1 Introduction
5G has been commercialized in 2020, and nonorthogonal multiple access (NOMA) plays a key role in this. NOMA has been used for many scenarios to solve the problems caused by the explosive growth of the number of mobile terminals [1]. Unlike the traditional orthogonal multiple access (OMA) system structure, powerdomain NOMA serves more users at the same time and frequency based on the power allocation of the transmitted signals. Consequently, NOMA can enhance the communication quality of users in poor channel conditions [2]. Recently, a multipleinput and multipleoutput (MIMO)NOMA system which uses maxmin transmit antenna selection strategy was proposed in [3]. In [4], the authors proposed a new hybrid decodeandforward (DF) and amplifyandforward (AF) transmission mode for a multiplerelay NOMA system. A NOMA system under Rician fading channels was studied in [5] and expressions of the average achievable rate were derived.
In wireless communication systems, the signals are broadcast so that physical layer security (PLS) has become a hot issue. The analysis of secrecy performance in various wireless systems is studied in the literature, such as dualhop RF/freespace optical (FSO) systems [6] and transmit antenna selection (TAS)/maximal ratio combining (MRC) systems [7]. Recently, many works about PLS for NOMA systems have been considered. For instance, in [8], PLS for cognitive radio inspired NOMA networks was investigated. PLS in a multiuser visible light communication (VLC) system with NOMA was considered in [9]. In [10], PLS for cooperative NOMA systems was investigated, where both AF and DF were considered.
Recently, a new material called reconfigurable intelligent surfaces (RIS) has been proposed. RISs have a large number of application scenarios in wireless communication, and even change the traditional communication structure [11]. So far, there are many works based on RISs have been reported in [1222]. For example, a mixed dualhop FSORF system through the RIS was proposed in [12]. An RISassisted dualhop UAV communication system was proposed in [13]. The authors in [14] quantitatively analyzed the coverage for an RISaided communication system. An RISaided downlink multiuser communication system was investigated in [15]. The authors in [16] proposed a deep learning method for deploying RISs in an indoor environment. Moreover, an important application of RISs is to combine with NOMA to further improve communication quality. For instance, the authors in [17] proposed an RISempowered NOMA network to introduce desirable channel gain differences by adjusting the phase shifts at the RISs. In [18], the authors conceived a system for serving paired powerdomain multiple NOMA users by designing the passive beamforming weights at the RISs. The authors in [19] proved that NOMA can achieve the capacity region when the channels are quasidegraded by using the RISs. In [20], the authors proposed a theoretical performance comparison between NOMA and OMA in the RISassisted downlink communication. The authors in [21] derived the bit error rate (BER) performance of the RISassisted power domain NOMA system. In [22], the authors studied both downlink and uplink RISaided NOMA and OMA networks. However, considering the PLS for the RISaided NOMA system is still not reported in the literature. Therefore, this is the main innovation of this work.
In this paper, we propose an RISassisted multiuser NOMA system. In particular, we assume that an eavesdropper in the considered network can receive signals from the RISs and source to affect the legitimate users. Based on this assumption, we intend to investigate whether the RIS always improves the secrecy performance. In particular, we derive analytical expressions for the secrecy outage probability (SOP). Also, the asymptotic SOP analysis at high signaltonoise ratio (SNR) condition is provided. Finally, some numerical results are presented to verify our analysis and investigate the effects of the number of reflecting surfaces in the RIS on the system secrecy performance.
2 System and channel models
As shown in Figure 1, consider an RISassisted NOMA system which includes a source (S), RISs, groups of NOMA users, and an eavesdropper (E), where near users are close to S, while far users have a long distance from S. Therefore, similar to [22] and [23], we utilize RISs to increase the signal coverage to improve the far users’ communication quality, while the near users directly communicate with S. We assume that RISs have the same reflecting elements. Furthermore, we assume the worst case that E can utilize the advantage of the RIS. Finally, we suppose that the channels in this system suffer from Rayleigh fading independently.
According to the NOMA protocol, we need to distribute the total transmit power to the NOMA users concurrently, but it is not preferable to group all the users in a NOMA system in practice [24]. Therefore, we choose one near user and one far user to constitute a NOMA group, and then in every group we use one RIS to improve the far user’s received SNR. Thus, users are divided into groups and only one group can be selected to communicate according to the criterion described later.
In particular, we assume that the far user with poor channel gains is defined as the weak user and the near user with good channel gains is defined as the strong user (). In order to enhance ’s communication quality, we set that , and let them satisfy [10], where is the power distribution coefficient (). First, the mixed signal, , is broadcast from S to the th RIS () and the near user, where is the unit signal needed by user . Then, passively reflects the signals to . Thus, the received signal by the far user can be written as
(1) 
where and are the channel gains for the S and  links. In (1), is the adjustable phase produced by the th reflecting element of (). Let and , where and are the distances for the S and  links, denotes the path loss coefficient, and denote the channels’ amplitudes, and are the phases of the fading channels. Similar to [11], we assume that has perfect knowledge of the channels phases of and .
For the near users, they receive mixed signals from S directly. Thus, the received signals by the near user can be written as
(2) 
where is the distance of the S link and is the average transmitted energy per symbol. In (1) and (2), and are the additive white Gaussian noise (AWGN) samples.
Since E receives the same signals from and S, the received signal at E can be expressed as
(3) 
where , and denote the distance for the E and SE links, and are the amplitude and phase of the fading channel, and
is the AWGN sample with variance
.According to [25], in NOMA systems, we can use successive interference cancellation (SIC) technology to decode the signals of different users. For the weak user , it has poor channel gains. normally decodes its own signal, but it has no power to remove the signal of from the mixed signals. Thus, suffers from slight extra interference from . Hence, the instantaneous signaltointerferencenoise ratio (SINR) for can be expressed as
(4) 
where .
For the legal far users, we assume that their channel state information (CSI) is known to the RIS. Like [11], the can use the phase shifting to maximize when . Therefore, the maximized can be written as
(5)  
where .
According to the center limit theorem (CLT),
is a Gaussian distributed random variable i.e.,
. Therefore,is a noncentral chisquare random variable with one degree of freedom [11]. From [26], the probability density function (PDF) of
can be written as(6) 
where is the first order modified Bessel function, , and .
On the other hand, the strong user also receives the mixed signal, and it has bigger channel gain than so that it can get more energy. Therefore, can decode the signal of first, and then use the complete mixed signal to reduce the interference signal of . Through this process, it can get a clean signal of its own and then decode it. Therefore, we have
(7) 
For the eavesdropper, we assume that the CSI of E is not known and can not maximize the eavesdropper’s SNR to protect the communication of legitimate users. Similar to [25], we assume that E has the multiuser detection ability, and it can use the parallel interference cancellation (PIC) technology to intercept the different users’ signal. Then, the received SNR at E is
(8)  
where and
follows the exponential distribution with parameter
[27] and its PDF can be written as(9) 
where , and are the average SNRs.
Finally, the secrecy rates of the group for two paired users can be expressed as
(10) 
(11) 
where .
To obtain the best secrecy performance, it is optimal to select the group with the maximum achievable secrecy rate as the intended pairing mechanism. For arbitrary group , when either or is lower than the legal users’ target rate, system outage appears. Therefore, the group selection policy is given by
(12) 
3 Secrecy Performance Analysis
In this section, we present the calculation of the SOP. To get more insights, an asymptotic SOP analysis is also presented.
3.1 SOP analysis
For paired groups, we assume that different groups are allocated with orthogonal bandwidth resources and have independent and identical distributions. With the group selection introduced in (12), the system achieves the best secrecy performance. Then, the system SOP in a multiuser scenario can be evaluated by
(13)  
Therefore, we need first to calculate . For the paired casual group , when either or is less than the legal users’ target rate, this group outage appears. Thus, the SOP of arbitrary group can be calculated as
(14) 
where . From (4), (7), (14), it is very difficult to obtain the exact analysis. Consequently, for tractable analysis, we consider a high SNR case and get the upper bound . Later in numerical results, we can see this upper bound is very tight to the exact simulation results. Then, can be expressed as
(15)  
where is the average SNR.
For notation simplicity, let , , and . Note that must be greater than zero, otherwise, . Since , we can obtain . Finally, can be further expressed as
(16)  
Therefore, with (14)(16), the SOP for the group can be given by
(17) 
Finally, with (13) and (17), the SOP of the whole system can be written as
(18) 
3.2 Asymptotic SOP analysis
The above analytical result is related to , , and
, which can not provide an explicit insight. Thus, we provide an asymptotic analysis. In particular,
and become zero when . Then, the SOP of the group can be asymptotically written as(19)  
With (13) and (19), the system SOP in a multiuser scenario can be asymptotically expressed as
(20) 
Above expression indicates that the asymptotic SOP is only related to and , and it tends to a constant when . Thus, at high SNRs, the secrecy performance is only related to the quality of eavesdropping link and . Interestingly, increasing results in poor secrecy performance, but increasing results in better secrecy performance.
4 Numerical results
In this section, some numerical results are provided to illustrate the secrecy performance of our proposed NOMA system. Meanwhile, MonteCarlo simulation results are provided to verify our analysis. Without loss of generality, we assume that = and = = for all .
In Fig.2, we plot the SOP curves for different when . We can see that our theoretical calculation and simulation are consistent. From Fig.2, it is shown that has a great impact on the system performance. The SOP becomes higher when increases. The reason is that although the RIS does not adjust the phase for the RISE channel to maximize , E still receives copies of the signals from the RIS. Thus, E also enjoys the advantage induced by the RIS. For large , from Eq.(4), we can see that is a constant. Thus, large results in a large and in turn results in a higher SOP since .
In Fig.3, we present a SOP comparison between different NOMA schemes. It is clearly observed that the system performance by using RISs is significantly improved compared to the directlink NOMA system and the relayaided NOMA system. In Fig.4, we plot a SOP comparison between the RISassisted NOMA system and the RISassisted OMA system. At low SNRs, the RISaided NOMA system has a better system performance than the RISaided OMA system. At high SNRs, the interference in NOMA users become dominant, which affects the system performance. However, OMA system has no interference between users, which in turn results in a good performance at high SNRs.
In Fig.5, we plot the SOP curves versus for different average SNRs of the wiretap link and target rates. In Fig.6, we plot the SOP curves for different . From Fig.5 and Fig.6, it is demonstrated that the SOP tends to a constant for large , which verifies our asymptotic analysis in Section 3.2. Also, we can see that large can improve the system performance.
5 Conclusions
In this paper, we analyzed the SOP of RISassisted NOMA systems. Results reveal that SOP tends to a constant at high SNRs. Moreover, increasing the number of intelligent elements has a negative impact on the system secrecy performance since E also takes advantages of the RIS. However, the secrecy performance can be improved by using the group selection.
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