draft-ietf-rmt-bb-fec-rs-02.txt   draft-ietf-rmt-bb-fec-rs-03.txt 
Reliable Multicast Transport J. Lacan Reliable Multicast Transport J. Lacan
Internet-Draft ENSICA/LAAS-CNRS Internet-Draft ENSICA/LAAS-CNRS
Intended status: Experimental V. Roca Intended status: Experimental V. Roca
Expires: June 25, 2007 INRIA Expires: November 8, 2007 INRIA
J. Peltotalo J. Peltotalo
S. Peltotalo S. Peltotalo
Tampere University of Technology Tampere University of Technology
December 22, 2006 May 7, 2007
Reed-Solomon Forward Error Correction (FEC) Reed-Solomon Forward Error Correction (FEC) Schemes
draft-ietf-rmt-bb-fec-rs-02.txt draft-ietf-rmt-bb-fec-rs-03.txt
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Copyright Notice Copyright Notice
Copyright (C) The IETF Trust (2006). Copyright (C) The IETF Trust (2007).
Abstract Abstract
This document describes a Fully-Specified FEC scheme for the Reed- This document describes a Fully-Specified FEC Scheme for the Reed-
Solomon forward error correction code and its application to the Solomon forward error correction codes over GF(2^^m), with m in
reliable delivery of data objects on the packet erasure channel. {2..16}, and its application to the reliable delivery of data objects
on the packet erasure channel.
This document also describes a Fully-Specified FEC Scheme for the
special case of Reed-Solomon codes over GF(2^^8) when there is no
encoding symbol group.
Finally, in the context of the Under-Specified Small Block Systematic
FEC Scheme (FEC Encoding ID 129), this document assigns an FEC
Instance ID to the special case of Reed-Solomon codes over GF(2^^8).
Reed-Solomon codes belong to the class of Maximum Distance Separable Reed-Solomon codes belong to the class of Maximum Distance Separable
(MDS) codes, i.e. they enable a receiver to recover the k source (MDS) codes, i.e. they enable a receiver to recover the k source
symbols from any set of k received symbols. symbols from any set of k received symbols. The schemes described
here are compatible with the implementation from Luigi Rizzo.
The implementation described here is compatible with the
implementation from Luigi Rizzo.
Table of Contents Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 4 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 4
2. Terminology . . . . . . . . . . . . . . . . . . . . . . . . . 5 2. Terminology . . . . . . . . . . . . . . . . . . . . . . . . . 5
3. Definitions Notations and Abbreviations . . . . . . . . . . . 6 3. Definitions Notations and Abbreviations . . . . . . . . . . . 6
3.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . 6 3.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . 6
3.2. Notations . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2. Notations . . . . . . . . . . . . . . . . . . . . . . . . 6
3.3. Abbreviations . . . . . . . . . . . . . . . . . . . . . . 7 3.3. Abbreviations . . . . . . . . . . . . . . . . . . . . . . 7
4. Formats and Codes . . . . . . . . . . . . . . . . . . . . . . 8 4. Formats and Codes with FEC Encoding ID 2 . . . . . . . . . . . 8
4.1. FEC Payload ID . . . . . . . . . . . . . . . . . . . . . . 8 4.1. FEC Payload ID . . . . . . . . . . . . . . . . . . . . . . 8
4.2. FEC Object Transmission Information . . . . . . . . . . . 9 4.2. FEC Object Transmission Information . . . . . . . . . . . 9
4.2.1. Mandatory Elements . . . . . . . . . . . . . . . . . . 9 4.2.1. Mandatory Elements . . . . . . . . . . . . . . . . . . 9
4.2.2. Common Elements . . . . . . . . . . . . . . . . . . . 9 4.2.2. Common Elements . . . . . . . . . . . . . . . . . . . 9
4.2.3. Scheme-Specific Elements . . . . . . . . . . . . . . . 9 4.2.3. Scheme-Specific Elements . . . . . . . . . . . . . . . 9
4.2.4. Encoding Format . . . . . . . . . . . . . . . . . . . 10 4.2.4. Encoding Format . . . . . . . . . . . . . . . . . . . 10
5. Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5. Formats and Codes with FEC Encoding ID 5 . . . . . . . . . . . 12
5.1. Determining the Maximum Source Block Length (B) . . . . . 12 5.1. FEC Payload ID . . . . . . . . . . . . . . . . . . . . . . 12
5.2. Determining the Number of Encoding Symbols of a Block . . 12 5.2. FEC Object Transmission Information . . . . . . . . . . . 12
6. Reed-Solomon Codes Specification for the Erasure Channel . . . 14 5.2.1. Mandatory Elements . . . . . . . . . . . . . . . . . . 12
6.1. Finite Field . . . . . . . . . . . . . . . . . . . . . . . 14 5.2.2. Common Elements . . . . . . . . . . . . . . . . . . . 12
6.2. Reed-Solomon Encoding Algorithm . . . . . . . . . . . . . 15 5.2.3. Scheme-Specific Elements . . . . . . . . . . . . . . . 13
6.2.1. Encoding Principles . . . . . . . . . . . . . . . . . 15 5.2.4. Encoding Format . . . . . . . . . . . . . . . . . . . 13
6.2.2. Encoding Complexity . . . . . . . . . . . . . . . . . 16 6. Procedures with FEC Encoding IDs 2 and 5 . . . . . . . . . . . 14
6.3. Reed-Solomon Decoding Algorithm . . . . . . . . . . . . . 16 6.1. Determining the Maximum Source Block Length (B) . . . . . 14
6.3.1. Decoding Principles . . . . . . . . . . . . . . . . . 16 6.2. Determining the Number of Encoding Symbols of a Block . . 14
6.3.2. Decoding Complexity . . . . . . . . . . . . . . . . . 17 7. Small Block Systematic FEC Scheme (FEC Encoding ID 129)
6.4. Implementation for the Packet Erasure Channel . . . . . . 17 and Reed-Solomon Codes over GF(2^^8) . . . . . . . . . . . . . 16
7. Security Considerations . . . . . . . . . . . . . . . . . . . 19 8. Reed-Solomon Codes Specification for the Erasure Channel . . . 17
8. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 20 8.1. Finite Field . . . . . . . . . . . . . . . . . . . . . . . 17
9. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . 21 8.2. Reed-Solomon Encoding Algorithm . . . . . . . . . . . . . 18
10. References . . . . . . . . . . . . . . . . . . . . . . . . . . 22 8.2.1. Encoding Principles . . . . . . . . . . . . . . . . . 18
10.1. Normative References . . . . . . . . . . . . . . . . . . . 22 8.2.2. Encoding Complexity . . . . . . . . . . . . . . . . . 19
10.2. Informative References . . . . . . . . . . . . . . . . . . 22 8.3. Reed-Solomon Decoding Algorithm . . . . . . . . . . . . . 19
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . . 24 8.3.1. Decoding Principles . . . . . . . . . . . . . . . . . 19
Intellectual Property and Copyright Statements . . . . . . . . . . 25 8.3.2. Decoding Complexity . . . . . . . . . . . . . . . . . 20
8.4. Implementation for the Packet Erasure Channel . . . . . . 20
9. Security Considerations . . . . . . . . . . . . . . . . . . . 23
10. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 24
11. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . 25
12. References . . . . . . . . . . . . . . . . . . . . . . . . . . 26
12.1. Normative References . . . . . . . . . . . . . . . . . . . 26
12.2. Informative References . . . . . . . . . . . . . . . . . . 26
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . . 28
Intellectual Property and Copyright Statements . . . . . . . . . . 29
1. Introduction 1. Introduction
The use of Forward Error Correction (FEC) codes is a classical The use of Forward Error Correction (FEC) codes is a classical
solution to improve the reliability of multicast and broadcast solution to improve the reliability of multicast and broadcast
transmissions. The [2] document describes a general framework to use transmissions. The [2] document describes a general framework to use
FEC in Content Delivery Protocols (CDP). The companion document [3] FEC in Content Delivery Protocols (CDP). The companion document [4]
describes some applications of FEC codes for content delivery. describes some applications of FEC codes for content delivery.
Recent FEC schemes like [6] and [7] proposed erasure codes based on Recent FEC schemes like [9] and [10] proposed erasure codes based on
sparse graphs/matrices. These codes are efficient in terms of sparse graphs/matrices. These codes are efficient in terms of
processing but not optimal in terms of correction capabilities when processing but not optimal in terms of correction capabilities when
dealing with "small" objects. dealing with "small" objects.
The FEC scheme described in this document belongs to the class of The FEC scheme described in this document belongs to the class of
Maximum Distance Separable codes that are optimal in terms of erasure Maximum Distance Separable codes that are optimal in terms of erasure
correction capability. In others words, it enables a receiver to correction capability. In others words, it enables a receiver to
recover the k source symbols from any set of exactly k encoding recover the k source symbols from any set of exactly k encoding
symbols. Even if the encoding/decoding complexity is larger than symbols. Even if the encoding/decoding complexity is larger than
that of [6] or [7], this family of codes is very useful. that of [9] or [10], this family of codes is very useful.
Many applications dealing with content transmission or content Many applications dealing with content transmission or content
storage already rely on packet-based Reed-Solomon codes. In storage already rely on packet-based Reed-Solomon codes. In
particular, many of them use the Reed-Solomon codec of Luigi Rizzo particular, many of them use the Reed-Solomon codec of Luigi Rizzo
[4]. The goal of the present document to specify an implementation [7]. The goal of the present document to specify an implementation
of Reed-Solomon codes that is compatible with this codec. of Reed-Solomon codes that is compatible with this codec.
The present document:
o introduces the Fully-Specified FEC Scheme with FEC Encoding ID 2
that specifies the use of Reed-Solomon codes over GF(2^^m), with m
in {2..16},
o introduces the Fully-Specified FEC Scheme with FEC Encoding ID 5
that focuses on the special case of Reed-Solomon codes over
GF(2^^8) and no encoding symbol group (i.e. exactly one symbol per
packet), and
o in the context of the Under-Specified Small Block Systematic FEC
Scheme (FEC Encoding ID 129) [3], assigns the FEC Instance ID 0 to
the special case of Reed-Solomon codes over GF(2^^8) and no
encoding symbol group.
2. Terminology 2. Terminology
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
document are to be interpreted as described in RFC 2119 [1]. document are to be interpreted as described in RFC 2119 [1].
3. Definitions Notations and Abbreviations 3. Definitions Notations and Abbreviations
3.1. Definitions 3.1. Definitions
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N denotes the number of source blocks into which the object shall N denotes the number of source blocks into which the object shall
be partitioned. be partitioned.
E denotes the encoding symbol length in bytes. E denotes the encoding symbol length in bytes.
S denotes the symbol size in units of m-bit elements. When m = 8, S denotes the symbol size in units of m-bit elements. When m = 8,
then S and E are equal. then S and E are equal.
m defines the length of the elements in the finite field, in bits. m defines the length of the elements in the finite field, in bits.
In this document, m belongs to {2..16}.
q defines the number of elements in the finite field. We have: q q defines the number of elements in the finite field. We have: q
= 2^^m in this specification. = 2^^m in this specification.
G denotes the number of encoding symbols per group, i.e. the G denotes the number of encoding symbols per group, i.e. the
number of symbols sent in the same packet. number of symbols sent in the same packet.
GM denotes the Generator Matrix of a Reed-Solomon code. GM denotes the Generator Matrix of a Reed-Solomon code.
rate denotes the "code rate", i.e. the k/n ratio. rate denotes the "code rate", i.e. the k/n ratio.
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FEC OTI stands for FEC Object Transmission Information. FEC OTI stands for FEC Object Transmission Information.
RS stands for Reed-Solomon. RS stands for Reed-Solomon.
MDS stands for Maximum Distance Separable code. MDS stands for Maximum Distance Separable code.
GF(q) denotes a finite field (A.K.A. Galois Field) with q GF(q) denotes a finite field (A.K.A. Galois Field) with q
elements. We assume that q = 2^^m in this document. elements. We assume that q = 2^^m in this document.
4. Formats and Codes 4. Formats and Codes with FEC Encoding ID 2
This section introduces the formats and codes associated to the
Fully-Specified FEC Scheme with FEC Encoding ID 2 that specifies the
use of Reed-Solomon codes over GF(2^^m).
4.1. FEC Payload ID 4.1. FEC Payload ID
The FEC Payload ID is composed of the Source Block Number and the The FEC Payload ID is composed of the Source Block Number and the
Encoding Symbol ID. The length of these two fields depends on the Encoding Symbol ID. The length of these two fields depends on the
parameter m (which is transmitted in the FEC OTI) as follows : parameter m (which is transmitted in the FEC OTI) as follows :
o The Source Block Number (field of size 32-m bits) identifies from o The Source Block Number (field of size 32-m bits) identifies from
which source block of the object the encoding symbol(s) in the which source block of the object the encoding symbol(s) in the
payload is (are) generated. There are a maximum of 2^^(32-m) payload is (are) generated. There are a maximum of 2^^(32-m)
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| Source Block Number (32-8=24 bits) | Enc. Symb. ID | | Source Block Number (32-8=24 bits) | Enc. Symb. ID |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
Figure 1: FEC Payload ID encoding format for m = 8 (default) Figure 1: FEC Payload ID encoding format for m = 8 (default)
0 1 2 3 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 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
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| Src Block Nb (32-16=16 bits) | Enc. Symbol ID (m=16 bits) | | Src Block Nb (32-16=16 bits) | Enc. Symbol ID (m=16 bits) |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
Figure 2: FEC Payload ID encoding format for m = 16 Figure 2: FEC Payload ID encoding format for m = 16
4.2. FEC Object Transmission Information 4.2. FEC Object Transmission Information
4.2.1. Mandatory Elements 4.2.1. Mandatory Elements
o FEC Encoding ID: the Fully-Specified FEC Scheme described in this o FEC Encoding ID: the Fully-Specified FEC Scheme described in this
document uses FEC Encoding ID 2. section uses FEC Encoding ID 2.
4.2.2. Common Elements 4.2.2. Common Elements
The following elements MUST be defined with the present FEC scheme: The following elements MUST be defined with the present FEC scheme:
o Transfer-Length (L): a non-negative integer indicating the length o Transfer-Length (L): a non-negative integer indicating the length
of the object in bytes. There are some restrictions on the of the object in bytes. There are some restrictions on the
maximum Transfer-Length that can be supported : maximum Transfer-Length that can be supported :
max_transfer_length = 2^^(32-m) * B * E max_transfer_length = 2^^(32-m) * B * E
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o Encoding-Symbol-Length (E): a non-negative integer indicating the o Encoding-Symbol-Length (E): a non-negative integer indicating the
length of each encoding symbol in bytes. length of each encoding symbol in bytes.
o Maximum-Source-Block-Length (B): a non-negative integer indicating o Maximum-Source-Block-Length (B): a non-negative integer indicating
the maximum number of source symbols in a source block. the maximum number of source symbols in a source block.
o Max-Number-of-Encoding-Symbols (max_n): a non-negative integer o Max-Number-of-Encoding-Symbols (max_n): a non-negative integer
indicating the maximum number of encoding symbols generated for indicating the maximum number of encoding symbols generated for
any source block. any source block.
Section 5 explains how to derive the values of each of these Section 6 explains how to derive the values of each of these
elements. elements.
4.2.3. Scheme-Specific Elements 4.2.3. Scheme-Specific Elements
The following element MUST be defined with the present FEC Scheme. The following element MUST be defined with the present FEC Scheme.
It contains two distinct pieces of information: It contains two distinct pieces of information:
o G: a non-negative integer indicating the number of encoding o G: a non-negative integer indicating the number of encoding
symbols per group used for the object. The default value is 1, symbols per group used for the object. The default value is 1,
meaning that each packet contains exactly one symbol. When no G meaning that each packet contains exactly one symbol. When no G
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o Finite Field parameter, m: The m parameter is the length of the o Finite Field parameter, m: The m parameter is the length of the
finite field elements, in bits. It also characterizes the number finite field elements, in bits. It also characterizes the number
of elements in the finite field: q = 2^^m elements. The default of elements in the finite field: q = 2^^m elements. The default
value is m = 8. When no finite field size parameter is value is m = 8. When no finite field size parameter is
communicated to the decoder, then this latter MUST assume that m = communicated to the decoder, then this latter MUST assume that m =
8. 8.
4.2.4. Encoding Format 4.2.4. Encoding Format
This section shows two possible encoding formats of the above FEC This section shows the two possible encoding formats of the above FEC
OTI. The present document does not specify when one encoding format OTI. The present document does not specify when one encoding format
or the other should be used. or the other should be used.
4.2.4.1. Using the General EXT_FTI Format 4.2.4.1. Using the General EXT_FTI Format
The FEC OTI binary format is the following, when the EXT_FTI The FEC OTI binary format is the following, when the EXT_FTI
mechanism is used (e.g. within the ALC [8] or NORM [9] protocols). mechanism is used (e.g. within the ALC [11] or NORM [12] protocols).
0 1 2 3 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 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 | | | HET = 64 | HEL = 4 | |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ + +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ +
| Transfer-Length (L) | | Transfer-Length (L) |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| m | G | Encoding Symbol Length (E) | | m | G | Encoding Symbol Length (E) |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| Max Source Block Length (B) | Max Nb Enc. Symbols (max_n) | | Max Source Block Length (B) | Max Nb Enc. Symbols (max_n) |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
Figure 3: EXT_FTI Header Format Figure 3: EXT_FTI Header Format
4.2.4.2. Using the FDT Instance (FLUTE specific) 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 When it is desired that the FEC OTI be carried in the FDT Instance of
a FLUTE session [10], the following XML attributes must be described a FLUTE session [13], the following XML attributes must be described
for the associated object: for the associated object:
o FEC-OTI-FEC-Encoding-ID o FEC-OTI-FEC-Encoding-ID
o FEC-OTI-Transfer-Length (L) o FEC-OTI-Transfer-Length (L)
o FEC-OTI-Encoding-Symbol-Length (E) o FEC-OTI-Encoding-Symbol-Length (E)
o FEC-OTI-Maximum-Source-Block-Length (B) o FEC-OTI-Maximum-Source-Block-Length (B)
o FEC-OTI-Max-Number-of-Encoding-Symbols (max_n) o FEC-OTI-Max-Number-of-Encoding-Symbols (max_n)
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The FEC-OTI-Scheme-Specific-Info contains the string resulting from The FEC-OTI-Scheme-Specific-Info contains the string resulting from
the Base64 encoding (in the XML Schema xs:base64Binary sense) of the the Base64 encoding (in the XML Schema xs:base64Binary sense) of the
following value: following value:
0 1 0 1
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| m | G | | m | G |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
Figure 4: FEC OTI Scheme Specific Information to be Included in the Figure 4: FEC OTI Scheme Specific Information to be included in the
FDT Instance FDT Instance
When no m parameter is to be carried in the FEC OTI, the m field is When no m parameter is to be carried in the FEC OTI, the m field is
set to 0 (which is not a valid seed value). Otherwise the m field set to 0 (which is not a valid seed value). Otherwise the m field
contains a valid value as explained in Section 4.2.3. Similarly, contains a valid value as explained in Section 4.2.3. Similarly,
when no G parameter is to be carried in the FEC OTI, the G field is when no G parameter is to be carried in the FEC OTI, the G field is
set to 0 (which is not a valid seed value). Otherwise the G field set to 0 (which is not a valid seed value). Otherwise the G field
contains a valid value as explained in Section 4.2.3. When neither m contains a valid value as explained in Section 4.2.3. When neither m
nor G are to be carried in the FEC OTI, then the sender simply omits nor G are to be carried in the FEC OTI, then the sender simply omits
the FEC-OTI-Scheme-Specific-Info attribute. the FEC-OTI-Scheme-Specific-Info attribute.
After Base64 encoding, the 2 bytes of the FEC OTI Scheme Specific After Base64 encoding, the 2 bytes of the FEC OTI Scheme Specific
Information are transformed into a string of 4 printable characters Information are transformed into a string of 4 printable characters
(in the 64-character alphabet) and added to the FEC-OTI-Scheme- (in the 64-character alphabet) and added to the FEC-OTI-Scheme-
Specific-Info attribute. Specific-Info attribute.
5. Procedures 5. Formats and Codes with FEC Encoding ID 5
5.1. Determining the Maximum Source Block Length (B) This section introduces the formats and codes associated to the
Fully-Specified FEC Scheme with FEC Encoding ID 5 that focuses on the
special case of Reed-Solomon codes over GF(2^^8) and no encoding
symbol group.
5.1. FEC Payload ID
The FEC Payload ID is composed of the Source Block Number and the
Encoding Symbol ID:
o The Source Block Number (24 bit field) identifies from which
source block of the object the encoding symbol in the payload is
generated. There are a maximum of 2^^24 blocks per object.
o The Encoding Symbol ID (8 bit field) identifies which specific
encoding symbol generated from the source block is carried in the
packet payload. There are a maximum of 2^^8 encoding symbols per
block. The first k values (0 to k - 1) identify source symbols,
the remaining n-k values identify repair symbols.
There MUST be exactly one FEC Payload ID per source or repair packet.
This FEC Payload ID refer to the one and only symbol of the packet.
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 (24 bits) | Enc. Symb. ID |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
Figure 5: FEC Payload ID encoding format with FEC Encoding ID 5
5.2. FEC Object Transmission Information
5.2.1. Mandatory Elements
o FEC Encoding ID: the Fully-Specified FEC Scheme described in this
section uses FEC Encoding ID 5.
5.2.2. Common Elements
The Common Elements are the same as those specified in Section 4.2.2
when m = 8 and G = 1.
5.2.3. Scheme-Specific Elements
No Scheme-Specific elements are defined by this FEC Scheme.
5.2.4. Encoding Format
This section shows the two possible encoding formats of the above FEC
OTI. The present document does not specify when one encoding format
or the other should be used.
5.2.4.1. 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 [11] or NORM [12] 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 = 3 | |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ +
| Transfer-Length (L) |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| Encoding Symbol Length (E) | MaxBlkLen (B) | max_n |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
Figure 6: EXT_FTI Header Format with FEC Encoding ID 5
5.2.4.2. Using the FDT Instance (FLUTE specific)
When it is desired that the FEC OTI be carried in the FDT Instance of
a FLUTE session [13], the following XML attributes must be described
for the associated object:
o FEC-OTI-FEC-Encoding-ID
o FEC-OTI-Transfer-Length (L)
o FEC-OTI-Encoding-Symbol-Length (E)
o FEC-OTI-Maximum-Source-Block-Length (B)
o FEC-OTI-Max-Number-of-Encoding-Symbols (max_n)
6. Procedures with FEC Encoding IDs 2 and 5
This section defines procedures that are common to FEC Encoding IDs 2
and 5. In case of FEC Encoding ID 5, m = 8 and G = 1. Note that the
block partitioning algorithm is defined in [2].
6.1. Determining the Maximum Source Block Length (B)
The finite field size parameter, m, defines the number of non zero The finite field size parameter, m, defines the number of non zero
elements in this field which is equal to: q - 1 = 2^^m - 1. Note elements in this field which is equal to: q - 1 = 2^^m - 1. Note
that q - 1 is also the theoretical maximum number of encoding symbols that q - 1 is also the theoretical maximum number of encoding symbols
that can be produced for a source block. For instance, when m = 8 that can be produced for a source block. For instance, when m = 8
(default): (default):
max1_B = 2^^8 - 1 = 255 max1_B = 2^^8 - 1 = 255
Additionally, a codec MAY impose other limitations on the maximum Additionally, a codec MAY impose other limitations on the maximum
skipping to change at page 12, line 30 skipping to change at page 14, line 34
implementation time, when the target use case is known. This results implementation time, when the target use case is known. This results
in a max2_B limitation. in a max2_B limitation.
Then, B is given by: Then, B is given by:
B = min(max1_B, max2_B) B = min(max1_B, max2_B)
Note that this calculation is only required at the coder, since the B Note that this calculation is only required at the coder, since the B
parameter is communicated to the decoder through the FEC OTI. parameter is communicated to the decoder through the FEC OTI.
5.2. Determining the Number of Encoding Symbols of a Block 6.2. Determining the Number of Encoding Symbols of a Block
The following algorithm, also called "n-algorithm", explains how to The following algorithm, also called "n-algorithm", explains how to
determine the actual number of encoding symbols for a given block. determine the actual number of encoding symbols for a given block.
AT A SENDER: AT A SENDER:
Input: Input:
B: Maximum source block length, for any source block. Section 5.1 B: Maximum source block length, for any source block. Section 6.1
explains how to determine its value. explains how to determine its value.
k: Current source block length. This parameter is given by the k: Current source block length. This parameter is given by the
block partitioning algorithm. block partitioning algorithm.
rate: FEC code rate, which is given by the user (e.g. when rate: FEC code rate, which is given by the user (e.g. when
starting a FLUTE sending application). It is expressed as a starting a FLUTE sending application). It is expressed as a
floating point value. floating point value.
Output: Output:
skipping to change at page 14, line 5 skipping to change at page 16, line 5
n = floor(k * max_n / B); n = floor(k * max_n / B);
Note that a Reed-Solomon decoder does not need to know the n value. Note that a Reed-Solomon decoder does not need to know the n value.
Therefore the receiver part of the "n-algorithm" is not necessary Therefore the receiver part of the "n-algorithm" is not necessary
from the Reed-Solomon decoder point of view. Yet a receiving from the Reed-Solomon decoder point of view. Yet a receiving
application using the Reed-Solomon FEC scheme will sometimes need to application using the Reed-Solomon FEC scheme will sometimes need to
know the n value used by the sender, for instance for memory know the n value used by the sender, for instance for memory
management optimizations. To that purpose, the FEC OTI carries all management optimizations. To that purpose, the FEC OTI carries all
the parameters needed for a receiver to execute the above algorithm. the parameters needed for a receiver to execute the above algorithm.
6. Reed-Solomon Codes Specification for the Erasure Channel 7. Small Block Systematic FEC Scheme (FEC Encoding ID 129) and Reed-
Solomon Codes over GF(2^^8)
In the context of the Under-Specified Small Block Systematic FEC
Scheme (FEC Encoding ID 129) [3], this document assigns the FEC
Instance ID 0 to the special case of Reed-Solomon codes over GF(2^^8)
and no encoding symbol group.
The FEC Instance ID 0 uses the Formats and Codes specified in [3].
The FEC Scheme with FEC Instance ID 0 MAY use the algorithm defined
in Section 9.1. of [3] to partition the file into source blocks.
This FEC Scheme MAY also use another algorithm. For instance the CDP
sender may change the length of each source block dynamically,
depending on some external criteria (e.g. to adjust the FEC coding
rate to the current loss rate experienced by NORM receivers) and
inform the CDP receivers of the current block length by means of the
EXT_FTI mechanism. This choice is out of the scope of the current
document.
8. Reed-Solomon Codes Specification for the Erasure Channel
Reed-Solomon (RS) codes are linear block codes. They also belong to Reed-Solomon (RS) codes are linear block codes. They also belong to
the class of MDS codes. A [n,k]-RS code encodes a sequence of k the class of MDS codes. A [n,k]-RS code encodes a sequence of k
source elements defined over a finite field GF(q) into a sequence of source elements defined over a finite field GF(q) into a sequence of
n encoding elements, where n is upper bounded by q - 1. The n encoding elements, where n is upper bounded by q - 1. The
implementation described in this document is based on a generator implementation described in this document is based on a generator
matrix built from a Vandermonde matrix put into systematic form. matrix built from a Vandermonde matrix put into systematic form.
Section 6.1 to Section 6.3 specify the [n,k]-RS codes when applied to Section 8.1 to Section 8.3 specify the [n,k]-RS codes when applied to
m-bit elements, and Section 6.4 the use of [n,k]-RS codes when m-bit elements, and Section 8.4 the use of [n,k]-RS codes when
applied to symbols composed of several m-bit elements, which is the applied to symbols composed of several m-bit elements, which is the
target of this specification. target of this specification.
6.1. Finite Field 8.1. Finite Field
A finite field GF(q) is defined as a finite set of q elements which A finite field GF(q) is defined as a finite set of q elements which
has a structure of field. It contains necessarily q = p^^m elements, has a structure of field. It contains necessarily q = p^^m elements,
where p is a prime number. With packet erasure channels, p is always where p is a prime number. With packet erasure channels, p is always
set to 2. The elements of the field GF(2^^m) can be represented by set to 2. The elements of the field GF(2^^m) can be represented by
polynomials with binary coefficients (i.e. over GF(2)) of degree polynomials with binary coefficients (i.e. over GF(2)) of degree
lower or equal than m-1. The polynomials can be associated to binary lower or equal than m-1. The polynomials can be associated to binary
vectors of length m. For example, the vector (11001) represents the vectors of length m. For example, the vector (11001) represents the
polynomial 1 + x + x^^4. This representation is often called polynomial 1 + x + x^^4. This representation is often called
polynomial representation. The addition between two elements is polynomial representation. The addition between two elements is
skipping to change at page 15, line 27 skipping to change at page 18, line 27
m = 15, "1100000000000001", (1+x+x^^15) m = 15, "1100000000000001", (1+x+x^^15)
m = 16, "11010000000010001", (1+x+x^^3+x^^12+x^^16) m = 16, "11010000000010001", (1+x+x^^3+x^^12+x^^16)
In order to facilitate the implementation, these polynomials are also In order to facilitate the implementation, these polynomials are also
primitive. This means that any element of GF(2^^m) can be expressed primitive. This means that any element of GF(2^^m) can be expressed
as a power of a given root of this polynomial. These polynomials are as a power of a given root of this polynomial. These polynomials are
also chosen so that they contain the minimum number of monomials. also chosen so that they contain the minimum number of monomials.
6.2. Reed-Solomon Encoding Algorithm 8.2. Reed-Solomon Encoding Algorithm
6.2.1. Encoding Principles 8.2.1. Encoding Principles
Let s = (s_0, ..., s_{k-1}) be a source vector of k elements over Let s = (s_0, ..., s_{k-1}) be a source vector of k elements over
GF(2^^m). Let e = (e_0, ..., e_{n-1}) be the corresponding encoding GF(2^^m). Let e = (e_0, ..., e_{n-1}) be the corresponding encoding
vector of n elements over GF(2^^m). Being a linear code, encoding is vector of n elements over GF(2^^m). Being a linear code, encoding is
performed by multiplying the source vector by a generator matrix, GM, performed by multiplying the source vector by a generator matrix, GM,
of k rows and n columns over GF(2^^m). Thus: of k rows and n columns over GF(2^^m). Thus:
e = s * GM. e = s * GM.
The definition of the generator matrix completely characterizes the The definition of the generator matrix completely characterizes the
RS code. RS code.
Let us consider that: n = 2^^m - 1 and: 0 < k <= n. Let us denote by Let us consider that: n = 2^^m - 1 and: 0 < k <= n. Let us denote by
alpha the root of the primitive polynomial of degree m chosen in the alpha the root of the primitive polynomial of degree m chosen in the
list of Section 6.1 for the corresponding value of m. Let us list of Section 8.1 for the corresponding value of m. Let us
consider a Vandermonde matrix of k rows and n columns, denoted by consider a Vandermonde matrix of k rows and n columns, denoted by
V_{k,n}, and built as follows: the {i, j} entry of V_{k,n} is v_{i,j} V_{k,n}, and built as follows: the {i, j} entry of V_{k,n} is v_{i,j}
= alpha^^(i*j), where 0 <= i <= k - 1 and 0 <= j <= n - 1. This = alpha^^(i*j), where 0 <= i <= k - 1 and 0 <= j <= n - 1. This
matrix generates a MDS code. However, this MDS code is not matrix generates a MDS code. However, this MDS code is not
systematic, which is a problem for many networking applications. To systematic, which is a problem for many networking applications. To
obtain a systematic matrix (and code), the simplest solution consists obtain a systematic matrix (and code), the simplest solution consists
in considering the matrix V_{k,k} formed by the first k columns of in considering the matrix V_{k,k} formed by the first k columns of
V_{k,n}, then to invert it and to multiply this inverse by V_{k,n}. V_{k,n}, then to invert it and to multiply this inverse by V_{k,n}.
Clearly, the product V_{k,k}^^-1 * V_{k,n} contains the identity Clearly, the product V_{k,k}^^-1 * V_{k,n} contains the identity
matrix I_k on its first k columns, meaning that the first k encoding matrix I_k on its first k columns, meaning that the first k encoding
skipping to change at page 16, line 19 skipping to change at page 19, line 19
Therefore, the generator matrix of the code considered in this Therefore, the generator matrix of the code considered in this
document is: document is:
GM = (V_{k,k}^^-1) * V_{k,n} GM = (V_{k,k}^^-1) * V_{k,n}
Note that, in practice, the [n,k]-RS code can be shortened to a Note that, in practice, the [n,k]-RS code can be shortened to a
[n',k]-RS code, where k <= n' < n, by considering the sub-matrix [n',k]-RS code, where k <= n' < n, by considering the sub-matrix
formed by the n' first columns of GM. formed by the n' first columns of GM.
6.2.2. Encoding Complexity 8.2.2. Encoding Complexity
Encoding can be performed by first pre-computing GM and by Encoding can be performed by first pre-computing GM and by
multiplying the source vector (k elements) by GM (k rows and n multiplying the source vector (k elements) by GM (k rows and n
columns). The complexity of the pre-computation of the generator columns). The complexity of the pre-computation of the generator
matrix can be estimated as the complexity of the multiplication of matrix can be estimated as the complexity of the multiplication of
the inverse of a Vandermonde matrix by n-k vectors (i.e. the last n-k the inverse of a Vandermonde matrix by n-k vectors (i.e. the last n-k
columns of V_{k,n}). Since the complexity of the inverse of a k*k- columns of V_{k,n}). Since the complexity of the inverse of a k*k-
Vandermonde matrix by a vector is O(k * log^^2(k)), the generator Vandermonde matrix by a vector is O(k * log^^2(k)), the generator
matrix can be computed in 0((n-k)* k * log^^2(k)) operations. When matrix can be computed in 0((n-k)* k * log^^2(k)) operations. When
the genarator matrix is pre-computed, the encoding needs k operations the generator matrix is pre-computed, the encoding needs k operations
per repair element (vector-matrix multiplication). per repair element (vector-matrix multiplication).
Encoding can also be performed by first computing the product s * Encoding can also be performed by first computing the product s *
V_{k,k}^^-1 and then by multiplying the result with V_{k,n}. The V_{k,k}^^-1 and then by multiplying the result with V_{k,n}. The
multiplication by the inverse of a square Vandermonde matrix is known multiplication by the inverse of a square Vandermonde matrix is known
as the interpolation problem and its complexity is O(k * log^^2 (k)). as the interpolation problem and its complexity is O(k * log^^2 (k)).
The multiplication by a Vandermonde matrix, known as the multipoint The multiplication by a Vandermonde matrix, known as the multipoint
evaluation problem, requires O((n-k) * log(k)) by using Fast Fourier evaluation problem, requires O((n-k) * log(k)) by using Fast Fourier
Transform, as explained in [11]. The total complexity of this Transform, as explained in [14]. The total complexity of this
encoding algorithm is then O((k/(n-k)) * log^^2(k) + log(k)) encoding algorithm is then O((k/(n-k)) * log^^2(k) + log(k))
operations per repair element. operations per repair element.
6.3. Reed-Solomon Decoding Algorithm 8.3. Reed-Solomon Decoding Algorithm
6.3.1. Decoding Principles 8.3.1. Decoding Principles
The Reed-Solomon decoding algorithm for the erasure channel allows The Reed-Solomon decoding algorithm for the erasure channel allows
the recovery of the k source elements from any set of k received the recovery of the k source elements from any set of k received
elements. It is based on the fundamental property of the generator elements. It is based on the fundamental property of the generator
matrix which is such that any k*k-submatrix is invertible (see [5]). matrix which is such that any k*k-submatrix is invertible (see [8]).
The first step of the decoding consists in extracting the k*k The first step of the decoding consists in extracting the k*k
submatrix of the generator matrix obtained by considering the columns submatrix of the generator matrix obtained by considering the columns
corresponding to the received elements. Indeed, since any encoding corresponding to the received elements. Indeed, since any encoding
element is obtained by multiplying the source vector by one column of element is obtained by multiplying the source vector by one column of
the generator matrix, the received vector of k encoding elements can the generator matrix, the received vector of k encoding elements can
be considered as the result of the multiplication of the source be considered as the result of the multiplication of the source
vector by a k*k submatrix of the generator matrix. Since this vector by a k*k submatrix of the generator matrix. Since this
submatrix is invertible, the second step of the algorithm is to submatrix is invertible, the second step of the algorithm is to
invert this matrix and to multiply the received vector by the invert this matrix and to multiply the received vector by the
obtained matrix to recover the source vector. obtained matrix to recover the source vector.
6.3.2. Decoding Complexity 8.3.2. Decoding Complexity
The decoding algorithm described previously includes the matrix The decoding algorithm described previously includes the matrix
inversion and the vector-matrix multiplication. With the classical inversion and the vector-matrix multiplication. With the classical
Gauss-Jordan algorithm, the matrix inversion requires O(k^^3) Gauss-Jordan algorithm, the matrix inversion requires O(k^^3)
operations and the vector-matrix multiplication is performed in operations and the vector-matrix multiplication is performed in
O(k^^2) operations. O(k^^2) operations.
This complexity can be improved by considering that the received This complexity can be improved by considering that the received
submatrix of GM is the product between the inverse of a Vandermonde submatrix of GM is the product between the inverse of a Vandermonde
matrix (V_(k,k)^^-1) and another Vandermonde matrix (denoted by V' matrix (V_(k,k)^^-1) and another Vandermonde matrix (denoted by V'
which is a submatrix of V_(k,n)). The decoding can be done by which is a submatrix of V_(k,n)). The decoding can be done by
multiplying the received vector by V'^^-1 (interpolation problem with multiplying the received vector by V'^^-1 (interpolation problem with
complexity O( k * log^^2(k)) ) then by V_{k,k} (multipoint evaluation complexity O( k * log^^2(k)) ) then by V_{k,k} (multipoint evaluation
with complexity O(k * log(k))). The global decoding complexity is with complexity O(k * log(k))). The global decoding complexity is
then O(log^^2(k)) operations per source element. then O(log^^2(k)) operations per source element.
6.4. Implementation for the Packet Erasure Channel 8.4. Implementation for the Packet Erasure Channel
In a packet erasure channel, each packet (and symbol(s) since packets In a packet erasure channel, each packet (and symbol(s) since packets
contain G >= 1 symbols) is either correctly received or erased. The contain G >= 1 symbols) is either correctly received or erased. The
location of the erased symbols in the sequence of symbols must be location of the erased symbols in the sequence of symbols MUST be
known. The following specification describes the use of Reed-Solomon known. The following specification describes the use of Reed-Solomon
codes for generating redundant symbols from the k source symbols and codes for generating redundant symbols from the k source symbols and
to recover the source symbols from any set of k received symbols. for recovering the source symbols from any set of k received symbols.
The k source symbols of a source block are assumed to be composed of The k source symbols of a source block are assumed to be composed of
S m-bit elements. Each m-bit element is associated to an element of S m-bit elements. Each m-bit element corresponds to an element of
the finite field GF(2^^m) through the polynomial representation the finite field GF(2^^m) through the polynomial representation
(Section 6.1). If some of the source symbols contain less than S (Section 8.1). If some of the source symbols contain less than S
elements, they are virtually padded with zero elements (it can be the elements, they MUST be virtually padded with zero elements (it can be
case for the last symbol of the last block of the object). the case for the last symbol of the last block of the object).
However, this padding need not be actually sent with the data to the
receivers.
The encoding process produces n-k repair symbols of size S m-bit The encoding process produces n encoding symbols of size S m-bit
elements, the k source symbols being also part of the n encoding elements, of which k are source symbols (this is a systematic code)
symbols (Figure 5). These repair symbols are created m-bit element and n-k are repair symbols (Figure 7). The m-bit elements of the
per m-bit element. More specifically, the j-th source vector is repair symbols are calculated using the corresponding m-bit elements
composed of the j-th element of each of the source symbols. of the source symbol set. A logical j-th source vector, comprised of
Similarly, the j-th encoding vector is composed of the j-th element the j-th elements from the set of source symbols, is used to
of each encoding symbol. calculate a j-th encoding vector. This j-th encoding vector then
provides the j-th elements for the set encoding symbols calculated
for the block. As a systematic code, the first k encoding symbols
are the same as the k source symbols and the last n-k repair symbols
are the result of the Reed Solomon encoding.
n Input: k source symbols
+-+-+----+-+ +---------------+ +-+-+-----------+-+
0 | | | | | | | | | | | |
| | | | | * | generator | = | | | | |
| | | | | | matrix | | | | | |
| | | | | | GM | | | | | |
source +--------------+ | (k x n) | +---------------------+
vector | | | | | | | +---------------+ ->| | | | | | |
j +--------------+ / +---------------------+
| | | | | / | | | | |
| | | | | encoding | | | | |
| | | | | vector | | | | |
| | | | | j | | | | |
| | | | | | | | | |
S-1 | | | | | | | | | |
+-+-+----+-+ +-+-+-----------+-+
k source symbols n encoding symbols
(source + repair)
Figure 5: Packet encoding scheme 0 j S-1
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| |X| | source symbol 0
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| |X| | source symbol 1
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
. . .
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| |X| | source symbol k-1
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
*
+----------------+
| generator |
| matrix |
| GM |
| (k x n) |
+----------------+
|
V
Output: n encoding symbols (source + repair)
0 j S-1
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| |X| | enc. symbol 0
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| |X| | enc. symbol 1
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
. . .
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| |Y| | enc. symbol n-1
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
Figure 7: Packet encoding scheme
An asset of this scheme is that the loss of some encoding symbols An asset of this scheme is that the loss of some encoding symbols
produces the same erasure pattern for each of the S encoding vectors. produces the same erasure pattern for each of the S encoding vectors.
It follows that the matrix inversion must be done only once and will It follows that the matrix inversion must be done only once and will
be used by all the S encoding vectors. For large S, this matrix be used by all the S encoding vectors. For large S, this matrix
inversion cost becomes negligible in front of the S matrix-vector inversion cost becomes negligible in front of the S matrix-vector
multiplications. multiplications.
Another asset is that the n-k repair symbols can be produced on Another asset is that the n-k repair symbols can be produced on
demand. For instance, a sender can start by producing a limited demand. For instance, a sender can start by producing a limited
number of repair symbols and later on, depending on the observed number of repair symbols and later on, depending on the observed
erasures on the channel, decide to produce additional repair symbols, erasures on the channel, decide to produce additional repair symbols,
up to the n-k upper limit. Indeed, to produce the repair symbol e_j, up to the n-k upper limit. Indeed, to produce the repair symbol e_j,
where k <= j < n, it is sufficient to multiply the S source vectors where k <= j < n, it is sufficient to multiply the S source vectors
with column j of GM. with column j of GM.
7. Security Considerations 9. Security Considerations
The security considerations for this document are the same as that of Data delivery can be subject to denial-of-service attacks by
[2]. attackers which send corrupted packets that are accepted as
legitimate by receivers. This is particularly a concern for
multicast delivery because a corrupted packet may be injected into
the session close to the root of the multicast tree, in which case
the corrupted packet will arrive at many receivers. This is
particularly a concern for the FEC building block because the use of
even one corrupted packet containing encoding data may result in the
decoding of an object that is completely corrupted and unusable. It
is thus RECOMMENDED that source authentication and integrity checking
are applied to decoded objects before delivering objects to an
application. For example, a SHA-1 hash [5] of an object may be
appended before transmission, and the SHA-1 hash is computed and
checked after the object is decoded but before it is delivered to an
application. Source authentication SHOULD be provided, for example
by including a digital signature verifiable by the receiver computed
on top of the hash value. It is also RECOMMENDED that a packet
authentication protocol such as TESLA [6] be used to detect and
discard corrupted packets upon arrival. Furthermore, it is
RECOMMENDED that Reverse Path Forwarding checks be enabled in all
network routers and switches along the path from the sender to
receivers to limit the possibility of a bad agent successfully
injecting a corrupted packet into the multicast tree data path.
8. IANA Considerations Another security concern is that some FEC information may be obtained
by receivers out-of-band in a session description, and if the session
description is forged or corrupted then the receivers will not use
the correct protocol for decoding content from received packets. To
avoid these problems, it is RECOMMENDED that measures be taken to
prevent receivers from accepting incorrect session descriptions,
e.g., by using source authentication to ensure that receivers only
accept legitimate session descriptions from authorized senders.
10. IANA Considerations
Values of FEC Encoding IDs and FEC Instance IDs are subject to IANA Values of FEC Encoding IDs and FEC Instance IDs are subject to IANA
registration. For general guidelines on IANA considerations as they registration. For general guidelines on IANA considerations as they
apply to this document, see [2]. This document assigns the Fully- apply to this document, see [2].
Specified FEC Encoding ID 2 under the ietf:rmt:fec:encoding name-
space to "Reed-Solomon Codes".
9. Acknowledgments This document assigns the Fully-Specified FEC Encoding ID 2 under the
"ietf:rmt:fec:encoding" name-space to "Reed-Solomon Codes over
GF(2^^m)".
The authors want to thank Luigi Rizzo for comments on the subject and This document assigns the Fully-Specified FEC Encoding ID 5 under the
for the design of the reference Reed-Solomon codec. "ietf:rmt:fec:encoding" name-space to "Reed-Solomon Codes over
GF(2^^8)".
10. References This document assigns the FEC Instance ID 0 scoped by the Under-
Specified FEC Encoding ID 129 to "Reed-Solomon Codes over GF(2^^8)".
More specifically, under the "ietf:rmt:fec:encoding:instance" sub-
name-space that is scoped by the "ietf:rmt:fec:encoding" called
"Small Block Systematic FEC Codes", this document assigns FEC
Instance ID 0 to "Reed-Solomon Codes over GF(2^^8)".
10.1. Normative References 11. Acknowledgments
The authors want to thank Brian Adamson for his valuable comments.
The authors also want to thank Luigi Rizzo for comments on the
subject and for the design of the reference Reed-Solomon codec.
12. References
12.1. Normative References
[1] Bradner, S., "Key words for use in RFCs to Indicate Requirement [1] Bradner, S., "Key words for use in RFCs to Indicate Requirement
Levels", RFC 2119. Levels", RFC 2119.
[2] Watson, M., Luby, M., and L. Vicisano, "Forward Error [2] Watson, M., Luby, M., and L. Vicisano, "Forward Error
Correction (FEC) Building Block", Correction (FEC) Building Block",
draft-ietf-rmt-fec-bb-revised-04.txt (work in progress), draft-ietf-rmt-fec-bb-revised-07.txt (work in progress),
September 2006. April 2007.
[3] Luby, M., Vicisano, L., Gemmell, J., Rizzo, L., Handley, M., [3] Watson, M., "Basic Forward Error Correction (FEC) Schemes",
draft-ietf-rmt-bb-fec-basic-schemes-revised-03.txt (work in
progress), February 2007.
[4] Luby, M., Vicisano, L., Gemmell, J., Rizzo, L., Handley, M.,
and J. Crowcroft, "The Use of Forward Error Correction (FEC) in and J. Crowcroft, "The Use of Forward Error Correction (FEC) in
Reliable Multicast", RFC 3453, December 2002. Reliable Multicast", RFC 3453, December 2002.
10.2. Informative References [5] "HMAC: Keyed-Hashing for Message Authentication", RFC 2104,
February 1997.
[4] Rizzo, L., "Reed-Solomon FEC codec (revised version of July [6] "Timed Efficient Stream Loss-Tolerant Authentication (TESLA):
Multicast Source Authentication Transform Introduction",
RFC 4082, June 2005.
12.2. Informative References
[7] Rizzo, L., "Reed-Solomon FEC codec (revised version of July
2nd, 1998), available at 2nd, 1998), available at
http://info.iet.unipi.it/~luigi/vdm98/vdm980702.tgz", http://info.iet.unipi.it/~luigi/vdm98/vdm980702.tgz, and
July 1998. mirrored at http://planete-bcast.inrialpes.fr/", July 1998.
[5] Mac Williams, F. and N. Sloane, "The Theory of Error Correcting [8] Mac Williams, F. and N. Sloane, "The Theory of Error Correcting
Codes", North Holland, 1977 . Codes", North Holland, 1977 .
[6] Luby, M., Shokrollahi, A., Watson, M., and T. Stockhammer, [9] Luby, M., Shokrollahi, A., Watson, M., and T. Stockhammer,
"Raptor Forward Error Correction Scheme", Internet "Raptor Forward Error Correction Scheme",
Draft draft-ietf-rmt-bb-fec-raptor-object-04 (work in draft-ietf-rmt-bb-fec-raptor-object-08 (work in progress),
progress), June 2006. April 2007.
[7] Roca, V., Neumann, C., and D. Furodet, "Low Density Parity [10] Roca, V., Neumann, C., and D. Furodet, "Low Density Parity
Check (LDPC) Forward Error Correction", Check (LDPC) Forward Error Correction",
draft-ietf-rmt-bb-fec-ldpc-04.txt (work in progress), draft-ietf-rmt-bb-fec-ldpc-06.txt (work in progress),
December 2006. May 2007.
[8] Luby, M., Watson, M., and L. Vicisano, "Asynchronous Layered [11] Luby, M., Watson, M., and L. Vicisano, "Asynchronous Layered
Coding (ALC) Protocol Instantiation", Coding (ALC) Protocol Instantiation",
draft-ietf-rmt-pi-alc-revised-03.txt (work in progress), draft-ietf-rmt-pi-alc-revised-04.txt (work in progress),
April 2006. February 2007.
[9] Adamson, B., Bormann, C., Handley, M., and J. Macker, [12] Adamson, B., Bormann, C., Handley, M., and J. Macker,
"Negative-acknowledgment (NACK)-Oriented Reliable Multicast "Negative-acknowledgment (NACK)-Oriented Reliable Multicast
(NORM) Protocol", draft-ietf-rmt-pi-norm-revised-03.txt (work (NORM) Protocol", draft-ietf-rmt-pi-norm-revised-04.txt (work
in progress), September 2006. in progress), March 2007.
[10] Paila, T., Walsh, R., Luby, M., Lehtonen, R., and V. Roca, [13] Paila, T., Walsh, R., Luby, M., Lehtonen, R., and V. Roca,
"FLUTE - File Delivery over Unidirectional Transport", "FLUTE - File Delivery over Unidirectional Transport",
draft-ietf-rmt-flute-revised-02.txt (work in progress), draft-ietf-rmt-flute-revised-03.txt (work in progress),
August 2006. January 2007.
[11] Gohberg, I. and V. Olshevsky, "Fast algorithms with [14] Gohberg, I. and V. Olshevsky, "Fast algorithms with
preprocessing for matrix-vector multiplication problems", preprocessing for matrix-vector multiplication problems",
Journal of Complexity, pp. 411-427, vol. 10, 1994 . Journal of Complexity, pp. 411-427, vol. 10, 1994 .
Authors' Addresses Authors' Addresses
Jerome Lacan Jerome Lacan
ENSICA/LAAS-CNRS ENSICA/LAAS-CNRS
1, place Emile Blouin 1, place Emile Blouin
Toulouse 31056 Toulouse 31056
France France
Email: jerome.lacan@ensica.fr Email: jerome.lacan@ensica.fr
URI: http://dmi.ensica.fr/auteur.php3?id_auteur=5 URI: http://dmi.ensica.fr/auteur.php3?id_auteur=5
Vincent Roca Vincent Roca
INRIA INRIA
655, av. de l'Europe 655, av. de l'Europe
Zirst; Montbonnot Inovallee; Montbonnot
ST ISMIER cedex 38334 ST ISMIER cedex 38334
France France
Email: vincent.roca@inrialpes.fr Email: vincent.roca@inrialpes.fr
URI: http://planete.inrialpes.fr/~roca/ URI: http://planete.inrialpes.fr/~roca/
Jani Peltotalo Jani Peltotalo
Tampere University of Technology Tampere University of Technology
P.O. Box 553 (Korkeakoulunkatu 1) P.O. Box 553 (Korkeakoulunkatu 1)
Tampere FIN-33101 Tampere FIN-33101
skipping to change at page 25, line 7 skipping to change at page 29, line 7
Tampere University of Technology Tampere University of Technology
P.O. Box 553 (Korkeakoulunkatu 1) P.O. Box 553 (Korkeakoulunkatu 1)
Tampere FIN-33101 Tampere FIN-33101
Finland Finland
Email: sami.peltotalo@tut.fi Email: sami.peltotalo@tut.fi
URI: http://atm.tut.fi/mad URI: http://atm.tut.fi/mad
Full Copyright Statement Full Copyright Statement
Copyright (C) The IETF Trust (2006). Copyright (C) The IETF Trust (2007).
This document is subject to the rights, licenses and restrictions This document is subject to the rights, licenses and restrictions
contained in BCP 78, and except as set forth therein, the authors contained in BCP 78, and except as set forth therein, the authors
retain all their rights. retain all their rights.
This document and the information contained herein are provided on an This document and the information contained herein are provided on an
"AS IS" basis and THE CONTRIBUTOR, THE ORGANIZATION HE/SHE REPRESENTS "AS IS" basis and THE CONTRIBUTOR, THE ORGANIZATION HE/SHE REPRESENTS
OR IS SPONSORED BY (IF ANY), THE INTERNET SOCIETY, THE IETF TRUST AND OR IS SPONSORED BY (IF ANY), THE INTERNET SOCIETY, THE IETF TRUST AND
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