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rfc:rfc5170

Network Working Group V. Roca Request for Comments: 5170 INRIA Category: Standards Track C. Neumann

                                                               Thomson
                                                            D. Furodet
                                                    STMicroelectronics
                                                             June 2008
       Low Density Parity Check (LDPC) Staircase and Triangle
               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.

Abstract

 This document describes two Fully-Specified Forward Error Correction
 (FEC) Schemes, Low Density Parity Check (LDPC) Staircase and LDPC
 Triangle, and their 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).  These systematic FEC codes belong to the well-
 known class of "Low Density Parity Check" codes, and are large block
 FEC codes in the sense of RFC 3453.

Roca, et al. Standards Track [Page 1] RFC 5170 LDPC Staircase and Triangle FEC June 2008

Table of Contents

 1.  Introduction . . . . . . . . . . . . . . . . . . . . . . . . .  3
 2.  Requirements Notation  . . . . . . . . . . . . . . . . . . . .  3
 3.  Definitions, Notations, and Abbreviations  . . . . . . . . . .  3
   3.1.  Definitions  . . . . . . . . . . . . . . . . . . . . . . .  3
   3.2.  Notations  . . . . . . . . . . . . . . . . . . . . . . . .  4
   3.3.  Abbreviations  . . . . . . . . . . . . . . . . . . . . . .  5
 4.  Formats and Codes  . . . . . . . . . . . . . . . . . . . . . .  6
   4.1.  FEC Payload IDs  . . . . . . . . . . . . . . . . . . . . .  6
   4.2.  FEC Object Transmission Information  . . . . . . . . . . .  6
     4.2.1.  Mandatory Element  . . . . . . . . . . . . . . . . . .  6
     4.2.2.  Common Elements  . . . . . . . . . . . . . . . . . . .  6
     4.2.3.  Scheme-Specific Elements . . . . . . . . . . . . . . .  7
     4.2.4.  Encoding Format  . . . . . . . . . . . . . . . . . . .  8
 5.  Procedures . . . . . . . . . . . . . . . . . . . . . . . . . .  9
   5.1.  General  . . . . . . . . . . . . . . . . . . . . . . . . .  9
   5.2.  Determining the Maximum Source Block Length (B)  . . . . . 11
   5.3.  Determining the Encoding Symbol Length (E) and Number
         of Encoding Symbols per Group (G)  . . . . . . . . . . . . 12
   5.4.  Determining the Maximum Number of Encoding Symbols
         Generated for Any Source Block (max_n) . . . . . . . . . . 13
   5.5.  Determining the Number of Encoding Symbols of a Block
         (n)  . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
   5.6.  Identifying the G Symbols of an Encoding Symbol Group  . . 14
   5.7.  Pseudo-Random Number Generator . . . . . . . . . . . . . . 17
 6.  Full Specification of the LDPC-Staircase Scheme  . . . . . . . 19
   6.1.  General  . . . . . . . . . . . . . . . . . . . . . . . . . 19
   6.2.  Parity Check Matrix Creation . . . . . . . . . . . . . . . 19
   6.3.  Encoding . . . . . . . . . . . . . . . . . . . . . . . . . 21
   6.4.  Decoding . . . . . . . . . . . . . . . . . . . . . . . . . 21
 7.  Full Specification of the LDPC-Triangle Scheme . . . . . . . . 22
   7.1.  General  . . . . . . . . . . . . . . . . . . . . . . . . . 22
   7.2.  Parity Check Matrix Creation . . . . . . . . . . . . . . . 22
   7.3.  Encoding . . . . . . . . . . . . . . . . . . . . . . . . . 23
   7.4.  Decoding . . . . . . . . . . . . . . . . . . . . . . . . . 23
 8.  Security Considerations  . . . . . . . . . . . . . . . . . . . 24
   8.1.  Problem Statement  . . . . . . . . . . . . . . . . . . . . 24
   8.2.  Attacks Against the Data Flow  . . . . . . . . . . . . . . 24
     8.2.1.  Access to Confidential Objects . . . . . . . . . . . . 24
     8.2.2.  Content Corruption . . . . . . . . . . . . . . . . . . 25
   8.3.  Attacks Against the FEC Parameters . . . . . . . . . . . . 26
 9.  IANA Considerations  . . . . . . . . . . . . . . . . . . . . . 27
 10. Acknowledgments  . . . . . . . . . . . . . . . . . . . . . . . 27
 11. References . . . . . . . . . . . . . . . . . . . . . . . . . . 27
   11.1. Normative References . . . . . . . . . . . . . . . . . . . 27
   11.2. Informative References . . . . . . . . . . . . . . . . . . 27
 Appendix A.  Trivial Decoding Algorithm (Informative Only) . . . . 30

Roca, et al. Standards Track [Page 2] RFC 5170 LDPC Staircase and Triangle FEC June 2008

1. Introduction

 [RFC3453] introduces large block FEC codes as an alternative to small
 block FEC codes like Reed-Solomon.  The main advantage of such large
 block codes is the possibility to operate efficiently on source
 blocks with a size of several tens of thousands (or more) of source
 symbols.  The present document introduces the Fully-Specified FEC
 Encoding ID 3 that is intended to be used with the LDPC-Staircase FEC
 codes, and the Fully-Specified FEC Encoding ID 4 that is intended to
 be used with the LDPC-Triangle FEC codes [RN04][MK03].  Both schemes
 belong to the broad class of large block codes.  For a definition of
 the term Fully-Specified Scheme, see Section 4 of [RFC5052].
 LDPC codes rely on a dedicated matrix, called a "parity check
 matrix", at the encoding and decoding ends.  The parity check matrix
 defines relationships (or constraints) between the various encoding
 symbols (i.e., source symbols and repair symbols), which are later
 used by the decoder to reconstruct the original k source symbols if
 some of them are missing.  These codes are systematic, in the sense
 that the encoding symbols include the source symbols in addition to
 the repair symbols.
 Since the encoder and decoder must operate on the same parity check
 matrix, information must be communicated between them as part of the
 FEC Object Transmission Information.
 A publicly available reference implementation of these codes is
 available and distributed under a GNU/LGPL (Lesser General Public
 License) [LDPC-codec].  Besides, the code extracts included in this
 document are directly contributed to the IETF process by the authors
 of this document and by Radford M. Neal.

2. Requirements Notation

 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 [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: a unit of data used during the encoding process

Roca, et al. Standards Track [Page 3] RFC 5170 LDPC Staircase and Triangle FEC June 2008

    Encoding Symbol: a unit of data generated by the encoding process
    Repair Symbol: an encoding symbol that is not a source symbol
    Code Rate: the k/n ratio, i.e., the ratio between the number of
    source symbols and the number of encoding symbols.  The code rate
    belongs to a ]0; 1] interval.  A code rate close to 1 indicates
    that a small number of repair symbols have been produced during
    the encoding process
    Systematic Code: FEC code in which the source symbols are part of
    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 object 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 denotes the object transfer length in bytes.
    k denotes the source block length in symbols, i.e., the number of
    source symbols of a source block.
    n denotes the encoding block length, i.e., the number of encoding
    symbols generated for a source block.
    E denotes the encoding symbol length in bytes.
    B denotes the maximum source block length in symbols, i.e., the
    maximum number of source symbols per source block.

Roca, et al. Standards Track [Page 4] RFC 5170 LDPC Staircase and Triangle FEC June 2008

    N denotes the number of source blocks into which the object shall
    be partitioned.
    G denotes the number of encoding symbols per group, i.e., the
    number of symbols sent in the same packet.
    CR denotes the "code rate", i.e., the k/n ratio.
    max_n denotes the maximum number of encoding symbols generated for
    any source block.  This is in particular the number of encoding
    symbols generated for a source block of size B.
    H denotes the parity check matrix.
    N1 denotes the target number of "1s" per column in the left side
    of the parity check matrix.
    N1m3 denotes the value N1 - 3, where N1 is the target number of
    "1s" per column in the left side of the parity check matrix.
    pmms_rand(m) denotes the pseudo-random number generator defined in
    Section 5.7 that returns a new random integer in [0; m-1] each
    time it is called.

3.3. Abbreviations

 This document uses the following abbreviations:
    ESI: Encoding Symbol ID
    FEC OTI: FEC Object Transmission Information
    FPI: FEC Payload ID
    LDPC: Low Density Parity Check
    PRNG: Pseudo-Random Number Generator

Roca, et al. Standards Track [Page 5] RFC 5170 LDPC Staircase and Triangle FEC June 2008

4. Formats and Codes

4.1. FEC Payload IDs

 The FEC Payload ID is composed of the Source Block Number and the
 Encoding Symbol ID:
    The Source Block Number (12-bit field) identifies from which
    source block of the object the encoding symbol(s) in the packet
    payload is(are) generated.  There is a maximum of 2^^12 blocks per
    object.  Source block numbering starts at 0.
    The Encoding Symbol ID (20-bit field) identifies which encoding
    symbol(s) generated from the source block is(are) carried in the
    packet payload.  There is a maximum of 2^^20 encoding symbols per
    block.  The first k values (0 to k-1) identify source symbols, the
    remaining n-k values (k to n-k-1) identify repair symbols.
 There MUST be exactly one FEC Payload ID per packet.  In the 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 thanks to a dedicated function, as explained in Section 5.6
  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  |      Encoding Symbol ID (20 bits)     |
 +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
 Figure 1: FEC Payload ID encoding format for FEC Encoding ID 3 and 4

4.2. FEC Object Transmission Information

4.2.1. Mandatory Element

 o  FEC Encoding ID: the LDPC-Staircase and LDPC-Triangle Fully-
    Specified FEC Schemes use the FEC Encoding ID 3 (Staircase) and 4
    (Triangle), respectively.

4.2.2. Common Elements

 The following elements MUST be defined with the present FEC Schemes:
 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:

Roca, et al. Standards Track [Page 6] RFC 5170 LDPC Staircase and Triangle FEC June 2008

       maximum transfer length = 2^^12 * B * E
    For instance, if B=2^^19 (because of a code rate of 1/2,
    Section 5.2), and if E=1024 bytes, then the maximum transfer
    length is 2^^41 bytes (or 2 TB).  The upper limit, with symbols of
    size 2^^16-1 bytes and a code rate larger or equal to 1/2, amounts
    to 2^^47 bytes (or 128 TB).
 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.  There are
    some restrictions on the maximum B value, as explained in
    Section 5.2.
 o  Max-Number-of-Encoding-Symbols (max_n): a non-negative integer
    indicating the maximum number of encoding symbols generated for
    any source block.  There are some restrictions on the maximum
    max_n value.  In particular max_n is at most equal to 2^^20.
 Section 5 explains how to define the values of each of these
 elements.

4.2.3. Scheme-Specific Elements

 The following elements MUST be defined with the present FEC Scheme:
 o  N1m3: an integer between 0 (default) and 7, inclusive.  The target
    number of "1s" per column in the left side of the parity check
    matrix, N1, is then equal to N1m3 + 3 (see Sections 6.2 and 7.2).
    Using the default value of 0 for N1m3 is recommended when the
    sender has no information on the decoding scheme used by the
    receivers.  A value greater than 0 for N1m3 can be a good choice
    in specific situations, e.g., with LDPC-staircase codes when the
    sender knows that all the receivers use a Gaussian elimination
    decoding scheme.  Nevertheless, the current document does not
    mandate any specific value.  This choice is left to the codec
    developer.
 o  G: an integer between 1 (default) and 31, inclusive, indicating
    the number of encoding symbols per group (i.e., per packet).  The
    default value is 1, meaning that each packet contains exactly one
    symbol.  Values greater than 1 can also be defined, as explained
    in Section 5.3.

Roca, et al. Standards Track [Page 7] RFC 5170 LDPC Staircase and Triangle FEC June 2008

 o  PRNG seed: the seed is a 32-bit unsigned integer between 1 and
    0x7FFFFFFE (i.e., 2^^31-2) inclusive.  This value is used to
    initialize the Pseudo-Random Number Generator (Section 5.7).

4.2.4. Encoding Format

 This section shows two possible encoding formats of the above FEC
 OTI.  The present document does not specify when or how these
 encoding formats 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 Asynchronous Layer Coding (ALC)
 [RMT-PI-ALC] or NACK-Oriented Reliable Multicast (NORM) [RMT-PI-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 = 5    |                               |
 +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+                               +
 |                      Transfer-Length (L)                      |
 +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
 |   Encoding Symbol Length (E)  | N1m3|    G    |   B (MSB)     |
 +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
 |        B (LSB)        |   Max Nb of Enc. Symbols  (max_n)     |
 +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
 |                           PRNG seed                           |
 +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
         Figure 2: EXT_FTI Header for FEC Encoding ID 3 and 4
 In particular:
 o  The Transfer-Length (L) field size (48 bits) is larger than the
    size required to store the maximum transfer length (Section 4.2.2)
    for field alignment purposes.
 o  The Maximum-Source-Block-Length (B) field (20 bits) is split into
    two parts: the 8 most significant bits (MSB) are in the third 32-
    bit word of the EXT_FTI, and the remaining 12 least significant
    bits (LSB) are in the fourth 32-bit word.

Roca, et al. Standards Track [Page 8] RFC 5170 LDPC Staircase and Triangle FEC June 2008

4.2.4.2. Using the FDT Instance (FLUTE-Specific)

 When it is desired that the FEC OTI be carried in the File Delivery
 Table (FDT) Instance of a File Delivery over Unidirectional Transport
 (FLUTE) session [RMT-FLUTE], the following XML attributes must be
 described for the associated object:
 o  FEC-OTI-FEC-Encoding-ID
 o  FEC-OTI-Transfer-length
 o  FEC-OTI-Encoding-Symbol-Length
 o  FEC-OTI-Maximum-Source-Block-Length
 o  FEC-OTI-Max-Number-of-Encoding-Symbols
 o  FEC-OTI-Scheme-Specific-Info
 The FEC-OTI-Scheme-Specific-Info contains the string resulting from
 the Base64 encoding [RFC4648] of the following value:
  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
 +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
 |                        PRNG seed                              |
 +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
 | N1m3|    G    |
 +-+-+-+-+-+-+-+-+
  Figure 3: FEC OTI Scheme-Specific Information to be Included in the
               FDT Instance for FEC Encoding ID 3 and 4
 During Base64 encoding, the 5 bytes of the FEC OTI Scheme-Specific
 Information are transformed into a string of 8 printable characters
 (in the 64-character alphabet) that is added to the FEC-OTI-Scheme-
 Specific-Info attribute.

5. Procedures

 This section defines procedures that are common to FEC Encoding IDs 3
 and 4.

5.1. General

 The B (maximum source block length in symbols), E (encoding symbol
 length in bytes), and G (number of encoding symbols per group)
 parameters are first determined.  The algorithms of Section 5.2 and

Roca, et al. Standards Track [Page 9] RFC 5170 LDPC Staircase and Triangle FEC June 2008

 Section 5.3 MAY be used to that purpose.  Using other algorithms is
 possible without compromising interoperability since the B, E, and G
 parameters are communicated to the receiver by means of the FEC OTI.
 Then, the source object MUST be partitioned using the block
 partitioning algorithm specified in [RFC5052].  To that purpose, the
 B, L (object transfer length in bytes), and E arguments are provided.
 As a result, the object is partitioned into N source blocks.  These
 blocks are numbered consecutively from 0 to N-1.  The first I source
 blocks consist of A_large source symbols, the remaining N-I source
 blocks consist of A_small source symbols.  Each source symbol is E
 bytes in length, except perhaps the last symbol, which may be
 shorter.
 Then, the max_n (maximum number of encoding symbols generated for any
 source block) parameter is determined.  The algorithm in Section 5.4
 MAY be used to that purpose.  Using another algorithm is possible
 without compromising interoperability since the max_n parameter is
 communicated to the receiver by means of the FEC OTI.
 For each block, the actual number of encoding symbols, n, MUST then
 be determined using the "n-algorithm" detailed in Section 5.5.
 Then, FEC encoding and decoding can be done block per block,
 independently.  To that purpose, a parity check matrix is created,
 that forms a system of linear equations between the source and repair
 symbols of a given block, where the basic operator is XOR.
 This parity check matrix is logically divided into two parts: the
 left side (from column 0 to k-1) describes the occurrences of each
 source symbol in the system of linear equations; the right side (from
 column k to n-1) describes the occurrences of each repair symbol in
 the system of linear equations.  The only difference between the
 LDPC-Staircase and LDPC-Triangle schemes is the construction of this
 right sub-matrix.  An entry (a "1") in the matrix at position (i,j)
 (i.e., at row i and column j) means that the symbol with ESI j
 appears in equation i of the system.
 When the parity symbols have been created, the sender transmits
 source and parity symbols.  The way this transmission occurs can
 largely impact the erasure recovery capabilities of the LDPC-* FEC.
 In particular, sending parity symbols in sequence is suboptimal.
 Instead, it is usually recommended to shuffle these symbols.  The
 interested reader will find more details in [NRFF05].

Roca, et al. Standards Track [Page 10] RFC 5170 LDPC Staircase and Triangle FEC June 2008

 The following sections detail how the B, E, G, max_n, and n
 parameters are determined (in Sections 5.2, 5.3, 5.4 and 5.5,
 respectively).  Section 5.6 details how Encoding Symbol Groups are
 created, and finally, Section 5.7 covers the PRNG.

5.2. Determining the Maximum Source Block Length (B)

 The B parameter (maximum source block length in symbols) depends on
 several parameters: the code rate (CR), the Encoding Symbol ID field
 length of the FEC Payload ID (20 bits), as well as possible internal
 codec limitations.
 The B parameter cannot be larger than the following values, derived
 from the FEC Payload ID limitations, for a given code rate:
    max1_B = 2^^(20 - ceil(Log2(1/CR)))
 Some common max1_B values are:
 o  CR == 1 (no repair symbol): max1_B = 2^^20 = 1,048,576
 o  1/2 <= CR < 1: max1_B = 2^^19 = 524,288 symbols
 o  1/4 <= CR < 1/2: max1_B = 2^^18 = 262,144 symbols
 o  1/8 <= CR < 1/4: max1_B = 2^^17 = 131,072 symbols
 Additionally, a codec MAY impose other limitations on the maximum
 block size.  For instance, this is the case when the codec uses
 internally 16-bit unsigned integers to store the Encoding Symbol ID,
 since it does not enable to store all the possible values of a 20-bit
 field.  In that case, if for instance, 1/2 <= CR < 1, then the
 maximum source block length is 2^^15.  Other limitations may also
 apply, for instance, because of a limited working memory size.  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.

Roca, et al. Standards Track [Page 11] RFC 5170 LDPC Staircase and Triangle FEC June 2008

5.3. Determining the Encoding Symbol Length (E) and Number of Encoding

    Symbols per Group (G)
 The E parameter usually depends on the maximum transmission unit on
 the path (PMTU) from the source to each receiver.  In order to
 minimize the protocol header overhead (e.g., the Layered Coding
 Transport (LCT), UDP, IPv4, or IPv6 headers in the case of ALC), E is
 chosen to be as large as possible.  In that case, E is chosen so that
 the size of a packet composed of a single symbol (G=1) remains below
 but close to the PMTU.
 However, other considerations can exist.  For instance, the E
 parameter can be made a function of the object transfer length.
 Indeed, LDPC codes are known to offer better protection for large
 blocks.  In the case of small objects, it can be advantageous to
 reduce the encoding symbol length (E) in order to artificially
 increase the number of symbols and therefore the block size.
 In order to minimize the protocol header overhead, several symbols
 can be grouped in the same Encoding Symbol Group (i.e., G > 1).
 Depending on how many symbols are grouped (G) and on the packet loss
 rate (G symbols are lost for each packet erasure), this strategy
 might or might not be appropriate.  A balance must therefore be
 found.
 The current specification does not mandate any value for either E or
 G.  The current specification only provides an example of possible
 choices for E and G.  Note that this choice is made by the sender,
 and the E and G parameters are then communicated to the receiver
 thanks to the FEC OTI.  Note also that the decoding algorithm used
 influences the choice of the E and G parameters.  Indeed, increasing
 the number of symbols will negatively impact the processing load when
 decoding is based (in part or totally) on Gaussian elimination,
 whereas the impacts will be rather low when decoding is based on the
 trivial algorithm sketched in Section 6.4.
 Example:
 Let us assume that the trivial decoding algorithm sketched in
 Section 6.4 is used.  First, define the target packet payload size,
 pkt_sz (at most equal to the PMTU minus the size of the various
 protocol headers).  The pkt_sz must be chosen in such a way that the
 symbol size is an integer.  This can require that pkt_sz be a
 multiple of 4, 8, or 16 (see the table below).  Then calculate the
 number of packets in the object: nb_pkts = ceil(L / pkt_sz).
 Finally, thanks to nb_pkts, use the following table to find a
 possible G value.

Roca, et al. Standards Track [Page 12] RFC 5170 LDPC Staircase and Triangle FEC June 2008

   +------------------------+----+-------------+-------------------+
   |    Number of packets   |  G | Symbol size |         k         |
   +------------------------+----+-------------+-------------------+
   |     4000 <= nb_pkts    |  1 |    pkt_sz   |     4000 <= k     |
   |                        |    |             |                   |
   | 1000 <= nb_pkts < 4000 |  4 |  pkt_sz / 4 | 4000 <= k < 16000 |
   |                        |    |             |                   |
   |  500 <= nb_pkts < 1000 |  8 |  pkt_sz / 8 |  4000 <= k < 8000 |
   |                        |    |             |                   |
   |   1 <= nb_pkts < 500   | 16 | pkt_sz / 16 |   16 <= k < 8000  |
   +------------------------+----+-------------+-------------------+

5.4. Determining the Maximum Number of Encoding Symbols Generated for

    Any Source Block (max_n)
 The following algorithm MAY be used by a sender to determine the
 maximum number of encoding symbols generated for any source block
 (max_n) as a function of B and the target code rate.  Since the max_n
 parameter is communicated to the decoder by means of the FEC OTI,
 another method MAY be used to determine max_n.
 Input:
    B: Maximum source block length, for any source block.  Section 5.2
    MAY be used to determine its value.
    CR: FEC code rate, which is provided by the user (e.g., when
    starting a FLUTE sending application).  It is expressed as a
    floating point value.  The CR value must be such that the
    resulting number of encoding symbols per block is at most equal to
    2^^20 (Section 4.1).
 Output:
    max_n: Maximum number of encoding symbols generated for any source
    block.
 Algorithm:
    max_n = ceil(B / CR);
    if (max_n > 2^^20), then return an error ("invalid code rate");
    (NB: if B has been defined as explained in Section 5.2, this error
    should never happen.)

Roca, et al. Standards Track [Page 13] RFC 5170 LDPC Staircase and Triangle FEC June 2008

5.5. Determining the Number of Encoding Symbols of a Block (n)

 The following algorithm, also called "n-algorithm", MUST be used by
 the sender and the receiver to determine the number of encoding
 symbols for a given block (n) as a function of B, k, and max_n.
 Input:
    B: Maximum source block length, for any source block.  At a
    sender, Section 5.2 MAY be used to determine its value.  At a
    receiver, this value MUST be extracted from the received FEC OTI.
    k: Current source block length.  At a sender or receiver, the
    block partitioning algorithm MUST be used to determine its value.
    max_n: Maximum number of encoding symbols generated for any source
    block.  At a sender, Section 5.4 MAY be used to determine its
    value.  At a receiver, this value MUST be extracted from the
    received FEC OTI.
 Output:
    n: Number of encoding symbols generated for this source block.
 Algorithm:
    n = floor(k * max_n / B);

5.6. Identifying the G Symbols of an Encoding Symbol Group

 When multiple encoding symbols are sent in the same packet, the FEC
 Payload ID information of the packet MUST refer to the first encoding
 symbol.  It MUST then be possible to identify each symbol from this
 single FEC Payload ID.  To that purpose, the symbols of an Encoding
 Symbol Group (i.e., packet):
 o  MUST all be either source symbols or repair symbols.  Therefore,
    only source packets and repair packets are permitted, not mixed
    ones.
 o  are identified by a function, sender(resp.
    receiver)_find_ESIs_of_group(), that takes as argument:
  • for a sender, the index of the Encoding Symbol Group (i.e.,

packet) that the application wants to create,

  • for a receiver, the ESI information contained in the FEC

Payload ID.

Roca, et al. Standards Track [Page 14] RFC 5170 LDPC Staircase and Triangle FEC June 2008

    and returns a list of G Encoding Symbol IDs.  In the case of a
    source packet, the G Encoding Symbol IDs are chosen consecutively,
    by incrementing the ESI.  In the case of a repair packet, the G
    repair symbols are chosen randomly, as explained below.
 o  are stored in sequence in the packet, without any padding.  In
    other words, the last byte of the i-th symbol is immediately
    followed by the first byte of (i+1)-th symbol.
 The system must first be initialized by creating a random permutation
 of the n-k indexes.  This initialization function MUST be called
 immediately after creating the parity check matrix.  More precisely,
 since the PRNG seed is not re-initialized, there must not have been a
 call to the PRNG function between the time the parity check matrix
 has been initialized and the time the following initialization
 function is called.  This is true both at a sender and at a receiver.
 int *txseqToID;
 int *IDtoTxseq;
 /*
  * Initialization function.
  * Warning: use only when G > 1.
  */
 void
 initialize_tables ()
 {
     int i;
     int randInd;
     int backup;
     txseqToID = malloc((n-k) * sizeof(int));
     IDtoTxseq = malloc((n-k) * sizeof(int));
     if (txseqToID == NULL || IDtoTxseq == NULL)
         handle the malloc failures as appropriate...
     /* initialize the two tables that map ID
      * (i.e., ESI-k) to/from TxSequence. */
     for (i = 0; i < n - k; i++) {
         IDtoTxseq[i] = i;
         txseqToID[i] = i;
     }
     /* now randomize everything */
     for (i = 0; i < n - k; i++) {
         randInd = pmms_rand(n - k);
         backup  = IDtoTxseq[i];
         IDtoTxseq[i] = IDtoTxseq[randInd];
         IDtoTxseq[randInd] = backup;
         txseqToID[IDtoTxseq[i]] =  i;

Roca, et al. Standards Track [Page 15] RFC 5170 LDPC Staircase and Triangle FEC June 2008

         txseqToID[IDtoTxseq[randInd]] = randInd;
     }
     return;
 }
 It is then possible, at the sender, to determine the sequence of G
 Encoding Symbol IDs that will be part of the group.
 /*
  * Determine the sequence of ESIs for the packet under construction
  * at a sender.
  * Warning: use only when G > 1.
  * PktIdx (IN):  index of the packet, in
  *               {0..ceil(k/G)+ceil((n-k)/G)} range
  * ESIs[] (OUT): list of ESIs for the packet
  */
 void
 sender_find_ESIs_of_group (int      PktIdx,
                            ESI_t    ESIs[])
 {
     int i;
     if (PktIdx < nbSourcePkts) {
         /* this is a source packet */
         ESIs[0] = PktIdx * G;
         for (i = 1; i < G; i++) {
                 ESIs[i] = (ESIs[0] + i) % k;
         }
     } else {
         /* this is a repair packet */
         for (i = 0; i < G; i++) {
             ESIs[i] =
                 k +
                 txseqToID[(i + (PktIdx - nbSourcePkts) * G)
                           % (n - k)];
         }
     }
     return;
 }
 Similarly, upon receiving an Encoding Symbol Group (i.e., packet), a
 receiver can determine the sequence of G Encoding Symbol IDs from the
 first ESI, esi0, that is contained in the FEC Payload ID.

Roca, et al. Standards Track [Page 16] RFC 5170 LDPC Staircase and Triangle FEC June 2008

 /*
  * Determine the sequence of ESIs for the packet received.
  * Warning: use only when G > 1.
  * esi0 (IN):  : ESI contained in the FEC Payload ID
  * ESIs[] (OUT): list of ESIs for the packet
  */
 void
 receiver_find_ESIs_of_group (ESI_t    esi0,
                              ESI_t    ESIs[])
 {
     int i;
     if (esi0 < k) {
         /* this is a source packet */
         ESIs[0] = esi0;
         for (i = 1; i < G; i++) {
             ESIs[i] = (esi0 + i) % k;
         }
     } else {
         /* this is a repair packet */
         for (i = 0; i < G; i++) {
             ESIs[i] =
                 k +
                 txseqToID[(i + IDtoTxseq[esi0 - k])
                           % (n - k)];
         }
     }
 }

5.7. Pseudo-Random Number Generator

 The FEC Encoding IDs 3 and 4 rely on a pseudo-random number generator
 (PRNG) that must be fully specified, in particular in order to enable
 the receivers and the senders to build the same parity check matrix.
 The Park-Miler "minimal standard" PRNG [PM88] MUST be used.  It
 defines a simple multiplicative congruential algorithm: Ij+1 = A * Ij
 (modulo M), with the following choices: A = 7^^5 = 16807 and M =
 2^^31 - 1 = 2147483647.  A validation criteria of such a PRNG is the
 following: if seed = 1, then the 10,000th value returned MUST be
 equal to 1043618065.
 Several implementations of this PRNG are known and discussed in the
 literature.  An optimized implementation of this algorithm, using
 only 32-bit mathematics, and which does not require any division, can
 be found in [rand31pmc].  It uses the Park and Miller algorithm
 [PM88] with the optimization suggested by D. Carta in [CA90].  The
 history behind this algorithm is detailed in [WI08].  Yet, any other

Roca, et al. Standards Track [Page 17] RFC 5170 LDPC Staircase and Triangle FEC June 2008

 implementation of the PRNG algorithm that matches the above
 validation criteria, like the ones detailed in [PM88], is
 appropriate.
 This PRNG produces, natively, a 31-bit value between 1 and 0x7FFFFFFE
 (2^^31-2) inclusive.  Since it is desired to scale the pseudo-random
 number between 0 and maxv-1 inclusive, one must keep the most
 significant bits of the value returned by the PRNG (the least
 significant bits are known to be less random, and modulo-based
 solutions should be avoided [PTVF92]).  The following algorithm MUST
 be used:
 Input:
    raw_value: random integer generated by the inner PRNG algorithm,
    between 1 and 0x7FFFFFFE (2^^31-2) inclusive.
    maxv: upper bound used during the scaling operation.
 Output:
    scaled_value: random integer between 0 and maxv-1 inclusive.
 Algorithm:
    scaled_value = (unsigned long) ((double)maxv * (double)raw_value /
    (double)0x7FFFFFFF);
    (NB: the above C type casting to unsigned long is equivalent to
    using floor() with positive floating point values.)
 In this document, pmms_rand(maxv) denotes the PRNG function that
 implements the Park-Miller "minimal standard" algorithm, defined
 above, and that scales the raw value between 0 and maxv-1 inclusive,
 using the above scaling algorithm.  Additionally, a function should
 be provided to enable the initialization of the PRNG with a seed
 (i.e., a 31-bit integer between 1 and 0x7FFFFFFE inclusive) before
 calling pmms_rand(maxv) the first time.

Roca, et al. Standards Track [Page 18] RFC 5170 LDPC Staircase and Triangle FEC June 2008

6. Full Specification of the LDPC-Staircase Scheme

6.1. General

 The LDPC-Staircase scheme is identified by the Fully-Specified FEC
 Encoding ID 3.
 The PRNG used by the LDPC-Staircase scheme must be initialized by a
 seed.  This PRNG seed is an instance-specific FEC OTI attribute
 (Section 4.2.3).

6.2. Parity Check Matrix Creation

 The LDPC-Staircase matrix can be divided into two parts: the left
 side of the matrix defines in which equations the source symbols are
 involved; the right side of the matrix defines in which equations the
 repair symbols are involved.
 The left side MUST be generated by using the following function:

/* * Initialize the left side of the parity check matrix. * This function assumes that an empty matrix of size n-k * k has * previously been allocated/reset and that the matrix_has_entry(), * matrix_insert_entry() and degree_of_row() functions can access it. * (IN): the k, n and N1 parameters. */ void left_matrix_init (int k, int n, int N1) {

  int i;      /* row index or temporary variable */
  int j;      /* column index */
  int h;      /* temporary variable */
  int t;      /* left limit within the list of possible choices u[] */
  int u[N1*MAX_K]; /* table used to have a homogeneous 1 distrib. */
  /* Initialize a list of all possible choices in order to
   * guarantee a homogeneous "1" distribution */
  for (h = N1*k-1; h >= 0; h--) {
      u[h] = h % (n-k);
  }

Roca, et al. Standards Track [Page 19] RFC 5170 LDPC Staircase and Triangle FEC June 2008

  /* Initialize the matrix with N1 "1s" per column, homogeneously */
  t = 0;
  for (j = 0; j < k; j++) { /* for each source symbol column */
      for (h = 0; h < N1; h++) { /* add N1 "1s" */
          /* check that valid available choices remain */
          for (i = t; i < N1*k && matrix_has_entry(u[i], j); i++);
          if (i < N1*k) {
              /* choose one index within the list of possible
               * choices */
              do {
                  i = t + pmms_rand(N1*k-t);
              } while (matrix_has_entry(u[i], j));
              matrix_insert_entry(u[i], j);
              /* replace with u[t] which has never been chosen */
              u[i] = u[t];
              t++;
          } else {
              /* no choice left, choose one randomly */
              do {
                  i = pmms_rand(n-k);
              } while (matrix_has_entry(i, j));
              matrix_insert_entry(i, j);
          }
      }
  }
  /* Add extra bits to avoid rows with less than two "1s".
   * This is needed when the code rate is smaller than 2/(2+N1) */
  for (i = 0; i < n-k; i++) { /* for each row */
      if (degree_of_row(i) == 0) {
          j = pmms_rand(k);
          matrix_insert_entry(i, j);
      }
      if (degree_of_row(i) == 1) {
          do {
              j = pmms_rand(k);
          } while (matrix_has_entry(i, j));
          matrix_insert_entry(i, j);
      }
  }

}

Roca, et al. Standards Track [Page 20] RFC 5170 LDPC Staircase and Triangle FEC June 2008

 The right side (the staircase) MUST be generated by using the
 following function:
 /*
  * Initialize the right side of the parity check matrix with a
  * staircase structure.
  * (IN): the k and n parameters.
  */
 void right_matrix_staircase_init (int k, int n)
 {
     int i;      /* row index */
     matrix_insert_entry(0, k);    /* first row */
     for (i = 1; i < n-k; i++) {   /* for the following rows */
         matrix_insert_entry(i, k+i);   /* identity */
         matrix_insert_entry(i, k+i-1); /* staircase */
     }
 }
 Note that just after creating this parity check matrix, when Encoding
 Symbol Groups are used (i.e., G > 1), the function initializing the
 two random permutation tables (Section 5.6) MUST be called.  This is
 true both at a sender and at a receiver.

6.3. Encoding

 Thanks to the staircase matrix, repair symbol creation is
 straightforward: each repair symbol is equal to the sum of all source
 symbols in the associated equation, plus the previous repair symbol
 (except for the first repair symbol).  Therefore, encoding MUST
 follow the natural repair symbol order: start with the first repair
 symbol and generate a repair symbol with ESI i before a symbol with
 ESI i+1.

6.4. Decoding

 Decoding basically consists in solving a system of n-k linear
 equations whose variables are the n source and repair symbols.  Of
 course, the final goal is to recover the value of the k source
 symbols only.
 To that purpose, many techniques are possible.  One of them is the
 following trivial algorithm [ZP74]: given a set of linear equations,
 if one of them has only one remaining unknown variable, then the
 value of this variable is that of the constant term.  So, replace
 this variable by its value in all the remaining linear equations and
 reiterate.  The value of several variables can therefore be found
 recursively.  Applied to LDPC FEC codes working over an erasure

Roca, et al. Standards Track [Page 21] RFC 5170 LDPC Staircase and Triangle FEC June 2008

 channel, the parity check matrix defines a set of linear equations
 whose variables are the source symbols and repair symbols.  Receiving
 or decoding a symbol is equivalent to having the value of a variable.
 Appendix A sketches a possible implementation of this algorithm.
 A Gaussian elimination (or any optimized derivative) is another
 possible decoding technique.  Hybrid solutions that start by using
 the trivial algorithm above and finish with a Gaussian elimination
 are also possible [CR08].
 Because interoperability does not depend on the decoding algorithm
 used, the current document does not recommend any particular
 technique.  This choice is left to the codec developer.
 However, choosing a decoding technique will have great practical
 impacts.  It will impact the erasure capabilities: a Gaussian
 elimination enables to solve the system with a smaller number of
 known symbols compared to the trivial technique.  It will also impact
 the CPU load: a Gaussian elimination requires more processing than
 the above trivial algorithm.  Depending on the target use case, the
 codec developer will favor one feature or the other.

7. Full Specification of the LDPC-Triangle Scheme

7.1. General

 LDPC-Triangle is identified by the Fully-Specified FEC Encoding ID 4.
 The PRNG used by the LDPC-Triangle scheme must be initialized by a
 seed.  This PRNG seed is an instance-specific FEC OTI attribute
 (Section 4.2.3).

7.2. Parity Check Matrix Creation

 The LDPC-Triangle matrix can be divided into two parts: the left side
 of the matrix defines in which equations the source symbols are
 involved; the right side of the matrix defines in which equations the
 repair symbols are involved.
 The left side MUST be generated by using the same left_matrix_init()
 function as with LDPC-Staircase (Section 6.2).

Roca, et al. Standards Track [Page 22] RFC 5170 LDPC Staircase and Triangle FEC June 2008

 The right side (the triangle) MUST be generated by using the
 following function:
 /*
  * Initialize the right side of the parity check matrix with a
  * triangle structure.
  * (IN): the k and n parameters.
  */
 void right_matrix_staircase_init (int k, int n)
 {
     int i;      /* row index */
     int j;      /* randomly chosen column indexes in 0..n-k-2 */
     int l;      /* limitation of the # of "1s" added per row */
     matrix_insert_entry(0, k);    /* first row */
     for (i = 1; i < n-k; i++) {   /* for the following rows */
         matrix_insert_entry(i, k+i);   /* identity */
         matrix_insert_entry(i, k+i-1); /* staircase */
         /* now fill the triangle */
         j = i-1;
         for (l = 0; l < j; l++) { /* limit the # of "1s" added */
             j = pmms_rand(j);
             matrix_insert_entry(i, k+j);
         }
     }
 }
 Note that just after creating this parity check matrix, when Encoding
 Symbol Groups are used (i.e., G > 1), the function initializing the
 two random permutation tables (Section 5.6) MUST be called.  This is
 true both at a sender and at a receiver.

7.3. Encoding

 Here also, repair symbol creation is straightforward: each repair
 symbol of ESI i is equal to the sum of all source and repair symbols
 (with ESI lower than i) in the associated equation.  Therefore,
 encoding MUST follow the natural repair symbol order: start with the
 first repair symbol, and generate repair symbol with ESI i before
 symbol with ESI i+1.

7.4. Decoding

 Decoding basically consists in solving a system of n-k linear
 equations, whose variables are the n source and repair symbols.  Of
 course, the final goal is to recover the value of the k source
 symbols only.  To that purpose, many techniques are possible, as
 explained in Section 6.4.

Roca, et al. Standards Track [Page 23] RFC 5170 LDPC Staircase and Triangle FEC June 2008

 Because interoperability does not depend on the decoding algorithm
 used, the current document does not recommend any particular
 technique.  This choice is left to the codec implementer.

8. Security Considerations

8.1. 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 an FEC building block
 independently of any particular CDP (even if ALC and NORM are two
 natural candidates), this section only discusses the additional
 threats that an arbitrary CDP may be exposed to when using this
 building block.
 More specifically, several kinds of attacks exist:
 o  those that are meant to give access to a confidential content
    (e.g., in case of a non-free content),
 o  those that try to corrupt the object being transmitted (e.g., to
    inject malicious code within an object, or to prevent a receiver
    from using an object), and
 o  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).

8.2. Attacks Against the Data Flow

 First of all, let us consider the attacks against the data flow.

8.2.1. Access to Confidential Objects

 Access control to a confidential 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/ESP is used [RFC4303]).  If confidentiality is a concern,

Roca, et al. Standards Track [Page 24] RFC 5170 LDPC Staircase and Triangle FEC June 2008

 it is RECOMMENDED that one of these solutions be used.  Even if we
 mention these attacks here, they are not related or facilitated by
 the use of FEC.

8.2.2. Content Corruption

 Protection against corruptions (e.g., after sending forged packets)
 is achieved by means of a content integrity verification/sender
 authentication scheme.  This service can be provided at the object
 level, but in that case a receiver has no way to identify which
 symbol(s) is(are) corrupted if the object is detected as corrupted.
 This service can also be provided at the packet level.  In this case,
 after removing all forged packets, the object may be, in some cases,
 recovered.  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 latter has been fully decoded.  Even if
    digital signatures are computationally expensive, this calculation
    occurs only once per object, which is usually acceptable;
 o  at the packet level, each packet can be digitally signed.  A major
    limitation is the high computational and transmission overheads
    that this solution requires (unless perhaps if 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-1
    with a secret key shared by all the group members, senders, and
    receivers.  This technique creates a cryptographically secured
    (thanks to the secret key) digest of a packet that is sent along
    with the packet.  The Group MAC scheme does not create a
    prohibitive processing load or 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 such as a
    pre-check);
 o  at the packet level, Timed Efficient Stream Loss-Tolerant
    Authentication (TESLA) [RFC4082] is an attractive solution that is
    robust to losses, provides a true authentication/integrity

Roca, et al. Standards Track [Page 25] RFC 5170 LDPC Staircase and Triangle FEC June 2008

    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 CDP developer, who knows 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.

8.3. 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.
 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 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) the latter SHOULD be protected, for instance, by
 digitally signing it with [RFC3852].
 The same considerations concerning the key management aspects apply
 here, also.

Roca, et al. Standards Track [Page 26] RFC 5170 LDPC Staircase and Triangle FEC June 2008

9. 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 3 under the
 "ietf:rmt:fec:encoding" name-space to "LDPC Staircase Codes".
 This document assigns the Fully-Specified FEC Encoding ID 4 under the
 "ietf:rmt:fec:encoding" name-space to "LDPC Triangle Codes".

10. Acknowledgments

 Section 5.5 is derived from an earlier document, and we would like to
 thank S. Peltotalo and J. Peltotalo for their contribution.  We would
 also like to thank Pascal Moniot, Laurent Fazio, Mathieu Cunche,
 Aurelien Francillon, Shao Wenjian, Magnus Westerlund, Brian
 Carpenter, Tim Polk, Jari Arkko, Chris Newman, Robin Whittle, and
 Alfred Hoenes for their comments.
 Last but not least, the authors are grateful to Radford M. Neal
 (University of Toronto) whose LDPC software
 (http://www.cs.toronto.edu/~radford/ldpc.software.html) inspired this
 work.

11. References

11.1. Normative References

 [RFC2119]      Bradner, S., "Key words for use in RFCs to Indicate
                Requirement Levels", RFC 2119, BCP 14, March 1997.
 [RFC5052]      Watson, M., Luby, M., and L. Vicisano, "Forward Error
                Correction (FEC) Building Block", RFC 5052,
                August 2007.

11.2. Informative References

 [ZP74]         Zyablov, V. and M. Pinsker, "Decoding Complexity of
                Low-Density Codes for Transmission in a Channel with
                Erasures", Translated from Problemy Peredachi
                Informatsii, Vol.10, No. 1, pp.15-28, January-
                March 1974.

Roca, et al. Standards Track [Page 27] RFC 5170 LDPC Staircase and Triangle FEC June 2008

 [RN04]         Roca, V. and C. Neumann, "Design, Evaluation and
                Comparison of Four Large Block FEC Codecs: LDPC, LDGM,
                LDGM-Staircase and LDGM-Triangle, Plus a Reed-Solomon
                Small Block FEC Codec", INRIA Research Report RR-5225,
                June 2004.
 [NRFF05]       Neumann, C., Roca, V., Francillon, A., and D. Furodet,
                "Impacts of Packet Scheduling and Packet Loss
                Distribution on FEC Performances: Observations and
                Recommendations", ACM CoNEXT'05 Conference, Toulouse,
                France (an extended version is available as INRIA
                Research Report RR-5578), October 2005.
 [CR08]         Cunche, M. and V. Roca, "Improving the Decoding of
                LDPC Codes for the Packet Erasure Channel with a
                Hybrid Zyablov Iterative Decoding/Gaussian Elimination
                Scheme", INRIA Research Report RR-6473, March 2008.
 [LDPC-codec]   Roca, V., Neumann, C., Cunche, M., and J. Laboure,
                "LDPC-Staircase/LDPC-Triangle Codec Reference
                Implementation", INRIA Rhone-Alpes and
                STMicroelectronics,
                <http://planete-bcast.inrialpes.fr/>.
 [MK03]         MacKay, D., "Information Theory, Inference and
                Learning Algorithms", Cambridge University
                Press, ISBN: 0-521-64298-1, 2003.
 [PM88]         Park, S. and K. Miller, "Random Number Generators:
                Good Ones are Hard to Find", Communications of the
                ACM, Vol. 31, No. 10, pp.1192-1201, 1988.
 [CA90]         Carta, D., "Two Fast Implementations of the Minimal
                Standard Random Number Generator", Communications of
                the ACM, Vol. 33, No. 1, pp.87-88, January 1990.
 [WI08]         Whittle, R., "Park-Miller-Carta Pseudo-Random Number
                Generator", January 2008,
                <http://www.firstpr.com.au/dsp/rand31/>.
 [rand31pmc]    Whittle, R., "31 bit pseudo-random number generator",
                September 2005, <http://www.firstpr.com.au/dsp/rand31/
                rand31-park-miller-carta.cc.txt>.
 [PTVF92]       Press, W., Teukolsky, S., Vetterling, W., and B.
                Flannery, "Numerical Recipes in C; Second Edition",
                Cambridge University Press, ISBN: 0-521-43108-5, 1992.

Roca, et al. Standards Track [Page 28] RFC 5170 LDPC Staircase and Triangle FEC June 2008

 [RMT-PI-ALC]   Luby, M., Watson, M., and L. Vicisano, "Asynchronous
                Layered Coding (ALC) Protocol Instantiation", Work
                in Progress, November 2007.
 [RMT-PI-NORM]  Adamson, B., Bormann, C., Handley, M., and J. Macker,
                "Negative-acknowledgment (NACK)-Oriented Reliable
                Multicast (NORM) Protocol", Work in Progress,
                January 2008.
 [RMT-FLUTE]    Paila, T., Walsh, R., Luby, M., Lehtonen, R., and V.
                Roca, "FLUTE - File Delivery over Unidirectional
                Transport", Work in Progress, October 2007.
 [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, D., Reagle, J., and D. Solo, "(Extensible
                Markup Language) XML-Signature Syntax and Processing",
                RFC 3275, March 2002.
 [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.
 [RFC3852]      Housley, R., "Cryptographic Message Syntax (CMS)",
                RFC 3852, July 2004.
 [RFC4648]      Josefsson, S., "The Base16, Base32, and Base64 Data
                Encodings", RFC 4648, October 2006.

Roca, et al. Standards Track [Page 29] RFC 5170 LDPC Staircase and Triangle FEC June 2008

Appendix A. Trivial Decoding Algorithm (Informative Only)

 A trivial decoding algorithm is sketched below (please see
 [LDPC-codec] for the details omitted here):
 Initialization: allocate a table partial_sum[n-k] of buffers, each
                 buffer being of size the symbol size.  There's one
                 entry per equation since the buffers are meant to
                 store the partial sum of each equation; Reset all
                 the buffers to zero;
 /*
  * For each newly received or decoded symbol, try to make progress
  * in the decoding of the associated source block.
  * NB: in case of a symbol group (G>1), this function is called for
  * each symbol of the received packet.
  * NB: a callback function indicates to the caller that new symbol(s)
  *     has(have) been decoded.
  * new_esi  (IN):  ESI of the new symbol received or decoded
  * new_symb (IN):  Buffer of the new symbol received or decoded
  */
 void
 decoding_step(ESI_t     new_esi,
               symbol_t  *new_symb)
 {
     If (new_symb is an already decoded or received symbol) {
         Return;        /* don't waste time with this symbol */
     }
     If (new_symb is the last missing source symbol) {
         Remember that decoding is finished;
         Return;        /* work is over now... */
     }
     Create an empty list of equations having symbols decoded
     during this decoding step;
     /*
      * First add this new symbol to the partial sum of all the
      * equations where the symbol appears.
      */
     For (each equation eq in which new_symb is a variable and
          having more than one unknown variable) {
         Add new_symb to partial_sum[eq];
         Remove entry(eq, new_esi) from the H matrix;

Roca, et al. Standards Track [Page 30] RFC 5170 LDPC Staircase and Triangle FEC June 2008

         If (the new degree of equation eq == 1) {
             /* a new symbol can be decoded, remember the
              * equation */
             Append eq to the list of equations having symbols
             decoded during this decoding step;
         }
     }
     /*
      * Then finish with recursive calls to decoding_step() for each
      * newly decoded symbol.
      */
     For (each equation eq in the list of equations having symbols
          decoded during this decoding step) {
         /*
          * Because of the recursion below, we need to check that
          * decoding is not finished, and that the equation is
          * __still__ of degree 1
          */
         If (decoding is finished) {
             break;        /* exit from the loop */
         }
         If ((degree of equation eq == 1) {
             Let dec_esi be the ESI of the newly decoded symbol in
             equation eq;
             Remove entry(eq, dec_esi);
             Allocate a buffer, dec_symb, for this symbol and
             copy partial_sum[eq] to dec_symb;
             Inform the caller that a new symbol has been
             decoded via a callback function;
             /* finally, call this function recursively */
             decoding_step(dec_esi, dec_symb);
         }
     }
     Free the list of equations having symbols decoded;
     Return;
 }

Roca, et al. Standards Track [Page 31] RFC 5170 LDPC Staircase and Triangle FEC June 2008

Authors' Addresses

 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/
 Christoph Neumann
 Thomson
 12, bd de Metz
 Rennes  35700
 France
 EMail: christoph.neumann@thomson.net
 URI:   http://planete.inrialpes.fr/people/chneuman/
 David Furodet
 STMicroelectronics
 12, Rue Jules Horowitz
 BP217
 Grenoble Cedex  38019
 France
 EMail: david.furodet@st.com
 URI:   http://www.st.com/

Roca, et al. Standards Track [Page 32] RFC 5170 LDPC Staircase and Triangle FEC June 2008

Full Copyright Statement

 Copyright (C) The IETF Trust (2008).
 This document is subject to the rights, licenses and restrictions
 contained in BCP 78, and except as set forth therein, the authors
 retain all their rights.
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 might or might not be available; nor does it represent that it has
 made any independent effort to identify any such rights.  Information
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Roca, et al. Standards Track [Page 33]

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