Network Working Group M. Luby
Request for Comments: 3453 Digital Fountain
Category: Informational L. Vicisano
Cisco
J. Gemmell
Microsoft
L. Rizzo
Univ. Pisa
M. Handley
ICIR
J. Crowcroft
Cambridge Univ.
December 2002
The Use of Forward Error Correction (FEC) in Reliable Multicast
Status of this Memo
This memo provides information for the Internet community. It does
not specify an Internet standard of any kind. Distribution of this
memo is unlimited.
Copyright Notice
Copyright (C) The Internet Society (2002). All Rights Reserved.
Abstract
This memo describes the use of Forward Error Correction (FEC) codes
to efficiently provide and/or augment reliability for one-to-many
reliable data transport using IP multicast. One of the key
properties of FEC codes in this context is the ability to use the
same packets containing FEC data to simultaneously repair different
packet loss patterns at multiple receivers. Different classes of FEC
codes and some of their basic properties are described and
terminology relevant to implementing FEC in a reliable multicast
protocol is introduced. Examples are provided of possible abstract
formats for packets carrying FEC.
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Table of Contents
1. Rationale and Overview . . . . . . . . . . . . . . . . . . . . 2
1.1. Application of FEC codes . . . . . . . . . . . . . . . . . 5
2. FEC Codes. . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1. Simple codes . . . . . . . . . . . . . . . . . . . . . . . 6
2.2. Small block FEC codes. . . . . . . . . . . . . . . . . . . 8
2.3. Large block FEC codes. . . . . . . . . . . . . . . . . . . 10
2.4. Expandable FEC codes . . . . . . . . . . . . . . . . . . . 11
2.5. Source blocks with variable length source symbols. . . . . 13
3. Security Considerations. . . . . . . . . . . . . . . . . . . . 14
4. Intellectual Property Disclosure . . . . . . . . . . . . . . . 14
5. Acknowledgments. . . . . . . . . . . . . . . . . . . . . . . . 15
6. References . . . . . . . . . . . . . . . . . . . . . . . . . . 15
7. Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . 17
8. Full Copyright Statement . . . . . . . . . . . . . . . . . . . 18
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,
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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.
Forward Error Correction (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
an IP multicast protocol, the network layers will detect corrupted
packets and discard them or the transport layers can use packet
authentication to discard corrupted packets. Therefore the primary
application of FEC codes to IP multicast protocols is as an erasure
code. The payloads are generated and processed using an FEC erasure
encoder and objects are reassembled from reception of packets
containing the generated encoding 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.
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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
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-k times
the length of the k source symbols, and decoding times proportional
to l times the length of the k source symbols, where l is the number
of missing source symbols among the k received encoding symbols and l
can be as large as k. Because of the growth rate of the encoding and
decoding times as a product of k and n-k, 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. We describe examples of
all of these types of codes later.
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
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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 codes
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 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
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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 additional reliable IP multicast protocols, the method to obtain
reliability is to generate enough encoding symbols so that each
encoding symbol is transmitted only once (at most). 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 applicable to streaming protocols, for
example, where the stream is partitioned into objects, the source
symbols for each object are encoded 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, 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. This is an example of 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
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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 correspond
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 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
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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 [16]. For example, [15]
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).
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.
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| 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. The encoding structure for an 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., 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.
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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
(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 [12], [13], [10], [11], [9] 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. The benefit of the small cost of non-zero reception
overhead is that the value of k may be on the order of tens of
thousands and still the encoding and decoding are efficient. 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, [4] describes
an implementation of Tornado codes where the encoding and decoding
speeds are tens of megabytes per second for Pentium class machines of
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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 [6], [7], [8], [5] are an example of large expandable FEC
codes with a small reception overhead. An LT encoder uses
randomization to generate each 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
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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 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
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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.
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
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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 discard the transferred
object. By injecting corrupted encoding symbols, they are accepted
as valid encoding symbols by a receiver, which at the very least, is
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.
It is recommended that a packet authentication scheme such as TESLA
[14] 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
The IETF has been notified of intellectual property rights claimed in
regard to some or all of the specification contained in this
document. For more information consult the online list of claimed
rights.
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5. Acknowledgments
Thanks to Vincent Roca and Hayder Radha for their detailed comments
on this document.
6. References
[1] Acharya, S., Franklin, M. and S. Zdonik, "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] Bradner, S., "The Internet Standards Process -- Revision 3", BCP
9, RFC 2026, October 1996.
[4] Byers, J.W., Luby, M., Mitzenmacher, M. and A. Rege, "A Digital
Fountain Approach to Reliable Distribution of Bulk Data",
Proceedings ACM SIGCOMM '98, Vancouver, Canada, Sept 1998.
[5] Haken, A., Luby, M., Horn, G., Hernek, D., Byers, J. and M.
Mitzenmacher, "Generating high weight encoding symbols using a
basis", U.S. Patent No. 6,411,223, June 25, 2002.
[6] Luby, M., "Information Additive Code Generator and Decoder for
Communication Systems", U.S. Patent No. 6,307,487, October 23,
2001.
[7] Luby, M., "Information Additive Group Code Generator and Decoder
for Communication Systems", U.S. Patent No. 6,320,520, November
20, 2001.
[8] Luby, M., "Information Additive Code Generator and Decoder for
Communication Systems", U.S. Patent No. 6,373,406, April 16,
2002.
[9] Luby, M. and M. Mitzenmacher, "Loss resilient code with double
heavy tailed series of redundant layers", U.S. Patent No.
6,195,777, February 27, 2001.
[10] Luby, M., Mitzenmacher, M., Shokrollahi, A., Spielman, D. and V.
Stemann, "Message encoding with irregular graphing", U.S. Patent
No. 6,163,870, December 19, 2000.
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RFC 3453 FEC in Reliable Multicast December 2002
[11] Luby, M., Mitzenmacher, M., Shokrollahi, A. and D. Spielman,
"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.
[12] Luby, M., Shokrollahi, A., Stemann, V., Mitzenmacher, M. and D.
Spielman, "Loss resilient decoding technique", U.S. Patent No.
6,073,250, June 6, 2000.
[13] Luby, M., Shokrollahi, A., Stemann, V., Mitzenmacher, M. and D.
Spielman, "Irregularly graphed encoding technique", U.S. Patent
No. 6,081,909, June 27, 2000.
[14] Perrig, A., Canetti, R., Song, D. and J.D. Tygar, "Efficient and
Secure Source Authentication for Multicast", Network and
Distributed System Security Symposium, NDSS 2001, pp. 35-46,
February 2001.
[15] Rizzo, L., "Effective Erasure Codes for Reliable Computer
Communication Protocols", ACM SIGCOMM Computer Communication
Review, Vol.27, No.2, pp.24-36, Apr 1997.
[16] 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
Digital Fountain
39141 Civic Center Drive
Suite 300
Fremont, CA 94538
EMail: luby@digitalfountain.com
Lorenzo Vicisano
Cisco Systems, Inc.
170 West Tasman Dr.,
San Jose, CA, USA, 95134
EMail: lorenzo@cisco.com
Jim Gemmell
Microsoft Research
455 Market St. #1690
San Francisco, CA, 94105
EMail: jgemmell@microsoft.com
Luigi Rizzo
Dip. di Ing. dell'Informazione
Universita` di Pisa
via Diotisalvi 2, 56126 Pisa, Italy
EMail:luigi@iet.unipi.it
Mark Handley
ICSI Center for Internet Research
1947 Center St.
Berkeley CA, USA, 94704
EMail: mjh@icir.org
Jon Crowcroft
Marconi Professor of Communications Systems
University of Cambridge
Computer Laboratory
William Gates Building
J J Thomson Avenue
Cambridge
CB3 0FD
EMail: Jon.Crowcroft@cl.cam.ac.uk
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8. Full Copyright Statement
Copyright (C) The Internet Society (2002). 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, except as needed for the purpose of
developing Internet standards in which case the procedures for
copyrights defined in the Internet Standards process must be
followed, or as required to translate it into languages other than
English.
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.
Acknowledgement
Funding for the RFC Editor function is currently provided by the
Internet Society.
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