# Secrecy Performance Analysis of a Cognitive Network for IoT overk-μ Channels

Authors: Junxia Li; Hui Zhao; Michael Johnson
Journal: Wireless Communications and Mobile Computing
 (2021)

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## Abstract

With the development of Internet of Things (IoTs), devices are now connecting and communicating together on a heretofore unheard-of scale, forming huge heterogeneous networks of mobile IoT-enabled devices. For beyond 5G- (B5G-) enabled networks, this raises concerns in terms of spectral resource allocation and associated security. Cognitive radio is one effective solution to such a spectrum sharing issue which can be adopted to these B5G networks, which works on the principle of sharing spectrum between primary and secondary users. In this paper, we develop the confidentiality of cognitive radio network (CRNs) for IoT overk-μ fading channels, with the information transmitted between secondary networks with multiple cooperative eavesdroppers, under the constraint of the maximum interference that the primary users can tolerate. All considered facilities use a single-antenna receiver. Of particular interest, the minimum limit values of secure outage probability (SOP) and the probability of strictly positive secrecy capacity (SPSC) are developed for this model in a concise form. Finally, the Monte Carlo simulations for the system are provided to support the theoretical analysis presented.


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## Body

## 1. Introduction

With the recent roll-out of 5G technology globally, an ever-increasing number of intelligent devices are now joining the Internet including mobile IoT devices in social, industrial, healthcare, and smart-grid nature [1]. With this rapid rise in connectivity between such devices, associated security threats and challenges are also on the increase, becoming more pressing, and need to be resolved urgently.Traditional network encryption mechanisms can resolve security problem through various encryption algorithms at the network layer and above. However, such encryption mechanisms can no longer provide perfect security for wireless communication networks due to the complexity and time-consuming nature of the problem. Fortunately, the influence of fading channel and noise actually provides the possibility for implementing physical layer security (PLS), which has been the subject of extensive research in the literature. On the basis of [2], Wyner first proposed a model to estimate the security of communication systems [3]. For the scenarios of active eavesdropping, Ai et al. provided another evaluation benchmark, namely, average secrecy capacity (ASC), over double-Rayleigh fading channels [4]. Referring to the classical Wyner eavesdropping model, SOP was given to study the security of the correlated Rician fading channels [5]. To minimize information leakage, a precoding scheme and the security of Rician fading channels were investigated by analyzing the SOP in [6]. Elsewhere, [7] studied the security capability of large-scale fading channels according to the probability of nonzero secrecy capacity (PNSC) and SOP.The generalized fading channel can model the real transmission environment. By changing its parameters, it can represent many channel models. To account for this, a large section of the literature has studied the transmission performance and security of the generalized fading channels [8–15]. In [8], Lei et al. employed two mathematical forms to complete the derivation of the lower limit of SOP and strictly positive secrecy capacity (SPSC). Elsewhere, the ability of such a channel to resist active eavesdropping was investigated by deriving the ASC in [9]. Using a system model of decode-and-forward (DF) relay cooperation over generalized-K channels, the exact and approximate theoretical expressions of SOP and ESC were evaluated in [10]. Using channels with the premise that the main link follows α-μ distribution and the eavesdropping link was modelled as k-μ distribution, the analytical expressions of ASC, SOP, and SPSC were derived in [11]. Sun et al. described the closed form of SOP and SPSC over other k-μ shadowed fading channels in a concise form [12] and gave an approximate analysis through the method of moment matching. The authors of [13] analyzed the security of Fox’s H-function fading channels by simulating the SOP and probability of nonzero secrecy capacity (PNZ). In real-world wireless communication networks (WCNs), the correlation between antennas cannot be ignored. Based on this, the security performance analysis of correlated systems over η-μ fading channels [14] and k-μ shadowed fading channels [15] has also been investigated.More recently, nonorthogonal multiple access (NOMA) and ambient backscatter communication technology have attracted more and more attention due to the high spectral and energy efficiency for the Internet of Things. In order to investigate the reliability and security of the ambient backscatter NOMA systems considering hardware damage, the outage probability (OP) and the intercept probability (IP) were studied [16]. More practically, the ambient backscatter NOMA system under in-phase and quadrature-phase imbalance (IQI) was taken into account in [17], where the expressions for the OP and the IP are derived in closed exact analytical form [18] and the secure performance for the future beyond 5G (B5G) networks in the presence of nonlinear energy harvesters and imperfect CSI and IQI in terms of the closed form of OP and IP was studied.Most recently, many scholars are interested in CRNs because they can make use of scarce spectrum resources without causing decoding errors to the primary user’s communication. Considering a multirelay network over Nakagami-m fading channels, the authors of [19] studied the effect of three different relay schemes on the security capacity of the channel. In [20], the SOP of the single-input multiple-output (SIMO) underlay CRNs over Rayleigh fading channels with imperfect CSI were derived and analyzed. Park et al. [21] proposed a CRN model composed of a multirelay primary network and a direct link secondary network, where the outage performance of the two networks was analyzed. The secrecy outage performance of DF-based multihop relay CRN under different parameters was investigated in the presence of imperfect CSI in [22]. The authors in [23] studied the energy distribution of CRN by analyzing spectrum sharing. Based on an underlying CRN, the derivation and analysis of SOP and SPSC are described in [24]. Combined with machine learning, a resource allocation protocol for CRN has also been proposed, and the influence of channel parameters on spectrum efficiency is presented in [25]. For conditions where the secondary network cannot interfere with the communication of the primary network, the authors in [26] took PNSC and SOP as the benchmark for studying CRN over Rayleigh fading channels. Recently, security issues are studied for popular applications such as relaying system a direct connection [27] and NOMA system [28].As a generalized channel,k-μ fading can be equated with Rayleigh, one-sided Gaussian, Nakagami-m, and Rician fading channel [29], which can be used to simulate many wireless communication scenarios, so it is of great value to explore transmission performance. Bhargav et al. [30] analyzed the security of the wiretap system over fading channels by deriving the SOP and SPSC of the considered system. The SOP was derived based on classical Wyner’s model over k-μ distribution [31]. The authors in [32] studied the statistical properties of k-μ distribution and obtained the probability density function (PDF) and cumulative distribution function (CDF) for multiple independent k-μ variables. Utilizing DF relay scheme, the SOP and SPSC in a relay system over k-μ channels were provided [33]. As an extension to [32], the authors of analyzed the secrecy outage performance of a SIMO wiretap system over k-μ channels.


### 1.1. Motivation and Contribution

To date, there is negligible work presented in the literature on the CRN security assessment of multiple eavesdroppers overk-μ fading channels. Motivated by the aforementioned discussions, this paper presents such an investigation into the secrecy outage performance of CRN under multiple eavesdroppers by deriving the SOP and SPSC.The main contributions of this paper are summarized as follows:(i)
The work presents a CRN security assessment of multiple eavesdroppers overk-μ fading channels, considering multiple eavesdroppers in the cognitive radio network. It provides theoretical analysis of SOP and studies the influence of channel parameters and other parameters on the secrecy outage performance(ii)
The paper also presents a derivation of SPSC for such a setup, from which it can be seen that SPSC is independent of the primary channel, and this conclusion is confirmed by simulation(iii)
To further evaluate the security for the considered system, the asymptotic analysis of SOP in the high signal-to-noise ratios (SNRs) is derived in this paper. The simulation results indicate that the secrecy diversity order is equal to the main channel parameters and is not influenced by the other parameters


### 1.2. Organization

The rest of this paper is organized as follows. Section2 illustrates the proposed system model. Section 3 presents the premise, including the PDFs and CDFs of the main, primary, and wiretap channel. The SOP is derived in Section 4, the asymptotic SOP is then considered in Section 5. Section 6 introduces the associated evaluation of the SPSC. In Section 7, numerical results are presented based on the Monte Carlo method to verify the theoretical analysis presented in the preceding sections. Finally, Section 8 provides the conclusion for this paper.


### 1.3. Notations

In this paper,Im⋅ is the modified Bessel function with order m and we present Γ⋅ as the Gamma function. γa,z denotes the Pochhammer symbol. Uk is the denotation of the generalized Laguerre polynomial. Lkm⋅ presents upper incomplete Gamma function. Ψa,b;z is the Tricomi confluent hypergeometric function defined in [33] (Equation (9.211.4)). F21a,b;c;z is the Gauss hypergeometric function of variable z with parameters of a, b, and c. In this paper, we represent the probability density function (PDF) in f⋅ and the cumulative distribution function (CDF) in F⋅.


## 1.1. Motivation and Contribution

To date, there is negligible work presented in the literature on the CRN security assessment of multiple eavesdroppers overk-μ fading channels. Motivated by the aforementioned discussions, this paper presents such an investigation into the secrecy outage performance of CRN under multiple eavesdroppers by deriving the SOP and SPSC.The main contributions of this paper are summarized as follows:(i)
The work presents a CRN security assessment of multiple eavesdroppers overk-μ fading channels, considering multiple eavesdroppers in the cognitive radio network. It provides theoretical analysis of SOP and studies the influence of channel parameters and other parameters on the secrecy outage performance(ii)
The paper also presents a derivation of SPSC for such a setup, from which it can be seen that SPSC is independent of the primary channel, and this conclusion is confirmed by simulation(iii)
To further evaluate the security for the considered system, the asymptotic analysis of SOP in the high signal-to-noise ratios (SNRs) is derived in this paper. The simulation results indicate that the secrecy diversity order is equal to the main channel parameters and is not influenced by the other parameters


## 1.2. Organization

The rest of this paper is organized as follows. Section2 illustrates the proposed system model. Section 3 presents the premise, including the PDFs and CDFs of the main, primary, and wiretap channel. The SOP is derived in Section 4, the asymptotic SOP is then considered in Section 5. Section 6 introduces the associated evaluation of the SPSC. In Section 7, numerical results are presented based on the Monte Carlo method to verify the theoretical analysis presented in the preceding sections. Finally, Section 8 provides the conclusion for this paper.


## 1.3. Notations

In this paper,Im⋅ is the modified Bessel function with order m and we present Γ⋅ as the Gamma function. γa,z denotes the Pochhammer symbol. Uk is the denotation of the generalized Laguerre polynomial. Lkm⋅ presents upper incomplete Gamma function. Ψa,b;z is the Tricomi confluent hypergeometric function defined in [33] (Equation (9.211.4)). F21a,b;c;z is the Gauss hypergeometric function of variable z with parameters of a, b, and c. In this paper, we represent the probability density function (PDF) in f⋅ and the cumulative distribution function (CDF) in F⋅.


## 2. System Model

The system model is presented in Figure1. It consists of one primary transmitter (PT), one primary receiver (PR), one secondary transmitter (ST), one secondary receiver (SR), and multiple eavesdroppers Ei,i=1,2,⋯,LE. An effective way to realize spectrum sharing is to use cognitive radio networks (CRN). There are three types of CRN: interweave, underlay, and overlay. The model in this paper adopts the underlying CRN. The system model and analysis method can also be applied to other wireless fading channels.Figure 1
System model.In this paper, we assume that the considered network functions in underlay mode, i.e., the secondary users (SUs), concurrently are entitled to use the resources of the primary network. In underlay mode, communication among the secondary networks of the CRN can be implemented, but it must be carried out under the limitation of guaranteeing the quantity of service (QoS) of the primary network.ST tries to transmit information to SR in the presence of multiple cooperative wiretappers, without reducing the communication quality of the primary network. Hence, the transmitter power Ps¯ of ST is written as
(1)Ps¯=minIphsp2,Pmax,where Ip is the maximum interference power at PR and Pmax represents the peak transmit power of S restricted by designed hardware. It is assumed that there are no direct links between PT and SR and that Eii=1,2,⋯,LE can only eavesdrop on the signal from ST. All links of the considered system are independent, nonidentity, frequency flat, and subject to k-μ fading, with the coefficients of the channel unchanging during a transmission block.Based on these assumptions,hs,v is the channel gain from S to v, v∈d,p,ei,i∈1,2,⋯,LE; the power gains can be denoted as a k-μ random variable with channel parameters kv,μv, assuming that all channel coefficients are integers; finally, LE is the number of wiretappers. Therefore, the received signals are
(2)ysp=hspxs+np,ysd=hsdxs+nd,ysei=hseixs+nei,where np,nd,nei are the additive white Gaussian noise with zero mean value and variance of σ2 on PR, SR, and Ei. From (2), the received instantaneous SNRs are
(3)γp=hsp2Ps¯σ2,γd=hsd2Ps¯σ2,γei=hsei2Ps¯σ2.For convenience, we definex=hsp2,yd=hsd2, and yei=hsei2. Taking into account that multiple eavesdroppers collaborate with maximal ratio combining (MRC) technology, the total received instantaneous SNR at E is
(4)γe=∑i=1LEγei=Ps¯σ2∑i=1LEhsei2.From this, one can get the total wiretapped channel power gain as(5)ye=∑i=1LEhsei2.


## 3. Statistical Characteristics ofk-μ Fading

Since all channels of the considered system experience the independent, nonidentityk-μ fading, from [26] (Equation (10)), the k-μ power probability density function (PDF) of the link from ST to SR or Ei can be expressed as
(6)fuz=μu1+kuμu+1/2kuμu‐1/2eμukuΩuzΩuμu‐1/2e−μu1+ku/Ωuz×Iμu−12μuku1+kuΩuz=1eμukuμu1+kuΩuμue−μu1+ku/Ωuz×∑i=0∞kuμuii!Γμu+iμu1+kuΩuizi+μu−1,where z∈yd,x, u∈p,d, and p, and d denote the subscripts of the channel coefficient from ST to PR or SR, respectively; yd,x is the channel power gain of the link from ST to SR or PR, respectively; Im⋅ is the modified Bessel function; and Γ⋅ is the Gamma function.Utilizing ([34] Equation (8.445)), we substitute λu=1+kuμu/Ωu into (6), and after some algebraic manipulations, (6) can be rewritten as
(7)fuz=λuμu+1/2zμu−1/2e−λuzμukuμu−1/2eμukuIμu−12μukuλuz=λuμueμuku∑i=0∞kuμuλuii!Γμu+izi+μu−1e−λuz.According to the relation between the CDF and PDF, the CDF of the channel gain can now be derived as(8)Fuz=1−1eμuku∑l=0∞kuμull!∑n=0μu+l−1λunn!zne−λuz=1eμuku∑l=0∞kuμull!Γμu+lγμu+l,λuz,where γμu+l,λuz denotes the lower incomplete Gamma function from ([34] Equation (8.350.1)).In the considered system, all eavesdropping links, though independent, are not necessarily identical, and the cooperative eavesdroppers all apply MRC techniques, such that the total channel gain of all of the wiretap links is written asye=∑l=1Lyei, where yei is the channel gain of the link from the transmitter to Ei and L is the number of eavesdroppers. Therefore, the PDF of ye is given by [33] (Equation (3))
(9)fye=e−y/2βyeU−12βUΓU∑k=0∞k!ckUkLkU−1Uye2βξ=e−ye/2β2βU∑k=0∞ck∑q=0k−kqyeq+U−1q!ΓU+qU2βξq,where Uk=ΓU+k/ΓU denotes the Pochhammer symbol [32], the series representation of the generalized Laguerre polynomial LkU−1 ([35] Equation (05.08.02.0001.01)). The efficient ck in fye can be calculated as
(10)c0=UξUexp−12∑i=1LχiaiU−ξβξ+aiU−ξ×∏i=1L1+aiβUξ−1−μei,(11)ck=1k∑j=0k−1cjdk−j,k≥1,(12)dj=−jβU2ξ∑i=1Lχiaiβ−aij−1ξβξ+aiU−ξj+1+∑i=1Lμei1−ai/β1+U/ξ−1ai/βjj≥1,where ai=Ωei/2μei1+kei;χei=2keiμei;U=∑i=0Lμi; L is the number of eavesdroppers; Ωei is the average power gain of the ith wiretap link; and kei,μei are the channel coefficients of the ith wiretap link. The parameters ξ and β must be carefully selected to guarantee the convergence of the series in (9). Specifically, when ξ<U/2 and β>0, (9) will converge in any finite interval; if ξ≥U/2,β must be chosen as β>2−U/ξan/2, to make certain the uniform convergence of (9) in any finite interval, where an=maxaii=1,⋯,L.


## 4. Analysis of Secrecy Outage Probability

According to information security theory, perfect secrecy connection can be guaranteed if the rate of encoding of the confidential data into code words is lower or equal to the secrecy capacity. Otherwise, the security of the information will be compromised. In this section, we focus on analyzing the SOP, which is an important performance metric of describing the security of the considered system; it denotes the maximum achievable rate. The secrecy capacity of CRN is(13)Cs=CD−CEγd>γe,0γd≤γe,where CD=log21+γD and CE=log21+γe are the instantaneous channel capacity of main and wiretap link (s), respectively.For a CRN working in underlay mode, in order to guarantee the quality of the service for the primary network, the transmitter power ofST must be constrained by the maximum interference threshold, Ip, that the primary user can tolerate and its maximum transmitter power, Pmax, as
(14)Ps¯=minIpx,Pmax.From the definition of SOP, it can be denoted as(15)SOP=PrCs≤Cth=PrCs≤Cth,x<IpPmax+PrCs≤Cth,x≥IpPmax=PrCs≤Cth,Ps=Pmax⏟SOP1+PrCs≤Cth,Ps=Ipx⏟SOP2.From expression (15), we can see that the SOP is composed of two components, with reference to SOP1 and SOP2. The remainder of this section will consider these 2 terms in greater detail.


### 4.1. SOP1 Analysis

WhenPs¯=Pmax, the work mode is the same as for a normal communication system. According to probability theory, SOP1 can be calculated as
(16)SOP1=PrCs≤Cth,Ps¯=Pmax=Prγd≤eCthγe+eCth−1⏟I1Prx≤IpPmax⏟I2.Next, we derive an expression forI1, from (16):
(17)I1=Prγd≤eCthγe+eCth−1,where γd=Ps/σ2yd, γe=Ps¯/σ2ye, and Ps¯=Pmax. Rearranging terms and using mathematical methods, we can obtain
(18)I1=Pryd≤Θye+Θ−1α=∫0∝FDΘye+Θ−1αfEyedye,where Θ=eCth and α=Pmax/σ2. Taking account of the fact that I1>∫0∝FDΘyefEyedye, here, we derive the lower bound of I1 as
(19)I1L=∫0∝FDΘyefEyedye.Substituting (8) and (9) into (19) and utilizing [38] (Equation (3.10.1.2)), we derived the expression of I1L as
(20)I1L=2βΘλdμdeμdkd∑k=0∞ck×∑q=0k−kqq!ΓU+qUξq×∑i=0∞2βΘλdkdμdii!Γμd+iΓμd+i+q+Uμd+i×F21μd+i,μd+i+q+U;μd+i+1;−2βΘλd.Applying (8) to this equation, we get
(21)I2=FPIpPmax=1eμpkp∑l=0∞Cl.From this, the lower bound ofSOP1 can be obtained by applying (20) and (21) as
(22)SOP1L=C0∑k=0∞∑l=0∞∑q=0k∑i=0∞ckCiClCqΓμd+i+q+UF21μd+i,μd+i+q+U;μd+i+1;−2βΘλd,where
(23)C0=1eμpkp2βΘλdμdeμdkd,Ci=2βΘλdkdμdii!μd+iΓμd+i,Cl=kpμpll!Γμp+lγμp+l,λpIpPmax,Cq=−kqq!ΓU+qUξq.


### 4.2. SOP2 Analysis

From (15), SOP2 can be expressed as
(24)SOP2=PrCs≤Cth,Ps=Ipx=Pr1+γd1+γe≤eCth,x>IpPmax=Prγd≤eCthγe+eCth−1,x>IpPmax.Letβ2=Ip/σ2, after some mathematical operations similar to those employed for I1, we obtain the following expression for SOP2:
(25)SOP2=Pryd≤Θye+Θ−1β2x,x>IpPmax=∫Ip/Pmax∞HxfPxdx,where Hx=∫0∞FDΘye+Θ−1/β2xfEyedye. Substituting (8) into this equation, after some mathematical derivation, we can obtain Hx as
(26)Hx=1−IHx,where
(27)IHx=1eμdkd∑i=0∞kdμdii!∑n=0μd+i−1λdnn!∫0∞e−λdΘye+Θ−1/β2xΘye+Θ−1β2xnfEyedye.Then, substituting (7) into (25), IHx can be given by
(28)IHx=φUekdμd∑g=0∞kdμdgg!∑n=0μd+g−11−φnn!∑k=0∞ck∑q=0k−kqq!ΓU+qφUξq×∑d=0nCndΘ−1β2dq+U−1+n−d!λd1−φdxde−λdΘ−1/β2x,where φ=1/2βλdΘ+1; substituting (26) into (25), SOP2 becomes
(29)SOP2=1−I2−∫Ip/Pmax∞IHxfPxdx⏟Ip2.From this, we can substitute (28) and (7) into IP2, and with the aid of binomial expansion and [34] (Equation (3.351.3)), IP2 can be shown to be
(30)Ip2=D0∑i=0∞∑g=0∞∑n=0μd+g−1∑k=0∞∑q=0k∑d=0nDiDgDnckDqDdCndq+U−1+n−d!×Γμp+i+d,λdΘ−1β2+λpIpPmax,where
(31)D0=φUψμpekdμd+μpkp,Di=kpμpψii!Γμp+i,Dg=kdμdgg!,Dn=1−φnn!,Dd=1−ψ1−φd,Dq=−kq/q!ΓU+qφU/ξq,ψ=λp/λdΘ−1/β2+λp.Γx,y is the upper incomplete Gamma function ([34] Equation (8.350.2)). Making use of the derivation result, the lower SOP can be obtained as
(32)SOPL=1−I2−Ip2+SOP1L.


## 4.1. SOP1 Analysis

WhenPs¯=Pmax, the work mode is the same as for a normal communication system. According to probability theory, SOP1 can be calculated as
(16)SOP1=PrCs≤Cth,Ps¯=Pmax=Prγd≤eCthγe+eCth−1⏟I1Prx≤IpPmax⏟I2.Next, we derive an expression forI1, from (16):
(17)I1=Prγd≤eCthγe+eCth−1,where γd=Ps/σ2yd, γe=Ps¯/σ2ye, and Ps¯=Pmax. Rearranging terms and using mathematical methods, we can obtain
(18)I1=Pryd≤Θye+Θ−1α=∫0∝FDΘye+Θ−1αfEyedye,where Θ=eCth and α=Pmax/σ2. Taking account of the fact that I1>∫0∝FDΘyefEyedye, here, we derive the lower bound of I1 as
(19)I1L=∫0∝FDΘyefEyedye.Substituting (8) and (9) into (19) and utilizing [38] (Equation (3.10.1.2)), we derived the expression of I1L as
(20)I1L=2βΘλdμdeμdkd∑k=0∞ck×∑q=0k−kqq!ΓU+qUξq×∑i=0∞2βΘλdkdμdii!Γμd+iΓμd+i+q+Uμd+i×F21μd+i,μd+i+q+U;μd+i+1;−2βΘλd.Applying (8) to this equation, we get
(21)I2=FPIpPmax=1eμpkp∑l=0∞Cl.From this, the lower bound ofSOP1 can be obtained by applying (20) and (21) as
(22)SOP1L=C0∑k=0∞∑l=0∞∑q=0k∑i=0∞ckCiClCqΓμd+i+q+UF21μd+i,μd+i+q+U;μd+i+1;−2βΘλd,where
(23)C0=1eμpkp2βΘλdμdeμdkd,Ci=2βΘλdkdμdii!μd+iΓμd+i,Cl=kpμpll!Γμp+lγμp+l,λpIpPmax,Cq=−kqq!ΓU+qUξq.


## 4.2. SOP2 Analysis

From (15), SOP2 can be expressed as
(24)SOP2=PrCs≤Cth,Ps=Ipx=Pr1+γd1+γe≤eCth,x>IpPmax=Prγd≤eCthγe+eCth−1,x>IpPmax.Letβ2=Ip/σ2, after some mathematical operations similar to those employed for I1, we obtain the following expression for SOP2:
(25)SOP2=Pryd≤Θye+Θ−1β2x,x>IpPmax=∫Ip/Pmax∞HxfPxdx,where Hx=∫0∞FDΘye+Θ−1/β2xfEyedye. Substituting (8) into this equation, after some mathematical derivation, we can obtain Hx as
(26)Hx=1−IHx,where
(27)IHx=1eμdkd∑i=0∞kdμdii!∑n=0μd+i−1λdnn!∫0∞e−λdΘye+Θ−1/β2xΘye+Θ−1β2xnfEyedye.Then, substituting (7) into (25), IHx can be given by
(28)IHx=φUekdμd∑g=0∞kdμdgg!∑n=0μd+g−11−φnn!∑k=0∞ck∑q=0k−kqq!ΓU+qφUξq×∑d=0nCndΘ−1β2dq+U−1+n−d!λd1−φdxde−λdΘ−1/β2x,where φ=1/2βλdΘ+1; substituting (26) into (25), SOP2 becomes
(29)SOP2=1−I2−∫Ip/Pmax∞IHxfPxdx⏟Ip2.From this, we can substitute (28) and (7) into IP2, and with the aid of binomial expansion and [34] (Equation (3.351.3)), IP2 can be shown to be
(30)Ip2=D0∑i=0∞∑g=0∞∑n=0μd+g−1∑k=0∞∑q=0k∑d=0nDiDgDnckDqDdCndq+U−1+n−d!×Γμp+i+d,λdΘ−1β2+λpIpPmax,where
(31)D0=φUψμpekdμd+μpkp,Di=kpμpψii!Γμp+i,Dg=kdμdgg!,Dn=1−φnn!,Dd=1−ψ1−φd,Dq=−kq/q!ΓU+qφU/ξq,ψ=λp/λdΘ−1/β2+λp.Γx,y is the upper incomplete Gamma function ([34] Equation (8.350.2)). Making use of the derivation result, the lower SOP can be obtained as
(32)SOPL=1−I2−Ip2+SOP1L.


## 5. Analysis of Secrecy Outage Probability Asymptotic Secure Outage Probability

Although the expressions of SOP can help us perform numerical analysis on the secrecy outage performance of the considered system, asymptotic analysis can also be used to further evaluate the system performance. Therefore, we focus on the derivation of an asymptotic expression of SOP in this section and study the impact of the maximum transmit powerPmax of ST and the maximum interference Ip that PU can tolerate on the secrecy communication with multiple eavesdroppers.In the high-SNR region, the asymptotic SOP can be defined as(33)SOP∞=GaΩd−Gd+οΩd−Gd,where Gd=μd denotes the secrecy diversity order and ο⋅ represents higher order terms. The secrecy array gain is
(34)Ga=11+kdμd12βΘUμdeμpkpΘ−1αU+μd∑l=0∞∑k=0∞∑q=0kA0ClckAq×Θ−1ΘαqΓq+UΨq+U,q+U+μd+1;Θ−12βΘα+2βΘμdμdeμpkp∑k=0∞∑q=0k∑d=0μd∑i=0∞A0ckAqAdAiΓμp+i+d,λpIpPmax−1/μd,where
(35)A0=1eμdkd,Aq=−kqq!ΓU+qUξq,Ad=CμddΘ−1β2d1λp2βΘdμd−d+q+U−1!,Al=kpμpλpll!Γμp+l.


## 6. Probability of Strictly Positive Secrecy Capacity

In information theory, the absolute security of communication can be guaranteed only when the instantaneous secrecy capacity exceeds zero. Thus, SPSC is considered to be an important indicator for measuring the secure communication system, which is given by the formula(36)SPSC=PrCs>0=1−PrCs≤0.SubstitutingCthinto (16), we can get
(37)SOPCth=0=∑k=0∞∑q=0k∑i=0∞B0ckAqAi2βi+q+μdΓμd+i+q+U×F21μd+i,μd+i+q+U;μd+i+1;−2λdβ,where λd=μd1+kd/Ωd;U=∑i=0Lμei;B0=λdμd/eμdkd;Ai=kdμdλdi/i!μd+i!; and Aq and A0 are the same as mentioned above. Then, SPSC can be obtained as
(38)SPSC=1−∑k=0∞∑q=0k∑i=0∞A0ckAqAi2βi+q+μdΓμd+i+q+U×F21μd+i,μd+i+q+U;μd+i+1;−2λdβ.From this expression ofSPSC, we can see that it does not rely on the primary channel gain but is only dependent on the gain of eavesdropping channel and main channel.


## 7. Numerical Results

In this section, the curves obtained by Monte Carlo simulation of SOP for the considered system are compared with the above mathematical analysis in order to consider the impact which the different related parameters have on the security of the cognitive networks. After verification, when the value of the variable reaches 50 times, it converges to a constant value. Infinite series does not affect the simulation results. The parameters utilised in this paper are set toPmax=1 and σ=1, and the other parameter settings are as shown in the relevant figure.Figure2 shows the curves of the SOP for different numbers of eavesdroppers. It can be seen that the analysis results are in agreement with the simulation curves across the entire range of SNRs. In addition, the approximate curve is the tangent line of the exact theoretical results. Moreover, we can also see that the SOP will increase as the number of eavesdroppers increases. This is due to the fact that all of the eavesdroppers cooperate with each other. The more eavesdroppers there are, the stronger the wiretapped signal strength, which means that the eavesdropper SNR increases, followed by an increase in the SOP.Figure 2
SOP with differentLe values, Ωp=2,Ωe=4,Ip=0.1,Cth=0.1;kp,μp=2,2;kd,μd=3,2;ke,μe=2,2.Figures3 and 4 provide the curves of SOP versus Ωd for different values of kd and μd. In order to observe the variation of SOP with kd more clearly, two groups of experiments with different parameters were carried out and the results are shown in Figure 3. It can be seen that SOP decreases with the increase of either kd or μd. Moreover, we can see that with this increase in the value ofkdorμdwhich means the SNR at the receiver increases, the secrecy performance will improve.Figure 3
SOP withkp,μp=2,2,μd=2;kei,μeii=1,2,3=3,25,24,1; red traces Ωp=1,Ωe=6; blue traces Ωp=3,Ωe=5.Figure 4
SOP with differentμd values Ωp=2,Ωe=6;Ip=0.1;Cth=0.01;kp,μp=1,1;kd=3;kei,μeii=1,2,3=2,23,12,1.Figures5 and 6 show the influence of channel parameters kd,μd on SOP. In Figure 6, we plot the different curves of SOP when kd and μd are varied. The blue curve is the difference between kd,μd=1,1 and kd,μd=1,2; the red curve shows the difference of SOP at kd,μd=3,1 and kd,μd=3,2. It is worth noting that the two curves coincide together in the low Ωd region; when Ωd is very high, the difference is very small. When Ωd=12, the difference of SOP is a negative peak; the corresponding positive maximum appears at Ωd=19,20. In other words, at these two points, the channel parameters kd,μd have the greatest impact on the channel security performance.Figure 5
SOP with differentμd and kd values and Ωp=2,Ωe=6;Ip=0.1,Cth=0.01;kp,μp=1,1;kei,μeii=1,2,3=2,23,12,1.Figure 6
The difference of SOP betweenkd,μd=1,1,kd,μd=1,2 and kd,μd=3,1,kd,μd=3,2.From Figure7, it can be seen that the higher the value of Ip, the better the security of the system, since ST can transmit more high power information. We can also see that there is a limitation for the SOP in the high Ip. This is attributed to the fact that the maximum transmit power of STPmax is equal to 1, while Ip⟶∞, as suggested in the above sections.Figure 7
SOP versusIp with Ωd=15,Ωe=1;Cth=0.1;kp,μp=1,1;kd,μd=2,1;kei,μeii=1,2,3=4,15,23,1.Next, without loss of generality, we plot a set of curves with different parameters to observe the effect ofCth on SOP. As we can see from Figure 8, the SOP for lower Cth outperforms the ones for the higher Cth. This is due to the fact that SOP is the probability that secrecy capacity Cs remains below the output threshold Cth. The lower the threshold Cth is, the smaller the corresponding probability obtained. In particular, the red curve of Cth=0.01 is almost identical to the blue curve of Cth=0.1.Figure 8
SOP versusΩd with Ip=0.1;kei,μeii=1,2,3=4,15,23,2; up ones Ωp=1,Ωe=6;kp,μpkd,μd=2,11,2; down ones Ωp=3,Ωe=5;kp,μpkd,μd=2,23,2.Figure9 shows the SPSC for different values of Ωp. From Figure 9, we can see that the SPSC decreases with the increasing values of ke,μe. This can be explained by noting that the larger ke,μe implies a stronger signal obtained by eavesdroppers; hence, the eavesdropper SNR increases which decreases the secrecy capacity and thereby increases the SOP. In addition, we can also observe that the SPSC does not change with the variation of kp,μp as discussed in (38).Figure 9
SPSC withΩp=2,Ωe=4;Ip=0.1;σ=1,Pmax=1;kp,μp=2,2;kd,μd=1,2.To sum up, the interesting conclusion can be obtained that the improvement of the confidentiality is manifested by a larger value of SPSC and a smaller value of SOP. Therefore, a smaller number of eavesdropping antennas, a largerkd, a larger μd, and a smaller Cth can improve the confidentiality of the CRN model.


## 8. Conclusions

In this paper, we have investigated the security performance for CRNs which operate in underlay mode with 5G, beyond 5G, and Internet of Things (IoT) technologies, where all channels experience independent, but not necessarily identically,k-μ fading. The exact and asymptotic theoretical expressions of SOP are derived for the considered system in the presence of multiple eavesdroppers. We also derive an equivalent expression for the SPSC of such a system. These resulting formulae show that the secrecy diversity order relies on the main channel parameter. This is corroborated with simulation results, which also prove this conclusion. Finally, Monte Carlo simulation results are presented to verify these analytical expressions and illustrate the influence which factors have on the secrecy performance versus the SNR ratio (s) of the channel (s).



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*Source: 5548428-2021-06-28.xml*