 + b'z^2 + d'
667
d = a_pow(bch, 2*l)^gf_mul(bch, b, f)^d;
668
b = gf_mul(bch, a, e)^b;
670
/* now, use Y=1/X to get Y^4 + b/dY^2 + a/dY + 1/d */
672
/* assume all roots have multiplicity 1 */
676
b2 = gf_div(bch, a, d);
677
a2 = gf_div(bch, b, d);
679
/* polynomial is already affine */
684
/* find the 4 roots of this affine polynomial */
685
if (find_affine4_roots(bch, a2, b2, c2, roots) == 4) {
686
for (i = 0; i < 4; i++) {
687
/* post-process roots (reverse transformations) */
688
f = a ? gf_inv(bch, roots[i]) : roots[i];
689
roots[i] = a_ilog(bch, f^e);
697
* build monic, log-based representation of a polynomial
699
static void gf_poly_logrep(struct bch_control *bch,
700
const struct gf_poly *a, int *rep)
702
int i, d = a->deg, l = GF_N(bch)-a_log(bch, a->c[a->deg]);
704
/* represent 0 values with -1; warning, rep[d] is not set to 1 */
705
for (i = 0; i < d; i++)
706
rep[i] = a->c[i] ? mod_s(bch, a_log(bch, a->c[i])+l) : -1;
710
* compute polynomial Euclidean division remainder in GF(2^m)[X]
712
static void gf_poly_mod(struct bch_control *bch, struct gf_poly *a,
713
const struct gf_poly *b, int *rep)
716
unsigned int i, j, *c = a->c;
717
const unsigned int d = b->deg;
722
/* reuse or compute log representation of denominator */
725
gf_poly_logrep(bch, b, rep);
728
for (j = a->deg; j >= d; j--) {
730
la = a_log(bch, c[j]);
732
for (i = 0; i < d; i++, p++) {
735
c[p] ^= bch->a_pow_tab[mod_s(bch,
741
while (!c[a->deg] && a->deg)
746
* compute polynomial Euclidean division quotient in GF(2^m)[X]
748
static void gf_poly_div(struct bch_control *bch, struct gf_poly *a,
749
const struct gf_poly *b, struct gf_poly *q)
751
if (a->deg >= b->deg) {
752
q->deg = a->deg-b->deg;
753
/* compute a mod b (modifies a) */
754
gf_poly_mod(bch, a, b, NULL);
755
/* quotient is stored in upper part of polynomial a */
756
memcpy(q->c, &a->c[b->deg], (1+q->deg)*sizeof(unsigned int));
764
* compute polynomial GCD (Greatest Common Divisor) in GF(2^m)[X]
766
static struct gf_poly *gf_poly_gcd(struct bch_control *bch, struct gf_poly *a,
771
dbg("gcd(%s,%s)=", gf_poly_str(a), gf_poly_str(b));
773
if (a->deg < b->deg) {
780
gf_poly_mod(bch, a, b, NULL);
786
dbg("%s\n", gf_poly_str(a));
792
* Given a polynomial f and an integer k, compute Tr(a^kX) mod f
793
* This is used in Berlekamp Trace algorithm for splitting polynomials
795
static void compute_trace_bk_mod(struct bch_control *bch, int k,
796
const struct gf_poly *f, struct gf_poly *z,
799
const int m = GF_M(bch);
802
/* z contains z^2j mod f */
805
z->c[1] = bch->a_pow_tab[k];
808
memset(out, 0, GF_POLY_SZ(f->deg));
810
/* compute f log representation only once */
811
gf_poly_logrep(bch, f, bch->cache);
813
for (i = 0; i < m; i++) {
814
/* add a^(k*2^i)(z^(2^i) mod f) and compute (z^(2^i) mod f)^2 */
815
for (j = z->deg; j >= 0; j--) {
816
out->c[j] ^= z->c[j];
817
z->c[2*j] = gf_sqr(bch, z->c[j]);
820
if (z->deg > out->deg)
825
/* z^(2(i+1)) mod f = (z^(2^i) mod f)^2 mod f */
826
gf_poly_mod(bch, z, f, bch->cache);
829
while (!out->c[out->deg] && out->deg)
832
dbg("Tr(a^%d.X) mod f = %s\n", k, gf_poly_str(out));
836
* factor a polynomial using Berlekamp Trace algorithm (BTA)
838
static void factor_polynomial(struct bch_control *bch, int k, struct gf_poly *f,
839
struct gf_poly **g, struct gf_poly **h)
841
struct gf_poly *f2 = bch->poly_2t[0];
842
struct gf_poly *q = bch->poly_2t[1];
843
struct gf_poly *tk = bch->poly_2t[2];
844
struct gf_poly *z = bch->poly_2t[3];
847
dbg("factoring %s...\n", gf_poly_str(f));
852
/* tk = Tr(a^k.X) mod f */
853
compute_trace_bk_mod(bch, k, f, z, tk);
856
/* compute g = gcd(f, tk) (destructive operation) */
858
gcd = gf_poly_gcd(bch, f2, tk);
859
if (gcd->deg < f->deg) {
860
/* compute h=f/gcd(f,tk); this will modify f and q */
861
gf_poly_div(bch, f, gcd, q);
862
/* store g and h in-place (clobbering f) */
863
*h = &((struct gf_poly_deg1 *)f)[gcd->deg].poly;
864
gf_poly_copy(*g, gcd);
871
* find roots of a polynomial, using BTZ algorithm; see the beginning of this
874
static int find_poly_roots(struct bch_control *bch, unsigned int k,
875
struct gf_poly *poly, unsigned int *roots)
878
struct gf_poly *f1, *f2;
881
/* handle low degree polynomials with ad hoc techniques */
883
cnt = find_poly_deg1_roots(bch, poly, roots);
886
cnt = find_poly_deg2_roots(bch, poly, roots);
889
cnt = find_poly_deg3_roots(bch, poly, roots);
892
cnt = find_poly_deg4_roots(bch, poly, roots);
895
/* factor polynomial using Berlekamp Trace Algorithm (BTA) */
897
if (poly->deg && (k <= GF_M(bch))) {
898
factor_polynomial(bch, k, poly, &f1, &f2);
900
cnt += find_poly_roots(bch, k+1, f1, roots);
902
cnt += find_poly_roots(bch, k+1, f2, roots+cnt);
909
#if defined(USE_CHIEN_SEARCH)
911
* exhaustive root search (Chien) implementation - not used, included only for
912
* reference/comparison tests
914
static int chien_search(struct bch_control *bch, unsigned int len,
915
struct gf_poly *p, unsigned int *roots)
918
unsigned int i, j, syn, syn0, count = 0;
919
const unsigned int k = 8*len+bch->ecc_bits;
921
/* use a log-based representation of polynomial */
922
gf_poly_logrep(bch, p, bch->cache);
923
bch->cache[p->deg] = 0;
924
syn0 = gf_div(bch, p->c[0], p->c[p->deg]);
926
for (i = GF_N(bch)-k+1; i <= GF_N(bch); i++) {
927
/* compute elp(a^i) */
928
for (j = 1, syn = syn0; j <= p->deg; j++) {
931
syn ^= a_pow(bch, m+j*i);
934
roots[count++] = GF_N(bch)-i;
939
return (count == p->deg) ? count : 0;
941
#define find_poly_roots(_p, _k, _elp, _loc) chien_search(_p, len, _elp, _loc)
942
#endif /* USE_CHIEN_SEARCH */
945
* decode_bch - decode received codeword and find bit error locations
946
* @bch: BCH control structure
947
* @data: received data, ignored if @calc_ecc is provided
948
* @len: data length in bytes, must always be provided
949
* @recv_ecc: received ecc, if NULL then assume it was XORed in @calc_ecc
950
* @calc_ecc: calculated ecc, if NULL then calc_ecc is computed from @data
951
* @syn: hw computed syndrome data (if NULL, syndrome is calculated)
952
* @errloc: output array of error locations
955
* The number of errors found, or -EBADMSG if decoding failed, or -EINVAL if
956
* invalid parameters were provided
958
* Depending on the available hw BCH support and the need to compute @calc_ecc
959
* separately (using encode_bch()), this function should be called with one of
960
* the following parameter configurations -
962
* by providing @data and @recv_ecc only:
963
* decode_bch(@bch, @data, @len, @recv_ecc, NULL, NULL, @errloc)
965
* by providing @recv_ecc and @calc_ecc:
966
* decode_bch(@bch, NULL, @len, @recv_ecc, @calc_ecc, NULL, @errloc)
968
* by providing ecc = recv_ecc XOR calc_ecc:
969
* decode_bch(@bch, NULL, @len, NULL, ecc, NULL, @errloc)
971
* by providing syndrome results @syn:
972
* decode_bch(@bch, NULL, @len, NULL, NULL, @syn, @errloc)
974
* Once decode_bch() has successfully returned with a positive value, error
975
* locations returned in array @errloc should be interpreted as follows -
977
* if (errloc[n] >= 8*len), then n-th error is located in ecc (no need for
980
* if (errloc[n] < 8*len), then n-th error is located in data and can be
981
* corrected with statement data[errloc[n]/8] ^= 1 << (errloc[n] % 8);
983
* Note that this function does not perform any data correction by itself, it
984
* merely indicates error locations.
986
int decode_bch(struct bch_control *bch, const uint8_t *data, unsigned int len,
987
const uint8_t *recv_ecc, const uint8_t *calc_ecc,
988
const unsigned int *syn, unsigned int *errloc)
990
const unsigned int ecc_words = BCH_ECC_WORDS(bch);
995
/* sanity check: make sure data length can be handled */
996
if (8*len > (bch->n-bch->ecc_bits))
999
/* if caller does not provide syndromes, compute them */
1002
/* compute received data ecc into an internal buffer */
1003
if (!data || !recv_ecc)
1005
encode_bch(bch, data, len, NULL);
1007
/* load provided calculated ecc */
1008
load_ecc8(bch, bch->ecc_buf, calc_ecc);
1010
/* load received ecc or assume it was XORed in calc_ecc */
1012
load_ecc8(bch, bch->ecc_buf2, recv_ecc);
1013
/* XOR received and calculated ecc */
1014
for (i = 0, sum = 0; i < (int)ecc_words; i++) {
1015
bch->ecc_buf[i] ^= bch->ecc_buf2[i];
1016
sum |= bch->ecc_buf[i];
1019
/* no error found */
1022
compute_syndromes(bch, bch->ecc_buf, bch->syn);
1026
err = compute_error_locator_polynomial(bch, syn);
1028
nroots = find_poly_roots(bch, 1, bch->elp, errloc);
1033
/* post-process raw error locations for easier correction */
1034
nbits = (len*8)+bch->ecc_bits;
1035
for (i = 0; i < err; i++) {
1036
if (errloc[i] >= nbits) {
1040
errloc[i] = nbits-1-errloc[i];
1041
errloc[i] = (errloc[i] & ~7)|(7-(errloc[i] & 7));
1044
return (err >= 0) ? err : -EBADMSG;
1046
EXPORT_SYMBOL_GPL(decode_bch);
1049
* generate Galois field lookup tables
1051
static int build_gf_tables(struct bch_control *bch, unsigned int poly)
1053
unsigned int i, x = 1;
1054
const unsigned int k = 1 << deg(poly);
1056
/* primitive polynomial must be of degree m */
1057
if (k != (1u << GF_M(bch)))
1060
for (i = 0; i < GF_N(bch); i++) {
1061
bch->a_pow_tab[i] = x;
1062
bch->a_log_tab[x] = i;
1064
/* polynomial is not primitive (a^i=1 with 0<i<2^m-1) */
1070
bch->a_pow_tab[GF_N(bch)] = 1;
1071
bch->a_log_tab[0] = 0;
1077
* compute generator polynomial remainder tables for fast encoding
1079
static void build_mod8_tables(struct bch_control *bch, const uint32_t *g)
1082
uint32_t data, hi, lo, *tab;
1083
const int l = BCH_ECC_WORDS(bch);
1084
const int plen = DIV_ROUND_UP(bch->ecc_bits+1, 32);
1085
const int ecclen = DIV_ROUND_UP(bch->ecc_bits, 32);
1087
memset(bch->mod8_tab, 0, 4*256*l*sizeof(*bch->mod8_tab));
1089
for (i = 0; i < 256; i++) {
1090
/* p(X)=i is a small polynomial of weight <= 8 */
1091
for (b = 0; b < 4; b++) {
1092
/* we want to compute (p(X).X^(8*b+deg(g))) mod g(X) */
1093
tab = bch->mod8_tab + (b*256+i)*l;
1097
/* subtract X^d.g(X) from p(X).X^(8*b+deg(g)) */
1098
data ^= g[0] >> (31-d);
1099
for (j = 0; j < ecclen; j++) {
1100
hi = (d < 31) ? g[j] << (d+1) : 0;
1102
g[j+1] >> (31-d) : 0;
1111
* build a base for factoring degree 2 polynomials
1113
static int build_deg2_base(struct bch_control *bch)
1115
const int m = GF_M(bch);
1117
unsigned int sum, x, y, remaining, ak = 0, xi[m];
1119
/* find k s.t. Tr(a^k) = 1 and 0 <= k < m */
1120
for (i = 0; i < m; i++) {
1121
for (j = 0, sum = 0; j < m; j++)
1122
sum ^= a_pow(bch, i*(1 << j));
1125
ak = bch->a_pow_tab[i];
1129
/* find xi, i=0..m-1 such that xi^2+xi = a^i+Tr(a^i).a^k */
1131
memset(xi, 0, sizeof(xi));
1133
for (x = 0; (x <= GF_N(bch)) && remaining; x++) {
1134
y = gf_sqr(bch, x)^x;
1135
for (i = 0; i < 2; i++) {
1137
if (y && (r < m) && !xi[r]) {
1141
dbg("x%d = %x\n", r, x);
1147
/* should not happen but check anyway */
1148
return remaining ? -1 : 0;
1151
static void *bch_alloc(size_t size, int *err)
1155
ptr = kmalloc(size, GFP_KERNEL);
1162
* compute generator polynomial for given (m,t) parameters.
1164
static uint32_t *compute_generator_polynomial(struct bch_control *bch)
1166
const unsigned int m = GF_M(bch);
1167
const unsigned int t = GF_T(bch);
1169
unsigned int i, j, nbits, r, word, *roots;
1173
g = bch_alloc(GF_POLY_SZ(m*t), &err);
1174
roots = bch_alloc((bch->n+1)*sizeof(*roots), &err);
1175
genpoly = bch_alloc(DIV_ROUND_UP(m*t+1, 32)*sizeof(*genpoly), &err);
1183
/* enumerate all roots of g(X) */
1184
memset(roots , 0, (bch->n+1)*sizeof(*roots));
1185
for (i = 0; i < t; i++) {
1186
for (j = 0, r = 2*i+1; j < m; j++) {
1188
r = mod_s(bch, 2*r);
1191
/* build generator polynomial g(X) */
1194
for (i = 0; i < GF_N(bch); i++) {
1196
/* multiply g(X) by (X+root) */
1197
r = bch->a_pow_tab[i];
1199
for (j = g->deg; j > 0; j--)
1200
g->c[j] = gf_mul(bch, g->c[j], r)^g->c[j-1];
1202
g->c[0] = gf_mul(bch, g->c[0], r);
1206
/* store left-justified binary representation of g(X) */
1211
nbits = (n > 32) ? 32 : n;
1212
for (j = 0, word = 0; j < nbits; j++) {
1214
word |= 1u << (31-j);
1216
genpoly[i++] = word;
1219
bch->ecc_bits = g->deg;
1229
* init_bch - initialize a BCH encoder/decoder
1230
* @m: Galois field order, should be in the range 5-15
1231
* @t: maximum error correction capability, in bits
1232
* @prim_poly: user-provided primitive polynomial (or 0 to use default)
1235
* a newly allocated BCH control structure if successful, NULL otherwise
1237
* This initialization can take some time, as lookup tables are built for fast
1238
* encoding/decoding; make sure not to call this function from a time critical
1239
* path. Usually, init_bch() should be called on module/driver init and
1240
* free_bch() should be called to release memory on exit.
1242
* You may provide your own primitive polynomial of degree @m in argument
1243
* @prim_poly, or let init_bch() use its default polynomial.
1245
* Once init_bch() has successfully returned a pointer to a newly allocated
1246
* BCH control structure, ecc length in bytes is given by member @ecc_bytes of
1249
struct bch_control *init_bch(int m, int t, unsigned int prim_poly)
1252
unsigned int i, words;
1254
struct bch_control *bch = NULL;
1256
const int min_m = 5;
1257
const int max_m = 15;
1259
/* default primitive polynomials */
1260
static const unsigned int prim_poly_tab[] = {
1261
0x25, 0x43, 0x83, 0x11d, 0x211, 0x409, 0x805, 0x1053, 0x201b,
1265
#if defined(CONFIG_BCH_CONST_PARAMS)
1266
if ((m != (CONFIG_BCH_CONST_M)) || (t != (CONFIG_BCH_CONST_T))) {
1267
printk(KERN_ERR "bch encoder/decoder was configured to support "
1268
"parameters m=%d, t=%d only!\n",
1269
CONFIG_BCH_CONST_M, CONFIG_BCH_CONST_T);
1273
if ((m < min_m) || (m > max_m))
1275
* values of m greater than 15 are not currently supported;
1276
* supporting m > 15 would require changing table base type
1277
* (uint16_t) and a small patch in matrix transposition
1282
if ((t < 1) || (m*t >= ((1 << m)-1)))
1283
/* invalid t value */
1286
/* select a primitive polynomial for generating GF(2^m) */
1288
prim_poly = prim_poly_tab[m-min_m];
1290
bch = kzalloc(sizeof(*bch), GFP_KERNEL);
1296
bch->n = (1 << m)-1;
1297
words = DIV_ROUND_UP(m*t, 32);
1298
bch->ecc_bytes = DIV_ROUND_UP(m*t, 8);
1299
bch->a_pow_tab = bch_alloc((1+bch->n)*sizeof(*bch->a_pow_tab), &err);
1300
bch->a_log_tab = bch_alloc((1+bch->n)*sizeof(*bch->a_log_tab), &err);
1301
bch->mod8_tab = bch_alloc(words*1024*sizeof(*bch->mod8_tab), &err);
1302
bch->ecc_buf = bch_alloc(words*sizeof(*bch->ecc_buf), &err);
1303
bch->ecc_buf2 = bch_alloc(words*sizeof(*bch->ecc_buf2), &err);
1304
bch->xi_tab = bch_alloc(m*sizeof(*bch->xi_tab), &err);
1305
bch->syn = bch_alloc(2*t*sizeof(*bch->syn), &err);
1306
bch->cache = bch_alloc(2*t*sizeof(*bch->cache), &err);
1307
bch->elp = bch_alloc((t+1)*sizeof(struct gf_poly_deg1), &err);
1309
for (i = 0; i < ARRAY_SIZE(bch->poly_2t); i++)
1310
bch->poly_2t[i] = bch_alloc(GF_POLY_SZ(2*t), &err);
1315
err = build_gf_tables(bch, prim_poly);
1319
/* use generator polynomial for computing encoding tables */
1320
genpoly = compute_generator_polynomial(bch);
1321
if (genpoly == NULL)
1324
build_mod8_tables(bch, genpoly);
1327
err = build_deg2_base(bch);
1337
EXPORT_SYMBOL_GPL(init_bch);
1340
* free_bch - free the BCH control structure
1341
* @bch: BCH control structure to release
1343
void free_bch(struct bch_control *bch)
1348
kfree(bch->a_pow_tab);
1349
kfree(bch->a_log_tab);
1350
kfree(bch->mod8_tab);
1351
kfree(bch->ecc_buf);
1352
kfree(bch->ecc_buf2);
1358
for (i = 0; i < ARRAY_SIZE(bch->poly_2t); i++)
1359
kfree(bch->poly_2t[i]);
1364
EXPORT_SYMBOL_GPL(free_bch);
1366
MODULE_LICENSE("GPL");
1367
MODULE_AUTHOR("Ivan Djelic <ivan.djelic@parrot.com>");
1368
MODULE_DESCRIPTION("Binary BCH encoder/decoder");