Note: Correction to the 1997 Tutorial on Reed-Solomon Coding

Transcription

1 Note: Correction to the 1997 utorial on Reed-Solomon Coding James S Plank Ying Ding University of ennessee Knoxville, N 3799 [plank,ying]@csutkedu echnical Report U-CS Department of Computer Science University of ennessee April 24, Introduction In 1997, SPE published a tutorial by Plank [19] on implementing Reed-Solomon codes for erasure correction in redundant data storage systems he motivation of this tutorial was to present these codes, which are typically described mathematically by coding theorists, in a way accessible to the programmers who need to implement them he tutorial as published presented an information dispersal matrix, which does not have the properties claimed that the deletion of any rows results in an invertable matrix he purpose of this note is to present a correct information dispersal matrix that has the desired properties, and to put the work in current context 2 he Continued Need or Erasure Correcting Codes As disk array technology continued to blossom in the 1990 s [4, 5], a need arose to tolerate a disk s failure without waiting for the disk to be repaired Straight replication performs this fault-tolerance, but at a high storage overhead RAID Level 5 encoding, termed N+1 Parity [5], reduces the storage overhead for fault-tolerance, and allows a parity disk to store redundancy for data disks in such a way that the failure of any single disk may be tolerated However, as the number of disks in a disk array grows, so does the the need to tolerate multiple simultaneous failures Reed- Solomon coding has the properties necessary to add arbitrary levels of fault-tolerance to disk array systems One may add coding disks to data disks so that the failure of any disks may be tolerated, and although none of the levels of RAID employs Reed-Solomon codes, the original work on disk arrays make note of the codes desirable properties [5] 1

2 As wide-area network computing has become more popular, the uses of erasure-correcting codes have broadened Rizzo employs them to avoid retransmission in point-to-point [20] and multicast [21] communication protocols his work has resulted in standardization efforts for such codes in multicast scenarios from the IE [15, 1] Additional uses of Reed-Solomon codes have been in cryptography [], distributed data structures [10], energy-efficient wireless communication [7] and distributed checkpointing [1] he advent of wide-area and peer-to-peer storage systems has further motivated the need for erasure-correcting codes or example, OceanStore employs Reed-Solomon coding for RAID-like fault-tolerance in a wide-area file system [9] More interestingly, several content dispersal systems have noted that erasure coding can be used for caching rather than for fault-tolerance [2, 3, 22] Specifically, suppose that blocks of a file need to be stored in a wide-area storage substrate, so that clients in all parts of the network may access it With replication, clients must find the closest copies of each of the blocks in order to retrieve the file However, with erasure-correcting codes, extra coding blocks may be distributed with the blocks of the file so that each client need only retrieve the closest blocks in order to reconstruct the file As files grow in size, the power of this application will be immense; hence the need to correct the error of the 1997 Reed-Solomon coding tutorial here are other erasure coding techniques in addition to the one which this tutorial addresses Examples are ornado codes [13, 14], Cauchy Reed-Solomon codes [1] and other parity-based schemes [] Of these, ornado codes are worth special mention, as they form the backbone of the Digital ountain content dispersal system [2] ornado codes have a randomized structure so that with the addition of extra parity blocks, a file may be reconstructed from any blocks he randomized structure ensures that should be small In a performance evaluation conducted by Luby [12], ornado codes display significantly better encoding and decoding performance than the codes in this paper (termed Vandermonde-based Reed-Solomon codes ) for large data sizes and large values of of or small values, a true comparison has yet to be performed Currently, there is no implementation guide for ornado codes akin to [19] As the knowledge of their performance advantages become more widespread, perhaps this will change 3 A Correct Information Dispersal Matrix he desired properties for the information dispersal matrix for Reed-Solomon coding is that: It is an matrix he matrix in the first rows are the identity matrix Any submatrix formed by deleting rows of the matrix is invertible We denote the correct information matrix is derived from an Vandermonde matrix using a sequence of elementary matrix transformations: 1 Any column may be swapped with column 2

3 2 Any column may be replaced by, where 3 Any column may be replaced by adding a multiple of another column to it:, where and Since arithmetic is over a Galois field, the addition operation is bitwise exclusive-or ( ( ( ( he -th element of a Vandermonde matrix is defined to be :!! "#"" $%&'! "#"" $%& "#"" $%& *),+-0/1# *),+-2/3 *),+-0/1# "#""4*),+-0/1# $% 57 By definition, this matrix has the property that any submatrix formed by deleting rows of this matrix is invertible [17] Moreover, any matrix derived from this matrix by a sequence of elementary matrix transformations maintains this property (since elementary matrix operations do not change the rank of a matrix [11]) herefore, constructing the matrix is a simple matter of performing elementary transformations on the Vandermonde matrix until the first rows are the identity matrix he algorithm for doing constructing is as follows: Suppose the first :9<; rows of the matrix are identity rows, and,= At each step, we will turn row into an identity row, without altering the other identity rows If the -th element of row is equal to zero, find a column such that?> and the -th element of row is non-zero, and swap columns and Such a column is guaranteed to exist; otherwise the first rows of the matrix would not compose an invertible matrix Moreover,, swapping columns and will not alter the first A9; rows of the matrix Let B C be the value of the -th element of row Let BEDG HC be the multiplicative inverse B HC In other words, B HCIB HC DJ K; Since B C, B C DG is guaranteed to exist if B C K;, replace column with B DJ L Now B C = 1 or all columns 0 and B C M, replace column with 9 B C, where B HC is the -th element in row At the end of this step, rows 0 through are identity rows, and the matrix still has the property that the deletion of any rows yields an invertible matrix Repeat this process until the first rows are identity rows, and the construction of is complete Example As an example, we construct for ONP ON, over G(QSR ) As detailed in [19], in G(QR ), addition is performed by exclusive-or, and multiplication/division may be performed using logarithm tables, reproduced in able 1 he N Vandermonde matrix over U,V WQR is: 3

4 Q B gflog[i] gfilog[i] able 1: Logarithm tables for U,V QR ; ; ; Q Q N N N Row 0 is already an identity row o convert row 1, we note that B with 9 and with 9 he resulting matrix is: All that is left is to convert row 2 irst, since B C hen we replace with 9'N and 57 ; N Q Q N 5 ; ; ; ; Q ; N ; N ; Q C 57 C B C ;, so we need to replace K;, we need to replace with DG = : ; N Q ; Q N ; with ; ; ; ; ; ; ; 5 9 Q to yield our desired : 5 4

5 References [1] J Blomer, M Kalfane, M Karpinski, R Karp, M Luby, and D Zuckerman An XOR-based erasure-resilient coding scheme echnical Report R-95-04, International Computer Science Institute, August 1995 [2] J Byers, M Luby, M Mitzenmacher, and A Rege A digital fountain approach to reliable distribution of bulk data In ACM SIGCOMM 9, pages 5 7, Vancouver, August 199 [3] J W Byers, M Luby, and M Mitzenmacher Accessing multiple mirror sites in parallel: Using tornado codes to speed up downloads In IEEE INOCOM, pages , New York, NY, March 1999 [4] P M Chen, E K Lee, G A Gibson, R H Katz, and D A Patterson RAID: High-performance, reliable secondary storage ACM Computing Surveys, 2(2):145 15, June 1994 [5] G A Gibson Redundant Disk Arrays: Reliable, Parallel Secondary Storage he MI Press, Cambridge, Massachusetts, 1992 [] G A Gibson, L Hellerstein, R M Karp, R H Katz, and D A Patterson ailure correction techniques for large disk arrays In hird International Conference on Architectural Support for Programming Languages and Operating Systems, pages , Boston, MA, April 199 [7] P J M Havinga Energy efficiency of error correction on wireless systems, 1999 [] C S Jutla Encryption modes with almost free message integrity Lecture Notes in Computer Science, 2045, 2001 [9] J Kubiatowicz, D Bindel, Y Chen, P Eaton, D Geels, R Gummadi, S Rhea, H Weatherspoon, W Weimer, C Wells, and B Zhao Oceanstore: An architecture for global-scale persistent storage In Proceedings of ACM ASPLOS ACM, November 2000 [10] W Litwin and Schwarz Lh*rs: a high-availability scalable distributed data structure using Reed Solomon codes In Proceedings of the 2000 ACM SIGMOD International Conference on Management of Data, pages ACM Press, 2000 [11] Lowenthal Linear Algebra with Linear Differential Equations John Wiley & Sons, Inc, New York, 1975 [12] M Luby Benchmark comparisons of erasure codes luby/ erasurehtml, 2002 [13] M Luby, M Mitzenmacher, and A Shokrollahi Analysis of random processes via and-or tree evaluation In 9th Annual ACM-SIAM Symposium on Discrete Algorithms, January 199 [14] M Luby, M Mitzenmacher, A Shokrollahi, D Spielman, and V Stemann Practical loss-resilient codes In 29th Annual ACM Symposium on heory of Computing,, pages ,

6 [15] M Luby, L Vicisano, J Gemmell, L Rizo, M Handley, and J Crowcroft orward error correction (EC) building block IE RC 3452 ( December 2002 [1] M Luby, L Vicisano, J Gemmell, L Rizo, M Handley, and J Crowcroft he use of forward error correction(ec) in reliable multicast IE RC 3453 ( December 2002 [17] J MacWilliams and NJA Sloane he heory of Error-Correcting Codes, Part I North-Holland Publishing Company, Amsterdam, New York, Oxford, 1977 [1] J S Plank Improving the performance of coordinated checkpointers on networks of workstations using RAID techniques In 15th Symposium on Reliable Distributed Systems, pages 7 5, October 199 [19] J S Plank A tutorial on Reed-Solomon coding for fault-tolerance in RAID-like systems Software Practice & Experience, 27(9): , September 1997 [20] L Rizzo Effective erasure codes for reliable computer communication protocols ACM SIGCOMM Computer Communication Review, 27(2):24 3, 1997 [21] L Rizzo and L Vicisano RMDP: an EC-based reliable multicast protocol for wireless environments Mobile Computer and Communication Review, 2(2), April 199 [22] A I Rowstron and P Druschel Storage management and caching in PAS, a large-scale, persistent peer-topeer storage utility In Symposium on Operating Systems Principles, pages 1 201, 2001