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 <[email protected]>
 *
 * 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);
}

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 */

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;
}

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);

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 <[email protected]>");
MODULE_DESCRIPTION("Binary BCH encoder/decoder");