 1/draftietfrmtbbfecrs02.txt 20070508 01:12:09.000000000 +0200
+++ 2/draftietfrmtbbfecrs03.txt 20070508 01:12:09.000000000 +0200
@@ 1,22 +1,22 @@
Reliable Multicast Transport J. Lacan
InternetDraft ENSICA/LAASCNRS
Intended status: Experimental V. Roca
Expires: June 25, 2007 INRIA
+Expires: November 8, 2007 INRIA
J. Peltotalo
S. Peltotalo
Tampere University of Technology
 December 22, 2006
+ May 7, 2007
 ReedSolomon Forward Error Correction (FEC)
 draftietfrmtbbfecrs02.txt
+ ReedSolomon Forward Error Correction (FEC) Schemes
+ draftietfrmtbbfecrs03.txt
Status of this Memo
By submitting this InternetDraft, each author represents that any
applicable patent or other IPR claims of which he or she is aware
have been or will be disclosed, and any of which he or she becomes
aware will be disclosed, in accordance with Section 6 of BCP 79.
InternetDrafts are working documents of the Internet Engineering
Task Force (IETF), its areas, and its working groups. Note that
@@ 27,101 +27,133 @@
and may be updated, replaced, or obsoleted by other documents at any
time. It is inappropriate to use InternetDrafts as reference
material or to cite them other than as "work in progress."
The list of current InternetDrafts can be accessed at
http://www.ietf.org/ietf/1idabstracts.txt.
The list of InternetDraft Shadow Directories can be accessed at
http://www.ietf.org/shadow.html.
 This InternetDraft will expire on June 25, 2007.
+ This InternetDraft will expire on November 8, 2007.
Copyright Notice
 Copyright (C) The IETF Trust (2006).
+ Copyright (C) The IETF Trust (2007).
Abstract
 This document describes a FullySpecified FEC scheme for the Reed
 Solomon forward error correction code and its application to the
 reliable delivery of data objects on the packet erasure channel.
+ This document describes a FullySpecified FEC Scheme for the Reed
+ Solomon forward error correction codes over GF(2^^m), with m in
+ {2..16}, and its application to the reliable delivery of data objects
+ on the packet erasure channel.
+
+ This document also describes a FullySpecified FEC Scheme for the
+ special case of ReedSolomon codes over GF(2^^8) when there is no
+ encoding symbol group.
+
+ Finally, in the context of the UnderSpecified Small Block Systematic
+ FEC Scheme (FEC Encoding ID 129), this document assigns an FEC
+ Instance ID to the special case of ReedSolomon codes over GF(2^^8).
ReedSolomon codes belong to the class of Maximum Distance Separable
(MDS) codes, i.e. they enable a receiver to recover the k source
 symbols from any set of k received symbols.

 The implementation described here is compatible with the
 implementation from Luigi Rizzo.
+ symbols from any set of k received symbols. The schemes described
+ here are compatible with the implementation from Luigi Rizzo.
Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 4
2. Terminology . . . . . . . . . . . . . . . . . . . . . . . . . 5
3. Definitions Notations and Abbreviations . . . . . . . . . . . 6
3.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . 6
3.2. Notations . . . . . . . . . . . . . . . . . . . . . . . . 6
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
4.2. FEC Object Transmission Information . . . . . . . . . . . 9
4.2.1. Mandatory Elements . . . . . . . . . . . . . . . . . . 9
4.2.2. Common Elements . . . . . . . . . . . . . . . . . . . 9
4.2.3. SchemeSpecific Elements . . . . . . . . . . . . . . . 9
4.2.4. Encoding Format . . . . . . . . . . . . . . . . . . . 10
 5. Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . 12
 5.1. Determining the Maximum Source Block Length (B) . . . . . 12
 5.2. Determining the Number of Encoding Symbols of a Block . . 12
 6. ReedSolomon Codes Specification for the Erasure Channel . . . 14
 6.1. Finite Field . . . . . . . . . . . . . . . . . . . . . . . 14
 6.2. ReedSolomon Encoding Algorithm . . . . . . . . . . . . . 15
 6.2.1. Encoding Principles . . . . . . . . . . . . . . . . . 15
 6.2.2. Encoding Complexity . . . . . . . . . . . . . . . . . 16
 6.3. ReedSolomon Decoding Algorithm . . . . . . . . . . . . . 16
 6.3.1. Decoding Principles . . . . . . . . . . . . . . . . . 16
 6.3.2. Decoding Complexity . . . . . . . . . . . . . . . . . 17
 6.4. Implementation for the Packet Erasure Channel . . . . . . 17
 7. Security Considerations . . . . . . . . . . . . . . . . . . . 19
 8. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 20
 9. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . 21
 10. References . . . . . . . . . . . . . . . . . . . . . . . . . . 22
 10.1. Normative References . . . . . . . . . . . . . . . . . . . 22
 10.2. Informative References . . . . . . . . . . . . . . . . . . 22
 Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . . 24
 Intellectual Property and Copyright Statements . . . . . . . . . . 25
+ 5. Formats and Codes with FEC Encoding ID 5 . . . . . . . . . . . 12
+ 5.1. FEC Payload ID . . . . . . . . . . . . . . . . . . . . . . 12
+ 5.2. FEC Object Transmission Information . . . . . . . . . . . 12
+ 5.2.1. Mandatory Elements . . . . . . . . . . . . . . . . . . 12
+ 5.2.2. Common Elements . . . . . . . . . . . . . . . . . . . 12
+ 5.2.3. SchemeSpecific Elements . . . . . . . . . . . . . . . 13
+ 5.2.4. Encoding Format . . . . . . . . . . . . . . . . . . . 13
+ 6. Procedures with FEC Encoding IDs 2 and 5 . . . . . . . . . . . 14
+ 6.1. Determining the Maximum Source Block Length (B) . . . . . 14
+ 6.2. Determining the Number of Encoding Symbols of a Block . . 14
+ 7. Small Block Systematic FEC Scheme (FEC Encoding ID 129)
+ and ReedSolomon Codes over GF(2^^8) . . . . . . . . . . . . . 16
+ 8. ReedSolomon Codes Specification for the Erasure Channel . . . 17
+ 8.1. Finite Field . . . . . . . . . . . . . . . . . . . . . . . 17
+ 8.2. ReedSolomon Encoding Algorithm . . . . . . . . . . . . . 18
+ 8.2.1. Encoding Principles . . . . . . . . . . . . . . . . . 18
+ 8.2.2. Encoding Complexity . . . . . . . . . . . . . . . . . 19
+ 8.3. ReedSolomon Decoding Algorithm . . . . . . . . . . . . . 19
+ 8.3.1. Decoding Principles . . . . . . . . . . . . . . . . . 19
+ 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
The use of Forward Error Correction (FEC) codes is a classical
solution to improve the reliability of multicast and broadcast
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.
 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
processing but not optimal in terms of correction capabilities when
dealing with "small" objects.
The FEC scheme described in this document belongs to the class of
Maximum Distance Separable codes that are optimal in terms of erasure
correction capability. In others words, it enables a receiver to
recover the k source symbols from any set of exactly k encoding
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
storage already rely on packetbased ReedSolomon codes. In
particular, many of them use the ReedSolomon 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 ReedSolomon codes that is compatible with this codec.
+ The present document:
+
+ o introduces the FullySpecified FEC Scheme with FEC Encoding ID 2
+ that specifies the use of ReedSolomon codes over GF(2^^m), with m
+ in {2..16},
+
+ o introduces the FullySpecified FEC Scheme with FEC Encoding ID 5
+ that focuses on the special case of ReedSolomon codes over
+ GF(2^^8) and no encoding symbol group (i.e. exactly one symbol per
+ packet), and
+
+ o in the context of the UnderSpecified Small Block Systematic FEC
+ Scheme (FEC Encoding ID 129) [3], assigns the FEC Instance ID 0 to
+ the special case of ReedSolomon codes over GF(2^^8) and no
+ encoding symbol group.
+
2. Terminology
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
document are to be interpreted as described in RFC 2119 [1].
3. Definitions Notations and Abbreviations
3.1. Definitions
@@ 170,20 +202,21 @@
N denotes the number of source blocks into which the object shall
be partitioned.
E denotes the encoding symbol length in bytes.
S denotes the symbol size in units of mbit elements. When m = 8,
then S and E are equal.
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
= 2^^m in this specification.
G denotes the number of encoding symbols per group, i.e. the
number of symbols sent in the same packet.
GM denotes the Generator Matrix of a ReedSolomon code.
rate denotes the "code rate", i.e. the k/n ratio.
@@ 202,21 +235,25 @@
FEC OTI stands for FEC Object Transmission Information.
RS stands for ReedSolomon.
MDS stands for Maximum Distance Separable code.
GF(q) denotes a finite field (A.K.A. Galois Field) with q
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
+ FullySpecified FEC Scheme with FEC Encoding ID 2 that specifies the
+ use of ReedSolomon codes over GF(2^^m).
4.1. FEC Payload ID
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
parameter m (which is transmitted in the FEC OTI) as follows :
o The Source Block Number (field of size 32m bits) identifies from
which source block of the object the encoding symbol(s) in the
payload is (are) generated. There are a maximum of 2^^(32m)
@@ 244,29 +281,28 @@
 Source Block Number (328=24 bits)  Enc. Symb. ID 
+++++++++++++++++++++++++++++++++
Figure 1: FEC Payload ID encoding format for m = 8 (default)
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
+++++++++++++++++++++++++++++++++
 Src Block Nb (3216=16 bits)  Enc. Symbol ID (m=16 bits) 
+++++++++++++++++++++++++++++++++

Figure 2: FEC Payload ID encoding format for m = 16
4.2. FEC Object Transmission Information
4.2.1. Mandatory Elements
o FEC Encoding ID: the FullySpecified FEC Scheme described in this
 document uses FEC Encoding ID 2.
+ section uses FEC Encoding ID 2.
4.2.2. Common Elements
The following elements MUST be defined with the present FEC scheme:
o TransferLength (L): a nonnegative integer indicating the length
of the object in bytes. There are some restrictions on the
maximum TransferLength that can be supported :
max_transfer_length = 2^^(32m) * B * E
@@ 285,21 +321,21 @@
o EncodingSymbolLength (E): a nonnegative integer indicating the
length of each encoding symbol in bytes.
o MaximumSourceBlockLength (B): a nonnegative integer indicating
the maximum number of source symbols in a source block.
o MaxNumberofEncodingSymbols (max_n): a nonnegative integer
indicating the maximum number of encoding symbols generated for
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.
4.2.3. SchemeSpecific Elements
The following element MUST be defined with the present FEC Scheme.
It contains two distinct pieces of information:
o G: a nonnegative integer indicating the number of encoding
symbols per group used for the object. The default value is 1,
meaning that each packet contains exactly one symbol. When no G
@@ 308,47 +344,47 @@
o Finite Field parameter, m: The m parameter is the length of the
finite field elements, in bits. It also characterizes the number
of elements in the finite field: q = 2^^m elements. The default
value is m = 8. When no finite field size parameter is
communicated to the decoder, then this latter MUST assume that m =
8.
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
or the other should be used.
4.2.4.1. Using the General EXT_FTI Format
The FEC OTI binary format is the following, when the EXT_FTI
 mechanism is used (e.g. within the ALC [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 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  
+++++++++++++++++ +
 TransferLength (L) 
+++++++++++++++++++++++++++++++++
 m  G  Encoding Symbol Length (E) 
+++++++++++++++++++++++++++++++++
 Max Source Block Length (B)  Max Nb Enc. Symbols (max_n) 
+++++++++++++++++++++++++++++++++
Figure 3: EXT_FTI Header Format
4.2.4.2. Using the FDT Instance (FLUTE specific)
When it is desired that the FEC OTI be carried in the FDT Instance of
 a FLUTE session [10], the following XML attributes must be described
+ a FLUTE session [13], the following XML attributes must be described
for the associated object:
o FECOTIFECEncodingID
o FECOTITransferLength (L)
o FECOTIEncodingSymbolLength (E)
o FECOTIMaximumSourceBlockLength (B)
o FECOTIMaxNumberofEncodingSymbols (max_n)
@@ 358,40 +394,132 @@
The FECOTISchemeSpecificInfo contains the string resulting from
the Base64 encoding (in the XML Schema xs:base64Binary sense) of the
following value:
0 1
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5
+++++++++++++++++
 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
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
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
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
nor G are to be carried in the FEC OTI, then the sender simply omits
the FECOTISchemeSpecificInfo attribute.
After Base64 encoding, the 2 bytes of the FEC OTI Scheme Specific
Information are transformed into a string of 4 printable characters
(in the 64character alphabet) and added to the FECOTIScheme
SpecificInfo 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
+ FullySpecified FEC Scheme with FEC Encoding ID 5 that focuses on the
+ special case of ReedSolomon 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 nk 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 FullySpecified 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. SchemeSpecific Elements
+
+ No SchemeSpecific 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  
+ +++++++++++++++++ +
+  TransferLength (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 FECOTIFECEncodingID
+
+ o FECOTITransferLength (L)
+
+ o FECOTIEncodingSymbolLength (E)
+
+ o FECOTIMaximumSourceBlockLength (B)
+
+ o FECOTIMaxNumberofEncodingSymbols (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
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 can be produced for a source block. For instance, when m = 8
(default):
max1_B = 2^^8  1 = 255
Additionally, a codec MAY impose other limitations on the maximum
@@ 400,30 +528,30 @@
implementation time, when the target use case is known. This results
in a max2_B limitation.
Then, B is given by:
B = min(max1_B, max2_B)
Note that this calculation is only required at the coder, since the B
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 "nalgorithm", explains how to
determine the actual number of encoding symbols for a given block.
AT A SENDER:
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.
k: Current source block length. This parameter is given by the
block partitioning algorithm.
rate: FEC code rate, which is given by the user (e.g. when
starting a FLUTE sending application). It is expressed as a
floating point value.
Output:
@@ 459,35 +588,55 @@
n = floor(k * max_n / B);
Note that a ReedSolomon decoder does not need to know the n value.
Therefore the receiver part of the "nalgorithm" is not necessary
from the ReedSolomon decoder point of view. Yet a receiving
application using the ReedSolomon FEC scheme will sometimes need to
know the n value used by the sender, for instance for memory
management optimizations. To that purpose, the FEC OTI carries all
the parameters needed for a receiver to execute the above algorithm.
6. ReedSolomon 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 UnderSpecified Small Block Systematic FEC
+ Scheme (FEC Encoding ID 129) [3], this document assigns the FEC
+ Instance ID 0 to the special case of ReedSolomon 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. ReedSolomon Codes Specification for the Erasure Channel
ReedSolomon (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
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
implementation described in this document is based on a generator
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
 mbit elements, and Section 6.4 the use of [n,k]RS codes when
+ Section 8.1 to Section 8.3 specify the [n,k]RS codes when applied to
+ mbit elements, and Section 8.4 the use of [n,k]RS codes when
applied to symbols composed of several mbit elements, which is the
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
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
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
lower or equal than m1. The polynomials can be associated to binary
vectors of length m. For example, the vector (11001) represents the
polynomial 1 + x + x^^4. This representation is often called
polynomial representation. The addition between two elements is
@@ 528,38 +677,38 @@
m = 15, "1100000000000001", (1+x+x^^15)
m = 16, "11010000000010001", (1+x+x^^3+x^^12+x^^16)
In order to facilitate the implementation, these polynomials are also
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
also chosen so that they contain the minimum number of monomials.
6.2. ReedSolomon Encoding Algorithm
+8.2. ReedSolomon Encoding Algorithm
6.2.1. Encoding Principles
+8.2.1. Encoding Principles
Let s = (s_0, ..., s_{k1}) be a source vector of k elements over
GF(2^^m). Let e = (e_0, ..., e_{n1}) be the corresponding encoding
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,
of k rows and n columns over GF(2^^m). Thus:
e = s * GM.
The definition of the generator matrix completely characterizes the
RS code.
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
 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
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
matrix generates a MDS code. However, this MDS code is not
systematic, which is a problem for many networking applications. To
obtain a systematic matrix (and code), the simplest solution consists
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}.
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
@@ 568,227 +717,308 @@
Therefore, the generator matrix of the code considered in this
document is:
GM = (V_{k,k}^^1) * V_{k,n}
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 submatrix
formed by the n' first columns of GM.
6.2.2. Encoding Complexity
+8.2.2. Encoding Complexity
Encoding can be performed by first precomputing GM and by
multiplying the source vector (k elements) by GM (k rows and n
columns). The complexity of the precomputation of the generator
matrix can be estimated as the complexity of the multiplication of
the inverse of a Vandermonde matrix by nk vectors (i.e. the last nk
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
matrix can be computed in 0((nk)* k * log^^2(k)) operations. When
 the genarator matrix is precomputed, the encoding needs k operations
+ the generator matrix is precomputed, the encoding needs k operations
per repair element (vectormatrix multiplication).
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
multiplication by the inverse of a square Vandermonde matrix is known
as the interpolation problem and its complexity is O(k * log^^2 (k)).
The multiplication by a Vandermonde matrix, known as the multipoint
evaluation problem, requires O((nk) * 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/(nk)) * log^^2(k) + log(k))
operations per repair element.
6.3. ReedSolomon Decoding Algorithm
+8.3. ReedSolomon Decoding Algorithm
6.3.1. Decoding Principles
+8.3.1. Decoding Principles
The ReedSolomon decoding algorithm for the erasure channel allows
the recovery of the k source elements from any set of k received
elements. It is based on the fundamental property of the generator
 matrix which is such that any k*ksubmatrix is invertible (see [5]).
+ matrix which is such that any k*ksubmatrix is invertible (see [8]).
The first step of the decoding consists in extracting the k*k
submatrix of the generator matrix obtained by considering the columns
corresponding to the received elements. Indeed, since any encoding
element is obtained by multiplying the source vector by one column of
the generator matrix, the received vector of k encoding elements can
be considered as the result of the multiplication of the source
vector by a k*k submatrix of the generator matrix. Since this
submatrix is invertible, the second step of the algorithm is to
invert this matrix and to multiply the received vector by the
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
inversion and the vectormatrix multiplication. With the classical
GaussJordan algorithm, the matrix inversion requires O(k^^3)
operations and the vectormatrix multiplication is performed in
O(k^^2) operations.
This complexity can be improved by considering that the received
submatrix of GM is the product between the inverse of a Vandermonde
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
multiplying the received vector by V'^^1 (interpolation problem with
complexity O( k * log^^2(k)) ) then by V_{k,k} (multipoint evaluation
with complexity O(k * log(k))). The global decoding complexity is
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
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 ReedSolomon
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
 S mbit elements. Each mbit element is associated to an element of
+ S mbit elements. Each mbit element corresponds to an element of
the finite field GF(2^^m) through the polynomial representation
 (Section 6.1). If some of the source symbols contain less than S
 elements, they are virtually padded with zero elements (it can be the
 case for the last symbol of the last block of the object).
+ (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
+ 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 nk repair symbols of size S mbit
 elements, the k source symbols being also part of the n encoding
 symbols (Figure 5). These repair symbols are created mbit element
 per mbit element. More specifically, the jth source vector is
 composed of the jth element of each of the source symbols.
 Similarly, the jth encoding vector is composed of the jth element
 of each encoding symbol.
+ The encoding process produces n encoding symbols of size S mbit
+ elements, of which k are source symbols (this is a systematic code)
+ and nk are repair symbols (Figure 7). The mbit elements of the
+ repair symbols are calculated using the corresponding mbit elements
+ of the source symbol set. A logical jth source vector, comprised of
+ the jth elements from the set of source symbols, is used to
+ calculate a jth encoding vector. This jth encoding vector then
+ provides the jth 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 nk repair symbols
+ are the result of the Reed Solomon encoding.
 n
 +++++ ++ +++++
 0            
      *  generator  =     
       matrix      
       GM      
 source ++  (k x n)  ++
 vector        ++ >      
 j ++ / ++
      /     
      encoding     
      vector     
      j     
          
 S1          
 +++++ +++++
 k source symbols n encoding symbols
 (source + repair)
+ Input: k source symbols
 Figure 5: Packet encoding scheme
+ 0 j S1
+ +++++++++++++++++++++++++++
+  X  source symbol 0
+ +++++++++++++++++++++++++++
+ +++++++++++++++++++++++++++
+  X  source symbol 1
+ +++++++++++++++++++++++++++
+ . . .
+ +++++++++++++++++++++++++++
+  X  source symbol k1
+ +++++++++++++++++++++++++++
+
+ *
+
+ ++
+  generator 
+  matrix 
+  GM 
+  (k x n) 
+ ++
+
+ 
+ V
+
+ Output: n encoding symbols (source + repair)
+
+ 0 j S1
+ +++++++++++++++++++++++++++
+  X  enc. symbol 0
+ +++++++++++++++++++++++++++
+ +++++++++++++++++++++++++++
+  X  enc. symbol 1
+ +++++++++++++++++++++++++++
+ . . .
+ +++++++++++++++++++++++++++
+  Y  enc. symbol n1
+ +++++++++++++++++++++++++++
+ Figure 7: Packet encoding scheme
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.
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
inversion cost becomes negligible in front of the S matrixvector
multiplications.
Another asset is that the nk repair symbols can be produced on
demand. For instance, a sender can start by producing a limited
number of repair symbols and later on, depending on the observed
erasures on the channel, decide to produce additional repair symbols,
up to the nk upper limit. Indeed, to produce the repair symbol e_j,
where k <= j < n, it is sufficient to multiply the S source vectors
with column j of GM.
7. Security Considerations
+9. Security Considerations
 The security considerations for this document are the same as that of
 [2].
+ Data delivery can be subject to denialofservice attacks by
+ 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 SHA1 hash [5] of an object may be
+ appended before transmission, and the SHA1 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 outofband 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
registration. For general guidelines on IANA considerations as they
 apply to this document, see [2]. This document assigns the Fully
 Specified FEC Encoding ID 2 under the ietf:rmt:fec:encoding name
 space to "ReedSolomon Codes".
+ apply to this document, see [2].
9. Acknowledgments
+ This document assigns the FullySpecified FEC Encoding ID 2 under the
+ "ietf:rmt:fec:encoding" namespace to "ReedSolomon Codes over
+ GF(2^^m)".
 The authors want to thank Luigi Rizzo for comments on the subject and
 for the design of the reference ReedSolomon codec.
+ This document assigns the FullySpecified FEC Encoding ID 5 under the
+ "ietf:rmt:fec:encoding" namespace to "ReedSolomon 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 "ReedSolomon Codes over GF(2^^8)".
+ More specifically, under the "ietf:rmt:fec:encoding:instance" sub
+ namespace that is scoped by the "ietf:rmt:fec:encoding" called
+ "Small Block Systematic FEC Codes", this document assigns FEC
+ Instance ID 0 to "ReedSolomon 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 ReedSolomon codec.
+
+12. References
+
+12.1. Normative References
[1] Bradner, S., "Key words for use in RFCs to Indicate Requirement
Levels", RFC 2119.
[2] Watson, M., Luby, M., and L. Vicisano, "Forward Error
Correction (FEC) Building Block",
 draftietfrmtfecbbrevised04.txt (work in progress),
 September 2006.
+ draftietfrmtfecbbrevised07.txt (work in progress),
+ April 2007.
 [3] Luby, M., Vicisano, L., Gemmell, J., Rizzo, L., Handley, M.,
+ [3] Watson, M., "Basic Forward Error Correction (FEC) Schemes",
+ draftietfrmtbbfecbasicschemesrevised03.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
Reliable Multicast", RFC 3453, December 2002.
10.2. Informative References
+ [5] "HMAC: KeyedHashing for Message Authentication", RFC 2104,
+ February 1997.
 [4] Rizzo, L., "ReedSolomon FEC codec (revised version of July
+ [6] "Timed Efficient Stream LossTolerant Authentication (TESLA):
+ Multicast Source Authentication Transform Introduction",
+ RFC 4082, June 2005.
+
+12.2. Informative References
+
+ [7] Rizzo, L., "ReedSolomon FEC codec (revised version of July
2nd, 1998), available at
 http://info.iet.unipi.it/~luigi/vdm98/vdm980702.tgz",
 July 1998.
+ http://info.iet.unipi.it/~luigi/vdm98/vdm980702.tgz, and
+ mirrored at http://planetebcast.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 .
 [6] Luby, M., Shokrollahi, A., Watson, M., and T. Stockhammer,
 "Raptor Forward Error Correction Scheme", Internet
 Draft draftietfrmtbbfecraptorobject04 (work in
 progress), June 2006.
+ [9] Luby, M., Shokrollahi, A., Watson, M., and T. Stockhammer,
+ "Raptor Forward Error Correction Scheme",
+ draftietfrmtbbfecraptorobject08 (work in progress),
+ 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",
 draftietfrmtbbfecldpc04.txt (work in progress),
 December 2006.
+ draftietfrmtbbfecldpc06.txt (work in progress),
+ 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",
 draftietfrmtpialcrevised03.txt (work in progress),
 April 2006.
+ draftietfrmtpialcrevised04.txt (work in progress),
+ February 2007.
 [9] Adamson, B., Bormann, C., Handley, M., and J. Macker,
+ [12] Adamson, B., Bormann, C., Handley, M., and J. Macker,
"Negativeacknowledgment (NACK)Oriented Reliable Multicast
 (NORM) Protocol", draftietfrmtpinormrevised03.txt (work
 in progress), September 2006.
+ (NORM) Protocol", draftietfrmtpinormrevised04.txt (work
+ 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",
 draftietfrmtfluterevised02.txt (work in progress),
 August 2006.
+ draftietfrmtfluterevised03.txt (work in progress),
+ January 2007.
 [11] Gohberg, I. and V. Olshevsky, "Fast algorithms with
+ [14] Gohberg, I. and V. Olshevsky, "Fast algorithms with
preprocessing for matrixvector multiplication problems",
Journal of Complexity, pp. 411427, vol. 10, 1994 .
Authors' Addresses
Jerome Lacan
ENSICA/LAASCNRS
1, place Emile Blouin
Toulouse 31056
France
Email: jerome.lacan@ensica.fr
URI: http://dmi.ensica.fr/auteur.php3?id_auteur=5
Vincent Roca
INRIA
655, av. de l'Europe
 Zirst; Montbonnot
+ Inovallee; Montbonnot
ST ISMIER cedex 38334
France
Email: vincent.roca@inrialpes.fr
URI: http://planete.inrialpes.fr/~roca/
Jani Peltotalo
Tampere University of Technology
P.O. Box 553 (Korkeakoulunkatu 1)
Tampere FIN33101
@@ 801,21 +1031,21 @@
Tampere University of Technology
P.O. Box 553 (Korkeakoulunkatu 1)
Tampere FIN33101
Finland
Email: sami.peltotalo@tut.fi
URI: http://atm.tut.fi/mad
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
contained in BCP 78, and except as set forth therein, the authors
retain all their rights.
This document and the information contained herein are provided on an
"AS IS" basis and THE CONTRIBUTOR, THE ORGANIZATION HE/SHE REPRESENTS
OR IS SPONSORED BY (IF ANY), THE INTERNET SOCIETY, THE IETF TRUST AND
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OR IMPLIED, INCLUDING BUT NOT LIMITED TO ANY WARRANTY THAT THE USE OF