1 /** \file fec.c Forward error correction based on Vandermonde matrices. */
5 * (C) 1997-98 Luigi Rizzo (luigi@iet.unipi.it)
7 * Portions derived from code by Phil Karn (karn@ka9q.ampr.org),
8 * Robert Morelos-Zaragoza (robert@spectra.eng.hawaii.edu) and Hari
9 * Thirumoorthy (harit@spectra.eng.hawaii.edu), Aug 1995
11 * Redistribution and use in source and binary forms, with or without
12 * modification, are permitted provided that the following conditions
15 * 1. Redistributions of source code must retain the above copyright
16 * notice, this list of conditions and the following disclaimer.
17 * 2. Redistributions in binary form must reproduce the above
18 * copyright notice, this list of conditions and the following
19 * disclaimer in the documentation and/or other materials
20 * provided with the distribution.
22 * THIS SOFTWARE IS PROVIDED BY THE AUTHORS ``AS IS'' AND
23 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO,
24 * THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A
25 * PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHORS
26 * BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY,
27 * OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO,
28 * PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA,
29 * OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
30 * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR
31 * TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT
32 * OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY
40 #include "portable_io.h"
44 #define GF_BITS 8 /* code over GF(256) */
45 #define GF_SIZE ((1 << GF_BITS) - 1)
48 * To speed up computations, we have tables for logarithm, exponent and inverse
49 * of a number. We use a table for multiplication as well (it takes 64K, no big
50 * deal even on a PDA, especially because it can be pre-initialized an put into
51 * a ROM!). The macro gf_mul(x,y) takes care of multiplications.
53 static unsigned char gf_exp
[2 * GF_SIZE
]; /* index->poly form conversion table */
54 static int gf_log
[GF_SIZE
+ 1]; /* Poly->index form conversion table */
55 static unsigned char inverse
[GF_SIZE
+ 1]; /* inverse of field elem. */
56 static unsigned char gf_mul_table
[GF_SIZE
+ 1][GF_SIZE
+ 1];
57 /* Multiply two numbers. */
58 #define gf_mul(x,y) gf_mul_table[x][y]
60 /* Compute x % GF_SIZE without a slow divide. */
61 static inline unsigned char modnn(int x
)
63 while (x
>= GF_SIZE
) {
65 x
= (x
>> GF_BITS
) + (x
& GF_SIZE
);
70 static void init_mul_table(void)
73 for (i
= 0; i
< GF_SIZE
+ 1; i
++)
74 for (j
= 0; j
< GF_SIZE
+ 1; j
++)
76 gf_exp
[modnn(gf_log
[i
] + gf_log
[j
])];
78 for (j
= 0; j
< GF_SIZE
+ 1; j
++)
79 gf_mul_table
[0][j
] = gf_mul_table
[j
][0] = 0;
82 static unsigned char *alloc_matrix(int rows
, int cols
)
84 return para_malloc(rows
* cols
);
88 * Initialize the data structures used for computations in GF.
90 * This generates GF(2**GF_BITS) from the irreducible polynomial p(X) in
94 * index->polynomial form gf_exp[] contains j= \alpha^i;
95 * polynomial form -> index form gf_log[ j = \alpha^i ] = i
96 * \alpha=x is the primitive element of GF(2^m)
98 * For efficiency, gf_exp[] has size 2*GF_SIZE, so that a simple
99 * multiplication of two numbers can be resolved without calling modnn
101 static void generate_gf(void)
104 unsigned char mask
= 1;
105 char *pp
= "101110001"; /* The primitive polynomial 1+x^2+x^3+x^4+x^8 */
106 gf_exp
[GF_BITS
] = 0; /* will be updated at the end of the 1st loop */
109 * first, generate the (polynomial representation of) powers of \alpha,
110 * which are stored in gf_exp[i] = \alpha ** i .
111 * At the same time build gf_log[gf_exp[i]] = i .
112 * The first GF_BITS powers are simply bits shifted to the left.
114 for (i
= 0; i
< GF_BITS
; i
++, mask
<<= 1) {
116 gf_log
[gf_exp
[i
]] = i
;
118 * If pp[i] == 1 then \alpha ** i occurs in poly-repr
119 * gf_exp[GF_BITS] = \alpha ** GF_BITS
122 gf_exp
[GF_BITS
] ^= mask
;
125 * now gf_exp[GF_BITS] = \alpha ** GF_BITS is complete, so can also
126 * compute its inverse.
128 gf_log
[gf_exp
[GF_BITS
]] = GF_BITS
;
130 * Poly-repr of \alpha ** (i+1) is given by poly-repr of \alpha ** i
131 * shifted left one-bit and accounting for any \alpha ** GF_BITS term
132 * that may occur when poly-repr of \alpha ** i is shifted.
134 mask
= 1 << (GF_BITS
- 1);
135 for (i
= GF_BITS
+ 1; i
< GF_SIZE
; i
++) {
136 if (gf_exp
[i
- 1] >= mask
)
138 gf_exp
[GF_BITS
] ^ ((gf_exp
[i
- 1] ^ mask
) << 1);
140 gf_exp
[i
] = gf_exp
[i
- 1] << 1;
141 gf_log
[gf_exp
[i
]] = i
;
144 * log(0) is not defined, so use a special value
147 /* set the extended gf_exp values for fast multiply */
148 for (i
= 0; i
< GF_SIZE
; i
++)
149 gf_exp
[i
+ GF_SIZE
] = gf_exp
[i
];
151 inverse
[0] = 0; /* 0 has no inverse. */
153 for (i
= 2; i
<= GF_SIZE
; i
++)
154 inverse
[i
] = gf_exp
[GF_SIZE
- gf_log
[i
]];
158 * Compute dst[] = dst[] + c * src[]
160 * This is used often, so better optimize it! Currently the loop is unrolled 16
161 * times. The case c=0 is also optimized, whereas c=1 is not.
164 static void addmul(unsigned char *dst1
, const unsigned char const *src1
,
165 unsigned char c
, int sz
)
169 unsigned char *dst
= dst1
, *lim
= &dst
[sz
- UNROLL
+ 1],
170 *col
= gf_mul_table
[c
];
171 const unsigned char const *src
= src1
;
173 for (; dst
< lim
; dst
+= UNROLL
, src
+= UNROLL
) {
174 dst
[0] ^= col
[src
[0]];
175 dst
[1] ^= col
[src
[1]];
176 dst
[2] ^= col
[src
[2]];
177 dst
[3] ^= col
[src
[3]];
178 dst
[4] ^= col
[src
[4]];
179 dst
[5] ^= col
[src
[5]];
180 dst
[6] ^= col
[src
[6]];
181 dst
[7] ^= col
[src
[7]];
182 dst
[8] ^= col
[src
[8]];
183 dst
[9] ^= col
[src
[9]];
184 dst
[10] ^= col
[src
[10]];
185 dst
[11] ^= col
[src
[11]];
186 dst
[12] ^= col
[src
[12]];
187 dst
[13] ^= col
[src
[13]];
188 dst
[14] ^= col
[src
[14]];
189 dst
[15] ^= col
[src
[15]];
192 for (; dst
< lim
; dst
++, src
++) /* final components */
197 * Compute C = AB where A is n*k, B is k*m, C is n*m
199 static void matmul(unsigned char *a
, unsigned char *b
, unsigned char *c
,
204 for (row
= 0; row
< n
; row
++) {
205 for (col
= 0; col
< m
; col
++) {
206 unsigned char *pa
= &a
[row
* k
], *pb
= &b
[col
], acc
= 0;
207 for (i
= 0; i
< k
; i
++, pa
++, pb
+= m
)
208 acc
^= gf_mul(*pa
, *pb
);
209 c
[row
* m
+ col
] = acc
;
214 #define FEC_SWAP(a,b) {typeof(a) tmp = a; a = b; b = tmp;}
217 * Compute the inverse of a matrix.
219 * k is the size of the matrix 'src' (Gauss-Jordan, adapted from Numerical
220 * Recipes in C). Returns negative on errors.
222 static int invert_mat(unsigned char *src
, int k
)
224 int irow
, icol
, row
, col
, ix
, error
;
225 int *indxc
= para_malloc(k
* sizeof(int));
226 int *indxr
= para_malloc(k
* sizeof(int));
227 int *ipiv
= para_malloc(k
* sizeof(int)); /* elements used as pivots */
228 unsigned char c
, *p
, *id_row
= alloc_matrix(1, k
),
229 *temp_row
= alloc_matrix(1, k
);
231 memset(id_row
, 0, k
);
232 memset(ipiv
, 0, k
* sizeof(int));
234 for (col
= 0; col
< k
; col
++) {
235 unsigned char *pivot_row
;
237 * Zeroing column 'col', look for a non-zero element.
238 * First try on the diagonal, if it fails, look elsewhere.
241 if (ipiv
[col
] != 1 && src
[col
* k
+ col
] != 0) {
246 for (row
= 0; row
< k
; row
++) {
247 if (ipiv
[row
] != 1) {
248 for (ix
= 0; ix
< k
; ix
++) {
250 if (src
[row
* k
+ ix
] != 0) {
255 } else if (ipiv
[ix
] > 1) {
256 error
= -E_FEC_PIVOT
;
262 error
= -E_FEC_PIVOT
;
268 * swap rows irow and icol, so afterwards the diagonal element
269 * will be correct. Rarely done, not worth optimizing.
272 for (ix
= 0; ix
< k
; ix
++)
273 FEC_SWAP(src
[irow
* k
+ ix
], src
[icol
* k
+ ix
]);
276 pivot_row
= &src
[icol
* k
];
277 error
= -E_FEC_SINGULAR
;
281 if (c
!= 1) { /* otherwise this is a NOP */
283 * this is done often , but optimizing is not so
284 * fruitful, at least in the obvious ways (unrolling)
288 for (ix
= 0; ix
< k
; ix
++)
289 pivot_row
[ix
] = gf_mul(c
, pivot_row
[ix
]);
292 * from all rows, remove multiples of the selected row to zero
293 * the relevant entry (in fact, the entry is not zero because
294 * we know it must be zero). (Here, if we know that the
295 * pivot_row is the identity, we can optimize the addmul).
298 if (memcmp(pivot_row
, id_row
, k
) != 0) {
299 for (p
= src
, ix
= 0; ix
< k
; ix
++, p
+= k
) {
303 addmul(p
, pivot_row
, c
, k
);
309 for (col
= k
- 1; col
>= 0; col
--) {
310 if (indxr
[col
] < 0 || indxr
[col
] >= k
)
311 PARA_CRIT_LOG("AARGH, indxr[col] %d\n", indxr
[col
]);
312 else if (indxc
[col
] < 0 || indxc
[col
] >= k
)
313 PARA_CRIT_LOG("AARGH, indxc[col] %d\n", indxc
[col
]);
314 else if (indxr
[col
] != indxc
[col
]) {
315 for (row
= 0; row
< k
; row
++) {
316 FEC_SWAP(src
[row
* k
+ indxr
[col
]],
317 src
[row
* k
+ indxc
[col
]]);
332 * Invert a Vandermonde matrix.
334 * It assumes that the matrix is not singular and _IS_ a Vandermonde matrix.
335 * Only uses the second column of the matrix, containing the p_i's.
337 * Algorithm borrowed from "Numerical recipes in C" -- sec.2.8, but largely
338 * revised for GF purposes.
340 static void invert_vdm(unsigned char *src
, int k
)
343 unsigned char *b
, *c
, *p
, t
, xx
;
345 if (k
== 1) /* degenerate */
348 * c holds the coefficient of P(x) = Prod (x - p_i), i=0..k-1
349 * b holds the coefficient for the matrix inversion
355 for (j
= 1, i
= 0; i
< k
; i
++, j
+= k
) {
360 * construct coeffs recursively. We know c[k] = 1 (implicit) and start
361 * P_0 = x - p_0, then at each stage multiply by x - p_i generating P_i
362 * = x P_{i-1} - p_i P_{i-1} After k steps we are done.
364 c
[k
- 1] = p
[0]; /* really -p(0), but x = -x in GF(2^m) */
365 for (i
= 1; i
< k
; i
++) {
366 unsigned char p_i
= p
[i
];
367 for (j
= k
- 1 - (i
- 1); j
< k
- 1; j
++)
368 c
[j
] ^= gf_mul(p_i
, c
[j
+ 1]);
372 for (row
= 0; row
< k
; row
++) {
374 * synthetic division etc.
378 b
[k
- 1] = 1; /* this is in fact c[k] */
379 for (i
= k
- 2; i
>= 0; i
--) {
380 b
[i
] = c
[i
+ 1] ^ gf_mul(xx
, b
[i
+ 1]);
381 t
= gf_mul(xx
, t
) ^ b
[i
];
383 for (col
= 0; col
< k
; col
++)
384 src
[col
* k
+ row
] = gf_mul(inverse
[t
], b
[col
]);
391 static int fec_initialized
;
393 static void init_fec(void)
400 /** Internal FEC parameters. */
402 /** Number of data slices. */
404 /** Number of slices (including redundant slices). */
406 /** The encoding matrix, computed by init_fec(). */
407 unsigned char *enc_matrix
;
411 * Deallocate a fec params structure.
413 * \param p The structure to free.
415 void fec_free(struct fec_parms
*p
)
424 * Create a new encoder and return an opaque descriptor to it.
426 * \param k Number of input slices.
427 * \param n Number of output slices.
428 * \param result On success the Fec descriptor is returned here.
432 * This creates the k*n encoding matrix. It is computed starting with a
433 * Vandermonde matrix, and then transformed into a systematic matrix.
435 int fec_new(int k
, int n
, struct fec_parms
**result
)
438 unsigned char *p
, *tmp_m
;
439 struct fec_parms
*parms
;
441 if (!fec_initialized
)
444 if (k
< 1 || k
> GF_SIZE
+ 1 || n
> GF_SIZE
+ 1 || k
> n
)
446 parms
= para_malloc(sizeof(struct fec_parms
));
449 parms
->enc_matrix
= alloc_matrix(n
, k
);
450 tmp_m
= alloc_matrix(n
, k
);
452 * fill the matrix with powers of field elements, starting from 0.
453 * The first row is special, cannot be computed with exp. table.
456 for (col
= 1; col
< k
; col
++)
458 for (p
= tmp_m
+ k
, row
= 0; row
< n
- 1; row
++, p
+= k
) {
459 for (col
= 0; col
< k
; col
++)
460 p
[col
] = gf_exp
[modnn(row
* col
)];
464 * quick code to build systematic matrix: invert the top
465 * k*k vandermonde matrix, multiply right the bottom n-k rows
466 * by the inverse, and construct the identity matrix at the top.
468 invert_vdm(tmp_m
, k
); /* much faster than invert_mat */
469 matmul(tmp_m
+ k
* k
, tmp_m
, parms
->enc_matrix
+ k
* k
, n
- k
, k
, k
);
471 * the upper matrix is I so do not bother with a slow multiply
473 memset(parms
->enc_matrix
, 0, k
* k
);
474 for (p
= parms
->enc_matrix
, col
= 0; col
< k
; col
++, p
+= k
+ 1)
482 * Compute one encoded slice of the given input.
484 * \param parms The fec parameters returned earlier by fec_new().
485 * \param src The \a k data slices to encode.
486 * \param dst Result pointer.
487 * \param idx The index of the slice to compute.
488 * \param sz The size of the input data packets.
490 * Encode the \a k slices of size \a sz given by \a src and store the output
491 * slice number \a idx in \a dst.
493 void fec_encode(struct fec_parms
*parms
, const unsigned char * const *src
,
494 unsigned char *dst
, int idx
, int sz
)
499 assert(idx
<= parms
->n
);
502 memcpy(dst
, src
[idx
], sz
);
505 p
= &(parms
->enc_matrix
[idx
* k
]);
507 for (i
= 0; i
< k
; i
++)
508 addmul(dst
, src
[i
], p
[i
], sz
);
511 /* Move src packets in their position. */
512 static int shuffle(unsigned char **data
, int *idx
, int k
)
516 for (i
= 0; i
< k
;) {
517 if (idx
[i
] >= k
|| idx
[i
] == i
)
519 else { /* put index and data at the right position */
522 if (idx
[c
] == c
) /* conflict */
523 return -E_FEC_BAD_IDX
;
524 FEC_SWAP(idx
[i
], idx
[c
]);
525 FEC_SWAP(data
[i
], data
[c
]);
532 * Construct the decoding matrix given the indices. The encoding matrix must
533 * already be allocated.
535 static int build_decode_matrix(struct fec_parms
*parms
, int *idx
,
536 unsigned char **result
)
538 int ret
= -E_FEC_BAD_IDX
, i
, k
= parms
->k
;
539 unsigned char *p
, *matrix
= alloc_matrix(k
, k
);
541 for (i
= 0, p
= matrix
; i
< k
; i
++, p
+= k
) {
542 if (idx
[i
] >= parms
->n
) /* invalid index */
548 memcpy(p
, &(parms
->enc_matrix
[idx
[i
] * k
]), k
);
550 ret
= invert_mat(matrix
, k
);
562 * Decode one slice from the group of received slices.
564 * \param parms Pointer to fec params structure.
565 * \param data Pointers to received packets.
566 * \param idx Pointer to packet indices (gets modified).
567 * \param sz Size of each packet.
569 * \return Zero on success, -1 on errors.
571 * The \a data vector of received slices and the indices of slices are used to
572 * produce the correct output slice. The data slices are modified in-place.
574 int fec_decode(struct fec_parms
*parms
, unsigned char **data
, int *idx
,
577 unsigned char *m_dec
, **slice
;
578 int ret
, row
, col
, k
= parms
->k
;
580 ret
= shuffle(data
, idx
, k
);
583 ret
= build_decode_matrix(parms
, idx
, &m_dec
);
586 /* do the actual decoding */
587 slice
= para_malloc(k
* sizeof(unsigned char *));
588 for (row
= 0; row
< k
; row
++) {
590 slice
[row
] = para_calloc(sz
);
591 for (col
= 0; col
< k
; col
++)
592 addmul(slice
[row
], data
[col
],
593 m_dec
[row
* k
+ col
], sz
);
596 /* move slices to their final destination */
597 for (row
= 0; row
< k
; row
++) {
599 memcpy(data
[row
], slice
[row
], sz
);