Reliable Multicast Transport J. Lacan Internet-Draft ENSICA/LAAS-CNRS Expires:~~August 27,~~December 25,2006 V. Roca INRIA J. Peltotalo S. Peltotalo Tampere University of Technology~~February~~June23, 2006 Reed-Solomon Forward Error Correction (FEC)~~draft-ietf-rmt-bb-fec-rs-00.txt~~draft-ietf-rmt-bb-fec-rs-01.txtStatus of this Memo By submitting this Internet-Draft, each author represents that any applicable patent or other IPR claims of which he or she is aware have been or will be disclosed, and any of which he or she becomes aware will be disclosed, in accordance with Section 6 of BCP 79. Internet-Drafts are working documents of the Internet Engineering Task Force (IETF), its areas, and its working groups. Note that other groups may also distribute working documents as Internet- Drafts. Internet-Drafts are draft documents valid for a maximum of six months and may be updated, replaced, or obsoleted by other documents at any time. It is inappropriate to use Internet-Drafts as reference material or to cite them other than as "work in progress." The list of current Internet-Drafts can be accessed at http://www.ietf.org/ietf/1id-abstracts.txt. The list of Internet-Draft Shadow Directories can be accessed at http://www.ietf.org/shadow.html. This Internet-Draft will expire on~~August 27,~~December 25,2006. Copyright Notice Copyright (C) The Internet Society (2006). Abstract This document describes a Fully-Specified FEC scheme for the Reed- Solomon forward error correction code and its application tothereliable delivery of data objects on the packet erasure channel.~~The~~Reed-Solomon codes belong to the class of Maximum Distance Separable (MDS) codes,~~i.e,~~i.e.they enable a receiver to recover the k source symbols from any set of k received symbols. The implementation described here is compatible with the~~IPR-free~~implementation from Luigi Rizzo. Table of Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 3 2. Terminology . . . . . . . . . . . . . . . . . . . . . . . . . 4 3. Definitions Notations and Abbreviations . . . . . . . . . . . 5 3.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . 5 3.2. Notations . . . . . . . . . . . . . . . . . . . . . . . . 5 3.3. Abbreviations . . . . . . . . . . . . . . . . . . . . . . 6 4. Formats and Codes . . . . . . . . . . . . . . . . . . . . . . 7 4.1. FEC Payload~~IDs~~ID .. . . . . . . . . . . . . . . . . . . . . 7 4.2. FEC Object Transmission Information . . . . . . . . . . .~~7~~84.2.1. Mandatory Elements . . . . . . . . . . . . . . . . . .~~7~~84.2.2. Common Elements . . . . . . . . . . . . . . . . . . .~~7~~84.2.3. Scheme-Specific Elements . . . . . . . . . . . . . . . 8 4.2.4. Encoding Format . . . . . . . . . . . . . . . . . . .~~8~~95. Procedures . . . . . . . . . . . . . . . . . . . . . . . . . .~~10~~115.1. Determining the Maximum Source Block Length (B) . . . . .~~10~~115.2. Determining the Number of Encoding Symbols of a Block . .~~10~~116. Reed-Solomon CodesSpecification for the Erasure Channel. . .13 6.1. Finite Field. . . . . . . . . . . . . . . . . . .~~12 6.1. Finite field . .~~. . . .13 6.2. Reed-Solomon Encoding Algorithm. . . . . . . . . . . . .14 6.2.1. Encoding Principles. . . .~~12 6.2. Reed-Solomon Encoding Algorithm~~. . . . . . . . . . . . .~~13 6.2.1.~~14 6.2.2.Encoding Complexity . . . . . . . . . . . . . . . . .~~14~~156.3. Reed-Solomon Decoding Algorithm~~for the Erasure Channel . 14 6.3.1. Decoding Complexity . . . . . . . . .~~. . . . . . . .~~14 6.4. Implementation~~. . . . .15 6.3.1. Decoding Principles. . . . . . . . . . . . . . . . . 15~~6.4.1. Implementation for the Packet Erasure Channel .~~6.3.2. Decoding Complexity. . .~~15 7. Security Considerations~~. . . . . . . . . . . . . .16 6.4. Implementation for the Packet Erasure Channel. . . . .~~17 8. Intellectual Property~~.16 7. Security Considerations. . . . . . . . . . . . . . . . . . . 18~~9.~~8.IANA Considerations . . . . . . . . . . . . . . . . . . . . . 19~~10.~~9.Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . 20~~11.~~10.References . . . . . . . . . . . . . . . . . . . . . . . . . . 21~~11.1.~~10.1.Normative References . . . . . . . . . . . . . . . . . . . 21~~11.2.~~10.2.Informative References . . . . . . . . . . . . . . . . . . 21 Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . .~~22~~23Intellectual Property and Copyright Statements . . . . . . . . . .~~23~~241. Introduction The use of Forward Error Correction (FEC) codes is a classical solution to improve the reliability of multicast and broadcast transmissions. The~~[RFC3452] and [draft-ietf-rmt-fec-bb-revised-03] documents describe~~[2] document describesa general framework to use FEC in Content Delivery Protocols (CDP). The companion document~~[RFC3453]~~[3]describes some applications of FEC codes for content delivery. Recent FEC schemes like~~[draft-ietf-rmt-bb-fec-raptor-object-03]~~[6]and~~[draft-ietf-rmt-bb-fec-ldpc-01]~~[7]proposed erasure codes based on sparse graphs/matrices. These codes are efficient in terms of~~CPU~~processingbut not optimal in terms of correction~~capabilities, at least for small~~capabilities when dealing with "small"objects. The FEC scheme~~presented~~describedin this document belongs to the class of~~Maximum-Distance~~Maximum DistanceSeparable~~codes, i.e., it is~~codes that areoptimal in terms of erasure correction capability. In others words, it enables~~the~~areceiver to recover the k source symbols from any set ofexactlyk encoding symbols. Even if the encoding/decoding complexity is larger than that of~~[draft-ietf-rmt-bb-fec-raptor-object-03]~~[6]or~~[draft-ietf-rmt-bb-fec-ldpc-01],~~[7],this family of codes is very~~useful for applications sending small objects (e.g., for video and audio streaming). Indeed many~~useful. Manyapplications dealing with content transmission or content storage already rely on packet-based Reed-Solomon codes. In particular, many of them~~are derived from~~usethe~~implementation~~Reed-Solomon codecof Luigi Rizzo~~[RS-Rizzo]. This latter is compatible with the Reed-Solomon codes specification~~[4]. The goalof the present~~document.~~document to specify an implementation of Reed-Solomon codes that is compatible with this codec.2. Terminology 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 RFC 2119~~[rfc2119].~~[1].3. Definitions Notations and Abbreviations 3.1. Definitions This document uses the same terms and definitions as those specified in~~[draft-ietf-rmt-fec-bb-revised-03].~~[2].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~~symbols~~symbolthat~~are~~isnotasource~~symbols.~~symbol.Systematic code:~~a~~FECcode in which the source symbols are part of the encoding~~symbols~~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,~~togetherwithin the same packet, and whose relationships to the source~~object~~blockcan be derived from a single Encoding Symbol ID. Source~~Packet~~Packet:a data packet containing only source symbols. Repair~~Packet~~Packet:a data packet containing only repair symbols. 3.2. Notations This document uses the following notations: L denotes the object transfer length in~~bytes~~bytes.k denotes the number of source symbols in a source~~block~~block.n_r denotes the number of repair symbols generated for a source~~block~~block.n denotes the encoding block length,~~i.e.,~~i.e.the number of encoding symbols generated for a source block.~~Then~~Therefore:n =~~k+ n_r~~k + n_r.max_n~~Maximum Number~~denotes the maximum numberof~~Encoding Symbols~~encoding symbolsgenerated for any source~~block~~block.B denotes the maximum source block length in symbols,~~i.e.,~~i.e.the maximum number of source symbols per source~~block~~block.N denotes the number of source blocks into which the object shall be~~partitioned~~partitioned.E denotes the encoding symbol length in~~bytes sz~~bytes. Sdenotes the symbol size in units of m bit~~elements~~elements. When m = 8, then S and E are equal.m defines the~~number~~lengthoftheelements in the finite field,~~namely~~in bits. q defines the number of elements in the finite field. We have:q~~2^^m.~~= 2^^m in this specification.G denotes the number of encoding symbols per group,~~i.e.,~~i.e.the number of symbols sent in the same~~packet~~packet. GM denotes the Generator Matrix of a Reed-Solomon code.rate denotes the~~so-called~~"code rate", i.e. the k/n~~ratio~~ratio.a^^b denotes a raised to the power~~b~~b.a^^-1 denotes the inverse of~~a~~a.I_k denotes the k*k identity~~matrix~~matrix.3.3. Abbreviations This document uses the following abbreviations: ESIstands forEncoding Symbol~~ID~~ID. FEC OTI stands for FEC Object Transmission Information.RS~~Reed-Solomon~~stands for Reed-Solomon.MDSstands forMaximum Distance Separable~~code~~code.GF(q)denotes afinite field (A.K.A. Galois Field) with q~~elements~~elements. We assume that q = 2^^m in this document.4. Formats and Codes 4.1. FEC Payload~~IDs~~IDThe FEC Payload ID is composed of the Source Block Number and the Encoding Symbol~~ID:~~ID. The length of these two fields depends on the parameter m (which is transmitted in the FEC OTI) as follows :o The Source Block Number~~(16~~(32-mbit field) identifies from which source block of the object the encoding symbol(s) in the payload is (are) generated. There~~is~~area maximum of~~2^^16~~2^^(32-m)blocks per object. o The Encoding Symbol ID~~(16~~(mbit field) identifies which specific encoding~~symbol~~symbol(s)generated from the source block~~is~~is(are)carried in the packet payload. There~~is~~area maximum of~~2^^16~~2^^mencoding symbols per block. The first k values (0 to~~k-1)~~k - 1)identify source symbols, the remaining n-k values identify repair symbols. There MUST be exactly one FEC Payload ID persource or repairpacket. 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 by incrementing sequentially the ESI.The format of the FEC Payload ID for m = 8 and m = 16 is illustrated in Figure 1 and Figure 2 respectively.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~~(16~~(32-8=24bits) |~~Encoding Symbol~~Enc. Symb.ID~~(16 bits)~~| +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ Figure 1: FEC Payload ID encoding format form = 8 (default) 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 +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ | Src Block Nb (32-16=16 bits) | Enc. Symbol ID (m=16 bits) | +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ Figure 2:FEC~~Encoding~~PayloadID~~XX~~encoding format for m = 164.2. FEC Object Transmission Information 4.2.1. Mandatory Elements o FEC Encoding ID: the Fully-Specified FEC Scheme described in this document~~use the~~usesFEC Encoding ID XX. 4.2.2. Common Elements The following elements MUST be defined with the present FEC~~Scheme:~~scheme: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:~~supported :max_transfer_length =~~2^^16~~2^^(32-m)* B * E For instance,~~if~~for m = 8, forB =~~2^^8-1~~2^^8 - 1(because the codec operates on a finite field with 2^^8~~elements),~~elements)and if E = 1024 bytes, then the maximum transfer length is~~2^^34~~approximately equal to 2^^42bytes~~(i.e., a bit more than 17 Giga~~(i.e. 4 TeraBytes).Similarly, for m = 16, for B = 2^^16 - 1 and if E = 1024 bytes, then the maximum transfer length is also approximately equal to 2^^42 bytes.For larger objects,~~it is expected that other FEC codes (e.g., LDPC codes) or~~another~~Reed-Solomon~~FEC~~Scheme~~scheme,with a larger Source Block Number field in the FEC Payload~~ID~~ID, couldbe~~used.~~defined. Another solution consists in fragmenting large objects into smaller objects, each of them complying with the above limits.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. o Max-Number-of-Encoding-Symbols (max_n): a non-negative integer indicating the maximum number of encoding symbols generated for any source block. Section 5 explains how to derive the values of each of these elements. 4.2.3. Scheme-Specific Elements The following element MUST be defined with the present FEC Scheme. It contains two distinct pieces of information: o G: a non-negative integer indicating the number of encoding symbols per group used for the object. The default value is 1, meaning that each packet contains exactly one symbol. When no G parameter is communicated to the decoder, then this latter MUST assume that G = 1. o Finite Field~~size~~parameter, m: The m parameter~~defines~~is the length ofthe finite field~~size equal to~~elements, in bits. It also characterizes the number of elements in the finite field:q =~~p^^m~~2^^melements. The default value is m=8. When no finite field size parameter is communicated to the decoder, then this latter MUST assume that m = 8. 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~~oneencoding~~formats~~format or the othershould 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.~~used (e.g. within the ALC [8] or NORM [9] 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 | | +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ + | Transfer-Length (L) | +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ | m | G | Encoding Symbol Length (E) | +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ | Max Source Block Length (B) | Max Nb Enc. Symbols (max_n) | +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+ Figure~~2:~~3:EXT_FTI Header Format 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,~~session [10],the following XML~~elements~~attributesmust be described for the associated object: o~~FEC-OTI-Transfer-length~~FEC-OTI-Transfer-Length (L)o FEC-OTI-Encoding-Symbol-Length(E)o FEC-OTI-Maximum-Source-Block-Length(B)o FEC-OTI-Max-Number-of-Encoding-Symbols(max_n)o~~FEC-OTI-Number-Encoding-Symbols-per-Group~~FEC-OTI-Number-of-Encoding-Symbols-per-Group(optional)(G)o~~FEC-OTI-Finite-Field-Size-Parameter~~FEC-OTI-Finite-Field-Parameter(optional)(m)When no~~finite field size~~Gparameter is to be carried in the FEC OTI, the sender simply omits the~~FEC-OTI-Finite-Field-Size-Parameter element.~~FEC-OTI-Number-of-Encoding-Symbols-per-Group attribute. When no Finite Field parameter is to be carried in the FEC OTI, the sender simply omits the FEC-OTI-Finite-Field-Parameter attribute.5. Procedures~~This section defines procedures for FEC Encoding ID XX.~~5.1. Determining the Maximum Source Block Length (B) The finite field size parameter, m, defines the number of non zero elements in this~~field,~~field which is equal to:q- 1=~~2^^m-1.~~2^^m - 1.Note that q- 1is also the theoretical maximum number of encoding symbols that can be produced for a source block. For instance, when m = 8 (default): max1_B =~~2^^8-1~~2^^8 - 1 = 255Additionally, a codec MAY impose other limitations on the maximum block size. Yet it is not expected that such limits exist when usingthe defaultm = 8~~(default).~~value.This decision SHOULD be clarified at implementation time, when the target use case is known. This results in a max2_B limitation. Then, B is given by: B = min(max1_B, max2_B) Note that this calculation is only required at the coder, since the B parameter is communicated to the decoder through the FEC OTI. 5.2. Determining the Number of Encoding Symbols of a Block The following algorithm, also called "n-algorithm", explains how to determine the actual number of encoding symbols for a given block. AT A SENDER: Input: B: Maximum source block length, for any source block. Section 5.1 explains how to determine its value. k: Current source block length. This parameter is given by the~~source blocking~~block partitioningalgorithm. rate: FEC code rate, which is given by the user~~(e.g.,~~(e.g.when starting a FLUTE sending~~application) for a given use case.~~application).It is expressed as a floating point value. Output: max_n: Maximum number of encoding symbols generated for any source block n: Number of encoding symbols generated for this source block Algorithm: max_n = floor(B / rate); if (max_n~~>= 2^^m)~~> 2^^m - 1)then return an error ("invalid code rate"); n = floor(k * max_n / B); AT A RECEIVER: Input:~~B~~B:Extracted from the received FEC OTI~~max_n~~max_n:Extracted from the received FEC OTI~~k~~k:Given by the~~source blocking~~block partitioningalgorithm Output: n Algorithm: n = floor(k * max_n / B); Note that a Reed-Solomon decoder does not need to know the n value. Therefore the receiver part of the "n-algorithm" is not necessary from the Reed-Solomon decoder point of view. Yet a receiving application using the Reed-Solomon FEC scheme will sometimes need to know the value of n used by the sender, for instance for memory management optimizations. To that purpose,the FEC OTI carriesall theparametersneeded~~information is carried in~~for a receiver to executethe~~FEC OTI.~~above algorithm.6. Reed-Solomon CodesSpecification for the Erasure ChannelReed-Solomon (RS) codes~~form a special class of~~arelinear block~~codes, which offer maximum erasure correction capability.~~codes. They also belong to the class of MDS codes.A [n,k]-RS code encodes a sequence of k source elements defined over a finite field GF(q) into a sequence of n encoding elements, where n is~~upperbounded~~upper boundedby~~q-1.~~q - 1.The implementation described in this document is based on a generator matrix built from a Vandermonde matrix put into systematic form.Section 6.1 to Section 6.3 specify the [n,k]-RS codes when applied to m-bit elements, and Section 6.4 the use of [n,k]-RS codes when applied to symbols composed of several m-bit elements, which is the target of this specification.6.1. Finite~~field~~FieldA finite field GF(q) is defined as a finite set of q elements which~~have~~hasa structure of field. It contains necessarily q = p^^m elements, where p is a prime number. With packet erasure channels, p is always set to 2. The elements of the field GF(2^^m) can be represented by polynomials with binary coefficients~~(i.e.,~~(i.e.over GF(2)) of degree less than m. The polynomials can be associated to binary vectors of length m. For example, the vector (11001) represents the polynomial 1 + x + x^^4. This representation is often called polynomial representation. The addition between two elements is defined as the addition of binary polynomials in GF(2) and the multiplication is the multiplication modulo a given irreducible~~(i.e., non-factorizable)~~polynomialover GF(2)of degree m with coefficients in GF(2).~~Since a~~Note that all the roots of this polynomial are in GF(2^^m) but not in GF(2). Afinite field GF(2^^m) is completely characterized by the irreducible~~polynomial, we propose the~~polynomial. Thefollowing polynomialsare chosento represent the field GF(2^^m), for m varying from 2 to 16: m = 2, "111" (1+x+x^^2) m = 3, "1101", (1+x+x^^3) m = 4, "11001", (1+x+x^^4) m = 5, "101001", (1+x^^2+x^^5) m = 6, "1100001", (1+x+x^^6) m = 7, "10010001", (1+x^^3+x^^7) m = 8, "101110001", (1+x^^2+x^^3+x^^4+x^^8) m = 9, "1000100001", (1+x^^4+x^^9) m = 10, "10010000001", (1+x^^3+x^^10) m = 11, "101000000001", (1+x^^2+x^^11) m = 12, "1100101000001", (1+x+x^^4+x^^6+x^^12) m = 13, "11011000000001", (1+x+x^^3+x^^4+x^^13) m = 14, "110000100010001", (1+x+x^^6+x^^10+x^^14) m = 15, "1100000000000001", (1+x+x^^15) m = 16, "11010000000010001", (1+x+x^^3+x^^12+x^^16)~~For implementation issues,~~In order to facilitate the implementation,these polynomials are also~~primitive elements of GF(2^^m), i.e.,~~primitive. This means thatany element of GF(2^^m) can be expressed as a power of agivenroot of this polynomial. These polynomialsarealsochosen so that theycontain the minimum number of monomials. 6.2. Reed-Solomon Encoding Algorithm~~The encoding algorithm produces a vector of n encoding elements e=(e_0,~~6.2.1. Encoding Principles Let s = (s_0,...,~~e_(n-1)) over GF(2^^m) from~~s_{k-1}) bea source vector of k elements~~s=(s_0,~~over GF(2^^m). Let e = (e_0,...,~~s_(k-1) )~~e_{n-1}) be the corresponding encoding vector of n elementsover GF(2^^m).~~The~~Being alinear~~codes can be encoded~~code, encoding is performedby multiplying the source vector by a generator~~matrix Gm~~matrix, GM,of k rows and n columns over GF(2^^m). Thus: e=s *~~Gm.~~GM.The definition of the generator matrix completely characterizes theRScode. Let us consider that: n = 2^^m - 1 and: 0 < k <= n. Let us denote alpha~~a~~theprimitive element of GF(2^^m)~~(i.e., any element~~chosen in the listof~~GF(2^^m) can be expressed as a power~~Section 6.1 for the corresponding valueof~~alpha). The generator matrix is build from~~m. Let us considera~~k*n-Vandermonde~~Vandermondematrixof k rows and n columns,denoted by~~V_{k,n}. The entries~~V_{k,n}, and built as follows: the {i, j} entryof V_{k,n}~~are~~isv_{i,j} = alpha^^(i*j), where 0 <= i <= k - 1 and 0 <= j <= n - 1. This matrix generates a MDS code. However,~~it~~this MDS codeis not~~systematic as required by most of network~~systematic, which is a problem for many networkingapplications. To obtain a systematic~~matrix,~~matrix (and code),the simplest solution~~is to consider~~consists in consideringthe matrix V_{k,k} formed by the first k columns of~~V_{k,n}~~V_{k,n},then to invert it and to multiply this inverse by V_{k,n}. Clearly, the product V_{k,k}^^-1 * V_{k,n} contains the identity matrix I_k on its first k~~columns and generates a~~columns, meaning that the first k encoding elements are equal to source elements. Besides the associated code keeps theMDS~~code. The product V_{k,k}^^-1 * V_{k,n} is denoted by Gm and is~~property. Therefore,the generator matrix of the code considered in this~~document.~~document is: GM = (V_{k,k}^^-1) * V_{k,n}Note that,~~for practical applications, the length of~~in practice,the[n,k]-RScode can be shortened toa [n',k]-RS code, wherek <= n' <~~n~~n,by considering the sub-matrix formed by the n' first columns of~~Gm. 6.2.1.~~GM. 6.2.2.Encoding Complexity~~The encoding process~~Encodingcan be~~done~~performedby first pre-computing~~G~~GMand by multiplying the source vector(k elements)by~~Gm.~~GM (k rows and n columns).The complexity~~is one~~of the pre-computation of the generator matrix can be estimated as the complexity of themultiplication~~s*Gm, where Gm is~~of the inverse ofa~~k*(n-k) matrix. The~~Vandermonde matrix by n-k vectors (i.e. the last n-k columns of V_{k,n}). Since thecomplexity of the~~vector-matrix multiplication~~inverse of a k*k- Vandermonde matrix by a vectoris~~then k*(n-k) (i.e.,~~O(k * log^^2(k)), the generator matrix can be computed in 0((n-k)* k * log^^2(k)) operations. When the genarator matrix is pre-computed, the encoding needsk operations per repair~~element). The encoding~~element (vector-matrix multiplication). Encodingcan also be~~processed~~performedby first computing the product~~s*~~s *V_{k,k}^^-1 and then by multiplying the result~~by~~withV_{k,n}. The multiplication by the inverse of a square Vandermonde matrix is known as the interpolation problem and its complexity is O(k*log^^2 (k)). The multiplication by a Vandermonde matrix, known as the multipoint evaluation problem, requires O((n-k)*log(k)) by using Fast Fourier Transform, as explained in~~[fastMatrix-vectorMultiplication].~~[11].The total complexity of this encoding algorithm is then~~O(k/(n-k) log^^2 (k)+~~O((k/(n-k)) * log^^2(k) +log(k)) operations per repair~~symbol.~~element.6.3. Reed-Solomon Decoding Algorithm~~for the Erasure Channel~~6.3.1. Decoding PrinciplesThe Reed-Solomon decoding algorithm for the erasure channel allows the recovery of the k source elements from any set of k received elements. It is based on the fundamental property of the generator matrix which is such that any k*k-submatrix is invertible (see~~[MWS]).~~[5]).The first step of the decoding consists in extracting the k*k submatrix of the generator matrix obtained by considering the columns corresponding to the received~~symbols.~~elements.Indeed, since any encoding element is obtained by multiplying the source vector by one column of the generator matrix, the received vector of k encoding~~symbols~~elementscan be considered as the result of the multiplication of the source vector by a k*k submatrix of the generator matrix. Since this submatrix is invertible, the second step of the algorithm is to invert this matrix and to multiply the received vector by the obtained matrix to recover the source vector.~~6.3.1.~~6.3.2.Decoding Complexity The decoding algorithm described previously includes the matrix inversion and the vector-matrix multiplication. With the classical Gauss-Jordan algorithm, the matrix inversion requires O(k^^3) operations and the vector-matrix multiplication is performed in O(k^^2) operations. This complexity can be improved by considering that the received submatrix of~~Gm~~GMis the product between the inverse of a Vandermonde matrix (V_(k,k)^^-1) and another Vandermonde matrix (denoted by V' which is a submatrix of V_(k,n)). The decoding can be done by multiplying the received vector by V'^^-1 (interpolation problem with complexity O( k*log^^2(k)) ) then by V_{k,k} (multipoint evaluation with complexity~~O( k log(k)) ).~~O(k * log(k))).The global decoding complexity is then O(log^^2(k)) operations per source~~symbol.~~element.6.4. Implementation~~6.4.1. Implementation~~for the Packet Erasure Channel In a packet erasure channel, each packet(and symbol(s) since packets contain G >= 1 symbols)is either received correctly or erased. The location of the erased~~packets~~symbolsin the sequence of~~packets~~symbolsmust be known. The following specification describes the use of Reed-Solomon codes for generating redundant~~packets~~symbolsfrom k source~~packets~~symbolsand to recover the source~~packets~~symbolsfromany set ofk received~~packets.~~symbols.The k source symbols of a source block are assumed to be composed of~~sz~~Sm-bit elements. Each m-bit element is associated to an element of the finite field GF(2^^m) through the polynomial representation (Section 6.1). If some of the source symbols contain less than~~sz~~Selements, they are virtually padded with zero elements (it can be the case for the last symbol of the last block of the object). The encoding~~processing~~processproduces n-k repair symbols of~~sz elements by encoding each~~size S m-bit elements, the k source symbols being also partof the~~sz~~nencoding~~vectors from the sz source vectors~~symbols(Figure~~3). The~~4). These repair symbols are created m-bit element per m-bit element. More specifically, thej-th source vector is composed of the j-th element of each of the source symbols. Similarly, the j-th encoding vector is composed of the j-th element of each encoding symbol. ------------ --------------- -------------------0| | | | | | | | | | | | | | | | | * | generator | = | | | | | | | | | | | matrix | | | | | | | | | | | |~~Gm~~GM| | | | | | source |--------------| | | |---------------------| vector | | | | | | | --------------- ->| | | | | | | j |--------------| / |---------------------| | | | | | / | | | | | | | | | | encoding | | | | | | | | | | vector | | | | | | | | | | j | | | | | | | | | | | | | | |S-1| | | | | | | | | | ------------ ------------------- k source symbols n encoding symbols(source + repair)Figure~~3:~~4:Packet encoding scheme An asset of this scheme is that the loss of some~~of~~encoding symbols~~produce~~producesthe same erasure pattern for each of the~~sz~~Sencoding vectors. It follows that the matrix inversion must be done only once and will be used by all the~~sz~~Sencoding vectors. For large~~sz,~~S,this~~complexity cost of the~~matrixinversioncostbecomes negligible~~compared to~~in front ofthe~~sz~~Smatrix-vector multiplications. Another asset is thatthe n-krepair symbols can be produced on~~demand, e.g.,~~demand. For instance, a sender can start by producing a limited number of repair symbols and later on,depending on the observed erasures on the~~channel. The only constraint~~channel, decide to produce additional repair symbols, up to the n-k upper limit. Indeed, to produce the repair symbol e_j, where k <= j < n, itissufficient to multiplythe~~finite field size (see Section 6.1)~~S source vectors with column j of GM.7. Security Considerations The security considerations for this document are the same as that of~~[RFC3452].~~[2].8.~~Intellectual Property To the best of our knowledge, there is no patent or patent application identified as being used in the Reed-Solomon FEC scheme. Yet other flavors of Reed-Solomon codes and associated techniques MAY be covered by Intellectual Property Rights. 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~~[draft-ietf-rmt-fec-bb-revised-03].~~[2].This document assigns the~~Fully-Specified~~Fully- SpecifiedFEC Encoding ID XX under the ietf:rmt:fec:encoding~~name-space~~name- spaceto "Reed-Solomon Codes".~~10.~~9.Acknowledgments~~11.~~The authors want to thank Luigi Rizzo for comments on the subject and for the design of the reference Reed-Solomon codec. 10.References~~11.1.~~10.1.Normative References~~[RFC3452]~~[1] Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", RFC 2119. [2] Watson, M.,Luby, M.,and L. Vicisano,"Forward Error Correction (FEC) Building Block",~~RFC 3452, December 2002. [RFC3453]~~draft-ietf-rmt-fec-bb-revised-03.txt (work in progress), January 2006. [3]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.~~[draft-ietf-rmt-fec-bb-revised-03] Watson, M., Luby, M., and L. Vicisano, "Forward Error Correction (FEC) Building Block", draft-ietf-rmt-fec-bb-revised-03.txt (work in progress), January 2006. [rfc2119] Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", RFC 2119. 11.2.~~10.2.Informative References~~[MWS]~~[4] Rizzo, L., "Reed-Solomon FEC codec (revised version of July 2nd, 1998), available at http://info.iet.unipi.it/~luigi/vdm98/vdm980702.tgz", July 1998. [5]Mac Williams, F. and N. Sloane, "The Theory of Error Correcting Codes", North Holland, 1977 .~~[RS-Rizzo] Rizzo, L., "New version of the FEC code (revised 2 july 98), available at http://info.iet.unipi.it/~luigi/vdm98/vdm980702.tgz", July 1998. [draft-ietf-rmt-bb-fec-ldpc-01]~~[6] Luby, M., Shokrollahi, A., Watson, M., and T. Stockhammer, "Raptor Forward Error Correction Scheme", Internet Draft draft-ietf-rmt-bb-fec-raptor-object-03 (work in progress), October 2005. [7]Roca, V., Neumann, C., and D. Furodet, "Low Density Parity Check (LDPC) Forward Error Correction", draft-ietf-rmt-bb-fec-ldpc-01.txt (work in progress), March 2006.~~[draft-ietf-rmt-bb-fec-raptor-object-03]~~[8]Luby, M.,~~"Raptor Forward Error Correction Scheme", Internet Draft (draft-ietf-rmt-bb-fec-raptor-object-03 : work~~Watson, M., and L. Vicisano, "Asynchronous Layered Coding (ALC) Protocol Instantiation", draft-ietf-rmt-pi-alc-revised-03.txt (workin progress),~~October 2005. [fastMatrix-vectorMultiplication]~~April 2006. [9] Adamson, B., Bormann, C., Handley, M., and J. Macker, "Negative-acknowledgment (NACK)-Oriented Reliable Multicast (NORM) Protocol", draft-ietf-rmt-pi-norm-revised-01.txt (work in progress), March 2006. [10] Paila, T., Walsh, R., Luby, M., Lehtonen, R., and V. Roca, "FLUTE - File Delivery over Unidirectional Transport", draft-ietf-rmt-flute-revised-01.txt (work in progress), January 2006. [11]Gohberg, I. and V. Olshevsky, "Fast algorithms with preprocessing for matrix-vector multiplication problems", Journal of Complexity, pp. 411-427, vol. 10, 1994 . Authors' Addresses Jerome Lacan ENSICA/LAAS-CNRS 1, place Emile Blouin Toulouse 31056 France Email: jerome.lacan@ensica.fr URI:http://dmi.ensica.fr/auteur.php3?id_auteur=5Vincent Roca INRIA 655, av. de l'Europe Zirst; Montbonnot ST ISMIER cedex 38334 France Email: vincent.roca@inrialpes.fr URI: http://planete.inrialpes.fr/~roca/ Jani Peltotalo Tampere University of Technology P.O. Box 553 (Korkeakoulunkatu 1) Tampere FIN-33101 Finland Email: jani.peltotalo@tut.fi URI:http://atm.tut.fi/madSami Peltotalo Tampere University of Technology P.O. 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