2 * fec.c -- forward error correction based on Vandermonde matrices
4 * (C) 1997-98 Luigi Rizzo (luigi@iet.unipi.it)
6 * Portions derived from code by Phil Karn (karn@ka9q.ampr.org),
7 * Robert Morelos-Zaragoza (robert@spectra.eng.hawaii.edu) and Hari
8 * Thirumoorthy (harit@spectra.eng.hawaii.edu), Aug 1995
10 * Redistribution and use in source and binary forms, with or without
11 * modification, are permitted provided that the following conditions
14 * 1. Redistributions of source code must retain the above copyright
15 * notice, this list of conditions and the following disclaimer.
16 * 2. Redistributions in binary form must reproduce the above
17 * copyright notice, this list of conditions and the following
18 * disclaimer in the documentation and/or other materials
19 * provided with the distribution.
21 * THIS SOFTWARE IS PROVIDED BY THE AUTHORS ``AS IS'' AND
22 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO,
23 * THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
24 * PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHORS
25 * BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY,
26 * OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
27 * PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA,
28 * OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
29 * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR
30 * TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT
31 * OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY
37 #include "portable_io.h"
41 #define GF_BITS 8 /* code over GF(256) */
42 #define GF_SIZE ((1 << GF_BITS) - 1)
45 * To speed up computations, we have tables for logarithm, exponent and inverse
46 * of a number. We use a table for multiplication as well (it takes 64K, no big
47 * deal even on a PDA, especially because it can be pre-initialized an put into
48 * a ROM!). The macro gf_mul(x,y) takes care of multiplications.
50 static unsigned char gf_exp[2 * GF_SIZE]; /* index->poly form conversion table */
51 static int gf_log[GF_SIZE + 1]; /* Poly->index form conversion table */
52 static unsigned char inverse[GF_SIZE + 1]; /* inverse of field elem. */
53 static unsigned char gf_mul_table[GF_SIZE + 1][GF_SIZE + 1];
54 /* Multiply two numbers. */
55 #define gf_mul(x,y) gf_mul_table[x][y]
57 /* Compute x % GF_SIZE without a slow divide. */
58 static inline unsigned char modnn(int x)
60 while (x >= GF_SIZE) {
62 x = (x >> GF_BITS) + (x & GF_SIZE);
67 static void init_mul_table(void)
70 for (i = 0; i < GF_SIZE + 1; i++)
71 for (j = 0; j < GF_SIZE + 1; j++)
73 gf_exp[modnn(gf_log[i] + gf_log[j])];
75 for (j = 0; j < GF_SIZE + 1; j++)
76 gf_mul_table[0][j] = gf_mul_table[j][0] = 0;
79 static unsigned char *alloc_matrix(int rows, int cols)
81 return para_malloc(rows * cols);
85 * Initialize the data structures used for computations in GF.
87 * This generates GF(2**GF_BITS) from the irreducible polynomial p(X) in
91 * index->polynomial form gf_exp[] contains j= \alpha^i;
92 * polynomial form -> index form gf_log[ j = \alpha^i ] = i
93 * \alpha=x is the primitive element of GF(2^m)
95 * For efficiency, gf_exp[] has size 2*GF_SIZE, so that a simple
96 * multiplication of two numbers can be resolved without calling modnn
98 static void generate_gf(void)
101 unsigned char mask = 1;
102 char *pp = "101110001"; /* The primitive polynomial 1+x^2+x^3+x^4+x^8 */
103 gf_exp[GF_BITS] = 0; /* will be updated at the end of the 1st loop */
106 * first, generate the (polynomial representation of) powers of \alpha,
107 * which are stored in gf_exp[i] = \alpha ** i .
108 * At the same time build gf_log[gf_exp[i]] = i .
109 * The first GF_BITS powers are simply bits shifted to the left.
111 for (i = 0; i < GF_BITS; i++, mask <<= 1) {
113 gf_log[gf_exp[i]] = i;
115 * If pp[i] == 1 then \alpha ** i occurs in poly-repr
116 * gf_exp[GF_BITS] = \alpha ** GF_BITS
119 gf_exp[GF_BITS] ^= mask;
122 * now gf_exp[GF_BITS] = \alpha ** GF_BITS is complete, so can also
123 * compute its inverse.
125 gf_log[gf_exp[GF_BITS]] = GF_BITS;
127 * Poly-repr of \alpha ** (i+1) is given by poly-repr of \alpha ** i
128 * shifted left one-bit and accounting for any \alpha ** GF_BITS term
129 * that may occur when poly-repr of \alpha ** i is shifted.
131 mask = 1 << (GF_BITS - 1);
132 for (i = GF_BITS + 1; i < GF_SIZE; i++) {
133 if (gf_exp[i - 1] >= mask)
135 gf_exp[GF_BITS] ^ ((gf_exp[i - 1] ^ mask) << 1);
137 gf_exp[i] = gf_exp[i - 1] << 1;
138 gf_log[gf_exp[i]] = i;
141 * log(0) is not defined, so use a special value
144 /* set the extended gf_exp values for fast multiply */
145 for (i = 0; i < GF_SIZE; i++)
146 gf_exp[i + GF_SIZE] = gf_exp[i];
148 inverse[0] = 0; /* 0 has no inverse. */
150 for (i = 2; i <= GF_SIZE; i++)
151 inverse[i] = gf_exp[GF_SIZE - gf_log[i]];
155 * Compute dst[] = dst[] + c * src[]
157 * This is used often, so better optimize it! Currently the loop is unrolled 16
158 * times. The case c=0 is also optimized, whereas c=1 is not.
161 static void addmul(unsigned char *dst1, const unsigned char const *src1,
162 unsigned char c, int sz)
166 unsigned char *dst = dst1, *lim = &dst[sz - UNROLL + 1],
167 *col = gf_mul_table[c];
168 const unsigned char const *src = src1;
170 for (; dst < lim; dst += UNROLL, src += UNROLL) {
171 dst[0] ^= col[src[0]];
172 dst[1] ^= col[src[1]];
173 dst[2] ^= col[src[2]];
174 dst[3] ^= col[src[3]];
175 dst[4] ^= col[src[4]];
176 dst[5] ^= col[src[5]];
177 dst[6] ^= col[src[6]];
178 dst[7] ^= col[src[7]];
179 dst[8] ^= col[src[8]];
180 dst[9] ^= col[src[9]];
181 dst[10] ^= col[src[10]];
182 dst[11] ^= col[src[11]];
183 dst[12] ^= col[src[12]];
184 dst[13] ^= col[src[13]];
185 dst[14] ^= col[src[14]];
186 dst[15] ^= col[src[15]];
189 for (; dst < lim; dst++, src++) /* final components */
194 * Compute C = AB where A is n*k, B is k*m, C is n*m
196 static void matmul(unsigned char *a, unsigned char *b, unsigned char *c,
201 for (row = 0; row < n; row++) {
202 for (col = 0; col < m; col++) {
203 unsigned char *pa = &a[row * k], *pb = &b[col], acc = 0;
204 for (i = 0; i < k; i++, pa++, pb += m)
205 acc ^= gf_mul(*pa, *pb);
206 c[row * m + col] = acc;
211 #define FEC_SWAP(a,b) {typeof(a) tmp = a; a = b; b = tmp;}
214 * Compute the inverse of a matrix.
216 * k is the size of the matrix 'src' (Gauss-Jordan, adapted from Numerical
217 * Recipes in C). Returns -1 if 'src' is singular.
219 static int invert_mat(unsigned char *src, int k)
221 int irow, icol, row, col, ix, error;
222 int *indxc = para_malloc(k * sizeof(int));
223 int *indxr = para_malloc(k * sizeof(int));
224 int *ipiv = para_malloc(k * sizeof(int)); /* elements used as pivots */
225 unsigned char c, *p, *id_row = alloc_matrix(1, k),
226 *temp_row = alloc_matrix(1, k);
228 memset(id_row, 0, k);
229 memset(ipiv, 0, k * sizeof(int));
231 for (col = 0; col < k; col++) {
232 unsigned char *pivot_row;
234 * Zeroing column 'col', look for a non-zero element.
235 * First try on the diagonal, if it fails, look elsewhere.
238 if (ipiv[col] != 1 && src[col * k + col] != 0) {
243 for (row = 0; row < k; row++) {
244 if (ipiv[row] != 1) {
245 for (ix = 0; ix < k; ix++) {
247 if (src[row * k + ix] != 0) {
252 } else if (ipiv[ix] > 1) {
253 error = -E_FEC_PIVOT;
259 error = -E_FEC_PIVOT;
265 * swap rows irow and icol, so afterwards the diagonal element
266 * will be correct. Rarely done, not worth optimizing.
269 for (ix = 0; ix < k; ix++)
270 FEC_SWAP(src[irow * k + ix], src[icol * k + ix]);
273 pivot_row = &src[icol * k];
274 error = -E_FEC_SINGULAR;
278 if (c != 1) { /* otherwise this is a NOP */
280 * this is done often , but optimizing is not so
281 * fruitful, at least in the obvious ways (unrolling)
285 for (ix = 0; ix < k; ix++)
286 pivot_row[ix] = gf_mul(c, pivot_row[ix]);
289 * from all rows, remove multiples of the selected row to zero
290 * the relevant entry (in fact, the entry is not zero because
291 * we know it must be zero). (Here, if we know that the
292 * pivot_row is the identity, we can optimize the addmul).
295 if (memcmp(pivot_row, id_row, k) != 0) {
296 for (p = src, ix = 0; ix < k; ix++, p += k) {
300 addmul(p, pivot_row, c, k);
306 for (col = k - 1; col >= 0; col--) {
307 if (indxr[col] < 0 || indxr[col] >= k)
308 PARA_CRIT_LOG("AARGH, indxr[col] %d\n", indxr[col]);
309 else if (indxc[col] < 0 || indxc[col] >= k)
310 PARA_CRIT_LOG("AARGH, indxc[col] %d\n", indxc[col]);
311 else if (indxr[col] != indxc[col]) {
312 for (row = 0; row < k; row++) {
313 FEC_SWAP(src[row * k + indxr[col]],
314 src[row * k + indxc[col]]);
329 * Invert a Vandermonde matrix.
331 * It assumes that the matrix is not singular and _IS_ a Vandermonde matrix.
332 * Only uses the second column of the matrix, containing the p_i's.
334 * Algorithm borrowed from "Numerical recipes in C" -- sec.2.8, but largely
335 * revised for GF purposes.
337 static void invert_vdm(unsigned char *src, int k)
340 unsigned char *b, *c, *p, t, xx;
342 if (k == 1) /* degenerate */
345 * c holds the coefficient of P(x) = Prod (x - p_i), i=0..k-1
346 * b holds the coefficient for the matrix inversion
352 for (j = 1, i = 0; i < k; i++, j += k) {
357 * construct coeffs recursively. We know c[k] = 1 (implicit) and start
358 * P_0 = x - p_0, then at each stage multiply by x - p_i generating P_i
359 * = x P_{i-1} - p_i P_{i-1} After k steps we are done.
361 c[k - 1] = p[0]; /* really -p(0), but x = -x in GF(2^m) */
362 for (i = 1; i < k; i++) {
363 unsigned char p_i = p[i];
364 for (j = k - 1 - (i - 1); j < k - 1; j++)
365 c[j] ^= gf_mul(p_i, c[j + 1]);
369 for (row = 0; row < k; row++) {
371 * synthetic division etc.
375 b[k - 1] = 1; /* this is in fact c[k] */
376 for (i = k - 2; i >= 0; i--) {
377 b[i] = c[i + 1] ^ gf_mul(xx, b[i + 1]);
378 t = gf_mul(xx, t) ^ b[i];
380 for (col = 0; col < k; col++)
381 src[col * k + row] = gf_mul(inverse[t], b[col]);
388 static int fec_initialized;
390 static void init_fec(void)
398 int k, n; /* parameters of the code */
399 unsigned char *enc_matrix;
403 * Deallocate a fec params structure.
405 * \param p The structure to free.
407 void fec_free(struct fec_parms *p)
416 * Create a new encoder and return an opaque descriptor to it.
418 * \param k Number of input slices.
419 * \param n Number of output slices.
420 * \param result On success the Fec descriptor is returned here.
424 * This creates the k*n encoding matrix. It is computed starting with a
425 * Vandermonde matrix, and then transformed into a systematic matrix.
427 int fec_new(int k, int n, struct fec_parms **result)
430 unsigned char *p, *tmp_m;
431 struct fec_parms *parms;
433 if (!fec_initialized)
436 if (k < 1 || k > GF_SIZE + 1 || n > GF_SIZE + 1 || k > n)
438 parms = para_malloc(sizeof(struct fec_parms));
441 parms->enc_matrix = alloc_matrix(n, k);
442 tmp_m = alloc_matrix(n, k);
444 * fill the matrix with powers of field elements, starting from 0.
445 * The first row is special, cannot be computed with exp. table.
448 for (col = 1; col < k; col++)
450 for (p = tmp_m + k, row = 0; row < n - 1; row++, p += k) {
451 for (col = 0; col < k; col++)
452 p[col] = gf_exp[modnn(row * col)];
456 * quick code to build systematic matrix: invert the top
457 * k*k vandermonde matrix, multiply right the bottom n-k rows
458 * by the inverse, and construct the identity matrix at the top.
460 invert_vdm(tmp_m, k); /* much faster than invert_mat */
461 matmul(tmp_m + k * k, tmp_m, parms->enc_matrix + k * k, n - k, k, k);
463 * the upper matrix is I so do not bother with a slow multiply
465 memset(parms->enc_matrix, 0, k * k);
466 for (p = parms->enc_matrix, col = 0; col < k; col++, p += k + 1)
474 * Compute one encoded slice of the given input.
476 * \param parms The fec parameters returned earlier by fec_new().
477 * \param src The \a k data slices to encode.
478 * \param dst Result pointer.
479 * \param idx The index of the slice to compute.
480 * \param sz The size of the input data packets.
482 * Encode the \a k slices of size \a sz given by \a src and store the output
483 * slice number \a idx in \dst.
485 void fec_encode(struct fec_parms *parms, const unsigned char * const *src,
486 unsigned char *dst, int idx, int sz)
491 assert(idx <= parms->n);
494 memcpy(dst, src[idx], sz);
497 p = &(parms->enc_matrix[idx * k]);
499 for (i = 0; i < k; i++)
500 addmul(dst, src[i], p[i], sz);
503 /* Move src packets in their position. */
504 static int shuffle(unsigned char **data, int *idx, int k)
508 for (i = 0; i < k;) {
509 if (idx[i] >= k || idx[i] == i)
511 else { /* put index and data at the right position */
514 if (idx[c] == c) /* conflict */
515 return -E_FEC_BAD_IDX;
516 FEC_SWAP(idx[i], idx[c]);
517 FEC_SWAP(data[i], data[c]);
524 * Construct the decoding matrix given the indices. The encoding matrix must
525 * already be allocated.
527 static int build_decode_matrix(struct fec_parms *parms, int *idx,
528 unsigned char **result)
530 int ret = -E_FEC_BAD_IDX, i, k = parms->k;
531 unsigned char *p, *matrix = alloc_matrix(k, k);
533 for (i = 0, p = matrix; i < k; i++, p += k) {
534 if (idx[i] >= parms->n) /* invalid index */
540 memcpy(p, &(parms->enc_matrix[idx[i] * k]), k);
542 ret = invert_mat(matrix, k);
554 * Decode one slice from the group of received slices.
556 * \param code Pointer to fec params structure.
557 * \param data Pointers to received packets.
558 * \param idx Pointer to packet indices (gets modified).
559 * \param sz Size of each packet.
561 * \return Zero on success, -1 on errors.
563 * The \a data vector of received slices and the indices of slices are used to
564 * produce the correct output slice. The data slices are modified in-place.
566 int fec_decode(struct fec_parms *parms, unsigned char **data, int *idx,
569 unsigned char *m_dec, **slice;
570 int ret, row, col, k = parms->k;
572 ret = shuffle(data, idx, k);
575 ret = build_decode_matrix(parms, idx, &m_dec);
578 /* do the actual decoding */
579 slice = para_malloc(k * sizeof(unsigned char *));
580 for (row = 0; row < k; row++) {
582 slice[row] = para_calloc(sz);
583 for (col = 0; col < k; col++)
584 addmul(slice[row], data[col],
585 m_dec[row * k + col], sz);
588 /* move slices to their final destination */
589 for (row = 0; row < k; row++) {
591 memcpy(data[row], slice[row], sz);