draft-ietf-rmt-bb-fec-rs-04.txt   draft-ietf-rmt-bb-fec-rs-05.txt 
Reliable Multicast Transport J. Lacan Reliable Multicast Transport J. Lacan
Internet-Draft ISAE Internet-Draft ISAE/LAAS-CNRS
Intended status: Standards Track V. Roca Intended status: Standards Track V. Roca
Expires: April 12, 2008 INRIA Expires: May 15, 2008 INRIA
J. Peltotalo J. Peltotalo
S. Peltotalo S. Peltotalo
Tampere University of Technology Tampere University of Technology
October 10, 2007 November 12, 2007
Reed-Solomon Forward Error Correction (FEC) Schemes Reed-Solomon Forward Error Correction (FEC) Schemes
draft-ietf-rmt-bb-fec-rs-04.txt draft-ietf-rmt-bb-fec-rs-05.txt
Status of this Memo Status of this Memo
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Copyright Notice Copyright Notice
Copyright (C) The IETF Trust (2007). Copyright (C) The IETF Trust (2007).
Abstract Abstract
This document describes a Fully-Specified Forward Error Correction This document describes a Fully-Specified Forward Error Correction
(FEC) Scheme for the Reed-Solomon FEC codes over GF(2^^m), with m in (FEC) Scheme for the Reed-Solomon FEC codes over GF(2^^m), with m in
{2..16}, and its application to the reliable delivery of data objects {2..16}, and its application to the reliable delivery of data objects
on the packet erasure channel. on the packet erasure channel (i.e., a communication path where
packets are either received without any corruption or discarded
during transmission).
This document also describes a Fully-Specified FEC Scheme for the 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 special case of Reed-Solomon codes over GF(2^^8) when there is no
encoding symbol group. encoding symbol group.
Finally, in the context of the Under-Specified Small Block Systematic Finally, in the context of the Under-Specified Small Block Systematic
FEC Scheme (FEC Encoding ID 129), this document assigns an FEC 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). 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. The schemes described symbols from any set of k received symbols. The schemes described
here are compatible with the implementation from Luigi Rizzo. here are compatible with the implementation from Luigi Rizzo.
Table of Contents Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 4 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 5
2. Terminology . . . . . . . . . . . . . . . . . . . . . . . . . 5 2. Terminology . . . . . . . . . . . . . . . . . . . . . . . . . 6
3. Definitions Notations and Abbreviations . . . . . . . . . . . 6 3. Definitions Notations and Abbreviations . . . . . . . . . . . 7
3.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . 6 3.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . 7
3.2. Notations . . . . . . . . . . . . . . . . . . . . . . . . 6 3.2. Notations . . . . . . . . . . . . . . . . . . . . . . . . 7
3.3. Abbreviations . . . . . . . . . . . . . . . . . . . . . . 7 3.3. Abbreviations . . . . . . . . . . . . . . . . . . . . . . 8
4. Formats and Codes with FEC Encoding ID 2 . . . . . . . . . . . 8 4. Formats and Codes with FEC Encoding ID 2 . . . . . . . . . . . 10
4.1. FEC Payload ID . . . . . . . . . . . . . . . . . . . . . . 8 4.1. FEC Payload ID . . . . . . . . . . . . . . . . . . . . . . 10
4.2. FEC Object Transmission Information . . . . . . . . . . . 9 4.2. FEC Object Transmission Information . . . . . . . . . . . 11
4.2.1. Mandatory Elements . . . . . . . . . . . . . . . . . . 9 4.2.1. Mandatory Elements . . . . . . . . . . . . . . . . . . 11
4.2.2. Common Elements . . . . . . . . . . . . . . . . . . . 9 4.2.2. Common Elements . . . . . . . . . . . . . . . . . . . 11
4.2.3. Scheme-Specific Elements . . . . . . . . . . . . . . . 9 4.2.3. Scheme-Specific Elements . . . . . . . . . . . . . . . 11
4.2.4. Encoding Format . . . . . . . . . . . . . . . . . . . 10 4.2.4. Encoding Format . . . . . . . . . . . . . . . . . . . 12
5. Formats and Codes with FEC Encoding ID 5 . . . . . . . . . . . 12 5. Formats and Codes with FEC Encoding ID 5 . . . . . . . . . . . 14
5.1. FEC Payload ID . . . . . . . . . . . . . . . . . . . . . . 12 5.1. FEC Payload ID . . . . . . . . . . . . . . . . . . . . . . 14
5.2. FEC Object Transmission Information . . . . . . . . . . . 12 5.2. FEC Object Transmission Information . . . . . . . . . . . 14
5.2.1. Mandatory Elements . . . . . . . . . . . . . . . . . . 12 5.2.1. Mandatory Elements . . . . . . . . . . . . . . . . . . 14
5.2.2. Common Elements . . . . . . . . . . . . . . . . . . . 12 5.2.2. Common Elements . . . . . . . . . . . . . . . . . . . 14
5.2.3. Scheme-Specific Elements . . . . . . . . . . . . . . . 13 5.2.3. Scheme-Specific Elements . . . . . . . . . . . . . . . 15
5.2.4. Encoding Format . . . . . . . . . . . . . . . . . . . 13 5.2.4. Encoding Format . . . . . . . . . . . . . . . . . . . 15
6. Procedures with FEC Encoding IDs 2 and 5 . . . . . . . . . . . 14 6. Procedures with FEC Encoding IDs 2 and 5 . . . . . . . . . . . 16
6.1. Determining the Maximum Source Block Length (B) . . . . . 14 6.1. Determining the Maximum Source Block Length (B) . . . . . 16
6.2. Determining the Number of Encoding Symbols of a Block . . 14 6.2. Determining the Number of Encoding Symbols of a Block . . 16
7. Small Block Systematic FEC Scheme (FEC Encoding ID 129) 7. Small Block Systematic FEC Scheme (FEC Encoding ID 129)
and Reed-Solomon Codes over GF(2^^8) . . . . . . . . . . . . . 16 and Reed-Solomon Codes over GF(2^^8) . . . . . . . . . . . . . 19
8. Reed-Solomon Codes Specification for the Erasure Channel . . . 17 8. Reed-Solomon Codes Specification for the Erasure Channel . . . 20
8.1. Finite Field . . . . . . . . . . . . . . . . . . . . . . . 17 8.1. Finite Field . . . . . . . . . . . . . . . . . . . . . . . 20
8.2. Reed-Solomon Encoding Algorithm . . . . . . . . . . . . . 18 8.2. Reed-Solomon Encoding Algorithm . . . . . . . . . . . . . 21
8.2.1. Encoding Principles . . . . . . . . . . . . . . . . . 18 8.2.1. Encoding Principles . . . . . . . . . . . . . . . . . 21
8.2.2. Encoding Complexity . . . . . . . . . . . . . . . . . 19 8.2.2. Encoding Complexity . . . . . . . . . . . . . . . . . 22
8.3. Reed-Solomon Decoding Algorithm . . . . . . . . . . . . . 19 8.3. Reed-Solomon Decoding Algorithm . . . . . . . . . . . . . 22
8.3.1. Decoding Principles . . . . . . . . . . . . . . . . . 19 8.3.1. Decoding Principles . . . . . . . . . . . . . . . . . 22
8.3.2. Decoding Complexity . . . . . . . . . . . . . . . . . 20 8.3.2. Decoding Complexity . . . . . . . . . . . . . . . . . 23
8.4. Implementation for the Packet Erasure Channel . . . . . . 20 8.4. Implementation for the Packet Erasure Channel . . . . . . 23
9. Security Considerations . . . . . . . . . . . . . . . . . . . 23 9. Security Considerations . . . . . . . . . . . . . . . . . . . 27
9.1. Problem Statement . . . . . . . . . . . . . . . . . . . . 23 9.1. Problem Statement . . . . . . . . . . . . . . . . . . . . 27
9.2. Attacks Against the Data Flow . . . . . . . . . . . . . . 23 9.2. Attacks Against the Data Flow . . . . . . . . . . . . . . 27
9.3. Attacks against the FEC parameters . . . . . . . . . . . . 25 9.2.1. Access to Confidential Objects . . . . . . . . . . . . 27
10. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 26 9.2.2. Content Corruption . . . . . . . . . . . . . . . . . . 28
11. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . 27 9.3. Attacks Against the FEC Parameters . . . . . . . . . . . . 29
12. References . . . . . . . . . . . . . . . . . . . . . . . . . . 28 10. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 30
12.1. Normative References . . . . . . . . . . . . . . . . . . . 28 11. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . 31
12.2. Informative References . . . . . . . . . . . . . . . . . . 28 12. References . . . . . . . . . . . . . . . . . . . . . . . . . . 32
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . . 30 12.1. Normative References . . . . . . . . . . . . . . . . . . . 32
Intellectual Property and Copyright Statements . . . . . . . . . . 31 12.2. Informative References . . . . . . . . . . . . . . . . . . 32
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . . 34
Intellectual Property and Copyright Statements . . . . . . . . . . 35
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 [RFC5052] document describes a general framework
FEC in Content Delivery Protocols (CDP). The companion document [4] to use FEC in Content Delivery Protocols (CDP). The companion
describes some applications of FEC codes for content delivery. document [RFC3453] describes some applications of FEC codes for
content delivery.
Recent FEC schemes like [9] and [8] proposed erasure codes based on Recent FEC schemes like [RFC5053] and [draft-ietf-rmt-bb-fec-ldpc]
sparse graphs/matrices. These codes are efficient in terms of proposed erasure codes based on sparse graphs/matrices. These codes
processing but not optimal in terms of correction capabilities when are efficient in terms of processing but not optimal in terms of
dealing with "small" objects. correction capabilities when 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 [9] or [8], this family of codes is very useful. that of [RFC5053] or [draft-ietf-rmt-bb-fec-ldpc], 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
[5]. The goal of the present document to specify an implementation [RS-codec] [Rizzo97]. The goal of the present document is to specify
of Reed-Solomon codes that is compatible with this codec. an implementation of Reed-Solomon codes that is compatible with this
codec.
The present document: The present document:
o introduces the Fully-Specified FEC Scheme with FEC Encoding ID 2 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 that specifies the use of Reed-Solomon codes over GF(2^^m), with m
in {2..16}, in {2..16},
o introduces the Fully-Specified FEC Scheme with FEC Encoding ID 5 o introduces the Fully-Specified FEC Scheme with FEC Encoding ID 5
that focuses on the special case of Reed-Solomon codes over that focuses on the special case of Reed-Solomon codes over
GF(2^^8) and no encoding symbol group (i.e., exactly one symbol GF(2^^8) and no encoding symbol group (i.e., exactly one symbol
per packet), and per packet), and
o in the context of the Under-Specified Small Block Systematic FEC 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 Scheme (FEC Encoding ID 129)
the special case of Reed-Solomon codes over GF(2^^8) and no [draft-ietf-rmt-bb-fec-basic-schemes-revised], assigns the FEC
encoding symbol group. Instance ID 0 to the special case of Reed-Solomon codes over
GF(2^^8) and no encoding symbol group.
For a definition of the terms Fully-Specified and Under-Specified FEC
Schemes, see [RFC5052], section 4.
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 [RFC2119].
3. Definitions Notations and Abbreviations 3. Definitions Notations and Abbreviations
3.1. Definitions 3.1. Definitions
This document uses the same terms and definitions as those specified This document uses the same terms and definitions as those specified
in [2]. Additionally, it uses the following definitions: in [RFC5052]. Additionally, it uses the following definitions:
Source symbol: unit of data used during the encoding process. Source symbol: unit of data used during the encoding process.
Encoding symbol: unit of data generated by the encoding process. Encoding symbol: unit of data generated by the encoding process.
Repair symbol: encoding symbol that is not a source symbol. 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 Systematic code: FEC code in which the source symbols are part of
the encoding symbols. the encoding symbols.
Source block: a block of k source symbols that are considered Source block: a block of k source symbols that are considered
together for the encoding. together for the encoding.
Encoding Symbol Group: a group of encoding symbols that are sent Encoding Symbol Group: a group of encoding symbols that are sent
together within the same packet, and whose relationships to the together within the same packet, and whose relationships to the
source block can be derived from a single Encoding Symbol ID. source block can be derived from a single Encoding Symbol ID.
Source Packet: a data packet containing only source symbols. Source Packet: a data packet containing only source symbols.
Repair Packet: a data packet containing only repair 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 3.2. Notations
This document uses the following notations: This document uses the following notations:
L denotes the object transfer length in bytes. L denotes the object transfer length in bytes.
k denotes the number of source symbols in a source block. k denotes the number of source symbols in a source block.
n_r denotes the number of repair symbols generated for a source n_r denotes the number of repair symbols generated for a source
block. block.
skipping to change at page 7, line 19 skipping to change at page 8, line 31
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}. 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. CR denotes the "code rate", i.e., the k/n ratio.
a^^b denotes a raised to the power b. a^^b denotes a raised to the power b.
a^^-1 denotes the inverse of a. a^^-1 denotes the inverse of a.
I_k denotes the k*k identity matrix. I_k denotes the k*k identity matrix.
3.3. Abbreviations 3.3. Abbreviations
This document uses the following abbreviations: This document uses the following abbreviations:
skipping to change at page 8, line 7 skipping to change at page 10, line 7
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 (also known as Galois Field) with q GF(q) denotes a finite field (also known as 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 with FEC Encoding ID 2 4. Formats and Codes with FEC Encoding ID 2
This section introduces the formats and codes associated to the This section introduces the formats and codes associated with the
Fully-Specified FEC Scheme with FEC Encoding ID 2 that specifies the Fully-Specified FEC Scheme with FEC Encoding ID 2 that specifies the
use of Reed-Solomon codes over GF(2^^m). 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
skipping to change at page 10, line 26 skipping to change at page 12, line 26
4.2.4. Encoding Format 4.2.4. Encoding Format
This section shows the 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 [10] or NORM [11] protocols). mechanism is used (e.g., within the ALC
[draft-ietf-rmt-pi-alc-revised] or NORM
[draft-ietf-rmt-pi-norm-revised] 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 = 4 | | | 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 (File When it is desired that the FEC OTI be carried in the FDT (File
Delivery Table) Instance of a FLUTE session [12], the following XML Delivery Table) Instance of a FLUTE session
attributes must be described for the associated object: [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-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)
o FEC-OTI-Scheme-Specific-Info o FEC-OTI-Scheme-Specific-Info
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
skipping to change at page 11, line 34 skipping to change at page 13, line 36
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 During 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) that is added to the FEC-OTI-Scheme-
Specific-Info attribute. Specific-Info attribute.
5. Formats and Codes with FEC Encoding ID 5 5. Formats and Codes with FEC Encoding ID 5
This section introduces the formats and codes associated to the This section introduces the formats and codes associated with the
Fully-Specified FEC Scheme with FEC Encoding ID 5 that focuses on 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 special case of Reed-Solomon codes over GF(2^^8) and no encoding
symbol group. symbol group.
5.1. FEC Payload ID 5.1. 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: Encoding Symbol ID:
o The Source Block Number (24 bit field) identifies from which o The Source Block Number (24 bit field) identifies from which
source block of the object the encoding symbol in the payload is source block of the object the encoding symbol in the payload is
generated. There is a maximum of 2^^24 blocks per object. generated. There is a maximum of 2^^24 blocks per object.
o The Encoding Symbol ID (8 bit field) identifies which specific o The Encoding Symbol ID (8 bit field) identifies which specific
encoding symbol generated from the source block is carried in the encoding symbol generated from the source block is carried in the
packet payload. There is a maximum of 2^^8 encoding symbols per packet payload. There is a maximum of 2^^8 encoding symbols per
block. The first k values (0 to k - 1) identify source symbols, block. The first k values (0 to k - 1) identify source symbols,
the remaining n-k values identify repair symbols. the remaining n-k values identify repair symbols.
There MUST be exactly one FEC Payload ID per source or repair packet. 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. This FEC Payload ID refers to the one and only symbol of the packet.
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
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| Source Block Number (24 bits) | Enc. Symb. ID | | Source Block Number (24 bits) | Enc. Symb. ID |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
Figure 5: FEC Payload ID encoding format with FEC Encoding ID 5 Figure 5: FEC Payload ID encoding format with FEC Encoding ID 5
5.2. FEC Object Transmission Information 5.2. FEC Object Transmission Information
skipping to change at page 13, line 18 skipping to change at page 15, line 18
5.2.4. Encoding Format 5.2.4. Encoding Format
This section shows the 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.
5.2.4.1. Using the General EXT_FTI Format 5.2.4.1. 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 [10] or NORM [11] protocols). mechanism is used (e.g., within the ALC
[draft-ietf-rmt-pi-alc-revised] or NORM
[draft-ietf-rmt-pi-norm-revised] 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 = 3 | | | HET = 64 | HEL = 3 | |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ + +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ +
| Transfer-Length (L) | | Transfer-Length (L) |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| Encoding Symbol Length (E) | MaxBlkLen (B) | max_n | | Encoding Symbol Length (E) | MaxBlkLen (B) | max_n |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
Figure 6: EXT_FTI Header Format with FEC Encoding ID 5 Figure 6: EXT_FTI Header Format with FEC Encoding ID 5
5.2.4.2. Using the FDT Instance (FLUTE specific) 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 When it is desired that the FEC OTI be carried in the FDT Instance of
a FLUTE session [12], the following XML attributes must be described a FLUTE session [draft-ietf-rmt-flute-revised], the following XML
for the associated object: attributes must be described 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)
6. Procedures with FEC Encoding IDs 2 and 5 6. Procedures with FEC Encoding IDs 2 and 5
This section defines procedures that are common to FEC Encoding IDs 2 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 and 5. In case of FEC Encoding ID 5, m = 8 and G = 1. The block
block partitioning algorithm is defined in [2]. partitioning algorithm that is defined in section 9.1 of [RFC5052]
MUST be used with FEC Encoding IDs 2 and 5.
6.1. Determining the Maximum Source Block Length (B) 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) there is a maximum of 2^^8 - 1 = 255 encoding symbols. (default) there is a maximum of 2^^8 - 1 = 255 encoding symbols.
Given the target FEC code rate (e.g., provided by the user when Given the target FEC code rate (e.g., provided by the user when
starting a FLUTE sending application), the sender calculates: starting a FLUTE sending application), the sender calculates:
max1_B = floor((2^^m - 1) * rate) max1_B = floor((2^^m - 1) * CR)
This max1_B value leaves enough room for the sender to produce the This max1_B value leaves enough room for the sender to produce the
desired number of parity symbols. desired number of parity symbols.
Additionally, a codec MAY impose other limitations on the maximum Additionally, a codec MAY impose other limitations on the maximum
block size. Yet it is not expected that such limits exist when using block size. Yet it is not expected that such limits exist when using
the default m = 8 value. This decision MUST be clarified at the default m = 8 value. This decision MUST be clarified at
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.
6.2. 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 maximum number of encoding symbols generated for any
source block (max_n) and the number of encoding symbols for a given
block (n) as a function of the target code rate.
AT A SENDER: AT A SENDER:
Input: Input:
B: Maximum source block length, for any source block. Section 6.1 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 CR: FEC code rate, which is given by the user (e.g., when starting
starting a FLUTE sending application). It is expressed as a a FLUTE sending application). It is expressed as a floating point
floating point value. value.
Output: Output:
max_n: Maximum number of encoding symbols generated for any source max_n: Maximum number of encoding symbols generated for any source
block. block.
n: Number of encoding symbols generated for this source block. n: Number of encoding symbols generated for this source block.
Algorithm: Algorithm:
max_n = ceil(B / rate); max_n = ceil(B / CR);
if (max_n > 2^^m - 1) then return an error ("invalid code rate"); if (max_n > 2^^m - 1) then return an error ("invalid code rate");
n = floor(k * max_n / B); n = floor(k * max_n / B);
AT A RECEIVER: AT A RECEIVER:
Input: Input:
B: Extracted from the received FEC OTI. B: Extracted from the received FEC OTI.
skipping to change at page 15, line 45 skipping to change at page 17, line 48
k: Given by the block partitioning algorithm. k: Given by the block partitioning algorithm.
Output: Output:
n n
Algorithm: Algorithm:
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. It is RECOMMENDED that the "n-algorithm" be used by a sender, but
Therefore the receiver part of the "n-algorithm" is not necessary other algorithms remain possible to determine max_n and/or n.
from the Reed-Solomon decoder point of view. Yet a receiving
application using the Reed-Solomon FEC scheme will sometimes need to At a receiver, the max_n value is extracted from the received FEC
know the n value used by the sender, for instance for memory OTI. Since the Reed-Solomon decoder does not need to know the actual
management optimizations. To that purpose, the FEC OTI carries all n value, using the receiver part of the "n-algorithm" is not
the parameters needed for a receiver to execute the above algorithm. necessary from a decoding point of view.
However a receiver may want to have an estimate of n for other
reasons (e.g., for memory management purposes). In that case, a
receiver knows that the number of encoding symbols of a block cannot
exceed max_n. Additionally, if a receiver believes that a sender
uses the "n-algorithm", this receiver MAY use the receiver part of
the "n-algorithm" to get a better estimate of n. When this is the
case, a receiver MUST be prepared to handle symbols with an Encoding
Symbol ID superior or equal to the computed n value (e.g., it can
choose to simply drop them).
7. Small Block Systematic FEC Scheme (FEC Encoding ID 129) and Reed- 7. Small Block Systematic FEC Scheme (FEC Encoding ID 129) and Reed-
Solomon Codes over GF(2^^8) Solomon Codes over GF(2^^8)
In the context of the Under-Specified Small Block Systematic FEC In the context of the Under-Specified Small Block Systematic FEC
Scheme (FEC Encoding ID 129) [3], this document assigns the FEC Scheme (FEC Encoding ID 129)
Instance ID 0 to the special case of Reed-Solomon codes over GF(2^^8) [draft-ietf-rmt-bb-fec-basic-schemes-revised], this document assigns
and no encoding symbol group. 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 Instance ID 0 uses the Formats and Codes specified in
[draft-ietf-rmt-bb-fec-basic-schemes-revised].
The FEC Scheme with FEC Instance ID 0 MAY use the algorithm defined The FEC Scheme with FEC Instance ID 0 MAY use the block partitioning
in Section 9.1. of [2] to partition the file into source blocks. algorithm defined in Section 9.1. of [RFC5052] to partition the
This FEC Scheme MAY also use another algorithm. For instance the CDP object into source blocks. This FEC Scheme MAY also use another
sender may change the length of each source block dynamically, algorithm. For instance the CDP sender may change the length of each
depending on some external criteria (e.g., to adjust the FEC coding source block dynamically, depending on some external criteria (e.g.,
rate to the current loss rate experienced by NORM receivers) and to adjust the FEC coding rate to the current loss rate experienced by
inform the CDP receivers of the current block length by means of the NORM receivers) and inform the CDP receivers of the current block
EXT_FTI mechanism. This choice is out of the scope of the current length by means of the EXT_FTI mechanism. This choice is out of the
document. scope of the current document.
8. Reed-Solomon Codes Specification for the Erasure Channel 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 8.1 to Section 8.3 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 8.4 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.
A reader who wants to understand the underlying theory is invited to
refer to references [Rizzo97] and [MWS77].
8.1. 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 to m-1. The polynomials can be associated with 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
defined as the addition of binary polynomials in GF(2) and the defined as the addition of binary polynomials in GF(2) and the
multiplication is the multiplication modulo a given irreducible multiplication is the multiplication modulo a given irreducible
polynomial over GF(2) of degree m with coefficients in GF(2). Note polynomial over GF(2) of degree m. Note that all the roots of this
that all the roots of this polynomial are in GF(2^^m) but not in polynomial are in GF(2^^m) but not in GF(2).
GF(2).
A finite field GF(2^^m) is completely characterized by the The chosen polynomial representation of the finite field GF(2^^m) is
irreducible polynomial. The following polynomials are chosen to completely characterized by the irreducible polynomial. The
represent the field GF(2^^m), for m varying from 2 to 16: following polynomials are chosen to represent the field GF(2^^m), for
m varying from 2 to 16:
m = 2, "111" (1+x+x^^2) m = 2, "111" (1+x+x^^2)
m = 3, "1101", (1+x+x^^3) m = 3, "1101", (1+x+x^^3)
m = 4, "11001", (1+x+x^^4) m = 4, "11001", (1+x+x^^4)
m = 5, "101001", (1+x^^2+x^^5) m = 5, "101001", (1+x^^2+x^^5)
m = 6, "1100001", (1+x+x^^6) m = 6, "1100001", (1+x+x^^6)
skipping to change at page 19, line 27 skipping to change at page 22, line 29
formed by the n' first columns of GM. formed by the n' first columns of GM.
8.2.2. 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 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 n-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 k*k-Vandermonde matrix by a vector is O(k * (log(k))^^2), the
matrix can be computed in 0((n-k)* k * log^^2(k)) operations. When generator matrix can be computed in 0((n-k)* k * (log(k))^^2))
the generator matrix is pre-computed, the encoding needs k operations operations. When the generator matrix is pre-computed, the encoding
per repair element (vector-matrix multiplication). needs k operations 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 *
The multiplication by a Vandermonde matrix, known as the multipoint (log(k))^^2). The multiplication by a Vandermonde matrix, known as
evaluation problem, requires O((n-k) * log(k)) by using Fast Fourier the multipoint evaluation problem, requires O((n-k) * log(k)) by
Transform, as explained in [7]. The total complexity of this using Fast Fourier Transform, as explained in [GO94]. The total
encoding algorithm is then O((k/(n-k)) * log^^2(k) + log(k)) complexity of this encoding algorithm is then O((k/(n-k)) *
operations per repair element. (log(k))^^2 + log(k)) operations per repair element.
8.3. Reed-Solomon Decoding Algorithm 8.3. Reed-Solomon Decoding Algorithm
8.3.1. 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 [6]). matrix which is such that any k*k-submatrix is invertible (see
The first step of the decoding consists in extracting the k*k [MWS77]). The first step of the decoding consists in extracting the
submatrix of the generator matrix obtained by considering the columns k*k submatrix of the generator matrix obtained by considering the
corresponding to the received elements. Indeed, since any encoding columns corresponding to the received elements. Indeed, since any
element is obtained by multiplying the source vector by one column of encoding element is obtained by multiplying the source vector by one
the generator matrix, the received vector of k encoding elements can column of the generator matrix, the received vector of k encoding
be considered as the result of the multiplication of the source elements can be considered as the result of the multiplication of the
vector by a k*k submatrix of the generator matrix. Since this source 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.
8.3.2. 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(k))^^2) ) then by V_{k,k} (multipoint
with complexity O(k * log(k))). The global decoding complexity is evaluation with complexity O(k * log(k))). The global decoding
then O(log^^2(k)) operations per source element. complexity is then O((log(k))^^2) operations per source element.
8.4. 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
for recovering 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 corresponds 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 8.1). 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 MUST be virtually padded with zero elements (it can be elements, they MUST be virtually padded with zero elements (this can
the case for the last symbol of the last block of the object). be the case for the last symbol of the last block of the object).
However, this padding does not need to be actually sent with the data However, this padding does not need to be actually sent with the data
to the receivers. to the receivers.
The encoding process produces n encoding symbols of size S m-bit The encoding process produces n encoding symbols of size S m-bit
elements, of which k are source symbols (this is a systematic code) elements, of which k are source symbols (this is a systematic code)
and n-k are repair symbols (Figure 7). The m-bit elements of the and n-k are repair symbols (Figure 7). The m-bit elements of the
repair symbols are calculated using the corresponding m-bit elements repair symbols are calculated using the corresponding m-bit elements
of the source symbol set. A logical j-th source vector, comprised of of the source symbol set. A logical u-th source vector, comprised of
the j-th elements from the set of source symbols, is used to the u-th elements from the set of source symbols, is used to
calculate a j-th encoding vector. This j-th encoding vector then calculate a u-th encoding vector. This u-th encoding vector then
provides the j-th elements for the set encoding symbols calculated provides the u-th elements for the set encoding symbols calculated
for the block. As a systematic code, the first k encoding symbols 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 same as the k source symbols and the last n-k repair symbols
are the result of the Reed-Solomon encoding. are the result of the Reed-Solomon encoding.
Input: k source symbols Input: k source symbols
0 j S-1 0 u S-1
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| |X| | source symbol 0 | |X| | source symbol 0
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| |X| | source symbol 1 | |X| | source symbol 1
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
. . . . . .
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| |X| | source symbol k-1 | |X| | source symbol k-1
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
skipping to change at page 21, line 39 skipping to change at page 25, line 32
| generator matrix | | generator matrix |
| GM | | GM |
| (k x n) | | (k x n) |
+--------------------+ +--------------------+
| |
V V
Output: n encoding symbols (source + repair) Output: n encoding symbols (source + repair)
0 j S-1 0 u S-1
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| |X| | enc. symbol 0 | |X| | enc. symbol 0
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| |X| | enc. symbol 1 | |X| | enc. symbol 1
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
. . . . . .
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| |Y| | enc. symbol n-1 | |Y| | enc. symbol n-1
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
skipping to change at page 23, line 40 skipping to change at page 27, line 40
making the decoding of an object computationally expensive). making the decoding of an object computationally expensive).
These attacks can be launched either against the data flow itself These attacks can be launched either against the data flow itself
(e.g. by sending forged symbols) or against the FEC parameters that (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- are sent either in-band (e.g., in an EXT_FTI or FDT Instance) or out-
of-band (e.g., in a session description). of-band (e.g., in a session description).
9.2. Attacks Against the Data Flow 9.2. Attacks Against the Data Flow
First of all, let us consider the attacks against the data flow. First of all, let us consider the attacks against the data flow.
Access control 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 [14]). If access
control 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.
Protection against corruptions (forged packets) is achieved by means 9.2.1. Access to Confidential Objects
of a content integrity verification/sender authentication scheme.
This service can be provided at the object level, but in that case a Access control to the object being transmitted is typically provided
receiver has no way to identify which symbol(s) is(are) corrupted if by means of encryption. This encryption can be done over the whole
the object is detected as corrupted. This service can also be object (e.g., by the content provider, before the FEC encoding
provided at the packet level, and after having removed all forged process), or be done on a packet per packet basis (e.g., when IPSec/
packets, the object can be recovered if the number of symbols ESP is used [RFC4303]). If access control is a concern, it is
remaining is sufficient. Several techniques can provide this source 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.
9.2.2. 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: authentication/content integrity service:
o at the object level, the object MAY be digitally signed (with o at the object level, the object MAY be digitally signed (with
public key cryptography) (e.g., using RSASSA-PKCS1-v1_5 [13]). public key cryptography), for instance by using RSASSA-PKCS1-v1_5
This signature enables a receiver to check the object, once this [RFC3447]. This signature enables a receiver to check the object
latter has been fully decoded. Even if digital signatures are integrity, once this latter has been fully decoded. Even if
computationally expensive, this calculation occurs only once per digital signatures are computationally expensive, this calculation
object, which is usually acceptable; occurs only once per object, which is usually acceptable;
o at the packet level, each packet can be digitally signed. A major o at the packet level, each packet can be digitally signed. A major
limitation is the high computational and transmission overheads limitation is the high computational and transmission overheads
that this solution incurs (unless ECC is used, but ECC is that this solution requires (unless Elliptic Curve Cryptography
protected by IPR). To avoid this problem, the signature may span (ECC) is used, but ECC is the subject of proprietary patents). To
a set of symbols in order to amortize the signature calculation, avoid this problem, the signature may span a set of symbols
but if a single symbol is missing, the integrity of the whole set (instead of a single one) in order to amortize the signature
cannot be checked; 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) o at the packet level, a Group Message Authentication Code (MAC)
[15] (e.g., using HMAC-SHA-1 with a secret key shared by all the [RFC2104] scheme can be used, for instance by using HMAC-SHA-1
group members, senders and receivers) scheme can be used. This with a secret key shared by all the group members, senders and
technique creates a cryptographically secured (thanks to the receivers. This technique creates a cryptographically secured
secret key) digest of a packet that is sent along with the packet. (thanks to the secret key) digest of a packet that is sent along
The Group MAC scheme does not incur prohibitive processing load with the packet. The Group MAC scheme does not create prohibitive
nor transmission overhead, but it has a major limitation: it only processing load nor transmission overhead, but it has a major
provides a group authentication/integrity service since all group limitation: it only provides a group authentication/integrity
members share the same secret group key, which means that each service since all group members share the same secret group key,
member can send a forged packet. It is therefore restricted to which means that each member can send a forged packet. It is
situations where group members are fully trusted (or in therefore restricted to situations where group members are fully
association with another technique as a pre-check); trusted (or in association with another technique as a pre-check);
o at the packet level, TESLA [16] is a very attractive and efficient
solution that is robust to losses, provides a true authentication/
integrity service, and does not incur any prohibitive processing
load or transmission overhead.
It is up to the developer, who knows the security requirements of the
target use-case, to define which solution is the most appropriate.
Nonetheless, it is RECOMMENDED that at least one of these techniques
be used.
o at the packet level, TESLA [RFC4082] is a very attractive and
efficient 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 Techniques relying on public key cryptography (digital signatures and
TESLA during the bootstrap process) require that public keys be TESLA during the bootstrap process, when used) require that public
securely associated to the entities. This can be achieved by a keys be securely associated to the entities. This can be achieved by
Public Key Infrastructure (PKI), or by a PGP Web of Trust, or by pre- a Public Key Infrastructure (PKI), or by a PGP Web of Trust, or by
distributing the public keys of each group member. It is up to the pre-distributing the public keys of each group member.
developer, who knows the features of the target use-case, to define
which solution to use.
Techniques relying on symmetric key cryptography (group MAC) require Techniques relying on symmetric key cryptography (group MAC) require
that a secret key be shared by all group members. This can be 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 achieved by means of a group key management protocol, or simply by
pre-distributing the secret key (but this manual solution has many pre-distributing the secret key (but this manual solution has many
limitations). Here also, it is up to the developer to define which limitations).
solution to use, taking into account the target use-case features.
9.3. Attacks against the FEC parameters It is up to the developer and deployer, who know 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.
9.3. Attacks Against the FEC Parameters
Let us now consider attacks against the FEC parameters (or FEC OTI). 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 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., FDT Instance containing FEC OTI for the object) or out-of-band (e.g.,
in a session description). Attacks on these FEC parameters can in a session description). Attacks on these FEC parameters can
prevent the decoding of the associated object: for instance modifying prevent the decoding of the associated object: for instance modifying
the B parameter will lead to a different block partitioning at a the B parameter will lead to a different block partitioning at a
receiver thereby compromising decoding; or setting the m parameter to receiver thereby compromising decoding; or setting the m parameter to
16 instead of 8 with FEC Encoding ID 2 will increase the processing 16 instead of 8 with FEC Encoding ID 2 will increase the processing
load while compromising decoding. load while compromising decoding.
It is therefore RECOMMENDED that security measures be taken to It is therefore RECOMMENDED that security measures be taken to
guarantee the FEC OTI integrity. To that purpose, the packets guarantee the FEC OTI integrity. To that purpose, the packets
carrying the FEC parameters sent in-band (i.e., in an EXT_FTI header carrying the FEC parameters sent in-band in an EXT_FTI header
extension or in an FDT Instance) may be protected by one of the per- extension SHOULD be protected by one of the per-packet techniques
packet techniques described above: TESLA, digital signature, or a described above: digital signature, group MAC, or TESLA. When FEC
group MAC. Alternatively, when FEC OTI is contained in an FDT OTI is contained in an FDT Instance, this object SHOULD be protected,
Instance, this object may be digitally signed. Finally, when FEC OTI for instance by digitally signing it with XML digital signatures
is sent out-of-band for instance in a session description, this [RFC3275]. Finally, when FEC OTI is sent out-of-band (e.g., in a
latter may be protected by a digital signature. session description) this latter SHOULD be protected, for instance by
digitally signing it.
The same considerations concerning the key management aspects apply The same considerations concerning the key management aspects apply
here also. here also.
10. IANA Considerations 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]. apply to this document, see [RFC5052].
This document assigns the Fully-Specified FEC Encoding ID 2 under the This document assigns the Fully-Specified FEC Encoding ID 2 under the
"ietf:rmt:fec:encoding" name-space to "Reed-Solomon Codes over "ietf:rmt:fec:encoding" name-space to "Reed-Solomon Codes over
GF(2^^m)". GF(2^^m)".
This document assigns the Fully-Specified FEC Encoding ID 5 under the This document assigns the Fully-Specified FEC Encoding ID 5 under the
"ietf:rmt:fec:encoding" name-space to "Reed-Solomon Codes over "ietf:rmt:fec:encoding" name-space to "Reed-Solomon Codes over
GF(2^^8)". GF(2^^8)".
This document assigns the FEC Instance ID 0 scoped by the Under- 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)". Specified FEC Encoding ID 129 to "Reed-Solomon Codes over GF(2^^8)".
More specifically, under the "ietf:rmt:fec:encoding:instance" sub- More specifically, under the "ietf:rmt:fec:encoding:instance" sub-
name-space that is scoped by the "ietf:rmt:fec:encoding" called name-space that is scoped by the "ietf:rmt:fec:encoding" called
"Small Block Systematic FEC Codes", this document assigns FEC "Small Block Systematic FEC Codes", this document assigns FEC
Instance ID 0 to "Reed-Solomon Codes over GF(2^^8)". Instance ID 0 to "Reed-Solomon Codes over GF(2^^8)".
11. Acknowledgments 11. Acknowledgments
The authors want to thank Brian Adamson, Igor Slepchin, Stephen Kent, The authors want to thank Brian Adamson, Igor Slepchin, Stephen Kent,
and Francis Dupont for their valuable comments. The authors also Francis Dupont, Elwyn Davies, Magnus Westerlund and Alfred Hoenes for
want to thank Luigi Rizzo for his comments and for the design of the their valuable comments. The authors also want to thank Luigi Rizzo
reference Reed-Solomon codec. for his comments and for the design of the reference Reed-Solomon
codec.
12. References 12. References
12.1. Normative References 12.1. Normative References
[1] Bradner, S., "Key words for use in RFCs to Indicate Requirement [RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Levels", RFC 2119. Requirement Levels", RFC 2119.
[2] Watson, M., Luby, M., and L. Vicisano, "Forward Error [RFC5052] Watson, M., Luby, M., and L. Vicisano, "Forward Error
Correction (FEC) Building Block", RFC 5052, August 2007. Correction (FEC) Building Block", RFC 5052, August 2007.
[3] Watson, M., "Basic Forward Error Correction (FEC) Schemes", [draft-ietf-rmt-bb-fec-basic-schemes-revised]
draft-ietf-rmt-bb-fec-basic-schemes-revised-03.txt (work in Watson, M., "Basic Forward Error Correction (FEC)
progress), February 2007. Schemes",
draft-ietf-rmt-bb-fec-basic-schemes-revised-03.txt (work
in progress), February 2007.
12.2. Informative References 12.2. Informative References
[4] Luby, M., Vicisano, L., Gemmell, J., Rizzo, L., Handley, M., [RFC3453] Luby, M., Vicisano, L., Gemmell, J., Rizzo, L., Handley,
and J. Crowcroft, "The Use of Forward Error Correction (FEC) in M., and J. Crowcroft, "The Use of Forward Error Correction
Reliable Multicast", RFC 3453, December 2002. (FEC) in Reliable Multicast", RFC 3453, December 2002.
[5] Rizzo, L., "Reed-Solomon FEC codec (revised version of July [RS-codec]
2nd, 1998), available at Rizzo, L., "Reed-Solomon FEC codec (revised version of
July 2nd, 1998), available at
http://info.iet.unipi.it/~luigi/vdm98/vdm980702.tgz and http://info.iet.unipi.it/~luigi/vdm98/vdm980702.tgz and
mirrored at http://planete-bcast.inrialpes.fr/", July 1998. mirrored at http://planete-bcast.inrialpes.fr/",
July 1998.
[6] Mac Williams, F. and N. Sloane, "The Theory of Error Correcting [Rizzo97] Rizzo, L., "Effective Erasure Codes for Reliable Computer
Codes", North Holland, 1977. Communication Protocols", ACM SIGCOMM Computer
Communication Review Vol.27, No.2, pp.24-36, April 1997.
[7] Gohberg, I. and V. Olshevsky, "Fast algorithms with [MWS77] Mac Williams, F. and N. Sloane, "The Theory of Error
Correcting Codes", North Holland, 1977.
[GO94] 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.
[8] Roca, V., Neumann, C., and D. Furodet, "Low Density Parity [draft-ietf-rmt-bb-fec-ldpc]
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-06.txt (work in progress), draft-ietf-rmt-bb-fec-ldpc-07.txt (work in progress),
May 2007. November 2007.
[9] Luby, M., Shokrollahi, A., Watson, M., and T. Stockhammer, [RFC5053] Luby, M., Shokrollahi, A., Watson, M., and T. Stockhammer,
"Raptor Forward Error Correction Scheme", "Raptor Forward Error Correction Scheme", RFC 5053,
draft-ietf-rmt-bb-fec-raptor-object-09 (work in progress),
June 2007. June 2007.
[10] Luby, M., Watson, M., and L. Vicisano, "Asynchronous Layered [draft-ietf-rmt-pi-alc-revised]
Coding (ALC) Protocol Instantiation", Luby, M., Watson, M., and L. Vicisano, "Asynchronous
Layered Coding (ALC) Protocol Instantiation",
draft-ietf-rmt-pi-alc-revised-04.txt (work in progress), draft-ietf-rmt-pi-alc-revised-04.txt (work in progress),
February 2007. February 2007.
[11] Adamson, B., Bormann, C., Handley, M., and J. Macker, [draft-ietf-rmt-pi-norm-revised]
"Negative-acknowledgment (NACK)-Oriented Reliable Multicast Adamson, B., Bormann, C., Handley, M., and J. Macker,
(NORM) Protocol", draft-ietf-rmt-pi-norm-revised-05.txt (work "Negative-acknowledgment (NACK)-Oriented Reliable
in progress), March 2007. Multicast (NORM) Protocol",
draft-ietf-rmt-pi-norm-revised-05.txt (work in progress),
March 2007.
[12] Paila, T., Walsh, R., Luby, M., Lehtonen, R., and V. Roca, [draft-ietf-rmt-flute-revised]
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-04.txt (work in progress), draft-ietf-rmt-flute-revised-05.txt (work in progress),
October 2007. October 2007.
[13] Jonsson, J. and B. Kaliski, "Public-Key Cryptography Standards [RFC3447] Jonsson, J. and B. Kaliski, "Public-Key Cryptography
(PKCS) #1: RSA Cryptography Specifications Version 2.1", Standards (PKCS) #1: RSA Cryptography Specifications
RFC 3447, February 2003. Version 2.1", RFC 3447, February 2003.
[14] Kent, S., "IP Encapsulating Security Payload (ESP)", RFC 4303, [RFC4303] Kent, S., "IP Encapsulating Security Payload (ESP)",
December 2005. RFC 4303, December 2005.
[15] "HMAC: Keyed-Hashing for Message Authentication", RFC 2104, [RFC2104] "HMAC: Keyed-Hashing for Message Authentication",
February 1997. RFC 2104, February 1997.
[16] "Timed Efficient Stream Loss-Tolerant Authentication (TESLA): [RFC4082] "Timed Efficient Stream Loss-Tolerant Authentication
Multicast Source Authentication Transform Introduction", (TESLA): Multicast Source Authentication Transform
RFC 4082, June 2005. 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.
Authors' Addresses Authors' Addresses
Jerome Lacan Jerome Lacan
ISAE ISAE/LAAS-CNRS
1, place Emile Blouin 1, place Emile Blouin
Toulouse 31056 Toulouse 31056
France France
Email: jerome.lacan@isae.fr Email: jerome.lacan@isae.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
Inovallee; Montbonnot Inovallee; Montbonnot
ST ISMIER cedex 38334 ST ISMIER cedex 38334
France France
Email: vincent.roca@inrialpes.fr Email: vincent.roca@inria.fr
URI: http://planete.inrialpes.fr/~roca/ URI: http://planete.inrialpes.fr/people/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
Finland Finland
Email: jani.peltotalo@tut.fi Email: jani.peltotalo@tut.fi
URI: http://atm.tut.fi/mad URI: http://atm.tut.fi/mad
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