Relay Selection to Improve Secrecy in Cooperative Threshold DecodeandForward Relaying
Abstract
In this paper, relay selection is considered to enhance security of a cooperative system with multiple thresholdselection decodeandforward (DF) relays. Thresholdselection DF relays are the relays in which a predefined signaltonoise ratio is set for the condition of successful decoding of the source message. We focus on the practical and general scenario where the channels suffer from independent nonidentical Rayleigh fading and where the direct links between the source and destination and source and eavesdropper are available. Based on channel state information knowledge, three relay selection strategies, namely traditional, improved traditional, and optimal, are studied. In particular, the secrecy outage probability of all three strategies are obtained in closedform. It is found that the diversity of secrecy outage probability of all strategies can improve with increasing the number of relays. It is also observed that the secrecy outage probability is limited by either the source to relay or relay to destination channel quality.
1 Introduction
To secure the broadcast nature of wireless communication against eavesdroppers, physical layer security has gained much prominence [1]. Motivated by recent advances in cooperative communication systems [2], employing the cooperative technique to enhance physical layer security of wireless systems has recently been receiving significant research interest. Compared to multirelay assisted transmission, relay selection in which a single relay among all possible candidates is selected for relaying a source’s signal has been shown to optimize system resource utilization, such as power and bandwidth, while maintaining the same diversity order.
Relay selection to improve secrecy in cooperative communication system has received considerable attention recently [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. The relays considered in these works are conventional amplifyandforward (AF) or decodeandforward (DF) relays [2]. In all these works, relay selection is performed depending on the availability of the instantaneous channel state information (ICSI) or statistical channel state information (SCSI) of the links. Based on the knowledge of ICSI or/and SCSI, the following three cases have been considered mostly for the relay selection problem. Case i): when the ICSI of the source to relay and relay to destination is known. In this case, the selected relay is the one which achieves the maximum rate through the sourcerelaydestination channel, which is described as the main channel in physical layer security. We refer to this as traditional relay selection (TS). Case ii): in addition to the ICSI of source to relay and relay to destination, the SCSI of the relay to eavesdropper channels are known. This is an improvement over the previous case and we refer to it as improved traditional selection (ITS). Case iii): when the ICSI of all links are known. In this case, the relay selected is the one which provides maximum secrecy rate. We refer to this as optimal selection (OS). In practise, the ICSI of the various links can be acquired using one of the techniques described in [14] and the references therein.
With the notable exception of [12], [13] and [15], most of the existing work on physical layer security in DF relay cooperative systems has only considered the high signaltonoise ratio (SNR) regime for the source to relay link. This is not very practical as fading can severely degrade the channel quality of a link in wireless communication systems. In [12], the source to relay link quality is taken into account by considering that the rate at the destination is limited by the minimum of the source to relay and relay to destination rate. In [13], the set of successful relays which recover the source symbol are those for which the source to relay link rate is above a minimum threshold rate. Furthermore, only [13] has considered the existence of direct links from source to destination and source to eavesdropper. However, in this work, all the links are assumed to experience independent and identical Rayleigh fading. Though the identical distribution assumption makes the analysis more tractable, it may not be valid for practical wireless communication applications because, in general, the relays are not closely placed in real environments. Moreover, only the TS scheme is studied in [13] and the relay selection problem is not tackled in [15].
With this motivation in mind, in this paper, we study relay selection to enhance security of a cooperative system with multiple thresholdselection DF relays. In thresholdselection relaying scheme for DF cooperation protocol, the possible candidate relays for selection are those for which the signaltonoise ratio (SNR) is above a predefined threshold [16]. We consider the more practical scenario where the direct links between the source to destination and source to eavesdropper exist and where the links experience independent but not necessarily identically distributed Rayleigh fading. The main contributions of the paper can be summarized as follows:

We study three relay selection strategies, depending on the ICSI and SCSI knowledge to enhance the secrecy outage probability of thresholdselection relaying.

We derive the secrecy outage probability in closedform for the most general case of independent but nonidentical channels.

We obtain the secrecy outage assuming direct links from source to destination and source to eavesdropper.
In detailing our contributions, we observe that the studied relay selection strategies can increase the diversity order of secrecy outage probability with increasing the number of relays. Interestingly, we also observe that the secrecy outage probability can not be increased beyond a certain level if either the source to relay or relay to destination channel quality is kept fixed while the other is increased.
The rest of the paper is organized as follows. Section 2 introduces the system model. Section 3 evaluates the secrecy outage probability for the relay selection strategies. Section 4 discusses the results, and finally, Section 5 concludes the paper.
Notation: is the probability of occurrence of an event, defines the expectation of its argument over the random variable (r.v) , and denotes the maximum of its argument, represents the cumulative distribution function (CDF) of the r.v , and is the corresponding probability density function (PDF).
2 System Model
The system model consists of one source (), one destination (), one passive eavesdropper (), and DF relays (, ), as shown in Fig. 1. All nodes are equipped with a single antenna. The relays are halfduplex in nature, and hence, complete information transmission takes place in two time slots. broadcasts its message in the first time slot. We assume that the relays are thresholdselection DF type [16]; in other words, they correctly decode the received message and retransmits in the second time slot only if their SNR is above a threshold, . The SNR threshold, , can be properly chosen to achieve the goal of correct decoding. The channels are modeled as independent nonidentically distributed flat Rayleigh fading. Both and utilize maximal ratio combining (MRC) technique to get the advantage of two copies of same signal from the direct transmission and the relayed transmission. The received SNR, , of any arbitrary  link from node to node can be expressed as , where and are from the set for any possible combination of , . is the transmit power from node and is the noise variance of the additive white Gaussian noise (AWGN) at node . As is assumed Rayleigh distributed with average power unity, i.e., , is exponentially distributed with mean . The CDF of can be written as .
For simplicity, we further denote the parameters of the  and  links which are terminating at as and , respectively. Parameters of the other links which are conveying messages towards , i.e.,  or , are denoted by and , respectively.
The achievable secrecy rate of the system is [1]
(1) 
where and are the main channel and eavesdropper channel SNR at and , respectively. The term reflects the fact that two time slots are required for information transfer.
The secrecy outage probability of the system represents the probability that the achievable secrecy rate is less than a target secrecy rate, , and is expressed as
(2) 
where .
3 Secrecy Outage of Relay Selection
Let us assume that is a set representing the relays which are able to decode successfully at the first stage. A relay is to be selected from this particular set by a relay selection rule in the second stage. The secrecy outage probability of the system with relay selection, , can be mathematically represented as
(3) 
where represents the probability of occurrence of a particular set containing relays, and represents the secrecy outage probability for a given relay selection rule. The second summation in (3) must be performed for possible combinations. It should be noted that and can be evaluated independently, as they are the result of independent events.
Now let us assume that the relays which are unable to exceed constitute the set . can be easily evaluated by multiplying the probability of occurrence of and . With the probability that a particular relay is in given by
(4) 
can be evaluated as
(5) 
When is the empty set, there exists only the direct link to and from . In this case, the secrecy outage probability can be obtained from (2) as
(6) 
When contains a single relay, the secrecy outage probability is obtained by using and . The and distributions can be readily found for different parameters of the exponential r.vs as [17]
(7) 
where , . and are the parameters of the two independent exponentially distributed r.vs, with . Thus, the secrecy outage probability can be obtained from (2) as
(8) 
3.1 Traditional Selection (TS)
The traditional relay selection rule does not take into account the  channel quality for all k. This scheme selects the relay which achieves the highest rate through the  link, as successful decoding has already been performed in the first stage. The highest rate is achievable on the link having highest instantaneous SNR.
The secrecy outage probability of the traditional rule corresponding to can be evaluated using the law of total probability as
(9) 
where is the secrecy outage probability for the th relay in , , where and , is the maximum SNR between all relays which are not selected from by the relay selection rule. The distribution is expressed as [12]
(10) 
where is defined as
(11) 
and . To further evaluate the probability in (3.1), let us assume a new r.v, . As all r.vs in (3.1) are independent, the solution can be written in integral form as
(12) 
where
(13)  
(14) 
and .
In deriving (12), , , and represent the realizations of , , and , respectively. The distribution is given in (10), while the distribution of the r.v is given in (7). The integration limits are due to following reasons: i) should be always less than ; hence, takes values from zero to , ii) none of the r.vs can take negative values, and hence, when exceeds , is higher than in (3.1), iii) when is below , has positive values; hence, the corresponding integral is from zero to infinity in (3.1). Final expressions of and are given in (5) and (5), respectively.
3.2 Improved Traditional Selection (ITS)
3.3 Optimal Selection (OS)
Optimal selection takes into account both main channel and channels quality. The relay is selected for which the secrecy rate is maximum, and
(17) 
The above probability can be evaluated first for given and , and then by averaging over them. In this case, (17) can be written as a product of individual probabilities
(18) 
where and are expressed in (30) and (32), respectively, with , and , and are realizations of the r.vs, , , and respectively. The solution of and are provided in (31) and (33), respectively, with defined as [12]
(19) 
We also define , and .
4 Numerical and Simulation Results
This section describes numerical and simulation results. Unless otherwise mentioned, dB, dB, dB, , and bits per channel use (bpcu).
In Fig. 2, the secrecy outage probabilities of the selection schemes TS, ITS and OS are plotted versus average SNR, , for different rate requirements, bpcu. Nonidentical link parameters are considered, with , , whereas dB, respectively, for . As expected, OS works the best, followed by ITS, and TS is the worst. Additionally, it is worth noting that as increases, the secrecy outage probability deteriorates.
Fig. 3 depicts the secrecy outage probabilities of the selection schemes TS, ITS and OS for two different values, i.e., dB. It is assumed that the  and  link qualities are identical, i.e., , while the  link qualities are nonidentical, i.e., dB for . The secrecy outage probability deteriorates with the increase in . Higher values of reduces the number of relays available for selection, hence the observation.
In Fig. 4, the secrecy outage probabilities of the selection schemes TS, ITS and OS are plotted versus the average SNR for increasing number of relays from to . Identical link qualities are assumed as and dB for all , respectively. It is clear that the performance improves as the number of relays increase. The slope of the curves also increases with increasing the number of relays, which means that the diversity order of the secrecy outage probability improves with . An important observation is that the improvement obtained by increasing the number of relays follows the laws of diminishing return. Furthermore, as the  link qualities are identical for all , the ITS selection scheme can not provide better performance than the TS selection scheme and merges with TS. It is worth noting that the performances of the TS and ITS schemes do not merge in Fig. 2 and Fig. 3, respectively.
Fig. 5 shows the secrecy outage probabilities of the selection schemes TS, ITS and OS when the links  and  are unbalanced. Two cases are considered, as follows: Case 1, when dB and , and Case 2, when dB and , for . The results are obtained assuming identical eavesdropper link qualities, dB, for all . It is observed that the secrecy outage probability saturates to a particular value depending on the values of or . This indicates that either link  or , for all , can limit the the secrecy outage probability. Furthermore, it can be observed that when dB, the performance of the relay selection schemes saturates to the same value, while the performance saturates to different values when dB. When the  link quality improves, the number of relays that exceed increases. As the relay selection schemes can take increased advantage when there are more relays to choose from, performance saturates to different values in Case 1 depending on the selection scheme.
It should be noted that, in all figures, simulation results are in agreement with numerical results. This validates our analysis in Section 3.
5 Conclusion
Three relay selection schemes, namely traditional, improved traditional, and optimal are proposed to enhance the secrecy outage probability using thresholdselection DF relays. The secrecy outage probability is derived in closedform assuming the most practical scenario of independent but nonidentical fading channels and including direct links from the source to destination and eavesdropper. It is found that by increasing the number of relays, the diversity gain of the secrecy outage probability can be increased. On the other hand, higher SNR threshold at the relays can decrease the secrecy performance. It is observed that the improved traditional relay selection can outperform the traditional relay selection only if the eavesdropper links are nonidentical. It is also noticed that the secrecy outage probability is limited by either the source to relay or the relay to destination link quality.

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