I Introduction
Ia Background and Related Works
Nonorthogonal multiple access (NOMA) has been considered as one of the most promising technologies to deal with the shortage of bandwidth resources and enhance user fairness in the fifth generation (5G) mobile network [1][9]. Compared to traditional orthogonal multiple access schemes, NOMA systems can obtain superior merits, such as high bandwidth efficiency, better user fairness, ultraconnectivity, and high flexibility. After receiving the superposition signal transmitted by the source node, the stronger user in NOMA systems firstly decodes the signal sent to the weaker user, cancels it from the received signals, and then decodes the signals for itself.
Cooperative communication is a particularly attractive technique that not only extends the network’s coverage but also enhances the system performance when diversity technology is utilized at the destinations. Since the stronger users in NOMA systems always firstly decode the messages for the weaker users, therefore the stronger users can be utilized as relays to improve the performance of those weaker users with poor channel conditions [10, 11]. Ding et al. analyzed the outage probability (OP), diversity order of cooperative NOMA systems, and proposed an approach based on user pairing to reduce system complexity in [10]. Furthermore, a new cooperative simultaneous wireless information and power transfer NOMA protocol was proposed and the closedform expressions for the OP and system throughput were derived in [11]. Recently, many literatures focused on the cooperative NOMA systems with dedicated relay nodes. For example, a dedicated amplifyandforward (AF) relay with multiple antennas was utilized in the cooperative NOMA systems and the closedform expressions for the exact and lower bound of OP were derived in [12, 13, 14]. Moreover, a novel power allocation (PA) scheme for dualhop relaying NOMA systems along with its ergodic sumrate and OP were investigated in [15].
Fullduplex (FD) relay also has been utilized in cooperative NOMA systems to obtain higher spectral efficiency and better performance [16]. For example, the stronger user working in the FD mode was utilized to forward signals to the weaker user in [17] and [18]. The closedform expression for OP was obtained and the results showed that FD relays outperform halfduplex (HD) relays. Liu et al. analyzed the PA problems for HD and FD cooperative NOMA systems thereby obtaining the closedform expression for the optimal power allocation policies in [18]. Since the weaker user decodes its signals by directly treating signals for the stronger user as interference, then the performance of the weaker users will become the bottleneck of NOMA systems in many scenarios. The dedicated FD relay was utilized to improve the performance of the weaker users in [19], [20]. The performance of FD cooperative NOMA systems over Nakagami channels was investigated in [19] and the analytical expressions for OP and ergodic rate were derived. In [20], a sharing FD relay was utilized in NOMA systems with two sourcedestination pairs and the ergodic rate, OP, and outage capacity were investigated while both perfect and imperfect selfinterference cancellation (SIC) schemes were considered, respectively.
Relay selection (RS) technique is an effective scheme in making full use of space diversity with low implementation complexity and it can straightforwardly improve the spectral efficiency of cooperative systems [21, 22]. Considering the quality of service (QoS) requirements for the two users are different, a twostage singlerelayselection strategy and dualrelayselection with fixed power allocation (FPA) was studied in [23] and [24], respectively, in order to obtain the OP. The results showed that the twostage strategy achieves a better OP and diversity gain. Combining DF and AF relaying, Yang et al. extended the twostage RS scheme with dynamic power allocation (DPA) and derived the exact and asymptotic analytical expressions for OP in [25]. In [26], considering whether the relay can correctly decode two users’ messages, two optimal RS schemes were proposed for cooperative NOMA systems with FPA and DPA at the relays, respectively. Yue et al., in [27], studied the performance of cooperative NOMA systems with two RS schemes in which the relay operates in either FD or HD mode. Modeling the spatial topology of relays with homogeneous poisson point process (PPP), the performance of cooperative NOMA systems with twostage RS scheme was analyzed in [28] and a closedform approximation for the OP was obtained.
To deal with the security issues followed by the explosive increase in cellular data, physical layer security (PLS) is emerging as one of the most promising ways to ensure secure communication basing on the time varying nature of the wireless channels [29], [30]. A new optimal PA strategy was proposed to maximize the secrecy sum rate of NOMA systems and the results confirmed that a significant secrecy performance for NOMA systems was obtained in [31]. In [32], the security performance of a NOMAbased largescale network is considered, where both the NOMA users and eavesdroppers were modeled by homogeneous PPPs. In [33], we investigated the secrecy outage performance of a multipleinput singleoutput (MISO) NOMA systems with transmit antenna selection (TAS) schemes and the closedform expressions for secrecy outage probability (SOP) were derived. The results demonstrated that the proposed DPA scheme can achieve nonzero secrecy diversity order (SDO) for all the TAS schemes. The secrecy outage performance of multipleinput multipleoutput (MIMO) NOMA systems with multiple legitimate and illegitimate receivers was studied in [34] and the closedform expressions for SOP were subsequently obtained. Feng et al. proposed in [35] a new PA to maximize the secrecy rate of the stronger user while guaranteeing the nonsecure transmission rate requirement to the weaker user, but the secrecy performance of the weaker user was not considered. Secrecy beamforming schemes were proposed for MISONOMA systems, cognitive MISONOMA systems, and MIMONOMA systems in [36], [37], and [38], respectively. The secrecy performance of a NOMA system with multiple eavesdroppers was investigated while zeroforcing and minimum meansquare error decoding schemes were utilized on the legitimate destinations in [39]. A new joint subcarrier (SC) assignment and PA scheme was proposed to improve the security of the twoway relay NOMA systems in [40].
The secrecy performance of a cooperative NOMA system with a dedicated AF/DF relay was investigated and the closedform expressions for the SOP were obtained in [41]. In some scenarios, the relay in cooperative NOMA systems is untrusted and curious to decode the users’ messages. Two new HD nonorthogonal AF schemes were proposed for cooperative NOMA systems with an untrusted relay and the secrecy rate maximizationbased PA scheme was discussed in [42]. Arafa et al. proposed two new relaying schemes to resist the untrusted relay and the secrecy performance of cooperative NOMA systems with new relaying schemes was analyzed and compared in [43]. Feng et al. proposed an artificialnoise (AN) aided scheme to build up secure cooperative NOMA systems with a FD relay and the closedform expression for the SOP was derived in [44]. The secrecy performance of cooperative NOMA systems in which the stronger user behaves as a FD relay was investigated in [45] and the SOP was analyzed with an assumption of the imperfect SIC. The securityreliability tradeoff for both cooperative and noncooperative NOMA schemes was analyzed and analytical expressions for SOP were derived in [46].
IB Contributions
In many works, such as [32], [34], [41], the worstcase scenario was considered, in which it is assumed that the eavesdroppers have powerful detection capabilities and they can extract the message from the signals transmitted from the resource. Under this assumption, one can realize that the secrecy capacity of both/all the users in NOMA systems are not independent [34]. The major contributions of this paper are as follows,

In this work, we analyze a cooperative NOMA system with multiple relays and an eavesdropper employing RS schemes to enhance the secrecy performance. Three RS (OSRS, TSRS, and ODRS) schemes are proposed. For the purpose of comparison, we also investigate the secrecy performance of the NOMA systems with traditional multiple relays combining (TMRC) scheme in which all the relays that successfully decode signals from the source forward signals to the NOMA users with equal power. Considering the correlation between the secrecy capacity of two users, the closedform expressions for the SOP under different RS schemes are derived and validated via simulations.

Different secrecy QoS is considered in our work, i.e., different secrecy threshold rates are requested for the two NOMA users. Although the weaker user cannot perform SIC, either the stronger or the weaker user will be the bottleneck of the cooperative NOMA system. Our results demonstrate that this depends on the power allocation parameters and secrecy rate thresholds.

To obtain more insights, we also derive the closedform expressions for the asymptotic SOP under different RS schemes with FPA and DPA. Furthermore, the SDO of cooperative NOMA systems are derived. The results demonstrate that all the RS schemes with FPA obtain zero SDO and the OSRS scheme achieves the same SDO as TMRC scheme.
IC Organization
This paper is organized as follows, in Section II, the system model and RS schemes are introduced. In Section III, we analyze the security outage performance of the proposed system. The asymptotic SOP for cooperative NOMA systems with FPA and DPA are derived in Sections IV and V, respectively. Numerical and simulation results are presented in Section VI to demonstrate the security performance of the system. Finally, Section VII concludes the paper.
Ii System Model
As shown in Fig. 1, we consider a cooperative downlink NOMA system that consists of a base station (), DF HD relays (, ), and two users ( and ). An eavesdropper () wants to wiretap the information through decoding the received signals. It is assumed that all nodes are equipped with a single antenna and the direct link between and both users are unavailable due to deep fading. The communication links between and all the receivers (, , and ) are relayed by .
The probability density function (PDF) and the cumulative distribution function (CDF) of the channel power gains of link between the source node
and the destination node can be expressed by(1) 
(2) 
where signifies the transmitter, denotes the receiver, is Gamma function, as defined by (8.310.1) of [47], and are the fading parameters and average channel power gains, respectively.
To make analysis simple, it is assumed that the links of the first hop are independent and identically distributed (i.i.d.), which means , . The same assumption is made to the links between and each receiver in the second hop, which means , , , , and .
Two time slots are assumed to utilize to the communication between and the NOMA users. In the first time slot, broadcasts the superposition signal to all relays by message mapping strategy, which is utilized and proved optimal to achieve the minimal common OP in [26]. During the second time slot, a relay is selected using the RS scheme proposed below to send superposition signal to the two NOMA users. We assume all the relays are close enough such that the relative distance between the two users and the relay is determined.
To make the representation clear and simple, we use . It is also assumed that has a better channel condition than , which is adopted in many NOMA studies, e.g., [5], [16], [17], [18], [26], [33]. Then the signaltointerferencenoise ratio (SINR) of the NOMA users can be written as
(3) 
(4) 
where represents the power allocation coefficients at the th relay, , , and signifies the transmit signal to noise ratio (SNR).
Iii Secrecy Outage Probability Analysis
The set of relays that can correctly decode mixed signals from can be expressed as
(5) 
where the factor arises from the fact that two slots are required to complete the data transmission, denotes the transmit SNR at , and is the data rate threshold for . Thus the SOP of cooperative NOMA systems can be written as
(6) 
where denotes the SOP under the condition that there are relays that correctly decode the mixed signals. One can easily have .
It is assumed that all the links between the source and the relays are i.i.d., so we have
(7)  
where , , , and .
Iiia Traditional Multiple Relays Combining Scheme
As a benchmark, the traditional multiple relays combining (TMRC) scheme is presented in this subsection, where all the relays can successfully decode and forward the signal to both the users with equal power. Both the legitimate and illegitimate receivers combine their received signals with maximal ratio combining (MRC) scheme to maximize their SINR. To make fair comparison, it is assumed that the total transmit power at these relays is given by . Then the with this scheme can be expressed as
(8) 
where means the secrecy capacity of , , , , , signifies the SNR at when is wiretapped, is the secrecy rate threshold for , , and is the noise power.
As similar to [32], [34], and [41], we assume that has enough capabilities to detect multiuser data. Then the SNR at can be expressed as
(10) 
where and . Similarly, the PDF of can be expressed as [48]
(11) 
where , , and .
We can express the secrecy connection probability (SCP) of , which is the complementary of SOP, as
(12) 
where , , and .
Similarly, we obtain the SCP of as
(13)  
where , , , , , and .
Remark 1: One can easily find that secrecy outage would occur at when . This means that in order to ensure a secure NOMA system (), there is a constraint for the power allocation coefficients, which is expressed as
(14) 
To facilitate the following analysis, we define
(16) 
To the authors’ best knowledge, it’s very difficult to obtain the closedform expression of . Here by making use of GaussianChebyshev quadrature from eq. (25.4.30) of [49], we obtain
(17) 
where is the number of terms, , is the th zero of Legendre polynomials, is the Gaussian weight, which is given in Table (25.4) of [49].
IiiB Optimal Single Relay Selection Scheme
In this subsection, we propose OSRS scheme to minimize the overall SOP of the proposed cooperative NOMA system. Based on (12), one can easily obtain the SOP for with th relay as
(19)  
where , , and .
Similarly, based on (13), we can express SOP for as
(20)  
where , , , , and . One can observe that when the secrecy outage would occur at , which means the cooperative NOMA system is not secure.
For the th relay, we define
(21) 
Then to maximize the secrecy performance the relay is selected with the following criterion
(22) 
Remark 2: It should be noted that the selection scheme in (22) is different from the maxmin transmit antenna selection scheme (MMTAS) proposed in [34], which is to maximize the minimum secrecy capacity of two users. Here the selection criterion is to minimize the SOP of the cooperative NOMA system.
Remark 3: There is another important difference between MMTAS and OSRS, i.e., it is assumed that in MMTAS. However, in the OSRS scheme, and can be different. MMTAS is a special case of OSRS when it is assumed that and the secrecy capacities of the two users are independent.
The SOP of the cooperative NOMA system conditioned on can be written as
(23)  
IiiC TwoStep Single Relay Selection Scheme
In some scenarios stated in [23, 24], the QoS requirements for the two users are different. We can easily obtain the similar conclusion: the secrecy QoS for one user is higher than that of the other user. In this subsection, a new twostep RS (TSRS) scheme is proposed for such scenarios and the follwing two purposes will be realized simultaneously. One is to ensure there is no secrecy outage for the user that has lower secrecy QoS requirement and the other is to serve the user with higher secrecy QoS requirement with a secrecy rate as large as possible. The TSRS scheme is presented as follows.
In the first step, the following subset is built in the relays by focusing on ’s target secrecy rate
(25) 
where signifying the secrecy capacity for from and .
At the second step, the relay to maximize the secrecy capacity of is selected, i.e.,
(26) 
where signifies the secrecy capacity for from .
The SOP with this scheme can be achieved as
(27)  
Remark 4: An interesting result can be observed from (27) that the expression for the SOP under TSRS scheme is the same as that under OSRS.
IiiD Optimal Dual Relay Selection Scheme
In this subsection, a new RS scheme named ODRS scheme is proposed, in which one relay is selected to transmit signals to NOMA users among the ones that can decode the mixed signals and another relay is selected to transmit AN among the ones that can not decode the source signals. In order to deteriorate the SINR at , we select the jamming relay with the following criterion as
(28) 
where means the complement of . For the sake of a fair comparison with TMRC and OSRS schemes, the total transmit power of all the relays is constrained to . It is assumed that portion is utilized to transmit the jamming signals, then the SINR at can be written as
(29) 
where , , , and .
We assume that both users are aware of the jamming signals, which means jamming signals do not influence the SINR at both users [32, 36, 37]. It must be noted that the special case of or , ODRS scheme reduces to OSRS scheme. Then the SOP with ODRS scheme is expressed as
(30) 
where is given in (23).
Similar to (27), the SOP of the cooperative NOMA system on condition that can be expressed as
(31)  
where , , , , and .
To achieve the analytical expression of , the PDF of and are given in Lemma 1 and Lemma 2, respectively.
Lemma 1: Given as , the PDF of is given by
(32) 
where , , , and .
Proof : See Appendix A.
Lemma 2: The PDF of is given by
(33) 
where , , , and .
Proof : See Appendix B.
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