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An Algebraic Watchdog for Wireless Network Coding
MinJi Kim†
Joint work with
Muriel Médard†, João Barros‡, Ralf Kötter*
†Massachusetts Institute of Technology‡University of Porto
*Technischen Universität München
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Background• Secure network coding
– Network error correction [Yeung et al. 2006]– Resilient coding in presence of Byzantine adversaries
[Jaggi et al. 2007]– Signature scheme [Charles et al. 2006][Zhao et al. 2007] – Locating attackers [Siavoshani et al. 2008]– NOTE: downstream nodes check for adversaries, the upstream
nodes unaware.
• Watchdog and pathrater [Marti et al. 2000]– Extensions of Dynamic Source Routing– Detect/mitigate misbehavior of the next node– Use wireless medium: promiscuous monitoring
• Combine the benefits of network coding and watchdog– Focus on two-hop network
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Problem Statement
Intended transmission in E1Intended transmission in E1
Overhearing with noise in E2Overhearing with noise in E2
• Wireless network G = (V, E1,E2).– V : Set of nodes in the network– E1: Set of hyperedges for connectivity/wireless links– E2: Set of hyperedges for interference• Transition probability known (Binary symmetric
channel)
Is v3 consistent with…• Overheard packets from v2 and v3?• Channel statistics?
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Problem Statement
Intended transmission in E1Intended transmission in E1
Overhearing with noise in E2Overhearing with noise in E2
• How can upstream nodes (v1 and v2) detect misbehaving node (v3) with high probability?
Routing: Packets individually recognizable
Network Coding: Packets are mixed
Errors from BSC channel : Probabilistic detection
Few bit errors can make dramatic change in the algebraic interpretation
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Packet Structure
• A node vi that receives messages xj ’s and transmits pi
– Note: hash is contained in one hop, dependent on in-degree• Goal:
If vi transmits xi = e + Σ αj xj where e≠0, detect it with high probability.
– Even if |e| small, the algebraic interpretation may change dramatically.
aj’s xicoding coefficients aj’s coded data xi = Σ αj xj
pi = h(xj)
hash of received messages h(xj)
h(xi)
hash of message h(xi)
aj’s h(xj) h(xi)
header: protected with error correction codes
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Algebraic Analysis
• v1 knows:
x1
h(x1)Estimate of x2: 2
h(x2)Estimate of x3: 3
h(x3)a1 and a2
Note: • h(x3) and x3 consistent• Errors in a1 and a2 translates to errors in x3
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Algebraic Analysis
• v1 knows:
x1
h(x1)Estimate of x2: 2
h(x2)Estimate of x3: 3
h(x3)a1 and a2
• v1 computed all “plausible” x3
• • • Intersect this with all typical x3
• v1 claims that v3 is misbehaving if this intersection is empty.
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Algebraic Analysis• Lemma 1: For n large enough, probability of false detection ≤ ε for
any constant ε.– If a neighbor sends valid packets, then the node overhears valid
information with noise introduced by the channel only.
• Lemma 2: P(A malicious v3 is undetected by v1) is
where ri→j is the radius such that the probability that the interference channel/noise from vi to vj is within a ball of radius ri→j is at least 1- ε.
• Using Lemma 2 (and equivalent result for v2), probability of misdetection is:Prob that v3
passes v2’s check
Prob that v3 passes v1’s check
Number of potential msgs v3 can send
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Graphical Model
• v1 knows:
x1
h(x1)Estimate of x2: 2
h(x2)Estimate of x3: 3
h(x3)a1 and a2
Layer 1: ( 2, h(x2)) Layer 2: x2
hash value: h(x2)
Layer 3: x3 Layer 4: ( 3, h(x3))
hash value: h(x3)a1 x1 + a2 x2
PermutationChannel Errors Channel Errors
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Graphical Model
• 4 Layers:– Layer 1 & 4: 2n+h vertices, representing [codeword, hash] pairs– Layer 2 & 3: 2n vertices, representing codewords
Layer 1: ( 2, h(x2)) Layer 2: x2 Layer 3: x3 Layer 4: ( 3, h(x3))
P(x2|Channel ∆( 2 , x2) & h(x2)) P(x3|Channel ∆( 2 , x3) & h(x3))
Compute x3 given x2
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Graphical Model
• Start & destination point in Layer 1 and 4: what v1 overhears.
• Computes the sum of the product of the weights of all possible paths from start to destination (= the probability that v3 is consistent)
• This model illustrates sequentially/visually the inference process.
Layer 1: ( 2, h(x2)) Layer 2: x2 Layer 3: x3 Layer 4: ( 3, h(x3))
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Summary• Probabilistically police downstream neighbors• Algebraic analysis:– Exact formulae for probabilities of misdetection and false-
detection• Graphical model: – Capture inference process– Compute/approximate probabilities of consistency within
the network
Future Work:– Generalize to multiple sources, multi-hop network– Combine with reputation based protocol and some practical
considerations
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Extra Slides
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Is v3 behaving?
Is v3 consistent with…• Overheard packets from v1 and v3?• Channel statistics?
Problem Statement
How to fool v2?• Insert errors without being noticed?• Lie about message from v1?
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Two-hop Network• Graphical model– Explains the decision process
• Algebraic analysis– Understand the performance of the protocol
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Graphical Model• 4 Layers:
– Layer 1 & 4: 2n+h vertices, representing [codeword, hash] pairs– Layer 2 & 3: 2n vertices, representing codewords
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Graphical Model
• Edges:– [v,u] in Layer 1 to w in Layer 2 iff h(w) = u .
Normalized, but edge weight proportional to:
– v in Layer 2 to w in Layer 3 iff All edge weights = 1.
– v in Layer 3 to [w,u] in Layer 4 iff h(v) = u . Normalized, but edge weight proportional to:
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Extensions
• More than 2 sources:– Generalized graphical model – Use Viterbi-like Algorithm to compute:• Most likely path (i.e. set of codewords)• Total probability of reaching a linear combination
• Multi-hop:– As long as not dominated by the adversaries– Hidden terminal problem: the probability of
detecting decreases, but still possible.
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Future Work
• Generalize to multiple sources, multi-hop network– Develop models/framework (cascading graphical model?)
• Develop inference methods/approximation algorithms to efficiently make decision regarding malicious neighbors
• Combine with reputation based protocol and some practical considerations
• Eventually, develop/analyze a protocol which allows nodes to probabilistically verify and locally police their neighbors (especially downstream)– Self-checking network