fit: Verify all configuration signatures

[u-boot] / lib / bch.c

// SPDX-License-Identifier: GPL-2.0
/*
 * Generic binary BCH encoding/decoding library
 *
 * Copyright © 2011 Parrot S.A.
 *
 * Author: Ivan Djelic <ivan.djelic@parrot.com>
 *
 * Description:
 *
 * This library provides runtime configurable encoding/decoding of binary
 * Bose-Chaudhuri-Hocquenghem (BCH) codes.
 *
 * Call init_bch to get a pointer to a newly allocated bch_control structure for
 * the given m (Galois field order), t (error correction capability) and
 * (optional) primitive polynomial parameters.
 *
 * Call encode_bch to compute and store ecc parity bytes to a given buffer.
 * Call decode_bch to detect and locate errors in received data.
 *
 * On systems supporting hw BCH features, intermediate results may be provided
 * to decode_bch in order to skip certain steps. See decode_bch() documentation
 * for details.
 *
 * Option CONFIG_BCH_CONST_PARAMS can be used to force fixed values of
 * parameters m and t; thus allowing extra compiler optimizations and providing
 * better (up to 2x) encoding performance. Using this option makes sense when
 * (m,t) are fixed and known in advance, e.g. when using BCH error correction
 * on a particular NAND flash device.
 *
 * Algorithmic details:
 *
 * Encoding is performed by processing 32 input bits in parallel, using 4
 * remainder lookup tables.
 *
 * The final stage of decoding involves the following internal steps:
 * a. Syndrome computation
 * b. Error locator polynomial computation using Berlekamp-Massey algorithm
 * c. Error locator root finding (by far the most expensive step)
 *
 * In this implementation, step c is not performed using the usual Chien search.
 * Instead, an alternative approach described in [1] is used. It consists in
 * factoring the error locator polynomial using the Berlekamp Trace algorithm
 * (BTA) down to a certain degree (4), after which ad hoc low-degree polynomial
 * solving techniques [2] are used. The resulting algorithm, called BTZ, yields
 * much better performance than Chien search for usual (m,t) values (typically
 * m >= 13, t < 32, see [1]).
 *
 * [1] B. Biswas, V. Herbert. Efficient root finding of polynomials over fields
 * of characteristic 2, in: Western European Workshop on Research in Cryptology
 * - WEWoRC 2009, Graz, Austria, LNCS, Springer, July 2009, to appear.
 * [2] [Zin96] V.A. Zinoviev. On the solution of equations of degree 10 over
 * finite fields GF(2^q). In Rapport de recherche INRIA no 2829, 1996.
 */

#ifndef USE_HOSTCC
#include <common.h>
#include <ubi_uboot.h>

#include <linux/bitops.h>
#else
#include <errno.h>
#if defined(__FreeBSD__)
#include <sys/endian.h>
#else
#include <endian.h>
#endif
#include <stdint.h>
#include <stdlib.h>
#include <string.h>

#undef cpu_to_be32
#define cpu_to_be32 htobe32
#define DIV_ROUND_UP(n,d) (((n) + (d) - 1) / (d))
#define kmalloc(size, flags)    malloc(size)
#define kzalloc(size, flags)    calloc(1, size)
#define kfree free
#define ARRAY_SIZE(arr) (sizeof(arr) / sizeof((arr)[0]))
#endif

#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];
};

#ifdef USE_HOSTCC
#if !defined(__DragonFly__) && !defined(__FreeBSD__)
static int fls(int x)
{
        int r = 32;

        if (!x)
                return 0;
        if (!(x & 0xffff0000u)) {
                x <<= 16;
                r -= 16;
        }
        if (!(x & 0xff000000u)) {
                x <<= 8;
                r -= 8;
        }
        if (!(x & 0xf0000000u)) {
                x <<= 4;
                r -= 4;
        }
        if (!(x & 0xc0000000u)) {
                x <<= 2;
                r -= 2;
        }
        if (!(x & 0x80000000u)) {
                x <<= 1;
                r -= 1;
        }
        return r;
}
#endif
#endif

/*
 * same as encode_bch(), but process input data one byte at a time
 */
static void encode_bch_unaligned(struct bch_control *bch,
                                 const unsigned char *data, unsigned int len,
                                 uint32_t *ecc)
{
        int i;
        const uint32_t *p;
        const int l = BCH_ECC_WORDS(bch)-1;

        while (len--) {
                p = bch->mod8_tab + (l+1)*(((ecc[0] >> 24)^(*data++)) & 0xff);

                for (i = 0; i < l; i++)
                        ecc[i] = ((ecc[i] << 8)|(ecc[i+1] >> 24))^(*p++);

                ecc[l] = (ecc[l] << 8)^(*p);
        }
}

/*
 * convert ecc bytes to aligned, zero-padded 32-bit ecc words
 */
static void load_ecc8(struct bch_control *bch, uint32_t *dst,
                      const uint8_t *src)
{
        uint8_t pad[4] = {0, 0, 0, 0};
        unsigned int i, nwords = BCH_ECC_WORDS(bch)-1;

        for (i = 0; i < nwords; i++, src += 4)
                dst[i] = (src[0] << 24)|(src[1] << 16)|(src[2] << 8)|src[3];

        memcpy(pad, src, BCH_ECC_BYTES(bch)-4*nwords);
        dst[nwords] = (pad[0] << 24)|(pad[1] << 16)|(pad[2] << 8)|pad[3];
}

/*
 * convert 32-bit ecc words to ecc bytes
 */
static void store_ecc8(struct bch_control *bch, uint8_t *dst,
                       const uint32_t *src)
{
        uint8_t pad[4];
        unsigned int i, nwords = BCH_ECC_WORDS(bch)-1;

        for (i = 0; i < nwords; i++) {
                *dst++ = (src[i] >> 24);
                *dst++ = (src[i] >> 16) & 0xff;
                *dst++ = (src[i] >>  8) & 0xff;
                *dst++ = (src[i] >>  0) & 0xff;
        }
        pad[0] = (src[nwords] >> 24);
        pad[1] = (src[nwords] >> 16) & 0xff;
        pad[2] = (src[nwords] >>  8) & 0xff;
        pad[3] = (src[nwords] >>  0) & 0xff;
        memcpy(dst, pad, BCH_ECC_BYTES(bch)-4*nwords);
}

/**
 * encode_bch - calculate BCH ecc parity of data
 * @bch:   BCH control structure
 * @data:  data to encode
 * @len:   data length in bytes
 * @ecc:   ecc parity data, must be initialized by caller
 *
 * The @ecc parity array is used both as input and output parameter, in order to
 * allow incremental computations. It should be of the size indicated by member
 * @ecc_bytes of @bch, and should be initialized to 0 before the first call.
 *
 * The exact number of computed ecc parity bits is given by member @ecc_bits of
 * @bch; it may be less than m*t for large values of t.
 */
void encode_bch(struct bch_control *bch, const uint8_t *data,
                unsigned int len, uint8_t *ecc)
{
        const unsigned int l = BCH_ECC_WORDS(bch)-1;
        unsigned int i, mlen;
        unsigned long m;
        uint32_t w, r[l+1];
        const uint32_t * const tab0 = bch->mod8_tab;
        const uint32_t * const tab1 = tab0 + 256*(l+1);
        const uint32_t * const tab2 = tab1 + 256*(l+1);
        const uint32_t * const tab3 = tab2 + 256*(l+1);
        const uint32_t *pdata, *p0, *p1, *p2, *p3;

        if (ecc) {
                /* load ecc parity bytes into internal 32-bit buffer */
                load_ecc8(bch, bch->ecc_buf, ecc);
        } else {
                memset(bch->ecc_buf, 0, sizeof(r));
        }

        /* process first unaligned data bytes */
        m = ((unsigned long)data) & 3;
        if (m) {
                mlen = (len < (4-m)) ? len : 4-m;
                encode_bch_unaligned(bch, data, mlen, bch->ecc_buf);
                data += mlen;
                len  -= mlen;
        }

        /* process 32-bit aligned data words */
        pdata = (uint32_t *)data;
        mlen  = len/4;
        data += 4*mlen;
        len  -= 4*mlen;
        memcpy(r, bch->ecc_buf, sizeof(r));

        /*
         * split each 32-bit word into 4 polynomials of weight 8 as follows:
         *
         * 31 ...24  23 ...16  15 ... 8  7 ... 0
         * xxxxxxxx  yyyyyyyy  zzzzzzzz  tttttttt
         *                               tttttttt  mod g = r0 (precomputed)
         *                     zzzzzzzz  00000000  mod g = r1 (precomputed)
         *           yyyyyyyy  00000000  00000000  mod g = r2 (precomputed)
         * xxxxxxxx  00000000  00000000  00000000  mod g = r3 (precomputed)
         * xxxxxxxx  yyyyyyyy  zzzzzzzz  tttttttt  mod g = r0^r1^r2^r3
         */
        while (mlen--) {
                /* input data is read in big-endian format */
                w = r[0]^cpu_to_be32(*pdata++);
                p0 = tab0 + (l+1)*((w >>  0) & 0xff);
                p1 = tab1 + (l+1)*((w >>  8) & 0xff);
                p2 = tab2 + (l+1)*((w >> 16) & 0xff);
                p3 = tab3 + (l+1)*((w >> 24) & 0xff);

                for (i = 0; i < l; i++)
                        r[i] = r[i+1]^p0[i]^p1[i]^p2[i]^p3[i];

                r[l] = p0[l]^p1[l]^p2[l]^p3[l];
        }
        memcpy(bch->ecc_buf, r, sizeof(r));

        /* process last unaligned bytes */
        if (len)
                encode_bch_unaligned(bch, data, len, bch->ecc_buf);

        /* store ecc parity bytes into original parity buffer */
        if (ecc)
                store_ecc8(bch, ecc, bch->ecc_buf);
}

static inline int modulo(struct bch_control *bch, unsigned int v)
{
        const unsigned int n = GF_N(bch);
        while (v >= n) {
                v -= n;
                v = (v & n) + (v >> GF_M(bch));
        }
        return v;
}

/*
 * shorter and faster modulo function, only works when v < 2N.
 */
static inline int mod_s(struct bch_control *bch, unsigned int v)
{
        const unsigned int n = GF_N(bch);
        return (v < n) ? v : v-n;
}

static inline int deg(unsigned int poly)
{
        /* polynomial degree is the most-significant bit index */
        return fls(poly)-1;
}

static inline int parity(unsigned int x)
{
        /*
         * public domain code snippet, lifted from
         * http://www-graphics.stanford.edu/~seander/bithacks.html
         */
        x ^= x >> 1;
        x ^= x >> 2;
        x = (x & 0x11111111U) * 0x11111111U;
        return (x >> 28) & 1;
}

/* Galois field basic operations: multiply, divide, inverse, etc. */

static inline unsigned int gf_mul(struct bch_control *bch, unsigned int a,
                                  unsigned int b)
{
        return (a && b) ? bch->a_pow_tab[mod_s(bch, bch->a_log_tab[a]+
                                               bch->a_log_tab[b])] : 0;
}

static inline unsigned int gf_sqr(struct bch_control *bch, unsigned int a)
{
        return a ? bch->a_pow_tab[mod_s(bch, 2*bch->a_log_tab[a])] : 0;
}

static inline unsigned int gf_div(struct bch_control *bch, unsigned int a,
                                  unsigned int b)
{
        return a ? bch->a_pow_tab[mod_s(bch, bch->a_log_tab[a]+
                                        GF_N(bch)-bch->a_log_tab[b])] : 0;
}

static inline unsigned int gf_inv(struct bch_control *bch, unsigned int a)
{
        return bch->a_pow_tab[GF_N(bch)-bch->a_log_tab[a]];
}

static inline unsigned int a_pow(struct bch_control *bch, int i)
{
        return bch->a_pow_tab[modulo(bch, i)];
}

static inline int a_log(struct bch_control *bch, unsigned int x)
{
        return bch->a_log_tab[x];
}

static inline int a_ilog(struct bch_control *bch, unsigned int x)
{
        return mod_s(bch, GF_N(bch)-bch->a_log_tab[x]);
}

/*
 * compute 2t syndromes of ecc polynomial, i.e. ecc(a^j) for j=1..2t
 */
static void compute_syndromes(struct bch_control *bch, uint32_t *ecc,
                              unsigned int *syn)
{
        int i, j, s;
        unsigned int m;
        uint32_t poly;
        const int t = GF_T(bch);

        s = bch->ecc_bits;

        /* make sure extra bits in last ecc word are cleared */
        m = ((unsigned int)s) & 31;
        if (m)
                ecc[s/32] &= ~((1u << (32-m))-1);
        memset(syn, 0, 2*t*sizeof(*syn));

        /* compute v(a^j) for j=1 .. 2t-1 */
        do {
                poly = *ecc++;
                s -= 32;
                while (poly) {
                        i = deg(poly);
                        for (j = 0; j < 2*t; j += 2)
                                syn[j] ^= a_pow(bch, (j+1)*(i+s));

                        poly ^= (1 << i);
                }
        } while (s > 0);

        /* v(a^(2j)) = v(a^j)^2 */
        for (j = 0; j < t; j++)
                syn[2*j+1] = gf_sqr(bch, syn[j]);
}

static void gf_poly_copy(struct gf_poly *dst, struct gf_poly *src)
{
        memcpy(dst, src, GF_POLY_SZ(src->deg));
}

static int compute_error_locator_polynomial(struct bch_control *bch,
                                            const unsigned int *syn)
{
        const unsigned int t = GF_T(bch);
        const unsigned int n = GF_N(bch);
        unsigned int i, j, tmp, l, pd = 1, d = syn[0];
        struct gf_poly *elp = bch->elp;
        struct gf_poly *pelp = bch->poly_2t[0];
        struct gf_poly *elp_copy = bch->poly_2t[1];
        int k, pp = -1;

        memset(pelp, 0, GF_POLY_SZ(2*t));
        memset(elp, 0, GF_POLY_SZ(2*t));

        pelp->deg = 0;
        pelp->c[0] = 1;
        elp->deg = 0;
        elp->c[0] = 1;

        /* use simplified binary Berlekamp-Massey algorithm */
        for (i = 0; (i < t) && (elp->deg <= t); i++) {
                if (d) {
                        k = 2*i-pp;
                        gf_poly_copy(elp_copy, elp);
                        /* e[i+1](X) = e[i](X)+di*dp^-1*X^2(i-p)*e[p](X) */
                        tmp = a_log(bch, d)+n-a_log(bch, pd);
                        for (j = 0; j <= pelp->deg; j++) {
                                if (pelp->c[j]) {
                                        l = a_log(bch, pelp->c[j]);
                                        elp->c[j+k] ^= a_pow(bch, tmp+l);
                                }
                        }
                        /* compute l[i+1] = max(l[i]->c[l[p]+2*(i-p]) */
                        tmp = pelp->deg+k;
                        if (tmp > elp->deg) {
                                elp->deg = tmp;
                                gf_poly_copy(pelp, elp_copy);
                                pd = d;
                                pp = 2*i;
                        }
                }
                /* di+1 = S(2i+3)+elp[i+1].1*S(2i+2)+...+elp[i+1].lS(2i+3-l) */
                if (i < t-1) {
                        d = syn[2*i+2];
                        for (j = 1; j <= elp->deg; j++)
                                d ^= gf_mul(bch, elp->c[j], syn[2*i+2-j]);
                }
        }
        dbg("elp=%s\n", gf_poly_str(elp));
        return (elp->deg > t) ? -1 : (int)elp->deg;
}

/*
 * solve a m x m linear system in GF(2) with an expected number of solutions,
 * and return the number of found solutions
 */
static int solve_linear_system(struct bch_control *bch, unsigned int *rows,
                               unsigned int *sol, int nsol)
{
        const int m = GF_M(bch);
        unsigned int tmp, mask;
        int rem, c, r, p, k, param[m];

        k = 0;
        mask = 1 << m;

        /* Gaussian elimination */
        for (c = 0; c < m; c++) {
                rem = 0;
                p = c-k;
                /* find suitable row for elimination */
                for (r = p; r < m; r++) {
                        if (rows[r] & mask) {
                                if (r != p) {
                                        tmp = rows[r];
                                        rows[r] = rows[p];
                                        rows[p] = tmp;
                                }
                                rem = r+1;
                                break;
                        }
                }
                if (rem) {
                        /* perform elimination on remaining rows */
                        tmp = rows[p];
                        for (r = rem; r < m; r++) {
                                if (rows[r] & mask)
                                        rows[r] ^= tmp;
                        }
                } else {
                        /* elimination not needed, store defective row index */
                        param[k++] = c;
                }
                mask >>= 1;
        }
        /* rewrite system, inserting fake parameter rows */
        if (k > 0) {
                p = k;
                for (r = m-1; r >= 0; r--) {
                        if ((r > m-1-k) && rows[r])
                                /* system has no solution */
                                return 0;

                        rows[r] = (p && (r == param[p-1])) ?
                                p--, 1u << (m-r) : rows[r-p];
                }
        }

        if (nsol != (1 << k))
                /* unexpected number of solutions */
                return 0;

        for (p = 0; p < nsol; p++) {
                /* set parameters for p-th solution */
                for (c = 0; c < k; c++)
                        rows[param[c]] = (rows[param[c]] & ~1)|((p >> c) & 1);

                /* compute unique solution */
                tmp = 0;
                for (r = m-1; r >= 0; r--) {
                        mask = rows[r] & (tmp|1);
                        tmp |= parity(mask) << (m-r);
                }
                sol[p] = tmp >> 1;
        }
        return nsol;
}

/*
 * this function builds and solves a linear system for finding roots of a degree
 * 4 affine monic polynomial X^4+aX^2+bX+c over GF(2^m).
 */
static int find_affine4_roots(struct bch_control *bch, unsigned int a,
                              unsigned int b, unsigned int c,
                              unsigned int *roots)
{
        int i, j, k;
        const int m = GF_M(bch);
        unsigned int mask = 0xff, t, rows[16] = {0,};

        j = a_log(bch, b);
        k = a_log(bch, a);
        rows[0] = c;

        /* buid linear system to solve X^4+aX^2+bX+c = 0 */
        for (i = 0; i < m; i++) {
                rows[i+1] = bch->a_pow_tab[4*i]^
                        (a ? bch->a_pow_tab[mod_s(bch, k)] : 0)^
                        (b ? bch->a_pow_tab[mod_s(bch, j)] : 0);
                j++;
                k += 2;
        }
        /*
         * transpose 16x16 matrix before passing it to linear solver
         * warning: this code assumes m < 16
         */
        for (j = 8; j != 0; j >>= 1, mask ^= (mask << j)) {
                for (k = 0; k < 16; k = (k+j+1) & ~j) {
                        t = ((rows[k] >> j)^rows[k+j]) & mask;
                        rows[k] ^= (t << j);
                        rows[k+j] ^= t;
                }
        }
        return solve_linear_system(bch, rows, roots, 4);
}

/*
 * compute root r of a degree 1 polynomial over GF(2^m) (returned as log(1/r))
 */
static int find_poly_deg1_roots(struct bch_control *bch, struct gf_poly *poly,
                                unsigned int *roots)
{
        int n = 0;

        if (poly->c[0])
                /* poly[X] = bX+c with c!=0, root=c/b */
                roots[n++] = mod_s(bch, GF_N(bch)-bch->a_log_tab[poly->c[0]]+
                                   bch->a_log_tab[poly->c[1]]);
        return n;
}

/*
 * compute roots of a degree 2 polynomial over GF(2^m)
 */
static int find_poly_deg2_roots(struct bch_control *bch, struct gf_poly *poly,
                                unsigned int *roots)
{
        int n = 0, i, l0, l1, l2;
        unsigned int u, v, r;

        if (poly->c[0] && poly->c[1]) {

                l0 = bch->a_log_tab[poly->c[0]];
                l1 = bch->a_log_tab[poly->c[1]];
                l2 = bch->a_log_tab[poly->c[2]];

                /* using z=a/bX, transform aX^2+bX+c into z^2+z+u (u=ac/b^2) */
                u = a_pow(bch, l0+l2+2*(GF_N(bch)-l1));
                /*
                 * let u = sum(li.a^i) i=0..m-1; then compute r = sum(li.xi):
                 * r^2+r = sum(li.(xi^2+xi)) = sum(li.(a^i+Tr(a^i).a^k)) =
                 * u + sum(li.Tr(a^i).a^k) = u+a^k.Tr(sum(li.a^i)) = u+a^k.Tr(u)
                 * i.e. r and r+1 are roots iff Tr(u)=0
                 */
                r = 0;
                v = u;
                while (v) {
                        i = deg(v);
                        r ^= bch->xi_tab[i];
                        v ^= (1 << i);
                }
                /* verify root */
                if ((gf_sqr(bch, r)^r) == u) {
                        /* reverse z=a/bX transformation and compute log(1/r) */
                        roots[n++] = modulo(bch, 2*GF_N(bch)-l1-
                                            bch->a_log_tab[r]+l2);
                        roots[n++] = modulo(bch, 2*GF_N(bch)-l1-
                                            bch->a_log_tab[r^1]+l2);
                }
        }
        return n;
}

/*
 * compute roots of a degree 3 polynomial over GF(2^m)
 */
static int find_poly_deg3_roots(struct bch_control *bch, struct gf_poly *poly,
                                unsigned int *roots)
{
        int i, n = 0;
        unsigned int a, b, c, a2, b2, c2, e3, tmp[4];

        if (poly->c[0]) {
                /* transform polynomial into monic X^3 + a2X^2 + b2X + c2 */
                e3 = poly->c[3];
                c2 = gf_div(bch, poly->c[0], e3);
                b2 = gf_div(bch, poly->c[1], e3);
                a2 = gf_div(bch, poly->c[2], e3);

                /* (X+a2)(X^3+a2X^2+b2X+c2) = X^4+aX^2+bX+c (affine) */
                c = gf_mul(bch, a2, c2);           /* c = a2c2      */
                b = gf_mul(bch, a2, b2)^c2;        /* b = a2b2 + c2 */
                a = gf_sqr(bch, a2)^b2;            /* a = a2^2 + b2 */

                /* find the 4 roots of this affine polynomial */
                if (find_affine4_roots(bch, a, b, c, tmp) == 4) {
                        /* remove a2 from final list of roots */
                        for (i = 0; i < 4; i++) {
                                if (tmp[i] != a2)
                                        roots[n++] = a_ilog(bch, tmp[i]);
                        }
                }
        }
        return n;
}

/*
 * compute roots of a degree 4 polynomial over GF(2^m)
 */
static int find_poly_deg4_roots(struct bch_control *bch, struct gf_poly *poly,
                                unsigned int *roots)
{
        int i, l, n = 0;
        unsigned int a, b, c, d, e = 0, f, a2, b2, c2, e4;

        if (poly->c[0] == 0)
                return 0;

        /* transform polynomial into monic X^4 + aX^3 + bX^2 + cX + d */
        e4 = poly->c[4];
        d = gf_div(bch, poly->c[0], e4);
        c = gf_div(bch, poly->c[1], e4);
        b = gf_div(bch, poly->c[2], e4);
        a = gf_div(bch, poly->c[3], e4);

        /* use Y=1/X transformation to get an affine polynomial */
        if (a) {
                /* first, eliminate cX by using z=X+e with ae^2+c=0 */
                if (c) {
                        /* compute e such that e^2 = c/a */
                        f = gf_div(bch, c, a);
                        l = a_log(bch, f);
                        l += (l & 1) ? GF_N(bch) : 0;
                        e = a_pow(bch, l/2);
                        /*
                         * use transformation z=X+e:
                         * z^4+e^4 + a(z^3+ez^2+e^2z+e^3) + b(z^2+e^2) +cz+ce+d
                         * z^4 + az^3 + (ae+b)z^2 + (ae^2+c)z+e^4+be^2+ae^3+ce+d
                         * z^4 + az^3 + (ae+b)z^2 + e^4+be^2+d
                         * z^4 + az^3 +     b'z^2 + d'
                         */
                        d = a_pow(bch, 2*l)^gf_mul(bch, b, f)^d;
                        b = gf_mul(bch, a, e)^b;
                }
                /* now, use Y=1/X to get Y^4 + b/dY^2 + a/dY + 1/d */
                if (d == 0)
                        /* assume all roots have multiplicity 1 */
                        return 0;

                c2 = gf_inv(bch, d);
                b2 = gf_div(bch, a, d);
                a2 = gf_div(bch, b, d);
        } else {
                /* polynomial is already affine */
                c2 = d;
                b2 = c;
                a2 = b;
        }
        /* find the 4 roots of this affine polynomial */
        if (find_affine4_roots(bch, a2, b2, c2, roots) == 4) {
                for (i = 0; i < 4; i++) {
                        /* post-process roots (reverse transformations) */
                        f = a ? gf_inv(bch, roots[i]) : roots[i];
                        roots[i] = a_ilog(bch, f^e);
                }
                n = 4;
        }
        return n;
}

/*
 * build monic, log-based representation of a polynomial
 */
static void gf_poly_logrep(struct bch_control *bch,
                           const struct gf_poly *a, int *rep)
{
        int i, d = a->deg, l = GF_N(bch)-a_log(bch, a->c[a->deg]);

        /* represent 0 values with -1; warning, rep[d] is not set to 1 */
        for (i = 0; i < d; i++)
                rep[i] = a->c[i] ? mod_s(bch, a_log(bch, a->c[i])+l) : -1;
}

/*
 * compute polynomial Euclidean division remainder in GF(2^m)[X]
 */
static void gf_poly_mod(struct bch_control *bch, struct gf_poly *a,
                        const struct gf_poly *b, int *rep)
{
        int la, p, m;
        unsigned int i, j, *c = a->c;
        const unsigned int d = b->deg;

        if (a->deg < d)
                return;

        /* reuse or compute log representation of denominator */
        if (!rep) {
                rep = bch->cache;
                gf_poly_logrep(bch, b, rep);
        }

        for (j = a->deg; j >= d; j--) {
                if (c[j]) {
                        la = a_log(bch, c[j]);
                        p = j-d;
                        for (i = 0; i < d; i++, p++) {
                                m = rep[i];
                                if (m >= 0)
                                        c[p] ^= bch->a_pow_tab[mod_s(bch,
                                                                     m+la)];
                        }
                }
        }
        a->deg = d-1;
        while (!c[a->deg] && a->deg)
                a->deg--;
}

/*
 * compute polynomial Euclidean division quotient in GF(2^m)[X]
 */
static void gf_poly_div(struct bch_control *bch, struct gf_poly *a,
                        const struct gf_poly *b, struct gf_poly *q)
{
        if (a->deg >= b->deg) {
                q->deg = a->deg-b->deg;
                /* compute a mod b (modifies a) */
                gf_poly_mod(bch, a, b, NULL);
                /* quotient is stored in upper part of polynomial a */
                memcpy(q->c, &a->c[b->deg], (1+q->deg)*sizeof(unsigned int));
        } else {
                q->deg = 0;
                q->c[0] = 0;
        }
}

/*
 * compute polynomial GCD (Greatest Common Divisor) in GF(2^m)[X]
 */
static struct gf_poly *gf_poly_gcd(struct bch_control *bch, struct gf_poly *a,
                                   struct gf_poly *b)
{
        struct gf_poly *tmp;

        dbg("gcd(%s,%s)=", gf_poly_str(a), gf_poly_str(b));

        if (a->deg < b->deg) {
                tmp = b;
                b = a;
                a = tmp;
        }

        while (b->deg > 0) {
                gf_poly_mod(bch, a, b, NULL);
                tmp = b;
                b = a;
                a = tmp;
        }

        dbg("%s\n", gf_poly_str(a));

        return a;
}

/*
 * Given a polynomial f and an integer k, compute Tr(a^kX) mod f
 * This is used in Berlekamp Trace algorithm for splitting polynomials
 */
static void compute_trace_bk_mod(struct bch_control *bch, int k,
                                 const struct gf_poly *f, struct gf_poly *z,
                                 struct gf_poly *out)
{
        const int m = GF_M(bch);
        int i, j;

        /* z contains z^2j mod f */
        z->deg = 1;
        z->c[0] = 0;
        z->c[1] = bch->a_pow_tab[k];

        out->deg = 0;
        memset(out, 0, GF_POLY_SZ(f->deg));

        /* compute f log representation only once */
        gf_poly_logrep(bch, f, bch->cache);

        for (i = 0; i < m; i++) {
                /* add a^(k*2^i)(z^(2^i) mod f) and compute (z^(2^i) mod f)^2 */
                for (j = z->deg; j >= 0; j--) {
                        out->c[j] ^= z->c[j];
                        z->c[2*j] = gf_sqr(bch, z->c[j]);
                        z->c[2*j+1] = 0;
                }
                if (z->deg > out->deg)
                        out->deg = z->deg;

                if (i < m-1) {
                        z->deg *= 2;
                        /* z^(2(i+1)) mod f = (z^(2^i) mod f)^2 mod f */
                        gf_poly_mod(bch, z, f, bch->cache);
                }
        }
        while (!out->c[out->deg] && out->deg)
                out->deg--;

        dbg("Tr(a^%d.X) mod f = %s\n", k, gf_poly_str(out));
}

/*
 * factor a polynomial using Berlekamp Trace algorithm (BTA)
 */
static void factor_polynomial(struct bch_control *bch, int k, struct gf_poly *f,
                              struct gf_poly **g, struct gf_poly **h)
{
        struct gf_poly *f2 = bch->poly_2t[0];
        struct gf_poly *q  = bch->poly_2t[1];
        struct gf_poly *tk = bch->poly_2t[2];
        struct gf_poly *z  = bch->poly_2t[3];
        struct gf_poly *gcd;

        dbg("factoring %s...\n", gf_poly_str(f));

        *g = f;
        *h = NULL;

        /* tk = Tr(a^k.X) mod f */
        compute_trace_bk_mod(bch, k, f, z, tk);

        if (tk->deg > 0) {
                /* compute g = gcd(f, tk) (destructive operation) */
                gf_poly_copy(f2, f);
                gcd = gf_poly_gcd(bch, f2, tk);
                if (gcd->deg < f->deg) {
                        /* compute h=f/gcd(f,tk); this will modify f and q */
                        gf_poly_div(bch, f, gcd, q);
                        /* store g and h in-place (clobbering f) */
                        *h = &((struct gf_poly_deg1 *)f)[gcd->deg].poly;
                        gf_poly_copy(*g, gcd);
                        gf_poly_copy(*h, q);
                }
        }
}

/*
 * find roots of a polynomial, using BTZ algorithm; see the beginning of this
 * file for details
 */
static int find_poly_roots(struct bch_control *bch, unsigned int k,
                           struct gf_poly *poly, unsigned int *roots)
{
        int cnt;
        struct gf_poly *f1, *f2;

        switch (poly->deg) {
                /* handle low degree polynomials with ad hoc techniques */
        case 1:
                cnt = find_poly_deg1_roots(bch, poly, roots);
                break;
        case 2:
                cnt = find_poly_deg2_roots(bch, poly, roots);
                break;
        case 3:
                cnt = find_poly_deg3_roots(bch, poly, roots);
                break;
        case 4:
                cnt = find_poly_deg4_roots(bch, poly, roots);
                break;
        default:
                /* factor polynomial using Berlekamp Trace Algorithm (BTA) */
                cnt = 0;
                if (poly->deg && (k <= GF_M(bch))) {
                        factor_polynomial(bch, k, poly, &f1, &f2);
                        if (f1)
                                cnt += find_poly_roots(bch, k+1, f1, roots);
                        if (f2)
                                cnt += find_poly_roots(bch, k+1, f2, roots+cnt);
                }
                break;
        }
        return cnt;
}

#if defined(USE_CHIEN_SEARCH)
/*
 * exhaustive root search (Chien) implementation - not used, included only for
 * reference/comparison tests
 */
static int chien_search(struct bch_control *bch, unsigned int len,
                        struct gf_poly *p, unsigned int *roots)
{
        int m;
        unsigned int i, j, syn, syn0, count = 0;
        const unsigned int k = 8*len+bch->ecc_bits;

        /* use a log-based representation of polynomial */
        gf_poly_logrep(bch, p, bch->cache);
        bch->cache[p->deg] = 0;
        syn0 = gf_div(bch, p->c[0], p->c[p->deg]);

        for (i = GF_N(bch)-k+1; i <= GF_N(bch); i++) {
                /* compute elp(a^i) */
                for (j = 1, syn = syn0; j <= p->deg; j++) {
                        m = bch->cache[j];
                        if (m >= 0)
                                syn ^= a_pow(bch, m+j*i);
                }
                if (syn == 0) {
                        roots[count++] = GF_N(bch)-i;
                        if (count == p->deg)
                                break;
                }
        }
        return (count == p->deg) ? count : 0;
}
#define find_poly_roots(_p, _k, _elp, _loc) chien_search(_p, len, _elp, _loc)
#endif /* USE_CHIEN_SEARCH */

/**
 * decode_bch - decode received codeword and find bit error locations
 * @bch:      BCH control structure
 * @data:     received data, ignored if @calc_ecc is provided
 * @len:      data length in bytes, must always be provided
 * @recv_ecc: received ecc, if NULL then assume it was XORed in @calc_ecc
 * @calc_ecc: calculated ecc, if NULL then calc_ecc is computed from @data
 * @syn:      hw computed syndrome data (if NULL, syndrome is calculated)
 * @errloc:   output array of error locations
 *
 * Returns:
 *  The number of errors found, or -EBADMSG if decoding failed, or -EINVAL if
 *  invalid parameters were provided
 *
 * Depending on the available hw BCH support and the need to compute @calc_ecc
 * separately (using encode_bch()), this function should be called with one of
 * the following parameter configurations -
 *
 * by providing @data and @recv_ecc only:
 *   decode_bch(@bch, @data, @len, @recv_ecc, NULL, NULL, @errloc)
 *
 * by providing @recv_ecc and @calc_ecc:
 *   decode_bch(@bch, NULL, @len, @recv_ecc, @calc_ecc, NULL, @errloc)
 *
 * by providing ecc = recv_ecc XOR calc_ecc:
 *   decode_bch(@bch, NULL, @len, NULL, ecc, NULL, @errloc)
 *
 * by providing syndrome results @syn:
 *   decode_bch(@bch, NULL, @len, NULL, NULL, @syn, @errloc)
 *
 * Once decode_bch() has successfully returned with a positive value, error
 * locations returned in array @errloc should be interpreted as follows -
 *
 * if (errloc[n] >= 8*len), then n-th error is located in ecc (no need for
 * data correction)
 *
 * if (errloc[n] < 8*len), then n-th error is located in data and can be
 * corrected with statement data[errloc[n]/8] ^= 1 << (errloc[n] % 8);
 *
 * Note that this function does not perform any data correction by itself, it
 * merely indicates error locations.
 */
int decode_bch(struct bch_control *bch, const uint8_t *data, unsigned int len,
               const uint8_t *recv_ecc, const uint8_t *calc_ecc,
               const unsigned int *syn, unsigned int *errloc)
{
        const unsigned int ecc_words = BCH_ECC_WORDS(bch);
        unsigned int nbits;
        int i, err, nroots;
        uint32_t sum;

        /* sanity check: make sure data length can be handled */
        if (8*len > (bch->n-bch->ecc_bits))
                return -EINVAL;

        /* if caller does not provide syndromes, compute them */
        if (!syn) {
                if (!calc_ecc) {
                        /* compute received data ecc into an internal buffer */
                        if (!data || !recv_ecc)
                                return -EINVAL;
                        encode_bch(bch, data, len, NULL);
                } else {
                        /* load provided calculated ecc */
                        load_ecc8(bch, bch->ecc_buf, calc_ecc);
                }
                /* load received ecc or assume it was XORed in calc_ecc */
                if (recv_ecc) {
                        load_ecc8(bch, bch->ecc_buf2, recv_ecc);
                        /* XOR received and calculated ecc */
                        for (i = 0, sum = 0; i < (int)ecc_words; i++) {
                                bch->ecc_buf[i] ^= bch->ecc_buf2[i];
                                sum |= bch->ecc_buf[i];
                        }
                        if (!sum)
                                /* no error found */
                                return 0;
                }
                compute_syndromes(bch, bch->ecc_buf, bch->syn);
                syn = bch->syn;
        }

        err = compute_error_locator_polynomial(bch, syn);
        if (err > 0) {
                nroots = find_poly_roots(bch, 1, bch->elp, errloc);
                if (err != nroots)
                        err = -1;
        }
        if (err > 0) {
                /* post-process raw error locations for easier correction */
                nbits = (len*8)+bch->ecc_bits;
                for (i = 0; i < err; i++) {
                        if (errloc[i] >= nbits) {
                                err = -1;
                                break;
                        }
                        errloc[i] = nbits-1-errloc[i];
                        errloc[i] = (errloc[i] & ~7)|(7-(errloc[i] & 7));
                }
        }
        return (err >= 0) ? err : -EBADMSG;
}

/*
 * generate Galois field lookup tables
 */
static int build_gf_tables(struct bch_control *bch, unsigned int poly)
{
        unsigned int i, x = 1;
        const unsigned int k = 1 << deg(poly);

        /* primitive polynomial must be of degree m */
        if (k != (1u << GF_M(bch)))
                return -1;

        for (i = 0; i < GF_N(bch); i++) {
                bch->a_pow_tab[i] = x;
                bch->a_log_tab[x] = i;
                if (i && (x == 1))
                        /* polynomial is not primitive (a^i=1 with 0<i<2^m-1) */
                        return -1;
                x <<= 1;
                if (x & k)
                        x ^= poly;
        }
        bch->a_pow_tab[GF_N(bch)] = 1;
        bch->a_log_tab[0] = 0;

        return 0;
}

/*
 * compute generator polynomial remainder tables for fast encoding
 */
static void build_mod8_tables(struct bch_control *bch, const uint32_t *g)
{
        int i, j, b, d;
        uint32_t data, hi, lo, *tab;
        const int l = BCH_ECC_WORDS(bch);
        const int plen = DIV_ROUND_UP(bch->ecc_bits+1, 32);
        const int ecclen = DIV_ROUND_UP(bch->ecc_bits, 32);

        memset(bch->mod8_tab, 0, 4*256*l*sizeof(*bch->mod8_tab));

        for (i = 0; i < 256; i++) {
                /* p(X)=i is a small polynomial of weight <= 8 */
                for (b = 0; b < 4; b++) {
                        /* we want to compute (p(X).X^(8*b+deg(g))) mod g(X) */
                        tab = bch->mod8_tab + (b*256+i)*l;
                        data = i << (8*b);
                        while (data) {
                                d = deg(data);
                                /* subtract X^d.g(X) from p(X).X^(8*b+deg(g)) */
                                data ^= g[0] >> (31-d);
                                for (j = 0; j < ecclen; j++) {
                                        hi = (d < 31) ? g[j] << (d+1) : 0;
                                        lo = (j+1 < plen) ?
                                                g[j+1] >> (31-d) : 0;
                                        tab[j] ^= hi|lo;
                                }
                        }
                }
        }
}

/*
 * build a base for factoring degree 2 polynomials
 */
static int build_deg2_base(struct bch_control *bch)
{
        const int m = GF_M(bch);
        int i, j, r;
        unsigned int sum, x, y, remaining, ak = 0, xi[m];

        /* find k s.t. Tr(a^k) = 1 and 0 <= k < m */
        for (i = 0; i < m; i++) {
                for (j = 0, sum = 0; j < m; j++)
                        sum ^= a_pow(bch, i*(1 << j));

                if (sum) {
                        ak = bch->a_pow_tab[i];
                        break;
                }
        }
        /* find xi, i=0..m-1 such that xi^2+xi = a^i+Tr(a^i).a^k */
        remaining = m;
        memset(xi, 0, sizeof(xi));

        for (x = 0; (x <= GF_N(bch)) && remaining; x++) {
                y = gf_sqr(bch, x)^x;
                for (i = 0; i < 2; i++) {
                        r = a_log(bch, y);
                        if (y && (r < m) && !xi[r]) {
                                bch->xi_tab[r] = x;
                                xi[r] = 1;
                                remaining--;
                                dbg("x%d = %x\n", r, x);
                                break;
                        }
                        y ^= ak;
                }
        }
        /* should not happen but check anyway */
        return remaining ? -1 : 0;
}

static void *bch_alloc(size_t size, int *err)
{
        void *ptr;

        ptr = kmalloc(size, GFP_KERNEL);
        if (ptr == NULL)
                *err = 1;
        return ptr;
}

/*
 * compute generator polynomial for given (m,t) parameters.
 */
static uint32_t *compute_generator_polynomial(struct bch_control *bch)
{
        const unsigned int m = GF_M(bch);
        const unsigned int t = GF_T(bch);
        int n, err = 0;
        unsigned int i, j, nbits, r, word, *roots;
        struct gf_poly *g;
        uint32_t *genpoly;

        g = bch_alloc(GF_POLY_SZ(m*t), &err);
        roots = bch_alloc((bch->n+1)*sizeof(*roots), &err);
        genpoly = bch_alloc(DIV_ROUND_UP(m*t+1, 32)*sizeof(*genpoly), &err);

        if (err) {
                kfree(genpoly);
                genpoly = NULL;
                goto finish;
        }

        /* enumerate all roots of g(X) */
        memset(roots , 0, (bch->n+1)*sizeof(*roots));
        for (i = 0; i < t; i++) {
                for (j = 0, r = 2*i+1; j < m; j++) {
                        roots[r] = 1;
                        r = mod_s(bch, 2*r);
                }
        }
        /* build generator polynomial g(X) */
        g->deg = 0;
        g->c[0] = 1;
        for (i = 0; i < GF_N(bch); i++) {
                if (roots[i]) {
                        /* multiply g(X) by (X+root) */
                        r = bch->a_pow_tab[i];
                        g->c[g->deg+1] = 1;
                        for (j = g->deg; j > 0; j--)
                                g->c[j] = gf_mul(bch, g->c[j], r)^g->c[j-1];

                        g->c[0] = gf_mul(bch, g->c[0], r);
                        g->deg++;
                }
        }
        /* store left-justified binary representation of g(X) */
        n = g->deg+1;
        i = 0;

        while (n > 0) {
                nbits = (n > 32) ? 32 : n;
                for (j = 0, word = 0; j < nbits; j++) {
                        if (g->c[n-1-j])
                                word |= 1u << (31-j);
                }
                genpoly[i++] = word;
                n -= nbits;
        }
        bch->ecc_bits = g->deg;

finish:
        kfree(g);
        kfree(roots);

        return genpoly;
}

/**
 * init_bch - initialize a BCH encoder/decoder
 * @m:          Galois field order, should be in the range 5-15
 * @t:          maximum error correction capability, in bits
 * @prim_poly:  user-provided primitive polynomial (or 0 to use default)
 *
 * Returns:
 *  a newly allocated BCH control structure if successful, NULL otherwise
 *
 * This initialization can take some time, as lookup tables are built for fast
 * encoding/decoding; make sure not to call this function from a time critical
 * path. Usually, init_bch() should be called on module/driver init and
 * free_bch() should be called to release memory on exit.
 *
 * You may provide your own primitive polynomial of degree @m in argument
 * @prim_poly, or let init_bch() use its default polynomial.
 *
 * Once init_bch() has successfully returned a pointer to a newly allocated
 * BCH control structure, ecc length in bytes is given by member @ecc_bytes of
 * the structure.
 */
struct bch_control *init_bch(int m, int t, unsigned int prim_poly)
{
        int err = 0;
        unsigned int i, words;
        uint32_t *genpoly;
        struct bch_control *bch = NULL;

        const int min_m = 5;
        const int max_m = 15;

        /* default primitive polynomials */
        static const unsigned int prim_poly_tab[] = {
                0x25, 0x43, 0x83, 0x11d, 0x211, 0x409, 0x805, 0x1053, 0x201b,
                0x402b, 0x8003,
        };

#if defined(CONFIG_BCH_CONST_PARAMS)
        if ((m != (CONFIG_BCH_CONST_M)) || (t != (CONFIG_BCH_CONST_T))) {
                printk(KERN_ERR "bch encoder/decoder was configured to support "
                       "parameters m=%d, t=%d only!\n",
                       CONFIG_BCH_CONST_M, CONFIG_BCH_CONST_T);
                goto fail;
        }
#endif
        if ((m < min_m) || (m > max_m))
                /*
                 * values of m greater than 15 are not currently supported;
                 * supporting m > 15 would require changing table base type
                 * (uint16_t) and a small patch in matrix transposition
                 */
                goto fail;

        /* sanity checks */
        if ((t < 1) || (m*t >= ((1 << m)-1)))
                /* invalid t value */
                goto fail;

        /* select a primitive polynomial for generating GF(2^m) */
        if (prim_poly == 0)
                prim_poly = prim_poly_tab[m-min_m];

        bch = kzalloc(sizeof(*bch), GFP_KERNEL);
        if (bch == NULL)
                goto fail;

        bch->m = m;
        bch->t = t;
        bch->n = (1 << m)-1;
        words  = DIV_ROUND_UP(m*t, 32);
        bch->ecc_bytes = DIV_ROUND_UP(m*t, 8);
        bch->a_pow_tab = bch_alloc((1+bch->n)*sizeof(*bch->a_pow_tab), &err);
        bch->a_log_tab = bch_alloc((1+bch->n)*sizeof(*bch->a_log_tab), &err);
        bch->mod8_tab  = bch_alloc(words*1024*sizeof(*bch->mod8_tab), &err);
        bch->ecc_buf   = bch_alloc(words*sizeof(*bch->ecc_buf), &err);
        bch->ecc_buf2  = bch_alloc(words*sizeof(*bch->ecc_buf2), &err);
        bch->xi_tab    = bch_alloc(m*sizeof(*bch->xi_tab), &err);
        bch->syn       = bch_alloc(2*t*sizeof(*bch->syn), &err);
        bch->cache     = bch_alloc(2*t*sizeof(*bch->cache), &err);
        bch->elp       = bch_alloc((t+1)*sizeof(struct gf_poly_deg1), &err);

        for (i = 0; i < ARRAY_SIZE(bch->poly_2t); i++)
                bch->poly_2t[i] = bch_alloc(GF_POLY_SZ(2*t), &err);

        if (err)
                goto fail;

        err = build_gf_tables(bch, prim_poly);
        if (err)
                goto fail;

        /* use generator polynomial for computing encoding tables */
        genpoly = compute_generator_polynomial(bch);
        if (genpoly == NULL)
                goto fail;

        build_mod8_tables(bch, genpoly);
        kfree(genpoly);

        err = build_deg2_base(bch);
        if (err)
                goto fail;

        return bch;

fail:
        free_bch(bch);
        return NULL;
}

/**
 *  free_bch - free the BCH control structure
 *  @bch:    BCH control structure to release
 */
void free_bch(struct bch_control *bch)
{
        unsigned int i;

        if (bch) {
                kfree(bch->a_pow_tab);
                kfree(bch->a_log_tab);
                kfree(bch->mod8_tab);
                kfree(bch->ecc_buf);
                kfree(bch->ecc_buf2);
                kfree(bch->xi_tab);
                kfree(bch->syn);
                kfree(bch->cache);
                kfree(bch->elp);

                for (i = 0; i < ARRAY_SIZE(bch->poly_2t); i++)
                        kfree(bch->poly_2t[i]);

                kfree(bch);
        }
}