MOBILITY AND SECURITY MODELS FOR WIRELESS SENSOR

NETWORKS USING VORONOI TESSELLATIONS

Manel Abdelkader, Mohamed Hamdi and Noureddine Boudriga

Communication Networks and Security Research Lab., University of Carthage, Carthage, Tunisia

Keywords:

Wireless sensor networks, Collaborative computing, Voronoi tessellation, Mobility models.

Abstract:

Recent advances in integrated electronic devices motivated the use of wireless sensor networks in many appli-

cations including target surveillance and tracking. A number of sensor nodes are scattered within a sensitive

region to detect the presence of intruders and forward subsequent events to the analysis center(s). Obviously,

the sensor deployment should guarantee and optimal event detection rate. This paper proposes a high-level

Voronoi-based technique to assess the area coverage based on information available locally for each sensor

node. We show that the proposed technique can be used to implement a coverage-preserving mobility process

to enhance the initial sensor deployment. We also highlight other potential applications of our approach.

1 INTRODUCTION

Wireless Sensor Networks (WSNs) are among the

technologies that will probably shape the ﬁrst decades

of the twenty ﬁrst century. These networks are mainly

cost-effective, easy to deploy, and multi-purpose. In

fact, WSNs have been used in various contexts in-

cluding mobile target detection, healthcare, water re-

source monitoring, and virtual reality. However, they

are also characterized by severe memory, CPU, and

(most importantly) energy limitations that hardens

their deployment in environments where voluminous

data should be processed and high-speed networks are

to be used to transmit these data. For instance, exist-

ing WSNs devised for military surveillance often pro-

vide coarse data about the hostile target(s) moving in

the battleﬁeld. This constraint, mainly ﬁxed by the

sensor node cost, considerably affects the efﬁciency

of the WSN.

The major WSN design issue that should be con-

sidered in target tracking applications is area cover-

age. This is because a sensor node detects the pres-

ence of hostile targets only if they are within its sens-

ing range. Therefore, the WSN detection perfor-

mance depends on how well the sensors observe the

physical space. In (Gui and Mohapatra, 2004), cov-

erage degree has been thought of as a measure of the

WSN quality of surveillance. A metric, called Aver-

age Linear Uncovered Length (ALUL), has been de-

veloped to estimate the average distance a mobile tar-

get can make before being detected by the sensor net-

work. Therefore, the ALUL can be used to assess the

detection efﬁciency of the WSN. However, the ma-

jor shortcoming of this approach is its heavy compu-

tational load making it non-conforming with the se-

vere processing and energy limitations characterizing

WSNs. This complexity is exacerbated when the met-

ric is extended to k-coverage assessment, where k > 1.

This paper proposes a coverage assessment ap-

proach amenable to implement advanced target track-

ing functionalities using a hybrid framework com-

posed of a large number of resource-impoverished

sensor nodes and a small number of powerful sen-

sor nodes. We rely on a WSN framework, called

WHOMoVeS (Wireless Hybrid Optimal Mobile Ve-

hicle Sensing), which has been introduced in (Obai-

dat, 2008) for military surveillance. To address this

issue, we build a higher-order Voronoi diagram of

the monitored region for an efﬁcient estimation of

the local coverage degree. A collaborative comput-

ing framework is set up to fulﬁll this task. Through-

out the paper, we give a general overview of the WSN

tasks that may take beneﬁt of the proposed coverage

assessment method (i.e., mobility modeling, activity

scheduling). To the best of our knowledge, this is the

ﬁrst time higher-order Voronoi tessellations are used

in the WSN context, even though simple Voronoi di-

agrams have already been investigated (Wang et al.,

2007; Stojmenovic et al., 2003; Vieira et al., 2003).

The major contributions of the paper are listed in

the following:

• The proposed cooperative coverage assessment

approach considerably reduces the computational

104

Abdelkader M., Hamdi M. and Boudriga N. (2010).

MOBILITY AND SECURITY MODELS FOR WIRELESS SENSOR NETWORKS USING VORONOI TESSELLATIONS.

In Proceedings of the International Conference on Data Communication Networking and Optical Communication Systems, pages 104-113

DOI: 10.5220/0002989501040113

Copyright

c

SciTePress

complexity with regard to existing methods

• Besides coverage optimization, higher-order

Voronoi tessellations can be useful for perform-

ing multiple tasks including activity scheduling or

distributed cryptographic protocols

• Local coverage information, gathered using the

Voronoi diagram, can be used to implementcover-

age preserving mobility models. A simple model

is presented in this paper to show that our idea

considerably enhances the WSN target detection

performance

The rest of the paper is structured as follows. Sec-

tion 2 gives an overview on the WHOMoVeS and

ALUL concepts, which are of utmost importance in

our work. A mathematical introduction to Voronoi

tessellations is presented in Section 2. Section 3

highlights the potential applications of k-Voronoi di-

agrams in WSNs. We particularly show in Section 4

how this tessellation is used to develop a coverage-

preserving mobility model. The higher-order Voronoi

diagram is also shown to be helpful in deﬁning ac-

tivity scheduling strategies and security schemes. Fi-

nally, Section 8 concludes the paper.

2 RELATED WORK

This section ﬁrst reviews a WSN framework that has

been introduced by the authors in (Obaidat, 2008).

Then, we give the basic deﬁnitions related to the

ALUL metric.

2.1 WHoMoVeS: a Framework for

Military Target Tracking

At this stage, the reader may wonder about the choice

of WHOMoVeS as a WSN framework. In fact, the

main reason motivating this choice is that WHO-

MoVeS builds upon a heterogeneous multi-layer ar-

chitecture enabling the support of advanced (image

and electromagnetic-based) target tracking function-

alities, which is very important for military applica-

tions. More accurately, Hamdi et al. (Obaidat, 2008)

present WHOMoVeS as an heterogeneous sensor net-

work composed of two layers:

• The core layer, consisting of sensor nodes

equipped with powerful data gathering and trans-

mission capabilities

• The sensing layer consisting of miniature devices

whose role is restricted to the detection of hostile

presence

Figure 1: Architecture of the proposed satellite-based com-

munication backbone.

Accurate tracking and long network lifetime are

achieved through a strong cooperation between those

layers.

According to this reasoning, the process of acquir-

ing and analyzing data related to mobile targets in the

battleﬁeld includes ﬁve steps as illustrated in Figure

1. These steps are brieﬂy described in the following:

1. Ground sensors detect the presence of a hostile

target in the monitoredﬁeld and store the events in

memory. The satellite periodically contacts sen-

sor nodes to download updates about target pres-

ence.

2. The satellite contacts the Uninhabited Aerial Ve-

hicles (UAVs) to acquire image data about the

scene where the intrusion has been detected.

3. The UAVs gather image data through the embed-

ded imaging sensors.

4. The UAVs establish connections with the satel-

lite communication backbone in order to transmit

high-quality multimedia data about the battleﬁeld.

5. Images related to multiple intrusion events are for-

warded through the broadband satellite backbone

to the analysis center where advanced tracking

functionalities are carried out.

2.2 The ALUL Metric

The Average Linear Uncovered Length (ALUL),

introduced in (Gui and Mohapatra, 2004), gives an

approximation of the distance that can be made by a

target before being detected by the sensor network.

The undetected path length of a target travel-

ing from location (x,y) with direction θ is given by

the the Linear Uncovered Length (LUL), denoted by

L ((x, y),θ). The average of target paths before detec-

tion at location x and over all directions is the Average

MOBILITY AND SECURITY MODELS FOR WIRELESS SENSOR NETWORKS USING VORONOI

TESSELLATIONS

105

Linear Uncovered Length (ALUL(x,y)) which means

the average distance can be traveled by a target at the

location (x,y) without be detected. Obviously, if (x,y)

is within the coverage of at least one sensor node then

ALUL(x,y) equals 0. More generally, ALUL(x) is cal-

culated as follows:

ALUL(x,y) ≡

(

0 : (x,y) is covered

R

2π

0

L ((x,y),θ)dθ

2π

: otherwise

(1)

The ALUL in an area A, denoted by ALUL(A) is

the mean uncovered distance that can be traveled by

a target without being detected by any of the nodes

deployed in the region of interest. The expression of

ALUL(A) is given by:

ALUL(A) ≡

R

(x,y)∈A

ALUL(x,y)dxdy

k A k

,

(2)

where k A k is the area of A.

3 HIGHER-ORDER VORONOI

TESSELLATIONS AND WSN

COVERAGE

The objective of this section is to provide a tool for

accurately gauging the coverage degree of the moni-

tored zone. To this purpose, we rely on higher-order

Voronoi diagrams to determine the sub-regions that

do not satisfy the k-coverage requirement. First, we

give a mathematical representation for higher order

Voronoi tessellation, which is a set of Voronoi cells.

Then, a parallel calculation framework allowing an

efﬁcient computation of this tessellation is provided.

3.1 Mathematical Modeling of

Higher-order Voronoi Diagrams

We start by the deﬁnition of the mathematical model

related to sensor nodes distribution. We identify the

groups of the k-nearest neighbors using the higher or-

der Voronoi model.

Let M be a metric space; δ : M × M → R de-

noting the Euclidean distance on M. We denote by

R = {p

i

,1 ≤ i ≤ N} ⊆ M , a set of N sensor nodes

having their coordinates in M.

The Voronoi diagram associated to R is the unique

subdivision deﬁned in M such that every part of the

subdivision contains the nearest neighbors deﬁned in

M for p

i

,1 ≤ i ≤ N, in R. Every subdivision part is

named a Voronoi Cell related to p

i

,1 ≤ i ≤ N, and is

determined using the following process.

For every p

i

, p

j

∈ R , we denote by H(p

i

, p

j

) the

half plane containing p

i

:

H(p

i

, p

j

) =

x ∈ M/δ(p

i

,x) < δ(p

j

,x)

. (3)

It can be noticed that H(p

i

, p

j

) is the half plane

delimited by the bisector of line segment [p

i

, p

j

] and

including p

i

.

The Voronoi cell related to p

i

is generated by the

deﬁnition of the common area between all half-planes

deﬁned aboveand containing p

i

. Therefore, a Voronoi

cell related to p

i

is expressed by:

V

R

(p

i

) =

\

p

j

∈R\{p

i

}

H(p

i,

p

j

). (4)

In our work, we are rather interested in partition-

ing M into isotopic cells according to k-nearest neigh-

bors for a given distribution of p

i

, p

j

∈ R. Start-

ing from a given sensor distribution, we search the

set P

(k)

i

= {p

i1

,..., p

ik

} containing the nearest sen-

sor neighbors. Such groups can be obtained using the

higher order-k Voronoi Diagram. The latter allows

deﬁning subsets of M containing the nearest elements

to P

(k)

i

. This can be performedby ﬁnding the elements

which are closer to the most distant neighboring of

P

(k)

i

than any other p

j

/∈ P

(k)

i

. In other terms, it can be

written that:

V(P

(k)

i

) =

(

x|∀p

j

∈ P\ P

(k)

i

, max

p

it

∈P

(k)

i

(δ(x, p

it

) ≤ δ(x, p

j

)

)

.

(5)

P

(k)

i

is called the generator of this Voronoi cell

V(P

(k)

i

). As for the order-1 voronoi cells, an order-k

cell is constructed using bisectors between its genera-

tors and the remaining of the metric space.

V(P

(k)

i

) =

T

p

j

∈R\P

(k)

i

[H(p

i1

, p

j

) ∩ ... ∩ H(p

ik

, p

j

)].

3.2 k-Voronoi Diagram Construction

In this section, we deﬁne the k-Voronoi diagram con-

struction model. The latter is based on the cooper-

ation of R elements. In the following, we present

the construction k-Voronoi diagram construction al-

gorithm.

Assumptions:

• Every iteration is related to an initial distribution

of R’s elements on M.

• Every sensor node present in the sensor layer

knows his “direct” neighbors (deﬁned in his de-

tection coverage or given by a core sensor).

• The presented algorithm is based on the parallel

PRAM algorithm. In this algorithm, we present

the parallel construction of k-Voronoi diagram.

DCNET 2010 - International Conference on Data Communication Networking

106

Algorithm 1: PRAM Algorithm.

Input:A set R of planar sensors, voronoi of order k−

1.

Output: the Voronoi diagram of order k.

1. Subdivide each region r

k−1

i

induced by P

(k−1)

t

⊂ R

into subregions according to V

1

(RP

(k−1)

t

)

2. Merge equivalent new subregions relevant to

neighboring r

k−1

i

.

3. Delete old edges and save the new vertices and

edges of each r

k

i

′

.

4 A K-VORONOI-BASED

MOBILITY MODEL

In this section, we show how higher-order Voronoi

diagrams can be used to implement sensor mobility

modeling. We consider two mobility models:

• The ﬁrst is an advanced model in which sensor

nodes move toward regions where the hostile tar-

get is supposed to be

• The second relies on estimating the uncovered

zones within a Voronoi cell and moving sensor

nodes toward the ’most uncovered region’

4.1 Advanced Mobility Model

Obviously, the ﬁrst model is more energy-consuming

since it encompasses the prediction of the target po-

sition. Therefore, we suppose that the second model

can be used when energy resources become scarce.

The performance of both models will be assessed in

the following sections.

Moreover, the prediction function is tightly related

to the coverage of the studied zone. In fact, the greater

is the number of target detection signals, the better is

the prediction precision. In the following, we distin-

guish both cases of a target crossing a k−covered and

a non k−covered zone.

For a Target Crossing a k−covered Zone

The mobility algorithm is triggered upon the detec-

tion of a target presence. Every ground sensor sends

his detection signal to the relevant intermediate sen-

sor. The latter collects all detection signals, veriﬁes

their integrity and deﬁnes the zones that might include

the target. The set of deﬁned zones are classiﬁed ac-

cording to the probability of presence of the target.

This probability reaches his maximum when a zone

is k−covered. The mobility algorithm is deﬁned as

follows:

1. The nearest k sensors s

i

, 1 ≤ i ≤ k, send their de-

tection signals to their intermediate sensors.

2. In the case where detection signals are sent to dif-

ferent intermediate sensors, the latters coordinate

to gather all signals at the IS with the highest num-

ber of detection signals.

3. IS veriﬁes the k−security of the received signals

and constructs the zone of presence of the target

z

t

.

(a) Let’s d

i

be the detection signal of the sen-

sor s

i

. d

i

= (r

ti

,α

ti

,θ

ti

,s

i

) where r

ti

=

p

(x

s

i

− x

t

i

)

2

+ (y

s

i

− y

t

i

)

2

, α

ti

= tan

−1

(

y

s

i

−y

t

i

x

s

i

−x

t

i

),

θ

ti

is the detection instant. For every s

i

, IS

computes the detection zone z

i

such that z

i

=

R

α

ti

+δα

α

ti

−δα

R

r

ti

+δr

r

ti

−δr

dαdr where δα, δr are the es-

timated detection error. The total target pres-

ence zone is resulted from the intersection be-

tween all the elementary detection zones. Thus,

z

t

= ∩

1≤i≤k

z

i

4. IS deﬁnes ∆Z as the zone surrounding z

t

and that

a target can not go beyond in the next mobility

step. IS computes the intersection between Z

T

=

z

t

+ ∆Z and the k−Voronoi diagram: ∪

p

(k)

i

∈S

∗

k

(Z

T

∩

V(P

(k)

i

)) = ∪

i

δV

(k)

i

, where δV

(k)

i

⊆ V

i

(P

(k)

i

) such

that V

i

(P

(k)

i

) is the Voronoi cell of the k−sensors

with index i.

5. To guarantee k−coverage in Z

T

, each δV

(k)

i

should be k−covered which means that δV

(k)

i

⊂

∩ Γ

1≤ j≤k

(s

j

,R

s

).

6. A mobility instruction is deﬁned by (r

i

,α

i

) where

r

i

≥ d(s

i

, p) such that ∃ p, ∀q ∈ δV

(k)

i

, d(s

i

, p) ≥

d(s

i

,q). and α

i

= argmax

d

xs

i

y where x,y ∈ v

i

and

v

i

is the set of the vertices of δV

i

.

For a Target Crossing a non k−covered Zone

In this case, only k

′

signed detection signals are re-

trieved by the intermediate sensors. IS proceeds at

the construction of the probable zone of presence of

the target as presented previously. In the same time, in

order to reﬁne the target presence zone, IS starts the

recovery of the remaining (k − k

′

) required signals.

For this purpose, IS proceeds as follows:

1. let’s z

i

be a probable zone of presence of a target,

p

i

be the probability of presence of a target where

p

i

= k

i

/k such that k

i

is the number of the veriﬁed

MOBILITY AND SECURITY MODELS FOR WIRELESS SENSOR NETWORKS USING VORONOI

TESSELLATIONS

107

detection signals received by the IS and k is the

minimum required number of signals.

2. IS deﬁnes the nearest k sensors to each part of

the zone z

i

. Thus, IS deﬁnes the intersection be-

tween z

i

and the k−Voronoi diagram and deduces

∪

i

δV

(k)

i

.

3. For each δV

(k)

i

, IS ascertain the sets of the nearest

ksensors, veriﬁes which sensors k

i

”, 0 ≤ k

i

” ≤ k

′

,

have sent detection signals. IS classiﬁes δV

(k)

i

ac-

cording to the value of k

i

”. The greater k

i

” is, the

most important is the probability of presence of

the target in δV

(k)

i

. A small value of k

j

” induces

that the target is going in or out δV

(k)

j

.

4. For each δV

(k)

i

, IS guides the (k− k”) nearest sen-

sors to move towards δV

(k)

i

. For that, he sends

them the mobility instruction including the proba-

bility of presence of a target A mobility instruc-

tion is deﬁned as.(r

i

,α

i

, p

i

) where r

i

≥ d(s

i

, p)

such that ∃ p, ∀q ∈ δV

(k)

i

, d(s

i

, p) ≥ d(s

i

,q). and

α

i

= argmax

d

xs

i

y where x, y ∈ v

i

and v

i

is the set of

the vertices of δV

i

, p

i

= k”/k is the probability of

presence of the target in δV

(k)

i

.

To enhance coverage while keeping more mobility

freedom, we suggest a group mobility model in which

ground sensors move in groups such that they pre-

serve a k−coverage. For this purpose, for each mo-

bility step, sensors deﬁne randomly groups of k mem-

bers for each, the latters are not required to be the

nearest neighbors. Each group deﬁnes randomly a

head which chooses the ﬁrst mobility step. The

remaining members of the group take into account

this choice to determine, in turn, their next mobil-

ity step. By this manner, each sensor’s mobility step

depends on his integrating group. Further, a sensor

may move from one group to another in each mobil-

ity step. This model enables the deﬁnition of overlap-

ping k−Voronoi groups which increases the guarantee

to have a k−coverage.

4.2 Simpliﬁed Mobility Model

We propose a mobility model which is only based on

the Voronoi diagram. The following proposition gives

a condition for a Voronoi cell to be partly uncovered.

Proposition 4.1. Let S be a set of sensor node posi-

tions and s

i

in S be a sensor node. If there exists n

j

in

N(s

i

,V(S)) such that d(s

i

,n

j

) > 2R

s

i

then V(s

i

) is not

fully covered.

Proof. We suppose that d(s

i

,n

j

) > 2R

s

i

. Let [v

p

,v

q

]

be the Voronoi edge deﬁned by n

j

and s

i

. The inter-

section of [v

p

,v

q

] and [s

i

,n

j

] is denoted by P. The

properties of the Voronoi diagram give that:

d(s

i

,P) = d(n

j

,P) =

d(s

i

,n

j

)

2

. (6)

Since d(s

i

,n

j

) > 2R

s

i

, we deduce from Equation 6

d(s

i

,P) > R

s

i

.

Consider the point Q ∈ [s

i

,P] such that d(s

i

,Q) =

R

s

i

. We can conclude that for every T ∈ [P, Q], T ∈

V(s

i

) and T /∈ Γ(s

i

,R

s

i

) (because d(s

i

,T) > R

s

i

). This

means that V(s

i

) is not totally covered.

This result can serve to implement a mobility al-

gorithm where a sensor node looks for one or more

neighbors that are at least 2R

s

i

-distant from it. If

such nodes exist, the sensor node moves toward the

most distant neighbor, denoted by n

f

, with a distance

d(s

i

,n

f

)−2R

s

i

2

.

Figure 2 illustrates this reasoning. In fact, we no-

tice that the disc centered in s

1

and having a radius

equal to R

s

i

does not cover the Voronoi cell generated

by s

1

. Hence, s

1

will move toward s

3

with a distance

d(Q,P).

Figure 2: Simpliﬁed mobility model.

The following corollaries extend this strategy to

the case where the monitored region is required to be

k-covered. For the sake of parsimony, we do not pro-

vide proofs for these corollaries in this paper.

Corollary 4.2. For s

i

in S, if |N(s

i

,V(S))| < k, where

|.| denotes set cardinality, then V(s

i

) is not k-covered.

Before giving the second corollary, we deﬁne, for

a sensor node s

i

in S, the set X(s

i

,V(S)) of intersec-

tion points expressed as follows:

X(s

i

,V(S)) =

^

V(S\ {s

i

})

\

Γ(s

i

,R

s

i

), (7)

where

e

P, for P ∈ R

2

denotes the boundary of P.

Informally speaking, X(s

i

,V(S)) denotes the in-

tersection of edges of the Voronoi diagramV(S\ {s

i

})

and the disk corresponding to the maximum sensing

coverage range of s

i

.

DCNET 2010 - International Conference on Data Communication Networking

108

Corollary 4.3. For s

i

in S, if |X(s

i

,V(S))| < k, then

V(s

i

) is not k-covered.

The major advantages of these results is that we

can rely on simple Voronoi diagrams to deal with k-

coverage while the advanced model proposed in the

previous subsection is based on k-Voronoi tessella-

tions which are more complex to build. A more ac-

curate comparison between the two models will be

carried out in the simulation section.

4.3 Group Mobility Modeling

To enhance coverage while keeping more mobility

freedom, we suggest a group mobility model in which

ground sensors move in groups such that they pre-

serve k-coverage. To this purpose, for each mobil-

ity step, sensors deﬁne randomly groups of k mem-

bers for each which are not required to be the near-

est neighbors. Each group has a leader which deﬁnes

mobility steps. The remaining members of the group

take into account this choice to determine, in turn,

their next mobility step. By this manner, each sen-

sor’s mobility step depends on his integrating group.

Further,a sensor may movefrom one group to another

in each mobility step. This model enables the deﬁni-

tion of overlapping k-Voronoi groups which increases

the guarantee to have a k-coverage. Thus, in the aim

to guarantee k-coverage all along the estimated target

path, the following model is deﬁned:

• A group leader is elected from the set of the near-

est nodes to the estimated target path and after re-

ceiving a mobility instruction. The group leader

follows the mobility instruction sent by IS. Other-

wise, groups will move away from the target path.

• Each group leader is in charge of gathering group

members. It searches increasingly in its neighbor-

hood.

• A member chooses to belong to a group as long

as it does not receive a mobility instruction from

an IS. Otherwise, a mobility instruction is priori-

tized.

• For a mobility step, a member could only belong

to a single group. It may then move to another

group for further mobility steps.

• A node may act as a group leader as long as it

receives mobility instructions from the IS.

5 EXTENSION TO

MULTI-TARGET TRACKING

The advanced mobility model have deﬁned the prob-

able zones of a target presence and drives sensors to-

wards these zones. Thus, a mobility instruction is

clearly deﬁned and weighted according to the prob-

ability of the target presence. In the case of mul-

tiple targets, a sensor may receive different mobil-

ity instructions and then it chooses which to follow.

Nevertheless, this may lead to uneven sensors distri-

butions. Hence, some targets may be not sufﬁciently

covered especially when the number of sensors is not

enough to cover all the targets’ estimated locations.

For these reasons, we propose in the following two

techniques enabling the extension of the advanced

mobility model for multi-target tracking.

5.1 Extended Advanced Mobility Model

To guarantee the coverage of multiple targets, we

present the modiﬁcations introduced to the advanced

mobility model. In the presented mobility model, sen-

sors are free to deﬁne their next movement, two main

situations may be deﬁned. In the ﬁrst one, sensors

follow the mobility instruction driving to the target;

so, they move towards the probable zones of target

presence. In the second, a sensor chooses another dif-

ferent direction taking him away from these zones.

The main idea introduced for multi-target tracking is

that even when sensors are in the second situation,

they remain in nearby locations increasing the proba-

bility to return to the target direction in next mobility

steps. This can be fulﬁlled through the customization

of velocity according to the chosen direction in the

sense that sensor’s mobility velocity increases when

sensors moves towards target location and inversely.

For this reason, we deﬁne two velocity ranges. The

ﬁrst, denoted by {V

hi

}, contains the high velocity val-

ues while the second, denoted by {V

li

}, contains the

low velocity values.

The extended advanced mobility model links the

sensor velocity to the chosen direction. Thus, mobil-

ity probability is deﬁned as follows.

• The probability that a node chooses a given ve-

locity is equal to the probability to choose target

direction.

• When receiving mobility instructions, the direc-

tion of the nearest targets have the higher proba-

bility. Consequently, they are assigned the higher

probability velocity values.

• Three subsets of velocity values may be deﬁned:

(1) a velocity valueV

t

enabling the sensor to reach

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109

the target position in the next mobility step; (2) a

velocity value from {V

hi

} when choosing the tar-

get direction but not sufﬁcient to reach the target;

(3) a velocity value from {V

li

} when choosing an

other direction.

The underlying probability distribution is deﬁned

as follows:

Pr

V

(v) =

P((r,α) = (r

t

,α

t

)) v = V

t

1

¯

V

hi

P((r,α) = (r

i

,α

i

)) v ∈ {V

hi

}

1

¯

V

li

(1− P((r,α) = (r

i

,α

i

)) v ∈ {V

li

}

5.2 Extended Mobility Model for

Multi-target Tracking

In the following, we divide the monitored zones into

regions related to the present targets. Let T be the

number of the tracked targets. In the following algo-

rithm, we identify the nearest sensors to each target.

Then, we move sensors such that target path remains

k-covered.

Algorithm 2: Extended group mobility model for multi-

target tracking.

While (number of sensors in target’s Delaunay trian-

gle ¿ threshold)

do

Compute T-Voronoi where T is the number of targets

Deﬁne the k-Voronoi in each T-cell

Apply the group mobility model in each cell

6 SENSOR ACTIVITY

SCHEDULING

In this section, we highlight the potential given by

Voronoi diagrams in implementing activity schedul-

ing strategies. We mainly show how sensors that do

not contribute effectively in enhancing the coverage

degree within a given zone can be detected and there-

fore turned-off for a laps of time.

Our idea is to exploit the properties of the Voronoi

tessellation to implement a distributed algorithm to

identify sensor nodes which sensing coverage is al-

ready covered by their neighbors. We ﬁrst give the

deﬁnition of a redundant sensor.

Deﬁnition 6.1. A sensor s

i

∈ S is said to be redundant

if, and only if:

/

0 6= Γ(s

i

,R

s

i

)

\

[

s

j

∈Σ(s

i

)

Γ(s

j

,R

s

j

)

= Γ(s

i

,R

s

i

),

(8)

where Σ(s

i

) = {s ∈ S : Γ(s,R

s

) ∩ Γ(s

i

,R

s

) 6=

/

0 ∧ s 6=

s

i

}.

The interest, from the energy consumption opti-

mization point of view, of identifying redundant sen-

sors is obvious since such nodes can be turned-off. In

fact, the information provided by a redundant sensor

about the presence of a hostile target can be obtained

from its neighbors. In the rest of the section, we look

for a characterization of redundant sensor nodes.

The following proposition gives a necessary and

sufﬁcient condition for node redundancy characteri-

zation.

Proposition 6.2. Let S be a set of sensor node po-

sitions in R

2

and X(s

i

) the set of intersection points

corresponding to s

i

∈ S. If s

i

is redundant if, and only

if:

X(s

i

,V(S)) ⊂

[

s

j

∈N(s

i

,V(S))

Γ(s

j

,R

s

j

). (9)

Proof. (i) Proof of ⇒: If a sensor s

i

is redundant, it

comes from Proposition 4.1 that:

N(s

i

,V(S)) ⊆ Σ(s

i

). (10)

Therefore, it can be written that:

[

s

j

∈N(s

i

,V(S))

Γ(s

j

,R

s

j

) ⊆

[

s

j

∈Σ(s

i

)

Γ(s

j

,R

s

j

). (11)

From Equations 8 and 11, it comes that if s

i

is redun-

dant then

Γ(s

i

,R

s

i

)

\

[

s

j

∈N(s

i

,V(S))

Γ(s

j

,R

s

j

)

= Γ(s

i

,R

s

i

).

Moreover, Equation 9 gives that X(s

i

,V(S)) ⊂

Γ(s

i

,R

s

i

). By transitivity of the inclusion operator, we

obtain:

X(s

i

,V(S)) ⊂

[

s

j

∈N(s

i

,V(S))

Γ(s

j

,R

s

j

).

(ii) Proof of ⇐: Trivial.

According to the proposition above, if there exists

x

j

∈ X(s

i

) such that x

j

/∈ Γ(s

i

,R

s

i

), then s

i

is not re-

dundant. Consequently, we propose an algorithm for

stating whether a node is redundant or not.

The following corollary extend the result of

Proposition 6.2 to the case where a k-coverage of the

monitored zone is needed. Obviously, the deﬁnition

of redundancyshould be slightly modiﬁed in this case

to encompass sensor nodes whose sensing coverage is

totally k-covered.

Corollary 6.3. Let s

i

in S be a sen-

sor node. For every x

j

in X(s

i

,V(S)), if

|{x

j

}

T

S

s

k

∈N(s

i

,V(S\{s

i

})

Γ(s

k

,R

s

i

)

| < k, then

s

i

is not redundant.

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110

Algorithm 3: Redundant sensor discovery.

∀s

i

∈ S

{ Compute N(s

i

,V(S));

Generate V(N(s

i

,V(S)) \ {s

i

}));

Compute X(s

i

,V(N(s

i

,V(S)) \ {s

i

})));

∀x

j

∈ X(s

i

,V(N(s

i

,V(S)) \ {s

i

})))

{ r:=0;

∀s

k

∈ N(s

i

,V(S))

{ if (x

j

/∈ Γ(s

k

,R

s

i

)) then

r:=1; } }

if (r=1) then

s

i

is not redundant;

else

s

i

is redundant; }

Using this corollary, the strategy deﬁned in the

above algorithm remains effective in sensitive con-

texts where the monitored area should be k-covered.

7 K-SECURITY IN WIRELESS

SENSOR NETWORKS

In this section, we will build on the security model

presented in (Sliti et al., 2008) to inroduce an opti-

mized k−security model. For this purpose, we use

higher-order Voronoi diagrams to develop a k out of

n threshold signature scheme. It allows any subset of

k sensor nodes deﬁned in a voronoi cell to generate a

valid signature. Conversely, any subset of k

′

individ-

ual signatures does not constitute a valid signature if

k

′

< k. In fact, since the monitored zone is subdivided

into Voronoi tesselation, each k−Voronoi cell deﬁnes

the nearest region to its k generators which means that

an eventualevent occuring in this region should be de-

tected by the k sensors.

In the following, we extend the well-known Shoup

threshold cryptosystem (Shoup, 2000) to the elliptic

curve context.

Let F

p

be a prime ﬁnite ﬁeld so that p is a prime

number and let a,b ∈ F

p

satisfying 4a

3

+ 27b

2

6= 0.

An elliptic curve E

p

(a,b) over F

p

is deﬁned by the set

of solutions of the following equation (called Weier-

strass equation):

y

2

≡ x

3

+ ax+ b(mod p), (12)

together with an extra point O called the point at

inﬁnity.

Cryptographic schemes based on ECC rely on

scalar multiplication of elliptic curve points. Given

an integer r and a point P ∈ E

p

(a,b), scalar multipli-

cation is the process of adding P to itself r times. The

result of this scalar multiplication is denoted r.P.

7.1 Distributed Key Management

Scheme

To implement the prposed k−security model, the sen-

sor network should be nriched by a Public Key In-

frastructure (PKI) that will be in charge of managing

the keys ans the certiﬁcates used in the various signa-

ture management phases. The basic PKI parameters,

including the number of CAs, the trust relationships

between these CAs, as well as the certiﬁcate lifetime

will obviously vary according to the features of the

monitored battleﬁeld and the nature of the military

mission. As far as we are concerned, we illustrate

the differentcryptographicfunctionnalitiesin the sim-

ple case where only one CA is considered. In fact,

these functionnalities should not change when being

applied in a generic context where multiple CAs may

be considered.

Key Generation

The key generation phase is performed by the CA as

follows:

- generate randomly two large primes p,q ∈ P,

where P is the set of prime integers, and p = 2p

′

+ 1,

q = 2q

′

+1, with p

′

and q

′

are themselves primes. De-

note t = pq and m = p

′

q

′

.

- generate the public exponent e ∈ P such that e >

n (n is the number of private keys). The public key is

π = (t,e).

- compute d ∈ Z such that de de ≡ 1modm.

- build the polynomial f(X) =

∑

k−1

i=0

a

i

X

i

∈ Z[X]

such that f(0) = d.

- compute, for 1 ≤ i ≤ n, κ

i

= f(i)modm. The

integer κ

i

is the private key of sensor node s

i

.

Key Revocation

Key revocation may occur in two cases:

1. The sensor node is physically accessed by unau-

thorized party. In this case, th sensor node should

ask for the revocation of its certiﬁcate before trig-

gering the tamper-proof functionnalities. To this

end, a secret parameter that should have been pre-

loaded by the CA before the deployment of the

WSN should have been sent to the CA in oreder

to revoke the corresponding certiﬁcate. Once this

task has been successfully conducted, the critical

components of the sensor node are intentionally

self-destructed.

2. The sensor node is suspected to be compromised.

The suspected node should ﬁrst be probed to state

whether it consists in an intruder node that has

spoofed the identity of the legitimate node or it

MOBILITY AND SECURITY MODELS FOR WIRELESS SENSOR NETWORKS USING VORONOI

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111

is a sensor node that has fallen under the con-

trol of non-authorized parties. To this purpose,

challenge-response authentication messages and

radio ﬁngerprinting can be used. If the suspected

node is a compromised one, a tolerance state,

where the sensor node can forward packets with-

out issuing alert messages, isinitiated. During this

state, the behavior of the potentially compromized

node is monitored to detect the anomalies that it

may generate. At the end of the tolerance inter-

val, the suspected node is either rehabilited or irre-

versibly discarded. From the certiﬁcate manage-

ment point of view, this pre-supposes that during

the tolerance state, the certiﬁcate of the suspected

node is suspended (i.e. subjected to a temporary

revocation). In addition, when the node is irre-

versibly de-activated, the certiﬁcate is deﬁnitively

revoked.

Key Renewal

It may occur that some sensor nodes are recuperated

and used for future surveillance missions. In such

cases, the key credentials are removedfrom the sensor

nodes and once the old certiﬁcate has been checked

for revocation, the CA generates a new private key

for the node of interest and uploads the correspond-

ing certiﬁcate in a physically protected zone of the

storage memory.

Key Distribution

Unlike traditional PKI frameworks, only one key dis-

tribution scheme is considered in our context. It

consists in the secure physical upload of the cryp-

tographic credentials by the CA. To implement this

scheme a secure procedure letting the CA access-

ing the sensitive zones of the non-volatile memory of

the sensor node should be available. This procedure

should be made available by the manufacturer and the

underlying access credentials should be submitted di-

rectly to the CA.

7.2 Intermediate Signature Generation

and Veriﬁcation Phase

An event V that may be related to the detection of a

hostile target must be converted to a point P = (v

1

,v

2

)

in E

m

(a,b). The conversion of a message to an ECC

point is done according to the approach proposed in

(Hankerson et al., 2004). Then, a sensor node s

i

can

generate its individual signature (σ

x

i

,σ

y

i

) using its pri-

vate key according to the following equation.

(σ

x

i

,σ

y

i

) = 2n!κ

i

(v

1

,v

2

)modm.

This signature will then be forwarded to the cor-

responding core sensor through a set of relay nodes

(belonging to the sensing layer) denoted by R =

{r

1

,..,r

u

} ⊆ {s

1

,..,s

n

}. An important functionality

that could be implemented by the relaying sensors is

to identify and withdraw false alert messages. This

feature is called intermediate veriﬁcation. We sup-

pose that a secret integer γ is broadcasted by the core

node (this broadcast should be enciphered using a

threshold encryption algorithm). Hence, node s

i

com-

putes the point

˜

P = γ.(σ

x

i

,σ

y

i

)mod t. Instead of send-

ing only the signed message to the core sensor, s

i

also

sends the integer ω, which stands for the order of

˜

P

(i.e., ω.

˜

P = O).

To check the validity of an individual signature,

node r

j

∈ R calculates the point Q as follows.

Q = ωγ.(σ

x

i

,σ

y

i

)mod t. (13)

The veriﬁcation succeeds if, and only if, Q = O.

Clearly, the robustness of the intermediate veriﬁ-

cation scheme builds upon the complexity, for the in-

truder, to determine γ. Effectively,computing γ know-

ing (σ

x

i

,σ

y

i

) and ω involves discrete logarithm compu-

tation within E

t

(a,b). This problem has been shown

to be intractable in (SEC 1, 2000).

7.3 k-Vornoi Cell Signature Veriﬁcation

Supposing that a core sensor has collected k signa-

tures generated by a subset of sensor nodes (s

i

)

i∈S

,

where S = {i

1

,..,i

k

}, the objectiveis to verifywhether

all these signatures are valid. To this purpose, we

compute:

w = Σ

k

i=1

2.λ

0,i

.(σ

x

i

,σ

y

i

)mod t,

where

λ

S

i, j

= n!

Π

j

′

∈S

(i− j

′

)

Π

j

′

∈S

( j − j

′

)

.

To verify the validity of the global signature, the

core sensor checks if:

ew = 4n!

2

(v

1

,v

2

)modt (14)

A proof of correctness is given in the following.

We ﬁrst develop the expression of w.

w = 4n!Σ

k

i=1

λ

0,i

j

( f(i)modm)(v

1

,v

2

)modt

From the properties of the determinant of the Van-

dermonde matrix, we obtain:

n! f(i) = Σ

j∈S

λ

S

i, j

f( j)modm.

Therefore, it can be written that:

w = 4n!

2

f(0)(v

1

,v

2

)modt = 4n!

2

d (v

1

,v

2

)modt

ew = 4n!

2

(v

1

,v

2

)modt.

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112

8 CONCLUSIONS AND

POTENTIAL EXTENSIONS

This paper presented two Voronoi-based mobility

models for target tracking using WSNs. The key ad-

vantage of these models is that they encompass the

potential target position in the construction of the mo-

bility instructions. This ensures that the locations

where the target is most probable to be are more cov-

ered than the rest of the monitored area. We also pro-

posed a redundancy discovery technique to enhance

the WSN cost-effectiveness. An enhancement of this

work to build a multi-target tracking framework is

currently under development.

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