| Rfc | 5510 |
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| Title | Reed-Solomon Forward Error Correction (FEC) Schemes |
|---|
| Author | J. Lacan, V.
Roca, J. Peltotalo, S. Peltotalo |
|---|
| Date | April 2009 |
|---|
| Format: | TXT, HTML |
|---|
| Status: | PROPOSED STANDARD |
|---|
|
Network Working Group J. Lacan
Request for Comments: 5510 ISAE/LAAS-CNRS
Category: Standards Track V. Roca
INRIA
J. Peltotalo
S. Peltotalo
Tampere University of Technology
April 2009
Reed-Solomon Forward Error Correction (FEC) Schemes
Status of This Memo
This document specifies an Internet standards track protocol for the
Internet community, and requests discussion and suggestions for
improvements. Please refer to the current edition of the "Internet
Official Protocol Standards" (STD 1) for the standardization state
and status of this protocol. Distribution of this memo is unlimited.
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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), where m is
in {2..16}, and its application to the reliable delivery of data
objects on the packet erasure channel (i.e., a communication path
where packets are either received without any corruption or discarded
during transmission). 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.
Table of Contents
1. Introduction ....................................................4
2. Terminology .....................................................5
3. Definitions Notations and Abbreviations .........................5
3.1. Definitions ................................................5
3.2. Notations ..................................................6
3.3. Abbreviations ..............................................7
4. Formats and Codes with FEC Encoding ID 2 ........................7
4.1. FEC Payload ID .............................................7
4.2. FEC Object Transmission Information ........................8
4.2.1. Mandatory Elements ..................................8
4.2.2. Common Elements .....................................8
4.2.3. Scheme-Specific Elements ............................9
4.2.4. Encoding Format .....................................9
5. Formats and Codes with FEC Encoding ID 5 .......................11
5.1. FEC Payload ID ............................................11
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 ...........................12
5.2.4. Encoding Format ....................................12
6. Procedures with FEC Encoding IDs 2 and 5 .......................13
6.1. Determining the Maximum Source Block Length (B) ...........13
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) ...........................15
8. Reed-Solomon Codes Specification for the Erasure Channel .......16
8.1. Finite Field ..............................................16
8.2. Reed-Solomon Encoding Algorithm ...........................17
8.2.1. Encoding Principles ................................17
8.2.2. Encoding Complexity ................................18
8.3. Reed-Solomon Decoding Algorithm ...........................18
8.3.1. Decoding Principles ................................18
8.3.2. Decoding Complexity ................................19
8.4. Implementation for the Packet Erasure Channel .............19
9. Security Considerations ........................................22
9.1. Problem Statement .........................................22
9.2. Attacks against the Data Flow .............................23
9.2.1. Access to Confidential Objects .....................23
9.2.2. Content Corruption .................................23
9.3. Attacks against the FEC Parameters ........................24
10. IANA Considerations ...........................................25
11. Acknowledgments ...............................................25
12. References ....................................................26
12.1. Normative References .....................................26
12.2. Informative References ...................................26
1. Introduction
The use of Forward Error Correction (FEC) codes is a classical
solution to improve the reliability of multicast and broadcast
transmissions. The [RFC5052] document describes a general framework
to use FEC in Content Delivery Protocols (CDPs). The companion
document [RFC3453] describes some applications of FEC codes for
content delivery.
Recent FEC schemes like [RFC5053] and [RFC5170] 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 schemes 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. They are also systematic codes, which means that the k
source symbols are part of the encoding symbols. Even if the
encoding/decoding complexity is larger than that of [RFC5053] or
[RFC5170], 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
[RS-codec] [Rizzo97]. The goal of the present document is 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,
which specifies the use of Reed-Solomon codes over GF(2^^m), where
m is in {2..16},
o introduces the Fully-Specified FEC Scheme with FEC Encoding ID 5,
which 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) [RFC5445], assigns the FEC 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
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 [RFC2119].
3. Definitions Notations and Abbreviations
3.1. Definitions
This document uses the same terms and definitions as those specified
in [RFC5052]. Additionally, it uses the following definitions:
Source symbol: unit of data used during the encoding process.
Encoding symbol: unit of data generated by the encoding process.
Repair symbol: encoding symbol that is not a source symbol.
Code rate: the k/n ratio, i.e., the ratio between the number of
source symbols and the number of encoding symbols. By
definition, the code rate is such that: 0 < code rate <= 1. A
code rate close to 1 indicates that a small number of repair
symbols have been produced during the encoding process.
Systematic code: FEC code in which the source symbols are part of
the encoding symbols.
Source block: a block of k source symbols that are considered
together for the encoding.
Encoding Symbol Group: a group of encoding symbols that are sent
together within the same packet, and whose relationships to the
source 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.
Packet Erasure Channel: a communication path where packets are
either dropped (e.g., by a congested router, or because the
number of transmission errors exceeds the correction
capabilities of the physical layer codes) or received. When a
packet is received, it is assumed that this packet is not
corrupted.
3.2. Notations
This document uses the following notations:
L the object transfer length in bytes.
k the number of source symbols in a source block.
n_r the number of repair symbols generated for a source block.
n the encoding block length, i.e., the number of encoding
symbols generated for a source block. Therefore: n = k +
n_r.
max_n the maximum number of encoding symbols generated for any
source block.
B the maximum source block length in symbols, i.e., the
maximum number of source symbols per source block.
N the number of source blocks into which the object shall be
partitioned.
E the encoding symbol length in bytes.
S the symbol size in units of m-bit elements. When m = 8,
then S and E are equal.
m the length of the elements in the finite field, in bits.
In this document, m belongs to {2..16}.
q the number of elements in the finite field. We have: q =
2^^m in this specification.
G the number of encoding symbols per group, i.e., the number
of symbols sent in the same packet.
GM the Generator Matrix of a Reed-Solomon code.
CR the "code rate", i.e., the k/n ratio.
a^^b a raised to the power b.
a^^-1 the inverse of a.
I_k the k*k identity matrix.
3.3. Abbreviations
This document uses the following abbreviations:
ESI Encoding Symbol ID.
FEC OTI FEC Object Transmission Information.
RS Reed-Solomon.
MDS Maximum Distance Separable code.
GF(q) a finite field (also known as Galois Field) with q
elements. We assume that q = 2^^m in this document.
4. Formats and Codes with FEC Encoding ID 2
This section introduces the formats and codes associated with the
Fully-Specified FEC Scheme with FEC Encoding ID 2, which 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 lengths of these two fields depend 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 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 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 formats of the FEC Payload ID for m = 8 and m = 16 are
illustrated 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 terabytes). 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:
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 the 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
the 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 [ALC] or NORM [NORM]
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 [FLUTE], 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)
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.
During 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) that is added to the FEC-OTI-Scheme-
Specific-Info attribute.
5. Formats and Codes with FEC Encoding ID 5
This section introduces the formats and codes associated with the
Fully-Specified FEC Scheme with FEC Encoding ID 5, which 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 refers to the one and only symbol of the packet.
0 1 2 3
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| Source Block Number (24 bits) | Enc. Symb. ID |
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
Figure 5: FEC Payload ID Encoding Format with FEC Encoding ID 5
5.2. FEC Object Transmission Information
5.2.1. Mandatory Elements
o FEC Encoding ID: the Fully-Specified FEC Scheme described in this
section uses FEC Encoding ID 5.
5.2.2. Common Elements
The Common elements are the same as those specified in Section 4.2.2
when m = 8 and G = 1.
5.2.3. Scheme-Specific Elements
No Scheme-Specific elements are defined by this FEC scheme.
5.2.4. Encoding Format
This section shows the two possible encoding formats of the above FEC
OTI. The present document does not specify when one encoding format
or the other should be used.
5.2.4.1. Using the General EXT_FTI Format
The FEC OTI binary format is the following, when the EXT_FTI
mechanism is used (e.g., within the ALC [ALC] or NORM [NORM]
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 [FLUTE], the following XML attributes must be
described for the associated object:
o FEC-OTI-FEC-Encoding-ID
o FEC-OTI-Transfer-Length (L)
o FEC-OTI-Encoding-Symbol-Length (E)
o FEC-OTI-Maximum-Source-Block-Length (B)
o FEC-OTI-Max-Number-of-Encoding-Symbols (max_n)
6. Procedures with FEC Encoding IDs 2 and 5
This section defines procedures that are common to FEC Encoding IDs 2
and 5. In case of FEC Encoding ID 5, m = 8 and G = 1. The block
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)
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) * CR)
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 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:
Input:
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.
CR: 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 / CR);
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);
It is RECOMMENDED that the "n-algorithm" be used by a sender, but
other algorithms remain possible to determine max_n and/or n.
At a receiver, the max_n value is extracted from the received FEC
OTI. Since the Reed-Solomon decoder does not need to know the actual
n value, using the receiver part of the "n-algorithm" is not
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-
Solomon Codes over GF(2^^8)
In the context of the Under-Specified Small Block Systematic FEC
Scheme (FEC Encoding ID 129) [RFC5445], 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
[RFC5445].
The FEC scheme with FEC Instance ID 0 MAY use the block partitioning
algorithm defined in Section 9.1 of [RFC5052] to partition the object
into source blocks. This FEC scheme MAY also use another algorithm.
For instance, the CDP sender may change the length of each source
block dynamically, depending on some external criteria (e.g., to
adjust the FEC coding rate to the current loss rate experienced by
NORM receivers) and inform the CDP receivers of the current block
length by means of the EXT_FTI mechanism. This choice is out of the
scope of the current document.
8. Reed-Solomon Codes Specification for the Erasure Channel
Reed-Solomon (RS) codes are linear block codes. They also belong to
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.
Sections 8.1 to 8.3 specify the [n,k]-RS codes when applied to m-bit
elements, and Section 8.4 specifies the use of [n,k]-RS codes when
applied to symbols composed of several m-bit elements. The use
described in Section 8.4 is the crux 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
A finite field GF(q) is defined as a finite set of q elements that
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 to m-1. The polynomials can be associated with 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. Note that all the roots of this
polynomial are in GF(2^^m) but not in GF(2).
The chosen polynomial representation of the 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)
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 that 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
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(k))^^2), the
generator matrix can be computed in 0((n-k)* k * (log(k))^^2))
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(k))^^2). 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 [GO94]. The total
complexity of this encoding algorithm is then O((k/(n-k)) *
(log(k))^^2 + 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
[MWS77]). 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.
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(k))^^2) ) then by V_{k,k} (multipoint
evaluation with complexity O(k * log(k))). The global decoding
complexity is then O((log(k))^^2) operations per source element.
8.4. Implementation for the Packet Erasure Channel
In a packet erasure channel, each packet (including its 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 (this 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
of the source symbol set. A logical u-th source vector, comprised of
the u-th elements from the set of source symbols, is used to
calculate a u-th encoding vector. This u-th encoding vector then
provides the u-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 u 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 u S-1
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| |X| | enc. symbol 0
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| |X| | enc. symbol 1
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
. . .
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
| |Y| | enc. symbol n-1
+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
Figure 7: Packet Encoding Scheme
An asset of this scheme is that the loss of some encoding symbols
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 vector-matrix
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.
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 confidential content (e.g.,
in case of 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.
9.2.1. Access to Confidential Objects
Access control to the object being transmitted is typically provided
by means of encryption. This encryption can be done over the whole
object (e.g., by the content provider, before the FEC encoding
process), or be done on a packet per-packet basis (e.g., when IPsec
Encapsulating Security Payload (ESP) is used [RFC4303]). 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.
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) 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 recovered
sometimes. Several techniques can provide this source
authentication/content integrity service:
o At the object level, the object MAY be digitally signed (with
public key cryptography), for instance by using RSASSA-PKCS1-v1_5
[RFC3447]. This signature enables a receiver to check the object
integrity, once the object has been fully decoded. Even if
digital signatures are computationally expensive, this calculation
occurs only once per object, which is usually acceptable.
o At the packet level, each packet can be digitally signed. A major
limitation is the high computational and transmission overheads
that this solution requires (unless Elliptic Curve Cryptography
(ECC) is used). To avoid this problem, the signature may span a
set of symbols (instead of a single one) in order to amortize the
signature calculation. But if a single symbol is missing, the
integrity of the whole set cannot be checked.
o At the packet level, a Group Message Authentication Code (MAC)
[RFC2104] scheme can be used; for instance, by using HMAC-SHA-256
with a secret key shared by all the group members (i.e., the
sender(s) and receivers). Thanks to the secret key, this
technique creates a cryptographically secured digest of a packet
that is sent along with the packet. The Group MAC scheme does not
create prohibitive processing load nor transmission overhead, but
it has a major limitation: it only provides a group
authentication/integrity service since all group members share the
same secret group key, which means that each member can send a
forged packet. It is therefore restricted to situations where
group members are fully trusted (or in association with another
technique as a pre-check).
o At the packet level, TESLA [RFC4082] is 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
TESLA during the bootstrap process, when used) 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.
Techniques relying on symmetric key cryptography (group MAC) require
that a secret key be shared by all group members. This can be
achieved by means of a group key management protocol, or simply by
pre-distributing the secret key (but this manual solution has many
limitations).
It is up to the 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).
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 in an EXT_FTI header
extension SHOULD be protected by one of the per-packet techniques
described above: digital signature, group MAC, or TESLA. When FEC
OTI is contained in an FDT Instance, this FDT Instance object SHOULD
be protected, for instance, by digitally signing it with XML digital
signatures [RFC3275]. Finally, when FEC OTI is sent out-of-band
(e.g., in a session description), this FEC OTI SHOULD be protected,
for instance, by digitally signing the object that includes this FEC
OTI.
The same considerations concerning the key management aspects apply
here also.
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 [RFC5052].
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)".
11. Acknowledgments
The authors want to thank Brian Adamson, Igor Slepchin, Stephen Kent,
Francis Dupont, Elwyn Davies, Magnus Westerlund, and Alfred Hoenes
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.
12. References
12.1. Normative References
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119, March 1997.
[RFC5052] Watson, M., Luby, M., and L. Vicisano, "Forward Error
Correction (FEC) Building Block", RFC 5052, August 2007.
[RFC5445] Watson, M., "Basic Forward Error Correction (FEC)
Schemes", RFC 5445, March 2009.
12.2. Informative References
[RFC3453] Luby, M., Vicisano, L., Gemmell, J., Rizzo, L., Handley,
M., and J. Crowcroft, "The Use of Forward Error
Correction (FEC) in Reliable Multicast", RFC 3453,
December 2002.
[RS-codec] Rizzo, L., "Reed-Solomon FEC codec", available at
http://info.iet.unipi.it/~luigi/vdm98/vdm980702.tgz and
mirrored at http://planete-bcast.inrialpes.fr/, revised
version of July 1998.
[Rizzo97] Rizzo, L., "Effective Erasure Codes for Reliable Computer
Communication Protocols", ACM SIGCOMM Computer
Communication Review Vol.27, No.2, pp.24-36, April 1997.
[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",
Journal of Complexity, pp. 411-427, vol. 10, 1994.
[RFC5170] Roca, V., Neumann, C., and D. Furodet, "Low Density
Parity Check (LDPC) Forward Error Correction", RFC 5170,
June 2008.
[RFC5053] Luby, M., Shokrollahi, A., Watson, M., and T.
Stockhammer, "Raptor Forward Error Correction Scheme",
RFC 5053, October 2007.
[ALC] Luby, M., Watson, M., and L. Vicisano, "Asynchronous
Layered Coding (ALC) Protocol Instantiation", Work
in Progress, November 2008.
[NORM] Adamson, B., Bormann, C., Handley, M., and J. Macker,
"NACK-Oriented Reliable Multicast Protocol", Work
in Progress, March 2009.
[FLUTE] Paila, T., Walsh, R., Luby, M., Lehtonen, R., and V.
Roca, "FLUTE - File Delivery over Unidirectional
Transport", Work in Progress, September 2008.
[RFC3447] Jonsson, J. and B. Kaliski, "Public-Key Cryptography
Standards (PKCS) #1: RSA Cryptography Specifications
Version 2.1", RFC 3447, February 2003.
[RFC4303] Kent, S., "IP Encapsulating Security Payload (ESP)",
RFC 4303, December 2005.
[RFC2104] "HMAC: Keyed-Hashing for Message Authentication",
RFC 2104, February 1997.
[RFC4082] "Timed Efficient Stream Loss-Tolerant Authentication
(TESLA): Multicast Source Authentication Transform
Introduction", RFC 4082, June 2005.
[RFC3275] Eastlake 3rd, D., Reagle, J., and D. Solo, "(Extensible
Markup Language) XML-Signature Syntax and Processing",
RFC 3275, March 2002.
Authors' Addresses
Jerome Lacan
ISAE/LAAS-CNRS
1, place Emile Blouin
Toulouse 31056
France
EMail: jerome.lacan@isae.fr
URI: http://pagespro.isae.fr/jerome-lacan/
Vincent Roca
INRIA
655, av. de l'Europe
Inovallee; Montbonnot
ST ISMIER cedex 38334
France
EMail: vincent.roca@inria.fr
URI: http://planete.inrialpes.fr/people/roca/
Jani Peltotalo
Tampere University of Technology
P.O. Box 553 (Korkeakoulunkatu 1)
Tampere FIN-33101
Finland
EMail: jani.peltotalo@tut.fi
URI: http://mad.cs.tut.fi/
Sami Peltotalo
Tampere University of Technology
P.O. Box 553 (Korkeakoulunkatu 1)
Tampere FIN-33101
Finland
EMail: sami.peltotalo@tut.fi
URI: http://mad.cs.tut.fi/