452907

8/8/2019 452907

http://slidepdf.com/reader/full/452907 1/11

Hindawi Publishing CorporationEURASIP Journal on Wireless Communications and NetworkingVolume 2009, Article ID 452907, 10 pagesdoi:10.1155/2009/452907

Research ArticlePhysical Layer Security Game: Interaction between Source,Eavesdropper, and Friendly Jammer

ZhuHan,1 NinoslavMarina,2MerouaneDebbah,3 and Are Hjørungnes2

1 Electrical and Computer Engineering Department, University of Houston, TX 77004, USA 2 UniK—University Graduate Center, University of Oslo, Gunnar Randers vei 19, P.O. Box 70,

NO-2027 Kjeller, Norway 3 SUPELEC, Plateau de Moulon, 3 rue Joliot-Curie, Bureau 5-24, 91192 Gif-sur-Yvette Cedex, France

Correspondence should be addressed to Zhu Han, [email protected]

Received 31 December 2008; Revised 4 August 2009; Accepted 9 November 2009

Recommended by Hesham El-Gamal

Physical layer security is an emerging security area that achieves perfect secrecy data transmission between intended networknodes, while malicious nodes that eavesdrop the communication obtain zero information. The so-called secrecy capacity can beimproved using friendly jammers that introduce extra interference to the eavesdroppers. We investigate the interaction betweenthe source that transmits the useful data and friendly jammers who assist the source by “masking” the eavesdropper. To obtaindistributed solution, we introduce a game theoretic approach. The game is defined such that the source pays the jammers tointerfere the eavesdropper, therefore, increasing the secrecy capacity. The friendly jammers charge the source with a certain pricefor the jamming, and there is a tradeoff for the price. If too low, the profit of the jammers is low; and if too high, the sourcewould not buy the “service” (jamming power) or would buy it from other jammers. To analyze the game outcome, we investigatea Stackelburg type of game and construct a distributed algorithm. Our analysis and simulation results show the eff ectiveness of friendly jamming and the tradeoff for setting the price. The distributed game solution is shown to have similar performances tothose of the centralized one.

Copyright © 2009 Zhu Han et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The future communication systems will be decentralizedand adhoc, therefore allowing various types of networkmobile terminals to join and leave. This aspect makes

the whole system vulnerable and susceptible to attacks.Anyone within communication range can listen and possibly extract information. While these days we have numerouscryptographic methods with high level security, there is nosystem with perfect security on physical layer. Therefore,the physical layer security is regaining a new attention.The main goal of this paper is to design a decentralizedsystem that will protect the broadcasted data and makeit impossible for the eavesdropper to receive the packetseven if it knows the encoding/decoding schemes used by the transmitter/receiver. In approaches where physical layersecurity is applied, the main objective is to maximize therate of reliable information from the source to the intended

destination, while all malicious nodes are kept as ignorant of that information as possible. This maximum reliable rate isknown as secrecy capacity .

This line of work was pioneered by Wyner, who definedthe wiretap channel and established the possibility to create

almost perfect secure communication links without relyingon private (secret) keys [1]. Wyner showed that whenthe eavesdropper channel is a degraded version of themain channel, the source and the destination can exchangeperfectly secure messages at a nonzero rate. The mainidea proposed by him is to exploit the additive noiseimpairing the eavesdropper by using a stochastic encoderthat maps each message to many codewords according toan appropriate probability distribution. With this scheme,a maximal equivocation (i.e., uncertainty) is induced at theeavesdropper. In other words, a maximal level of secrecy isobtained. By ensuring that the equivocation rate is arbitrarily close to the message rate, one can achieve perfect secrecy

8/8/2019 452907


2 EURASIP Journal on Wireless Communications and Networking

SC1

C2

D

J 2

J 1

M

J J

S: Source

D: Destination

M : Malicious node (eavesdropper)

J 1, . . . , J J : J friendly jammers

Useful data

Interference

Payment

Figure 1: System model for the proposed physical layer security game.

in the sense that the eavesdropper is now limited to learnalmost nothing about the source-destination messages fromits observations. Follow-up work by Leung-Yan-Cheong andHellman characterized the secrecy capacity of the additivewhite Gaussian noise (AWGN) wiretap channel [2]. In theirlandmark paper, Csiszar and Korner generalized Wyner’sapproach by considering the transmission of confidentialmessages over broadcast channels [3]. Recently, there havebeen considerable eff orts on generalizing these studies to thewireless channel and multiuser scenarios (see [2, 4–11] andreferences therein). Jamming [12–14] has been studied fora long time to analyze the hostile behaviors of maliciousnodes. Recently, jamming has been employed to physicallayer security to reduce the eavesdropper’s ability to decodethe source’s information [15]. In other words, the jammingis friendly in this context. Moreover, the friendly helper canassist the secrecy by sending codewords, and bring furthergains relative to unstructured Gaussian noise [15–17].

Game theory [18] is a formal framework with a setof mathematical tools to study some complex interactionsamong interdependent rational players. During the pastdecade, there has been a surge in research activities that

employ game theory to model and analyze modern dis-tributed communication systems. Most of these works [19–22] concentrate on the distributed resource allocation forwireless networks. As far as the authors’ knowledge, the gametheory has not yet been used in the physical layer security.

In this paper, we investigate the interaction between thesource and its friendly jammers using game theory. Althoughthe friendly jammers help the source by reducing the datarate that is “leaking” from the source to the malicious node,at the same time they also reduce the useful data rate fromthe source to the destination. Using well chosen amounts of power from the friendly jammers, the secrecy capacity canbe maximized. In the game that we define here, the source

0

0.1

0.2

0.3

0.4

0.50.6

0.7

0.8

0.9

1

0 0.005 0.01 0.015 0.02

Jamming power

Jammer location (50, 75)


S e c r e c y c a p a c

i t y C s

Secrecy capacity as a function of jammer power

Figure 2: Secrecy capacity versus the power of the single jammer.

pays the jammers to interfere the malicious eavesdropper,and therefore, to increase the secrecy capacity. The friendly jammers charge the source with a certain price for theirservice of jamming the eavesdropper. One could notice thatthere is a tradeoff for the proposed price. If the price of acertain jammer is too low, its profit is also low; if its priceis too high, the source will buy from the other jammers.In modeling the outcome of the above games our analysisuses the Stackelberg type of game. Initially, the existence of

equilibrium will be studied. Then, a distributed algorithmwill be proposed and its convergence will be investigated. Theoutcome of the distributed algorithm will be compared tothe centralized genie aided solution. Some implementationconcerns are also discussed. From the simulation results, wecan see the efficiency of friendly jamming and tradeoff forsetting the price, the source prefers buying service from only one jammer, and the centralized scheme and the proposedgame scheme have similar performance.

The rest of the paper is organized as follows. In Section 2,the system model of physical layer security with friendly jamming users is described. In Section 3, the game modelsare formulated, and the outcomes as well as properties of the

game are analyzed. Simulation results are shown in Section4,and conclusions are drawn in Section 5.

2. SystemModel

We consider a network with a source, a destination, amalicious eavesdropper node, and J friendly jammer nodesas shown in Figure 1. The malicious node tries to eavesdropthe transmitted data coming from the source node. Whenthe eavesdropper channel from the source to the maliciousnode is a degraded version of the main source-destinationchannel, the source and destination can exchange perfectly secure messages at a nonzero rate. By transmitting a message

8/8/2019 452907


EURASIP Journal on Wireless Communications and Networking 3

at a rate higher than the rate of the malicious node, themalicious node can learn almost nothing about the messagesfrom its observations. The maximum rate of secrecy infor-mation from the source to its intended destination is definedby the term secrecy capacity.

Suppose the source transmits with power P 0. The channel

gains from the source to the destination and from the sourceto the malicious node are Gsd and Gsm, respectively. Eachfriendly jammer i, i = 1, . . . , J , transmits with power P i andthe channelgains from it to the destination and the maliciousnode are Gid and Gim, respectively. We denote by J the setof indices {1,2, . . . , J }. If the path loss model is used, thechannel gain is given by the distance to the negative power of the path loss coefficient. The thermal noise for each channelis σ 2 and the bandwidth is W . The channel capacity for thesource to the destination is

C1 = W log2

1 +

P 0Gsd

σ 2 +

i∈J P iGid

. (1)

The channel capacity from the source to the malicious nodeis

C2 = W log2

1 +

P 0Gsm

σ 2 +

i∈J P iGim

. (2)

The secrecy capacity is

Cs = (C1 − C2)+, (3)

where (·)+= max(·,0). Both C1 and C2 are decreasing and

convex functions of jamming power P i. However, Cs=

C1−

C2 might not be a monotonous and convex function.( Minusof two convex functions is not a convex function anymore.)This is because the jamming power might decrease C1 fasterthan C2. Asa result, Cs might increase in some region of valueP i. When P i further increases, both C1 and C2 approach zero.As a result, Cs approaches zero. So, the questions are whetheror not Cs can be increased, and how to control the jammingpower in a distributed manner so as to achieve the maximalCs. We will try to solve the problems in the following sectionusing a game theoretical approach.

3. Game for Physical Layer Security

In this section, we study how to use game theory to analyzethe physical layer security. First, we define the game betweenthe source and friendly jammers. Next, we optimize thesource and jammer sides, respectively. Then, we prove someproperties of the proposed game. Furthermore, a comparisonwith the centralized scheme is constructed. Finally, wediscuss some implementation concerns.

3.1. Game Definition. The source can be modeled as abuyer who wants to optimize its secrecy capacity minus costby modifying the “service” (jamming power P i) from thefriendly jammers, that is,

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0 50 100 150 200

Jammer price



A m o u n t o f p o w

e r b o u g h t

How much power bought versus jammer price

Figure 3: How much power the source buys as a function of theprice.

Source’s Game: max U s = max(aCs − M ), (4)

s.t. P i ≤ P max, (5)

where a is the gain per unit capacity, P max is the maximalpower that a jammer can provide, and M is the cost to pay for the other friendly jamming nodes. Here

M = i∈J

piP i, (6)

where pi is the price per unit power for the friendly jammer,P i is the friendly jammer’s power, and J is the set of friendly jammers. From (4) we note that the source will notparticipate in the game if C1 < C2, or in other words, thesecrecy capacity is zero. For each jammer, U i( pi, P i( pi)) is theutility function of the price and power bought by the source.For the jammer’s (seller’s) utility, in this paper we define thefollowing utility:

U i = piP cii , (7)

where ci ≥ 1 is a constant to balance from the payment

piP i from the source and the transmission cost P i. Withdiff erent values of ci, jammers have diff erent strategies forasking the price pi. Notice that P i is also a function of thevector of prices ( p1, . . . pN ), since the power that the sourcewill buy also depends on the price that the friendly jammersask. Hence, for each friendly jammer, the optimizationproblem is

Friendly Jammer’s Game: max pi

U i. (8)

In the next two subsections, we analyze the optimalstrategies for the source and friendly jammers to maximizetheir own utilities.

8/8/2019 452907



3.2. Source (Buyer) Side Analysis. Introducing A =

P 0Gsd /σ 2, B = P 0Gsm /σ 2, ui = Gid /σ 2, and vi = Gim /σ 2, i ∈

J, we have

U s = aW

log

1+

A

1+ j∈J u j P j

−log

1+

B

1+ j∈J v j P j

+

−

j∈J

p j P j .

(9)

For the source (buyer) size, we analyze the case C1 > C2.By diff erentiating (4), we have

∂U s∂P i

= −aWAui / ln 2

1 + A +

j∈J u jP j

1 +

j∈J u j P j

+aWBvi / ln 2

1 + B +

j∈J v j P j1 +

j∈J v j P j− pi = 0.

(10)

Rearranging the above equation, we have

P 4i + F i,3P 3i + F i,2 pi

P 2i + F i,1

pi

P i + F i,0

pi

= 0, (11)

where

F i,3 = (2 + 2αi + A)2 +

2 + 2 βi + B2

,

F i,2 pi

=

(2 + 2αi + A)

2 + 2 βi + B

uivi

+Li

v2

i

+K i

u2

i

−aW

piuivi

B

vi

−A

ui,

F i,1 pi

=

LiCi + K iDi

u2i v2

i

+aW ( ADi − BCi)

piu2i v2

i

,

F i,0 pi

=

K iLi

u2i v2

i

+aW ( AuiLi − BviK i)

piu2i v2

i

,

αi = j / = i

G jd P j ,

βi = j / = i

G jm P j ,

K i = (1 + αi)(1 + αi + A),

Li =

1 + βi

1 + βi + B

,

Ci = ui(2 + 2αi + A),

Di = vi

2 + 2 βi + B

.

(12)

The solutions of the quartic equation (11) can be expressedin closed form but this is not the primary goal here. It isimportant that the solution we are interested in is given by the following function:

P ∗i = P ∗i

pi, A, B,

u j

,

v j

,

P j

j / = i

. (13)

0.2

0.4

0.6

0.8

1

0

100

200

300150

10050

0

Source (0,0), dest. (100,0), malic. node (50,90),

user 1 (50,50), user 2 (50,75)

S o u r c e U s

U s e r 1 p r i c e p

2 U s e r 2

p r i c ep 1

Figure 4: U s versus the prices of both users.

Note that 0 ≤ P i ≤ P max. Since P i satisfies the polynomialfunction, we can have the optimal strategy as

P ∗i = min[max(P i, 0), P max]. (14)

Because of the complexity of the closed form solutionof the quartic equation in (14), we also consider two specialcases: low interference case and high interference case.

3.2.1. Interference at the Destination Is Much Smaller than

the Noise. Remember the definitions: A = P 0Gsd /σ 2, B =P 0Gsm /σ 2, ui = Gid /σ 2, and vi = Gim /σ 2. Imagine a situationin which all jammers are close to the malicious node and farfrom the destination node. In that case the interference fromthe jammers to the destination is very small in comparisonto the additive noise and therefore we have

U s ≈ aW

log(1 + A) − log

1 +

B

1 +

j∈J v j P j

+

−

j∈J

p j P j .

(15)

Then

∂U s∂P i

=aWBvi / ln 2

1 + B +

j∈J v j P j

1 +

j∈J v j P j − pi = 0.

(16)

Rearranging we get

P 2i +2 + 2 βi + B

viP i +

1 + βi

1 + B + βi

v2

i

−aWB

pivi ln 2= 0.

(17)

8/8/2019 452907



0

1

2

3

×10−7

300250

200150

10050

0

50

100

150


user 1 (50,50), user 2 (50,75)

S o u r c e U 1

U s e r 1 p r i c e p 2

U s e r

2 p r i c e

p 1

Figure 5: U 1 versus the prices of both users.

Solving the above equation we obtain a closed-form solution

P ∗i = −2 + 2 βi + B

2vi,

+

2 + 2 βi + B2

4v2i

−

1 + βi

1 + B + βi

v2

i

+aWB

pivi ln 2,

= qi +

wi +

z i pi

,

(18)

where

qi = −2 + 2 βi + B

2vi

wi =

2 + 2 βi + B

2

4v2i

−

1 + βi

1 + B + βi

v2

i

z i =aWB

vi ln 2.

(19)

Finally, by comparing P ∗i with the power under theboundary conditions (P i = 0, P i = P max, and Cs = 0), theoptimal P ∗i in the low SNR region can be obtained.

3.2.2. One Jammer with Interference That Is Much Higher than the Noise but Much Smaller than the Received Power at the Destination and the Malicious Node. In this case theinterference from the jammer is much higher than theadditive noise but much smaller than the power of thereceived signal at the destination and the malicious node. Inother words, that means 1 u1P 1 A and 1 v1P 1 B.Therefore the utility function of the source is given by

U s ≈ aW

log

1 +

A

u1P 1

− log

1 +

B

v1P 1

− p1P 1

≈aWA

u1P 1−

aWB

v1P 1− p1P 1.

(20)

If (B/v1) − ( A/u1) ≤ 0, U s is a decreasing function of P 1. As a result, P s is optimized when P 1 = 0, that is, the jammer would not participate the game. On the other hand,if (B/v1)−( A/u1) > 0, in order to find the maximizing powerswe have to calculate

∂U s∂P i = −

aWA

u1P 21 +

aWB

v1P 21 − p1 = 0. (21)

Hence

P ∗1 =

aW

p1

B

v1−

A

u1

=

D1

p1. (22)

From this equation we get the optimal closed-form solutionP ∗i , and similarly by comparing P ∗1 with the power under theboundary conditions (P 1 = 0, P 1 = P max, and Cs = 0), wecan obtain the optimal solution for the this special case.

3.3. Friendly Jammer (Seller) Side Analysis. In this subsec-

tion, we study how the friendly jammers can set the optimalprice to maximize its utility. By diff erentiating the utility in(7) and setting it to zero, we have

∂U i∂p i

=

P ∗ici + pici

P ∗ici−1 ∂P ∗i

∂p i= 0. (23)

This is equivalent to

P ∗ici−1

P ∗i + pici ·

∂P ∗i∂p i

= 0. (24)

This happens either if P ∗i = 0 or if

P ∗i + pici · ∂P ∗

i

∂pi= 0. (25)

From the closed form solution of P ∗i the solution of p∗i will

be a function given as

p∗i = p∗

i

σ 2; Gsd ; Gsm; {Gid }; {Gim}

. (26)

Notice that p∗i should be positive. Otherwise, the friendly

jammer would not play.

3.4. Properties. In this subsection, we prove some propertiesof the proposed game. First, we prove that the power ismonotonous function of the price under the two extremecases. The properties can help for the proof of equilibriumexistence in the later part of this subsection.

Property 1. Under the two special cases, the optimal power consumption P ∗i for friendly jammer i is monotonous with its price pi , when the other friendly jammers prices are fixed. The proof is straightforward from (18) and (22).

We investigate the following analysis of the relationbetween the price and the power. We find out that thefriendly jammer power P i bought from the source is convexin its own price pi under some conditions. To prove this weneed to check whether the second derivative ∂2P i /∂p2

i < 0.

8/8/2019 452907



In the first special case in which the interference is small

∂P ∗i∂pi

= −z i

2 p2i

wi +

z i / pi

,

∂2P ∗i

∂p2i

=z i

p3i

wi +

z i / pi

1 / 21 −

1

4

piwi /z i

+ 1.

(27)

The above equation is greater than zero when pi is small. Thismeans when the interference is small and the price is small,the power is convex as a function of the price.

In the second special case in which the interference issevere

∂P ∗i∂p i

= −1

2

D1 p

−3 / 21 ,

∂2P ∗i∂p 2

i

=3

4

D1 p

−5 / 21 > 0.

(28)

This means when the interference is severe, the power is aconvex function of the price.

Next, we investigate the equilibrium of the proposedgame. At the equilibrium, no user can improve its utility by changing its own strategy only. We first define the Stackelbergequilibrium as follows.

Definition 1. P SEi and pSE

i are the Stackelberg equilibrium of the proposed game, if when pi is fixed,

U s

P SEi

= sup

P max ≥{P SEi }≥0, ∀i

U s({P i}), ∀i ∈ J (29)

and when P i is fixed,

U i pSE

i

= sup

pi

U i pi

, ∀i ∈ J. (30)

Finally, from the analysis in the previous two subsections,we can show the following property for the proposed game.

Property 2. The pair of {P ∗i }N i=1 in (14) and { p∗

i }N i=1 in (26)

is the Stackelberg equilibrium for the proposed game.

Notice that there might be multiple roots in (11), as aresult, there might be multiple Stackelberg equilibria. In thesimulation results shown in later section, we will show that

the proposed scheme can still achieve the equilibria withbetter performances than those of the no-jammer case.

3.5. Distributed Algorithm and Convergence. In this subsec-tion, we study how the distributed game can converge tothe Stackelberg equilibrium defined in the above subsection.After rearranging (23), we have

pi = I i

p

= −

P ∗i

ci

∂P ∗i /∂ pi

, (31)

where p = [ p1, . . . , pN ]T and I i(p) is the price update

function. Notice that P ∗i is a function of p. The information

0

1

2

3

4

×10−6

300

200

100

050

100150


user 1 (50,50), user 2 (50,75)

S o u r c e

U 2

U s e r

1 p

r i c e p

2

U se r 2 p r ice p1

Figure 6: U 2 versus the prices of both users.

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

−50 0 50 100

Jammer 2’s location

Game

Centralized

S o u r c e u t i l i t y

Two user case, U s versus jammer 2 location

Figure 7: U s versus the location of the second jammer.

for the update can be obtained from the source node. This issimilar to the distributed power control [25]. The update of the friendly jammers’ prices can be written in a vector formas

Distributed Algorithm: p(t + 1) = I

p(t )

, (32)

where I = [I 1, . . . , I N ]T , and the iteration is from time t to

time t +1. Next we show that the convergence of the proposedscheme by proving that the price update function in (32) is astandard function [23] defined as follows.

Definition 2. A function I(p) is standard, if for all p ≥ 0, thefollowing properties are satisfied

8/8/2019 452907



(1) Positivity: p > 0.

(2) Monotonicity: if p ≥ p, then I(p) ≥ I(p

), or I(p) ≤

I(p

).

(3) Scalability: for all η > 1, ηI(p) ≥ I (ηp).

In [23], it has been proved that the price will converge to

the fixed point (i.e., the Stackelberg equilibrium in our case)from any feasible initial price vector. The positivity is very easy to prove. If the price pi goes up, the source would buy less from the ith friendly jammer. As a result, (∂P ∗i /∂ pi) in(23) is negative, and we prove positivity pi = I i(p) > 0.

The monotonicity and scalability can only be shown inthe two special cases. For the low interference case, from (18)it is obvious that

I i

p

= −

P ∗i

ci

∂P ∗i /∂ pi

=2

wi p

2i + z i pi

qi pi +

wi p

2i + z i pi

ciz i

(33)

which is monotonically increasing in pi. For scalability, wehave

I i

ηp

ηI i

p =

wi p

2i +

z i pi /η

qi pi +

wi p

2i +

z i pi /η

wi p2i + z i pi

qi pi +

wi p

2i + z i pi

< 1,

(34)

since η > 1.For the large interference case, from (22) we have

I ip

= −

P ∗i

ci∂P ∗i /∂ pi=

2 pi

ci(35)

which is monotonically increasing in pi and scalable.For more general cases, the analysis is tractable. In the

simulation section later, we employ the general simulationsetups. The simulation results show that the proposedscheme can converge and outperform the no-jammer case.

3.6. Centralized Scheme. Traditionally, the centralizedscheme is employed assuming that all channel informationis known. The objective is to optimize the secrecy capacity under the constraints of maximal jamming power.

maxP i

Cs = max⎡⎣W log2⎛⎝

1+

P 0Gsd /

σ 2 +

i∈J P iGid 1+P 0Gsm /

σ 2 +

i∈J P iGim

⎞⎠, 0⎤⎦.

s.t. 0 ≤ P i ≤ P max, ∀i.

(36)

The centralized solution is found by maximizing thesecrecy capacity only. If we do not consider the constraint,we have

∂Cs

∂P i= −

AWui

(1 + αi + uiP i)(1 + A + αi + uiP i)

+BW vi

1 + βi + uiP i1 + B + βi + uiP i= 0.

(37)

0

0.005

0.01

0.015

0.02

0.025

−50 0 50 100

Jammer 2 location

Jammer 1 power

Jammer 2 power

J a m m i n g p

o w e r

Two user case, jamming power versus jammer 2 location

Figure 8: Power versus the location of the second jammer.

Rearranging we get

P 2i +Au2

i

2 + B + 2 βi

− Bv2

i (2 + A + 2αi)

Au3i − Bv3

i

P i

+Aui

1 + βi

1 + B + βi

− Bvi(1 + αi)(1 + A + αi)

Au3i − Bv3

i

= 0.

(38)

Using the KKT condition theorem [24], the final solutionwould be obtained by comparing the boundary conditions(i.e., P i = 0, P i = P max, and Cs = 0).

Notice that our proposed algorithm is distributive,in the sense that only the pricing information needs tobe exchanged. In the simulation results, we compare theproposed game theoretical approach with this centralizedscheme.

Finally, from the simulation results in the next section, wesee that the distributed solution and the centralized solutionare asymptotically the same if a is sufficiently large (thesource cares more about the secrecy capacity than for the

payment, i.e., the source is sufficiently rich).

3.7. Implementation Discussion. There are several implemen-tation concerns for the proposed scheme. First, the channelinformation from the source to the malicious eavesdroppermight not be known or accurately known. Under thiscondition, the secrecy capacity formula should be rewrittenconsidering the uncertainty. If the direction of arrival isknown, multiple antenna techniques can be employed suchas in [11]. Second, the proposed scheme needs to iteratively update the price and power information. A natural questionarises if the distributed scheme has less signalling than thecentralized scheme. The comparison is similar to distributed

8/8/2019 452907



0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

×10−5

−50 0 50 100

Jammer 2 location

Jammer 1 utility

Jammer 2 utility

J a m m e r u t i l i t

y

Two user case, jamming utility versus jammer 2 location

Figure 9: Utility versus the location of the second jammer.

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

2 2.5 3 3.5 4 4.5 5

Factor a

Optimal solution

Game result

S

e c r e c y c a p a c i t y

Eff ect of parameter a: jammer 2 location (0, 75)

Figure 10: Eff ect of the parameter a on the game.

and centralized power control in the literature [23, 25].Since the channel condition is continuously changing, thedistributed solution only needs to update the diff erence of the parameters such as power and price to be adaptive, whilethe centralized scheme requires all channel information ineach time period. As a result, the distributed solution hasa clear advantage and dominates the current and futurewireless network designs. For example, the power control forcellular networks, the open loop power control is done only once during the link initialization, while the close loop powercontrol (distributed power allocation such as [23]) is per-formed 1500 times for UMTS and 800 times for CDMA2000.

Finally, for the multisource multidestination case, there aretwo possible choices to solve the problem. First, we can useclustering method to divide the network into sub-networks,and then employ the single-source-destination pair andmultiple-friendly-jammer solution proposed in this paper.If we believe that the jamming power can be useful for

multiple eavesdroppers, some techniques such as doubleauction could be investigated. The detailed discussion isbeyond the scope of this paper and would be considered inour future research.

4. Simulation Results

The simulation is set up as follows. The source and friendly jammer have power of 0.02, the bandwidth is 1, thenoise level is 10−8, the propagation loss factor is 3, andAWGN channel is assumed. The source, destination, andeavesdropper are located at the coordinates (0,0), (100,0),and (50,50), respectively. Here we select a = 2 for the friendly

jammer utility in (7).For single friendly jammer case, we show the simulation

with the friendly jammer at the location of (50,75) and(10,75). In Figure 2, we show the secrecy capacity as afunction of the jamming power. We can see that with theincrease of the jamming power, the secrecy capacity firstincreases and then decreases. This is because the jammingpower has diff erent eff ects on C1 and C2. So there is anoptimal point for the jamming power. Also the optimal pointdepends on the location of the friendly jammer, and thefriendly jammer close to the eavesdropper is more eff ectiveto improve the secrecy capacity. Moreover, notice that thecurve is neither convex nor concave. Figure 3 shows how the

amount of the power bought by the source from the jammerdepends on the requested price. We can see that the poweris reduced if the price goes high. At some point, the sourcewould stop buying the power. So there is a tradeoff for settingthe price, that is, if the price too high, the source would buy less power or even stop buying.

For the two-jammer case, we set up the followingsimulations. Malicious node is located at (50,90), the firstfriendly jammer is located at (50,50), and the second friendly jammer is located at (50,75). In Figures 4, 5 and 6, thesource’s utility U s, the first jammer’s utility U 1, and thesecond jammer’s utility U 2 as function of both users’ price,are shown respectively. We can see that the source would buy

service from only one of the friendly jammers. If the friendly jammer asks too low price, the jammer’s utility is very low.On the other hand, if the jammer asks too high price, it risksthe situation in which the source would buy the service fromthe other friendly jammer. There is an optimal price for eachfriendly jammer to ask, and the source would always selectthe one that can provide the best performance improvement.

Next, we set up a simulation of mobility. The first friendly jammer is fixed at (50,50), while the second friendly jammermoves from (−50,75) to (100,75). In Figure 7, we show thesource utilities of the centralized scheme and the proposedgame. We can see that the centralized scheme serves asa performance upper bound. The game result is not far

8/8/2019 452907



away from the upper bound, while the game solution canbe implemented in a distributed manner. The performancegame is trivial when the friendly jammer 2 is close to themalicious eavesdropper from (20,75) to (70,75). In Figure 8,we show the jammers’ power as a function of jammer2’s location. We can see that depending on the jammers’

location, the source switches between two jammers for thebest performance. Moreover, the source also buys the optimalamount of jamming power: when the jammer is close tothe malicious eavesdropper, the source would buy less powersince the jammer is more eff ective to improve the secrecy capacity. In Figure 9, we show the corresponding friendly jammers’ utilities of the proposed game.

Finally, we show the eff ect of parameter a for the friendly jammer utility in (7). When a is large, the friendly jammer’sutility reduces quick if the source does not buy the service. Asa result, the friendly jammer would not ask arbitrary price,and performance gap to the optima solution is small. InFigure 10, we show the secrecy capacity as a function of awhen the second jammer is located at (0,75). We can see thatthe performance gap is shrinking whena is increasing. Noticethat for the condition in which the game almost converges tothe optimal solution, most value of a > 1 will achieve goodsolution, for example, the second friendly jammer located at(50,75).

5. Conclusions

Physical layer security is an emerging security techniquethat is an alternative for traditional cryptographic-basedprotocols to achieve perfect secrecy capacity as eavesdroppersobtain zero information. Jamming has been shown in theliterature to eff ectively improve secrecy capacity. In this

paper, we investigate the interaction between the sourceand friendly jammers using the game theory in order tohave a distributed solution. The source pays the friendly jammers to interfere the malicious eavesdropper such thatthe secrecy capacity is increased, and therefore the security of the network. The friendly jammers charge the source witha price for the jamming. To analyze the game outcome, weinvestigate the Stackelburg game and construct a distributedalgorithm. Some properties such as equilibrium and conver-gence are analyzed. From the simulation results, we concludethe following. First, there is a tradeoff for the price: If theprice is too low, the profit is low; and if the price is toohigh, the source would not buy or buy from other jammers.

Second, for the multiple jammer case, the source would buy service from only one jammer. Third, the centralized schemeand distributed scheme have similar performance, especially when a is sufficiently large. Overall, the proposed gametheoretical scheme can achieve a comparable performancewith distributed implementation.

Acknowledgments

This work was supported by NSF CNS-0910461 and NSFCNS-0905556 and was supported by the Research Councilof Norway through the project entitled “Mobile-to-MobileCommunication Systems (M2M).”

References

[1] A. D. Wyner, “The wire-tap channel,” Bell System Technical Journal , vol. 54, no. 8, pp. 1355–1387, 1975.

[2] S. K. Leung-Yan-Cheong and M. E. Hellman, “The Gaussianwiretap channel,” IEEE Transactions on Information Theory ,vol. 24, pp. 451–456, 1978.

[3] I. Csiszar and J. Korner, “Broadcast channels with confidentialmessages,” IEEE Transactions on Information Theory , vol. 24,no. 3, pp. 339–348, 1978.

[4] A. O. Hero III, “Secure space-time communication,” IEEETransactions on Information Theory , vol. 49, no. 12, pp. 3235–3249+3351, 2003.

[5] Z. Li, W. Trappe, and R. Yates, “Secret communication viamulti-antenna transmission,” in Proceedings of the 41st Annual Conference on Information Sciences and Systems (CISS ’07), pp.905–910, Baltimore, Md, USA, March 2007.

[6] R. Negi and S. Goelm, “Secret communication using artificialnoise,” in Proceedings of IEEE Vehicular Technology Conference,vol. 3, pp. 1906–1910, September 2005.

[7] P. Parada and R. Blahut, “Secrecy capacity of SIMO and

slow fading channels,” in Proceedings of IEEE International Symposium on Information Theory (ISIT ’05), pp. 2152–2155,Adelaide, South Australia, September 2005.

[8] S. Shafiee and S. Ulukus, “Achievable rates in Gaussian MISOchannels with secrecy constraints,” in Proceedings of IEEEInternational Symposium on Information Theory , pp. 2466–2470, Nice, France, June 2007.

[9] Y. Liang, H. V. Poor, and S. Shamai, “Secure communicationover fading channels,” IEEE Transactions on InformationTheory , vol. 54, no. 6, pp. 2470–2492, 2008.

[10] P. K. Gopala, L. Lai, and H. El Gamal, “On the secrecy capacity of fading channels,” IEEE Transactions on Information Theory ,vol. 54, no. 10, pp. 4687–4698, 2008.

[11] L. Dong, Z. Han, A. P. Petropulu, and H. V. Poor, “Securecollaborative beamforming,” in Proceedings of Allerton Confer-ence on Communication, Control, and Computing , Allerton, Ill,USA, October 2008.

[12] A. Kashyap, T. Basar, and R. Srikant, “Correlated jammingon MIMO Gaussian fading channels,” IEEE Transactions onInformation Theory , vol. 50, no. 9, pp. 2119–2123, 2004.

[13] S. Shafiee and S. Ulukus, “Mutual Information Games inMulti-user Channels with Correlated Jamming,” IEEE Trans.on Information Theory , vol. 55, no. 10, pp. 4598–4607, 2009.

[14] M. H. Brady, M. Mohseni, and J. M. Cioffi, “Spatially-correlated jamming in Gaussian multiple access and broadcastchannels,” in Proceedings of the 40th IEEE Conference onInformation Sciences and Systems (CISS ’06), pp. 1635–1639,Princeton, NJ, USA, March 2006.

[15] L. Lai and H. El Gamal, “The relay-eavesdropper channel:

cooperation for secrecy,” IEEE Transactions on InformationTheory , vol. 54, no. 9, pp. 4005–4019, 2008.

[16] X. Tang, R. Liu, P. Spasojevic, and H. V. Poor, “The Gaussianwiretap channel with a helping interferer,” in Proceedings of IEEE International Symposium on Information Theory (ISIT ’08), pp. 389–393, Toronto, Canada, July 2008.

[17] X. Tang, R. Liu, P. Spasojevic, and H. V. Poor, “Interference-assisted secret communication,” in Proceedings of the IEEEInformation Theory Workshop (ITW ’08), pp. 164–168, Porto,Portugal, May 2008.

[18] D. Fudenberg and J. Tirole, Game Theory , MIT Press, Cam-bridge, Mass, USA, 1991.

[19] C. U. Saraydar, N. B. Mandayam, and D. J. Goodman,“Efficient power control via pricing in wireless data networks,”

8/8/2019 452907



IEEE Transactions on Communications, vol. 50, no. 2, pp. 291–303, 2002.

[20] G. Scutari, S. Barbarossa, and D. P. Palomar, “Potential games:a framework for vector power control problems with coupledconstraints,” in Proceedings of IEEE International Conference on

Acoustics, Speech and Signal Processing (ICASSP ’06), vol. 4, pp.241–244, Toulouse, France, May 2006.

[21] B. Wang, Z. Han, and K. J. R. Liu, “Distributed relay selectionand power control for multiuser cooperative communicationnetworks using buyer/ seller game,” in Proceedings of Annual IEEE Conference on Computer Communications (INFOCOM ’07), pp. 544–552, Anchorage, Alaska, USA, May 2007.

[22] N. Bonneau, M. Debbah, E. Altman, and A. Hjørungnes,“Non-atomic games for multi-user systems,” IEEE Journal onSelected Areas in Communications, vol. 26, no. 7, pp. 1047–1058, 2008.

[23] R. D. Yates, “A framework for uplink power control incellular radio systems,” IEEE Journal on Selected Areas inCommunications, vol. 13, no. 7, pp. 1341–1347, 1995.

[24] S. Boyd and L. Vandenberghe, Convex Optimization, Cam-bridge University Press, Cambridge, UK, 2006.

[25] Z. Han and K. J. R. Liu, Resource Allocation for Wireless Networks: Basics, Techniques, and Applications, CambridgeUniversity Press, Cambridge, UK, 2008.

8/8/2019 452907


Photograph©TurismedeBarcelona/ J.Trullàs

Preliminarycallforpapers

The 2011 European Signal Processing Conference (EUSIPCO2011) is thenineteenth in a series of conferences promoted by the European Association for

Signal Processing (EURASIP, www.eurasip.org). This year edition will take placein Barcelona, capital city of Catalonia (Spain), and will be jointly organized by theCentre Tecnològic de Telecomunicacions de Catalunya (CTTC) and theUniversitat Politècnica de Catalunya (UPC).

EUSIPCO2011 will focus on key aspects of signal processing theory and

OrganizingCommittee

HonoraryChair MiguelA.Lagunas(CTTC)

GeneralChair

Ana

I.

Pérez

Neira

(UPC)GeneralViceChair CarlesAntónHaro(CTTC)

TechnicalProgramChair XavierMestre(CTTC)

Technical Pro ram CoChairsapp c a o ns as s e e ow. ccep ance o su m ss ons w e ase on qua y,relevance and originality. Accepted papers will be published in the EUSIPCOproceedings and presented during the conference. Paper submissions, proposalsfor tutorials and proposals for special sessions are invited in, but not limited to,the following areas of interest.

Areas of Interest

• Audio and electroacoustics.

• Design, implementation, and applications of signal processing systems.

JavierHernando(UPC)MontserratPardàs(UPC)

PlenaryTalksFerranMarqués(UPC)YoninaEldar(Technion)

SpecialSessionsIgnacioSantamaría(Unversidad

deCantabria)MatsBengtsson(KTH)

Finances

• Multimedia signal processing and coding.• Image and multidimensional signal processing.• Signal detection and estimation.• Sensor array and multichannel signal processing.• Sensor fusion in networked systems.• Signal processing for communications.• Medical imaging and image analysis.

• Nonstationary, nonlinear and nonGaussian signal processing.

TutorialsDanielP.Palomar(HongKongUST)BeatricePesquetPopescu(ENST)

Publicity

StephanPfletschinger(CTTC)MònicaNavarro(CTTC)

PublicationsAntonioPascual(UPC)CarlesFernández(CTTC)

Procedures to submit a paper and proposals for special sessions and tutorials will be detailed at www.eusipco2011.org. Submitted papers must be cameraready, nomore than 5 pages long, and conforming to the standard specified on the

EUSIPCO 2011 web site. First authors who are registered students can participatein the best student paper competition.

ImportantDeadlines:

IndustrialLiaison&ExhibitsAngelikiAlexiou

(UniversityofPiraeus)AlbertSitjà(CTTC)

InternationalLiaison

JuLiu

(Shandong

University

China)

JinhongYuan(UNSWAustralia)TamasSziranyi(SZTAKIHungary)RichStern(CMUUSA)RicardoL.deQueiroz(UNBBrazil)

Webpage:www.eusipco2011.org

roposa s orspec a sess ons ec

Proposalsfortutorials 18Feb 2011

Electronicsubmissionoffullpapers 21Feb 2011

Notificationofacceptance 23May 2011

Submissionofcamerareadypapers 6 Jun 2011

Documents

452907