GENWiki

Premier IT Outsourcing and Support Services within the UK

User Tools

Site Tools


rfc:rfc5510

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.

Copyright Notice

 Copyright (c) 2009 IETF Trust and the persons identified as the
 document authors.  All rights reserved.
 This document is subject to BCP 78 and the IETF Trust's Legal
 Provisions Relating to IETF Documents in effect on the date of
 publication of this document (http://trustee.ietf.org/license-info).
 Please review these documents carefully, as they describe your rights
 and restrictions with respect to this document.
 This document may contain material from IETF Documents or IETF
 Contributions published or made publicly available before November
 10, 2008.  The person(s) controlling the copyright in some of this
 material may not have granted the IETF Trust the right to allow
 modifications of such material outside the IETF Standards Process.
 Without obtaining an adequate license from the person(s) controlling
 the copyright in such materials, this document may not be modified
 outside the IETF Standards Process, and derivative works of it may
 not be created outside the IETF Standards Process, except to format
 it for publication as an RFC or to translate it into languages other
 than English.

Lacan, et al. Standards Track [Page 1] RFC 5510 Reed-Solomon Forward Error Correction April 2009

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.

Lacan, et al. Standards Track [Page 2] RFC 5510 Reed-Solomon Forward Error Correction April 2009

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

Lacan, et al. Standards Track [Page 3] RFC 5510 Reed-Solomon Forward Error Correction April 2009

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.

Lacan, et al. Standards Track [Page 4] RFC 5510 Reed-Solomon Forward Error Correction April 2009

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.

Lacan, et al. Standards Track [Page 5] RFC 5510 Reed-Solomon Forward Error Correction April 2009

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.

Lacan, et al. Standards Track [Page 6] RFC 5510 Reed-Solomon Forward Error Correction April 2009

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.

Lacan, et al. Standards Track [Page 7] RFC 5510 Reed-Solomon Forward Error Correction April 2009

  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.

Lacan, et al. Standards Track [Page 8] RFC 5510 Reed-Solomon Forward Error Correction April 2009

 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).

Lacan, et al. Standards Track [Page 9] RFC 5510 Reed-Solomon Forward Error Correction April 2009

  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,

Lacan, et al. Standards Track [Page 10] RFC 5510 Reed-Solomon Forward Error Correction April 2009

 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

Lacan, et al. Standards Track [Page 11] RFC 5510 Reed-Solomon Forward Error Correction April 2009

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

Lacan, et al. Standards Track [Page 12] RFC 5510 Reed-Solomon Forward Error Correction April 2009

 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.

Lacan, et al. Standards Track [Page 13] RFC 5510 Reed-Solomon Forward Error Correction April 2009

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.

Lacan, et al. Standards Track [Page 14] RFC 5510 Reed-Solomon Forward Error Correction April 2009

 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.

Lacan, et al. Standards Track [Page 15] RFC 5510 Reed-Solomon Forward Error Correction April 2009

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)

Lacan, et al. Standards Track [Page 16] RFC 5510 Reed-Solomon Forward Error Correction April 2009

    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

Lacan, et al. Standards Track [Page 17] RFC 5510 Reed-Solomon Forward Error Correction April 2009

 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

Lacan, et al. Standards Track [Page 18] RFC 5510 Reed-Solomon Forward Error Correction April 2009

 [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.

Lacan, et al. Standards Track [Page 19] RFC 5510 Reed-Solomon Forward Error Correction April 2009

 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.

Lacan, et al. Standards Track [Page 20] RFC 5510 Reed-Solomon Forward Error Correction April 2009

        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.

Lacan, et al. Standards Track [Page 21] RFC 5510 Reed-Solomon Forward Error Correction April 2009

 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).

Lacan, et al. Standards Track [Page 22] RFC 5510 Reed-Solomon Forward Error Correction April 2009

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

Lacan, et al. Standards Track [Page 23] RFC 5510 Reed-Solomon Forward Error Correction April 2009

    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

Lacan, et al. Standards Track [Page 24] RFC 5510 Reed-Solomon Forward Error Correction April 2009

 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.

Lacan, et al. Standards Track [Page 25] RFC 5510 Reed-Solomon Forward Error Correction April 2009

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.

Lacan, et al. Standards Track [Page 26] RFC 5510 Reed-Solomon Forward Error Correction April 2009

 [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.

Lacan, et al. Standards Track [Page 27] RFC 5510 Reed-Solomon Forward Error Correction April 2009

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/

Lacan, et al. Standards Track [Page 28]

/data/webs/external/dokuwiki/data/pages/rfc/rfc5510.txt · Last modified: 2009/04/27 15:27 by 127.0.0.1

Donate Powered by PHP Valid HTML5 Valid CSS Driven by DokuWiki