Network RS Codes for Efficient Network Adversary Localization

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

2. Disease Localization Heart 2

3. Network Adversary Localization 001001 • Adversarial errors: The corrupted packets are carefully chosen by the enemies for specific reasons. • Our object: Locating network adversaries. 3

4. Network coding S • 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]. m1 m2 m1 m2 am1+bm2 m1+m2 m1 m2 r1 r2 5

5. back e1 x x x x x=2 . 3+2 2 e1 e3 Network Coding Aids Localization • Random Network coding (RLNC): x(e3)=x(e1)+2x(e2), x(e4)=x(e1)+x(e2). • Routing scheme is used by u:x(e3)=x(e1), x(e4)=x(e2). Probe messages: M=[1, 2] e1 e3 3 1 3 2x 7 3 x YE=[3, 2] YM=[1,2] E=YE-YM=[2,0] YE=[7, 5] YM=[5,3] E=YE-YM=[2,2] s r 2 2 u 2 2 x 5 0 x[1,1] x[2,1] x[0,1] x[1,0] 3+2 e2 e4 • Network coding scheme is enough for r to locate adversarial edge e1. • Routing scheme is not enough for r to locate adversarial edge e1. 7

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

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

8. Our contribution: Network Reed-Solomon Codes • 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. 8

9. Concept: IRV 0 0 Edge Impulse Response Vector (IRV): The linear transform from the edge to the receiver. UsingIRVswe and locate failures. 1 [2 9 6] e1 [0 3 2] 3 1 2 e3 3 1 3 1 1. Relation between IRVs and network structure: 2 3 4 2 1 3 9 IRV(e1) is in the linear space spanned by IRV(e2) and IRV(e3). [1 0 0] 6 2 e2 2 1 0 9 6 0 2. Unique mapping from edge to IRV: Two independent edges can have independentIRVs. 9

10. 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(e1), IRV(e2),…,IRV(em)}. That is • E=c1 IRV(e1) + c2 IRV(e2) + … + cm IRV(em). • In particular, only the IRVs of adversarial edges has nonzero coefficients to E.

11. Adversary Localization by IRV • Without loss of generality, assume e1, e1, …, ez are the adversarial edges. • Thus, E=c1 IRV(e1) + c2 IRV(e2) + … + cz IRV(ez). • 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.

12. 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].

13. Key idea of Network RS Codes • Motivated by classical Reed Solomon (RS) codes [MS77]. • We want the IRV of ei to be its RS IRV IRV’(ei), which is a randomly chosed column of RS parity check matrix. Parity Check Matrix H of a RS code.

14. 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.

15. To achieve RS IRVs • Each node, say u, performs local coding as follows. • Node u assume e1 and e2 have RS IRVs, i.e., IRV(e1)=IRV’(e1) and IRV(e2)=IRV’(e2). • Recall that the IRV of e3 is in the span of IRV(e1) and IRV(e2). • Node u chooses the coding coefficients such that IRV(e3)=IRV’(e3). e3 u e1 e2

16. 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.

17. Summary of our contribution

18. Network Coding Tomography forNetwork Failures • Thanks! • Questions? 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: 0908-0711 14