The 2011 Military Communications Conference - Track 1 - Waveforms and Signal Processing

Eavesdropping and Jamming in Next-Generation Wireless

Networks: A Game-Theoretic Approach Quanyan Zhut, Walid Saad+, Zhu Han**, H. Vincent Poor* and Tamer Ba�art

tCoordinated Science Laboratory, University of Illinois at Urbana-Champaign, IL, USA, E-mail: {zhu31, basarl }@illinois.edu tElectrical and Computer Engineering Department, University of Miami, Coral Gables, FL, USA, E-mail: {w.saad}@miami.edu

'Electrical Engineering Department, Princeton University, Princeton, NJ, USA, E-mail: {poor}@princeton.edu

" Electrical and Computer Engineering Department, University of Houston, TX, USA, E-mail: zhan2@mail.uh.edu

Abstract- The efficient design of next-generation wireless sys- their perceived quality-of-service but also the security of their tems faces multifaceted problems involving a variety of node types transmission. These challenges have led to significant research such as wireless users, relay stations, base stations, eavesdroppers, works that address the physical layer security of wireless net­and jammers. In this paper, the complex decision making pro-cesses between a network of wireless users that perform uplink works as well as the challenges of deploying relay stations in

transmission via relay stations and an active malicious node, that next-generation networks. From a security perspective, using

is able to act as an eavesdropper and as a jammer, is studied. A information-theoretic techniques, the authors in [4] analyze the noncooperative game in which the users and the malicious node capacity region for the Gaussian and fading broadcast channels, are the players is formulated. On the one hand, the users seek to in the presence of eavesdroppers. The work in [5] studies the choose the relay station that maximizes their utilities which reflect their mutual interference as well as the security of the chosen path. interactions between a jammer and a multiple antenna system

On the other hand, the objective of the malicious node is to choose using game-theoretic techniques. The achievable rate region

whether to eavesdrop, jam, or use a combination of both strategies, and the optimal power allocation techniques for improving the in a way to reduce the total network capacity. To solve the game, physical layer security of a multiple antenna wireless network a fictitious play-based algorithm is proposed using which the users are studied in [6]. Other studies on the physical layer security and the malicious node reach a mixed-strategy Nash equilibrium. Simulation results show that the proposed approach improves the of wireless networks are given in [7] and [8] (For a review, see

average expected utility per user up to 49.4% relative to a nearest [9].).

neighbor algorithm. The results also show how the malicious node From a relaying perspective, different possible relay archi­can strategically decide on whether to jam or eavesdrop depending tectures for an LTE-advanced network are discussed in [2]. on its capabilities and objectives. The work in [10] considers the optimal deployment of a single

I. INTRODUCTION relay station for two-hop transmission in a multi-hop WiMAX With the recent advances in communication technologies, it is network. In [11], the authors propose a resource management al­

expected that next-generation wireless systems will be character- gorithm for optimizing the bandwidth reservation and admission ized by their heterogeneity due to the presence of different node control in the presence of relay station nodes. types such as wireless users, relay stations, eavesdroppers, and Briefly, the majority of the existing literature is focused either jammers. The emergence of cooperation as a novel networking on the study of the various performance aspects of deploying paradigm has motivated the deployment of relay station nodes in relay stations in next-generation wireless networks or on the many future wireless systems such as 3GPP's long term evolution information theoretic link-level aspects of wireless physical layer advanced (LTE-Advanced) [1], [2]. This deployment is motivated security, separately. Very few existing work seems to investigate by the significant performance gains that the deployment of one the impact of the presence of a malicious node on the relay or multiple relays can entail at different levels such as bit error station choices of the wireless nodes. In particular, it is of interest rate reduction or energy savings [2], [3]. Reaping the benefits of to study the dynamic interactions between an active malicious these advanced wireless communication techniques faces major node, which can act either as an eavesdropper or as a jammer, security threats in the presence of malicious nodes such as and a next-generation wireless network in which the wireless eavesdroppers or jammers. In consequence, one prominent chal- users must select their serving relay station so as to improve the lenge facing the design of future wireless systems is to develop rate and security of their transmission.

novel models and algorithms that enable secure and reliable The main contribution of this paper is to analyze, using game-

communications under heterogeneous network configurations. theoretic techniques, the complex decision making processes and

Communicating efficiently and securely in such heterogeneous interactions between an active malicious node and a network network conditions faces several challenges. On the one hand, the of wireless users that are seeking to choose their serving relay presence of relay station nodes imposes a multi-hop transmission station for uplink transmission. In this context, we formulate mode in which a number of mobile users must select at least a noncooperative strategic game in which the players are the one relay station to use for their transmission. On the other wireless users and the malicious node. In this game, the strategy hand, the presence of eavesdroppers or jammers requires the of the wireless users is to choose a serving relay station in a wireless users to select the station that can improve not only way to maximize a utility function that captures their mutual

This research was supported, in part, by AFOSR under Grant FA9550-09-1-0249 and MURI Grant FA9550-09-1-0643, by the Boeing Company, and in part by NSF under Grant CNS-09-05398.

978-1-4673-0081-0/11/$26.00 ©2011 IEEE 119

interference as well as the security of the chosen path for

data transmission. Conversely, the goal of the malicious node

is to reduce the overall network's rate by choosing whether

to eavesdrop, jam, or use a combination of both strategies,

depending on various parameters such as the network structure,

the location of the nodes, and the malicious node's jamming

power. We solve this game using a fictitious play-based algorithm

in which the wireless users and the malicious node can update

their strategies using their best responses, until they reach a

mixed-strategy Nash equilibrium (MSNE). Through simulations,

we study the properties of the resulting equilibrium and we

show that, using the proposed approach, the wireless users

can improve their average utility while the malicious node can

strategically decide on whether to eavesdrop or jam, depending

on its capabilities (e.g., its jamming power and cost).

The rest of this paper is organized as follows. In Section II,

we describe the system model and formulate a noncooperative

game between the users and the malicious node. In Section III,

we introduce the solution concepts for the game as well as the

fictitious play-based algorithm to find the Nash equilibrium in

mixed strategies. Simulation results are analyzed and discussed

in Section IV. Finally, conclusions are drawn in Section V.

II. S YSTEM MODEL AND GAME FORMULATION

A. Network Model

Consider the uplink transmission of a wireless system com­

prised of N wireless users that need to transmit their data to

a common base station (BS) via a single relay station node

chosen out of M relay stations that are in the coverage area

of the base station. We let N = {nl' n2,'" ,nN} and M = {ml' m2, ... ,mM} denote, respectively, the set of all N users

and the set of all M relay stations. Every user in the network

attempts to choose a serving relay station given the potential

interference that will be experienced at this relay station as well

as the channel conditions. Note that, hereinafter, we are mainly

interested in the case in which all the users will connect to

the base station through one of the relay stations, i.e., using

a maximum of two hops. Nonetheless, the proposed approach

can equally be applied to any multi-hop network [12].

In the considered network, an active malicious node, i.e.,

an attacker A, that is able to act as an eavesdropper or as a

jammer, is present and seeks to compromise the security of the

users' wireless transmissions. In essence, when eavesdropping,

the attacker would act passively by turning on its receiver

and attempting to tap into the users' transmissions. By doing

so, the malicious node threatens the physical layer security of

the network by reducing the so-called secrecy capacity of the

users. This secrecy capacity is defined as the maximum rate of

information that can be sent, securely, from a wireless node to its

destination in the presence of an eavesdropper [4]. Ultimately,

reducing the secrecy capacity of the users maps into reducing

the overall secure transmission rate of these users. Alternatively,

the malicious node can decide to act as a jammer. In this case, it

can reduce the overall transmission rate of the users by causing

harmful interference in the network, e.g., by transmitting with a

certain power PJ so as to reduce the overall rate of the users. The

goal of an attacker is thus to compromise the well-being of the

network by reducing as much as possible the total transmission

rate of the wireless network by eavesdropping, jamming, or using

a combination of both techniques.

120

Relay Stations

Attacker

Mobile Nodes Fig. I. An illustration of the system model for a wireless network consisting of M relay stations and N wireless users subject to eavesdropping and jamming from an active malicious node.

In the presence of the malicious node, the goal of a wireless

user is to seek an optimal configuration, i.e., pick the best relay

station node, so as to escape from the attacker as well as optimize

the quality-of-service experienced over its transmission path to

the BS via the relay stations. In Fig. 1, we illustrate the system

model in a hierarchical manner showing the interconnections

between the users, relay stations, and the BS. Our goal is

thus to study the complex interactions between the users and

a malicious node for a network such as in Fig. 1 in an effort to

better understand how the malicious node chooses its attacking

techniques as well as how the users react to this attack as well

as to the actions of other users (e.g., to the mutual interference).

Formally, each wireless user ni E N competes with the other

users over the choice of its serving uplink relay station mj E M while taking into account three key metrics: (i)- the potential

channel conditions on the path towards the BS, (ii)- the amount

of potential interference from other users, and (iii)- the expected

level of security for the transmission. The users also compete

with the malicious node whose strategic behavior, i.e., choice

of whether to eavesdrop or jam, will impact the overall secure

transmission rate of the users. To capture these objectives, next,

we introduce a noncooperative game-theoretic framework that is

capable of modeling both the competition among the users as

well as between the users and the malicious node.

B. Noncooperative Game Formulation

For the network model previously defined, we formulate an

N + 1-person noncooperative game with the players being the

wireless users in N and a single malicious node (attacker)

denoted by A. Each user ni E N selects an action ai from its

action space, which is the set of relay station nodes M, whereas

the malicious node chooses an action 8 from its action space

A := {E, J} in which a choice (8 = E) corresponds to the act

of eavesdropping while an action choice (8 = J) corresponds to

the act of jamming. The users and the attacker need to choose

their strategies so as to optimize utility or cost functions that

account for the two-hop structure in the system model. Such

functions will be composed of two independent components.

One part is attributed to the communication channel between

the wireless users and the relay stations and the other one is to

the communications between relay stations and the BS.

From the users' perspective, their overall uplink transmission

rate depends on the chosen relay station and on the type of attack

chosen by the malicious node. Under an eavesdropping attacker

(s = E), the maximum achievable rate for the transmission

between any user ni E N and its chosen relay station ai is

given by the secrecy capacity V; : MN X A --+ R+:

(1)

where (x)+ : = max(O, x) , for x E R; a-i = {aj,nj E N,nj i­ni} represents the relay station choices of the other users; Ci,a; is the capacity of the direct transmission between user ni E N and its relay station ai E M while Cj is the capacity of user i as received at the eavesdropper. The direct transmission capacity

Ci,ai is given by

( Pi9i a . ) Ci,ai = log2 1 + 2 " ,

a + Ji,ai (2)

where gi,ai = � ( �)" is the channel gain between user ni and relay station at�ith do a reference distance (typically 1 -10 meters), di,ai ?: do the distance between ni and ai, � the path

loss constant, and a the path loss exponent. Pi is the transmit

power of user ni, a2

is the variance of the Gaussian noise, and

Ji,a; is the interference experienced by user ni at relay station

ai given by

Ji,a; = L Pjgj,a;, njEN;

(3)

with Ni � N is a set of users connecting to the same relay

station ai chosen by user ni, i.e., Ni = {nj EN : aj = ai, nj i­ni}. Note that, here, we assume that the users that transmit to

two different relay stations mi i- mj, mi, mj E M do not

interfere. This assumption is motivated by the fact that, in next­

generation wireless networks, the relay stations will often adopt

advanced interference cancelation techniques (e.g., using smart

antenna techniques or by coordination) as discussed in [2], [3]

and references therein.

The capacity Cj of user ni as received at the malicious node

is given by:

Ce = 1 ( 1 + Pigi,A ) (4) • og2

a2 + Ji,A '

where gi,A = � ( d��A)" is the channel gain between the at­

tacker and user ni with di,A the distance between them and

Ji,A = 2:njEN-{ni} Pjgj,A . is the interference experienced at the eavesdropper when user ni is transmitting.

When the malicious node decides to act as a jammer, instead of

passively listening to the users' transmissions, it will attempt to

jam all receivers in the network, i.e., cause harmful interference

at all relay stations and the BS, by transmitting with a certain

jamming power PJ• In this case, for any user ni E N, the

channel capacity under a jamming attacker (s = J) is given by:

V;(ai' a_i, J) = log2 (1 + 2 Pigi,ai ) ,

ai + J.,ai + PJga;,A (5)

where gai,A is the channel gain between the jammer and the

chosen relay station ai. In (1) and (5), we were mainly interested

in the first transmission hop, however, the attacker will also affect

the performance of the wireless transmission at the second hop,

i.e., from the relay stations to the BS. Under different attacks,

similar to (1) and (5), the capacity of the communication link

121

from any relay station ai E M chosen by a user ni to the BS

is given by:

{ (Ca; - C�J+, Vai(s) =

log2 (1 + 2 Pa;9a;,BS a ) , s = J, a +Iai,Bs+PJgai,A

s=E, (6)

where gai,BS is the channel gain between ai and the BS, Ca; is

the capacity for direct transmission from ai to the BS, and C�i is the rate of ai transmission as received by the eavesdropper.

These capacities are given by:

C 1 (1 + Pa;gai,BS ) a; Og2 2 J ' a + a;,BS

( Pa·ga· A ) log2 1 + 2 ' J" , ai E M

a + a;,A

(7)

(8)

where ga;,A is the channel gain between the attacker and ai, Ja;,BS = 2:mjEM-{a;} Pmjgmj,A is the interference experi­

enced by ai at the BS, and Ja;,A = 2:mjEM-{a;} Pmjgmj,A is the interference at the eavesdropper when overhearing ai's

transmission.

Using (1), (5) and (6), we define the utility Ui of any user

ni E N over the two-hop communication link between ni and

the BS via the chosen relay station ai as follows:

Ui(ai,a-i,s) = min(V;(ai,a_i,s), Vai(s)). (9)

This utility represents the bottleneck transmission capacity for

the uplink transmission from user ni to the BS using its chosen

relay station ai E M in the presence of the malicious node.

The objective of each user ni is to choose the relay station

ai that maximizes its utility in (9) given the presence of the

attacker. We note that the payoff functions as defined in (9) are

strongly coupled in the actions of the users through the mutual

interference as well as in the action of the malicious node.

From the malicious node's perspective, the goal is to compro­

mise the overall well-being of the network and, thus, the attacker

seeks to minimize a cost function J A : M N X A --+ R, given by

JA(a, s) =

s=E s= J

(10)

The cost in (10) represents the total network's rate that the

attacker seeks to minimize. We note that, while eavesdropping

is a passive form of attack with no cost, jamming with a certain

power PJ entails an extra cost for the attacker captured, in (10),

by the term cJPJ in which CJ E R+ represents a per unit cost on

the power level used by the attacker while jamming the network.

By choosing a proper action (eavesdrop, jam, or a combination),

the attacker seeks to minimize the overall network's transmission

rate as captured by (10).

Given the users' and the attacker's payoffs as ex­

pressed, respectively, by (9) and (10), we formulate a

:=: = {N, M, {A}, {U;}n;EN, JA} noncooperative nonzero-sum

game between the users and the malicious node and our objective

is to study a solution of this game using the concept of Nash

equilibrium, as discussed in the next section.

III. GAME SOLU TION AND FICTITIOUS-PLAY ALGORITHM

The game S formulated in Section II-B is a nonzero-sum

finite game in its strategic form [13]. A commonly used solution

concept to strategic games is the Nash equilibrium. The Nash

equilibrium is a game state in which no player can benefit (e.g.,

improve its utility or reduce its cost) from unilaterally deviating

from its equilibrium strategy. In this section, we study the Nash

equilibrium in mixed strategies of the game S, in which the

users and the attacker choose to randomize over their action

spaces. Mixed strategies here for the attacker can be interpreted

as the fraction of time that an attacker uses to launch an attack.

We propose a 3-phase fictitious-play-based algorithm to find the

mixed equilibrium strategies online in an iterative manner.

Let 7rA = [7rA,E,7rA,J] E PA be a mixed strategy of the

attacker A. In other words, the attacker chooses to jam with

probability (or frequency) 7rA,J and chooses to eavesdrop with

probability 7r A,E, where PAis the set of admissible mixed strate­

gies of the attacker, defined by PA := {7rA E [0,1]2 : 7rA,J + 7rA,E = I}. Similarly, we let Pi,m, m E M, be the probability of

user ni choosing a relay station m and Pi = [Pi, m]mEM E Pi be

the mixed strategy of node ni in the set of all feasible strategies Pi, defined by Pi := {Pi,m E R+ : LmEM Pi,m = I } . Let

P-i := {Pil,nil E N\{ni}} and P := {pi,ni E N}. Then,

using (9), the average utility of ni is defined by

Ui(Pi,P-i,7rA) = Ep,7rA [Ui(ai,a-i,s)]. (11)

Using (10), the malicious node's average cost that it seeks to

minimize is defined by

Ep,7rA [JA(a, s)] (12)

Ep,7rA [L Ui(ai, a_i, S)] + 7rA,JCJPJ. niEN

From (11) and (12), we note that the jamming cost term in

(12) essentially renders the game S a nonzero-sum game. Using

Theorem 3.2 in [13], we can arrive at the following result on

existence of mixed-strategy Nash equilibrium (MSNE).

Proposition 1: Let S = {N, M, {A}, {U;}niEN, JA} be the

game described in Section II-B, with Ui and JA defined in (9) and (10), respectively. The finite N + 1 nonzero-sum game admits

a Nash equilibrium (p', 7r:4J in mixed strategies, which satisfies

the following set of inequalities:

JA(p',7r:4J::; JA(P',7rA), V7rA EPA, (13)

Ui(p',7r:4J ?: Ui(Pi, P�i' 7r:4J, VPi E Pi, ni EN. (14)

The MSNE of the game defines a state in which no user has

an incentive to change the probabilities for selection of its

serving relay station while the malicious node has no incentive

of changing the frequency with which it jams and/or eavesdrop.

In general, multiple MSNEs can exist; however, for the scope

of this work, we are interested in the MSNE that results from

a certain practical initial network state. The analysis for other

MSNE points follows a similar reasoning and is omitted due to

space limitations.

In order to find an MSNE of the N + I-person game S, we

propose an algorithm that enables the users and the malicious

node to make strategic decisions, in a distributed manner, given

122

their past observations on the state of the network. The proposed

algorithm considers that the users and the malicious node are

myopic, in the sense that they aim to maximize their payoffs

or minimize their costs considering only the current and pre­

vious states of the network. The proposed algorithm consists

of three phases: Network discovery, fictitious play, and network

operation. In the first phase, starting from a certain initial state,

the users and the malicious node discover one another using

well-known neighbor discovery techniques such as those used

in ad hoc routing [14]. On the one hand, before attacking, the

malicious node can observe the transmissions of the users to

estimate their locations and channel conditions. On the other

hand, the users can monitor their environment for any suspicious

malicious behavior as explained in [8] (and references therein).

Notably, when the malicious node attempts to jam, the users

can further improve their knowledge on the malicious node

and its possible location. Note that, the details of the discovery

techniques are outside the scope of this paper and are discussed

in [6--8], [14] and references therein.

Once the discovery phase is complete, the users and the ma­

licious node engage in the noncooperative game using fictitious

play [15], which is a practical approach that enables to find, in

a distributed manner, an MSNE for a noncooperative game [15].

Let p� E Pi be the mixed strategy of user ni at time k E Z+ and 7r� E PA be the attacker's mixed strategy at time k. At

each step k, user ni chooses an action a� E M, ni E N, to

maximize the expected utility in best response to other players'

mixed strategies p�i 1 := {p�,-l, ni' E N\ {ni}}, 7r�-1 at the

previous time step k - 1. Likewise, the attacker A chooses an

action sk E A to minimize his cost in response to the users'

mixed strategies pk-1 := {p�-l, ni EN}. More precisely,

E arg max Epk�l 7rk-1 [Ui(ai, a-i, s)], ni EN, (15) aiEM -1 ' A

E arg min Epk-l [JA(a, s)]. (16) sEA

Let v� = [Vf,j]�l be an M -dimensional vector with vf,j = 1 ·f k - d k - a th

. L·k · I k 1 ai - mj an Vi,j - , 0 erwlse. 1 eWlse, we et W A =

[w�,J' W�,E]' be a 2-dimensional vector with w� = [1, 0]' if

s� = J and w� = [0,1]' if s� = E. The mixed strategies are

updated at time k as follows:

(17)

In (17), at each step, the update on the empirical frequencies is

a linear combination between the ones in the previous step and

the elementary vectors w� and v� .

Starting with initial network conditions, i.e., initial mixed­

strategies, pO and 7r�, during the fictitious play phase, iteratively,

each player (users and the malicious node) needs to update its

knowledge of the empirical frequencies of the other players and

choose an action based on the empirical frequencies estimated at

the current state. The chosen action is essentially a best response

to the perceived network state, i.e., the action that maximizes the

utility (for the users) or minimizes the cost (for the malicious

node), at any given iteration. This iterative fictitious play process

TABLE I PROPOSED NONCOOPERATIVE GAME APPROACH

Initial State

The users choose their serving relay stations with initial probabil­ities pO and the malicious node chooses its attack type (eavesdrop or jam) with initial mixed strategy 7r�. Noncooperative game solution algorithm in three phases

Phase 1 - Network Discovery:

The users and the malicious node discover the network state using well-known techniques such as in [6-8], [14].

Phase 2 - Fictitious Play:

repeat (Iteration k)

a) Each user ni E N and the malicious node A choose their best response based on the empirical probabilities and payoffs of the other players as observed in iteration k - 1 as per (15). b) Each user ni and the malicious node update their mixed strategies based on (17).

until convergence to the MSNE. Phase 3 - Network Operation

a) Each user transmits its uplink data based on its Nash equilibrium relay station selections. b) The malicious node jams and eavesdrops based on its Na�h equilibrium strategies.

continues, with the users and malicious node interacting until all

empirical frequencies converge to the MSNE. It is well-known

that whenever fictitious play converges, it converges to a mixed

Nash equilibrium of the game [15]. Hence, in our model, by

using fictitious play for solving the noncooperative game, our

algorithm reaches the MSNE between the users and the malicious

node when it reaches steady-state. In general, fictitious play

algorithms have been proven to converge in almost all cases,

and several modification schemes have also been proposed to

ensure convergence [15], [16].

Once the network reaches an MSNE, the final phase of the

algorithm involves the uplink data transmission of the users,

based on their mixed strategies under the chosen attack strategies

for eavesdropping and jamming by the malicious node. Note

that, as the network is in an MSNE, no user can improve its

transmission capacity by unilaterally changing its choices over

the relay stations, given the current actions of the other users and

the attacker. Similarly, the attacker has no incentive to change

its eavesdropping and jamming frequencies given the current

network state. This algorithm is summarized in Table I.

IV. SIMULATION RESULTS AND ANALYSIS

For our simulations, we set up a wireless network composed

of a square area of 1.2 km x 1.2 km with the base station located

at the center and the malicious node randomly deployed within

the area. Further, inside this area, the relay stations are randomly

deployed within a square of 600 m x 600 m around the base

station while the mobile users are randomly deployed outside the

relay station area. The number of relay stations is set to M = 3 while the transmit power of all non-malicious nodes (users and

relay stations) is set to Pi = 10 mW. Unless stated otherwise, the

jamming power of the malicious node is set to Pj = 100 mW

while the cost for jamming is set to Cj = 1. The noise level is

set to (72 = -90 dBm, the reference distance is do = 1 m, the

path loss exponent is set to a: = 3, and /'i, = 1. We assume that

the malicious node initializes its attack strategies to 7r� = 7r� = 0.5, i.e., it starts by using eavesdropping and jamming with a

123

0.4r-�--�--�--�--�--�----'

� �

0.35

g 0.25

"5 " � 0.2

� 0.'5

0.'

-6- Proposed game-theoretic approach

� Nearest neighbor scheme

0.05"---:-----:----8:------:,70----,,72----,''"'4----' Number of users (N)

Fig. 2. Average expected utility (secure capacity) per user achieved for a network with M = 3 relay stations a� the number of wireless users N varies.

.� 0.9

� 0.8 'g. " � 0.7

>. u � 0.3

.g: 0.2 " '" � �

___ Jamming cost C J = 0

-e- Jamming cost cJ = 1 -a- Jamming cost cJ = 5

50 '00 '50 200 250 300 350 400 450 500 Power PJ used for jamming (mW)

Fig. 3. Strategic behavior of the malicious node shown through the average frequency (mixed strategy) of a jamming attack at the Nash equilibrium as the jamming power PJ and the jamming cost CJ vary for a network with N =

8 users and M = 3 relay stations.

similar frequency. All statistical results are averaged over random

positions of the users, the relay stations, and the malicious node.

In Fig. 2, we show the average expected utility achieved

per wireless user as the number of users, N, increases. The

performance is compared to that of a classical nearest neighbor

algorithm in which each user selects the relay station that

provides the best channel gain while the malicious node uses

its equilibrium strategies. The initial starting point of the users

is also the nearest neighbor network, i.e., the users start by

connecting to the relay station having the best channel conditions

with a probability of 1. First, in Fig. 2, we can see that as the

number of users N increases, the average expected utility per

user decreases for both the proposed approach and the nearest

neighbor case. This is a direct consequence of the fact that, as

N increases, the interference in the network increases and, thus,

the capacity decreases. Fig. 2 clearly demonstrates that, at all

network sizes N, the proposed approach presents a significant

performance gain relative to the nearest neighbor algorithm. This

performance gain of the proposed algorithm is increasing with

the number of the users N and reaches an improvement of up

to 49.4% relative to the nearest neighbor at N = 15 users.

In order to assess the strategic behavior of the malicious

node, we show, in Fig. 3 the average probability with which

the malicious node chooses to jam at the Nash equilibrium as

the jamming power Pj and the jamming cost Cj vary for a

network with N = 8 users and M = 3 relay stations. First,

Fig. 3 shows that, in the case with no jamming cost, i.e.,

CJ = 0, although the frequency with which eavesdropping is

chosen is higher than jamming at low jamming powers, i.e.,

for PJ < 50 mW, the attacker starts to prefer jamming over

eavesdropping as its jamming power increases. However, as the

cost for jamming increases, the malicious node would have an

incentive to spend less power for jamming, and, thus, it starts

to eavesdrop more frequently at the equilibrium. For example,

in the case in which CJ = 5, for all jamming power values,

the malicious node chooses more frequently to eavesdrop rather

than to jam. In contrast, for CJ = 1, the frequencies of jamming

and eavesdropping are somewhat balanced, depending on the

jamming power. For instance, Fig. 3 shows that, for CJ = 1 and

CJ = 5, as the jamming power increases, the average frequency

with which the malicious node selects to jam at the Nash

equilibrium starts by increasing. However, this average jamming

probability starts to decrease once the jamming power reaches

around PJ = 200 mW and PJ = 100 mW, for CJ = 1 and

CJ = 5, respectively. This decrease is due to the fact that the cost

for jamming increases with the chosen power value as the energy

used for jamming becomes more costly than its associated gains,

i.e., capacity reductions. In a nutshell, Fig. 3 shows interesting

insights on the strategic decisions and frequency of attack types

chosen by the malicious node at the equilibrium.

Fig. 4 shows the average and maximum number of iterations

needed for convergence of the fictitious play phase of the algo­

rithm in Table I as the number of users N varies. Fig. 4 shows

that both the average and the maximum number of iterations

increase with the network size N. This average and maximum

number of iterations vary, respectively, from about 18.5 and

35 iterations at N = 3 to around 26.8 and 48 iterations at

N = 15 users. Clearly, the increase slope of the average number

of iterations is quite reasonable as increasing the number of users

of about five times (i.e., from N = 3 to N = 15) yields only

an extra 8 iterations on the average. In summary, Fig. 4 shows

that the proposed fictitious play algorithm presents a reasonable

convergence time and its complexity grows relatively slowly with

the network size.

V. CONCLUSIONS

In this paper, we have analyzed, using game-theoretic tech­

niques, the complex interactions between wireless users and a

malicious node in the context of relay station-enabled wireless

networks. For this purpose, we have formulated a noncooperative

game between the users and the malicious node. In the proposed

game, the goal of each wireless user is to choose the serving relay

station that can optimize the security and quality-of-service of

its transmission, given the presence of interference as well as of

a malicious node. Conversely, the malicious node's objective is

to decide on whether to eavesdrop, jam, or use a combination

of both strategies, in a way to reduce the overall transmission

rate of the network. To solve this game, we have adopted an

algorithm based on fictitious play using which the users and the

malicious node can reach a mixed-strategy Nash equilibrium.

Simulation results have shown that the proposed algorithm

allows the attacker to make a strategic decision on whether to

eavesdrop or jam depending on its capabilities (e.g., its jamming

124

5o,--.-----.-----.-----.-----.--------,

45 g " e> 40 � 8 '" c o � 30 .�

� 25 § z

-e-- Maximum number of iterations

__ Average number of iterations

153�-L----�----�----�1�0-----1�2------�15· Number of users

Fig. 4. Average and maximum number of iterations needed till convergence of the fictitious play phase of the algorithm to an MSNE for a network with M =

3 relay stations as the number of users varies.

power). The results have also shown that the proposed approach

enables the wireless users to significantly improve their average

expected utilities which reflect the security and the quality-of­

service (in terms of transmission rate) perceived over the chosen

relay stations.

REFERENCES

[I] 3GPP TR 36. 814 Technical Specification Group Radio Access Network, "Further advancements for E-UTRA. physical layer aspects." Tech. Rep.

[2] S. W. Peters. A. Panah, K. Truong, and R. W. Heath, "Relay architectures for 3GPP LTE-Advanced," EURASIP Journal on Wireless Communication and Networking, vol. 2009, May 2009.

[3] J. Laneman, D. Tse, and G. Womell, "Cooperative diversity in wireless networks: Efficient protocols and outage behavior," IEEE Trans. Inform. Theory, vol. 50, pp. 3062-3080, Dec. 2004.

[4] Y. Liang, H. V. Poor, and S. Shamai, "Secure communication over fading channels," IEEE Trans. Inform. Theory, vol. 54, no. 6, pp. 2470-2492, Jun. 2008.

[5] A. Kashyap, T. Ba�ar, and R. Srikant, "Correlated jamming on MIMO Gaussian fading channels," IEEE Trans. Inform. Theory, vol. 50, pp. 2119-2123, Sep. 2004.

[6] X. Zhou and M. R. McKay, "Secure transmission with artificial noise over fading channels: Achievable rate and optimal power allocation," IEEE Trans. Veh. Techno!., vol. 59, no. 8, pp. 3831 - 3842, Oct. 2010.

[7] A. Mukherjee and A. L. Swindlehurst, "Equilibrium outcomes of dynamic games in MIMO channels with active eavesdroppers," in Proc. IEEE Int. Co'!r on Communications, Cape Town, South Africa, May 2010.

[8] L. Dong, Z. Han, A. Petropulu, and H. V. Poor, "Secure wireless communi­cations via cooperation," in Proc. Allerton Conference on Communication, Control, and Computing, Monticello, IL, USA, Sep. 2008.

[9] Y. Liang, H. V. Poor, and S. Shamai, Information Theoretic Security. Hanover, MA, USA: Now Publisher, 2009.

[10] B. Lin, P. Ho, L. Xie, and X. Shen, "Optimal relay station placement in IEEE 802.16j networks," in Proc. International Conference on Communi­cations and Mobile Computing, Hawaii, USA, Aug. 2007.

[II] D. Niyato, E. Hossain, D. I. Kim, and Z. Han, "Relay-centric radio resource management and network planning in ieee 802.16j mobile multi hop relay networks," IEEE Trans. Wireless Commun., vol. 8, no. 12, Dec. 2009.

[12] W. Saad, Q. Zhu, T. Ba�ar, Z. Han, and A. Hji1lrungnes, "Hierarchical network formation games in the uplink of multi-hop wireless networks," in Proc. IEEE Global Communication Co'!{erence, Honolulu, Hawaii, USA, Dec. 2009.

[13] T. Ba�ar and G. J. Olsder, Dynamic Noncooperative Game Theory. Philadelphia, PA, USA: SIAM Series in Classics in Applied Mathematics, 1999.

[14] C. S. R. Murthy and B. Manoj, Ad Hoc Wireless Networks: Architectures and Protocols. Upper Saddle River, NJ, USA: Prentice Hall, Jun. 2004.

[15] D. Fudenberg and D. K. Levine, The Theory of Learning in Games. Cambridge, MA, USA: MIT Press, 1998.

[16] J. Shamma and G. Arslan, "Unified convergence proofs of continuous-time fictitious play," IEEE Trans. Automatic Control, vol. 49, no. 7, pp. 1137-1142, Jul. 2004.