Reliable Multicast Transport J. Lacan
Internet-Draft ISAE
Intended status: Standards Track V. Roca
Expires: April 12, 2008 INRIA
J. Peltotalo
S. Peltotalo
Tampere University of Technology
October 10, 2007
Reed-Solomon Forward Error Correction (FEC) Schemes
draft-ietf-rmt-bb-fec-rs-04.txt
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Abstract
This document describes a Fully-Specified Forward Error Correction
(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
on the packet erasure channel.
This document also describes a Fully-Specified FEC Scheme for the
special case of Reed-Solomon codes over GF(2^^8) when there is no
encoding symbol group.
Finally, in the context of the Under-Specified Small Block Systematic
FEC Scheme (FEC Encoding ID 129), this document assigns an FEC
Instance ID to the special case of Reed-Solomon codes over GF(2^^8).
Reed-Solomon codes belong to the class of Maximum Distance Separable
(MDS) codes, i.e., they enable a receiver to recover the k source
symbols from any set of k received symbols. The schemes described
here are compatible with the implementation from Luigi Rizzo.
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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 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. Scheme-Specific Elements . . . . . . . . . . . . . . . 9
4.2.4. Encoding Format . . . . . . . . . . . . . . . . . . . 10
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. Scheme-Specific 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 Reed-Solomon Codes over GF(2^^8) . . . . . . . . . . . . . 16
8. Reed-Solomon Codes Specification for the Erasure Channel . . . 17
8.1. Finite Field . . . . . . . . . . . . . . . . . . . . . . . 17
8.2. Reed-Solomon Encoding Algorithm . . . . . . . . . . . . . 18
8.2.1. Encoding Principles . . . . . . . . . . . . . . . . . 18
8.2.2. Encoding Complexity . . . . . . . . . . . . . . . . . 19
8.3. Reed-Solomon 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
9.1. Problem Statement . . . . . . . . . . . . . . . . . . . . 23
9.2. Attacks Against the Data Flow . . . . . . . . . . . . . . 23
9.3. Attacks against the FEC parameters . . . . . . . . . . . . 25
10. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 26
11. Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . 27
12. References . . . . . . . . . . . . . . . . . . . . . . . . . . 28
12.1. Normative References . . . . . . . . . . . . . . . . . . . 28
12.2. Informative References . . . . . . . . . . . . . . . . . . 28
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . . 30
Intellectual Property and Copyright Statements . . . . . . . . . . 31
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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 [4]
describes some applications of FEC codes for content delivery.
Recent FEC schemes like [9] and [8] 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 [9] or [8], this family of codes is very useful.
Many applications dealing with content transmission or content
storage already rely on packet-based Reed-Solomon codes. In
particular, many of them use the Reed-Solomon codec of Luigi Rizzo
[5]. The goal of the present document to specify an implementation
of Reed-Solomon codes that is compatible with this codec.
The present document:
o introduces the Fully-Specified FEC Scheme with FEC Encoding ID 2
that specifies the use of Reed-Solomon codes over GF(2^^m), with m
in {2..16},
o introduces the Fully-Specified FEC Scheme with FEC Encoding ID 5
that focuses on the special case of Reed-Solomon codes over
GF(2^^8) and no encoding symbol group (i.e., exactly one symbol
per packet), and
o in the context of the Under-Specified Small Block Systematic FEC
Scheme (FEC Encoding ID 129) [3], assigns the FEC Instance ID 0 to
the special case of Reed-Solomon codes over GF(2^^8) and no
encoding symbol group.
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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].
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3. Definitions Notations and Abbreviations
3.1. Definitions
This document uses the same terms and definitions as those specified
in [2]. Additionally, it uses the following definitions:
Source symbol: unit of data used during the encoding process.
Encoding symbol: unit of data generated by the encoding process.
Repair symbol: encoding symbol that is not a source symbol.
Systematic code: FEC code in which the source symbols are part of
the encoding symbols.
Source block: a block of k source symbols that are considered
together for the encoding.
Encoding Symbol Group: a group of encoding symbols that are sent
together within the same packet, and whose relationships to the
source block can be derived from a single Encoding Symbol ID.
Source Packet: a data packet containing only source symbols.
Repair Packet: a data packet containing only repair symbols.
3.2. Notations
This document uses the following notations:
L denotes the object transfer length in bytes.
k denotes the number of source symbols in a source block.
n_r denotes the number of repair symbols generated for a source
block.
n denotes the encoding block length, i.e., the number of encoding
symbols generated for a source block. Therefore: n = k + n_r.
max_n denotes the maximum number of encoding symbols generated for
any source block.
B denotes the maximum source block length in symbols, i.e., the
maximum number of source symbols per source block.
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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 m-bit 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 Reed-Solomon code.
rate denotes the "code rate", i.e., the k/n ratio.
a^^b denotes a raised to the power b.
a^^-1 denotes the inverse of a.
I_k denotes the k*k identity matrix.
3.3. Abbreviations
This document uses the following abbreviations:
ESI stands for Encoding Symbol ID.
FEC OTI stands for FEC Object Transmission Information.
RS stands for Reed-Solomon.
MDS stands for Maximum Distance Separable code.
GF(q) denotes a finite field (also known as Galois Field) with q
elements. We assume that q = 2^^m in this document.
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4. Formats and Codes with FEC Encoding ID 2
This section introduces the formats and codes associated to the
Fully-Specified FEC Scheme with FEC Encoding ID 2 that specifies the
use of Reed-Solomon codes over GF(2^^m).
4.1. FEC Payload ID
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 32-m bits) identifies from
which source block of the object the encoding symbol(s) in the
payload is(are) generated. There is a maximum of 2^^(32-m) blocks
per object.
o The Encoding Symbol ID (field of size m bits) identifies which
specific encoding symbol(s) generated from the source block
is(are) carried in the packet payload. There is a maximum of 2^^m
encoding symbols per block. The first k values (0 to k - 1)
identify source symbols, the remaining n-k values identify repair
symbols.
There MUST be exactly one FEC Payload ID per source or repair packet.
In case of an Encoding Symbol Group, when multiple encoding symbols
are sent in the same packet, the FEC Payload ID refers to the first
symbol of the packet. The other symbols can be deduced from the ESI
of the first symbol by incrementing sequentially the ESI.
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 (32-8=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 (32-16=16 bits) | Enc. Symbol ID (m=16 bits) |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
Figure 2: FEC Payload ID encoding format for m = 16
The format of the FEC Payload ID for m = 8 and m = 16 is illustrated
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in Figure 1 and Figure 2 respectively.
4.2. FEC Object Transmission Information
4.2.1. Mandatory Elements
o FEC Encoding ID: the Fully-Specified FEC Scheme described in this
section uses FEC Encoding ID 2.
4.2.2. Common Elements
The following elements MUST be defined with the present FEC scheme:
o Transfer-Length (L): a non-negative integer indicating the length
of the object in bytes. There are some restrictions on the
maximum Transfer-Length that can be supported:
max_transfer_length = 2^^(32-m) * B * E
For instance, for m = 8, for B = 2^^8 - 1 (because the codec
operates on a finite field with 2^^8 elements) and if E = 1024
bytes, then the maximum transfer length is approximately equal to
2^^42 bytes (i.e., 4 Tera Bytes). Similarly, for m = 16, for B =
2^^16 - 1 and if E = 1024 bytes, then the maximum transfer length
is also approximately equal to 2^^42 bytes. For larger objects,
another FEC scheme, with a larger Source Block Number field in the
FEC Payload ID, could be defined. Another solution consists in
fragmenting large objects into smaller objects, each of them
complying with the above limits.
o Encoding-Symbol-Length (E): a non-negative integer indicating the
length of each encoding symbol in bytes.
o Maximum-Source-Block-Length (B): a non-negative integer indicating
the maximum number of source symbols in a source block.
o Max-Number-of-Encoding-Symbols (max_n): a non-negative integer
indicating the maximum number of encoding symbols generated for
any source block.
Section 6 explains how to derive the values of each of these
elements.
4.2.3. Scheme-Specific Elements
The following element MUST be defined with the present FEC Scheme.
It contains two distinct pieces of information:
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o G: a non-negative 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
parameter is communicated to the decoder, then this latter MUST
assume that G = 1.
o 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 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 [10] or NORM [11] 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 = 4 | |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ +
| Transfer-Length (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 (File
Delivery Table) Instance of a FLUTE session [12], the following XML
attributes must be described for the associated object:
o FEC-OTI-FEC-Encoding-ID
o FEC-OTI-Transfer-Length (L)
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o FEC-OTI-Encoding-Symbol-Length (E)
o FEC-OTI-Maximum-Source-Block-Length (B)
o FEC-OTI-Max-Number-of-Encoding-Symbols (max_n)
o FEC-OTI-Scheme-Specific-Info
The FEC-OTI-Scheme-Specific-Info 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
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 FEC-OTI-Scheme-Specific-Info 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 64-character alphabet) and added to the FEC-OTI-Scheme-
Specific-Info attribute.
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5. Formats and Codes with FEC Encoding ID 5
This section introduces the formats and codes associated to the
Fully-Specified FEC Scheme with FEC Encoding ID 5 that focuses on the
special case of Reed-Solomon codes over GF(2^^8) and no encoding
symbol group.
5.1. FEC Payload ID
The FEC Payload ID is composed of the Source Block Number and the
Encoding Symbol ID:
o The Source Block Number (24 bit field) identifies from which
source block of the object the encoding symbol in the payload is
generated. There is 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 is a maximum of 2^^8 encoding symbols per
block. The first k values (0 to k - 1) identify source symbols,
the remaining n-k values identify repair symbols.
There MUST be exactly one FEC Payload ID per source or repair packet.
This FEC Payload ID refer to the one and only symbol of the packet.
0 1 2 3
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| Source Block Number (24 bits) | Enc. Symb. ID |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
Figure 5: FEC Payload ID encoding format with FEC Encoding ID 5
5.2. FEC Object Transmission Information
5.2.1. Mandatory Elements
o FEC Encoding ID: the Fully-Specified FEC Scheme described in this
section uses FEC Encoding ID 5.
5.2.2. Common Elements
The Common Elements are the same as those specified in Section 4.2.2
when m = 8 and G = 1.
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5.2.3. Scheme-Specific Elements
No Scheme-Specific elements are defined by this FEC Scheme.
5.2.4. Encoding Format
This section shows the two possible encoding formats of the above FEC
OTI. The present document does not specify when one encoding format
or the other should be used.
5.2.4.1. Using the General EXT_FTI Format
The FEC OTI binary format is the following, when the EXT_FTI
mechanism is used (e.g., within the ALC [10] or NORM [11] protocols).
0 1 2 3
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| HET = 64 | HEL = 3 | |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ +
| Transfer-Length (L) |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| Encoding Symbol Length (E) | MaxBlkLen (B) | max_n |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
Figure 6: EXT_FTI Header Format with FEC Encoding ID 5
5.2.4.2. Using the FDT Instance (FLUTE specific)
When it is desired that the FEC OTI be carried in the FDT Instance of
a FLUTE session [12], the following XML attributes must be described
for the associated object:
o FEC-OTI-FEC-Encoding-ID
o FEC-OTI-Transfer-Length (L)
o FEC-OTI-Encoding-Symbol-Length (E)
o FEC-OTI-Maximum-Source-Block-Length (B)
o FEC-OTI-Max-Number-of-Encoding-Symbols (max_n)
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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) there is a maximum of 2^^8 - 1 = 255 encoding symbols.
Given the target FEC code rate (e.g., provided by the user when
starting a FLUTE sending application), the sender calculates:
max1_B = floor((2^^m - 1) * rate)
This max1_B value leaves enough room for the sender to produce the
desired number of parity symbols.
Additionally, a codec MAY impose other limitations on the maximum
block size. Yet it is not expected that such limits exist when using
the default m = 8 value. This decision MUST be clarified at
implementation time, when the target use case is known. This results
in a max2_B limitation.
Then, B is given by:
B = min(max1_B, max2_B)
Note that this calculation is only required at the coder, since the B
parameter is communicated to the decoder through the FEC OTI.
6.2. Determining the Number of Encoding Symbols of a Block
The following algorithm, also called "n-algorithm", 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 6.1
explains how to determine its value.
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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:
max_n: Maximum number of encoding symbols generated for any source
block.
n: Number of encoding symbols generated for this source block.
Algorithm:
max_n = ceil(B / rate);
if (max_n > 2^^m - 1) then return an error ("invalid code rate");
n = floor(k * max_n / B);
AT A RECEIVER:
Input:
B: Extracted from the received FEC OTI.
max_n: Extracted from the received FEC OTI.
k: Given by the block partitioning algorithm.
Output:
n
Algorithm:
n = floor(k * max_n / B);
Note that a Reed-Solomon decoder does not need to know the n value.
Therefore the receiver part of the "n-algorithm" is not necessary
from the Reed-Solomon decoder point of view. Yet a receiving
application using the Reed-Solomon 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.
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7. Small Block Systematic FEC Scheme (FEC Encoding ID 129) and Reed-
Solomon Codes over GF(2^^8)
In the context of the Under-Specified Small Block Systematic FEC
Scheme (FEC Encoding ID 129) [3], this document assigns the FEC
Instance ID 0 to the special case of Reed-Solomon codes over GF(2^^8)
and no encoding symbol group.
The FEC Instance ID 0 uses the Formats and Codes specified in [3].
The FEC Scheme with FEC Instance ID 0 MAY use the algorithm defined
in Section 9.1. of [2] 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.
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8. Reed-Solomon Codes Specification for the Erasure Channel
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
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 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
applied to symbols composed of several m-bit elements, which is the
target of this specification.
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 m-1. 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
defined as the addition of binary polynomials in GF(2) and the
multiplication is the multiplication modulo a given irreducible
polynomial over GF(2) of degree m with coefficients in GF(2). Note
that all the roots of this polynomial are in GF(2^^m) but not in
GF(2).
A finite field GF(2^^m) is completely characterized by the
irreducible polynomial. The 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 = 3, "1101", (1+x+x^^3)
m = 4, "11001", (1+x+x^^4)
m = 5, "101001", (1+x^^2+x^^5)
m = 6, "1100001", (1+x+x^^6)
m = 7, "10010001", (1+x^^3+x^^7)
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m = 8, "101110001", (1+x^^2+x^^3+x^^4+x^^8)
m = 9, "1000100001", (1+x^^4+x^^9)
m = 10, "10010000001", (1+x^^3+x^^10)
m = 11, "101000000001", (1+x^^2+x^^11)
m = 12, "1100101000001", (1+x+x^^4+x^^6+x^^12)
m = 13, "11011000000001", (1+x+x^^3+x^^4+x^^13)
m = 14, "110000100010001", (1+x+x^^6+x^^10+x^^14)
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.
8.2. Reed-Solomon Encoding Algorithm
8.2.1. Encoding Principles
Let s = (s_0, ..., s_{k-1}) be a source vector of k elements over
GF(2^^m). Let e = (e_0, ..., e_{n-1}) be the corresponding encoding
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 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
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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
elements are equal to source elements. Besides the associated code
keeps the MDS property.
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 sub-matrix
formed by the n' first columns of GM.
8.2.2. Encoding Complexity
Encoding can be performed by first pre-computing GM and by
multiplying the source vector (k elements) by GM (k rows and n
columns). The complexity of the pre-computation of the generator
matrix can be estimated as the complexity of the multiplication of
the inverse of a Vandermonde matrix by n-k vectors (i.e., the last
n-k 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((n-k)* k * log^^2(k)) operations. When
the generator matrix is pre-computed, the encoding needs k operations
per repair element (vector-matrix 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((n-k) * log(k)) by using Fast Fourier
Transform, as explained in [7]. The total complexity of this
encoding algorithm is then O((k/(n-k)) * log^^2(k) + log(k))
operations per repair element.
8.3. Reed-Solomon Decoding Algorithm
8.3.1. Decoding Principles
The Reed-Solomon 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*k-submatrix is invertible (see [6]).
The first step of the decoding consists in extracting the k*k
submatrix of the generator matrix obtained by considering the columns
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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.
8.3.2. Decoding Complexity
The decoding algorithm described previously includes the matrix
inversion and the vector-matrix multiplication. With the classical
Gauss-Jordan algorithm, the matrix inversion requires O(k^^3)
operations and the vector-matrix 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.
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
known. The following specification describes the use of Reed-Solomon
codes for generating redundant symbols from the k source symbols and
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 m-bit elements. Each m-bit element corresponds to an element of
the finite field GF(2^^m) through the polynomial representation
(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 does not need to be actually sent with the data
to the receivers.
The encoding process produces n encoding symbols of size S m-bit
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
repair symbols are calculated using the corresponding m-bit elements
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of the source symbol set. A logical j-th source vector, comprised of
the j-th elements from the set of source symbols, is used to
calculate a j-th encoding vector. This j-th encoding vector then
provides the j-th elements for the set encoding symbols calculated
for the block. As a systematic code, the first k encoding symbols
are the same as the k source symbols and the last n-k repair symbols
are the result of the Reed-Solomon encoding.
Input: k source symbols
0 j S-1
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| |X| | source symbol 0
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| |X| | source symbol 1
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
. . .
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| |X| | source symbol k-1
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
*
+--------------------+
| generator matrix |
| GM |
| (k x n) |
+--------------------+
|
V
Output: n encoding symbols (source + repair)
0 j S-1
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| |X| | enc. symbol 0
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| |X| | enc. symbol 1
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
. . .
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| |Y| | enc. symbol n-1
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
Figure 7: Packet encoding scheme
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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 matrix-vector
multiplications.
Another asset is that the n-k 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 n-k upper limit. Indeed, to produce the repair symbol e_j,
where k <= j < n, it is sufficient to multiply the S source vectors
with column j of GM.
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9. Security Considerations
9.1. Problem Statement
A content delivery system is potentially subject to many attacks:
some of them target the network (e.g., to compromise the routing
infrastructure, by compromising the congestion control component),
others target the Content Delivery Protocol (CDP) (e.g., to
compromise its normal behavior), and finally some attacks target the
content itself. Since this document focuses on a FEC building block
independently of any particular CDP (even if ALC and NORM are two
natural candidates), this section only discusses the additional
threats that an arbitrary CDP may be exposed to when using this
building block.
More specifically, several kinds of attacks exist:
o those that are meant to give access to a confidential content
(e.g., in case of a non-free content),
o those that try to corrupt the object being transmitted (e.g., to
inject malicious code within an object, or to prevent a receiver
from using an object),
o and those that try to compromise the receiver's behavior (e.g., by
making the decoding of an object computationally expensive).
These attacks can be launched either against the data flow itself
(e.g. by sending forged symbols) or against the FEC parameters that
are sent either in-band (e.g., in an EXT_FTI or FDT Instance) or out-
of-band (e.g., in a session description).
9.2. 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
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
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provided at the packet level, and after having removed all forged
packets, the object can be recovered if the number of symbols
remaining is sufficient. Several techniques can provide this source
authentication/content integrity service:
o at the object level, the object MAY be digitally signed (with
public key cryptography) (e.g., using RSASSA-PKCS1-v1_5 [13]).
This signature enables a receiver to check the object, once this
latter has been fully decoded. Even if digital signatures are
computationally expensive, this calculation occurs only once per
object, which is usually acceptable;
o at the packet level, each packet can be digitally signed. A major
limitation is the high computational and transmission overheads
that this solution incurs (unless ECC is used, but ECC is
protected by IPR). To avoid this problem, the signature may span
a set of symbols in order to amortize the signature calculation,
but if a single symbol is missing, the integrity of the whole set
cannot be checked;
o at the packet level, a Group Message Authentication Code (MAC)
[15] (e.g., using HMAC-SHA-1 with a secret key shared by all the
group members, senders and receivers) scheme can be used. This
technique creates a cryptographically secured (thanks to the
secret key) digest of a packet that is sent along with the packet.
The Group MAC scheme does not incur prohibitive processing load
nor transmission overhead, but it has a major limitation: it only
provides a group authentication/integrity service since all group
members share the same secret group key, which means that each
member can send a forged packet. It is therefore restricted to
situations where group members are fully trusted (or in
association with another technique as a pre-check);
o at the packet level, TESLA [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.
Techniques relying on public key cryptography (digital signatures and
TESLA during the bootstrap process) require that public keys be
securely associated to the entities. This can be achieved by a
Public Key Infrastructure (PKI), or by a PGP Web of Trust, or by pre-
distributing the public keys of each group member. It is up to the
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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
that a secret key be shared by all group members. This can be
achieved by means of a group key management protocol, or simply by
pre-distributing the secret key (but this manual solution has many
limitations). Here also, it is up to the developer to define which
solution to use, taking into account the target use-case features.
9.3. Attacks against the FEC parameters
Let us now consider attacks against the FEC parameters (or FEC OTI).
The FEC OTI can either be sent in-band (i.e., in an EXT_FTI or in an
FDT Instance containing FEC OTI for the object) or out-of-band (e.g.,
in a session description). Attacks on these FEC parameters can
prevent the decoding of the associated object: for instance modifying
the B parameter will lead to a different block partitioning at a
receiver thereby compromising decoding; or setting the m parameter to
16 instead of 8 with FEC Encoding ID 2 will increase the processing
load while compromising decoding.
It is therefore RECOMMENDED that security measures be taken to
guarantee the FEC OTI integrity. To that purpose, the packets
carrying the FEC parameters sent in-band (i.e., in an EXT_FTI header
extension or in an FDT Instance) may be protected by one of the per-
packet techniques described above: TESLA, digital signature, or a
group MAC. Alternatively, when FEC OTI is contained in an FDT
Instance, this object may be digitally signed. Finally, when FEC OTI
is sent out-of-band for instance in a session description, this
latter may be protected by a digital signature.
The same considerations concerning the key management aspects apply
here also.
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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 "Reed-Solomon Codes over
GF(2^^m)".
This document assigns the Fully-Specified FEC Encoding ID 5 under the
"ietf:rmt:fec:encoding" name-space to "Reed-Solomon Codes over
GF(2^^8)".
This document assigns the FEC Instance ID 0 scoped by the Under-
Specified FEC Encoding ID 129 to "Reed-Solomon Codes over GF(2^^8)".
More specifically, under the "ietf:rmt:fec:encoding:instance" sub-
name-space that is scoped by the "ietf:rmt:fec:encoding" called
"Small Block Systematic FEC Codes", this document assigns FEC
Instance ID 0 to "Reed-Solomon Codes over GF(2^^8)".
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11. Acknowledgments
The authors want to thank Brian Adamson, Igor Slepchin, Stephen Kent,
and Francis Dupont for their valuable comments. The authors also
want to thank Luigi Rizzo for his comments and for the design of the
reference Reed-Solomon codec.
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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", RFC 5052, August 2007.
[3] Watson, M., "Basic Forward Error Correction (FEC) Schemes",
draft-ietf-rmt-bb-fec-basic-schemes-revised-03.txt (work in
progress), February 2007.
12.2. Informative References
[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.
[5] Rizzo, L., "Reed-Solomon FEC codec (revised version of July
2nd, 1998), available at
http://info.iet.unipi.it/~luigi/vdm98/vdm980702.tgz and
mirrored at http://planete-bcast.inrialpes.fr/", July 1998.
[6] Mac Williams, F. and N. Sloane, "The Theory of Error Correcting
Codes", North Holland, 1977.
[7] Gohberg, I. and V. Olshevsky, "Fast algorithms with
preprocessing for matrix-vector multiplication problems",
Journal of Complexity, pp. 411-427, vol. 10, 1994.
[8] Roca, V., Neumann, C., and D. Furodet, "Low Density Parity
Check (LDPC) Forward Error Correction",
draft-ietf-rmt-bb-fec-ldpc-06.txt (work in progress),
May 2007.
[9] Luby, M., Shokrollahi, A., Watson, M., and T. Stockhammer,
"Raptor Forward Error Correction Scheme",
draft-ietf-rmt-bb-fec-raptor-object-09 (work in progress),
June 2007.
[10] Luby, M., Watson, M., and L. Vicisano, "Asynchronous Layered
Coding (ALC) Protocol Instantiation",
draft-ietf-rmt-pi-alc-revised-04.txt (work in progress),
February 2007.
[11] Adamson, B., Bormann, C., Handley, M., and J. Macker,
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"Negative-acknowledgment (NACK)-Oriented Reliable 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,
"FLUTE - File Delivery over Unidirectional Transport",
draft-ietf-rmt-flute-revised-04.txt (work in progress),
October 2007.
[13] Jonsson, J. and B. Kaliski, "Public-Key Cryptography Standards
(PKCS) #1: RSA Cryptography Specifications Version 2.1",
RFC 3447, February 2003.
[14] Kent, S., "IP Encapsulating Security Payload (ESP)", RFC 4303,
December 2005.
[15] "HMAC: Keyed-Hashing for Message Authentication", RFC 2104,
February 1997.
[16] "Timed Efficient Stream Loss-Tolerant Authentication (TESLA):
Multicast Source Authentication Transform Introduction",
RFC 4082, June 2005.
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Authors' Addresses
Jerome Lacan
ISAE
1, place Emile Blouin
Toulouse 31056
France
Email: jerome.lacan@isae.fr
URI: http://dmi.ensica.fr/auteur.php3?id_auteur=5
Vincent Roca
INRIA
655, av. de l'Europe
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 FIN-33101
Finland
Email: jani.peltotalo@tut.fi
URI: http://atm.tut.fi/mad
Sami Peltotalo
Tampere University of Technology
P.O. Box 553 (Korkeakoulunkatu 1)
Tampere FIN-33101
Finland
Email: sami.peltotalo@tut.fi
URI: http://atm.tut.fi/mad
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