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Versions: 00 01 02 03 04 05 06 07 08 RFC 5170

RMT                                                              V. Roca
Internet-Draft                                                     INRIA
Intended status: Standards Track                              C. Neumann
Expires: May 19, 2008                                            Thomson
                                                              D. Furodet
                                                      STMicroelectronics
                                                       November 16, 2007


  Low Density Parity Check (LDPC) Staircase and Triangle Forward Error
                        Correction (FEC) Schemes
                   draft-ietf-rmt-bb-fec-ldpc-07.txt

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Copyright Notice

   Copyright (C) The IETF Trust (2007).










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Abstract

   This document describes two Fully-Specified FEC Schemes, 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''
   (LDPC) codes, and are large block FEC codes in the sense of RFC3453.


Table of Contents

   1.  Introduction . . . . . . . . . . . . . . . . . . . . . . . . .  4
   2.  Requirements notation  . . . . . . . . . . . . . . . . . . . .  5
   3.  Definitions, Notations and Abbreviations . . . . . . . . . . .  6
     3.1.  Definitions  . . . . . . . . . . . . . . . . . . . . . . .  6
     3.2.  Notations  . . . . . . . . . . . . . . . . . . . . . . . .  6
     3.3.  Abbreviations  . . . . . . . . . . . . . . . . . . . . . .  7
   4.  Formats and Codes  . . . . . . . . . . . . . . . . . . . . . .  8
     4.1.  FEC Payload IDs  . . . . . . . . . . . . . . . . . . . . .  8
     4.2.  FEC Object Transmission Information  . . . . . . . . . . .  8
       4.2.1.  Mandatory Element  . . . . . . . . . . . . . . . . . .  8
       4.2.2.  Common Elements  . . . . . . . . . . . . . . . . . . .  8
       4.2.3.  Scheme-Specific Elements . . . . . . . . . . . . . . .  9
       4.2.4.  Encoding Format  . . . . . . . . . . . . . . . . . . .  9
   5.  Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . 12
     5.1.  General  . . . . . . . . . . . . . . . . . . . . . . . . . 12
     5.2.  Determining the Maximum Source Block Length (B)  . . . . . 13
     5.3.  Determining the Encoding Symbol Length (E) and Number
           of Encoding Symbols per Group (G)  . . . . . . . . . . . . 14
     5.4.  Determining the  Maximum Number of Encoding Symbols
           Generated for Any Source Block (max_n) . . . . . . . . . . 15
     5.5.  Determining the Number of Encoding Symbols of a Block
           (n)  . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
     5.6.  Identifying the G Symbols of an Encoding Symbol Group  . . 16
     5.7.  Pseudo Random Number Generator . . . . . . . . . . . . . . 20
   6.  Full Specification of the LDPC-Staircase Scheme  . . . . . . . 22
     6.1.  General  . . . . . . . . . . . . . . . . . . . . . . . . . 22
     6.2.  Parity Check Matrix Creation . . . . . . . . . . . . . . . 22
     6.3.  Encoding . . . . . . . . . . . . . . . . . . . . . . . . . 24
     6.4.  Decoding . . . . . . . . . . . . . . . . . . . . . . . . . 24
   7.  Full Specification of the LDPC-Triangle Scheme . . . . . . . . 26
     7.1.  General  . . . . . . . . . . . . . . . . . . . . . . . . . 26
     7.2.  Parity Check Matrix Creation . . . . . . . . . . . . . . . 26
     7.3.  Encoding . . . . . . . . . . . . . . . . . . . . . . . . . 26
     7.4.  Decoding . . . . . . . . . . . . . . . . . . . . . . . . . 27
   8.  Security Considerations  . . . . . . . . . . . . . . . . . . . 28



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     8.1.  Problem Statement  . . . . . . . . . . . . . . . . . . . . 28
     8.2.  Attacks Against the Data Flow  . . . . . . . . . . . . . . 28
       8.2.1.  Access to Confidential Objects . . . . . . . . . . . . 28
       8.2.2.  Content Corruption . . . . . . . . . . . . . . . . . . 29
     8.3.  Attacks Against the FEC Parameters . . . . . . . . . . . . 30
   9.  IANA Considerations  . . . . . . . . . . . . . . . . . . . . . 31
   10. Acknowledgments  . . . . . . . . . . . . . . . . . . . . . . . 32
   11. References . . . . . . . . . . . . . . . . . . . . . . . . . . 33
     11.1. Normative References . . . . . . . . . . . . . . . . . . . 33
     11.2. Informative References . . . . . . . . . . . . . . . . . . 33
   Appendix A.  Pseudo Random Number Generator Example
                Implementation (Informative Only) . . . . . . . . . . 35
   Appendix B.  Trivial Decoding Algorithm (Informative Only) . . . . 37
   Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . . 39
   Intellectual Property and Copyright Statements . . . . . . . . . . 40




































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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 of size several tens of thousands (or more) 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 [RFC5052], section 4.

   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), that 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 license [LDPC-codec].
   Besides, the code extracts included in this document (except
   Appendix A that is only provided as an example) are directly
   contributed to the IETF process by the authors of this document and
   by Radford M. Neal.



















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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].














































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3.  Definitions, Notations and Abbreviations

3.1.  Definitions

   This document uses the same terms and definitions as those specified
   in [RFC5052].  Additionally, it uses the following definitions:

      Source symbol: unit of data used during the encoding process

      Encoding symbol: unit of data generated by the encoding process

      Repair symbol: encoding symbol that is not a source symbol

      Code rate: the k/n ratio, i.e., the ratio between the number of
      source symbols and the number of encoding symbols.  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




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      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

      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

      srand(s) denotes the initialization function of the pseudo-random
      number generator, where s is the seed (s > 0)

      rand(m) denotes a pseudo-random number generator 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 are 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 are 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 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 respectively the FEC Encoding ID 3
      (Staircase) and 4 (Triangle).

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  G: a non-negative integer 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.

   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 ALC
   [draft-ietf-rmt-pi-alc-revised] or NORM



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   [draft-ietf-rmt-pi-norm-revised] 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)  |       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.

4.2.4.2.  Using the FDT Instance (FLUTE specific)

   When it is desired that the FEC OTI be carried in the FDT Instance of
   a FLUTE session [draft-ietf-rmt-flute-revised], 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



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   the Base64 encoding (in the XML Schema xs:base64Binary sense) 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                              |
   +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
   |       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.

































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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
   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 of 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



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   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 the shuffle these symbols.  The
   interested reader will find more details in [NRFF05].

   The following sections detail how the B, E, G, max_nand n parameters
   are determined (respectively in Section 5.2, Section 5.3, Section 5.4
   and Section 5.5), how encoding symbol groups are created
   (Section 5.6), and finally Section 5.7 details 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



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   parameter is communicated to the decoder through the FEC OTI.

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 LCT/UDP/IPv4 or IPv6
   headers in case of ALC), E is chosen 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 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 done 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 case of a source
      packet, the G Encoding Symbol IDs are chosen consecutively, by
      incrementing the ESI.  In 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, no call to the PRNG
   function must have happened 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.



































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   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));
       /* 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 = rand(n - k);
           backup  = IDtoTxseq[i];
           IDtoTxseq[i] = IDtoTxseq[randInd];
           IDtoTxseq[randInd] = backup;
           txseqToID[IDtoTxseq[i]] =  i;
           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.















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   /*
    * 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 minimal standard generator [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.  Several implementations of this PRNG are known and
   discussed in the literature.  All of them provide the same sequence
   of pseudo random numbers.  A validation criteria of such a PRNG is
   the following: if seed = 1, then the 10,000th value returned MUST be
   equal to 1043618065.

   An optimized implementation of this algorithm, using only 32 bit
   mathematics which does not require any division, is provided, as an
   example, in Appendix A.  Yet any other implementation of the PRNG
   algorithm that matches the above validation criteria is appropriate.



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   This PRNG produces a 31 bit value between 1 and 0x7FFFFFFE (2^^31-2)
   inclusive.  When 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)


























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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 is generated with the following algorithm:
































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      /* initialize a list of all possible choices in order to
       * guarantee a homogeneous "1" distribution */
      for (h = 3*k-1; h >= 0; h--) {
          u[h] = h % (n-k);
      }
      /* left limit within the list of possible choices, u[] */
      t = 0;

      for (j = 0; j < k; j++) { /* for each source symbol column */
          for (h = 0; h < 3; h++) { /* add 3 "1s" */
              /* check that valid available choices remain */
              for (i = t; i < 3*k && matrix_has_entry(u[i], j); i++);

              if (i < 3*k) {
                  /* choose one index within the list of possible
                   * choices */
                  do {
                      i = t + rand(3*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 = 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 2/5. */
      for (i = 0; i < n-k; i++) { /* for each row */
          if (degree_of_row(i) == 0) {
              j = rand(k);
              matrix_insert_entry(i, j);
          }
          if (degree_of_row(i) == 1) {
              do {
                  j = rand(k);
              } while (matrix_has_entry(i, j));
              matrix_insert_entry(i, j);
          }
      }



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   The right side (the staircase) is generated by the following
   algorithm:

      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 repair symbol with ESI i before 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
   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 B 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.

   Because interoperability does not depend on the decoding algorithm



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   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.









































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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 is generated with the same algorithm as that of LDPC-
   Staircase (Section 6.2).

   The right side (the triangle) is generated with the following
   algorithm:

      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 = 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.



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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.

   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.








































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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 a FEC building block
   independently of any particular CDP (even if ALC and NORM are two
   natural candidates), this section only discusses the additional
   threats that an arbitrary CDP may be exposed to when using this
   building block.

   More specifically, several kinds of attacks exist:

   o  those that are meant to give access to a confidential content
      (e.g., in case of a non-free content),

   o  those that try to corrupt the object being transmitted (e.g., to
      inject malicious code within an object, or to prevent a receiver
      from using an object),

   o  and those that try to compromise the receiver's behavior (e.g., by
      making the decoding of an object computationally expensive).

   These attacks can be launched either against the data flow itself
   (e.g., by sending forged symbols) or against the FEC parameters that
   are sent either in-band (e.g., in an EXT_FTI or FDT Instance) or out-
   of-band (e.g., in a session description).

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 the object being transmitted is typically provided
   by means of encryption.  This encryption can be done over the whole
   object (e.g., by the content provider, before the FEC encoding
   process), or be done on a packet per packet basis (e.g., when IPSec/
   ESP is used [RFC4303]).  If access control is a concern, it is
   RECOMMENDED that one of these solutions be used.  Even if we mention
   these attacks here, they are not related nor facilitated by the use
   of FEC.





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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 case
   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 this latter has been fully decoded.  Even if
      digital signatures are computationally expensive, this calculation
      occurs only once per object, which is usually acceptable;

   o  at the packet level, each packet can be digitally signed.  A major
      limitation is the high computational and transmission overheads
      that this solution requires (unless Elliptic Curve Cryptography
      (ECC) is used).  To avoid this problem, the signature may span a
      set of symbols (instead of a single one) in order to amortize the
      signature calculation.  But if a single symbol is missing, the
      integrity of the whole set cannot be checked;

   o  at the packet level, a Group Message Authentication Code (MAC)
      [RFC2104] scheme can be used, for instance by using HMAC-SHA-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 prohibitive
      processing load nor transmission overhead, but it has a major
      limitation: it only provides a group authentication/integrity
      service since all group members share the same secret group key,
      which means that each member can send a forged packet.  It is
      therefore restricted to situations where group members are fully
      trusted (or in association with another technique as a pre-check);

   o  at the packet level, TESLA [RFC4082] is a very attractive and
      efficient solution that is robust to losses, provides a true
      authentication/integrity service, and does not create any
      prohibitive processing load or transmission overhead.  Yet
      checking a packet requires a small delay (a second or more) after
      its reception;

   Techniques relying on public key cryptography (digital signatures and



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   TESLA during the bootstrap process, when used) require that public
   keys be securely associated to the entities.  This can be achieved by
   a Public Key Infrastructure (PKI), or by a PGP Web of Trust, or by
   pre-distributing the public keys of each group member.

   Techniques relying on symmetric key cryptography (group MAC) require
   that a secret key be shared by all group members.  This can be
   achieved by means of a group key management protocol, or simply by
   pre-distributing the secret key (but this manual solution has many
   limitations).

   It is up to the developer and deployer, who know the security
   requirements and features of the target application area, to define
   which solution is the most appropriate.  Nonetheless, in case there
   is any concern of the threat of object corruption, it is RECOMMENDED
   that at least one of these techniques be used.

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) this latter SHOULD be protected, for instance by
   digitally signing it.

   The same considerations concerning the key management aspects apply
   here also.












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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".








































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10.  Acknowledgments

   Section 5.5 is derived from a previous Internet-Draft, and we would
   like to thank S. Peltotalo and J. Peltotalo for their contribution.
   We would also like to thank Pascal Moniot, Laurent Fazio, Aurelien
   Francillon, Shao Wenjian, Brian Carpenter, Magnus Westerlund, 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.







































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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.

   [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.

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.

   [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.

   [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.




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   [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.

   [PTVF92]   Press, W., Teukolsky, S., Vetterling, W., and B. Flannery,
              "Numerical Recipies in C; Second Edition", Cambridge
              University Press, ISBN: 0-521-43108-5, 1992.

   [draft-ietf-rmt-pi-alc-revised]
              Luby, M., Watson, M., and L. Vicisano, "Asynchronous
              Layered Coding (ALC) Protocol Instantiation",
               draft-ietf-rmt-pi-alc-revised-04.txt (work in progress),
              February 2007.

   [draft-ietf-rmt-pi-norm-revised]
              Adamson, B., Bormann, C., Handley, M., and J. Macker,
              "Negative-acknowledgment (NACK)-Oriented Reliable
              Multicast (NORM) Protocol",
               draft-ietf-rmt-pi-norm-revised-05.txt (work in progress),
              March 2007.

   [draft-ietf-rmt-flute-revised]
              Paila, T., Walsh, R., Luby, M., Lehtonen, R., and V. Roca,
              "FLUTE - File Delivery over Unidirectional Transport",
               draft-ietf-rmt-flute-revised-05.txt (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.







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Appendix A.  Pseudo Random Number Generator Example Implementation
             (Informative Only)

   The following is an implementation of the minimal standard generator
   defined in Section 5.7 that scales the result between 0 and maxv-1
   inclusive.  It uses the Park and Miller algorithm [PM88] with the
   optimization suggested by D. Carta in [CA90].  The inner algorithm
   relies on 32 bit mathematics only and does not require any division.











































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   unsigned long           seed;

   /*
    * Initialize the PRNG with a seed between
    * 1 and 0x7FFFFFFE (i.e., 2^^31-2) inclusive.
    */
   void srand (unsigned long s)
   {
           if ((s > 0) && (s < 0x7FFFFFFF))
                   seed = s;
           else
                   exit(-1);
   }

   /*
    * Returns a random integer in [0; maxv-1]
    * Derived from rand31pmc, Robin Whittle,
    * September 20th, 2005.
    * http://www.firstpr.com.au/dsp/rand31/
    *      16807           multiplier constant (7^^5)
    *      0x7FFFFFFF      modulo constant (2^^31-1)
    * The inner PRNG produces a value between 1 and
    * 0x7FFFFFFE (2^^31-2) inclusive.
    * This value is then scaled between 0 and maxv-1
    * inclusive.
    */
   unsigned long
   rand (unsigned long maxv)
   {
           unsigned long   hi, lo;

           lo = 16807 * (seed & 0xFFFF);
           hi = 16807 * (seed >> 16);  /* binary shift to right */
           lo += (hi & 0x7FFF) << 16;  /* binary shift to left */
           lo += hi >> 15;
           if (lo > 0x7FFFFFFF)
                   lo -= 0x7FFFFFFF;
           seed = lo;
           /* don't use modulo, least significant bits are less random
            * than most significant bits [PTVF92] */
           return ((unsigned long)
                   ((double)maxv * (double)seed / (double)0x7FFFFFFF));
   }








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Appendix B.  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;
   }







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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/



















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Full Copyright Statement

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   This document is subject to the rights, licenses and restrictions
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