Relay Selection for Secured Communication in ...

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Relay Selection for Secured Communication in Interference-limited Networks Tran Trung Duy1 and Vo Nguyen Quoc Bao1 School of Telecommunications, Posts and Telecommunications Institute of Technology, Vietnam [email protected], [email protected]

Abstract. In this paper, we analyze the secrecy performance of the relay selection methods in interference-limited networks. In particular, we propose the optimal and suboptimal relay selection methods to enhance the secrecy rate. For performance evaluation, we derive exact and asymptotic closed-form expressions of secrecy outage probability over Rayleigh fading channel. Monte Carlo simulations are performed to validate our derivations. Keywords: Relay selection, physical layer security, secrecy outage probability, Rayleigh fading channel, co-channel interference.

1

Introduction

In wireless networks, diversity relay selection [1-5] is one of important methods to enhance the wireless system performance in terms of outage probability, error rate and channel capacity. Recently, the new relay selection methods have been proposed to enhance the secrecy performance at physical layer. In physicallayer security [6-8], the physical characteristics of wireless channels are exploited to obtain the secure transmission, which is built on the concept of perfect secrecy. In [9-10], the authors proposed the optimal and suboptimal strategies to maximize the secrecy rate. In [11-14], the joint relay and jammer selection schemes were proposed and analyzed. Although these methods significantly enhance the security of wireless networks, their implementation which requires perfect synchronization between nodes is a difficult work. Recently, the impact of co-channel interference on the performance of wireless systems was investigated. In [15], the authors evaluated the performance of a dual-hop amplify-and-forward (AF) system when the relay is impaired by the co-channel interference. In [16], the performance of two-way AF relaying systems in interference-limited networks was studied. To the best of our knowledge, there has been no published work studying the relay selection methods in physical-layer security and in the presence of the co-channel interference. In this paper, we propose two relay selection methods to enhance the secrecy performance in interference-limited networks. In the

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first scheme, the best relay is selected to maximize the secrecy rate. In the second scheme, a suboptimal relay selection method is proposed. For performance evaluation and comparison, we derive the exact and asymptotic closed-form expressions of the secrecy outage probability over Rayleigh fading channels. Then, Monte Carlo simulations are performed to verify our derivations. Results show that the proposed methods obtain higher performance as compared with the method in which the relay is randomly selected. The rest of the paper is organized as follows. The system model is described in Section 2. In Section 3, the performance evaluation is analyzed. The simulation results are presented in Section 4, and Section 5 concludes the paper.

2

System Model

R1 Data

D

Interference

I

Rn Eavesdropping

E

Interference

RN Fig. 1. Secured communication in interference-limited networks.

In Fig. 1, we present the system model of the proposed scheme. Similar to the scheme in [10], we consider the cooperative phase where N potential relays are already to transmit the source data to the destination D. It is assumed that these relays obtain the source data in the broadcast phase. For the conciseness, we skip presenting the operation of the broadcast phase in which the source transmits its data to all relays. In this network, the eavesdropper E attempts to decode the source signal transmitted from relays to the destination. In addition, while the relays forward the data to the destination, another source I in the network is also transmitting their data and hence causing the interferences to the destination and eavesdropper. We assume that all of the transmitters such as the relays and the source I have the same the transmit power P . We also assume that additive noises at all of the receivers such as the destination D, the eavesdropper E are Gaussian random variables (RVs) with zero mean and variance of N0 . Let us denote hn , gn , k and l as the channel coefficients of the Rn → D, Rn → E, I → D and I → E

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links, respectively, where n = 1, 2, ..., N . It is assumed that all of the channels follow Rayleigh distribution. Hence, the channel gains |hn |2 , |gn |2 , |k|2 and |l|2 are exponential RVs with parameters ω1 , ω2 , ξ1 and ξ2 , respectively [17]. All of the terminals are equipped with a single antenna and operate on halfduplex mode. In the proposed scheme, only a relay is selected to transmit the source data to the destination. Assume that the relay Rn is the chosen relay, the instantaneous signal-to-noise ratio (SNR) received at the destination and the eavesdropper can be given respectively as ψR n D =

Xn P |hn |2 = , 2 N0 + P |k| 1+Y

(1)

ψR n E =

P |gn |2 Zn = , N0 + P |l|2 1+T

(2)

where Xn = P |hn |2 /N0 , Y = P |k|2 /N0 , Zn = P |gn |2 /N0 and T = P |l|2 /N0 . Since |hn |2 , |gn |2 , |k|2 and |l|2 follow exponential distribution, Xn , Y , Zn and T are also exponential RVs with parameters λ1 , Ω1 , λ2 and Ω2 , respectively, where λ1 = N0 ω1 /P , λ2 = N0 ω2 /P , Ω1 = N0 ξ1 /P and Ω2 = N0 ξ2 /P . As formulated in [10], the secrecy capacity is expressed as     1 + ψR n D ,0 . (3) Csec = max log2 1 + ψR n E In this paper, we propose two relay selection methods. In the first one, named OPTimal scheme (OPT), the best relay is selected by the following strategy:   1 + ψR n D Rb = arg max . (4) n=1,2,...,N 1 + ψR n E However, to realize the strategy (4), it is required the channel state information (CSI) of all of the links, which is a very difficult work. Hence, a SUB-optimal relay selection scheme, named SUB, is proposed as follows: Rb = arg

max

n=1,2,...,N

Xn .

(5)

In the second method, the best relay is selected by using the CSIs between the relays and the destination. Because this information can be easily obtained from the local control message, the implementation of the second method is more feasible than that of the first one. To show the advantages of the proposed schemes, we compare the performance of the proposals with that of a scheme, named RAN, in which a relay is RANdomly selected to forward the source data to the destination. Next, the secrecy outage probability is defined as the probability that the secrecy capacity is below a positive threshold as [10]:
X X Pout = Pr Csec < Rth , (6) where X ∈ {OPT, SUB, RAN}.

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3 3.1

Performance Evaluation Exact Secrecy Outage Probability

In this section, the exact closed-form expressions of the secrecy outage probability for the considered schemes are derived. At first, for the RAN protocol, the secrecy outage probability can be formulated as follows: RAN Pout = Pr (ψRn D < ρ − 1 + ρψRn E ) Z +∞ FψRn D (ρ − 1 + ρy) fψRn E (y) dy, =

(7)

0

where ρ = 2Rth , FψRn D (.) and fψRn E (.) are the cumulative density function (CDF) and the probability density function (PDF) of ψRn D and ψRn E , respectively. From (1), we can write the CDF FψRn D (.) under the following form: FψRn D (x) = Pr (Xn < x + xY ) Z +∞ FXn (x + xy) fY (y) dy. =

(8)

0

Since FXn (t) = 1 − exp (−λ1 t) and fY (y) = Ω1 exp (−Ω1 y), from (8), the CDF FψRn D (.) can be easily obtained by the following expression: Ω1 exp (−λ1 x) Ω1 + λ 1 x α1 exp (−λ1 x) , =1− x + α1

FψRn D (x) = 1 −

(9)

where α1 = Ω1 /λ1 . With the same manner, we easily obtain the CDF of ψRn E . Then, the corresponding PDF of ψRn E can be given as fψRn E (y) =

α2 exp (−λ2 y) (y + α2 )

2

+

λ2 α2 exp (−λ2 y) , y + α2

(10)

where α2 = Ω2 /λ2 . Substituting (9) and (10) into (7), yields RAN Pout = 1 − α1 α2 exp (−λ1 (ρ − 1)) /ρ # Z +∞ " 1 λ2 × 2 + (y + α ) (y + α ) 3 2 (y + α3 ) (y + α2 ) 0

× exp (− (ρλ1 + λ2 ) x) dy, where α3 = (α1 + ρ − 1)/ρ.

(11)

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For ease of presentation, we assume that α3 6= α2 . In this case, we first expand the integrands in (11) into partial expansions as follows:   1 λ2 1 − λ2 (α3 − α2 ) 1 1 = − + 2 2 (y + α3 ) (y + α2 ) y + α3 y + α2 (y + α3 ) (y + α2 ) (α3 − α2 ) 1 1 . . (12) + α3 − α2 (y + α2 )2 By applying the following integrals Z +∞ exp (−λx) dx = exp (λα) E1 (λα) , x+α 0 Z +∞ exp (−λx) 1 2 dx = α − λ exp (λα) E1 (λα) , (x + α) 0

(13)

(14)

RAN can be expressed by a closed-form into (11), the secrecy outage probability Pout expression as RAN Pout = 1 − α1 α2 exp (−λ1 (ρ − 1)) /ρ         (1−λ2 (α3 −α2 2 )) exp ((λ1 + λ2 ) α3 ) E1 ((λ1 + λ2 ) α3 ) (α3 −α2 ) − exp ((λ1 + λ2 ) α2 ) E1 ((λ1 + λ2 ) α2 ) × i , h   1  + 1 α3 −α2 α2 − (λ1 + λ2 ) exp ((λ1 + λ2 ) α2 ) E1 ((λ1 + λ2 ) α2 )

(15)

where E1 (.) is exponential integral function [18]. Next, let consider the OPT protocol, from (4), we can easily obtain the secrecy outage probability of this protocol as follows:     1 + ψR n D OPT < Rth Pout = Pr max n=1,2,...,N 1 + ψR n E   N 1 + ψR n D = Pr < Rth 1 + ψR n E
RAN N . (16) = Pout

Now, for the SUB protocol, similar to (7), the secrecy outage probability can be formulated as Z +∞ SUB (17) FψRb D (ρ − 1 + y) fψRb E (y) dy. Pout = 0

In (17), the CDF FψRb D (z) can be formulated as FψRb D (x) = Pr (Xb < x + xY ) Z +∞ FXb (x + xy) fY (y) dy. = 0

(18)

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where the CDF of Xb is given by  max FXb (z) = Pr

n=1,2,...,N

Xn < z N



= (1 − exp (−λ1 z))   N X N n (−1) exp (−nλ1 z) . =1+ n

(19)

n=1

Plugging fY (y) = Ω1 exp (−Ω1 y), (18) and (19) together, which yields FψRb D (x) = 1 +

N X

n=1

(−1)

n



N n



βn exp (−Φn x) , x + βn

(20)

where βn = Ω1 /nλ1 and Φn = nλ1 . It is noted that the PDF fψRb E (.) in (17) is same with fψRn E (.) in (10). Hence, with the same manner as above, we can obtain the secrecy outage probability of the SUB protocol as follows: SUB Pout

=1+

N X

n=1

(−1)

n



N n



βn α2 exp (−λ1 (ρ − 1)) /ρ

      (1−λ2 (ϕn −α2 2 )) exp ((Φn ρ + λ2 ) ϕn ) E1 ((Φn ρ + λ2 ) ϕn )  (ϕn −α2 ) − exp ((Φ ρ + λ ) α ) E ((Φ ρ + λ ) α ) n 2 2 1 n 2 2 × i , (21) h   1 + 1  ϕn −α2 α2 − (Φn ρ + λ2 ) exp ((Φn ρ + λ2 ) α2 ) E1 ((Φn ρ + λ2 ) α2 )

where ϕn = (βn + ρ − 1) /ρ and ϕn 6= α2 for all n. 3.2

Asymptotic Secrecy Outage Probability

In this subsection, we consider the secrecy outage probability at high transmit SNR, i.e., Ψ = P/N0 → +∞. At first, we can rewrite (1) and (2) at high Ψ value as Ψ →+∞ Xn , (22) ψR n D ≈ Y Ψ →+∞ Zn . (23) ψR n E ≈ T Then, using the methods presented above, we respectively obtain the asymptotic expression of the secrecy outage probability for the RAN, OPT and SUB schemes at high Ψ region as follows: Z +∞ α1 α2 RAN Ψ →+∞ ≈ 1− Pout 2 dy ρ (y + α3 ) (y + α2 ) 0    Ψ →+∞ α1 α2 α3 − α2 α3 ≈ 1− − ln , (24) 2 α2 α2 ρ(α3 − α2 )

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OPT Ψ →+∞ ≈ Pout

RAN Pout

Ψ →+∞

≈

1+

N X

n=1

4

(

1−

(−1)

n

α1 α2 ρ(α3 − α2 ) 

N n



2



α3 − α2 − ln α2

βn α2 ρ(ϕn − α2 )

2





α3 α2

)N

ϕn − α 2 − ln α2



,

ϕn α2

(25)



.

(26)

Simulation Results

In this section, Monte Carlo simulations are performed to verify our derivations and to compare the performance of the considered protocols. In all simulations, we fix the parameters as follows: ω1 = ω2 = 1, ξ1 = 1.5 and ξ2 = 2. In Fig. 1, we present the secrecy outage probability as a function of transmit SNR (Ω = P/N0 ) in dB. In this simulation, the number of relays N and the threshold value γth are fixed by N = 2 and Rth = 0.5. It can be seen from this figure that the OPT protocol obtains the best performance while that of the RAN protocol is worst. It is also seen that the secrecy outage probability of all of the protocols decreases when increasing the transmit SNR Ω. However, at very high Ω region, the outage performances converge to the asymptotic values which does not depend on Ω. Figure 2 presents the secrecy outage probability as a function of the number of relays. In this figure, we assign values of Ω and Rth by 10dB and 1, respectively. As can be seen from Fig. 2 that the secrecy outage probability decreases when increasing the number of relays. It can also be observed from this figure that the secrecy outage probability of the RAN protocol does not depend on the number of relays due to the random relay selection. From Figs. 1- 2, it is worth noting that the simulation results match very well with the theoretical results, which validates our derivations.

5

Conclusion

In this paper, we proposed two relay selection methods to mitigate the effect of co-channel interference and enhance the secrecy performance. Results presented that two proposed schemes obtains better performance as compared with the random relay selection protocol, in terms of the secrecy outage probability. Finally, Monte Carlo simulations match very well with theoretical results which verify our derivations.

Acknowledgment This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 102.04-2012.20.

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RAN−Sim SUB−Sim OPT−Sim Theory−Exact Theory−Asymptotic

Secrecy Outage Probability

0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35

0

5

10

15

Ψ

20

25

30

Fig. 2. Secrecy outage probability as a function of Ψ in dB when N = 2 and Rth = 0.5.

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0

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10

RAN−Sim OPT−Sim SUB−Sim Theory−Exact −1

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N

Fig. 3. Secrecy outage probability as a function of N in dB when P = 10 dB and Rth = 1.

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