Data Consistency in Sensor Networks: Secure Agreement

Fatemeh Borran

Supervised by: Panos Papadimitratos, Marcin PoturalskiProf. Jean-Pierre Hubaux

IC-29 Self-Organised Wireless and Sensor Networks

Outline

• Introduction

• Problem Statement

• Assumptions

• System Model

• Algorithms

• Results

• Conclusion

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Introduction

• Classical Sensor Networks– centralized and reliable base station– one-to-many association

• Distributed Sensor Networks– decentralized architecture– every node could be faulty or malicious– many-to-many association

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Problem Statement

• Environment produces single actual value α

• Each sensor node measures the noisy environment

• Measurement error is bounded by ε

• All sensor nodes don’t behave correctly

• incorrect measurement or malicious behavior

Problem: value of single sensor node is not reliable

Goal: ensure data consistency among sensor nodes

Approach: agreement on actual value α

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Fault Model

Correct Sensor:• behave according to the protocol specification• measurement error is bounded by ε

Faulty Sensor:• measurement error is not bounded• follow assigned protocol

Byzantine Sensor:• under control of a unique adversary• behave arbitrary (crash-failure, omission-failure,…)

|F|≤ k

|B|≤ t

|C|≥ n-k-t

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System Model

System• Synchronous: transmission delay and process speed are

bounded and known• Asynchronous: slow process is not detectable

Authentication• Unique identity and signature• A modified message is detectable

Communication Channels• Integrity: every received message was previously sent• No-duplication: each message is received at most once• Reliability: messages sent by a correct node are received by all

nodes and are not modified.

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Secure Agreement Problem

Properties:• Validity: if si decides v, then |v-vi|≤ε and vi is initial value of

some non-Byzantine node

• Strong Validity: if si decides v, then |v-α|≤ε

• Agreement: if si decides vi and sj decides vj then |vi-vj|≤Φ

• Termination: every non-Byzantine node eventually decides

Primitives:• broadcast(vi)

• decide(v)

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Algorithm I: Synchronous One-hop

Vp := <p,xp>

r := 1

while r < t+1 do

broadcast(Vp) to all nodes

Vp := Vp U {Vq | Vq is received from q}

r := r + 1

end while

T := all duplicated values in Vp

Vp := Vp - T

decide(f(Vp))

f: trimming and averaging function

Wp := reduce(Vp,k+t-|T|/2)f(Vp) := mean(Wp)

r ≤ 1

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Theorem I

Theorem I: Algorithm I solves secure agreement for one-hop synchronous sensor networks with authenticated messages.

Lemma I: After t+1 rounds, all nodes have the same set.

Lemma II: All nodes apply the same deterministic function: f.

Communication complexity: O((t+1)n2)

S

P Q

x

<S, x>

Round 1

S

P Q

x

<S, x>

Round 2

<S, x>

S

P Q

x

<S, x>

Round 1

S

P Q

x

<S, x><S, y>

Round 2

<S, y><S, x>

y y

<S, y>

S is Byzantine

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Algorithm II: Synchronous One-hop

r := 1

while true do

broadcast(xp) to all nodes

Vp := U{<q,xq> | xq is received from q}

Wp := reduce(Vp,t+k)

xp := median(Wp)

if (δ(Wp) < Φ) then

decide(xp)

end if

r := r + 1

end while

Φ = ε => one round is requiredΦ < ε => two rounds are required

δ(Wp):= max(Wp) – min(Wp)

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Theorem II

Theorem II: Algorithm II solves secure agreement for one-hop synchronous sensor networks with authenticated messages.

Lemma I: Wp contains only the values from correct nodes.Lemma II: Every faulty node corrects its value after first

round.

Communication complexity: O(n)

Question: Is it possible to achieve O(c)complexity?

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Algorithm III: Synchronous One-hop

r := 1

S := arbitrary set of 2t+2k+1 nodes

while true do

if p in S then

broadcast(xp) to all nodes

end if

… // same as Algorithm II

r := r + 1

end while

Communication complexity: O(2t+2k+1)

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Modified Algorithm II: Asynchronous One-hop

r := 1

while true do

broadcast(xp) to all nodes

Vp := U{<q,xq> | xq is received from q}

if (|Vp | ≥ n-t) then

Wp := reduce(Vp,t+k)

xp := median(Wp)

if (δ(Wp) < Φ) then

decide(xp)

end if

end if

r := r + 1

end while

|V|: cardinality of VΦ = ε => one round is required in best caseΦ < ε => t rounds are required in best case

Multi-hop Communication

Connectivity: there is a path between each pair of non-Byzantine nodes in the network.

t-connectivity: there are no t nodes whose removal disconnects the network

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Correct node

Faulty node

Byzantine node

Communication range

unconnected network

Multi-hop Communication

Connectivity: there is a path between each pair of non-Byzantine nodes in the network.

t-connectivity: there are no t nodes whose removal disconnects the network

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Correct node

Faulty node

Byzantine node

Communication range

unconnected network

Multi-hop Communication

Connectivity: there is a path between each pair of non-Byzantine nodes in the network.

t-connectivity: there are no t nodes whose removal disconnects the network

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connected network

Correct node

Faulty node

Byzantine node

Communication range

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Modified Algorithm I: Synchronous Multi-hop

Vp := <p,xp>

r := 1

while r < t+d+1 do

broadcast(Vp) to all nodes

Vp := Vp U {Vq | Vq is received from q}

r := r + 1

end while

T := all duplicated values in Vp

Vp := Vp – T

decide(f(Vp))

f: trimming and averaging function

Wp := reduce(Vp,k+t-|T|/2)f(Vp) := mean(Wp)

d: network diameter

r < d+1

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Theorem III

Theorem III: Algorithm I solves secure agreement for multi-hop synchronous sensor networks with authenticated messages.

Lemma I: After t+d+1 rounds, all nodes have the same set.

Lemma II: All nodes apply the same deterministic function: f.

Lemma III: t-connectivity ensures agreement and termination.

Communication complexity: O((t+d+1)n2)

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Algorithm IV: Asynchronous Multi-hop

Vp := <p,xp>

r := 1

while true do

broadcast(Vp) to all nodes

Vp := Vp U {Vq | Vq is received from q}

if (|Vp| > 2(t+k)) then

Wp := reduce(Vp,t+k)

xp := median(Wp)

if (δ(Wp) < Φ) then

decide(xp)

end if

end if

Vp := <p,xp>

r := r + 1

end while

Φ = ε => one round is required in best caseΦ < ε => n-2t-2k rounds are required in best case

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Theorem IV

Theorem IV: Algorithm IV solves secure agreement for multi-hop asynchronous sensor networks with authenticated messages.

Lemma I: Within 2(t+k)+1 values, t+k+1 values are correct.

Lemma II: All nodes apply the same deterministic function: f.

Lemma III: t-connectivity ensures termination.

Communication complexity: O(2(t+k)n)

Results: One-hop

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Algorithm System Assumption Communication complexity

Algorithm I Synchronous n > 2t+2k O(n2)

Algorithm II Synchronous n > 2t+2k O(n)

Algorithm III Synchronous n > 2t+2k O(2(t+k))

Algorithm II’ Asynchronous n > 3t+2k O(n) *

Table I: Secure Agreement with Strong Validity

* best case results

Results: One-hop

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Algorithm System Assumption Communication complexity

Algorithm I Synchronous n > 2t O(tn2)

Algorithm II Synchronous n > 2t O(tn)

Algorithm III Synchronous n > 2t O(2t(t+k))

Algorithm II’ Asynchronous n > 3t O(tn) *

Table II: Secure Agreement with Validity

* best case results

Results: Multi-hop

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Algorithm System Assumption Communication complexity

Algorithm I’ Synchronous n > 2t+2k O(dn2)

Algorithm IV Asynchronous n > 2t+2k O(2(t+k)n) *

Table III: Secure Agreement with Strong Validity

* best case results

Results: Multi-hop

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Algorithm System Assumption Communication complexity

Algorithm I’ Synchronous n > 2t O((t+d+1)n2)

Algorithm IV Asynchronous n > 2t O(2(t+k)(n-2t-2k)n)*

Table IV: Secure Agreement with Validity

* best case results

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Conclusion

• Distributed sensor networks vs. Classical sensor networks.• save communication bandwidth• provide redundancy• eliminate single-point of failure• use broadcast instead of unicast• inform quickly and easily the end-user

• Data consistency as agreement problem. • New variant of agreement problem: secure agreement.• Φ can be chosen arbitrarily small to get as close to

consensus as desired.• t-connectivity is not required to be held in every round.

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Future works

• Strong validity requires n>2(t+k)Impossibility results with n≤2(t+k)?

• Asynchronous algorithm with constant communication complexity?

• Analyse communication complexity of worst case in asynchronous algorithms?

• Simulation results