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Network Coding and Reliable
Communications Group
A Multi-hop Multi-source Algebraic Watchdog
Muriel Médard†
Joint work with MinJi Kim†, João Barros‡
†Massachusetts Institute of Technology
‡University of Porto
Network Coding and Reliable
Communications Group
Background
• Secure network coding
– Network error correction [Yeung et al. 2006]
– Resilient coding in presence of Byzantine adversaries
[Jaggi et al. 2007]
– Confidential coding scheme [Vilela et al. 2008]
– 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
• Algebraic Watchdog [Kim et al. 2009]
– Combine the benefits of network coding and watchdog
– Extend to multi-hop, multi-source setting
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Problem Statement
Is vm+1 consistent
with…
• Overheard
packets from v2 ,
v3 ,… vm?
• Channel
statistics?
• Wireless network G = (V, E1,E2).
Overhearing with noise in E2
Intended transmission in E1
– 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)
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Problem Statement
Overhearing with noise in E2
Intended transmission in E1
• How can upstream nodes (v1, v2, …,vm) detect misbehaving
node (vm+1) 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
coded data xi = Σ αj xj with errorcorrecting code Ci = (n, ki, di)
coding coefficients aj’s
pi =
aj’s
h(xj)
h(xi)
hash of received messages h(xj)
xi
hash of message h(xi)
header: protected with error correction codes
• 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.
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Threat Model
• Adversary
–
–
–
–
Eavesdrops its neighbors’ transmissions
Injects/corrupts packets
Computationally unbounded
Knows the channel statistics, but does not know the
specific realization of the channel errors
• Adversary’s objective: Corrupt information flow
without being detected by other nodes
• Our objective: limit errors introduced by the
adversaries to be at most that of the channel
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Algebraic Watchdog
• Focus on v1
– Listens to neighbors and infer the messages: Using
transition matrix T
– Combines the inferred messages to “guess” what
the next hop node should transmit: Watchdog
trellis & Viterbi-like algorithm
– Check the “guessed message” with next-hop
node’s transmission: Inverse transition matrix T-1
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Transition Matrix/List T
• Relates the overheard
information
from
source vi to list of candidates
(inferred list of xi)
Overheard information
Start state
Overheard information
Inferred information
xi
y
Edge iff
Edge weight
probability of receiving
message:
proportional to
given y is original
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Watchdog Trellis
• Uses overheard & inferred
information (candidates)
to generate a list of
“guesses” on what vm+1
should send
Combine information from v2
What v1
already has
Layer 1
α1x1
Start
state
Combine infor- Combine information from vm-1 mation from vm
Layer 2
Layer 3
Layer m-1 Layer m
α1x1 +α2x2 α1x1 +α2x2 +α3x3 Σ1≤i≤m-1 αixi Σ1≤i≤m αixi
“guesses” are
states with positive
weight at Layer m
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Inverse Transition Matrix T-1
Overheard information
• Using the “guesses”
generated, checks that
vm+1 is well-behaving
• Same as T, just inverse
Guesses
Σ1≤i≤m αixi
Overheard information
[xm
̃ +1,h(xm+1)]
Edge iff
Inferred linear
combinations
(guesses)
Σ1≤i≤m αixi
y
Edge weight
End
node
probability of receiving
original message:
proportional to
given y is
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Decision Making
Layer 1
α1x1
Layer 2
Layer 3
Layer m-1 Layer m
α1x1 +α2x2 α1x1 +α2x2 +α3x3 Σ1≤i≤m-1 αixi Σ1≤i≤m αixi
Overheard information
[xm
̃ +1,h(xm+1)]
End
state
Start
state
“Guesses”
• Total weight of end state = p* = probability of overhearing
given channel statistics
• Can use various decision policy, such as threshold decision
rule p*>t
– Depending on the rule, different false positive/false negative
probabilities
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Simulation Results: Varying adversarial attack
Adversarial relay (flips
bit with probability padv)
•
•
•
•
All channel noise: 10%, i.e. BSC(0.1)
3 sources
10-bit field size
2-bit hash size
Honest relay (does not
inject errors)
When adversary injects
more than channel
noise (10%), the p*adv
and p*relay have different
distribution!
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Conclusions
• Probabilistically police downstream neighbors in a multi-hop,
multi-source network using network coding
– Only discussed multi-source, two-hop setting
• Trellis-like graphical model:
– Capture inference process
– Compute/approximate probabilities of consistency within the network
(Viterbi-like algorithm)
• Preliminary simulation results agree with the intuition
Future Work:
– Combine with reputation based protocol and some practical
considerations
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EXTRA SLIDES
Network Coding and Reliable
Communications Group
Multi-hop Algebraic Watchdog
• As long as the min-cut to any node from the
source is not dominated by adversarial node,
can detect malicious behavior
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Multi-hop Algebraic Watchdog
S0 v1
S2
v4
S0 monitors v5
S1 monitors v7
S1 monitors v8
v2
v3
v5
v6
S2 monitors v4
v7
v8
S1
edges in E1
• As long as the min-cut to any node from the
source is not dominated by adversarial node,
can detect malicious behavior
Network Coding and Reliable
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Simulation Results: Varying hash size
Hash size
(in bits)
Adversarial relay (flips bit
with probability 10%)
Honest relay (does not
inject errors)
Hash size > 1 bit sufficient
• All channel noise & adversarial attack level: 10%, i.e. BSC(0.1)
• 3 sources
• 10-bit field size
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Simulation Results: Varying channel noise
Channel noise
between sources
•
•
•
•
Adversarial relay (flips bit
with probability 10%)
Honest relay (does not
inject errors)
When channel noise >
Adversarial attack level: 10%, i.e. BSC(0.1) 10% (adversarial attack
3 sources
level), then may not be
10-bit field size
able to detect the
2-bit hash size
adversary!
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Simulation results: Varying number of sources
Number of
sources
Adversarial relay (flips bit
with probability 10%)
• All channel noise &
adversarial attack level:
10%, i.e. BSC(0.1)
• 3 sources
• 10-bit field size
• 2-bit hash size
Honest relay (does not
inject errors)
When only one source, v1 can detect (even by itself)
v1 can detect adversary when there are moderate
number of sources
with high probability
v1 can not detect by itself when many sources
• Need more hash or better overhearing channel
• Does not take into account other nodes vi’s
independent watchdog