043a1f21869617869d2c6d0573bf71777611e2a3
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 \a 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 parms 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
);