Correction of Adversarial Errors in NetworksSidharth JaggiMichael Langberg Tracey HoMichelle Effros

Submitted to ISIT 2005

Greater throughputRobust against random errorsAha!Network Coding!!!

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XavierYvonneZorba???

BackgroundNoisy channel models (Shannon,)Binary Symmetric Channelp (Noise parameter)011C (Capacity)01H(p)0.5

BackgroundNoisy channel models (Shannon,)Binary Symmetric ChannelBinary Erasure Channelp (Noise parameter)011C (Capacity)0E1-p0.5

BackgroundAdversarial channel modelsLimited-flip adversary (Hamming,Gilbert-Varshanov,McEliece et al) Shared randomness, private key, computationally bounded adversaryp (Noise parameter)011C (Capacity)010.5

Model 1XavierYvonne?Zorba??|E| directed unit-capacity linksZorba (hidden to Xavier/Yvonne) controls |Z| links Z. p = |Z|/|E|Xavier and Yvonne share no resources (private key, randomness)Zorba computationally unbounded; Xavier and Yvonne can only perform simple computations. Zorba knows protocols and already knows almost all of Xaviers message (except Xaviers private coin tosses)

Model 1 Xavier/Yvonnes GoalXavierYvonne?Zorba??Knowing |Z| but not Z, to come up with an encoding/decoding scheme thatallows a maximal rate of information to be decoded correctly with high probability.

Normalized rate (divide by number of links |E|)

Model 1 - Resultsp (Noise parameter)011C (Capacity)0.50.5

Model 1 - Resultsp (Noise parameter)011C (Capacity)0.50.5

Model 1 - Resultsp (Noise parameter)011C (Capacity)0.50.5???Probability of error = 0.5

Model 1 - Resultsp (Noise parameter)011C (Capacity)0.50.5

Model 1 - Resultsp (Noise parameter)011C (Capacity)0.50.5Eureka

Model 1 - Encoding|E|-|Z||E||E|-|Z|

Model 1 - Encoding|E|-|Z||E|MDSCodeX|E|-|Z|Block length n over finite field Fq|E|-|Z|n(1-)x1nVandermonde matrixT|E||E|n(1-)T1. . .nRate fudge-factorEasy to use consistency informationnSymbol from Fq

Model 1 - EncodingT|E||E|n(1-)T1. . .Easy to use consistency informationn

Model 1 - Encoding T|E| T1. . .r1r|E|nD11D1|E|D|E|1D|E||E|Dij=Tj(1).1+Tj(2).ri++Tj(n(1- )).rin(1- ) TjriDijj

Model 1 - Encoding T|E| T1. . .r1r|E|nD11D1|E|D|E|1D|E||E|Dij=Tj(1).1+Tj(2).ri++Tj(n(1- )).rin(1- ) TjriDiji

Model 1 - Transmission T|E| T1. . .r1r|E|D11D1|E|D|E|1D|E||E| T|E| T1. . .r1r|E|D11D1|E|D|E|1D|E||E|

Model 1 - Decoding T|E| T1. . .r1r|E|D11D1|E|D|E|1D|E||E|Quick consistency checkDij=Tj(1).1+Tj(2).ri++Tj(n(1- )).rin(1- ) ? TjriDij

Model 1 - Decoding T|E| T1. . .r1r|E|D11D1|E|D|E|1D|E||E|Quick consistency checkDij=Tj(1).1+Tj(2).ri++Tj(n(1- )).rin(1- ) ? TjriDijDji=Ti(1).1+Ti(2).rj++Ti(n(1- )).rjn(1- ) ?

Model 1 - Decoding T|E| T1. . .r1r|E|D11D1|E|D|E|1D|E||E|Edge i consistent with edge jDij=Tj(1).1+Tj(2).ri++Tj(n(1- )).rin(1- ) Dji=Ti(1).1+Ti(2).rj++Ti(n(1- )).rjn(1- ) Consistency graph

Model 1 - Decoding T|E| T1. . .r1r|E|D11D1|E|D|E|1D|E||E|Edge i consistent with edge jDij=Tj(1).1+Tj(2).ri++Tj(n(1- )).rin(1- ) Dji=Ti(1).1+Ti(2).rj++Ti(n(1- )).rjn(1- ) Consistency graph12453(Self-loops not important)T r,DT r,DT r,D12345

Model 1 - Decoding T|E| T1. . .r1r|E|D11D1|E|D|E|1D|E||E|12453T r,DT r,DT r,D12345Consistency graphDetection select vertices connected to at least |E|/2 other vertices in the consistency graph.Decode using Tis on corresponding edges.

Model 1 - Proof T|E| T1. . .r1r|E|D11D1|E|D|E|1D|E||E|12453T r,DT r,DT r,D12345Consistency graphDij=Tj(1).1+Tj(2).ri++Tj(n(1- )).rin(1- ) Dij=Tj(1).1+Tj(2).ri++Tj(n(1- )).rin(1- ) k(Tj(k)-Tj(k)).rik=0 Polynomial in ri of degree n over Fq,value of ri unknown to ZorbaProbability of error < n/q

Variations - FeedbackCp011

Variations Know thy enemyCp011Cp011

Variations Random NoiseCp0CN1SEPARATION

Model 2 - Multicast???

Model 2 - Resultsp = |Z|/h011C (Normalized by h)0.50.5hZSR1R|T|

Model 2 - Resultsp = |Z|/h011C (Normalized by h)0.50.5R1R|T|S

Model 2 Sketch of ProofR1R|T|SS|Z|S2S1Lemma 1: There exists an easy random design of network codes such that for any Z of size < h/2,if Z is known, each decoder can decode.Lemma 2: Using similar consistencycheck arguments as inModel 1, Z can be detected.EasyHard

THE END