[paraslash.git] / fec.c

/*
 * fec.c -- forward error correction based on Vandermonde matrices
 * 980624
 * (C) 1997-98 Luigi Rizzo (luigi@iet.unipi.it)
 *
 * Portions derived from code by Phil Karn (karn@ka9q.ampr.org),
 * Robert Morelos-Zaragoza (robert@spectra.eng.hawaii.edu) and Hari
 * Thirumoorthy (harit@spectra.eng.hawaii.edu), Aug 1995
 *
 * Redistribution and use in source and binary forms, with or without
 * modification, are permitted provided that the following conditions
 * are met:
 *
 * 1. Redistributions of source code must retain the above copyright
 *    notice, this list of conditions and the following disclaimer.
 * 2. Redistributions in binary form must reproduce the above
 *    copyright notice, this list of conditions and the following
 *    disclaimer in the documentation and/or other materials
 *    provided with the distribution.
 *
 * THIS SOFTWARE IS PROVIDED BY THE AUTHORS ``AS IS'' AND
 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO,
 * THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
 * PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHORS
 * BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY,
 * OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
 * PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA,
 * OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
 * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR
 * TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT
 * OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY
 * OF SUCH DAMAGE.
 */

#include "para.h"
#include "error.h"
#include "portable_io.h"
#include "string.h"
#include "fec.h"

#define GF_BITS  8 /* code over GF(256) */
#define GF_SIZE ((1 << GF_BITS) - 1)

/*
 * To speed up computations, we have tables for logarithm, exponent and inverse
 * of a number. We use a table for multiplication as well (it takes 64K, no big
 * deal even on a PDA, especially because it can be pre-initialized an put into
 * a ROM!). The macro gf_mul(x,y) takes care of multiplications.
 */
static unsigned char gf_exp[2 * GF_SIZE]; /* index->poly form conversion table */
static int gf_log[GF_SIZE + 1]; /* Poly->index form conversion table    */
static unsigned char inverse[GF_SIZE + 1]; /* inverse of field elem. */
static unsigned char gf_mul_table[GF_SIZE + 1][GF_SIZE + 1];
/* Multiply two numbers. */
#define gf_mul(x,y) gf_mul_table[x][y]

/* Compute x % GF_SIZE without a slow divide. */
static inline unsigned char modnn(int x)
{
        while (x >= GF_SIZE) {
                x -= GF_SIZE;
                x = (x >> GF_BITS) + (x & GF_SIZE);
        }
        return x;
}

static void init_mul_table(void)
{
        int i, j;
        for (i = 0; i < GF_SIZE + 1; i++)
                for (j = 0; j < GF_SIZE + 1; j++)
                        gf_mul_table[i][j] =
                                gf_exp[modnn(gf_log[i] + gf_log[j])];

        for (j = 0; j < GF_SIZE + 1; j++)
                gf_mul_table[0][j] = gf_mul_table[j][0] = 0;
}

static unsigned char *alloc_matrix(int rows, int cols)
{
        return para_malloc(rows * cols);
}

/*
 * Initialize the data structures used for computations in GF.
 *
 * This generates GF(2**GF_BITS) from the irreducible polynomial p(X) in
 * p[0]..p[m].
 *
 * Lookup tables:
 *     index->polynomial form           gf_exp[] contains j= \alpha^i;
 *     polynomial form -> index form    gf_log[ j = \alpha^i ] = i
 * \alpha=x is the primitive element of GF(2^m)
 *
 * For efficiency, gf_exp[] has size 2*GF_SIZE, so that a simple
 * multiplication of two numbers can be resolved without calling modnn
 */
static void generate_gf(void)
{
        int i;
        unsigned char mask = 1;
        char *pp = "101110001"; /* The primitive polynomial 1+x^2+x^3+x^4+x^8 */
        gf_exp[GF_BITS] = 0; /* will be updated at the end of the 1st loop */

        /*
         * first, generate the (polynomial representation of) powers of \alpha,
         * which are stored in gf_exp[i] = \alpha ** i .
         * At the same time build gf_log[gf_exp[i]] = i .
         * The first GF_BITS powers are simply bits shifted to the left.
         */
        for (i = 0; i < GF_BITS; i++, mask <<= 1) {
                gf_exp[i] = mask;
                gf_log[gf_exp[i]] = i;
                /*
                 * If pp[i] == 1 then \alpha ** i occurs in poly-repr
                 * gf_exp[GF_BITS] = \alpha ** GF_BITS
                 */
                if (pp[i] == '1')
                        gf_exp[GF_BITS] ^= mask;
        }
        /*
         * now gf_exp[GF_BITS] = \alpha ** GF_BITS is complete, so can also
         * compute its inverse.
         */
        gf_log[gf_exp[GF_BITS]] = GF_BITS;
        /*
         * Poly-repr of \alpha ** (i+1) is given by poly-repr of \alpha ** i
         * shifted left one-bit and accounting for any \alpha ** GF_BITS term
         * that may occur when poly-repr of \alpha ** i is shifted.
         */
        mask = 1 << (GF_BITS - 1);
        for (i = GF_BITS + 1; i < GF_SIZE; i++) {
                if (gf_exp[i - 1] >= mask)
                        gf_exp[i] =
                                gf_exp[GF_BITS] ^ ((gf_exp[i - 1] ^ mask) << 1);
                else
                        gf_exp[i] = gf_exp[i - 1] << 1;
                gf_log[gf_exp[i]] = i;
        }
        /*
         * log(0) is not defined, so use a special value
         */
        gf_log[0] = GF_SIZE;
        /* set the extended gf_exp values for fast multiply */
        for (i = 0; i < GF_SIZE; i++)
                gf_exp[i + GF_SIZE] = gf_exp[i];

        inverse[0] = 0; /* 0 has no inverse. */
        inverse[1] = 1;
        for (i = 2; i <= GF_SIZE; i++)
                inverse[i] = gf_exp[GF_SIZE - gf_log[i]];
}

/*
 * Compute dst[] = dst[] + c * src[]
 *
 * This is used often, so better optimize it! Currently the loop is unrolled 16
 * times. The case c=0 is also optimized, whereas c=1 is not.
 */
#define UNROLL 16
static void addmul(unsigned char *dst1, const unsigned char const *src1,
        unsigned char c, int sz)
{
        if (c == 0)
                return;
        unsigned char *dst = dst1, *lim = &dst[sz - UNROLL + 1],
                *col = gf_mul_table[c];
        const unsigned char const *src = src1;

        for (; dst < lim; dst += UNROLL, src += UNROLL) {
                dst[0] ^= col[src[0]];
                dst[1] ^= col[src[1]];
                dst[2] ^= col[src[2]];
                dst[3] ^= col[src[3]];
                dst[4] ^= col[src[4]];
                dst[5] ^= col[src[5]];
                dst[6] ^= col[src[6]];
                dst[7] ^= col[src[7]];
                dst[8] ^= col[src[8]];
                dst[9] ^= col[src[9]];
                dst[10] ^= col[src[10]];
                dst[11] ^= col[src[11]];
                dst[12] ^= col[src[12]];
                dst[13] ^= col[src[13]];
                dst[14] ^= col[src[14]];
                dst[15] ^= col[src[15]];
        }
        lim += UNROLL - 1;
        for (; dst < lim; dst++, src++) /* final components */
                *dst ^= col[*src];
}

/*
 * Compute C = AB where A is n*k, B is k*m, C is n*m
 */
static void matmul(unsigned char *a, unsigned char *b, unsigned char *c,
                int n, int k, int m)
{
        int row, col, i;

        for (row = 0; row < n; row++) {
                for (col = 0; col < m; col++) {
                        unsigned char *pa = &a[row * k], *pb = &b[col], acc = 0;
                        for (i = 0; i < k; i++, pa++, pb += m)
                                acc ^= gf_mul(*pa, *pb);
                        c[row * m + col] = acc;
                }
        }
}

#define FEC_SWAP(a,b) {typeof(a) tmp = a; a = b; b = tmp;}

/*
 * Compute the inverse of a matrix.
 *
 * k is the size of the matrix 'src' (Gauss-Jordan, adapted from Numerical
 * Recipes in C). Returns -1 if 'src' is singular.
 */
static int invert_mat(unsigned char *src, int k)
{
        int irow, icol, row, col, ix, error;
        int *indxc = para_malloc(k * sizeof(int));
        int *indxr = para_malloc(k * sizeof(int));
        int *ipiv = para_malloc(k * sizeof(int)); /* elements used as pivots */
        unsigned char c, *p, *id_row = alloc_matrix(1, k),
                *temp_row = alloc_matrix(1, k);

        memset(id_row, 0, k);
        memset(ipiv, 0, k * sizeof(int));

        for (col = 0; col < k; col++) {
                unsigned char *pivot_row;
                /*
                 * Zeroing column 'col', look for a non-zero element.
                 * First try on the diagonal, if it fails, look elsewhere.
                 */
                irow = icol = -1;
                if (ipiv[col] != 1 && src[col * k + col] != 0) {
                        irow = col;
                        icol = col;
                        goto found_piv;
                }
                for (row = 0; row < k; row++) {
                        if (ipiv[row] != 1) {
                                for (ix = 0; ix < k; ix++) {
                                        if (ipiv[ix] == 0) {
                                                if (src[row * k + ix] != 0) {
                                                        irow = row;
                                                        icol = ix;
                                                        goto found_piv;
                                                }
                                        } else if (ipiv[ix] > 1) {
                                                error = -E_FEC_PIVOT;
                                                goto fail;
                                        }
                                }
                        }
                }
                error = -E_FEC_PIVOT;
                if (icol == -1)
                        goto fail;
found_piv:
                ++(ipiv[icol]);
                /*
                 * swap rows irow and icol, so afterwards the diagonal element
                 * will be correct. Rarely done, not worth optimizing.
                 */
                if (irow != icol)
                        for (ix = 0; ix < k; ix++)
                                FEC_SWAP(src[irow * k + ix], src[icol * k + ix]);
                indxr[col] = irow;
                indxc[col] = icol;
                pivot_row = &src[icol * k];
                error = -E_FEC_SINGULAR;
                c = pivot_row[icol];
                if (c == 0)
                        goto fail;
                if (c != 1) { /* otherwise this is a NOP */
                        /*
                         * this is done often , but optimizing is not so
                         * fruitful, at least in the obvious ways (unrolling)
                         */
                        c = inverse[c];
                        pivot_row[icol] = 1;
                        for (ix = 0; ix < k; ix++)
                                pivot_row[ix] = gf_mul(c, pivot_row[ix]);
                }
                /*
                 * from all rows, remove multiples of the selected row to zero
                 * the relevant entry (in fact, the entry is not zero because
                 * we know it must be zero).  (Here, if we know that the
                 * pivot_row is the identity, we can optimize the addmul).
                 */
                id_row[icol] = 1;
                if (memcmp(pivot_row, id_row, k) != 0) {
                        for (p = src, ix = 0; ix < k; ix++, p += k) {
                                if (ix != icol) {
                                        c = p[icol];
                                        p[icol] = 0;
                                        addmul(p, pivot_row, c, k);
                                }
                        }
                }
                id_row[icol] = 0;
        }
        for (col = k - 1; col >= 0; col--) {
                if (indxr[col] < 0 || indxr[col] >= k)
                        PARA_CRIT_LOG("AARGH, indxr[col] %d\n", indxr[col]);
                else if (indxc[col] < 0 || indxc[col] >= k)
                        PARA_CRIT_LOG("AARGH, indxc[col] %d\n", indxc[col]);
                else if (indxr[col] != indxc[col]) {
                        for (row = 0; row < k; row++) {
                                FEC_SWAP(src[row * k + indxr[col]],
                                        src[row * k + indxc[col]]);
                        }
                }
        }
        error = 0;
fail:
        free(indxc);
        free(indxr);
        free(ipiv);
        free(id_row);
        free(temp_row);
        return error;
}

/*
 * Invert a Vandermonde matrix.
 *
 * It assumes that the matrix is not singular and _IS_ a Vandermonde matrix.
 * Only uses the second column of the matrix, containing the p_i's.
 *
 * Algorithm borrowed from "Numerical recipes in C" -- sec.2.8, but largely
 * revised for GF purposes.
 */
static void invert_vdm(unsigned char *src, int k)
{
        int i, j, row, col;
        unsigned char *b, *c, *p, t, xx;

        if (k == 1) /* degenerate */
                return;
        /*
         * c holds the coefficient of P(x) = Prod (x - p_i), i=0..k-1
         * b holds the coefficient for the matrix inversion
         */
        c = para_malloc(k);
        b = para_malloc(k);
        p = para_malloc(k);

        for (j = 1, i = 0; i < k; i++, j += k) {
                c[i] = 0;
                p[i] = src[j];
        }
        /*
         * construct coeffs recursively. We know c[k] = 1 (implicit) and start
         * P_0 = x - p_0, then at each stage multiply by x - p_i generating P_i
         * = x P_{i-1} - p_i P_{i-1} After k steps we are done.
         */
        c[k - 1] = p[0]; /* really -p(0), but x = -x in GF(2^m) */
        for (i = 1; i < k; i++) {
                unsigned char p_i = p[i];
                for (j = k - 1 - (i - 1); j < k - 1; j++)
                        c[j] ^= gf_mul(p_i, c[j + 1]);
                c[k - 1] ^= p_i;
        }

        for (row = 0; row < k; row++) {
                /*
                 * synthetic division etc.
                 */
                xx = p[row];
                t = 1;
                b[k - 1] = 1; /* this is in fact c[k] */
                for (i = k - 2; i >= 0; i--) {
                        b[i] = c[i + 1] ^ gf_mul(xx, b[i + 1]);
                        t = gf_mul(xx, t) ^ b[i];
                }
                for (col = 0; col < k; col++)
                        src[col * k + row] = gf_mul(inverse[t], b[col]);
        }
        free(c);
        free(b);
        free(p);
}

static int fec_initialized;

static void init_fec(void)
{
        generate_gf();
        init_mul_table();
        fec_initialized = 1;
}

struct fec_parms {
        int k, n; /* parameters of the code */
        unsigned char *enc_matrix;
};

/**
 * Deallocate a fec params structure.
 *
 * \param p The structure to free.
 */
void fec_free(struct fec_parms *p)
{
        if (!p)
                return;
        free(p->enc_matrix);
        free(p);
}

/**
 * Create a new encoder and return an opaque descriptor to it.
 *
 * \param k Number of input slices.
 * \param n Number of output slices.
 * \param result On success the Fec descriptor is returned here.
 *
 * \return Standard.
 *
 * This creates the k*n encoding matrix.  It is computed starting with a
 * Vandermonde matrix, and then transformed into a systematic matrix.
 */
int fec_new(int k, int n, struct fec_parms **result)
{
        int row, col;
        unsigned char *p, *tmp_m;
        struct fec_parms *parms;

        if (!fec_initialized)
                init_fec();

        if (k < 1 || k > GF_SIZE + 1 || n > GF_SIZE + 1 || k > n)
                return -E_FEC_PARMS;
        parms = para_malloc(sizeof(struct fec_parms));
        parms->k = k;
        parms->n = n;
        parms->enc_matrix = alloc_matrix(n, k);
        tmp_m = alloc_matrix(n, k);
        /*
         * fill the matrix with powers of field elements, starting from 0.
         * The first row is special, cannot be computed with exp. table.
         */
        tmp_m[0] = 1;
        for (col = 1; col < k; col++)
                tmp_m[col] = 0;
        for (p = tmp_m + k, row = 0; row < n - 1; row++, p += k) {
                for (col = 0; col < k; col++)
                        p[col] = gf_exp[modnn(row * col)];
        }

        /*
         * quick code to build systematic matrix: invert the top
         * k*k vandermonde matrix, multiply right the bottom n-k rows
         * by the inverse, and construct the identity matrix at the top.
         */
        invert_vdm(tmp_m, k); /* much faster than invert_mat */
        matmul(tmp_m + k * k, tmp_m, parms->enc_matrix + k * k, n - k, k, k);
        /*
         * the upper matrix is I so do not bother with a slow multiply
         */
        memset(parms->enc_matrix, 0, k * k);
        for (p = parms->enc_matrix, col = 0; col < k; col++, p += k + 1)
                *p = 1;
        free(tmp_m);
        *result = parms;
        return 0;
}

/**
 * Compute one encoded slice of the given input.
 *
 * \param parms The fec parameters returned earlier by fec_new().
 * \param src The \a k data slices to encode.
 * \param dst Result pointer.
 * \param idx The index of the slice to compute.
 * \param sz The size of the input data packets.
 *
 * Encode the \a k slices of size \a sz given by \a src and store the output
 * slice number \a idx in \a dst.
 */
void fec_encode(struct fec_parms *parms, const unsigned char * const *src,
                unsigned char *dst, int idx, int sz)
{
        int i, k = parms->k;
        unsigned char *p;

        assert(idx <= parms->n);

        if (idx < k) {
                memcpy(dst, src[idx], sz);
                return;
        }
        p = &(parms->enc_matrix[idx * k]);
        memset(dst, 0, sz);
        for (i = 0; i < k; i++)
                addmul(dst, src[i], p[i], sz);
}

/* Move src packets in their position. */
static int shuffle(unsigned char **data, int *idx, int k)
{
        int i;

        for (i = 0; i < k;) {
                if (idx[i] >= k || idx[i] == i)
                        i++;
                else { /* put index and data at the right position */
                        int c = idx[i];

                        if (idx[c] == c) /* conflict */
                                return -E_FEC_BAD_IDX;
                        FEC_SWAP(idx[i], idx[c]);
                        FEC_SWAP(data[i], data[c]);
                }
        }
        return 0;
}

/*
 * Construct the decoding matrix given the indices. The encoding matrix must
 * already be allocated.
 */
static int build_decode_matrix(struct fec_parms *parms, int *idx,
                unsigned char **result)
{
        int ret = -E_FEC_BAD_IDX, i, k = parms->k;
        unsigned char *p, *matrix = alloc_matrix(k, k);

        for (i = 0, p = matrix; i < k; i++, p += k) {
                if (idx[i] >= parms->n) /* invalid index */
                        goto err;
                if (idx[i] < k) {
                        memset(p, 0, k);
                        p[i] = 1;
                } else
                        memcpy(p, &(parms->enc_matrix[idx[i] * k]), k);
        }
        ret = invert_mat(matrix, k);
        if (ret < 0)
                goto err;
        *result = matrix;
        return 0;
err:
        free(matrix);
        *result = NULL;
        return ret;
}

/**
 * Decode one slice from the group of received slices.
 *
 * \param parms Pointer to fec params structure.
 * \param data Pointers to received packets.
 * \param idx Pointer to packet indices (gets modified).
 * \param sz Size of each packet.
 *
 * \return Zero on success, -1 on errors.
 *
 * The \a data vector of received slices and the indices of slices are used to
 * produce the correct output slice. The data slices are modified in-place.
 */
int fec_decode(struct fec_parms *parms, unsigned char **data, int *idx,
                int sz)
{
        unsigned char *m_dec, **slice;
        int ret, row, col, k = parms->k;

        ret = shuffle(data, idx, k);
        if (ret < 0)
                return ret;
        ret = build_decode_matrix(parms, idx, &m_dec);
        if (ret < 0)
                return ret;
        /* do the actual decoding */
        slice = para_malloc(k * sizeof(unsigned char *));
        for (row = 0; row < k; row++) {
                if (idx[row] >= k) {
                        slice[row] = para_calloc(sz);
                        for (col = 0; col < k; col++)
                                addmul(slice[row], data[col],
                                        m_dec[row * k + col], sz);
                }
        }
        /* move slices to their final destination */
        for (row = 0; row < k; row++) {
                if (idx[row] >= k) {
                        memcpy(data[row], slice[row], sz);
                        free(slice[row]);
                }
        }
        free(slice);
        free(m_dec);
        return 0;
}