Network RS Codes for Efficient Network Adversary Localization Sidharth Jaggi Minghua Chen Hongyi Yao

Disease Localization Heart 2

Network Adversary Localization Adversarial errors: The corrupted packets are carefully chosen by the enemies for specific reasons. Our object: Locating network adversaries

Network coding Network coding suffices to achieve to the optimal throughput for multicast[RNSY00]. Random linear network coding suffices, in addition to its distributed feature and low design complexity[TMJMD03]. S r1r1 r2r2 m1m1 m1m1 m2m2 m2m2 m2m2 m1m1 m 1 +m 2 am 1 +bm 2 5

Network Coding Aids Localization Routing scheme is used by u: x(e 3 )=x(e 1 ), x(e 4 )=x(e 2 ). Probe messages: M=[1, 2] s u r e1e1 e2e2 e3e3 e4e4 1 2 x= Y E =[3, 2] Y M =[1,2] E=Y E -Y M =[2,0] x[1,0]x[0,1] e3e3 e1e Y E =[7, 5] Y M =[5,3] E=Y E -Y M =[2,2] 3 2 e1e1 Random Network coding (RLNC): x(e 3 )=x(e 1 )+2x(e 2 ), x(e 4 )=x(e 1 )+x(e 2 ). x[1,1]x[2,1] x x x x Routing scheme is not enough for r to locate adversarial edge e 1. Network coding scheme is enough for r to locate adversarial edge e 1. 7 x 0x 2x back

RLNC for Adversary Localization [YSM10] 8 Desired features of RLNC  Distributed Implementation.  Achieving communication capacity.  Locate maximum number of adversaries.

RLNC for Adversary Localization [YSM10] 8 Drawbacks of RLNC  Require topology information.  Locating adversaries is a computational hard problem.

Our contribution: Network Reed-Solomon Codes 8 Network RS Codes preserves all the desired features of RLNC.  Distributed Implementation.  Achieving communication capacity.  Locate maximum number of adversaries. Furthermore, Network RS Codes Do not require topology information. Locate network adversaries efficiently.

Concept: IRV Edge Impulse Response Vector (IRV): The linear transform from the edge to the receiver. Using IRVs we and locate failures [1 0 0] [0 3 2] [2 9 6] Relation between IRVs and network structure: e1e1 e2e2 e3e3 IRV(e 1 ) is in the linear space spanned by IRV(e 2 ) and IRV(e 3 ) Unique mapping from edge to IRV: Two independent edges can have independent IRVs.

Adversary Localization by IRV Using network error correction codes [JLKHKM07], error vector E can be decoded at the receiver. Error E is in fact a linear combination of IRVs={IRV(e 1 ), IRV(e 2 ),…,IRV(e m )}. That is E=c 1 IRV(e 1 ) + c 2 IRV(e 2 ) + … + c m IRV(e m ). In particular, only the IRVs of adversarial edges has nonzero coefficients to E.

Adversary Localization by IRV Without loss of generality, assume e 1, e 1, …, e z are the adversarial edges. Thus, E=c 1 IRV(e 1 ) + c 2 IRV(e 2 ) + … + c z IRV(e z ). The adversarial edge number z is much smaller than the total edge number m. Therefore, locating adversaries is mathematically equivalent with sparsely decomposing E into IRVs.

Why RLNC is not good? Locating adversaries is mathematically equivalent with sparsely decomposing E into IRVs. For RLNC, IRVs are sensitive to network topology… For RLNC, IRVs are randomized chosen. Sparse decomposition into randomized vectors are hard [V97].

Key idea of Network RS Codes Motivated by classical Reed Solomon (RS) codes [MS77]. We want the IRV of e i to be its RS IRV IRV’(e i ), which is a randomly chosed column of RS parity check matrix. IRV’(e 2 )IRV’(e 4 )IRV’(e 1 )IRV’(e 5 )IRV’(e 3 ) (A 1 ) 1 (A 2 ) 1 (A 3 ) 1 (A 4 ) 1 (A 5 ) 1 (A 1 ) 2 (A 2 ) 2 (A 3 ) 2 (A 4 ) 2 (A 5 ) 2 (A 1 ) 3 (A 2 ) 3 (A 3 ) 3 (A 4 ) 3 (A 5 ) 3 (A 1 ) 4 (A 2 ) 4 (A 3 ) 4 (A 4 ) 4 (A 5 ) 4 Parity Check Matrix H of a RS code.

Nice properties of RS parity check matrix H Assume E is a sparse linear combination of the columns of H. We can decompose E into sparse columns of H in a computational efficient manner. Thus, if all edge IRVs equal their RS IRVs, we can locate network adversaries efficiently.

To achieve RS IRVs Each node, say u, performs local coding as follows.  Node u assume e 1 and e 2 have RS IRVs, i.e., IRV(e 1 )=IRV’(e 1 ) and IRV(e 2 )=IRV’(e 2 ).  Recall that the IRV of e 3 is in the span of IRV(e 1 ) and IRV(e2).  Node u chooses the coding coefficients such that IRV(e 3 )=IRV’(e 3 ). u e3e3 e2e2 e1e1

To achieve RS IRVs Surprisingly, previous local node scheme guarantees the desired global performance: each user’s IRV equals the corresponding RS IRV. Distributed Implementation. No topology information is needed.

Summary of our contribution Code TypeImplementCommunication Capacity Number of locatable Adversaries RLNC [YJM10]DistributedAchievedMaximum Network RS Codes DistributedAchievedMaximum Code TypeTopology Information Computational Complexity RLNC [YJM10]RequiredExponential time Network RS Codes Not neededPolynomial time

Network Coding Tomography for Network Failures Thanks! Questions? 14 Details in: Hongyi Yao and Sidharth Jaggi and Minghua Chen, Passive network tomography: A network coding approach, under submission to IEEE Trans. on Information Theory, and arxiv: