    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.

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    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

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    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

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          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.

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          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

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    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:

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             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.

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       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.

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    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

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       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].

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       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.

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    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.

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         +------------------------+----+-------------+-------------------+
         |    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.)

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    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.

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          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;

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               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.

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       /*
        * 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

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       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]

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    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);
        }

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        /* 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);
            }
        }
    }

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       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

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       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]

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       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]

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       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,

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       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

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          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]

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    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;

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               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
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    Roca, et al.                Standards Track                    [Page 33]