RMT V. Roca
Internet-Draft INRIA
Intended status: Standards Track C. Neumann
Expires: July 26, 2008 Thomson
D. Furodet
STMicroelectronics
January 23, 2008
Low Density Parity Check (LDPC) Staircase and Triangle Forward Error
Correction (FEC) Schemes
draft-ietf-rmt-bb-fec-ldpc-08.txt
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Abstract
This document describes two Fully-Specified FEC Schemes, LDPC-
Staircase and LDPC-Triangle, and their application to the reliable
delivery of data objects on the packet erasure channel (i.e., a
communication path where packets are either received without any
corruption or discarded during transmission). These systematic FEC
codes belong to the well known class of ``Low Density Parity Check''
(LDPC) codes, and are large block FEC codes in the sense of RFC3453.
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 5
2. Requirements notation . . . . . . . . . . . . . . . . . . . . 6
3. Definitions, Notations and Abbreviations . . . . . . . . . . . 7
3.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . 7
3.2. Notations . . . . . . . . . . . . . . . . . . . . . . . . 7
3.3. Abbreviations . . . . . . . . . . . . . . . . . . . . . . 8
4. Formats and Codes . . . . . . . . . . . . . . . . . . . . . . 9
4.1. FEC Payload IDs . . . . . . . . . . . . . . . . . . . . . 9
4.2. FEC Object Transmission Information . . . . . . . . . . . 9
4.2.1. Mandatory Element . . . . . . . . . . . . . . . . . . 9
4.2.2. Common Elements . . . . . . . . . . . . . . . . . . . 9
4.2.3. Scheme-Specific Elements . . . . . . . . . . . . . . . 10
4.2.4. Encoding Format . . . . . . . . . . . . . . . . . . . 10
5. Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5.1. General . . . . . . . . . . . . . . . . . . . . . . . . . 13
5.2. Determining the Maximum Source Block Length (B) . . . . . 14
5.3. Determining the Encoding Symbol Length (E) and Number
of Encoding Symbols per Group (G) . . . . . . . . . . . . 15
5.4. Determining the Maximum Number of Encoding Symbols
Generated for Any Source Block (max_n) . . . . . . . . . . 16
5.5. Determining the Number of Encoding Symbols of a Block
(n) . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5.6. Identifying the G Symbols of an Encoding Symbol Group . . 17
5.7. Pseudo Random Number Generator . . . . . . . . . . . . . . 21
6. Full Specification of the LDPC-Staircase Scheme . . . . . . . 23
6.1. General . . . . . . . . . . . . . . . . . . . . . . . . . 23
6.2. Parity Check Matrix Creation . . . . . . . . . . . . . . . 23
6.3. Encoding . . . . . . . . . . . . . . . . . . . . . . . . . 25
6.4. Decoding . . . . . . . . . . . . . . . . . . . . . . . . . 25
7. Full Specification of the LDPC-Triangle Scheme . . . . . . . . 27
7.1. General . . . . . . . . . . . . . . . . . . . . . . . . . 27
7.2. Parity Check Matrix Creation . . . . . . . . . . . . . . . 27
7.3. Encoding . . . . . . . . . . . . . . . . . . . . . . . . . 28
7.4. Decoding . . . . . . . . . . . . . . . . . . . . . . . . . 28
8. Security Considerations . . . . . . . . . . . . . . . . . . . 29
8.1. Problem Statement . . . . . . . . . . . . . . . . . . . . 29
8.2. Attacks Against the Data Flow . . . . . . . . . . . . . . 29
8.2.1. Access to Confidential Objects . . . . . . . . . . . . 29
8.2.2. Content Corruption . . . . . . . . . . . . . . . . . . 30
8.3. Attacks Against the FEC Parameters . . . . . . . . . . . . 31
9. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 32
10. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . 33
11. References . . . . . . . . . . . . . . . . . . . . . . . . . . 34
11.1. Normative References . . . . . . . . . . . . . . . . . . . 34
11.2. Informative References . . . . . . . . . . . . . . . . . . 34
Appendix A. Trivial Decoding Algorithm (Informative Only) . . . . 37
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Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . . 39
Intellectual Property and Copyright Statements . . . . . . . . . . 40
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1. Introduction
[RFC3453] introduces large block FEC codes as an alternative to small
block FEC codes like Reed-Solomon. The main advantage of such large
block codes is the possibility to operate efficiently on source
blocks of size several tens of thousands (or more) source symbols.
The present document introduces the Fully-Specified FEC Encoding ID 3
that is intended to be used with the LDPC-Staircase FEC codes, and
the Fully-Specified FEC Encoding ID 4 that is intended to be used
with the LDPC-Triangle FEC codes [RN04][MK03]. Both schemes belong
to the broad class of large block codes. For a definition of the
term Fully-Specified Scheme, see [RFC5052], section 4.
LDPC codes rely on a dedicated matrix, called a "Parity Check
Matrix", at the encoding and decoding ends. The parity check matrix
defines relationships (or constraints) between the various encoding
symbols (i.e., source symbols and repair symbols), that are later
used by the decoder to reconstruct the original k source symbols if
some of them are missing. These codes are systematic, in the sense
that the encoding symbols include the source symbols in addition to
the repair symbols.
Since the encoder and decoder must operate on the same parity check
matrix, information must be communicated between them as part of the
FEC Object Transmission Information.
A publicly available reference implementation of these codes is
available and distributed under a GNU/LGPL license [LDPC-codec].
Besides, the code extracts included in this document are directly
contributed to the IETF process by the authors of this document and
by Radford M. Neal.
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2. Requirements notation
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
document are to be interpreted as described in [RFC2119].
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3. Definitions, Notations and Abbreviations
3.1. Definitions
This document uses the same terms and definitions as those specified
in [RFC5052]. Additionally, it uses the following definitions:
Source symbol: unit of data used during the encoding process
Encoding symbol: unit of data generated by the encoding process
Repair symbol: encoding symbol that is not a source symbol
Code rate: the k/n ratio, i.e., the ratio between the number of
source symbols and the number of encoding symbols. The code rate
belongs to a ]0; 1] interval. A code rate close to 1 indicates
that a small number of repair symbols have been produced during
the encoding process
Systematic code: FEC code in which the source symbols are part of
the encoding symbols
Source block: a block of k source symbols that are considered
together for the encoding
Encoding Symbol Group: a group of encoding symbols that are sent
together, within the same packet, and whose relationships to the
source object can be derived from a single Encoding Symbol ID
Source Packet: a data packet containing only source symbols
Repair Packet: a data packet containing only repair symbols
Packet Erasure Channel: a communication path where packets are
either dropped (e.g., by a congested router, or because the number
of transmission errors exceeds the correction capabilities of the
physical layer codes) or received. When a packet is received, it
is assumed that this packet is not corrupted
3.2. Notations
This document uses the following notations:
L denotes the object transfer length in bytes
k denotes the source block length in symbols, i.e., the number of
source symbols of a source block
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n denotes the encoding block length, i.e., the number of encoding
symbols generated for a source block
E denotes the encoding symbol length in bytes
B denotes the maximum source block length in symbols, i.e., the
maximum number of source symbols per source block
N denotes the number of source blocks into which the object shall
be partitioned
G denotes the number of encoding symbols per group, i.e., the
number of symbols sent in the same packet
CR denotes the "code rate", i.e., the k/n ratio
max_n denotes the maximum number of encoding symbols generated for
any source block. This is in particular the number of encoding
symbols generated for a source block of size B
H denotes the parity check matrix
pmms_rand(m) denotes the pseudo-random number generator defined in
Section 5.7 that returns a new random integer in [0; m-1] each
time it is called
3.3. Abbreviations
This document uses the following abbreviations:
ESI: Encoding Symbol ID
FEC OTI: FEC Object Transmission Information
FPI: FEC Payload ID
LDPC: Low Density Parity Check
PRNG: Pseudo Random Number Generator
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4. Formats and Codes
4.1. FEC Payload IDs
The FEC Payload ID is composed of the Source Block Number and the
Encoding Symbol ID:
The Source Block Number (12 bit field) identifies from which
source block of the object the encoding symbol(s) in the packet
payload is(are) generated. There are a maximum of 2^^12 blocks
per object. Source block numbering starts at 0.
The Encoding Symbol ID (20 bit field) identifies which encoding
symbol(s) generated from the source block is(are) carried in the
packet payload. There are a maximum of 2^^20 encoding symbols per
block. The first k values (0 to k-1) identify source symbols, the
remaining n-k values (k to n-k-1) identify repair symbols.
There MUST be exactly one FEC Payload ID per packet. In case of an
Encoding Symbol Group, when multiple encoding symbols are sent in the
same packet, the FEC Payload ID refers to the first symbol of the
packet. The other symbols can be deduced from the ESI of the first
symbol thanks to a dedicated function, as explained in Section 5.6
0 1 2 3
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| Source Block Number | Encoding Symbol ID (20 bits) |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
Figure 1: FEC Payload ID encoding format for FEC Encoding ID 3 and 4
4.2. FEC Object Transmission Information
4.2.1. Mandatory Element
o FEC Encoding ID: the LDPC-Staircase and LDPC-Triangle Fully-
Specified FEC Schemes use respectively the FEC Encoding ID 3
(Staircase) and 4 (Triangle).
4.2.2. Common Elements
The following elements MUST be defined with the present FEC Schemes:
o Transfer-Length (L): a non-negative integer indicating the length
of the object in bytes. There are some restrictions on the
maximum Transfer-Length that can be supported:
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maximum transfer length = 2^^12 * B * E
For instance, if B=2^^19 (because of a code rate of 1/2,
Section 5.2), and if E=1024 bytes, then the maximum transfer
length is 2^^41 bytes (or 2 TB). The upper limit, with symbols of
size 2^^16-1 bytes and a code rate larger or equal to 1/2, amounts
to 2^^47 bytes (or 128 TB).
o Encoding-Symbol-Length (E): a non-negative integer indicating the
length of each encoding symbol in bytes.
o Maximum-Source-Block-Length (B): a non-negative integer indicating
the maximum number of source symbols in a source block. There are
some restrictions on the maximum B value, as explained in
Section 5.2.
o Max-Number-of-Encoding-Symbols (max_n): a non-negative integer
indicating the maximum number of encoding symbols generated for
any source block. There are some restrictions on the maximum
max_n value. In particular max_n is at most equal to 2^^20.
Section 5 explains how to define the values of each of these
elements.
4.2.3. Scheme-Specific Elements
The following elements MUST be defined with the present FEC Scheme:
o G: a non-negative integer indicating the number of encoding
symbols per group (i.e., per packet). The default value is 1,
meaning that each packet contains exactly one symbol. Values
greater than 1 can also be defined, as explained in Section 5.3.
o PRNG seed: the seed is a 32 bit unsigned integer between 1 and
0x7FFFFFFE (i.e., 2^^31-2) inclusive. This value is used to
initialize the Pseudo Random Number Generator (Section 5.7).
4.2.4. Encoding Format
This section shows two possible encoding formats of the above FEC
OTI. The present document does not specify when or how these
encoding formats should be used.
4.2.4.1. Using the General EXT_FTI Format
The FEC OTI binary format is the following, when the EXT_FTI
mechanism is used (e.g., within the ALC
[draft-ietf-rmt-pi-alc-revised] or NORM
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[draft-ietf-rmt-pi-norm-revised] protocols).
0 1 2 3
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| HET = 64 | HEL = 5 | |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ +
| Transfer-Length (L) |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| Encoding Symbol Length (E) | G | B (MSB) |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| B (LSB) | Max Nb of Enc. Symbols (max_n) |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| PRNG seed |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
Figure 2: EXT_FTI Header for FEC Encoding ID 3 and 4.
In particular:
o The Transfer-Length (L) field size (48 bits) is larger than the
size required to store the maximum transfer length (Section 4.2.2)
for field alignment purposes.
o The Maximum-Source-Block-Length (B) field (20 bits) is split into
two parts: the 8 most significant bits (MSB) are in the third 32-
bit word of the EXT_FTI, and the remaining 12 least significant
bits (LSB) are in the fourth 32-bit word.
4.2.4.2. Using the FDT Instance (FLUTE specific)
When it is desired that the FEC OTI be carried in the FDT Instance of
a FLUTE session [draft-ietf-rmt-flute-revised], the following XML
attributes must be described for the associated object:
o FEC-OTI-FEC-Encoding-ID
o FEC-OTI-Transfer-length
o FEC-OTI-Encoding-Symbol-Length
o FEC-OTI-Maximum-Source-Block-Length
o FEC-OTI-Max-Number-of-Encoding-Symbols
o FEC-OTI-Scheme-Specific-Info
The FEC-OTI-Scheme-Specific-Info contains the string resulting from
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the Base64 encoding (in the XML Schema xs:base64Binary sense)
[RFC4648] of the following value:
0 1 2 3
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| PRNG seed |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| G |
+-+-+-+-+-+-+-+-+
Figure 3: FEC OTI Scheme Specific Information to be Included in the
FDT Instance for FEC Encoding ID 3 and 4.
During Base64 encoding, the 5 bytes of the FEC OTI Scheme Specific
Information are transformed into a string of 8 printable characters
(in the 64-character alphabet) that is added to the FEC-OTI-Scheme-
Specific-Info attribute.
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5. Procedures
This section defines procedures that are common to FEC Encoding IDs 3
and 4.
5.1. General
The B (maximum source block length in symbols), E (encoding symbol
length in bytes) and G (number of encoding symbols per group)
parameters are first determined. The algorithms of Section 5.2 and
Section 5.3 MAY be used to that purpose. Using other algorithms is
possible without compromising interoperability since the B, E and G
parameters are communicated to the receiver by means of the FEC OTI.
Then, the source object MUST be partitioned using the block
partitioning algorithm specified in [RFC5052]. To that purpose, the
B, L (object transfer length in bytes), and E arguments are provided.
As a result, the object is partitioned into N source blocks. These
blocks are numbered consecutively from 0 to N-1. The first I source
blocks consist of A_large source symbols, the remaining N-I source
blocks consist of A_small source symbols. Each source symbol is E
bytes in length, except perhaps the last symbol which may be shorter.
Then, the max_n (maximum number of encoding symbols generated for any
source block) parameter is determined. The algorithm of Section 5.4
MAY be used to that purpose. Using another algorithm is possible
without compromising interoperability since the max_n parameter is
communicated to the receiver by means of the FEC OTI.
For each block, the actual number of encoding symbols, n, MUST then
be determined using the "n-algorithm" detailed in Section 5.5.
Then, FEC encoding and decoding can be done block per block,
independently. To that purpose, a parity check matrix is created,
that forms a system of linear equations between the source and repair
symbols of a given block, where the basic operator is XOR.
This parity check matrix is logically divided into two parts: the
left side (from column 0 to k-1) describes the occurrences of each
source symbol in the system of linear equations; the right side (from
column k to n-1) describes the occurrences of each repair symbol in
the system of linear equations. The only difference between the
LDPC-Staircase and LDPC-Triangle schemes is the construction of this
right sub-matrix. An entry (a "1") in the matrix at position (i,j)
(i.e., at row i and column j) means that the symbol with ESI j
appears in equation i of the system.
When the parity symbols have been created, the sender transmits
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source and parity symbols. The way this transmission occurs can
largely impact the erasure recovery capabilities of the LDPC-* FEC.
In particular, sending parity symbols in sequence is suboptimal.
Instead it is usually recommended the shuffle these symbols. The
interested reader will find more details in [NRFF05].
The following sections detail how the B, E, G, max_nand n parameters
are determined (respectively in Section 5.2, Section 5.3, Section 5.4
and Section 5.5), how encoding symbol groups are created
(Section 5.6), and finally Section 5.7 details the PRNG.
5.2. Determining the Maximum Source Block Length (B)
The B parameter (maximum source block length in symbols) depends on
several parameters: the code rate (CR), the Encoding Symbol ID field
length of the FEC Payload ID (20 bits), as well as possible internal
codec limitations.
The B parameter cannot be larger than the following values, derived
from the FEC Payload ID limitations, for a given code rate:
max1_B = 2^^(20 - ceil(Log2(1/CR)))
Some common max1_B values are:
o CR == 1 (no repair symbol): max1_B = 2^^20 = 1,048,576
o 1/2 <= CR < 1: max1_B = 2^^19 = 524,288 symbols
o 1/4 <= CR < 1/2: max1_B = 2^^18 = 262,144 symbols
o 1/8 <= CR < 1/4: max1_B = 2^^17 = 131,072 symbols
Additionally, a codec MAY impose other limitations on the maximum
block size. For instance, this is the case when the codec uses
internally 16 bit unsigned integers to store the Encoding Symbol ID,
since it does not enable to store all the possible values of a 20 bit
field. In that case, if for instance 1/2 <= CR < 1, then the maximum
source block length is 2^^15. Other limitations may also apply, for
instance because of a limited working memory size. This decision
MUST be clarified at implementation time, when the target use case is
known. This results in a max2_B limitation.
Then, B is given by:
B = min(max1_B, max2_B)
Note that this calculation is only required at the coder, since the B
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parameter is communicated to the decoder through the FEC OTI.
5.3. Determining the Encoding Symbol Length (E) and Number of Encoding
Symbols per Group (G)
The E parameter usually depends on the maximum transmission unit on
the path (PMTU) from the source to each receiver. In order to
minimize the protocol header overhead (e.g., the LCT/UDP/IPv4 or IPv6
headers in case of ALC), E is chosen as large as possible. In that
case, E is chosen so that the size of a packet composed of a single
symbol (G=1) remains below but close to the PMTU.
However other considerations can exist. For instance, the E
parameter can be made a function of the object transfer length.
Indeed, LDPC codes are known to offer better protection for large
blocks. In case of small objects, it can be advantageous to reduce
the encoding symbol length (E) in order to artificially increase the
number of symbols, and therefore the block size.
In order to minimize the protocol header overhead, several symbols
can be grouped in the same Encoding Symbol Group (i.e., G > 1).
Depending on how many symbols are grouped (G) and on the packet loss
rate (G symbols are lost for each packet erasure), this strategy
might or might not be appropriate. A balance must therefore be
found.
The current specification does not mandate any value for either E or
G. The current specification only provides an example of possible
choices for E and G. Note that this choice is done by the sender, and
the E and G parameters are then communicated to the receiver thanks
to the FEC OTI. Note also that the decoding algorithm used
influences the choice of the E and G parameters. Indeed, increasing
the number of symbols will negatively impact the processing load when
decoding is based (in part or totally) on Gaussian elimination,
whereas the impacts will be rather low when decoding is based on the
trivial algorithm sketched in Section 6.4.
Example:
Let us assume that the trivial decoding algorithm sketched in
Section 6.4 is used. First define the target packet payload size,
pkt_sz (at most equal to the PMTU minus the size of the various
protocol headers). The pkt_sz must be chosen in such a way that the
symbol size is an integer. This can require that pkt_sz be a
multiple of 4, 8 or 16 (see the table below). Then calculate the
number of packets in the object: nb_pkts = ceil(L / pkt_sz).
Finally, thanks to nb_pkts, use the following table to find a
possible G value.
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+------------------------+----+-------------+-------------------+
| Number of packets | G | Symbol size | k |
+------------------------+----+-------------+-------------------+
| 4000 <= nb_pkts | 1 | pkt_sz | 4000 <= k |
| | | | |
| 1000 <= nb_pkts < 4000 | 4 | pkt_sz / 4 | 4000 <= k < 16000 |
| | | | |
| 500 <= nb_pkts < 1000 | 8 | pkt_sz / 8 | 4000 <= k < 8000 |
| | | | |
| 1 <= nb_pkts < 500 | 16 | pkt_sz / 16 | 16 <= k < 8000 |
+------------------------+----+-------------+-------------------+
5.4. Determining the Maximum Number of Encoding Symbols Generated for
Any Source Block (max_n)
The following algorithm MAY be used by a sender to determine the
maximum number of encoding symbols generated for any source block
(max_n) as a function of B and the target code rate. Since the max_n
parameter is communicated to the decoder by means of the FEC OTI,
another method MAY be used to determine max_n.
Input:
B: Maximum source block length, for any source block. Section 5.2
MAY be used to determine its value.
CR: FEC code rate, which is provided by the user (e.g., when
starting a FLUTE sending application). It is expressed as a
floating point value. The CR value must be such that the
resulting number of encoding symbols per block is at most equal to
2^^20 (Section 4.1).
Output:
max_n: Maximum number of encoding symbols generated for any source
block.
Algorithm:
max_n = ceil(B / CR);
if (max_n > 2^^20) then return an error ("invalid code rate");
(NB: if B has been defined as explained in Section 5.2, this error
should never happen)
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5.5. Determining the Number of Encoding Symbols of a Block (n)
The following algorithm, also called "n-algorithm", MUST be used by
the sender and the receiver to determine the number of encoding
symbols for a given block (n) as a function of B, k, and max_n.
Input:
B: Maximum source block length, for any source block. At a
sender, Section 5.2 MAY be used to determine its value. At a
receiver, this value MUST be extracted from the received FEC OTI.
k: Current source block length. At a sender or receiver, the
block partitioning algorithm MUST be used to determine its value.
max_n: Maximum number of encoding symbols generated for any source
block. At a sender, Section 5.4 MAY be used to determine its
value. At a receiver, this value MUST be extracted from the
received FEC OTI.
Output:
n: Number of encoding symbols generated for this source block.
Algorithm:
n = floor(k * max_n / B);
5.6. Identifying the G Symbols of an Encoding Symbol Group
When multiple encoding symbols are sent in the same packet, the FEC
Payload ID information of the packet MUST refer to the first encoding
symbol. It MUST then be possible to identify each symbol from this
single FEC Payload ID. To that purpose, the symbols of an Encoding
Symbol Group (i.e., packet):
o MUST all be either source symbols, or repair symbols. Therefore
only source packets and repair packets are permitted, not mixed
ones.
o are identified by a function, sender(resp.
receiver)_find_ESIs_of_group(), that takes as argument:
* for a sender, the index of the Encoding Symbol Group (i.e.,
packet) that the application wants to create,
* for a receiver, the ESI information contained in the FEC
Payload ID.
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and returns a list of G Encoding Symbol IDs. In case of a source
packet, the G Encoding Symbol IDs are chosen consecutively, by
incrementing the ESI. In case of a repair packet, the G repair
symbols are chosen randomly, as explained below.
o are stored in sequence in the packet, without any padding. In
other words, the last byte of the i-th symbol is immediately
followed by the first byte of (i+1)-th symbol.
The system must first be initialized by creating a random permutation
of the n-k indexes. This initialization function MUST be called
immediately after creating the parity check matrix. More precisely,
since the PRNG seed is not re-initialized, no call to the PRNG
function must have happened between the time the parity check matrix
has been initialized and the time the following initialization
function is called. This is true both at a sender and at a receiver.
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int *txseqToID;
int *IDtoTxseq;
/*
* Initialization function.
* Warning: use only when G > 1.
*/
void
initialize_tables ()
{
int i;
int randInd;
int backup;
txseqToID = malloc((n-k) * sizeof(int));
IDtoTxseq = malloc((n-k) * sizeof(int));
if (txseqToID == NULL || IDtoTxseq == NULL)
handle the malloc failures as appropriate...
/* initialize the two tables that map ID
* (i.e., ESI-k) to/from TxSequence. */
for (i = 0; i < n - k; i++) {
IDtoTxseq[i] = i;
txseqToID[i] = i;
}
/* now randomize everything */
for (i = 0; i < n - k; i++) {
randInd = pmms_rand(n - k);
backup = IDtoTxseq[i];
IDtoTxseq[i] = IDtoTxseq[randInd];
IDtoTxseq[randInd] = backup;
txseqToID[IDtoTxseq[i]] = i;
txseqToID[IDtoTxseq[randInd]] = randInd;
}
return;
}
It is then possible, at the sender, to determine the sequence of G
Encoding Symbol IDs that will be part of the group.
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/*
* Determine the sequence of ESIs for the packet under construction
* at a sender.
* Warning: use only when G > 1.
* PktIdx (IN): index of the packet, in
* {0..ceil(k/G)+ceil((n-k)/G)} range
* ESIs[] (OUT): list of ESIs for the packet
*/
void
sender_find_ESIs_of_group (int PktIdx,
ESI_t ESIs[])
{
int i;
if (PktIdx < nbSourcePkts) {
/* this is a source packet */
ESIs[0] = PktIdx * G;
for (i = 1; i < G; i++) {
ESIs[i] = (ESIs[0] + i) % k;
}
} else {
/* this is a repair packet */
for (i = 0; i < G; i++) {
ESIs[i] =
k +
txseqToID[(i + (PktIdx - nbSourcePkts) * G)
% (n - k)];
}
}
return;
}
Similarly, upon receiving an Encoding Symbol Group (i.e., packet), a
receiver can determine the sequence of G Encoding Symbol IDs from the
first ESI, esi0, that is contained in the FEC Payload ID.
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/*
* Determine the sequence of ESIs for the packet received.
* Warning: use only when G > 1.
* esi0 (IN): : ESI contained in the FEC Payload ID
* ESIs[] (OUT): list of ESIs for the packet
*/
void
receiver_find_ESIs_of_group (ESI_t esi0,
ESI_t ESIs[])
{
int i;
if (esi0 < k) {
/* this is a source packet */
ESIs[0] = esi0;
for (i = 1; i < G; i++) {
ESIs[i] = (esi0 + i) % k;
}
} else {
/* this is a repair packet */
for (i = 0; i < G; i++) {
ESIs[i] =
k +
txseqToID[(i + IDtoTxseq[esi0 - k])
% (n - k)];
}
}
}
5.7. Pseudo Random Number Generator
The FEC Encoding IDs 3 and 4 rely on a pseudo-random number generator
(PRNG) that must be fully specified, in particular in order to enable
the receivers and the senders to build the same parity check matrix.
The Park-Miler "minimal standard" PRNG [PM88] MUST be used. It
defines a simple multiplicative congruential algorithm: Ij+1 = A * Ij
(modulo M), with the following choices: A = 7^^5 = 16807 and M =
2^^31 - 1 = 2147483647. A validation criteria of such a PRNG is the
following: if seed = 1, then the 10,000th value returned MUST be
equal to 1043618065.
Several implementations of this PRNG are known and discussed in the
literature. An optimized implementation of this algorithm, using
only 32 bit mathematics and which does not require any division, can
be found in [rand31pmc]. It uses the Park and Miller algorithm
[PM88] with the optimization suggested by D. Carta in [CA90]. The
history behind this algorithm is detailed in [WI08]. Yet any other
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implementation of the PRNG algorithm that matches the above
validation criteria, like the ones detailed in [PM88], is
appropriate.
This PRNG produces natively a 31 bit value between 1 and 0x7FFFFFFE
(2^^31-2) inclusive. Since it is desired to scale the pseudo random
number between 0 and maxv-1 inclusive, one must keep the most
significant bits of the value returned by the PRNG (the least
significant bits are known to be less random and modulo based
solutions should be avoided [PTVF92]). The following algorithm MUST
be used:
Input:
raw_value: random integer generated by the inner PRNG algorithm,
between 1 and 0x7FFFFFFE (2^^31-2) inclusive.
maxv: upper bound used during the scaling operation.
Output:
scaled_value: random integer between 0 and maxv-1 inclusive.
Algorithm:
scaled_value = (unsigned long) ((double)maxv * (double)raw_value /
(double)0x7FFFFFFF);
(NB: the above C type casting to unsigned long is equivalent to
using floor() with positive floating point values)
In this document, pmms_rand(maxv) denotes the PRNG function that
implements the Park-Miller "minimal standard" algorithm defined above
and that scales the raw value between 1 and maxv-1 inclusive, using
the above scaling algorithm. Additionally, a function should be
provided to enable the initialization of the PRNG with a seed (i.e.,
a 31 bit interger between 1 and 0x7FFFFFFE inclusive) before calling
pmms_rand(maxv) the first time.
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6. Full Specification of the LDPC-Staircase Scheme
6.1. General
The LDPC-Staircase scheme is identified by the Fully-Specified FEC
Encoding ID 3.
The PRNG used by the LDPC-Staircase scheme must be initialized by a
seed. This PRNG seed is an instance-specific FEC OTI attribute
(Section 4.2.3).
6.2. Parity Check Matrix Creation
The LDPC-Staircase matrix can be divided into two parts: the left
side of the matrix defines in which equations the source symbols are
involved; the right side of the matrix defines in which equations the
repair symbols are involved.
The left side MUST be generated by using the following function:
/*
* Initialize the left side of the parity check matrix.
* This function assumes that an empty matrix of size n-k * k has
* previously been allocated/reset and that the matrix_has_entry(),
* matrix_insert_entry() and degree_of_row() functions can access it.
* (IN): the k and n parameters.
*/
void left_matrix_init (int k, int n)
{
int i; /* row index or temporary variable */
int j; /* column index */
int h;
int t; /* left limit within the list of possible choices u[] */
int u[3*MAX_K]; /* table used to have a homogeneous 1 distrib. */
/* Initialize a list of all possible choices in order to
* guarantee a homogeneous "1" distribution */
for (h = 3*k-1; h >= 0; h--) {
u[h] = h % (n-k);
}
/* Initialize the matrix with 3 "1s" per column, homogeneously */
t = 0;
for (j = 0; j < k; j++) { /* for each source symbol column */
for (h = 0; h < 3; h++) { /* add 3 "1s" */
/* check that valid available choices remain */
for (i = t; i < 3*k && matrix_has_entry(u[i], j); i++);
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if (i < 3*k) {
/* choose one index within the list of possible
* choices */
do {
i = t + pmms_rand(3*k-t);
} while (matrix_has_entry(u[i], j));
matrix_insert_entry(u[i], j);
/* replace with u[t] which has never been chosen */
u[i] = u[t];
t++;
} else {
/* no choice left, choose one randomly */
do {
i = pmms_rand(n-k);
} while (matrix_has_entry(i, j));
matrix_insert_entry(i, j);
}
}
}
/* Add extra bits to avoid rows with less than two "1s".
* This is needed when the code rate is smaller than 2/5. */
for (i = 0; i < n-k; i++) { /* for each row */
if (degree_of_row(i) == 0) {
j = pmms_rand(k);
matrix_insert_entry(i, j);
}
if (degree_of_row(i) == 1) {
do {
j = pmms_rand(k);
} while (matrix_has_entry(i, j));
matrix_insert_entry(i, j);
}
}
}
The right side (the staircase) MUST be generated by using the
following function:
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/*
* Initialize the right side of the parity check matrix with a
* staircase structure.
* (IN): the k and n parameters.
*/
void right_matrix_staircase_init (int k, int n)
{
int i; /* row index */
matrix_insert_entry(0, k); /* first row */
for (i = 1; i < n-k; i++) { /* for the following rows */
matrix_insert_entry(i, k+i); /* identity */
matrix_insert_entry(i, k+i-1); /* staircase */
}
}
Note that just after creating this parity check matrix, when encoding
symbol groups are used (i.e., G > 1), the function initializing the
two random permutation tables (Section 5.6) MUST be called. This is
true both at a sender and at a receiver.
6.3. Encoding
Thanks to the staircase matrix, repair symbol creation is
straightforward: each repair symbol is equal to the sum of all source
symbols in the associated equation, plus the previous repair symbol
(except for the first repair symbol). Therefore encoding MUST follow
the natural repair symbol order: start with the first repair symbol,
and generate repair symbol with ESI i before symbol with ESI i+1.
6.4. Decoding
Decoding basically consists in solving a system of n-k linear
equations whose variables are the n source and repair symbols. Of
course, the final goal is to recover the value of the k source
symbols only.
To that purpose, many techniques are possible. One of them is the
following trivial algorithm [ZP74]: given a set of linear equations,
if one of them has only one remaining unknown variable, then the
value of this variable is that of the constant term. So, replace
this variable by its value in all the remaining linear equations and
reiterate. The value of several variables can therefore be found
recursively. Applied to LDPC FEC codes working over an erasure
channel, the parity check matrix defines a set of linear equations
whose variables are the source symbols and repair symbols. Receiving
or decoding a symbol is equivalent to having the value of a variable.
Appendix A sketches a possible implementation of this algorithm.
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A Gaussian elimination (or any optimized derivative) is another
possible decoding technique. Hybrid solutions that start by using
the trivial algorithm above and finish with a Gaussian elimination
are also possible.
Because interoperability does not depend on the decoding algorithm
used, the current document does not recommend any particular
technique. This choice is left to the codec developer.
However choosing a decoding technique will have great practical
impacts. It will impact the erasure capabilities: a Gaussian
elimination enables to solve the system with a smaller number of
known symbols compared to the trivial technique. It will also impact
the CPU load: a Gaussian elimination requires more processing than
the above trivial algorithm. Depending on the target use case, the
codec developer will favor one feature or the other.
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7. Full Specification of the LDPC-Triangle Scheme
7.1. General
LDPC-Triangle is identified by the Fully-Specified FEC Encoding ID 4.
The PRNG used by the LDPC-Triangle scheme must be initialized by a
seed. This PRNG seed is an instance-specific FEC OTI attribute
(Section 4.2.3).
7.2. Parity Check Matrix Creation
The LDPC-Triangle matrix can be divided into two parts: the left side
of the matrix defines in which equations the source symbols are
involved; the right side of the matrix defines in which equations the
repair symbols are involved.
The left side MUST be generated by using the same left_matrix_init()
function as with LDPC-Staircase (Section 6.2).
The right side (the triangle) MUST be generated by using the
following function:
/*
* Initialize the right side of the parity check matrix with a
* triangle structure.
* (IN): the k and n parameters.
*/
void right_matrix_staircase_init (int k, int n)
{
int i; /* row index */
int j; /* randomly chosen column indexes in 0..n-k-2 */
int l; /* limitation of the # of "1s" added per row */
matrix_insert_entry(0, k); /* first row */
for (i = 1; i < n-k; i++) { /* for the following rows */
matrix_insert_entry(i, k+i); /* identity */
matrix_insert_entry(i, k+i-1); /* staircase */
/* now fill the triangle */
j = i-1;
for (l = 0; l < j; l++) { /* limit the # of "1s" added */
j = pmms_rand(j);
matrix_insert_entry(i, k+j);
}
}
}
Note that just after creating this parity check matrix, when encoding
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symbol groups are used (i.e., G > 1), the function initializing the
two random permutation tables (Section 5.6) MUST be called. This is
true both at a sender and at a receiver.
7.3. Encoding
Here also repair symbol creation is straightforward: each repair
symbol of ESI i is equal to the sum of all source and repair symbols
(with ESI lower than i) in the associated equation. Therefore
encoding MUST follow the natural repair symbol order: start with the
first repair symbol, and generate repair symbol with ESI i before
symbol with ESI i+1.
7.4. Decoding
Decoding basically consists in solving a system of n-k linear
equations, whose variables are the n source and repair symbols. Of
course, the final goal is to recover the value of the k source
symbols only. To that purpose, many techniques are possible, as
explained in Section 6.4.
Because interoperability does not depend on the decoding algorithm
used, the current document does not recommend any particular
technique. This choice is left to the codec implementer.
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8. Security Considerations
8.1. Problem Statement
A content delivery system is potentially subject to many attacks:
some of them target the network (e.g., to compromise the routing
infrastructure, by compromising the congestion control component),
others target the Content Delivery Protocol (CDP) (e.g., to
compromise its normal behavior), and finally some attacks target the
content itself. Since this document focuses on a FEC building block
independently of any particular CDP (even if ALC and NORM are two
natural candidates), this section only discusses the additional
threats that an arbitrary CDP may be exposed to when using this
building block.
More specifically, several kinds of attacks exist:
o those that are meant to give access to a confidential content
(e.g., in case of a non-free content),
o those that try to corrupt the object being transmitted (e.g., to
inject malicious code within an object, or to prevent a receiver
from using an object),
o and those that try to compromise the receiver's behavior (e.g., by
making the decoding of an object computationally expensive).
These attacks can be launched either against the data flow itself
(e.g., by sending forged symbols) or against the FEC parameters that
are sent either in-band (e.g., in an EXT_FTI or FDT Instance) or out-
of-band (e.g., in a session description).
8.2. Attacks Against the Data Flow
First of all, let us consider the attacks against the data flow.
8.2.1. Access to Confidential Objects
Access control to a confidential object being transmitted is
typically provided by means of encryption. This encryption can be
done over the whole object (e.g., by the content provider, before the
FEC encoding process), or be done on a packet per packet basis (e.g.,
when IPsec/ESP is used [RFC4303]). If confidentiality is a concern,
it is RECOMMENDED that one of these solutions be used. Even if we
mention these attacks here, they are not related nor facilitated by
the use of FEC.
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8.2.2. Content Corruption
Protection against corruptions (e.g., after sending forged packets)
is achieved by means of a content integrity verification/sender
authentication scheme. This service can be provided at the object
level, but in that case a receiver has no way to identify which
symbol(s) is(are) corrupted if the object is detected as corrupted.
This service can also be provided at the packet level. In this case,
after removing all forged packets, the object may be in some case
recovered. Several techniques can provide this source
authentication/content integrity service:
o at the object level, the object MAY be digitally signed (with
public key cryptography), for instance by using RSASSA-PKCS1-v1_5
[RFC3447]. This signature enables a receiver to check the object
integrity, once this latter has been fully decoded. Even if
digital signatures are computationally expensive, this calculation
occurs only once per object, which is usually acceptable;
o at the packet level, each packet can be digitally signed. A major
limitation is the high computational and transmission overheads
that this solution requires (unless perhaps if Elliptic Curve
Cryptography (ECC) is used). To avoid this problem, the signature
may span a set of symbols (instead of a single one) in order to
amortize the signature calculation. But if a single symbol is
missing, the integrity of the whole set cannot be checked;
o at the packet level, a Group Message Authentication Code (MAC)
[RFC2104] scheme can be used, for instance by using HMAC-SHA-1
with a secret key shared by all the group members, senders and
receivers. This technique creates a cryptographically secured
(thanks to the secret key) digest of a packet that is sent along
with the packet. The Group MAC scheme does not create prohibitive
processing load nor transmission overhead, but it has a major
limitation: it only provides a group authentication/integrity
service since all group members share the same secret group key,
which means that each member can send a forged packet. It is
therefore restricted to situations where group members are fully
trusted (or in association with another technique as a pre-check);
o at the packet level, TESLA [RFC4082] is an attractive solution
that is robust to losses, provides a true authentication/integrity
service, and does not create any prohibitive processing load or
transmission overhead. Yet checking a packet requires a small
delay (a second or more) after its reception;
Techniques relying on public key cryptography (digital signatures and
TESLA during the bootstrap process, when used) require that public
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keys be securely associated to the entities. This can be achieved by
a Public Key Infrastructure (PKI), or by a PGP Web of Trust, or by
pre-distributing the public keys of each group member.
Techniques relying on symmetric key cryptography (group MAC) require
that a secret key be shared by all group members. This can be
achieved by means of a group key management protocol, or simply by
pre-distributing the secret key (but this manual solution has many
limitations).
It is up to the CDP developer, who knows the security requirements
and features of the target application area, to define which solution
is the most appropriate. Nonetheless, in case there is any concern
of the threat of object corruption, it is RECOMMENDED that at least
one of these techniques be used.
8.3. Attacks Against the FEC Parameters
Let us now consider attacks against the FEC parameters (or FEC OTI).
The FEC OTI can either be sent in-band (i.e., in an EXT_FTI or in an
FDT Instance containing FEC OTI for the object) or out-of-band (e.g.,
in a session description). Attacks on these FEC parameters can
prevent the decoding of the associated object: for instance modifying
the B parameter will lead to a different block partitioning.
It is therefore RECOMMENDED that security measures be taken to
guarantee the FEC OTI integrity. To that purpose, the packets
carrying the FEC parameters sent in-band in an EXT_FTI header
extension SHOULD be protected by one of the per-packet techniques
described above: digital signature, group MAC, or TESLA. When FEC
OTI is contained in an FDT Instance, this object SHOULD be protected,
for instance by digitally signing it with XML digital signatures
[RFC3275]. Finally, when FEC OTI is sent out-of-band (e.g., in a
session description) this latter SHOULD be protected, for instance by
digitally signing it with [RFC3852].
The same considerations concerning the key management aspects apply
here also.
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9. IANA Considerations
Values of FEC Encoding IDs and FEC Instance IDs are subject to IANA
registration. For general guidelines on IANA considerations as they
apply to this document, see [RFC5052].
This document assigns the Fully-Specified FEC Encoding ID 3 under the
"ietf:rmt:fec:encoding" name-space to "LDPC Staircase Codes".
This document assigns the Fully-Specified FEC Encoding ID 4 under the
"ietf:rmt:fec:encoding" name-space to "LDPC Triangle Codes".
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10. Acknowledgments
Section 5.5 is derived from a previous Internet-Draft, and we would
like to thank S. Peltotalo and J. Peltotalo for their contribution.
We would also like to thank Pascal Moniot, Laurent Fazio, Aurelien
Francillon, Shao Wenjian, Magnus Westerlund, Brian Carpenter, Tim
Polk, Jari Arkko, Chris Newman, Robin Whittle and Alfred Hoenes for
their comments.
Last but not least, the authors are grateful to Radford M. Neal
(University of Toronto) whose LDPC software
(http://www.cs.toronto.edu/~radford/ldpc.software.html) inspired this
work.
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11. References
11.1. Normative References
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", RFC 2119, BCP 14, March 1997.
[RFC5052] Watson, M., Luby, M., and L. Vicisano, "Forward Error
Correction (FEC) Building Block", RFC 5052, August 2007.
[RFC3453] Luby, M., Vicisano, L., Gemmell, J., Rizzo, L., Handley,
M., and J. Crowcroft, "The Use of Forward Error Correction
(FEC) in Reliable Multicast", RFC 3453, December 2002.
11.2. Informative References
[ZP74] Zyablov, V. and M. Pinsker, "Decoding Complexity of Low-
Density Codes for Transmission in a Channel with
Erasures", Translated from Problemy Peredachi
Informatsii, Vol.10, No. 1, pp.15-28, January-March 1974.
[RN04] Roca, V. and C. Neumann, "Design, Evaluation and
Comparison of Four Large Block FEC Codecs: LDPC, LDGM,
LDGM-Staircase and LDGM-Triangle, Plus a Reed-Solomon
Small Block FEC Codec", INRIA Research Report RR-5225,
June 2004.
[NRFF05] Neumann, C., Roca, V., Francillon, A., and D. Furodet,
"Impacts of Packet Scheduling and Packet Loss Distribution
on FEC Performances: Observations and Recommendations",
ACM CoNEXT'05 Conference, Toulouse, France (an extended
version is available as INRIA Research Report RR-5578),
October 2005.
[LDPC-codec]
Roca, V., Neumann, C., Cunche, M., and J. Laboure, "LDPC-
Staircase/LDPC-Triangle Codec Reference Implementation",
INRIA Rhone-Alpes and STMicroelectronics,
http://planete-bcast.inrialpes.fr/.
[MK03] MacKay, D., "Information Theory, Inference and Learning
Algorithms", Cambridge University Press, ISBN: 0-521-
64298-1, 2003.
[PM88] Park, S. and K. Miller, "Random Number Generators: Good
Ones are Hard to Find", Communications of the ACM, Vol.
31, No. 10, pp.1192-1201, 1988.
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[CA90] Carta, D., "Two Fast Implementations of the Minimal
Standard Random Number Generator", Communications of the
ACM, Vol. 33, No. 1, pp.87-88, January 1990.
[WI08] Whittle, R., "Park-Miller-Carta Pseudo-Random Number
Generator", http://www.firstpr.com.au/dsp/rand31/,
January 2008.
[rand31pmc]
Whittle, R., "31 bit pseudo-random number generator", htt
p://www.firstpr.com.au/dsp/rand31/
rand31-park-miller-carta.cc.txt, September 2005.
[PTVF92] Press, W., Teukolsky, S., Vetterling, W., and B. Flannery,
"Numerical Recipies in C; Second Edition", Cambridge
University Press, ISBN: 0-521-43108-5, 1992.
[draft-ietf-rmt-pi-alc-revised]
Luby, M., Watson, M., and L. Vicisano, "Asynchronous
Layered Coding (ALC) Protocol Instantiation",
draft-ietf-rmt-pi-alc-revised-04.txt (work in progress),
February 2007.
[draft-ietf-rmt-pi-norm-revised]
Adamson, B., Bormann, C., Handley, M., and J. Macker,
"Negative-acknowledgment (NACK)-Oriented Reliable
Multicast (NORM) Protocol",
draft-ietf-rmt-pi-norm-revised-05.txt (work in progress),
March 2007.
[draft-ietf-rmt-flute-revised]
Paila, T., Walsh, R., Luby, M., Lehtonen, R., and V. Roca,
"FLUTE - File Delivery over Unidirectional Transport",
draft-ietf-rmt-flute-revised-05.txt (work in progress),
October 2007.
[RFC3447] Jonsson, J. and B. Kaliski, "Public-Key Cryptography
Standards (PKCS) #1: RSA Cryptography Specifications
Version 2.1", RFC 3447, February 2003.
[RFC4303] Kent, S., "IP Encapsulating Security Payload (ESP)",
RFC 4303, December 2005.
[RFC2104] "HMAC: Keyed-Hashing for Message Authentication",
RFC 2104, February 1997.
[RFC4082] "Timed Efficient Stream Loss-Tolerant Authentication
(TESLA): Multicast Source Authentication Transform
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Introduction", RFC 4082, June 2005.
[RFC3275] Eastlake, D., Reagle, J., and D. Solo, "(Extensible Markup
Language) XML-Signature Syntax and Processing", RFC 3275,
March 2002.
[RFC3852] Housley, R., "Cryptographic Message Syntax (CMS)",
RFC 3852, July 2004.
[RFC4648] Josefsson, S., "The Base16, Base32, and Base64 Data
Encodings", RFC 4648, October 2006.
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Appendix A. Trivial Decoding Algorithm (Informative Only)
A trivial decoding algorithm is sketched below (please see
[LDPC-codec] for the details omitted here):
Initialization: allocate a table partial_sum[n-k] of buffers, each
buffer being of size the symbol size. There's one
entry per equation since the buffers are meant to
store the partial sum of each equation; Reset all
the buffers to zero;
/*
* For each newly received or decoded symbol, try to make progress
* in the decoding of the associated source block.
* NB: in case of a symbol group (G>1), this function is called for
* each symbol of the received packet.
* NB: a callback function indicates to the caller that new symbol(s)
* has(have) been decoded.
* new_esi (IN): ESI of the new symbol received or decoded
* new_symb (IN): Buffer of the new symbol received or decoded
*/
void
decoding_step(ESI_t new_esi,
symbol_t *new_symb)
{
If (new_symb is an already decoded or received symbol) {
Return; /* don't waste time with this symbol */
}
If (new_symb is the last missing source symbol) {
Remember that decoding is finished;
Return; /* work is over now... */
}
Create an empty list of equations having symbols decoded
during this decoding step;
/*
* First add this new symbol to the partial sum of all the
* equations where the symbol appears.
*/
For (each equation eq in which new_symb is a variable and
having more than one unknown variable) {
Add new_symb to partial_sum[eq];
Remove entry(eq, new_esi) from the H matrix;
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If (the new degree of equation eq == 1) {
/* a new symbol can be decoded, remember the
* equation */
Append eq to the list of equations having symbols
decoded during this decoding step;
}
}
/*
* Then finish with recursive calls to decoding_step() for each
* newly decoded symbol.
*/
For (each equation eq in the list of equations having symbols
decoded during this decoding step) {
/*
* Because of the recursion below, we need to check that
* decoding is not finished, and that the equation is
* __still__ of degree 1
*/
If (decoding is finished) {
break; /* exit from the loop */
}
If ((degree of equation eq == 1) {
Let dec_esi be the ESI of the newly decoded symbol in
equation eq;
Remove entry(eq, dec_esi);
Allocate a buffer, dec_symb, for this symbol and
copy partial_sum[eq] to dec_symb;
Inform the caller that a new symbol has been
decoded via a callback function;
/* finally, call this function recursively */
decoding_step(dec_esi, dec_symb);
}
}
Free the list of equations having symbols decoded;
Return;
}
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Authors' Addresses
Vincent Roca
INRIA
655, av. de l'Europe
Inovallee; Montbonnot
ST ISMIER cedex 38334
France
Email: vincent.roca@inria.fr
URI: http://planete.inrialpes.fr/people/roca/
Christoph Neumann
Thomson
12, bd de Metz
Rennes 35700
France
Email: christoph.neumann@thomson.net
URI: http://planete.inrialpes.fr/people/chneuman/
David Furodet
STMicroelectronics
12, Rue Jules Horowitz
BP217
Grenoble Cedex 38019
France
Email: david.furodet@st.com
URI: http://www.st.com/
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