fec-0.1: zfec/fec.c
/**
* zfec -- fast forward error correction library with Python interface
*/
#include "fec.h"
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <assert.h>
/*
* Primitive polynomials - see Lin & Costello, Appendix A,
* and Lee & Messerschmitt, p. 453.
*/
static const char*const Pp="101110001";
/*
* 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!), otherwhise we use a table of
* logarithms. In any case the macro gf_mul(x,y) takes care of
* multiplications.
*/
static gf gf_exp[510]; /* index->poly form conversion table */
static int gf_log[256]; /* Poly->index form conversion table */
static gf inverse[256]; /* inverse of field elem. */
/* inv[\alpha**i]=\alpha**(GF_SIZE-i-1) */
/*
* modnn(x) computes x % GF_SIZE, where GF_SIZE is 2**GF_BITS - 1,
* without a slow divide.
*/
static gf
modnn(int x) {
while (x >= 255) {
x -= 255;
x = (x >> 8) + (x & 255);
}
return x;
}
#define SWAP(a,b,t) {t tmp; tmp=a; a=b; b=tmp;}
/*
* gf_mul(x,y) multiplies two numbers. It is much faster to use a
* multiplication table.
*
* USE_GF_MULC, GF_MULC0(c) and GF_ADDMULC(x) can be used when multiplying
* many numbers by the same constant. In this case the first call sets the
* constant, and others perform the multiplications. A value related to the
* multiplication is held in a local variable declared with USE_GF_MULC . See
* usage in _addmul1().
*/
static gf gf_mul_table[256][256];
#define gf_mul(x,y) gf_mul_table[x][y]
#define USE_GF_MULC register gf * __gf_mulc_
#define GF_MULC0(c) __gf_mulc_ = gf_mul_table[c]
#define GF_ADDMULC(dst, x) dst ^= __gf_mulc_[x]
/*
* Generate GF(2**m) 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
_init_mul_table(void) {
int i, j;
for (i = 0; i < 256; i++)
for (j = 0; j < 256; j++)
gf_mul_table[i][j] = gf_exp[modnn (gf_log[i] + gf_log[j])];
for (j = 0; j < 256; j++)
gf_mul_table[0][j] = gf_mul_table[j][0] = 0;
}
#define NEW_GF_MATRIX(rows, cols) \
(gf*)malloc(rows * cols)
/*
* initialize the data structures used for computations in GF.
*/
static void
generate_gf (void) {
int i;
gf mask;
mask = 1; /* x ** 0 = 1 */
gf_exp[8] = 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 8 powers are simply bits shifted to the left.
*/
for (i = 0; i < 8; 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[8] = \alpha ** 8
*/
if (Pp[i] == '1')
gf_exp[8] ^= mask;
}
/*
* now gf_exp[8] = \alpha ** 8 is complete, so can also
* compute its inverse.
*/
gf_log[gf_exp[8]] = 8;
/*
* Poly-repr of \alpha ** (i+1) is given by poly-repr of
* \alpha ** i shifted left one-bit and accounting for any
* \alpha ** 8 term that may occur when poly-repr of
* \alpha ** i is shifted.
*/
mask = 1 << 7;
for (i = 9; i < 255; i++) {
if (gf_exp[i - 1] >= mask)
gf_exp[i] = gf_exp[8] ^ ((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] = 255;
/* set the extended gf_exp values for fast multiply */
for (i = 0; i < 255; i++)
gf_exp[i + 255] = gf_exp[i];
/*
* again special cases. 0 has no inverse. This used to
* be initialized to 255, but it should make no difference
* since noone is supposed to read from here.
*/
inverse[0] = 0;
inverse[1] = 1;
for (i = 2; i <= 255; i++)
inverse[i] = gf_exp[255 - gf_log[i]];
}
/*
* Various linear algebra operations that i use often.
*/
/*
* addmul() computes dst[] = dst[] + c * src[]
* This is used often, so better optimize it! Currently the loop is
* unrolled 16 times, a good value for 486 and pentium-class machines.
* The case c=0 is also optimized, whereas c=1 is not. These
* calls are unfrequent in my typical apps so I did not bother.
*/
#define addmul(dst, src, c, sz) \
if (c != 0) _addmul1(dst, src, c, sz)
#define UNROLL 16 /* 1, 4, 8, 16 */
static void
_addmul1(register gf*restrict dst, const register gf*restrict src, gf c, size_t sz) {
USE_GF_MULC;
const gf* lim = &dst[sz - UNROLL + 1];
GF_MULC0 (c);
#if (UNROLL > 1) /* unrolling by 8/16 is quite effective on the pentium */
for (; dst < lim; dst += UNROLL, src += UNROLL) {
GF_ADDMULC (dst[0], src[0]);
GF_ADDMULC (dst[1], src[1]);
GF_ADDMULC (dst[2], src[2]);
GF_ADDMULC (dst[3], src[3]);
#if (UNROLL > 4)
GF_ADDMULC (dst[4], src[4]);
GF_ADDMULC (dst[5], src[5]);
GF_ADDMULC (dst[6], src[6]);
GF_ADDMULC (dst[7], src[7]);
#endif
#if (UNROLL > 8)
GF_ADDMULC (dst[8], src[8]);
GF_ADDMULC (dst[9], src[9]);
GF_ADDMULC (dst[10], src[10]);
GF_ADDMULC (dst[11], src[11]);
GF_ADDMULC (dst[12], src[12]);
GF_ADDMULC (dst[13], src[13]);
GF_ADDMULC (dst[14], src[14]);
GF_ADDMULC (dst[15], src[15]);
#endif
}
#endif
lim += UNROLL - 1;
for (; dst < lim; dst++, src++) /* final components */
GF_ADDMULC (*dst, *src);
}
/*
* computes C = AB where A is n*k, B is k*m, C is n*m
*/
static void
_matmul(gf * a, gf * b, gf * c, unsigned n, unsigned k, unsigned m) {
unsigned row, col, i;
for (row = 0; row < n; row++) {
for (col = 0; col < m; col++) {
gf *pa = &a[row * k];
gf *pb = &b[col];
gf acc = 0;
for (i = 0; i < k; i++, pa++, pb += m)
acc ^= gf_mul (*pa, *pb);
c[row * m + col] = acc;
}
}
}
/*
* _invert_mat() takes a matrix and produces its inverse
* k is the size of the matrix.
* (Gauss-Jordan, adapted from Numerical Recipes in C)
* Return non-zero if singular.
*/
static void
_invert_mat(gf* src, unsigned k) {
gf c, *p;
unsigned irow = 0;
unsigned icol = 0;
unsigned row, col, i, ix;
unsigned* indxc = (unsigned*) malloc (k * sizeof(unsigned));
unsigned* indxr = (unsigned*) malloc (k * sizeof(unsigned));
unsigned* ipiv = (unsigned*) malloc (k * sizeof(unsigned));
gf *id_row = NEW_GF_MATRIX (1, k);
memset (id_row, '\0', k * sizeof (gf));
/*
* ipiv marks elements already used as pivots.
*/
for (i = 0; i < k; i++)
ipiv[i] = 0;
for (col = 0; col < k; col++) {
gf *pivot_row;
/*
* Zeroing column 'col', look for a non-zero element.
* First try on the diagonal, if it fails, look elsewhere.
*/
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
assert (ipiv[ix] <= 1);
}
}
}
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++)
SWAP (src[irow * k + ix], src[icol * k + ix], gf);
indxr[col] = irow;
indxc[col] = icol;
pivot_row = &src[icol * k];
c = pivot_row[icol];
assert (c != 0);
if (c != 1) { /* otherwhise 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 * sizeof (gf)) != 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;
} /* done all columns */
for (col = k; col > 0; col--)
if (indxr[col-1] != indxc[col-1])
for (row = 0; row < k; row++)
SWAP (src[row * k + indxr[col-1]], src[row * k + indxc[col-1]], gf);
}
/*
* fast code for inverting a vandermonde matrix.
*
* NOTE: 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 my purposes.
* p = coefficients of the matrix (p_i)
* q = values of the polynomial (known)
*/
void
_invert_vdm (gf* src, unsigned k) {
unsigned i, j, row, col;
gf *b, *c, *p;
gf t, xx;
if (k == 1) /* degenerate case, matrix must be p^0 = 1 */
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 = NEW_GF_MATRIX (1, k);
b = NEW_GF_MATRIX (1, k);
p = NEW_GF_MATRIX (1, k);
for (j = 1, i = 0; i < k; i++, j += k) {
c[i] = 0;
p[i] = src[j]; /* p[i] */
}
/*
* 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++) {
gf p_i = p[i]; /* see above comment */
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 - 1; i > 0; i--) {
b[i-1] = c[i] ^ gf_mul (xx, b[i]);
t = gf_mul (xx, t) ^ b[i-1];
}
for (col = 0; col < k; col++)
src[col * k + row] = gf_mul (inverse[t], b[col]);
}
free (c);
free (b);
free (p);
return;
}
static int fec_initialized = 0;
static void
init_fec (void) {
generate_gf();
_init_mul_table();
fec_initialized = 1;
}
/*
* This section contains the proper FEC encoding/decoding routines.
* The encoding matrix is computed starting with a Vandermonde matrix,
* and then transforming it into a systematic matrix.
*/
#define FEC_MAGIC 0xFECC0DEC
void
fec_free (fec_t *p) {
assert (p != NULL && p->magic == (((FEC_MAGIC ^ p->k) ^ p->n) ^ (unsigned long) (p->enc_matrix)));
free (p->enc_matrix);
free (p);
}
fec_t *
fec_new(unsigned k, unsigned n) {
unsigned row, col;
gf *p, *tmp_m;
fec_t *retval;
if (fec_initialized == 0)
init_fec ();
retval = (fec_t *) malloc (sizeof (fec_t));
retval->k = k;
retval->n = n;
retval->enc_matrix = NEW_GF_MATRIX (n, k);
retval->magic = ((FEC_MAGIC ^ k) ^ n) ^ (unsigned long) (retval->enc_matrix);
tmp_m = NEW_GF_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, retval->enc_matrix + k * k, n - k, k, k);
/*
* the upper matrix is I so do not bother with a slow multiply
*/
memset (retval->enc_matrix, '\0', k * k * sizeof (gf));
for (p = retval->enc_matrix, col = 0; col < k; col++, p += k + 1)
*p = 1;
free (tmp_m);
return retval;
}
/* To make sure that we stay within cache in the inner loops of fec_encode()
and fec_decode(). */
#define STRIDE 8192
void
fec_encode(const fec_t* code, const gf*restrict const*restrict const src, gf*restrict const*restrict const fecs, const unsigned*restrict const block_nums, size_t num_block_nums, size_t sz) {
unsigned char i, j;
size_t k;
unsigned fecnum;
const gf* p;
for (k = 0; k < sz; k += STRIDE) {
size_t stride = ((sz-k) < STRIDE)?(sz-k):STRIDE;
for (i=0; i<num_block_nums; i++) {
fecnum=block_nums[i];
assert (fecnum >= code->k);
memset(fecs[i]+k, 0, stride);
p = &(code->enc_matrix[fecnum * code->k]);
for (j = 0; j < code->k; j++)
addmul(fecs[i]+k, src[j]+k, p[j], stride);
}
}
}
/**
* Build decode matrix into some memory space.
*
* @param matrix a space allocated for a k by k matrix
*/
void
build_decode_matrix_into_space(const fec_t*restrict const code, const unsigned*const restrict index, const unsigned k, gf*restrict const matrix) {
unsigned char i;
gf* p;
for (i=0, p=matrix; i < k; i++, p += k) {
if (index[i] < k) {
memset(p, 0, k);
p[i] = 1;
} else {
memcpy(p, &(code->enc_matrix[index[i] * code->k]), k);
}
}
_invert_mat (matrix, k);
}
void
fec_decode(const fec_t* code, const gf*restrict const*restrict const inpkts, gf*restrict const*restrict const outpkts, const unsigned*restrict const index, size_t sz) {
gf* m_dec = (gf*)alloca(code->k * code->k);
unsigned char outix=0;
unsigned char row=0;
unsigned char col=0;
build_decode_matrix_into_space(code, index, code->k, m_dec);
for (row=0; row<code->k; row++) {
assert ((index[row] >= code->k) || (index[row] == row)); /* If the block whose number is i is present, then it is required to be in the i'th element. */
if (index[row] >= code->k) {
memset(outpkts[outix], 0, sz);
for (col=0; col < code->k; col++)
addmul(outpkts[outix], inpkts[col], m_dec[row * code->k + col], sz);
outix++;
}
}
}
/**
* zfec -- fast forward error correction library with Python interface
*
* Copyright (C) 2007 Allmydata, Inc.
* Author: Zooko Wilcox-O'Hearn
*
* This file is part of zfec.
*
* See README.txt for licensing information.
*/
/*
* This work is derived from the "fec" software by Luigi Rizzo, et al., the
* copyright notice and licence terms of which are included below for reference.
* 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
*
* Modifications by Dan Rubenstein (see Modifications.txt for
* their description.
* Modifications (C) 1998 Dan Rubenstein (drubenst@cs.umass.edu)
*
* 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.
*/