multihop properties of distinct di erence con gurations properties of distinct di erence con...

Multihop Properties of Distinct DifferenceConfigurations

Maura B. Paterson

Department of Economics, Mathematics and StatisticsBirkbeck, University of London

20 January 2010

M.B. Paterson Multihop Properties of DDCs 1/9

Distinct Difference Configurations

zz

z z z

Definition (DD(m, r))

I m dots are placed in a square grid

I the distance between any two dots is at most r

I vector differences between dots are all distinct

(examples: Costas arrays, Golomb rulers, sonar sequences)


Security in Wireless Sensor Networks

Jovanov et al. Journal of NeuroEngineering and

Rehabilitation 2005 2:6

I battery-operated sensingdevices deployed in largenumbers

I form an ad hoc networkto transmit collected data

I applications can becommercial, military,humanitarian...


Key Predistribution for Wireless Sensor Networks

Definition (key predistribution scheme (KPS))

I nodes are assigned keys before deployment

I nodes that share keys can communicate securely

~ ~{k1, k5, k7} {k3, k5, k12}k5 !!!!!aaaaa~ ~~

{k1, k5} {k3, k4}

{k1, k3}k1 k3

two-hop path: nodes communicate via intermediate node(generalises to k-hop path)


Grid-Based Wireless Sensor Networks

f f f f f f f f ff f f f f f f f ff f f f f f f f ff f f f f f f f ff f f f f f f f ff f f f f f f f ff f f f f f f f ff f f f f f f f ff f f f f f f f f

v v v v v v vv v v v vv v v v vv v v v v

v v v v v

v

vv -� r

I maximise number of pairs of neighbours that cancommunicate securely (directly or using a two-hop path)

I minimise storage

I ensure resilience against node compromise

Observation: it is not necessary for two nodes to share more thanone key


Translated DDCs Overlap in at Most One Point

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Key Predistribution Using a DD(m, r)

I each sensor stores m keys

I each key is assigned to msensors

I two sensors share at mostone key

I the distance between twosensors that share a key isat most r •

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Two-hop Coverage

I Each node shares a key with m(m − 1) neighbours.

I The number of nodes that a given node can reach via a securetwo-hop path depends on the choice of DD(m, r).

Definition (Two-hop coverage of a DD(m, r))

The number of nodes that a given node can reach via a securetwo-hop path when keys are distributed based on the DD(m, r).

vv vfvvv vv

vv v vvv v v v vv

vv(2-hop coverage=18)


Questions

I Given m, what is the maximum possible 2-hop coverage of aDD(m, r)?

I Can you explicitly construct DD(m, r) achieving the maximumpossible 2-hop coverage?

yes!

I What is the smallest r for which there exists a DD(m, r)achieving the maximum possible 2-hop coverage?


Questions

I Given m, what is the maximum possible 2-hop coverage of aDD(m, r)?

I Can you explicitly construct DD(m, r) achieving the maximumpossible 2-hop coverage? yes!

I What is the smallest r for which there exists a DD(m, r)achieving the maximum possible 2-hop coverage?


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