Download - Multihop Properties of Distinct Di erence Con gurations Properties of Distinct Di erence Con gurations Maura B. Paterson ... vvvvvvv vvvvv vvvvv vvvvv vvvvv ... two-hop path depends

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

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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?


Download - Multihop Properties of Distinct Di erence Con gurations Properties of Distinct Di erence Con gurations Maura B. Paterson ... vvvvvvv vvvvv vvvvv vvvvv vvvvv ... two-hop path depends

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