Network RS Codes for Efficient Network Adversary Localization

Sidharth JaggiMinghua Chen

Hongyi Yao

Disease Localization

Heart

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Network Adversary Localization

•Adversarial errors: The corrupted packets are carefully chosen by the enemies for specific reasons.

•Our object: Locating network adversaries.

001001

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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].

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r1 r2

m1

m1

m2

m2

m2m1

m1+m2am1+bm2

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Network Coding Aids Localization

Routing scheme is used by u: x(e3)=x(e1), x(e4)=x(e2).

Probe messages: M=[1, 2]

su

r

e1

e2

e3

e4

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x=2

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7

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YE=[3, 2] YM=[1,2]E=YE-YM=[2,0]x[1,0]x[0,1]

e3e1

3+2

3+2 2.YE=[7, 5] YM=[5,3]E=YE-YM=[2,2]

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e1

Random Network coding (RLNC): x(e3)=x(e1)+2x(e2), x(e4)=x(e1)+x(e2).

x[1,1]x[2,1]

x

xx

x

Routing scheme is not enough for r to locate adversarial edge e1.

Network coding scheme is enough for r to locate adversarial edge e1.

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x

0x

2x

back

RLNC for Adversary Localization [YSM10]

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Desired features of RLNC

Distributed Implementation.

Achieving communication capacity.

Locate maximum number of adversaries.

RLNC for Adversary Localization [YSM10]

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Drawbacks of RLNC

Require topology information.

Locating adversaries is a computational hard problem.

Our contribution: Network Reed-Solomon Codes

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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.

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[1 0 0]

1 00

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[0 3 2]

[2 9 6]

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2 41. Relation between IRVs and network structure:

e1

e2

e3

IRV(e1) is in the linear space spanned by IRV(e2) and IRV(e3).

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2. 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(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.

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.

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 ei to be its RS IRV IRV’(ei), which is a randomly chosed column of RS parity check matrix.

IRV’(e2) IRV’(e4) IRV’(e1) IRV’(e5) IRV’(e3)

(A1)1 (A2) 1 (A3) 1 (A4) 1 (A5) 1

(A1)2 (A2) 2 (A3) 2 (A4) 2 (A5) 2

(A1)3 (A2) 3 (A3) 3 (A4) 3 (A5) 3

(A1)4 (A2) 4 (A3) 4 (A4) 4 (A5) 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 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).

u

e3

e2e1

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 Type Implement Communication Capacity

Number of locatable

Adversaries RLNC [YJM10] Distributed Achieved MaximumNetwork RS

CodesDistributed Achieved Maximum

Code Type Topology Informatio

n

Computational Complexity

RLNC [YJM10]

Required Exponential time

Network RS Codes

Not needed Polynomial time

Network Coding Tomography for Network Failures

Thanks!

Questions?

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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