Physical layer security in wireless sensors networks: secrecy outage probability analysis

ABSTRACT

The secrecy performance of wireless networks powered up by a dedicated power beacon (PB) is addressed in this work. Particularly, the closed-form expression of the secrecy outage probability (SOP) under the presence of multiple eavesdroppers is given. The considered networks are asymmetric since there are multiple eavesdroppers but only a single legitimate receiver. As a consequence, two wiretap schemes are employed, namely, non-colluding and colluding schemes, to take advantage of multiple eavesdroppers. To draw insights from the derived framework, the asymptotic framework of the SOP under the high transmit power regime is also provided. Next, Monte Carlo simulations are provided to validate the theoretical foundations of the proposed approach. Our findings show that by increasing the transmit power from −10 dB to 30 dB, the SOP ameliorates almost a thousand times. Additionally, the SOP is a monotonic decreasing function with a time-switching ratio. We also make comparisons with work in the literature in which the full-duplex relay is employed to help the legitimate link. Numerical results illustrate that the proposed network outperforms those described in the literature. Particularly, through reliance on the setup, the SOP under the proposed system is better than those of the literature by about twofold.

KEYWORDS:

• Colluding eavesdroppers
• energy harvesting
• non-colluding eavesdroppers
• secrecy outage probability

1. Introduction

Smart cities, multimodal virtual reality, and mobile streaming media are just a few examples among numerous applications supported by next-generation communication (5G and beyond) and the Internet of Things (IoT) networks (T. T. H. Nguyen et al., 2023, november). Yet, as IoT devices are typically battery-limited, one of the most popular recognized difficulties is how to autonomously maintain connectivity and the network lifespan. Although the battery can be recharged or replaced, some deployments/specific conditions such as in hazardous environments, underground or isolated areas, and areas of catastrophic devastation make these approaches impractical, expensive, and/or impossible (Al Hajj et al., 2020). Recently, wireless power transfer (WPT) has emerged as a potential technique to extend the lifetime of energy-constrained wireless networks such as wireless sensor networks (WSN) and/or postdisaster emergency communications (Zhao et al., 2019). There are two different types of WPT-based networks. The first is the simultaneous wireless information and power transfer (SWIPT) (Q. S. Nguyen et al., 2023), in which radio frequency (RF) signals are used not only for bearing information but also for charging the energy-harvesting (EH) receiver. This kind of SWIPT-enabled receiver exhibits a symmetric structure since the transceiver comprises the EH circuit and decoding circuit. Additionally, to realize such a technique, three popular protocols are proposed in the literature including time switching (TS), power splitting (PS), and antenna-splitting (if the receiver has at least two antennae) protocols. The second type of this network is called the wireless powered communications network (WPCN) in which the battery of the EH-based devices is solely powered by dedicated power stations such as a power beacon (PB) or a hybrid access point (HAP). Additionally, information security has become a considerable concern in wireless communications due to the nature of broadcasting which introduces vulnerability to many types of wiretap activity. The security issue in IoT networks is even more serious. This is because most IoT networks, such as LoRa, SigFox, and WiFi HaLow, are employed in unlicensed bands. Additionally, as the number of IoT devices connecting to the wireless network increases exponentially year by year, they become more vulnerable to wiretap activities even from the intranetworks. Thus, it is important to examine the secrecy performance in IoT networks. Recently, the topic of physical layer security (PLS) has gained increased attention since it exploits the characteristics of the wireless channel instead of cryptography to secure communications (Duc et al., 2022; Tin et al., 2017). As a result, even if eavesdroppers have advanced computing capabilities, secure communications can still be assured. The performance of the wireless-powered transfer-based networks and PLS is outlined in the next literature review section.

2. State of the art

The performance of two WPT models, namely, PB-based and SWIPT, has been studied widely in the literature. For SWIPT, the authors in Cao et al. (2022) investigated the maximization problem of energy efficiency (EE) of the system under the constraints of quality of service (QoS) and transmission power. Additionally, the performance of the wireless network under both TS and PS protocols was addressed in Do et al. (2020). Moreover, performance evaluations and optimizations of SWIPT-based networks combined with cooperative relaying (Chen et al., 2020), cognitive radio (Tin et al., 2021), cellular networks (Aravanis et al., 2019), device-to-device (D2D) communications (H. N. Nguyen et al., 2018), and distributed antenna systems (Huang et al., 2018) were also thoroughly investigated. The authors in T. N. Nguyen et al. (2016) investigated the performance of random networks with three user selection protocols under the PS protocol. In Tin et al. (2021), the outage probability (OP) of the cooperative cognitive radio network (CCRN) under the PS protocol was studied. Furthermore, Tan and other authors derived the throughput of the SWIPT-enabled IoT networks over the independent and nonidentical distribution (i.n.i.d.) Rayleigh fading (T. N. Nguyen et al., 2021). In Huynh et al. (2017), the authors studied throughput analysis of the best relay selections network with in-phase and Quadrature-phase Imbalances (IQI) under TS protocol. For WPCN, it is suitable for long-distance communications (T. H. Nguyen et al., 2020). Moreover, a PB-based WPCN is suggested as a potential contender for powering up wireless networks and for enabling a huge number of applications requiring high EE (Tin et al., 2020). In T. T. Nguyen, Nguyen et al. (2019), a hybrid access point employed PB to provide data transmission in WPCN, with the channel coefficient between users being symmetrical. The OP and ergodic capacity of the energy harvesting-enabled full-duplex (FD) nonorthogonal multiple access (NOMA) networks were addressed in Toan et al. (2020). To be more precise, the relay was operated in the FD mode instead of in the half-duplex mode as in conventional relaying. Moreover, the impact of the near user on the relay was taken into account. The authors, however, did not consider the secrecy performance of the considered networks. The related works for WPCN focused on the applications to relaying networks (Zhong et al., 2018), cognitive radio networks (Zhai et al., 2018), and cellular networks (Lam & Di Renzo, 2020). More specifically, the authors in Lam and Di Renzo (2020) investigated the EE of the two-tier cellular networks where a tier of the base station (BS) relies only on renewable sources to operate. Both BSs and end users are modelled according to the Poison Point Process (PPP) to capture the randomness of the wireless networks. Their results show that there exists an optimal value of the transmit power for maximizing the multitier cellular networks.

On the other hand, the secrecy performance from the physical layer point of view was addressed in Wang et al. (2017), Le et al. (2023), Wang et al. (2017), Lai et al. (2019), T. N. Nguyen, Tran, Van Chien, Phan, Voznak et al. (2022), Zou et al. (2015), Tu and Bradai (2021), Wang et al. (2015), Park et al. (2013), Oleiwi and Al-Raweshidy (2022), T. T. Nguyen et al. (2022), Tu et al. (2023) and Mitev et al. (2022). More precisely, a link selection strategy for a buffer-aided relaying network was discussed in Wang et al. (2017). The secrecy performance of an ambient backscatter communication (ABC) system was significantly enhanced by employing reconfigurable intelligent surfaces (RIS) (Le et al., 2023). By utilizing cooperative jamming and superposition coding techniques, secrecy outage probability (SOP) and feasible secrecy rate for a vehicle relay network were examined in Wang et al. (2017). A symmetrically distributed secure switch-and-stay combined strategy to protect communications was put forth in Lai et al. (2019). In T. N. Nguyen, Tran, Van Chien, Phan, Voznak et al. (2022), the author investigated the security and reliability of the satellite network with a friendly jammer applied from the satellite to the relay to improve the security of the transmission. The PLS was enhanced by applying a diversity of techniques in Zou et al. (2015). Several methods were proposed to deteriorate the quality of wiretapped signals at the eavesdroppers, including RIS (Tu & Bradai, 2021) and artificial noise (Wang et al., 2015). The authorized receiver selects a relay to weaken eavesdropper signals by transmitting jamming signals (Park et al., 2013). Cooperative jamming and cooperative relaying (Oleiwi & Al-Raweshidy, 2022) have been proven to be effective ways to boost the secrecy capability in cooperative networks. Moreover, the works in T. T. Nguyen et al. (2022) and Tu et al. (2023) examined the PLS of cooperative systems in the nonorthogonal multiple access networks. The authors in Mitev et al. (2022) studied the PLS via the physical unclonable functions, RF fingerprinting/proximity-based authentication, and secret key generation (SKG), while the present work focuses on SOP. Moreover, these authors only considered a single eavesdropper, while we study the scenario where multiple eavesdroppers are present. Additionally, their work was merely suitable for indoor environments, while we include arbitrary environments.

There is limited work that on the combination of PLS and energy harvesting (Ghosh et al., 2023; T. N. Nguyen, Tran, Van Chien, Phan, Voznak, Tin et al., 2022; Viet Tuan et al., 2020). More specifically, the authors in Ghosh et al. (2023) investigated the secrecy performance of an underwater optical communication (UOWC)-radio frequency (RF) network, while we studied the case of fully wireless networks. Additionally, they considered the relay in order to help transform the optical signals into electrical signals, while we do not employ relays to transform the signal form. Furthermore, we apply EH-enabled communications at the source node to improve the EE of the whole network, while they do not consider this technique in their work. On the other hand, Nguyen and other authors in T. N. Nguyen, Tran, Van Chien, Phan, Voznak, Tin et al. (2022) derived the OP and intercept probability (IP), while in the present work, we study the SOP, which is more challenging since it involves many correlated variables from both the legitimate and wiretap links. Moreover, they solely consider a single eavesdropper, while we consider multiple ones. Another divergent point between the two works is that they employed SWIPT, while we consider the PB to power the wireless devices. The secrecy performance of a two-way relaying SWIPT network with a hidden eavesdropper was investigated in Viet Tuan et al. (2020). In particular, they jointly optimized the downlink beamforming vectors, power splitting ratio, and uplink transmission power. However, they considered a single eavesdropper scenario which is much easier than using several eavesdropper scenarios. Additionally, they aimed to minimize the number of transmissions at both the uplink and downlink, while we concentrated on the SOP and its insights when the system is on the high transmit power regime.

Based on these motivations, in this paper, we investigate the performance of EH-based asymmetric wireless networks where a source is charged by harvested energy from a PB in the presence of multiple eavesdroppers. This network topology is asymmetric, as there is a single legitimate user versus many eavesdroppers. Table 1 compares the uniqueness of our research to related articles. This paper's main contributions and novelties are summarized as follows:

• We considered an EH-based wireless network comprising a source, a destination, a PB, and multiple eavesdroppers. More precisely, the operations of the source solely reply to the harvested energy from the PB, thus significantly enhancing the EE of the considered networks. Additionally, we consider two eavesdropper schemes, i.e. colluding and noncolluding schemes.

• We analyzed the security performance of the considered networks. The closed-form expression of the SOP under both schemes is provided. To provide insight into the proposed networks, the asymptotic framework of the SOP is also given.

• Monte Carlo simulations are provided to evaluate the validity of our framework and findings. The impact of some key parameters on the performance of the SOP is also discussed.

Table 1. Comparison of the uniqueness of our research to related articles.

References	PLS	EH	Eavesdropper	Wiretap scheme	SOP analysis	Asymptotic
This work	x	x	Multiple	Non-colluding and colluding scheme	x	x
Ssettumba et al. (2019)	x	None	Multiple	Non-colluding and colluding scheme	x	x
Yan et al. (2020)	x	x	Single	None	x	x
T. N. Nguyen, Tran, Van Chien, Phan, Voznak, Tin et al. (2022)	x	x	Single	None	None	None
Phan et al. (2023)	x	x	Single	None	None	None

Compared with the related works in the literature, our paper's strengths and weaknesses can be briefly summarized in view of the advantages of the considered networks:

• We compute the SOP in the closed-form expression. It should be noted that it is not a trivial task since the SOP involves many correlated variables, i.e. the channel gain from the PB to the source node, in both the legitimate and eavesdropper receivers. Additionally, many works in the literature have only examined the IP, which is far easier compared to computing the SOP. The main rationale behind this statement is that under the IP, one only considers the random variables relevant to the eavesdropper links, while in the SOP, both the main and wiretap links are taken into consideration via a rational function.

• We take into account two kinds of combinations of multiple eavesdroppers cases.

• We apply the energy-harvesting technique to improve the EE of the whole network. However, regarding the expense of such techniques compared with conventional communications (nonenergy harvesting), under EH-enabled communications, the transmit power is no longer a constant number but a random variable. As a result, it is extremely difficult to derive the mathematical framework to derive the closed-form expression.

• We provide an asymptotic framework of the SOP under the high transmit power regime which provides several insights into the considered networks. For example, the SOP under the colluding scheme scales with the number of eavesdroppers.

• We provide simulation results based on the Monte Carlo approach to consolidate the accuracy of the derived framework as well as to highlight the impact of some vital parameters on the performance metric.

• We compare our work with other work in the literature to highlight the advantages of the considered networks.

A few drawbacks of the considered networks are as follows:

• We do not apply the reconfigurable intelligent surfaces technique to actively control the channel coefficient so that we can both improve the channel gain of the legitimated link and suppress the eavesdropper link (provided that the global channel state information (CSI) is available as the source node), thus significantly enhancing the system performance. However, estimating the channel coefficient in RIS-assisted systems is a nontrivial task and still in its infancy since the RIS generally comprises passive devices, and thus we leave this technique to future work.

• Another minor weakness of the proposed networks is the application of tools from stochastic geometry (SG) to better capture the position of all nodes. However, its mathematical framework can not generally be obtained in the closed-form expression and is thus not able to reveal certain engineering insights. We therefore also leave this issue to our future work.

A concise summary of the contributions and novelties of the present work compared to existing literature is provided in Table 1. The remainder of this paper is organized as follows: Section 3 presents the system model, Section 4 outlines the derivation of the secrecy performance analysis, Section 5 provides the numerical results, and Section 6 concludes the paper.

3. System model

In this paper, we consider a wirelessly powered network that includes a source (denoted by S), a destination (denoted by D), and N eavesdroppers denoted by $E_n,1⩽n⩽N$ as shown in Figure 1. Moreover, we also have a PB (denoted by P) which is replenished by the battery of the source node S. We assume that all transmission links are modelled by a complex Gaussian random variable (RV) with zero mean and Ω variance, $\mathcal{CN}(0,\mathrm{\Omega })$. Particularly, $h_{\mathrm{PS}}\sim \mathcal{CN}(0,{\mathrm{\Omega }}_{\mathrm{PS}})$ is the channel coefficient from P to S, $h_{\mathrm{SD}}\sim \mathcal{CN}(0,{\mathrm{\Omega }}_{\mathrm{SD}})$ is the channel coefficient from S to D, and $h_{S{E}_n}\sim \mathcal{CN}(0,{\mathrm{\Omega }}_{S{E}_n})$ is the channel coefficient from S to $E_n$. ${\mathrm{\Omega }}_{\mathrm{PS}},{\mathrm{\Omega }}_{\mathrm{SD}},\mathrm{and}{\mathrm{\Omega }}_{S{E}_n}$ are the variance of the corresponding channels. As a consequence, the channel gain from $P\to S$, $S\to D$, and $S\to E_n,\mathrm{\forall n}$ denoted as $|h_{\mathrm{PS}}{|}^2,|h_{\mathrm{SD}}{|}^2$ and $|h_{S{E}_n}{|}^2,\mathrm{\forall n}$ are followed by an exponential RV. Additionally, we assume that all nodes are only equipped with a single antenna. The extension of multiple antennae is a promising extension of the current work. Furthermore, we also consider the scenario where the fading changes independently between each transmission and remains constant during each transmission. Additionally, the flowchart in Figure 2 illustrates the sequence describing the proposed system.

Figure 1. The considered EH-based wireless networks.

Figure 2. The flowchart describes the sequence of the proposed system.

The whole transmission occurs in two phases. In the first phase, the source S harvests energy from the PB signals, and in the second phase, the source node spends all the harvested energy at the first phase to send data to the destination by virtue of the harvest-then-use protocol being employed at the source node.

3.1. Energy harvesting

In the EH phase, P transmits energy signals to S for a period of time αT where $\alpha \in (0,1)$ is the time-switching ratio and T is the time duration for the whole procedure. Without loss of generality, we assume that T = 1 second. The harvested energy at S is given as follows Tin et al. (2020): (1) $E_S=\mathrm{\alpha T\eta }{P}_P{\left|h_{\mathrm{PS}}\right|}^2,$(1) where $\eta \in (0,1)$ denotes the energy conversion efficiency, and $P_P$ is the transmit power of P. As a result, the transmit power of S at the second phase is computed as (2) $P_S=\frac{E_S}{\left(1-\alpha \right)T}=\frac{\mathrm{\alpha \eta P}{\left|h_{\mathrm{PS}}\right|}^2}{\left(1-\alpha \right)}=\chi P_P{\left|h_{\mathrm{PS}}\right|}^2,$(2) where $\chi =\frac{\mathrm{\alpha \eta }}{\left(1-\alpha \right)}$. We observe that if $\alpha =\frac{1}{2}$, we have a symmetrical setup where half the duration is for harvesting energy and the other is for decoding information.

3.2. Data transmission

In the data transmission phase, S transmits unit power signals x to D, i.e. $\mathbb{E}\left\{\right|x{|}^2\}=1$ where $\mathbb{E}\{.\}$ is the expectation operator. Therefore, the received signals at D are given as follows: (3) $y_D=\sqrt{P_S}{h}_{\mathrm{SD}}x+n_D.$(3) Here, $n_D\sim \mathcal{CN}(0,N_0)$ is the additive white Gaussian noise (AWGN) of the destination and $N_0$ is the noise variance. The signal-to-noise ratio (SNR) at D is then given as follows: (4) ${\gamma }_D=\frac{P_S{\left|h_{\mathrm{SD}}\right|}^2}{N_0}.$(4) With the help of (2(2) $P_S=\frac{E_S}{\left(1-\alpha \right)T}=\frac{\mathrm{\alpha \eta P}{\left|h_{\mathrm{PS}}\right|}^2}{\left(1-\alpha \right)}=\chi P_P{\left|h_{\mathrm{PS}}\right|}^2,$(2) ), the SNR at D is rewritten as (5) ${\gamma }_D=\frac{P_P\chi {\left|h_{\mathrm{SD}}\right|}^2{\left|h_{\mathrm{PS}}\right|}^2}{N_0}=\mathrm{\rho \chi }{\left|h_{\mathrm{SD}}\right|}^2{\left|h_{\mathrm{PS}}\right|}^2,$(5) where $\rho =\frac{P_P}{N_0}$. Moreover, the received signals at ${\mathrm{E}}_n$, $n\in \{1,\dots ,N\}$ are given as (6) $y_{E_n}=\sqrt{P_S}{h}_{S{E}_n}x+n_{E_n},$(6) where $n_{E_n}\sim \mathcal{CN}(0,N_0)$ is the AWGN at $E_n,\mathrm{\forall n}$. In this paper, we consider two wiretap schemes: (1) the non-colluding scheme where each eavesdropper decodes messages independently and (2) the colluding scheme where all eavesdroppers' signals are processed by a single central node. Under the non-colluding scheme, the eavesdropper who experiences the best channel gain is selected to evaluate the secrecy performance because the systems will be secured if the best eavesdropper is not able to wiretap the secure information (Wang et al., 2017). As a result, the SNR under the non-colluding scheme denoted by ${\gamma }_E^{\mathrm{non}}$ is formulated as (7) $\begin{array}{rl}{\gamma }_E^{\mathrm{non}}& =\frac{P_S\underset{1⩽n⩽N}{max}\left[{\left|h_{S{E}_n}\right|}^2\right]}{N_0}\\ & =\frac{P_P\chi Z_{\mathrm{SE}}{\left|h_{\mathrm{PS}}\right|}^2}{N_E}={\rho }_E\chi Z_{\mathrm{SE}}{\left|h_{\mathrm{PS}}\right|}^2,\end{array}$(7) where ${\rho }_E=\frac{P_P}{N_E}$ and $Z_{\mathrm{SE}}={max}_{1⩽n⩽N}\left[|h_{S{E}_n}{|}^2\right]$. On the other hand, under the colluding scheme, a central node combines all the eavesdropper's signals to maximize the wiretap probability (Tin et al., 2017). Thus, the SNR under the colluding scheme denoted by ${\gamma }_E^{\mathrm{con}}$ and is computed as (8) $\begin{array}{rl}{\gamma }_E^{\mathrm{con}}& =\frac{P_S\sum _{n=1}^N\left[{\left|h_{S{E}_n}\right|}^2\right]}{N_0}\\ & =\frac{P_P\chi {\tilde{Z}}_{\mathrm{SE}}{\left|h_{\mathrm{PS}}\right|}^2}{{\sigma }_E^2}={\rho }_E\chi {\tilde{Z}}_{\mathrm{SE}}{\left|h_{\mathrm{PS}}\right|}^2,\end{array}$(8) where ${\tilde{Z}}_{\mathrm{SE}}=\sum _{n=1}^N\left[|h_{S{E}_n}{|}^2\right]$. The instantaneous capacity of the main and eavesdropper links under the $v\in \left\{\mathrm{non},\mathrm{con}\right\}$ scheme is given by (9) $C_D={\log }_2\left[1+{\gamma }_D\right],$(9) and (10) $C_{E^v}={\log }_2\left[1+{\gamma }_{E^v}\right].$(10) With the instantaneous capacity of both links, we can derive the secrecy performance of the considered systems, as presented in the next section.

4. Secrecy performance analysis

In this section, we address the performance of the SOP under both the non-colluding and colluding schemes. The SOP is the probability that the instantaneous rate of the main link divided by the instantaneous rate of the eavesdropper link is smaller than a given threshold. For deriving the SOP, two Lemmas 4.1 and 4.2 can be used.

Lemma 4.1

Consider a set of independent and identically distributed (i.i.d.) exponential RVs $X_n$, $n\in \{1,\dots ,N\}$ with parameter Ω. The cumulative distribution function (CDF) and probability density function (PDF) of the maximum of these RVs, $Z={max}_{n\in \left\{1,\dots ,N\right\}}\left\{X_n\right\}$, are then given as (11) $\begin{array}{rl}{F}_Z\left(x\right)& =1-\sum _{a=1}^N\left(\begin{array}{c}N\\ a\end{array}\right){\left(-1\right)}^{a-1}{e}^{-\frac{\mathrm{ax}}{\mathrm{\Omega }}},\end{array}$(11) (12) $\begin{array}{rl}{f}_Z\left(x\right)& =\sum _{a=1}^N\left(\begin{array}{c}N\\ a\end{array}\right)\frac{a{\left(-1\right)}^{a-1}}{\mathrm{\Omega }}{e}^{-\frac{\mathrm{ax}}{\mathrm{\Omega }}}.\end{array}$(12)

Proof.

The definition of the CDF of the maximal RV $Z={max}_{n\in \left\{1,\dots ,N\right\}}\left\{X_n\right\}$ is as follows: (13) $\begin{array}{rl}{F}_Z\left(x\right)& =Pr\left\{\underset{n\in \left\{1,\dots ,N\right\}}{max}\left\{X_n\right\}<x\right\}\stackrel{\left(a\right)}{=}\prod _{n=1}^N{F}_{X_n}\left(x\right)\\ & \stackrel{\left(b\right)}{=}{\left(1-\exp \left(-\frac{x}{\mathrm{\Omega }}\right)\right)}^N=1-\sum _{a=1}^N\left(\begin{array}{c}N\\ a\end{array}\right){\left(-1\right)}^{a-1}{e}^{-\frac{\mathrm{ax}}{\mathrm{\Omega }}}.\end{array}$(13) Here, $\left(a\right)$ and $\left(b\right)$ are obtained via the i.i.d. property of these RVs combining with the CDF of an exponential RV with parameter Ω, and the last equation is obtained by applying the binomial theorem. To derive the PDF, we simply take the first-order derivative of the CDF with respect to x. Mathematical speaking, we have the following (14) $f_Z\left(x\right)=\frac{d{F}_Z\left(x\right)}{\mathrm{d}x}=\sum _{a=1}^N\left(\begin{array}{c}N\\ a\end{array}\right)\frac{a{\left(-1\right)}^{a-1}}{\mathrm{\Omega }}{e}^{-\frac{\mathrm{ax}}{\mathrm{\Omega }}}.$(14) This completes the proof.

Lemma 4.2

Consider a set of i.i.d. exponential RVs $X_n$, $n\in \{1,\dots ,N\}$ with parameter Ω. The cumulative distribution function and probability density function of the sum of these RVs, $Z=\sum _{n=1}^N{X}_n$, are given as follows: (15) $F_{\tilde{Z}}\left(x\right)=\frac{1}{\mathrm{\Gamma }\left(N\right)}\gamma \left(N,\frac{x}{\mathrm{\Omega }}\right),$(15) (16) $f_{\tilde{Z}}\left(x\right)=\frac{x^{N-1}}{{\mathrm{\Omega }}^N\mathrm{\Gamma }\left(N\right)}{e}^{-\frac{x}{\mathrm{\Omega }}},$(16) where $\mathrm{\Gamma }(.)$ and $\gamma (a,b)$ are the Gamma function and the lower incomplete Gamma function, respectively (Gradshteyn & Ryzhik, 2014).

Proof.

The proof is immediately obtained by employing the results in T. N. Nguyen, Nguyen et al. (2022).

In the next section, we present the derivation of the security aspect of the considered networks via the SOP metric with the help of the above Lemmas.

4.1. Secrecy outage probability analysis

The SOP under both schemes, i.e. colluding and non-colluding schemes, are computed in this section.¹ We begin with the case of the non-colluding eavesdroppers.

4.1.1. Non-colluding scheme

Under the non-colluding scheme, each eavesdropper will individually attempt to wiretap the secure information from S to D. As a result, the system will be called in an outage event if the instantaneous rate of the legitimate link is lower than the best eavesdropper link. Mathematically speaking, the SOP under the non-colluding scheme is formulated and computed as follows: (17) $\begin{array}{rl}{\mathrm{SOP}}^{\mathrm{non}}& =Pr\left({\left[C_D-C_{E^{\mathrm{non}}}\right]}^{+}<R_{\mathrm{th}}\right)\\ & =Pr\left({\log }_2\left[\frac{1+{\gamma }_D}{1+{\gamma }_E^{\mathrm{non}}}\right]<R_{\mathrm{th}}\right)\\ & =1-\sum _{a=1}^N\left(\begin{array}{c}N\\ a\end{array}\right)\frac{2\mathrm{a\rho }{\mathrm{\Omega }}_{\mathrm{SD}}{\left(-1\right)}^{a-1}}{{\gamma }_{\mathrm{th}}{\rho }_E{\mathrm{\Omega }}_{\mathrm{SE}}+\mathrm{a\rho }{\mathrm{\Omega }}_{\mathrm{SD}}}\\ & \times \sqrt{\frac{{\overline{\gamma }}_{\mathrm{th}}}{\mathrm{\rho \chi }{\mathrm{\Omega }}_{\mathrm{PS}}{\mathrm{\Omega }}_{\mathrm{SD}}}}{K}_1\left(2\sqrt{\frac{{\overline{\gamma }}_{\mathrm{th}}}{\mathrm{\rho \chi }{\mathrm{\Omega }}_{\mathrm{PS}}{\mathrm{\Omega }}_{\mathrm{SD}}}}\right),\end{array}$(17) where $K_v(\bullet )$ is the modified Bessel function of the second kind with vth order, $[x{]}^{+}=max(0,x)$, and $R_{\mathrm{th}}$ (in bits/s/Hz) is the target secrecy rate.

Proof.

The proof begins with the definition of the SOP as follows Lysiak (2023): (18) $\begin{array}{rl}{\mathrm{SOP}}^{\mathrm{non}}& =Pr\left({\log }_2\left[\frac{1+{\gamma }_D}{1+{\gamma }_E^{\mathrm{non}}}\right]<R_{\mathrm{th}}\right)\\ & =Pr\left(\frac{1+{\gamma }_D}{1+{\gamma }_E^{\mathrm{non}}}<{\gamma }_{\mathrm{th}}\right)\\ & =Pr\left(\frac{1+\mathrm{\rho \chi }{\left|h_{\mathrm{SD}}\right|}^2{\left|h_{\mathrm{PS}}\right|}^2}{1+{\rho }_E\chi Z_{\mathrm{SE}}{\left|h_{\mathrm{PS}}\right|}^2}<{\gamma }_{\mathrm{th}}\right)\end{array}$(18) (19) $=1-Pr\left({\left|h_{\mathrm{PS}}\right|}^2>\frac{{\overline{\gamma }}_{\mathrm{th}}}{\mathrm{\rho \chi }{\left|h_{\mathrm{SD}}\right|}^2-{\gamma }_{\mathrm{th}}{\rho }_E\chi Z_{\mathrm{SE}}}\right)$(19) (20) $\begin{array}{rl}& =1-{\int }_0^{\mathrm{\infty }}{f}_{Z_{\mathrm{SE}}}\left(x\right){\int }_{\frac{{\gamma }_{\mathrm{th}}{\rho }_{E}x}{\rho }}^{\mathrm{\infty }}{f}_{{\left|h_{\mathrm{SD}}\right|}^2}\left(y\right)\\ & \times \left[1-F_{{\left|h_{\mathrm{PS}}\right|}^2}\left(\frac{{\overline{\gamma }}_{\mathrm{th}}}{\mathrm{\rho \chi y}-{\gamma }_{\mathrm{th}}{\rho }_E\mathrm{\chi x}}\right)\right]\mathrm{d}y\mathrm{d}x,\end{array}$(20) where ${\gamma }_{\mathrm{th}}=2^{R_{\mathrm{th}}}$, ${\overline{\gamma }}_{\mathrm{th}}={\gamma }_{\mathrm{th}}-1$. With the help of Lemma 4.1 and the definition of the PDF of an exponential RV, the ${\mathrm{SOP}}^{\mathrm{non}}$ is expressed as follows: (21) $\begin{array}{rl}\mathrm{SO}{\mathrm{P}}^{\mathrm{non}}=& 1-\sum _{a=1}^N\left(c\mathrm{Na}\right){\displaystyle \frac{a{\left(-1\right)}^{a-1}}{{\mathrm{\Omega }}_{\mathrm{SE}}}}\\ & \times {\int }_0^{\mathrm{\infty }}{e}^{-{\displaystyle \frac{\mathrm{ax}}{{\mathrm{\Omega }}_{\mathrm{SE}}}}}{\int }_{{\displaystyle \frac{{\gamma }_{\mathrm{th}}{\rho }_{E}x}{\rho }}}^{\mathrm{\infty }}{\displaystyle \frac{1}{{\mathrm{\Omega }}_{\mathrm{SD}}}{e}^{-{\displaystyle \frac{y}{{\mathrm{\Omega }}_{\mathrm{SD}}}}}}\\ & \times e^{-{\displaystyle \frac{{\overline{\gamma }}_{\mathrm{th}}}{\mathrm{\rho \chi y}{\mathrm{\Omega }}_{\mathrm{PS}}-{\gamma }_{\mathrm{th}}{\rho }_E\mathrm{\chi x}{\mathrm{\Omega }}_{\mathrm{PS}}}}}\mathrm{d}y\mathrm{d}x.\end{array}$(21) To compute (21(21) $\begin{array}{rl}\mathrm{SO}{\mathrm{P}}^{\mathrm{non}}=& 1-\sum _{a=1}^N\left(c\mathrm{Na}\right){\displaystyle \frac{a{\left(-1\right)}^{a-1}}{{\mathrm{\Omega }}_{\mathrm{SE}}}}\\ & \times {\int }_0^{\mathrm{\infty }}{e}^{-{\displaystyle \frac{\mathrm{ax}}{{\mathrm{\Omega }}_{\mathrm{SE}}}}}{\int }_{{\displaystyle \frac{{\gamma }_{\mathrm{th}}{\rho }_{E}x}{\rho }}}^{\mathrm{\infty }}{\displaystyle \frac{1}{{\mathrm{\Omega }}_{\mathrm{SD}}}{e}^{-{\displaystyle \frac{y}{{\mathrm{\Omega }}_{\mathrm{SD}}}}}}\\ & \times e^{-{\displaystyle \frac{{\overline{\gamma }}_{\mathrm{th}}}{\mathrm{\rho \chi y}{\mathrm{\Omega }}_{\mathrm{PS}}-{\gamma }_{\mathrm{th}}{\rho }_E\mathrm{\chi x}{\mathrm{\Omega }}_{\mathrm{PS}}}}}\mathrm{d}y\mathrm{d}x.\end{array}$(21) ), we change the variable as follows: $t=\mathrm{\rho \chi }{\mathrm{\Omega }}_{\mathrm{PS}}y-{\gamma }_{\mathrm{th}}{\rho }_E\chi {\mathrm{\Omega }}_{\mathrm{PS}}\Rightarrow y=\frac{t+{\gamma }_{\mathrm{th}}{\rho }_E\chi {\mathrm{\Omega }}_{\mathrm{PS}}x}{\mathrm{\rho \chi }{\mathrm{\Omega }}_{\mathrm{PS}}}\Rightarrow \mathrm{dt}=\mathrm{\rho \chi }{\mathrm{\Omega }}_{\mathrm{PS}}\mathrm{dy}$. Then, (21(21) $\begin{array}{rl}\mathrm{SO}{\mathrm{P}}^{\mathrm{non}}=& 1-\sum _{a=1}^N\left(c\mathrm{Na}\right){\displaystyle \frac{a{\left(-1\right)}^{a-1}}{{\mathrm{\Omega }}_{\mathrm{SE}}}}\\ & \times {\int }_0^{\mathrm{\infty }}{e}^{-{\displaystyle \frac{\mathrm{ax}}{{\mathrm{\Omega }}_{\mathrm{SE}}}}}{\int }_{{\displaystyle \frac{{\gamma }_{\mathrm{th}}{\rho }_{E}x}{\rho }}}^{\mathrm{\infty }}{\displaystyle \frac{1}{{\mathrm{\Omega }}_{\mathrm{SD}}}{e}^{-{\displaystyle \frac{y}{{\mathrm{\Omega }}_{\mathrm{SD}}}}}}\\ & \times e^{-{\displaystyle \frac{{\overline{\gamma }}_{\mathrm{th}}}{\mathrm{\rho \chi y}{\mathrm{\Omega }}_{\mathrm{PS}}-{\gamma }_{\mathrm{th}}{\rho }_E\mathrm{\chi x}{\mathrm{\Omega }}_{\mathrm{PS}}}}}\mathrm{d}y\mathrm{d}x.\end{array}$(21) ) can be rewritten as (22(22) $\begin{array}{rl}\mathrm{SO}{\mathrm{P}}^{\mathrm{non}}& =1-{\displaystyle \frac{1}{\mathrm{\rho \chi }{\mathrm{\Omega }}_{\mathrm{PS}}{\mathrm{\Omega }}_{\mathrm{SD}}}\sum _{a=1}^N\left(\mathrm{cN}a\right){\displaystyle \frac{a{\left(-1\right)}^{a-1}}{{\mathrm{\Omega }}_{\mathrm{SE}}}{\int }_0^{\mathrm{\infty }}{e}^{-{\displaystyle \frac{{\gamma }_{\mathrm{th}}{\rho }_{E}x}{\rho {\mathrm{\Omega }}_{\mathrm{SD}}}}}{\int }_0^{\mathrm{\infty }}{e}^{-{\displaystyle \frac{t}{\mathrm{\rho \chi }{\mathrm{\Omega }}_{\mathrm{PS}}{\mathrm{\Omega }}_{\mathrm{SD}}}}}{e}^{-{\displaystyle \frac{{\overline{\gamma }}_{\mathrm{th}}}{t}}}\mathrm{d}t\mathrm{d}x}}\\ & \stackrel{\left(a\right)}{=}1-\sum _{a=1}^N\left(\mathrm{cN}a\right){\displaystyle \frac{a{\left(-1\right)}^{a-1}}{{\mathrm{\Omega }}_{\mathrm{SE}}}2\sqrt{{\displaystyle \frac{{\overline{\gamma }}_{\mathrm{th}}}{\mathrm{\rho \chi }{\mathrm{\Omega }}_{\mathrm{PS}}{\mathrm{\Omega }}_{\mathrm{SD}}}}}{K}_1\left(2\sqrt{{\displaystyle \frac{{\overline{\gamma }}_{\mathrm{th}}}{\mathrm{\rho \chi }{\mathrm{\Omega }}_{\mathrm{PS}}{\mathrm{\Omega }}_{\mathrm{SD}}}}}\right){\int }_0^{\mathrm{\infty }}{e}^{-{\displaystyle \frac{\mathrm{ax}}{{\mathrm{\Omega }}_{\mathrm{SE}}}}}{e}^{-{\displaystyle \frac{{\gamma }_{\mathrm{th}}{\rho }_{E}x}{\rho {\mathrm{\Omega }}_{\mathrm{SD}}}}}\mathrm{d}x}\\ & =1-\sum _{a=1}^N\left(\mathrm{cN}a\right){\displaystyle \frac{2\mathrm{a\rho }{\mathrm{\Omega }}_{\mathrm{SD}}{\left(-1\right)}^{a-1}}{{\gamma }_{\mathrm{th}}{\rho }_E{\mathrm{\Omega }}_{\mathrm{SE}}+\mathrm{a\rho }{\mathrm{\Omega }}_{\mathrm{SD}}}\sqrt{{\displaystyle \frac{{\overline{\gamma }}_{\mathrm{th}}}{\mathrm{\rho \chi }{\mathrm{\Omega }}_{\mathrm{PS}}{\mathrm{\Omega }}_{\mathrm{SD}}}}}{K}_1\left(2\sqrt{{\displaystyle \frac{{\overline{\gamma }}_{\mathrm{th}}}{\mathrm{\rho \chi }{\mathrm{\Omega }}_{\mathrm{PS}}{\mathrm{\Omega }}_{\mathrm{SD}}}}}\right),}\end{array}$(22) ). (22) $\begin{array}{rl}\mathrm{SO}{\mathrm{P}}^{\mathrm{non}}& =1-{\displaystyle \frac{1}{\mathrm{\rho \chi }{\mathrm{\Omega }}_{\mathrm{PS}}{\mathrm{\Omega }}_{\mathrm{SD}}}\sum _{a=1}^N\left(\mathrm{cN}a\right){\displaystyle \frac{a{\left(-1\right)}^{a-1}}{{\mathrm{\Omega }}_{\mathrm{SE}}}{\int }_0^{\mathrm{\infty }}{e}^{-{\displaystyle \frac{{\gamma }_{\mathrm{th}}{\rho }_{E}x}{\rho {\mathrm{\Omega }}_{\mathrm{SD}}}}}{\int }_0^{\mathrm{\infty }}{e}^{-{\displaystyle \frac{t}{\mathrm{\rho \chi }{\mathrm{\Omega }}_{\mathrm{PS}}{\mathrm{\Omega }}_{\mathrm{SD}}}}}{e}^{-{\displaystyle \frac{{\overline{\gamma }}_{\mathrm{th}}}{t}}}\mathrm{d}t\mathrm{d}x}}\\ & \stackrel{\left(a\right)}{=}1-\sum _{a=1}^N\left(\mathrm{cN}a\right){\displaystyle \frac{a{\left(-1\right)}^{a-1}}{{\mathrm{\Omega }}_{\mathrm{SE}}}2\sqrt{{\displaystyle \frac{{\overline{\gamma }}_{\mathrm{th}}}{\mathrm{\rho \chi }{\mathrm{\Omega }}_{\mathrm{PS}}{\mathrm{\Omega }}_{\mathrm{SD}}}}}{K}_1\left(2\sqrt{{\displaystyle \frac{{\overline{\gamma }}_{\mathrm{th}}}{\mathrm{\rho \chi }{\mathrm{\Omega }}_{\mathrm{PS}}{\mathrm{\Omega }}_{\mathrm{SD}}}}}\right){\int }_0^{\mathrm{\infty }}{e}^{-{\displaystyle \frac{\mathrm{ax}}{{\mathrm{\Omega }}_{\mathrm{SE}}}}}{e}^{-{\displaystyle \frac{{\gamma }_{\mathrm{th}}{\rho }_{E}x}{\rho {\mathrm{\Omega }}_{\mathrm{SD}}}}}\mathrm{d}x}\\ & =1-\sum _{a=1}^N\left(\mathrm{cN}a\right){\displaystyle \frac{2\mathrm{a\rho }{\mathrm{\Omega }}_{\mathrm{SD}}{\left(-1\right)}^{a-1}}{{\gamma }_{\mathrm{th}}{\rho }_E{\mathrm{\Omega }}_{\mathrm{SE}}+\mathrm{a\rho }{\mathrm{\Omega }}_{\mathrm{SD}}}\sqrt{{\displaystyle \frac{{\overline{\gamma }}_{\mathrm{th}}}{\mathrm{\rho \chi }{\mathrm{\Omega }}_{\mathrm{PS}}{\mathrm{\Omega }}_{\mathrm{SD}}}}}{K}_1\left(2\sqrt{{\displaystyle \frac{{\overline{\gamma }}_{\mathrm{th}}}{\mathrm{\rho \chi }{\mathrm{\Omega }}_{\mathrm{PS}}{\mathrm{\Omega }}_{\mathrm{SD}}}}}\right),}\end{array}$(22) Where (a) is obtained by borrowing the results from (Gradshteyn & Ryzhik, 2014, 3.324.1), the last equation is derived by computing the integration, thus concluding the proof.

4.1.2. Colluding scheme

Under the colluding scheme, there exists a central eavesdropper that collects information from all eavesdroppers and then decodes the data sent by S to D. Mathematical speaking, the SOP under the colluding scheme is computed as follows: (23) $\begin{array}{rl}{\mathrm{SOP}}^{\mathrm{con}}& =1-2{\left(\frac{\rho {\mathrm{\Omega }}_{\mathrm{SD}}}{\rho {\mathrm{\Omega }}_{\mathrm{SD}}+{\gamma }_{\mathrm{th}}{\rho }_E{\mathrm{\Omega }}_{\mathrm{SE}}}\right)}^N\\ & \times \sqrt{\frac{{\overline{\gamma }}_{\mathrm{th}}}{\mathrm{\rho \chi }{\mathrm{\Omega }}_{\mathrm{PS}}{\mathrm{\Omega }}_{\mathrm{SD}}}}{K}_1\left(2\sqrt{\frac{{\overline{\gamma }}_{\mathrm{th}}}{\mathrm{\rho \chi }{\mathrm{\Omega }}_{\mathrm{PS}}{\mathrm{\Omega }}_{\mathrm{SD}}}}\right).\end{array}$(23)

Proof.

Let us start the derivation by rewriting the definition under the colluding scheme as follows Guerraiche et al. (2023): (24) $\begin{array}{rl}{\mathrm{SOP}}^{\mathrm{con}}& =Pr\left({\log }_2\left[\frac{1+{\gamma }_D}{1+{\gamma }_E^{\mathrm{con}}}\right]<R_{\mathrm{th}}\right)\\ & =Pr\left(\frac{1+{\gamma }_D}{1+{\gamma }_E^{\mathrm{con}}}<{\gamma }_{\mathrm{th}}\right)\\ & =Pr\left(\frac{1+\mathrm{\rho \chi }{\left|h_{\mathrm{SD}}\right|}^2{\left|h_{\mathrm{PS}}\right|}^2}{1+{\rho }_E\chi {\tilde{Z}}_{\mathrm{SE}}{\left|h_{\mathrm{PS}}\right|}^2}<{\gamma }_{\mathrm{th}}\right)\\ & =1-Pr\left({\left|h_{\mathrm{PS}}\right|}^2>\frac{{\overline{\gamma }}_{\mathrm{th}}}{\mathrm{\rho \chi }{\left|h_{\mathrm{SD}}\right|}^2-{\gamma }_{\mathrm{th}}{\rho }_E\chi {\tilde{Z}}_{\mathrm{SE}}}\right)\\ & =1-{\int }_0^{\mathrm{\infty }}{f}_{{\tilde{Z}}_{\mathrm{SE}}}\left(x\right){\int }_{\frac{{\gamma }_{\mathrm{th}}{\rho }_{E}x}{\rho }}^{\mathrm{\infty }}{f}_{{\left|h_{\mathrm{SD}}\right|}^2}\left(y\right)\\ & \times \left[1-F_{{\left|h_{\mathrm{PS}}\right|}^2}\left(\frac{{\overline{\gamma }}_{\mathrm{th}}}{\mathrm{\rho \chi y}-{\gamma }_{\mathrm{th}}{\rho }_E\mathrm{\chi x}}\right)\right]\mathrm{d}y\mathrm{d}x,\end{array}$(24) With the help of Lemma 4.2, (24(24) $\begin{array}{rl}{\mathrm{SOP}}^{\mathrm{con}}& =Pr\left({\log }_2\left[\frac{1+{\gamma }_D}{1+{\gamma }_E^{\mathrm{con}}}\right]<R_{\mathrm{th}}\right)\\ & =Pr\left(\frac{1+{\gamma }_D}{1+{\gamma }_E^{\mathrm{con}}}<{\gamma }_{\mathrm{th}}\right)\\ & =Pr\left(\frac{1+\mathrm{\rho \chi }{\left|h_{\mathrm{SD}}\right|}^2{\left|h_{\mathrm{PS}}\right|}^2}{1+{\rho }_E\chi {\tilde{Z}}_{\mathrm{SE}}{\left|h_{\mathrm{PS}}\right|}^2}<{\gamma }_{\mathrm{th}}\right)\\ & =1-Pr\left({\left|h_{\mathrm{PS}}\right|}^2>\frac{{\overline{\gamma }}_{\mathrm{th}}}{\mathrm{\rho \chi }{\left|h_{\mathrm{SD}}\right|}^2-{\gamma }_{\mathrm{th}}{\rho }_E\chi {\tilde{Z}}_{\mathrm{SE}}}\right)\\ & =1-{\int }_0^{\mathrm{\infty }}{f}_{{\tilde{Z}}_{\mathrm{SE}}}\left(x\right){\int }_{\frac{{\gamma }_{\mathrm{th}}{\rho }_{E}x}{\rho }}^{\mathrm{\infty }}{f}_{{\left|h_{\mathrm{SD}}\right|}^2}\left(y\right)\\ & \times \left[1-F_{{\left|h_{\mathrm{PS}}\right|}^2}\left(\frac{{\overline{\gamma }}_{\mathrm{th}}}{\mathrm{\rho \chi y}-{\gamma }_{\mathrm{th}}{\rho }_E\mathrm{\chi x}}\right)\right]\mathrm{d}y\mathrm{d}x,\end{array}$(24) ) is then computed as follows: (25) $\begin{array}{rl}\mathrm{SO}{P}^{\mathrm{con}}& =1-\frac{1}{{{\mathrm{\Omega }}_{\mathrm{SE}}}^N\mathrm{\Gamma }\left(N\right)}{\int }_0^{\mathrm{\infty }}{x}^{N-1}{e}^{-\frac{x}{{\mathrm{\Omega }}_{\mathrm{SE}}}}\\ & \times {\int }_{\frac{{\gamma }_{\mathrm{th}}{\rho }_{E}x}{\rho }}^{\mathrm{\infty }}\frac{1}{{\mathrm{\Omega }}_{\mathrm{SD}}}{e}^{-\frac{y}{{\mathrm{\Omega }}_{\mathrm{SD}}}}{e}^{-\frac{{\overline{\gamma }}_{\mathrm{th}}}{\mathrm{\rho \chi y}{\mathrm{\Omega }}_{\mathrm{PS}}-{\gamma }_{\mathrm{th}}{\rho }_E\mathrm{\chi x}{\mathrm{\Omega }}_{\mathrm{PS}}}}\mathrm{d}y\mathrm{d}x\\ & =1-\frac{2}{{\left[{\mathrm{\Omega }}_{\mathrm{SE}}\right]}^N\mathrm{\Gamma }\left(N\right)}\sqrt{\frac{{\overline{\gamma }}_{\mathrm{th}}}{\mathrm{\rho \chi }{\mathrm{\Omega }}_{\mathrm{PS}}{\mathrm{\Omega }}_{\mathrm{SD}}}}{K}_1\left(2\sqrt{\frac{{\overline{\gamma }}_{\mathrm{th}}}{\mathrm{\rho \chi }{\mathrm{\Omega }}_{\mathrm{PS}}{\mathrm{\Omega }}_{\mathrm{SD}}}}\right)\\ & \times {\int }_0^{\mathrm{\infty }}{x}^{N-1}{e}^{-\frac{\rho {\mathrm{\Omega }}_{\mathrm{SD}}+{\gamma }_{\mathrm{th}}{\rho }_E{\mathrm{\Omega }}_{\mathrm{SE}}}{\rho {\mathrm{\Omega }}_{\mathrm{SD}}{\mathrm{\Omega }}_{\mathrm{SE}}}x}\mathrm{d}x\\ & =1-2{\left(\frac{\rho {\mathrm{\Omega }}_{\mathrm{SD}}}{\rho {\mathrm{\Omega }}_{\mathrm{SD}}+{\gamma }_{\mathrm{th}}{\rho }_E{\mathrm{\Omega }}_{\mathrm{SE}}}\right)}^N\\ & \times \sqrt{\frac{{\overline{\gamma }}_{\mathrm{th}}}{\mathrm{\rho \chi }{\mathrm{\Omega }}_{\mathrm{PS}}{\mathrm{\Omega }}_{\mathrm{SD}}}}{K}_1\left(2\sqrt{\frac{{\overline{\gamma }}_{\mathrm{th}}}{\mathrm{\rho \chi }{\mathrm{\Omega }}_{\mathrm{PS}}{\mathrm{\Omega }}_{\mathrm{SD}}}}\right).\end{array}$(25) The last equation is held by applying (Gradshteyn & Ryzhik, 2014, Equation (3.351.3)), thus concluding the proof.

The complexity of the considered networks primarily depends on the evaluation of the modified Bessel function of the second kind with 1st order, i.e. $K_1\left(x\right)$. Fortunately, this function is readily available in a variety of numerical computation software packages such as Matlab, Mathematica, and Python Numpy. In these environments, the computation time typically falls below 50 milliseconds (Takekawa, 2022).

4.2. Secrecy outage probability asymptotic framework

In this section, we provide insights into the considered networks when the transmit power of the PB goes without bound, i.e. $\rho \to \mathrm{\infty }$. In particular, we derive the asymptotic framework of SOP under both schemes, i.e. the non-colluding and colluding schemes, given as follows Pandey and Yadav (2018): (26) $\begin{array}{rl}\mathrm{SO}{P}^{\mathrm{non},\mathrm{\infty }}\approx & 1-\sum _{a=1}^N\left(c\mathrm{Na}\right){\displaystyle \frac{\mathrm{a\rho }{\mathrm{\Omega }}_{\mathrm{SD}}{\left(-1\right)}^{a-1}}{\mathrm{a\rho }{\mathrm{\Omega }}_{\mathrm{SD}}+{\gamma }_{\mathrm{th}}{\rho }_E{\mathrm{\Omega }}_{\mathrm{SE}}},}\\ \approx 1-{\left({\displaystyle \frac{\rho {\mathrm{\Omega }}_{\mathrm{SD}}}{\rho {\mathrm{\Omega }}_{\mathrm{SD}}+{\gamma }_{\mathrm{th}}{\rho }_E{\mathrm{\Omega }}_{\mathrm{SE}}}}\right)}^N,\end{array}$(26)

Proof.

Let us start the proof with the non-colluding scheme as follows: (27) $\begin{array}{rl}\mathrm{SO}{P}^{\mathrm{non},\mathrm{\infty }}& \approx Pr\left({\displaystyle \frac{{\gamma }_D}{{\gamma }_E^{\mathrm{non}}}<{\gamma }_{\mathrm{th}}}\right)\\ & =1-Pr\left({\left|h_{\mathrm{SD}}\right|}^2>{\displaystyle \frac{{\gamma }_{\mathrm{th}}{\rho }_E{Z}_{\mathrm{SE}}}{\rho }}\right)=1-{\int }_0^{\mathrm{\infty }}{f}_{Z_{\mathrm{SE}}}\left(x\right)\left[1-F_{{\left|h_{\mathrm{SD}}\right|}^2}\left({\displaystyle \frac{{\gamma }_{\mathrm{th}}{\rho }_{E}x}{\rho }}\right)\right]\mathrm{d}x\\ & =1-\sum _{a=1}^N\left(\mathrm{cN}a\right){\displaystyle \frac{a{\left(-1\right)}^{a-1}}{{\mathrm{\Omega }}_{\mathrm{SE}}}{\int }_0^{\mathrm{\infty }}{e}^{-{\displaystyle \frac{\mathrm{ax}}{{\mathrm{\Omega }}_{\mathrm{SE}}}-{\displaystyle \frac{{\gamma }_{\mathrm{th}}{\rho }_{E}x}{{\mathrm{\Omega }}_{\mathrm{SD}}\rho }}}}\mathrm{d}x=1-\sum _{a=1}^N\left(\mathrm{cN}a\right){\displaystyle \frac{\mathrm{a\rho }{\mathrm{\Omega }}_{\mathrm{SD}}{\left(-1\right)}^{a-1}}{\mathrm{a\rho }{\mathrm{\Omega }}_{\mathrm{SD}}+{\gamma }_{\mathrm{th}}{\rho }_E{\mathrm{\Omega }}_{\mathrm{SE}}}.}}\end{array}$(27) Next, by following the same steps as those in the non-colluding scheme, the closed-form expression under the colluding scheme is computed as follows: (28) $\mathrm{SO}{P}^{\mathrm{con},\mathrm{\infty }}=1-{\left(\frac{\rho {\mathrm{\Omega }}_{\mathrm{SD}}}{\rho {\mathrm{\Omega }}_{\mathrm{SD}}+{\gamma }_{\mathrm{th}}{\rho }_E{\mathrm{\Omega }}_{\mathrm{SE}}}\right)}^N.$(28) The proof is thus completed.

5. Numerical results

In this section, simulation results based on the Monte Carlo method (N. T. Nguyen et al., 2018; T. N. Nguyen, Minh et al., 2019; T. N. Nguyen, Tran et al., 2019; Phu et al., 2020) are provided to verify the accuracy of the derived mathematical frameworks as well as to identify the insights of the SOP with respect to some key parameters. Without loss of generality, the following parameters are employed through this section: the target secrecy rate $R_{\mathrm{th}}=0.1$, $\alpha =0.2$, $\eta =0.8$, ${\mathrm{\Omega }}_{\mathrm{PS}}={\mathrm{\Omega }}_{\mathrm{SD}}=1$, ${\mathrm{\Omega }}_{\mathrm{SE}}=0.1$, and ${\rho }_E=10$ [dB]. The short-hand Sim. and Ana. are used to denote the results from simulations and mathematical framework.

In Figure 3, SOP is plotted versus ρ [dB] with two schemes, non-colluding and colluding, respectively. The exact analytical curves are generated from the expression derived in (17(17) $\begin{array}{rl}{\mathrm{SOP}}^{\mathrm{non}}& =Pr\left({\left[C_D-C_{E^{\mathrm{non}}}\right]}^{+}<R_{\mathrm{th}}\right)\\ & =Pr\left({\log }_2\left[\frac{1+{\gamma }_D}{1+{\gamma }_E^{\mathrm{non}}}\right]<R_{\mathrm{th}}\right)\\ & =1-\sum _{a=1}^N\left(\begin{array}{c}N\\ a\end{array}\right)\frac{2\mathrm{a\rho }{\mathrm{\Omega }}_{\mathrm{SD}}{\left(-1\right)}^{a-1}}{{\gamma }_{\mathrm{th}}{\rho }_E{\mathrm{\Omega }}_{\mathrm{SE}}+\mathrm{a\rho }{\mathrm{\Omega }}_{\mathrm{SD}}}\\ & \times \sqrt{\frac{{\overline{\gamma }}_{\mathrm{th}}}{\mathrm{\rho \chi }{\mathrm{\Omega }}_{\mathrm{PS}}{\mathrm{\Omega }}_{\mathrm{SD}}}}{K}_1\left(2\sqrt{\frac{{\overline{\gamma }}_{\mathrm{th}}}{\mathrm{\rho \chi }{\mathrm{\Omega }}_{\mathrm{PS}}{\mathrm{\Omega }}_{\mathrm{SD}}}}\right),\end{array}$(17) ) and (23(23) $\begin{array}{rl}{\mathrm{SOP}}^{\mathrm{con}}& =1-2{\left(\frac{\rho {\mathrm{\Omega }}_{\mathrm{SD}}}{\rho {\mathrm{\Omega }}_{\mathrm{SD}}+{\gamma }_{\mathrm{th}}{\rho }_E{\mathrm{\Omega }}_{\mathrm{SE}}}\right)}^N\\ & \times \sqrt{\frac{{\overline{\gamma }}_{\mathrm{th}}}{\mathrm{\rho \chi }{\mathrm{\Omega }}_{\mathrm{PS}}{\mathrm{\Omega }}_{\mathrm{SD}}}}{K}_1\left(2\sqrt{\frac{{\overline{\gamma }}_{\mathrm{th}}}{\mathrm{\rho \chi }{\mathrm{\Omega }}_{\mathrm{PS}}{\mathrm{\Omega }}_{\mathrm{SD}}}}\right).\end{array}$(23) ), and the asymptotic curves are plotted from (26(26) $\begin{array}{rl}\mathrm{SO}{P}^{\mathrm{non},\mathrm{\infty }}\approx & 1-\sum _{a=1}^N\left(c\mathrm{Na}\right){\displaystyle \frac{\mathrm{a\rho }{\mathrm{\Omega }}_{\mathrm{SD}}{\left(-1\right)}^{a-1}}{\mathrm{a\rho }{\mathrm{\Omega }}_{\mathrm{SD}}+{\gamma }_{\mathrm{th}}{\rho }_E{\mathrm{\Omega }}_{\mathrm{SE}}},}\\ \approx 1-{\left({\displaystyle \frac{\rho {\mathrm{\Omega }}_{\mathrm{SD}}}{\rho {\mathrm{\Omega }}_{\mathrm{SD}}+{\gamma }_{\mathrm{th}}{\rho }_E{\mathrm{\Omega }}_{\mathrm{SE}}}}\right)}^N,\end{array}$(26) ). As shown in the figure, the gap between the exact and asymptotic framework is negligible when $\rho \gg 1$ acts as a lower bound of the exact expressions. Accordingly, it is possible to closely monitor the impact of network parameters under high SNR regimes via easy-to-understand approximations. Furthermore, we notice that the larger the ρ is, the smaller the SOP. Additionally, the SOP of the non-colluding scheme is better than that of the other.

Figure 3. The SOP versus ρ [dB].

Figure 4 shows the SOP versus ρ [dB] with different ${\rho }_E$ of the eavesdroppers. We can straightforwardly observe that increasing ${\rho }_E$ will lead to the increase of the SOP, thus reducing the security of the considered systems.

Figure 4. The SOP of the proposed system versus ρ [dB] with different ${\rho }_E$ [dB].

Figure 5 illustrates the performance of the SOP with respect to α with different values of ρ [dB]. We see that the SOP is a monotonic decrease function of α. Moreover, when α approaches 1, we achieve the best performance of the SOP.

Figure 5. The SOP of the proposed system versus α with different ρ[dB].

Figure 6 illustrates the performance of the SOP concerning ${\rho }_E$ with various values of ρ. It is evident that the larger ${\rho }_E$ is, the worse the SOP becomes. This is due to the fact that the received SNR of the eavesdropper is proportional to ${\rho }_E$, hence, the higher the SNR at the eavesdropper, the higher the probability that the eavesdropper successfully wiretaps the secure information. Fortunately, this issue can be effectively mitigated by increasing ρ. We observe that the larger ρ is, the lower the SOP, and thus, the better the system performance.

Figure 6. The SOP versus ${\rho }_E$ with different ρ.

Figure 7 illustrates the impact of energy efficiency conversion, denoted as η, on SOP performance. It is evident that increasing η benefits the SOP. However, this improvement necessitates greater expense and complexity at the receiver. Thus, there exists a trade-off between system performance and receiver complexity in practice. Nonetheless, this intriguing issue remains open for future investigation. Additionally, this figure also reveals that a longer time interval for energy harvesting leads to better SOP performance. Further insights into the impact of α on SOP performance are provided in Figure 5.

Figure 7. The SOP versus η with different α.

The performance of the SOP regarding the ratio of the channel gain of the legitimate and wiretap links, $\frac{{\mathrm{\Omega }}_{\mathrm{SD}}}{{\mathrm{\Omega }}_{\mathrm{SE}}}$, are supplied in Figure 8. We observe that if the channel gain of the main link is favorable compared to the eavesdropper links, the SOP is significantly improved; otherwise, the SOP is almost 1, and the system is not able to transmit secure information. Nonetheless, the performance can be ameliorated by simply adding some artificial noise at the eavesdropper as proposed in Wang et al. (2015).

Figure 8. The SOP versus $\frac{{\mathrm{\Omega }}_{\mathrm{SD}}}{{\mathrm{\Omega }}_{\mathrm{SE}}}$.

Comparison with the state-of-the-art

Figure 9 addresses the performance of the SOP with respect to the targeted rate $R_{\mathrm{th}}$. It is evident that increasing $R_{\mathrm{th}}$ will simply increase SOP, thus degrading the system performance. In this figure, we also compare the performance of the proposed systems with the ones in Tin et al. (2020). Particularly, curves from Tin et al. (2020) are denoted by ‘Simu-Non-Lit.’ and ‘Simu-Con-Lit.’ for the non-colluding and colluding schemes. We observe that there is a large gap between the proposed networks and the ones in Tin et al. (2020). More precisely, when $R_{\mathrm{th}}$ is equal to 0.4, the SOP under the non-colluding scheme of the proposed networks is just above 0.4, while for the work in Tin et al. (2020), the SOP is around 0.8, which is two-fold higher than that of the proposed system. Additionally, we confirm again that the SOP under the colluding scheme is worse than its counterpart for both systems.

Figure 9. The SOP versus $R_{\mathrm{th}}$ [bits/s/Hz]. The curves ‘Simu-Non-Lit.’ and ‘Simu-Con-Lit.’ are plotted by employing Monte Carlo simulations of the works in Tin et al. (2020) with the following setup parameters: ${\eta }_R$ = η = 0.8, ${\mathrm{\Omega }}_{\mathrm{SR}}={\mathrm{\Omega }}_{\mathrm{RD}}=2$, ${\mathrm{\Omega }}_{\mathrm{RR}}=0.1$, and ${\mathrm{\Omega }}_{\mathrm{RE}}=0.3$. The power splitting ratio is 0.5.

The impact of the number of eavesdroppers N on the performance of the SOP is given in Figure 10 under the proposed networks and Tin et al. (2020). It is certain that scaling up N will have a negative effect on the system performance regardless of the networks and schemes. However, the increasing pace of the SOP is divergent between the considered networks and Tin et al. (2020). Specifically, we see that when N goes from 2 to 8 eavesdroppers, the SOP under the colluding scheme of the considered networks only goes up from less than 0.05 to under 0.1, while in Tin et al. (2020), the SOP rises from above 0.15 to above 0.25. Additionally, the impact of N on the non-colluding scheme is almost constant for both proposed systems and Tin et al. (2020).

Figure 10. The SOP versus N. The curves ‘Simu-Non-Lit.’ and ‘Simu-Con-Lit.’ are plotted by employing Monte Carlo simulations of the works in Tin et al. (2020) with the following setup parameters: ${\eta }_R$ = η = 0.8, ${\mathrm{\Omega }}_{\mathrm{SR}}={\mathrm{\Omega }}_{\mathrm{RD}}=2$, ${\mathrm{\Omega }}_{\mathrm{RR}}=0.1$, and ${\mathrm{\Omega }}_{\mathrm{RE}}=0.3$. The power splitting ratio is 0.5.

6. Conclusions

In this paper, we discussed an EH-based wireless network composed of a source node, which is operated by the harvested energy from the power beacon and sends secure information to a destination in the presence of multiple eavesdroppers. Particularly, we considered two wiretap schemes, i.e. the colluding and non-colluding schemes. In this context, we derived the secrecy outage probability at the receiver for both schemes. Numerical results were also provided to demonstrate the accuracy of our studies. Moreover, we performed a comparison with the state-of-the-art approaches, and the numerical results showed that the proposed networks outperformed those described in the literature.

The findings of this research can inspire intriguing follow-up works in several directions. One of the feasible directions is to employ relaying networks where the relay not only helps to forward information to the main destination but also to create a jamming signal to all eavesdroppers so that the secrecy performance can be enhanced dramatically. Beside the help from relays, it is also promising to use RIS to both strengthen the main link and suppress the wiretap link provided that the global channel state information is available at the source node. Another well-known approach is to deploy multiple antennae at the source and/or destination so that a diversity of transmitting and receiving techniques can be used to boost the secrecy performance as well.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research is funded by Ton Duc Thang University under grant number FOSTECT.2023.31.

Notes

1 The adopted methodology can be readily applied to other significant metrics such as average secrecy rate and intercept probability. Nonetheless, we defer the exploration of these interesting problems to future work, as the present manuscript is focused on comprehensively investigating the secrecy outage probability.