Erasure Codes for Reading and Writing

1. Erasure Codes for Reading and Writing Mario Vodisek ( joint work with AG Schindelhauer)

2. Agenda • Erasure (Resilient) Codes in storage networks • The Read-Write-Coding-System • A Lower Bound and Perfect Codes • Requirements and Techniques

3. Erasure (Resilient) Coding • n-symbol message x with symbols from alphabet  • m-symbol encoding y with symbols from  (m > n) • erasure coding provides mapping: n!m such that • reading any n·r < m symbols of y are sufficient for recovery • (mostly: r = n )optimal for reading) • advantages: • bm-rcerasures can be tolerated • storage overhead is a factor of • Generally, erasure codes are used to guarantee information recovery for data transmission over unreliable channels (RS-, Turbo-, LT-Codes, …) • Lots of research in code properties such as • scalability • encoding/decoding speed-up • rateless-ness • Attractive also to storage networks: downloads (P2P) and fault-tolerance coding

4. Erasure Codes for Storage (Area) Networks • SANs require • high system availability • disks fail or be blocked (probability $ size) • efficient modification handling • Slow devices ) expensive I/O-operations • Properties: • a fixed set E of existing errors can be considered at encoding time • E can have changed to E‘ at decoding time • Additional requirements to erasure codes: • tolerate some certain number of erasures • ensure modification of codeword even if erasures occur • consider E at encoding time and E‘ at decoding time Network

5. The Read-Write-Coding-System • An (n, r, w, m)b-Read-Write-Coding System (RWC) is defined as follows: • The base b : b-symbol alphabet b as the set of all used items • n 1 blocks of information x1, …, xnb • mn code blocks y1, …, ym b • any nrmcode words sufficient to read the information • any nwmcode words sufficient to change the information by 1, …, n • (In the language of Coding Theory) : given m, n, r, w, our RW-Codes provide: • a (linear) code of dimension n and block length m such that for n·r,w·m: • the minimum distance of the code is at least m-r+1 • any two codewords y1, y2 are within a distance of at most w from another • distance(x, y):=|{1· i·m: xiyi }| m, r, w n coding

6. A Lower Bound for RW-Codes n m |S| = WR {n, n-1} w r Theorem: For r+w<n+mand any base b there does not exist any (n, r, w, m)b-RWC system ! Proof: • We know: nr,wm • Assume: r = w = nmn+1 • Write and subsequent read Index Sets (W, R): • |W| = w • |R| = r Assume: |S| = n • there are bn possible change vectors to be encoded by `write` into S; only basis for reading with r= n (notice: R\S code words remain unchanged) Assume: |S| <n = n-1 at most bn-1 possible change vectors for S can be encoded by `write` ´read´ will produce faulty output

7. Codes at Lower Bound: Perfect Codes • In the best case (n, r, w, m)b-RWC have parameters r + w = n + m (perfect Codes) • Unfortunately, perfect RWC do not always exist !! • E.g. there is no (1, 2, 2, 3)2-RWC but there exists a (1, 2, 2, 3)3-RWC ! • But: all perfect RW-Codes exist if the alphabet is sufficiently large ! • Notice to RAID: • Definition of parity RAID (RAID 4/5) corresponds to an (n, n, n+1, n+1)2-RWC • From the lower bounds it follows: there is no (n, n, n, n+1)2-RWC • ) there is no RAID-system with improved access properties !

8. The Model: Operations • Given: • X=x1,…, xnthe n-symbol information vectorover a finite alphabet . • Y=y1,…, yn the m-symbol code over  • b=||. • P(M) : the power set of M, Pk(M):={S2 P(M): |S|=k} • Define [m]:={1,…,m} • An (n, r, w, m)b-RWC-system consists of the following operations: • Inital state:X02n, Y02m • Read function: f: Pr([m]) £r!m • Write function: g: Pr([m]) £r£Pw([m]) £n!w • Differential write function: : Pw([m]) £n!w

9. Initialization: Compute the Encoding Y0 • RW-Codes are closely related to Reed-Solomon-Codes ! Given (in general): • the information vector X= x1, …, xnb • the encoded vector Y= y1, …, ynb • internal variables V = v1, …, vk for k =m-w = r-n, with no particular information • set of functions M=M1,…,Mn for encoding • Compute yi from X and V by function Mi ; define Mi as linear combination of X and V • yi = Mi(x1,…,xn,v1,…,vk) = j=1nxjMi,j + l=1k vlMi,l • ( Define M as some m£r matrix; Mi as rows. It follows: M(XV = Y )

10. The Matrix Approach: (n, r, w, m)b-RWC Consider: • the information vector X= x1, …, xnb • the encoded vector Y= y1, …, ymb • internal slack variables V = v1, …, vk for k =m-w = r-n • Further: • an mr generator matrix M: Mi,jb • the submatrix (Mi,j)i [m], j{n+1, …, r} is called the variable matrix =

11. Efficient Encoding: b = F[b] (Finite Fields) • RWC requires efficient arithmetic on elements of b for encoding • )set b = F[b] (finite field with b elements (formerly: GF(b))) • b = pn for some prime number p and integer n) F[pn] always exists • Computation of binary words of length v: b = 2v, F[2v] = {0,…,2v-1} • Features: • F[b] is closed under addition, multiplication • )exact computation on field elements )not more than v bits for representiation of results • Addition, subtraction via XOR (avoids rounding, no carryover) • Multiplication, division via mapping tables (analogous to logarithm tables for real numbers) • T : table mapping an integer to its logarithm in F[2v] • IT: table mapping an integer to its inverse logarithm in F[2v] • )multiplication, division by • adding/subtracting the logs • taking the inverse log

12. The Vandermonde Matrix • Consider M as m£r Vandermonde matrix Mi,j = ji-1: • X, Y, V 2 F[b] • Mi,j2 F[b] and all elements are different • The Vandermonde matrix is non-singular ) invertible • Any k‘ £k‘ submatix M‘ is also invertible = Consider: each device i in the SAN corresponds to a row of M and element yi

13. Reading (or Recovery) • Read: Given any r code entries from Y, compute X • Rearrange rows of M and Y such that first r entries of Y are available • (any r rows of M are linear independent in a Vandermonde matrix) • M!M‘ and Y!Y‘ • The first r rows of M‘ describe an invertible r£r matrix M‘‘ • X is computed by: (X | V)T = (M‘‘)-1Y (X | V) M Y r M‘ Y‘ m

14. Differential Write • Given: • The change vector =1,…,n and w code entries from Y • X‘ = X +is new information vector ) change X without reading • entries (XOR) • Compute the difference for the w code entries of Y • Further: • Only choices w < r make sense • Rearrange m£r matrix M and Y as follows: y1,…,yw (denote M‘ and Y‘) • k = r-n (slack vector V)

15. Differential Write (con‘t) n+1…r n • Define following sub-matrices: • MÃ" = (M‘i,j)i2[w], j2[n] • M"! = (M’i,j)i2[w], j2{n+1,…,r} • MÃ# = (M’i,j)i2{w+1,…,m}, j2[n] • M#! = (M’i,j)i2{w+1,…,m}, j2{n+1,…,r} w M"! MÃ" M#! MÃ# w+1…m • M#!is k£k = m-w£r-n matrix) M#! invertible • The vector Y can then be updated by a vector =1,…,w: • = ((MÃ") – (M"!)(M#!)-1(MÃ#)) ¢

16. Differential Write: Proof • Use: • Vector  = 1,…,kthe change of vector V • Vector  = 1,…,w the change of vector Y • X’ =X+ • V’ =V+ • Y’ =Y+ Correctness follows by combining: + M + M = = M = M This equation is equivalent to: (M#!) + (MÃ#) = 0, (MÃ") + (M"!) =  Since  is given,  is obained as follows:  = (M#!)-1(-MÃ#) ¢ M"! MÃ" M#! MÃ#

17. Thank you for your attention! Heinz Nixdorf Institute & Computer Science Institute University of Paderborn Fürstenallee 11 33102 Paderborn, Germany Tel.: +49 (0) 52 51/60 64 51 Fax: +49 (0) 52 51/62 64 82 E-Mail: vodisek@upb.de http://www.upb.de/cs/ag-madh