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Internet Engineering Task Force                                 RMT WG
INTERNET-DRAFT                                  M. Luby/Digital Fountain
draft-ietf-rmt-info-fec-01.txt                   L. Vicisano/Cisco
                                                    J. Gemmell/Microsoft
                                           L. Rizzo/ACIRI and Univ. Pisa
                                                        M. Handley/ACIRI
                                                        J. Crowcroft/UCL
                                                         18 October 2001
                                                     Expires: April 2002


       The use of Forward Error Correction in Reliable Multicast


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                                Abstract


     This memo describes the use of Forward Error Correction (FEC)
     codes within the context of reliable IP multicast transport
     and provides an introduction to some commonly-used FEC codes.






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1.  Rationale and Overview

There are many ways to provide reliability for transmission protocols.
A common method is to use ARQ, automatic request for retransmission.
With ARQ, receivers use a back channel to the sender to send requests
for retransmission of lost packets.  ARQ works well for one-to-one
reliable protocols, as evidenced by the pervasive success of TCP/IP.
ARQ has also been an effective reliability tool for one-to-many
reliability protocols, and in particular for some reliable IP multicast
protocols.  However, for one-to-very-many reliability protocols, ARQ has
limitations, including the feedback implosion problem because many
receivers are transmitting back to the sender, and the need for a back
channel to send these requests from the receiver.  Another limitation is
that receivers may experience different loss patterns of packets, and
thus receivers may be delayed by retransmission of packets that other
receivers have lost that but they have already received.  This may also
cause wasteful use of bandwidth used to retransmit packets that have
already been received by many of the receivers.

In environments where ARQ is either costly or impossible because there
is either a very limited capacity back channel or no back channel at
all, such as satellite transmission, a Data Carousel approach to
reliability is sometimes used [1]. With a Data Carousel, the sender
partitions the object into equal length pieces of data, which we
hereafter call source symbols, places them into packets, and then
continually cycles through and sends these packets. Receivers
continually receive packets until they have received a copy of each
packet. Data Carousel has the advantage that it requires no back
channel because there is no data that flows from receivers to the
sender. However, Data Carousel also has limitations. For example, if a
receiver loses a packet in one round of transmission it must wait an
entire round before it has a chance to receive that packet again.  This
may also cause wasteful use of bandwidth, as the sender continually
cycles through and transmits the packets until no receiver is missing a
packet.

FEC codes provide a reliability method that can be used to augment or
replace other reliability methods, especially for one-to-many
reliability protocols such as reliable IP multicast.  We first briefly
review some of the basic properties and types of FEC codes before
reviewing their uses in the context of reliable IP multicast.  Later, we
provide a more detailed description of some of FEC codes.

In the general literature, FEC refers to the ability to overcome both
erasures (losses) and bit-level corruption. However, in the case of IP
multicast, lower network layers will detect corrupted packets and
discard them. Therefore, an IP multicast protocol need not be concerned
with corruption; the focus is solely on erasure codes.  The payloads are



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generated and processed using an FEC erasure encoder and objects are
reassembled from reception of packets using the corresponding FEC
erasure decoder.

The input to an FEC encoder is some number k of equal length source
symbols.  The FEC encoder generates some number of encoding symbols that
are of the same length as the source symbols.  The chosen length of the
symbols can vary upon each application of the FEC encoder, or it can be
fixed.  These encoding symbols are placed into packets for transmission.
The number of encoding symbols placed into each packet can vary on a per
packet basis, or a fixed number of symbols (often one) can be placed
into each packet.  Also, in each packet is placed enough information to
identify the particular encoding symbols carried in that packet.  Upon
receipt of packets containing encoding symbols, the receiver feeds these
encoding symbols into the corresponding FEC decoder to recreate an exact
copy of the k source symbols.  Ideally, the FEC decoder can recreate an
exact copy from any k of the encoding symbols.

In a later section, we describe a technique for using FEC codes as
described above to handle blocks with variable length source symbols.

Block FEC codes work as follows.  The input to a block FEC encoder is k
source symbols and a number n.  The encoder generates a total of n
encoding symbols.  The encoder is systematic if it generates n-k
redundant symbols yielding an encoding block of n encoding symbols in
total composed of the k source symbols and the n-k redundant symbols.  A
block FEC decoder has the property that any k of the n encoding symbols
in the encoding block is sufficient to reconstruct the original k source
symbols.

Expandable FEC codes work as follows.  An expandable FEC encoder takes
as input k source symbols and generates as many unique encoding symbols
as requested on demand, where the amount of time for generating each
encoding symbol is the same independent of how many encoding symbols are
generated.  An expandable FEC decoder has the property that any k of the
unique encoding symbols is sufficient to reconstruct the original k
source symbols.

The above definitions explain the ideal situation when the reception of
any k encoding symbols is sufficient to recover the k source symbols, in
which case the reception overhead is 0%.  For some practical FEC codes,
slightly more than k encoding symbols are needed to recover the k source
symbols.  If k*(1+ep) encoding symbols are needed, we say the reception
overhead is ep*100%, e.g., if k*1.05 encoding symbols are needed then
the reception overhead is 5%.

Along a different dimension, we classify FEC codes loosely as being
either small or large.  A small FEC code is efficient in terms of



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processing time requirements for encoding and decoding for small values
of k, and a large FEC code is efficient for encoding and decoding for
much large values of k. There are implementations of block FEC codes
that have encoding times proportional to n times the length of the k
source symbols, and decoding times proportional l times the length of
the k source symbols, where l is the number of missing source symbols
among the k received encoding symbols.  Because of the growth rate of
the encoding and decoding times as a product of k and n, these are
typically considered to be small block FEC codes.  There are block FEC
codes with a small reception overhead that can generate n encoding
symbols and can decode the k source symbols in time proportional to the
length of the n encoding symbols.  These codes are considered to be
large block FEC codes.  There are expandable FEC codes with a small
reception overhead that can generate each encoding symbol in time
roughly proportional to its length, and can decode all k source symbols
in time roughly proportional to their length.  These are considered to
be large expandable FEC codes.

Ideally, FEC codes in the context of IP multicast can be used to
generate encoding symbols that are transmitted in packets in such a way
that each received packet is fully useful to a receiver to reassemble
the object regardless of previous packet reception patterns. Thus, if
some packets are lost in transit between the sender and the receiver,
instead of asking for specific retransmission of the lost packets or
waiting till the packets are resent using Data Carousel, the receiver
can use any other subsequent equal number of packets that arrive to
reassemble the object.  These packets can either be proactively
transmitted or they can be explicitly requested by receivers.  This
implies that the same packet is fully useful to all receivers to
reassemble the object, even though the receivers may have previously
experienced different packet loss patterns.  This property can reduce or
even eliminate the problems mentioned above associated with ARQ and Data
Carousel and thereby dramatically increase the scalability of the
protocol to orders of magnitude more receivers.


1.1.  Application of FEC codecs

For some reliable IP multicast protocols, FEC codes are used in
conjunction with ARQ to provide reliability.  For example, a large
object could be partitioned into a number of source blocks consisting of
a small number of source symbols each, and in a first round of
transmission all of the source symbols for all the source blocks could
be transmitted. Then, receivers could report back to the sender the
number of source symbols they are missing from each source block.  The
sender could then compute the maximum number of missing source symbols
from each source block among all receivers.  Based on this, a small
block FEC encoder could be used to generate for each source block a



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number of redundant symbols equal to the computed maximum number of
missing source symbols from the block among all receivers, as long as
this maximum maximum for each block does not exceed the number of
redundant symbols that can be generated efficiently.  In a second round
of transmission, the server would then send all of these redundant
symbols for all blocks. In this example, if there are no losses in the
second round of transmission then all receivers will be able to recreate
an exact copy of each original block.  In this case, even if different
receivers are missing different symbols in different blocks, transmitted
redundant symbols for a given block are useful to all receivers missing
symbols from that block in the second round.

For other reliable IP multicast protocols, FEC codes are used in a Data
Carousel fashion to provide reliability, which we call an FEC Data
Carousel.  For example, an FEC Data Carousel using a large block FEC
code could work as follows.  The large block FEC encoder produces n
encoding symbols considering all the k source symbols of an object as
one block. The sender cycles through and transmits the n encoding
symbols in packets in the same order in each round.  An FEC Data
Carousel can have much better protection against packet loss than a Data
Carousel.  For example, a receiver can join the transmission at any
point in time, and, as long as the receiver receives at least k encoding
symbols during the transmission of the next n encoding symbols, the
receiver can completely recover the object.  This is true even if the
receiver starts receiving packets in the middle of a pass through the
encoding symbols.  This method can also be used when the object is
partitioned into blocks and a short block FEC code is applied to each
block separately.  In this case, as we explain in more detail below, it
is useful to interleave the symbols from the different blocks when they
are transmitted.

Since any number of encoding symbols can be generated using an
expandable FEC encoder, reliable IP multicast protocols that use
expandable FEC codes generally rely solely on these codes for
reliability.  For example, when an expandable FEC code is used in a FEC
Data Carousel application, the encoding packets never repeat, and thus
any k of the encoding symbols in the potentially unbounded number of
encoding symbols are sufficient to recover the original k source
symbols.

For yet other reliable IP multicast protocols the method to obtain
reliability is to generate enough encoding symbols so that each encoding
symbol is transmitted at most once.  For example, the sender can decide
a priori how many encoding symbols it will transmit, use an FEC code to
generate that number of encoding symbols from the object, and then
transmit the encoding symbols to all receivers. This method is for
example applicable to streaming protocols, where the stream is
partitioned into objects, the source symbols for each object are encoded



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into encoding symbols using an FEC code, and then the sets of encoding
symbols for each object are transmitted one after the other using IP
multicast.


2.  FEC Codes


2.1.  Simple codes

There are some very simple codes that are effective for repairing packet
loss under very low loss conditions.  For example, one simple way to
provide protection from a single loss is to partition the object into
fixed size source symbols and then add a redundant symbol that is the
parity (XOR) of all the source symbols. The size of a source symbol is
chosen so that it fits perfectly into the payload of a packet, i.e. if
the packet payload is 512 bytes then each source symbol is 512 bytes.
The header of each packet contains enough information to identify the
payload.  In this case, this is an encoding symbol ID.  The encoding
symbol IDs can consist of two parts in this example.  The first part is
an encoding flag that is equal to 1 if the encoding symbol is a source
symbol and is equal to 0 if the encoding symbol is a redundant symbol.
The second part of the encoding symbol ID is a source symbol ID if the
encoding flag is 1 and a redundant symbol ID if the encoding flag is 0.
The source symbol IDs can be numbered from 0 to k-1 and the redundant
symbol ID can be 0.  For example, if the object consists of four source
symbols that have values a, b, c and d, then the value of the redundant
symbol is e = a XOR b XOR c XOR d.  Then, the packets carrying these
symbols look like
         (1, 0: a), (1, 1: b), (1, 2: c), (1, 3: d), (0, 0: e).

In this example, the encoding symbol ID consists of the first two
values, where the first value is the encoding flag and the second value
is either a source symbol ID or the redundant symbol ID. The portion of
the packet after the colon is the value of the encoding symbol. Any
single source symbol of the object can be recovered as the parity of all
the other symbols.  For example, if packets
               (1, 0: a), (1, 1: b), (1, 3: d), (0, 0: e)

are received then the missing source symbol value with source symbol ID
= 2 can be recovered by computing a XOR b XOR d XOR e = c.

Another way of forming the encoding symbol ID is to let values 0,...,k-1
correspond to the k source symbols and value k corresponds to the
redundant symbol that is the XOR of the k source symbols.

When the number of source symbols in the object is large, a simple block
code variant of the above can be used.  In this case, the source symbols



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are grouped together into source blocks of some number k of consecutive
symbols each, where k may be different for different blocks.  If a block
consists of k source symbols then a redundant symbol is added to form an
encoding block consisting of k+1 encoding symbols.  Then, a source block
consisting of k source symbols can be recovered from any k of the k+1
encoding symbols from the associated encoding block.

Slightly more sophisticated ways of adding redundant symbols using
parity can also be used. For example, one can group a block consisting
of k source symbols in an object into a p x p square matrix, where p =
sqrt(k).  Then, for each row a redundant symbol is added that is the
parity of all the source symbols in the row.  Similarly, for each column
a redundant symbol is added that is the parity of all the source symbols
in the column.  Then, any row of the matrix can be recovered from any p
of the p+1 symbols in the row, and similarly for any column.  Higher
dimensional product codes using this technique can also be used.
However, one must be wary of using these constructions without some
thought towards the possible loss patterns of symbols.  Ideally, the
property that one would like to obtain is that if k source symbols are
encoded into n encoding symbols (the encoding symbols consist of the
source symbols and the redundant symbols) then the k source symbols can
be recovered from any k of the n encoding symbols.  Using the simple
constructions described above does not yield codes that come close to
obtaining this ideal behavior.


2.2.  Small block FEC codes

Reliable IP multicast protocols may use a block (n, k) FEC code [2]. For
such codes, k source symbols are encoded into n > k encoding symbols,
such that any k of the encoding symbols can be used to reassemble the
original k source symbols.  Thus, these codes have no reception overhead
when used to encode the entire object directly. Block codes are usually
systematic, which means that the n encoding symbols consist of the k
source symbols and n-k redundant symbols generated from these k source
symbols, where the size of a redundant symbol is the same as that for a
source symbol.  For example, the first simple code (XOR) described in
the previous subsection is a (k+1, k) code.  In general, the freedom to
choose n larger than k+1 is desirable, as this can provide much better
protection against losses.  A popular example of these types of codes is
a class of Reed-Solomon codes, which are based on algebraic methods
using finite fields.  Implementations of (n, k) FEC erasure codes are
efficient enough to be used by personal computers [8]. For example, [7]
describes an implementation where the encoding and decoding speeds decay
as C/j, where the constant C is on the order of 10 to 80 Mbytes/second
for Pentium class machines of various vintages and j is upper bounded by
min(k, n-k).




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In practice, the values of k and n must be small (for example below 256)
for such FEC codes as large values make encoding and decoding
prohibitively expensive.  As many objects are longer than k symbols for
reasonable values of k and the symbol length (e.g. larger than 16
kilobyte for k = 16 using 1 kilobyte symbols), they can be divided into
a number of source blocks.  Each source block consists of some number k
of source symbols, where k may vary between different source blocks.
The FEC encoder is used to encode a k source symbol source block into a
n encoding symbol encoding block, where the number n of encoding symbols
in the encoding block may vary for each source block.  For a receiver to
completely recover the object, for each source block consisting of k
source symbols, k distinct encoding symbols (i.e., with different
encoding symbol IDs) must be received from the corresponding encoding
block.  For some encoding blocks, more encoding symbols may be received
than there are source symbols in the corresponding source block, in
which case the excess encoding symbols are discarded.  An example
encoding structure is shown in Figure 1.



    |   source symbol IDs   |   source symbols IDs  |
    |   of source block 0   |   of source block 1   |
                 |                        |
                 v                        v
    +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
    |0 |1 |2 |3 |4 |5 |6 |7 |0 |1 |2 |3 | 4|5 |6 |7 |
    +--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
                            |
                        FEC encoder
                            |
                            v
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
|0 |1 |2 |3 | 4| 5| 6| 7| 8| 9| 0| 1| 2| 3| 4| 5| 6| 7| 8| 9|
+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+--+
               ^                             ^
               |                             |
|  encoding symbol IDs        | encoding symbol IDs         |
|  of encoding block 0        | of encoding block 1         |


Figure 1. Encoding structure for object divided into two source
blocks consisting of 8 source symbols each, and the FEC encoder is used to
generate 2 additional redundant symbols (10 encoding symbols in total)
for each of the two source blocks.


In many cases, an object is partitioned into equal length source blocks
each consisting of k contiguous source symbols of the object, i.e.,



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block c consists of the range of source symbols [ck, (c+1)k-1]. This
ensures that the FEC encoder can be optimized to handle a particular
number k of source symbols.  This also ensures that memory references
are local when the sender reads source symbols to encode, and when the
receiver reads encoding symbols to decode.  Locality of reference is
particularly important when the object is stored on disk, as it reduces
the disk seeks required.  The block number and the source symbol ID
within that block can be used to uniquely specify a source symbol within
the object. If the size of the object is not a multiple of k source
symbols, then the last source block will contain less than k symbols.

The block numbers can be numbered consecutively starting from zero.
Encoding symbols within a block can be uniquely identified by an
encoding symbol ID.  One way of identifying encoding symbols within a
block is to use the combination of an encoding flag that identifies the
symbol as either a source symbol or a redundant symbol together with
either a source symbol ID or a redundant symbol ID.  For example, an
encoding flag value of 1 can indicate that the encoding symbol is a
source symbol and 0 can indicate that it is a redundant symbol. The
source symbol IDs can be numbered from 0 to k-1 and the redundant symbol
IDs can be numbered from 0 to n-k-1.

For example, if the object consists 10 source symbols with values a, b,
c, d, e, f, g, h, i, and j, and k = 5 and n = 8, then there are two
source blocks consisting of 5 symbols each, and there are two encoding
blocks consisting of 8 symbols each.  Let p, q and r be the values of
the redundant symbols for the first encoding block, and let x, y and z
be the values of the redundant symbols for the second encoding block.
Then the encoding symbols together with their identifiers are

(0, 1, 0: a), (0, 1, 1: b), (0, 1, 2: c), (0, 1, 3: d), (0, 1, 4: e),
(0, 0, 0: p), (0, 0, 1: q), (0, 0, 2: r),
(1, 1, 0: f), (1, 1, 1: g), (1, 1, 2: h), (1, 1, 3: i), (1, 1, 4: j),
(1, 0, 0: x), (1, 0, 1: y), (1, 0, 2: z).


In this example, the first value identifies the block number and the
second two values together identify the encoding symbol within the
block, i.e, the encoding symbol ID consists of the encoding flag
together with either the source symbol ID or the redundant symbol ID
depending on the value of the encoding flag.  The value of the encoding
symbol is written after the colon.  Each block can be recovered from any
5 of the 8 encoding symbols associated with that block. For example,
reception of

  (0, 1, 1: b), (0, 1, 2: c), (0, 1, 3: d), (0, 0, 0: p), (0, 0, 1: q)

 is sufficient to recover the first source block, and reception of



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  (1, 1, 0: f), (1, 1, 1: g), (1, 0, 0: x), (1, 0, 1: y), (1, 0, 2: z)


is sufficient to recover the second source block.

Another way of uniquely identifying encoding symbols within a block is
to let the encoding symbol IDs for source symbols be 0,...,k-1 and to
let the encoding symbol IDs for redundant symbols be k,...,n-1.


2.3.  Large block FEC codes

Tornado codes [4] are large block FEC codes that provide an alternative
to small block FEC codes.  An (n, k) Tornado code requires slightly more
than k out of n encoding symbols to recover k source symbols, i.e.,
there is a small reception overhead.  However, the advantage is that the
value of k may be on the order of tens of thousands and still run
efficiently.  Because of memory considerations, in practice the value of
n is restricted to be a small multiple of k, e.g., n = 2k.  For example,
[3] describes an implementation of Tornado codes where the encoding and
decoding speeds are tens of megabytes per second for Pentium class
machines of various vintages when k is in the tens of thousands and n =
2k.  The reception overhead for such values of k and n is in the 5-10%
range.  Tornado codes require a large amount of out of band information
to be communicated to all senders and receivers for each different
object length, and require an amount of memory on the encoder and
decoder which is proportional to the object length times 2n/k.

Tornado codes are designed to have low reception overhead on average
with respect to reception of a random portion of the encoding packets.
Thus, to ensure that a receiver can reassemble the object with low
reception overhead, the packets are permuted into a random order before
transmission.


2.4.  Expandable FEC codes

All of the FEC codes described up to this point are block codes.  There
is a different type of FEC codes that we call expandable FEC codes.
Like block codes, an expandable FEC encoder operates on an object of
known size that is partitioned into equal length source symbols.  Unlike
block codes, there is no predetermined number of encoding symbols that
can be generated for expandable FEC codes. Instead, an expandable FEC
encoder can generate as few or as many unique encoding symbols as
required on demand.

LT codes [5] are an example of large expandable FEC codes with a small
reception overhead.  An LT encoder uses randomization to generate each



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encoding symbol randomly and independently of all other encoding
symbols.  Like Tornado codes, the number of source symbols k may be very
large for LT codes, i.e., on the order of tens to hundreds of thousands,
and the encoder and decoder run efficiently in software. For example the
encoding and decoding speeds for LT codes are in the range 3-20
megabytes per second for Pentium class machines of various vintages when
k is in the high tens of thousands.  An LT encoder can generate as few
or as many encoding symbols as required on demand.  When a new encoding
symbol is to be generated by an LT encoder, it is based on a randomly
chosen encoding symbol ID that uniquely describes how the encoding
symbol is to be generated from the source symbols. In general, each
encoding symbol ID value corresponds to a unique encoding symbol, and
thus the space of possible encoding symbols is approximately four
billion if for example the encoding symbol ID is 4 bytes.  Thus, the
chance that a particular encoding symbol is the same as any other
particular encoding symbol is inversely proportional to the number of
possible encoding symbol IDs.  An LT decoder has the property that with
very high probability the receipt of any set of slightly more than k
randomly and independently generated encoding symbols is sufficient to
reassemble the k source symbols.  For example, when k is on the order of
tens to hundreds of thousands the reception overhead is less than 5%
with no failures in hundreds of millions of trials under any loss
conditions.

Because encoding symbols are randomly and independently generated by
choosing random encoding symbol IDs, LT codes have the property that
encoding symbols for the same k source symbols can be generated and
transmitted from multiple senders and received by a receiver and the
reception overhead and the decoding time is the same as if though all
the encoding symbols were generated by a single sender. The only
requirement is that the senders choose their encoding symbol IDs
randomly and independently of one another.

There is a weak tradeoff between the number of source symbols and the
reception overhead for LT codes, and the larger the number of source
symbols the smaller the reception overhead.  Thus, for shorter objects,
it is sometimes advantageous to partition the object into many short
source symbols and include multiple encoding symbols in each packet.  In
this case, a single encoding symbol ID is used to identify the multiple
encoding symbols contained in a single packet.

There are a couple of factors for choosing the appropriate symbol
length/ number of source symbols tradeoff. The primary consideration is
that there is a fixed overhead per symbol in the overall processing
requirements of the encoding and decoding, independent of the number of
source symbols. Thus, using shorter symbols means that this fixed
overhead processing per symbol will be a larger component of the overall
processing requirements, leading to larger overall processing



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requirements.  A second much less important consideration is that there
is a component of the processing per symbol that depends logarithmically
on the number of source symbols, and thus for this reason there is a
slight preference towards fewer source symbols.

Like small block codes, there is a point when the object is large enough
that it makes sense to partition it into blocks when using LT codes.
Generally the object is partitioned into blocks whenever the number of
source symbols times the packet payload length is less than the size of
the object.  Thus, if the packet payload is 1024 bytes and the maximum
number of source symbols is 128,000 then any object over 128 megabytes
will be partitioned into more than one block.  One can choose the
maximum number of source symbols to use, depending on the desired
encoding and decoding speed versus reception overhead tradeoff desired.
Encoding symbols can be uniquely identified by a block number (when the
object is large enough to be partitioned into more than one block) and
an encoding symbol ID.  The block numbers, if they are used, are
generally numbered consecutively starting from zero within the object.
The block number and the encoding symbol ID are both chosen uniformly
and randomly from their range when an encoding symbol is to be
transmitted. For example, suppose the number of source symbols is
128,000 and the number of blocks is 2.  Then, each packet generated by
the LT encoder could be of the form (b, x: y).  In this example, the
first value identifies the block number and the second value identifies
the encoding symbol within the block.  In this example, the block number
b is either 0 or 1, and the encoding symbol ID x might be a 32-bit
value.  The value y after the colon is the value of the encoding symbol.


2.5.  Source blocks with variable length source symbols

For all the FEC codes described above, all the source symbols in the
same source block are all of the same length.  In this section, we
describe a general technique to handle the case when it is desirable to
use source symbols of varying lengths in a single source block. This
technique is applicable to block FEC codes.

Let l_1, l_2, ... , l_k be the lengths in bytes of k varying length
source symbols to be considered part of the same source block.  Let lmax
be the maximum over i = 1, ... , k of l_i.  To prepare the source block
for the FEC encoder, pad each source symbol i out to length lmax with a
suffix of lmax-l_i zeroes, and then prepend to the beginning of this the
value l_i.  Thus, each padded source symbol is of length x+lmax,
assuming that it takes x bytes to store an integer with possible values
0,...,lmax, where x is a protocol constant known to both the encoder and
the decoder.  These padded source symbols, each of length x+lmax, are
the input to the encoder, together with the value n.  The encoder then
generates n-k redundant symbols, each of length x+lmax.



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The encoding symbols that are placed into packets consist of the
original k varying length source symbols and n-k redundant symbols, each
of length x+lmax.  From any k of the received encoding symbols, the FEC
decoder recreates the k original source symbols as follows.  If all k
original source symbols are received, then no decoding is necessary.
Otherwise, at least one redundant symbol is received, from which the
receiver can easily determine if the block is composed of variable-
length source symbols: if the redundant symbol(s) is longer than the
source symbols then the source symbols are variable-length. Note that in
a variable-length block the redundant symbols are always longer than the
longest source symbol, due to the presence of the prepended symbol-
length. The receiver can determine the value of lmax by subtracting x
from the length of a received redundant symbol. For each of the
received original source symbols, the receiver can generate the
corresponding padded source symbol as described above.  Then, the input
to the FEC decoder is the set of received redundant symbols, together
with the set of padded source symbols constructed from the received
original symbols.  The FEC decoder then produces the set of k padded
source symbols. Once the k padded source symbols have been recovered,
the length l_i of original source symbol i can be recovered from the
first x bytes of the ith padded source symbol, and then original source
symbol i is obtained by taking the next l_i bytes following the x bytes
of the length field.


3.  Security Considerations

The use of FEC, in and of itself, imposes no additional security
considerations versus sending the same information without FEC.
However, just like for any transmission system, a malicious sender may
try to inject packets carrying corrupted encoding symbols.  If a
receiver accepts one or more corrupted encoding symbol in place of
authentic ones then such a receiver may reconstruct a corrupted object.

Application-level transmission object authentication can detect the
corrupted transfer, but the receiver must then discard the transferred
object. Thus, injecting corrupted encoding symbols they are accepted as
valid encoding symbols by a receiver is at the very least an effective
denial of service attack.

In light of this possibility, FEC receivers may screen the source
address of a received symbol against a list of authentic transmitter
addresses.  Since source addresses may be spoofed, transport protocols
using FEC may provide mechanisms for robust source authentication of
each encoding symbol. Multicast routers along the path of a FEC transfer
may provide the capability of discarding multicast packets that
originated on that subnet, and whose source IP address does not
correspond with that subnet.



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It is recommended that a packet authentication scheme such as TESLA [6]
be used in conjunction with FEC codes.  Then, packets that cannot be
authenticated can be discarded and the object can be reliably recovered
from the received authenticated packets.


4.  Intellectual Property Disclosure

Tornado codes [4] have both patents issued and patents pending. There
is an issued patent for LT codes [5].

5.  Acknowledgments

Thanks to Vincent Roca and Hayder Radha for their detailed comments on
this draft.


6.  References

[1] Acharya, S., Franklin, M., and Zdonik, S., ``Dissemination- Based
Data Delivery Using Broadcast Disks'', IEEE Personal Communications,
pp.50-60, Dec 1995.

[2] Blahut, R.E., ``Theory and Practice of Error Control Codes'',
Addison Wesley, MA 1984.

[3] Byers, J.W., Luby, M., Mitzenmacher, M., and Rege, A., ``A Digital
Fountain Approach to Reliable Distribution of Bulk Data'', Proceedings
ACM SIGCOMM '98, Vancouver, Canada, Sept 1998.

[4] Luby, M., Mitzenmacher, M., Shokrollahi, A., Spielman, D.,
``Efficient Erasure Correcting Codes'', IEEE Transactions on Information
Theory, Special Issue: Codes on Graphs and Iterative Algorithms, pp.
569-584, Vol. 47, No. 2, February 2001.

[5] Luby, M., "Information Additive Code Generator and Decoder for
Communication Systems", U.S. Patent No. 6,307,487, October 23, 2001.

[6] Perrig, A., Canetti, R., Song, D., Tygar, J.D., "Efficient and
Secure Source Authentication for Multicast", Network and Distributed
System Security Symposium, NDSS 2001, pp. 35-46, February 2001.

[7] Rizzo, L., ``Effective Erasure Codes for Reliable Computer
Communication Protocols'', ACM SIGCOMM Computer Communication Review,
Vol.27, No.2, pp.24-36, Apr 1997.

[8] Rizzo, L., ``On the Feasibility of Software FEC'', DEIT Tech Report,
http://www.iet.unipi.it/~luigi/softfec.ps, Jan 1997.



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7.  Authors' Addresses

   Michael Luby
   luby@digitalfountain.com
   Digital Fountain
   39141 Civic Center Drive
   Suite 300
   Fremont, CA  94538

   Lorenzo Vicisano
   lorenzo@cisco.com
   cisco Systems, Inc.
   170 West Tasman Dr.,
   San Jose, CA, USA, 95134

   Jim Gemmell
   jgemmell@microsoft.com
   Microsoft Research
   301 Howard St., #830
   San Francisco, CA, USA, 94105

   Luigi Rizzo
   luigi@iet.unipi.it
   ACIRI, 1947 Center St., Berkeley CA 94704
   and
   Dip. di Ing. dell'Informazione
   Universita` di Pisa
   via Diotisalvi 2, 56126 Pisa, Italy

   Mark Handley
   mjh@aciri.org
   ACIRI
   1947 Center St.
   Berkeley CA, USA, 94704

   Jon Crowcroft
   J.Crowcroft@cs.ucl.ac.uk
   Department of Computer Science
   University College London
   Gower Street,
   London WC1E 6BT, UK










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8.  Full Copyright Statement

Copyright (C) The Internet Society (2001).  All Rights Reserved.

This document and translations of it may be copied and furnished to
others, and derivative works that comment on or otherwise explain it or
assist in its implementation may be prepared, copied, published and
distributed, in whole or in part, without restriction of any kind,
provided that the above copyright notice and this paragraph are included
on all such copies and derivative works. However, this document itself
may not be modified in any way, such as by removing the copyright notice
or references to the Internet Society or other Internet organizations,
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The limited permissions granted above are perpetual and will not be
revoked by the Internet Society or its successors or assigns.

This document and the information contained herein is provided on an "AS
IS" basis and THE INTERNET SOCIETY AND THE INTERNET ENGINEERING TASK
FORCE DISCLAIMS ALL WARRANTIES, EXPRESS OR IMPLIED, INCLUDING BUT NOT
LIMITED TO ANY WARRANTY THAT THE USE OF THE INFORMATION HEREIN WILL NOT
INFRINGE ANY RIGHTS OR ANY IMPLIED WARRANTIES OF MERCHANTABILITY OR
FITNESS FOR A PARTICULAR PURPOSE."


























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