Cooperative Jamming-Assisted Untrusted Relaying Based on Game Theory for Next-Generation Communication Systems

Abstract

In this contribution, we investigate the performance of an untrusted relaying system when Cooperative Jammers (CJs) are available. We propose two scenarios, Untrusted Relaying-aided Multiple Cooperative-Jammers-based Simultaneous Transmission (UR-MCJST) and an Untrusted Relaying-aided Multiple Cooperative Jammers-based Time-Division Transmission (UR-MCJTDT). The performances of both UR-MCJST and UR-MCJTDT schemes are investigated. The source node is the primary user (PU) that has access to a transmission bandwidth. An untrusted relay is employed for improving the reliability of the PU transmission, while CJs are invoked for improving the secrecy rate of the PU transmission. The transmission strategy for the scheme proposed in this paper is split into three unique phases. The first phase is called the broadcast-and-jamming phase, while the second phase is called the relaying phase, and the final phase is called CJ’s secondary phase, which is used for transmitting the secondary user message. More explicitly, CJs transmit noise for impairing the untrusted relay node’s ability to decode the message during the broadcast-and-jamming phase. In compensation for their help, CJs are allocated part of the available spectrum by the PU for their transmission during the secondary transmission phase. A Stackelberg leader–follower game was considered between the PU and the CJs, while a power control game that is based on the well-known Nash equilibrium is employed by the CJs. Furthermore, an investigation into the effect of invoking simultaneous transmission and time-division-based transmission scenarios during the CJs’ secondary transmission phase was carried out. Finally, we evaluated the achievable secrecy rate of the PU and the utility rate of the CJs in our proposed scheme.

1. Introduction

Security and reliability are crucial for any wireless communication system, especially in the case of ultra-dense 5G and beyond networks. The wireless medium has a broadcast nature, which makes it a major challenge in ensuring security while maintaining reliability in the presence of unauthorized users [1]. Shannon was the first to provide the foundations for secrecy in communication between legitimate parties over a noiseless channel [2], while Wyner presented the discrete memory-less wiretap channel, ensuring secrecy and reliability among the legitimate parties in the presence of an eavesdropper over the noisy communication medium [3]. It was found in [4] that both reliability and confidentiality can be ensured for the broadcast channel with the aid of two receivers.

To properly scale, adapt, and deliver quantum-safe security solutions for 6G connectivity, the security of ultra-dense networks is crucial. Hence, there arises the need for a bottom-up strategy using all security planes over the general communication stack. In this regard, physical-layer security (PLS) is a leading contender [5]. PLS has attracted a lot of attention because it is capable of using the channel’s impairment, including noise, fading, interference, synchronization and location, to provide security against an eavesdropper without data encryption at the upper layers [1,6,7,8]. More specifically, PLS takes the advantage of the random nature of noise and fading channel for minimizing the leaking of information to an eavesdropper. Explicitly employing the Cooperative/Friendly Jammer (CJ) has been known to be one of the most important PLS methods, where legitimate nodes communicate secretly with the help of an external jammer that transmits a jamming signal to perturb the eavesdropper [9,10]. Recently, unmanned aerial vehicles have also been investigated as CJs for improving PLS [11,12,13].

Cooperative communications, as mentioned in the cited work [14,15], present an appealing approach to accomplish reliable and efficient transmission. The wireless transmission’s broadcast nature enables relay terminals to receive signals without incurring additional costs. Moreover, cooperative communications generally take advantage of decreased path loss, resulting in enhanced power efficiency, as emphasized in the studies [16,17].

The introduction of 6G presents a complex dilemma. Since the majority of connecting devices are anticipated to utilize the existing spectrum, the number of networking devices is predicted to increase at an accelerated rate, raising concerns about spectrum scarcity. This issue calls into question 6G’s viability as an upcoming networking paradigm. Cognitive radio networks (CRNs) are seen as the most promising solution for resolving spectrum shortages in 6G networks [18,19]. The licensed/primary users (PUs) and unlicensed/secondary users (SUs) of the network can agree on interference limits that allow a variety of devices from the primary and secondary networks to share the same frequency band. More specifically, CRN allows the SUs to use its spectrum based on certain conditions, which can be occupied by the PUs, or their slot might be empty, hence accommodating more users [20,21,22]. Dynamic Spectrum Access (DSA) [20] technology, which divides the currently unoccupied spectrum between the licensed PUs and unlicensed SUs in a dynamically changing radio environment, is the foundation of the CRN concept. Based on how the SUs acquire and use the primary spectrum, CRN transmissions can be divided into three main branches: interweave, underlay, and overlay DSA methods [23,24]. In both the interweave and underlay schemes, the PU has no knowledge of the secondary transmission because the SU transmits in a spectrum hole in the interweave DSA mechanism, while the SU transmits its information within the interference threshold of the primary transmitter in an underlay scheme. Both interweave and underlay will introduce slight degradation to the primary transmission. While cognition is employed to reduce interference at the destination of both the PU’s and SU’s destination in the CRN, the overlay transmission model focuses on the interference reduction or cancellation method. The SU should have access to the PU’s codebooks in order to cancel out the interference [23].

A two-stage Stackelberg game was proposed in [25] to analyze the participation level of the mobile users and the optimal incentive mechanism of the CSP using backward induction. Similarly, ref. [26] proposed a two-stage Stackelberg game-based computational offloading mechanism in which the mobile edge nodes (leaders) set the appropriate price for their computational resources and the industrial IoT devices (followers) formulate their utility functions by considering the social interaction among each other. The work proposed a dynamic iterative algorithm for Nash equilibrium. In [27], the authors exploited the concept of a social tie in CJs, where the source and jammers are smart phones carried by humans who have diverse social relationships. The jammers would be more willing to become the CJs for the source–destination pair that has a closer tie with them.

Literature Review Comparison

Most of the works consider the relaying terminal as a trusted node, while the eavesdropper is considered as an outsider that is not part of the communication infrastructure [28,29,30]. Furthermore, in cooperative cognitive radio networks [31], the spectrum is shared among the licensed and unlicensed users, where the relay terminals can be part of the unlicensed users. In such a scenario, the relay node may not be trustworthy and could be a potential eavesdropper [32,33,34]

The concept of untrusted relays was initially explored in [35], where the destination played the role of a jammer during the broadcast phase to ensure secrecy for the source and destination pair. Osorio et al. [36] investigated the secrecy rate performance of an amplify-and-forward (AF) relaying network with an untrusted relay, employing destination-based jamming to enable secure transmissions. Ku et al. [37] considered a two-way relaying scheme involving a large-scale multiple-antenna base station (BS), a single-antenna mobile user (MU), and a multiple-antenna external jammer, analyzing the ergodic secrecy sum rate, which was found to improve with an increasing number of untrusted relays.

To enhance the secrecy rate performance of systems using an untrusted relay and a multiple-antenna destination, Zhao et al. [38] proposed a novel full-duplex destination jamming scheme with optimal antenna selection. The analytical curves presented in [38] were verified to align well with the simulation results. Jiang et al. [39] considered an untrusted relay-based network where the source transmitted a confidential message to both the relay and the destination, while the destination transmitted a jamming signal to the relay. The Non-orthogonal AF (NAF) and Orthogonal AF (OAF) protocols were examined in [39], with the source-destination (S-D) link being considered in both the source-broadcasting and relaying phases in the NAF scenario, whereas in the OAF scenario, the S-D link was only involved in the source-broadcasting phase.

The topic of secrecy transmission with multiple untrusted relays was explored in [40]. Specifically, the study focused on the use of cooperative jammers to safeguard the transmitted confidential signal against eavesdropping. These cooperative jammers employed techniques such as transmitting noise [9,41] or sending codewords [42,43] to counteract eavesdroppers. Most of the existing works in this area have investigated jamming methods involving the destination or utilizing altruistic nodes [44,45].

Game theoretic secrecy maximization techniques have been investigated in [34,46,47]. Jamming through non-altruistic nodes was considered in [48], where the communication is secured through these nodes, and in return, they are considered as the SUs in the network. Specifically, the jammers would be given access to the spectrum of the PUs for their secondary transmissions. Furthermore, a Stackelberg-based game was presented in [46], for the enhancement of the PU’s secrecy rate. In the study by Al et al. in [49], a cognitive radio network (CRN) was examined, consisting of a primary transmission involving a PU transmitter (PT) and PU receiver (PR), as well as a secondary transmission involving an SU transmitter (ST) and SU receiver (SR). The ST in [49] was considered as a trusted relay for the PT, and the cooperation between the PT and ST was utilized to establish physical layer security using a multilevel Stackelberg game. Throughout the network, all transmissions were susceptible to potential eavesdropping attacks.

The study demonstrated that employing spectrum leasing, where secondary access is traded for cooperation, is a promising framework for enhancing the secrecy rate in CRNs. This finding was highlighted in [49].

In Zhang et al. [47], a two-way untrusted relay-based network was studied, incorporating a Stackelberg game between the sources and friendly jammers. The objective was to investigate the cooperation dynamics and strategies between the sources and jammers to optimize the network performance.

In H et al. [50], the focus was on a cooperation scenario between primary users (PUs) and untrusted secondary users (SUs) in cognitive radio networks. The SUs were willing to assist the PUs by relaying their messages in exchange for sharing the PUs’ spectrum. However, a concern arose regarding the PUs’ reluctance to accept this assistance due to the untrustworthiness of the SUs, which may attempt unauthorized decoding of the PUs’ messages. The study aimed to address the challenges and trade-offs associated with the cooperation between PUs and untrusted SUs in order to establish secure and efficient communication in cognitive radio networks.

A distributed mechanism to improve the secrecy rate using CJs without a relaying network was investigated in [48], where the eavesdropper is considered to be an outside entity. CJs access the spectrum allocated to them simultaneously, which may cause significant interference to all CJs. The scheme in [48] was extended to the case where the eavesdropper is considered to be an untrusted AF relay (UR) in [51] and a single CJ is used. We will refer to the scheme in [48] as the Cooperative Jamming-based Simultaneous Transmission (CJST) scheme and to the untrusted relaying-aided CJST scheme in [51] as UR-CJST throughout the paper. A secure cooperative network consisting of a source, a destination, a group of untrusted AF relays and a passive eavesdropper whose location was unknown was proposed in [52]. The joint cooperative beamforming, jamming and power allocation strategy was considered to protect the confidential information while at the same time satisfying the required QoS at the destination. In [53], an untrusted relay is considered for enabling secure device-to-device communication by deploying a directional-dedicated jammer. As the jammer is directional, its impact is only on the untrusted relay but not on the destination device. In [54], a trusted relay employing an external eavesdropper is considered, and the impact of the relay, jammer and eavesdropper locations on the secrecy capacity is presented. Dedicated jamming nodes are beneficial in terms of improving the secrecy rate, but it would incur an additional burden on the existing infrastructure; therefore, non-altruistic solutions are financially more beneficial.

Against this backdrop, in this contribution, we investigate the employment of non-altruistic multiple CJs for an untrusted relaying case and further propose a time-division transmission scenario, where all CJs access their dedicated channels in a time-division fashion. A literature review comparison is summarized in

Table 1. Literature review comparison table (Here ✓ and ✗ refers to consideration and non-consideration of the scenario in the proposed work respectively).

S/No.	Related Works	Relaying	Untrusted Relay $\left(\mathit{UR}\right)$/ Eve $\left(\mathit{E}\right)$	No. of Relays ($\mathit{S}/\mathit{M}$)	Cognitive Radio $\left(\mathit{CR}\right)$	Simultaneous Transmission $\left(\mathit{MCJST}\right)$	Time-Division Transmission $\left(\mathit{MCJTDT}\right)$	${\mathit{Y}}_{\left\{{\mathit{CJ}}_{\mathit{i},{\mathit{t}}_1}\right\}}$ Is Known to the D	Jamming Node $(\mathit{D}/\mathit{R}/\mathit{CJ}/\mathit{S})$
1	Krikidis et al. 2009 [28]	✓	E	S	✗	✗	✗	✗	R
2	Dong et al. 2010 [30]	✓	E	M	✗	✗	✗	✓	R
3	He et al. 2010 [35]	✓	$UR$	S	✗	✗	✗	$N/A$	✗
4	Ekrem et al. 2011 [43]	✓	E	S	✗	✗	✗	$N/A$	✗
5	Wu et al. 2011 [46]	✗	E	✗	✓	✓	✗	✓	$CJ$
6	Sun et al. 2012 [40]	✓	$UR$	S	✗	✗	✗	✓	D
7	Zhang et al. 2012 [47]	✓	$UR$	S	✗	✗	✗	✓	$CJ$
8	Saad et al. 2012 [34]	✗	E	✗	✗	✗	✗	✗	$N/A$
9	Khodakarami et al. 2013 [44]	✓	$UR$	M	✗	✗	✗	$N/A$	✗
10	Bao et al. 2013 [29]	✓	E	S	✗	✗	✗	$N/A$	✗
11	Stanojev et al. 2013 [48]	✓	E	S	✓	✓	✗	✓	$CJ$
12	Jeon et al. 2014 [50]	✓	$SU$ act as a $UR$	S	✓	✗	✗	$N/A$	✗
13	Sun et al. 2015 [45]	✓	$UR$	S	✗	✗	✗	✓	D
14	Al-Talabani et al. 2016 [49]	✓	E	S	✓	✓$\left(SCJST\right)$	✗	✓	$CJ$
15	Zamir et al. 2016 [51]	✓	$UR$	S	✓	✓$\left(SCJCT\right)$	✗	✓	$CJ$
16	Ali et al. 2017 [33]	✓	$UR$	S	✗	✗	✗	✓	$CJ$
17	Zhou et al. 2018 [11]	✗	E	✗	✗	✗	✗	✗	$CJ$
18	Li et al. 2018 [12]	✗	E	✗	✗	✗	✗	✗	$CJ$
19	Kuhestani et al. 2018 [37]	✓	$UR$	M	✗	✗	✗	✓	$CJ$
20	Zhao et al. 2018 [38]	✓	$UR$	S	✗	✗	✗	✓	$CJ$
21	Jiang et al. 2018 [39]	✓	$UR$	S	✗	✗	✗	✓	D
22	Yan et al. 2018 [41]	✗	E	✗	✗	✗	✗	$N/A$	S
23	Osorio et al. 2020 [36]	✓	$UR$	S	✗	✗	✗	✓	D
24	Izanlou et al. 2021 [53]	✓	$UR$	S	✗	✗	✗	✓	$CJ$
25	Hayajneh et al. 2023 [54]	✓	E	S	✗	✗	✗	✓	$CJ$
26	Our proposed scheme	✓	$UR$	S	✓	✓	✓	✓	$CJ$

. In addition, we present a classification structure in Figure 1, which classifies the cited contributions for security in the communication networks presented in Table 1. More explicitly, our contributions are stated in the novelty statement below:

• We investigate the secrecy rate of the PU and the utility rate of the CJs/SUs, based on the locations of the CJs for the AF-based untrusted relaying case.
• The model of the Stackelberg game between PU and CJs, as well as that of the power control game among CJs, are derived and investigated when an untrusted relaying terminal is employed.
• We extend the CJST scheme in [48] to the case where the eavesdropper is considered to be an untrusted relay. We further extend the single CJ-assisted UR-CJST scheme [51] to multiple CJs, which we refer to as Untrusted Relaying-aided Multiple Cooperative Jammers-based Simultaneous Transmission (UR-MCJST).
• Finally, an Untrusted Relaying-aided Multiple Cooperative Jammers-based Time-Division Transmission (UR-MCJTDT) scheme is proposed, where time division is considered in the utility phase. The performances of both UR-MCJST and UR-MCJTDT schemes are investigated.

The outline of the paper is as follows. Section 2 presents the system model of our proposed schemes, and Section 3 presents the proposed transmission scenarios, while Section 4 shows the results of the proposed scheme in addition to their discussions. Finally, we present the conclusions in Section 5.

2. System Model

In this section, we describe the system parameters, the cooperative jamming model used in our system, the Stackelberg game between PU and CJs, and the power control game between CJs.

2.1. Cooperative Jamming Model

We consider an AF-based cooperative communication system, which consists of a source node (S), a destination node (D), an untrusted relay node (R), N number of cooperative jammer nodes ($C{J}_i$) and N number of CJ’s destination nodes ($D{J}_i$), where $i=\{1,\cdots ,N\}$ [55]. It is assumed that each node has a single antenna. Our suggested system is divided into three phases depending on time slots. In more detail, the phases are as follows: the source information transmission denoted as the first phase in this paper, the relaying and jamming signal transmission denoted as the second phase, while in the end, the cooperative jammer will transmit its own information/message denoted as the third phase, as is depicted in Figure 2. During the first phase, the CJs will also transmit a jamming signal in addition to the source data transmission.

The information symbols conveyed by the PU, cooperative jammers, and untrusted relay are denoted by the symbols $X\{.\}$, $Y\{.\}$ and $Z\{.\}$, respectively. The first phase, second phase, and third phase transmissions are denoted by the subscripts $t_1$, $t_2$, and $t_3$, respectively. The eavesdropper is an outside entity as was presented in the two-phase model provided in [48]. In this paper, we consider the eavesdropper as an untrusted relay, which is not an outside entity and hence poses a different challenge. The time slot T of the overall transmission is scaled to unity. The primary transmission is kept as a time slot fraction $\alpha \le 1$. That time slot will be equally divided into the first and second phase, i.e., we have a $\frac{\alpha }{2}$ duration for each of the two phases.

In the third phase, the CJs will receive a reward, with the remaining $(1-\alpha )$ period being devoted to their non-altruistic jamming contribution. As a reward, we take into account two transmission strategies for the SUs throughout their designated third slot. These are the UR-MCJST scenario described in Section 3.1 and the UR-MCJTDT scenario described in Section 3.2. In the UR-MCJST scenario, all jammers simultaneously access the allocated bandwidth during the third phase time slot of duration $(1-\alpha )$. However, for the UR-MCJTDT scenario, the access time is equally divided to $\frac{1-\alpha }{N}$ for each jammer during the third phase of transmission.

The wireless channel considered between different transceiver pairs represented as a pair $(k,r)$ is denoted by $h_{kr}=\sqrt{G_{kr}}.{\overline{h}}_{kr}$, where ${\overline{h}}_{kr}$ is the Rayleigh fading coefficient, $G_{kr}={\left(\frac{d_{SD}}{d_{kr}}\right)}^{\rho }$ is the reduced distance related path gain [17,56], and $\rho$ is the pathloss exponent, while the distance between the two transceivers is represented as $d_{kr}$. The received SNR at node r will be defined as ${\gamma }_{kr}=\frac{|h_{kr}{|}^2}{N_0}$, where $N_0/2$ is the noise variance per dimension. Furthermore, $P_s$ is the source’s power, and $P_{C{J}_i}$ is the ith jammer’s power. The ratio of the power invested for transmitting jamming signals to that invested for the CJ’s secondary transmission is given by $\beta$.

2.2. Stackelberg Game Model

The Stackelberg game involves a leader and a follower, where the follower acts according to the strategy chosen by the leader. In our case, the PU or the source will act as the leader, while CJs will act as the followers. The PU’s optimal strategy (${\alpha }^{*},{\beta }^{*}$) would maximize its secrecy rate $R_s$, while the power level of the $i^{th}$ jammer $P_{C{J}_i}^{*}$ is selected to maximize its utility rate $U_{C{J}_i}$, as is illustrated in Figure 3. The PU’s optimal strategy and the CJ’s power choice are jointly referred to as the Stackelberg equilibrium. We assume that CJs are honest and trustworthy, where they do not deviate from transmitting jamming signals based on a power ratio of $\beta$ determined by the PU, and they do not turn malicious. All PUs and CJs are also assumed to be selfish and rational, which mimic a non-altruistic behavior.

Once the PU has decided the optimal strategy (${\alpha }^{*},{\beta }^{*}$), the CJs will respond with a set of power choices $P_{C{J}_i}^{*}$, where $\mathcal{J}\subseteq \{1,\cdots ,N\}$ is the set of chosen jammers. Since there are multiple CJs in our scheme, all chosen $\mathcal{J}$ CJs will need to compete among themselves. The outcome of the competition is a set of power choices that can be described by the game-theoretical concept of the Nash equilibrium (NE), where any unilateral deviation in the player’s strategy would not produce any gain [57]. (The NE considered in this paper is based on [48], which considers all the values of $\alpha$ and $\beta$ for its calculation and hence gives the best results. In comparison, iterative algorithms for NE have been presented in [25,26], which will give optimum results, as they incur lower computational complexity.) The power control game of [48] was modified and adapted to our relaying-based cooperative jamming system, as will be described in the following section.

3. Transmission Scenarios

In this section, the power control game is presented for both the simultaneous trans-mission-based UR-MCJST scenario and the time-division transmission-based UR-MCJTDT scenario.

3.1. UR-MCJST Scenario

The S and CJs play the game known as the leader–follower Stackelberg game. In this game, the source acts as the leader, and CJs are assigned the role of the followers, as shown in Figure 3. The S, which is the leader, will first allocate the time fraction $\alpha$ to be used during the first and second phases, then the chosen set of CJs $\mathcal{J}\subseteq \{1,\cdots ,N\}$, where N is the total number of available CJs, as well as the power ratio $\beta$. Here, $\alpha$ is normalized to one. The power ratio $\beta$ will be defined as in [48]:

$$\beta =\frac{𝔼\left[Y_{\{C{J}_{i,t_1},\frac{\alpha }{2}\}}^2\right]}{𝔼\left[Y_{\{C{J}_{i,t_3},1-\alpha \}}^2\right]},$$

(1)

where $𝔼\left[Y_{\{C{J}_{i,t_1},\frac{\alpha }{2}\}}^2\right]$ is the average power for the ith CJ denoted as $C{J}_i$ for transmitting a jamming signal during the first phase, while $𝔼\left[Y_{\{C{J}_{i,t_3},1-\alpha \}}^2\right]$ will be $C{J}_i$’s average power for its own transmission during the third phase. K is a factor that represents the number of transmitted symbols by the PU denoted as S for the complete normalized interval. S transmits the symbols $X_{\{t_1,\frac{\alpha }{2}\}}$ during the first time slot only, i.e., we have $X_{\{t_2,\frac{\alpha }{2}\}}=0$ and $X_{\{t_3,1-\alpha \}}=0$. Consequently, the total power that is represented as $P_s$ can be calculated by employing the following equation, which is the revised form of Equation (1) in [48]:

$$\begin{array}{ccc}\hfill P_s& =& \frac{1}{K}\sum _{t_1=1}^{\lfloor \frac{\alpha }{2}K\rfloor }𝔼\left[X_{\{t_1,\frac{\alpha }{2}\}}^2\right]+\frac{1}{K}\sum _{t_2=\lfloor \frac{\alpha }{2}K+1\rfloor }^{\lfloor \alpha K\rfloor }𝔼\left[X_{\{t_2,\frac{\alpha }{2}\}}^2\right]\hfill \\ & & +\frac{1}{K}\sum _{t_3=\lfloor \alpha K+1\rfloor }^K𝔼\left[X_{\{t_3,1-\alpha \}}^2\right]\hfill \\ & =& \frac{1}{K}\sum _{t_1=1}^{\lfloor \frac{\alpha }{2}K\rfloor }𝔼\left[X_{\{t_1,\frac{\alpha }{2}\}}^2\right]=\frac{\alpha }{2}𝔼\left[X_{\{t_1,\frac{\alpha }{2}\}}^2\right],\hfill \end{array}$$

(2)

and thus, we have:

$$𝔼\left[X_{\{t_1,\frac{\alpha }{2}\}}^2\right]=\frac{2P_s}{\alpha }.$$

(3)

Furthermore, the CJs will not be transmitting during the second transmission phase, i.e., $𝔼\left[Y_{\{C{J}_{i,t_2},\frac{\alpha }{2}\}}^2\right]=0$. Therefore, the ith CJ’s transmission power will be given by:

$$\begin{array}{ccc}\hfill P_{C{J}_i}& =& \frac{1}{K}\sum _{t_1=1}^{\lfloor \frac{\alpha }{2}K\rfloor }𝔼\left[Y_{\{C{J}_{i,t_1},\frac{\alpha }{2}\}}^2\right]+\hfill \\ & & \frac{1}{K}\sum _{t_2=\lfloor \frac{\alpha }{2}K+1\rfloor }^{\lfloor \alpha K\rfloor }𝔼\left[Y_{\{C{J}_{i,t_2},\frac{\alpha }{2}\}}^2\right]+\hfill \\ & & \frac{1}{K}\sum _{t_3=\lfloor \alpha K+1\rfloor }^{\lfloor K\rfloor }𝔼\left[Y_{\{C{J}_{i,t_3},1-\alpha \}}^2\right]\hfill \\ & =& \frac{\alpha }{2}𝔼\left[Y_{\{C{J}_{i,t_1},\frac{\alpha }{2}\}}^2\right]+(1-\alpha )𝔼\left[Y_{\{C{J}_{i,t_3},1-\alpha \}}^2\right].\hfill \end{array}$$

(4)

By integrating the power ratio $\beta$ from Equation (1) into Equation (4), we obtain the following expression:

$$𝔼\left[Y_{\{C{J}_{i,t_1},\frac{\alpha }{2}\}}^2\right]={\kappa }_1{P}_{C{J}_i},$$

(5)

where ${\kappa }_1=\frac{\beta }{\frac{\alpha \beta }{2}+1-\alpha }$ and

$$𝔼\left[Y_{\{C{J}_{i,t_3},1-\alpha \}}^2\right]=\frac{{\kappa }_1}{\beta }{P}_{C{J}_i}.$$

(6)

During the first phase, S sends the signal $X_{\{t_1,\frac{\alpha }{2}\}}$ as its message to the receiver denoted by D and an untrusted relay R. At the same time, non-altruistic CJs transmit the jamming signal $Y_{\{C{J}_{i,t_1},\frac{\alpha }{2}\}}\sim \mathcal{N}(0,1)$ to ensure that a secure communication between the S and D has been carried out over the wireless channel as shown in Figure 2. During the first phase, the signals that will be received at R and D can be represented, respectively, as

$$\begin{array}{ccc}\hfill y_r& =& h_{sr}\sqrt{\frac{2P_s}{\alpha }}{X}_{\{t_1,\frac{\alpha }{2}\}}+\hfill \\ & & \sum _{i\in \mathcal{J}}{h}_{C{J}_{i}r}\sqrt{{\kappa }_1{P}_{C{J}_i}}{Y}_{\{C{J}_{i,t_1},\frac{\alpha }{2}\}}+w_r\hfill \end{array}$$

(7)

and

$$\begin{array}{ccc}\hfill y_d^{\left(1\right)}& =& h_{sd}\sqrt{\frac{2P_s}{\alpha }}{X}_{\{t_1,\frac{\alpha }{2}\}}+\hfill \\ & & \sum _{i\in \mathcal{J}}{h}_{C{J}_{i}d}\sqrt{{\kappa }_1{P}_{C{J}_i}}{Y}_{\{C{J}_{i,t_1},\frac{\alpha }{2}\}}+w_d^{\left(1\right)}\hfill \end{array}$$

(8)

where $w_r$ and $w_d^{\left(1\right)}$ are the AWGN noise at R and D, respectively. Afterward, R will relay the received signal $y_r$ to D using an amplify and forward strategy during the second phase; consequently, the signal received at the D can be represented as:

$$\begin{array}{ccc}\hfill y_d^{\left(2\right)}& =& h_{rd}{\eta }_r{y}_r+w_d^{\left(2\right)},\hfill \end{array}$$

(9)

where $w_d^{\left(2\right)}$ denotes the AWGN noise at the D. The amplification factor ${\eta }_r$ can be computed as:

$$\begin{array}{ccc}\hfill {\eta }_r& =& \sqrt{\frac{1}{\frac{2P_s}{\alpha }|h_{sr}{|}^2+{\sum }_{i\in \mathcal{J}}{\kappa }_1{P}_{C{J}_i}{\left|h_{C{J}_{i}r}\right|}^2+N_0}}.\hfill \end{array}$$

(10)

By substituting (7) and (10) into (9), we obtain:

$$\begin{array}{ccc}\hfill y_d^{\left(2\right)}& =& {\eta }_r{h}_{rd}{h}_{sr}\sqrt{\frac{2P_s}{\alpha }}{X}_{\{t_1,\frac{\alpha }{2}\}}+\hfill \\ & & \sum _{i\in \mathcal{J}}{\eta }_r{h}_{rd}{h}_{C{J}_{i}r}\sqrt{{\kappa }_1{P}_{C{J}_i}}{Y}_{\{C{J}_{i,t_3},\frac{\alpha }{2}\}}+\hfill \\ & & {\eta }_r{h}_{rd}{w}_r+w_d^{\left(2\right)}.\hfill \end{array}$$

(11)

By summing the two received signals, specifically $y_d^{\left(1\right)}$ from Equation (8) and $y_d^{\left(2\right)}$ from Equation (11), we obtain the following expression:

$$\begin{array}{ccc}\hfill y_d& =& h_{sd}\sqrt{\frac{2P_s}{\alpha }}{X}_{\{t_1,\frac{\alpha }{2}\}}+\sum _{i\in \mathcal{J}}{h}_{C{J}_{i}d}\sqrt{{\kappa }_1{P}_{C{J}_i}}{Y}_{\{C{J}_{i,t_1},\frac{\alpha }{2}\}}\hfill \\ & & +\sum _{i\in \mathcal{J}}{\eta }_r{h}_{rd}{h}_{C{J}_{i}r}\sqrt{{\kappa }_1{P}_{C{J}_i}}{Y}_{\{C{J}_{i,t_3},\frac{\alpha }{2}\}}+\hfill \\ & & {\eta }_r{h}_{rd}{h}_{sr}\sqrt{\frac{2P_s}{\alpha }}{X}_{\{t_1,\frac{\alpha }{2}\}}+{\eta }_r{h}_{rd}{w}_r+w_d,\hfill \end{array}$$

(12)

Here, we define $w_d$ as the sum of $w_d^{\left(1\right)}$ and $w_d^{\left(2\right)}$, and ${\eta }_r$ is provided in Equation (10). Since $Y_{C{J}_{i,t_1},\frac{\alpha }{2}}$ is typically assumed to be known at receiver D, we can subtract the second and third terms from $y_d$ in Equation (12) to yield:

$$\begin{array}{ccc}\hfill y_d& =& h_{sd}\sqrt{\frac{2P_s}{\alpha }}{X}_{\{t_1,\frac{\alpha }{2}\}}+\hfill \\ & & {\eta }_r{h}_{rd}{h}_{sr}\sqrt{\frac{2P_s}{\alpha }}{X}_{\{t_1,\frac{\alpha }{2}\}}+{\eta }_r{h}_{rd}{w}_r+w_d.\hfill \end{array}$$

(13)

We can calculate the $SNR$ at D as follows:

$$\begin{array}{ccc}\hfill {\gamma }_D& =& \frac{{\left|h_{sd}\right|}^2\frac{2P_s}{\alpha }+|{\eta }_r{|}^2|h_{rd}{|}^2{\left|h_{sr}\right|}^2\frac{2P_s}{\alpha }}{|{\eta }_r{|}^2{\left|h_{rd}\right|}^2{N}_0+2N_0}\hfill \\ & =& \frac{\frac{2}{\alpha }{\gamma }_{sd}{\gamma }_{sr}+{\gamma }_{sd}{\kappa }_1{\sum }_{i\in \mathcal{J}}{\gamma }_{C{J}_{i}r}+{\gamma }_{sr}{\gamma }_{rd}+{\gamma }_{sd}}{\frac{\alpha }{2}{\gamma }_{rd}+{\gamma }_{sr}+\frac{\alpha }{2}{\kappa }_1{\sum }_{i\in \mathcal{J}}{\gamma }_{C{J}_{i}r}+\alpha }\hfill \end{array}$$

(14)

where ${\gamma }_{sr}=\frac{P_s{\left|h_{sr}\right|}^2}{N_0}$, ${\gamma }_{sd}=\frac{P_s{\left|h_{sd}\right|}^2}{N_0}$, ${\gamma }_{C{J}_{i}r}=\frac{P_{C{J}_{i}r}{\left|h_{C{J}_{i}r}\right|}^2}{N_0}$ and ${\gamma }_{rd}=\frac{|h_{rd}{|}^2}{N_0}$. Similarly, from (7), we can derive the received $SNR$ at the relay as:

$$\begin{array}{ccc}\hfill {\gamma }_{R;C{J}_i}& =& \frac{|h_{sr}{|}^2\frac{2P_s}{\alpha }}{{\sum }_{i\in \mathcal{J}}{\left|h_{C{J}_{i}r}\right|}^2{\kappa }_1{P}_{C{J}_i}+N_0}\hfill \\ & & =\frac{{\gamma }_{sr}}{\frac{\alpha }{2}{\kappa }_1{\sum }_{i\in \mathcal{J}}{\gamma }_{C{J}_{i}r}+\frac{\alpha }{2}}.\hfill \end{array}$$

(15)

The achievable rate at the destination $R_D$ for the multiple jammers case can be computed as:

$$\begin{array}{ccc}\hfill R_D& =& \frac{\alpha }{2}log(1+{\gamma }_D),\hfill \end{array}$$

(16)

Similarly, the achievable rate at the untrusted relay is given by:

$$\begin{array}{ccc}\hfill R_R& =& \frac{\alpha }{2}log(1+\sum _{i\in \mathcal{J}}{\gamma }_{R;C{J}_i}),\hfill \end{array}$$

(17)

where ${\gamma }_{R;C{J}_i}$ is the $SNR$ at the ith relay given by (15). The achievable S-D secrecy rate is the measure of how much information is transmitted from S to D without being revealed to the eavesdropper, which is given by:

$$\begin{array}{ccc}\hfill R_s& =& max\{0,(R_D-R_R)\}.\hfill \end{array}$$

(18)

Each CJ transmits an average power of $\frac{{\kappa }_1}{\beta }$, as given by (6), during the third phase. Hence, the SNR at the CJ’s destination is given by:

$$\begin{array}{ccc}\hfill {\gamma }_{C{J}_{i}D{J}_i}& =& \frac{\frac{{\kappa }_1}{\beta }{P}_{C{J}_i}{\left|h_{C{J}_{i}D{J}_i}\right|}^2}{\frac{{\kappa }_1}{\beta }{\sum }_{j\in \mathcal{J},j\ne i}{\left|h_{C{J}_{j}D{J}_i}\right|}^2{P}_{C{J}_j}+N_0}\hfill \\ & =& \frac{|h_{C{J}_{i}D{J}_i}{|}^2{P}_{C{J}_i}}{{\sum }_{j\in \mathcal{J},j\ne i}{\left|h_{C{J}_{j}D{J}_i}\right|}^2{P}_{C{J}_j}+\frac{\beta }{{\kappa }_1}{N}_0},\hfill \end{array}$$

(19)

Here, $h_{C{J}_{i}D{J}_i}$ denotes the fading channel coefficient between the ith jammer and the corresponding destination $D{J}_i$ of that jammer. Additionally, the power of the ith jammer, represented as $P_{C{J}_i}$, is subject to the power budget constraint $\overline{P}C{J}_i$, which ensures that $PC{J}_i$ does not exceed ${\overline{P}}_{C{J}_i}$. Notice that there is interference from other CJs, who transmit to their destinations at the same time. (The utility rate of a node functioning as a cooperative jammer is determined by its attainable reliable communication rate toward its destination $D{J}_i$, considering the time-bandwidth fraction $(1-\alpha )$. This utility rate is evaluated in relation to the cost associated with the total transmission power.) The utility rate of the ith jammer can then be computed as:

$$\begin{array}{ccc}\hfill U_{C{J}_i}(\alpha ,\beta ,\mathcal{J};P_{C{J}_i})& =& (1-\alpha ){\log }_2\left(1+{\gamma }_{C{J}_{i}D{J}_i}\right)\hfill \\ & & -c_i{P}_{C{J}_i}\hfill \end{array}$$

(20)

where the per unit transmission power cost for the ith jammer is denoted by $c_i$. The followers’ response refers to the power levels selected by the jammers for transmission. A Nash equilibrium occurs when no jammer has motivation to alter their power level, considering the power levels chosen by the other jammers [48]:

$$\begin{array}{ccc}\hfill P_{C{J}_{i\in \mathcal{J}}}^{*}(\alpha ,\beta ;\mathcal{J})& =& arg\underset{P_{C{J}_{i\in \mathcal{J}}}(\alpha ,\beta ;\mathcal{J})}{max}{U}_{C{J}_i}(\alpha ,\beta ,\mathcal{J};P_{C{J}_i})\hfill \\ & s.t.& 0<P_{CJ}<{\overline{P}}_{C{J}_i}.\hfill \end{array}$$

(21)

By equating the derivative of the utility rate $U_{C{J}_i}$ with respect to $P_{C{J}_i}$ to zero, we can find the optimal power allocation for each jammer, which is expressed as follows:

$$\frac{\partial U_{C{J}_i}(\alpha ,\beta ,\mathcal{J};P_{C{J}_i})}{\partial P_{C{J}_i}}{|}_{P_{C{J}_{j\in \mathcal{J}},j\ne i}=P_{C{J}_{j\in \mathcal{J},j\ne i}}^{*}}=0,\forall i\in \mathcal{J}$$

(22)

By deriving (22) using (20), we can determine the optimal power distribution for each jammer, which was determined to be:

$$\begin{array}{ccc}\hfill P_{C{J}_{i\in \mathcal{J}}}^{*}(\alpha ,\beta ;\mathcal{J})& =& \left[\frac{1-\alpha }{c_i\ln \left(2\right)}-\frac{\beta }{{\kappa }_1{\left|h_{C{J}_{i}D{J}_i}\right|}^2}-\right.\hfill \\ & & {\left.\sum _{j\in \mathcal{J},j\ne i}\frac{|h_{C{J}_{j}D{J}_i}{|}^2}{|h_{C{J}_{i}D{J}_i}{|}^2}{P}_{C{J}_j}^{*}(\alpha ,\beta ;\mathcal{J})\right]}_0^{{\overline{P}}_{C{J}_i}}.\hfill \end{array}$$

(23)

Note that in the situation of weak interference, NE exists and is distinct [58], i.e., for the case where the interference matrix H is defined as ${\left[\mathbf{H}\right]}_{C{J}_{j}D{J}_i}={\left|h_{C{J}_{j}D{J}_i}\right|}^2$, it will be strictly diagonally dominant, i.e., ${\sum }_{j\in \mathcal{J},j\ne i}\frac{|h_{C{J}_{j}D{J}_i}{|}^2}{|h_{C{J}_{i}D{J}_i}{|}^2}<1$. The $D{J}_i$ in our simulation is nearer to the $C{J}_i$ than other $C{J}_j$ where $i\ne j$. The interference matrix $\mathbf{H}$ exhibits strict diagonal dominance, indicating that the diagonal elements are significantly larger than the off-diagonal elements, and hence, this property leads to an apparent NE.

Figure 3 presents the Stackelberg game established for optimizing the values of ${\alpha }^{*}$ and ${\beta }^{*}$, which can be calculated from:

$$\begin{array}{ccc}\hfill ({\alpha }^{*},{\beta }^{*})& =& arg\underset{\alpha ,\beta }{max}{R}_s(\alpha ,\beta ;P_{C{J}_{i\in \mathcal{J}}}^{*}(\alpha ,\beta ;\mathcal{J})\hfill \\ & s.t.& 0<\alpha \le 1,0<\beta <\infty .\hfill \end{array}$$

(24)

As shown in Figure 3, the source (leader) provides the cooperative jammer (follower) the initial information of $\alpha$ and $\beta$. The cooperative jammer calculates the optimized power using (21). The optimized power is used by the source to maximize its secrecy rate. The iterations between the source and cooperative jammer continue until optimized values of ${\alpha }^{*}$ and ${\beta }^{*}$ are attained for providing the maximum secrecy rate.

3.2. UR-MCJTDT Scenario

Note that the UR-MCJST scenario described in Section 3.1 requires weak interference for the NE to exist. It is hence rational to derive an interference-free scenario. More specifically, in the time-division-based UR-MCJTDT scenario, each of the selected CJs is allocated a time-bandwidth fraction of $\frac{1-\alpha }{N}$, for an interference-free transmission during the third phase. In this interference-free UR-MCJTDT scenario, NE will always exist.

In a similar fashion to the UR-MCJST scenario, each jammer in this case will emit a noise signal during the initial $\alpha /2$ period. This behavior of the jammers remains consistent across both scenarios. Then, the relay node will transmit during the second $\alpha /2$ duration as shown in Figure 2. The remaining $(1-\alpha )$ time-bandwidth fraction will then be divided for all N selected CJs, where each CJ can enjoy an interference-free allocation of $\frac{1-\alpha }{N}$ during the third phase. The power ratio $\beta$ for the UR-MCJTDT scenario can be calculated as:

$$\tilde{\beta }=\frac{\widetilde{𝔼}\left[Y_{\{C{J}_{i,t_1},\frac{\alpha }{2}\}}^2\right]}{\widetilde{𝔼}\left[Y_{\{C{J}_{i,t_3},\frac{1-\alpha }{N}\}}^2\right]},$$

(25)

where the mean power used by the ith jammer for transmitting the jamming signal in the first phase and consequently its own information in the third phase will be denoted by $\widetilde{𝔼}\left[Y_{C{J}_{i,t_1},\frac{\alpha }{2}}^2\right]$ and $\widetilde{𝔼}\left[Y_{C{J}_{i,t_3},\frac{1-\alpha }{N}}^2\right]$, respectively. These represent the power levels associated with the jamming signal and the CJ’s own data transmission during the respective phases.

The total power utilized by the S represented as $P_s$ will be similar to (2). However, the transmission power for the ith CJ is given by:

$$\begin{array}{ccc}\hfill {\tilde{P}}_{C{J}_i}& =& \frac{1}{K}\sum _{t_1=1}^{\lfloor \frac{\alpha }{2}K\rfloor }\widetilde{𝔼}\left[Y_{\{C{J}_{i,t_1},\frac{\alpha }{2}\}}^2\right]+\hfill \\ & & \frac{1}{K}\sum _{t_2=\lfloor \frac{\alpha }{2}K+1\rfloor }^{\lfloor \alpha K\rfloor }\widetilde{𝔼}\left[Y_{\{C{J}_{i,t_2},\frac{\alpha }{2}\}}^2\right]+\hfill \\ & & \frac{1}{K}\sum _{t_3=\lfloor \left(\alpha +\frac{1-\alpha }{N}(i-1)\right)K+1\rfloor }^{\lfloor \left(\alpha +\frac{1-\alpha }{N}i\right)K\rfloor }\widetilde{𝔼}\left[Y_{\{C{J}_{i,t_3},\frac{1-\alpha }{N}\}}^2\right]\hfill \\ & =& \frac{\alpha }{2}\widetilde{𝔼}\left[Y_{\{C{J}_{i,t_1},\frac{\alpha }{2}\}}^2\right]+\frac{1-\alpha }{N}\widetilde{𝔼}\left[Y_{\{C{J}_{i,t_3},\frac{1-\alpha }{N}\}}^2\right].\hfill \end{array}$$

(26)

By integrating the power ratio $\tilde{\beta }$ from Equation (25) into Equation (26), we obtain:

$$\widetilde{𝔼}\left[Y_{\{C{J}_{i,t_1},\frac{\alpha }{2}\}}^2\right]={\kappa }_2{\tilde{P}}_{C{J}_i},$$

(27)

where ${\kappa }_2=\frac{\tilde{\beta }}{\frac{\alpha \tilde{\beta }}{2}+\frac{1-\alpha }{N}}$ and

$$\widetilde{𝔼}\left[Y_{\{C{J}_{i,t_3},\frac{1-\alpha }{N}\}}^2\right]=\frac{N{\kappa }_2}{\tilde{\beta }}{\tilde{P}}_{C{J}_i}.$$

(28)

The SNR at D and R are similar to those given by (14) and (15), except that ${\kappa }_2$ is used instead of ${\kappa }_1$, i.e., they are given by:

$$\begin{array}{c}\hfill \widetilde{{\gamma }_D}=\frac{\frac{2}{\alpha }{\gamma }_{sd}{\gamma }_{sr}+{\gamma }_{sd}{\kappa }_2{\sum }_{i\in \mathcal{J}}{\tilde{\gamma }}_{C{J}_{i}r}+{\gamma }_{rd}{\gamma }_{sr}+{\gamma }_{sd}}{\frac{\alpha }{2}{\gamma }_{rd}+{\gamma }_{sr}+\frac{\alpha }{2}{\kappa }_2{\sum }_{i\in \mathcal{J}}{\tilde{\gamma }}_{C{J}_{i}r}+\frac{\alpha }{2}}\end{array}$$

(29)

and

$$\begin{array}{c}\hfill {\tilde{\gamma }}_{R;C{J}_i}=\frac{{\gamma }_{sr}}{\frac{\alpha }{2}{\kappa }_2{\sum }_{i\in \mathcal{J}}{\tilde{\gamma }}_{C{J}_{i}r}+\frac{\alpha }{2}},\end{array}$$

(30)

respectively. Based on these SNR values, the achievable rates $R_D$ and $R_R$ as well as the secrecy rate $\widetilde{R_s}$, can be computed from (16), (17) and (18), respectively.

By contrast, the SNR at the CJ’s destination for this orthogonal transmission scenario is computed as:

$$\begin{array}{ccc}\hfill {\tilde{\gamma }}_{C{J}_{i}D{J}_i}& =& \frac{\frac{{\kappa }_2}{\tilde{\beta }}{\tilde{P}}_{C{J}_i}{\left|h_{C{J}_{i}D{J}_i}\right|}^2}{N_0}\hfill \\ & =& \frac{|h_{C{J}_{i}D{J}_i}{|}^2{\tilde{P}}_{C{J}_i}}{\frac{\tilde{\beta }}{{\kappa }_2}{N}_0},\hfill \end{array}$$

(31)

The utility rate of the selected jammers $C{J}_{i\in \mathcal{J}}$ in the orthogonal scenario can be derived as follows:

$$\begin{array}{ccc}\hfill {\tilde{U}}_{C{J}_i}(\alpha ,\tilde{\beta },\mathcal{J};{\tilde{P}}_{C{J}_i})& =& \frac{1-\alpha }{N}{\log }_2\left(1+{\tilde{\gamma }}_{C{J}_{i}D{J}_i}\right)\hfill \\ & & -c_i{\tilde{P}}_{C{J}_i}.\hfill \end{array}$$

(32)

By comparing the SNR formulas in (31) and (19), we see that the UR-MCJTDT scenario has a higher SNR because it has no interference at the denominator. However, the time-bandwidth of the UR-MCJTDT scenario is reduced by N times, compared to those of the UR-MCJST scenario, as seen in (32) and (20). Based on a partial derivative of ${\tilde{U}}_{C{J}_i}$ in (32), The optimal power allocation for each jammer can be calculated in a similar manner as Equation (22), yielding the following expression:

$$\begin{array}{c}\hfill {\tilde{P}}_{C{J}_{i\in \mathcal{J}}}^{*}(\alpha ,\tilde{\beta };\mathcal{J})={\left[\frac{1-\alpha }{c_i\ln \left(2\right)}-\frac{\tilde{\beta }}{{\kappa }_2{\left|h_{C{J}_{i}D{J}_i}\right|}^2}\right]}_0^{{\overline{P}}_{C{J}_i}},\end{array}$$

(33)

where NE always exists because there is no interference term from other CJs; hence, it can be decoupled (p. 141 [48]). Based on the best power allocation in (33), the optimized values of ${\alpha }^{*}$ and ${\beta }^{*}$ can be calculated from (24) by the leader/source.

4. Results and Discussions

Having investigated the game theoretical aspect of the proposed schemes in Section 2, let us now evaluate our simulation results. The first set of simulations consider $N=3$ CJs, where the positions of $C{J}_2$ and $C{J}_3$ are fixed, while the impact of changing the location of $C{J}_1$ (mobile jammer) is investigated. The (x, y) location coordinates of of S, R, $C{J}_2$, $C{J}_3$ and D are given at the caption of the figures. In this scenario, the pathloss exponent was chosen to be $\rho =4$, with $P_s/N_0={\overline{P}}_{C{J}_i}=10$ (mW) and $c=0.25$ (bit/s/Hz/mW). The jammers’ destinations are considered to be a fraction of the distance between the S and D, i.e., $d_j=d_{sd}/4$ from its jammer transmitter location, where $d_{sd}$ is the distance between S to D and is measures in meters. In order to compare UR-MCJST and UR-MCJTDT scenarios when the $\alpha$ and $\beta$ values are fixed, we set the $\alpha$ and $\beta$ values to 0.65 and 1, respectively, in this first set of simulations.

Figure 4 illustrates the secrecy rate and the corresponding utility rate for the sender S and the cooperative jammer $C{J}_i$ in the case of the UR-MCJST scenario employed by the jammers. On the other hand, Figure 5 shows the secrecy rate and the corresponding utility rate when the jammers adopt the UR-MCJTDT scenario. We found from Figure 4a that the secrecy rate drops to $1.2042$ (bit/s/Hz) and $1.43$ (bit/s/Hz) when $C{J}_1$ gets closer to $C{J}_2$ at (22,43) and $C{J}_3$ at (26,50), respectively, in the UR-MCJST scenario. This is due to high interference levels at the jammers’ destination, i.e., when the interference matrix H is no longer strictly diagonally dominant. Since the power used during the third phase has to be small when two CJs are close to each other, the jamming power used during the first phase also needs to be small because we have $\beta =1$. When the jamming power used during the first phase is small, the achievable rate by the untrusted relay would be higher, which would result in a low secrecy rate for the PU. Similarly, the utility rate of $C{J}_1$ also drops when it is closer to $C{J}_2$ or $C{J}_3$ due to the use of lower power during the third phase. As expected, this effect is mitigated in the UR-MCJTDT scenario, as seen in Figure 5a, because the transmission power of each CJ causes no interference to other CJs during the third transmission phase.

Note that there is another drop in the secrecy rate when $C{J}_1$ gets closer to R, which gives a secrecy rate of $1.3951$ (bit/s/Hz) and $1.3983$ (bit/s/Hz) at location (25,25) for Figure 4a and Figure 5a, respectively. However, after considering all factors including the effect of the locations of CJ’s destination nodes, a local minimum of the secrecy rate (of value 1.3883 (bit/s/Hz)) is located at (23,23) in Figure 4a. Similarly, Figure 5a also shows a minimum value of 1.3901 (bit/s/Hz) at location (23,23). This is due to the fact that the jamming signals would form part of the amplification factor, as seen in (10), which will be forwarded to the destination node, as shown in (13). As we note from the secrecy rate region of the UR-MCJTDT scenario depicted in Figure 5a, the best location for $C{J}_1$ is farther away from the relay node, while the effect of the other CJs’ locations is negligible compared to that of the UR-MCJST scenario shown in Figure 4a. Furthermore, the utility rate of the UR-MCJTDT scenario is not affected by the relay location because the relayed signal during the second phase is independent of the third-phase transmission. Note that the utility rate of $C{J}_1$ in the UR-MCJTDT scenario is constant and is higher than that of the UR-MCJST scenario, as seen in Figure 4b and Figure 5b.

Our second set of simulations explores the effect of increasing the number of CJs, i.e., N, when both the Stackelberg game and power control game were invoked for optimal performance. We compare our results with the baseline Stackelberg and the power-control-based algorithms in [46,48]. The optimum values of $\alpha$ and $\beta$ are determined by the source, while the optimum jammer power values are determined by the jammer, as seen in the Stackelberg leader–follower game in Figure 3. In the non-relaying scheme of CJST [48], both the secrecy rate and utility rate would improve as the number of CJs increases. However, when relaying is invoked in our UR-MCJST scenario, it is interesting to see from Figure 6 that as N increases, the secrecy rate drops. Hence, the CJST scenario is not desirable when an untrusted relay is involved. By contrast, our proposed UR-MCJTDT scenario works well in the untrusted relaying system, where the secrecy rate improves as N increases, as seen in Figure 6. However, it appears that we only need $N=3$ CJs in our case, when the SNR used is 10 dBm. We can also see from Figure 6 that our proposed algorithms outperform the baseline Stackelberg in [46] and the power control algorithm in [48]. The optimized ${\alpha }^{*}$ and ${\beta }^{*}$ chosen by the source, as well as the average optimized CJ power of all N CJs given by $P_{CJ}^{*}=\frac{1}{N}{\sum }_i^N{P}_{C{J}_i}^{*}$, are shown in Figure 6 and Figure 7. For example, when $N=2$, we have ${\alpha }^{*}=0.93$, ${\beta }^{*}=6$ and ${\tilde{P}}_{CJ}^{*}=0.41$ (mWatts) for the orthogonal scenario. Furthermore, as seen in Figure 7, the per user utility rates of the UR-MCJTDT and UR-MCJST scenarios are similar. However, the CJs in the UR-MCJTDT scenario can achieve the same utility rate with less power, when compared to the UR-MCJST scenario. For example, as seen in Figure 7 when $N=3$, the average CJ power is $P_{CJ}^{*}=0.18$ (mWatts) and ${\tilde{P}}_{CJ}^{*}=0.93$ (mWatts) for the UR-MCJST and UR-MCJTDT scenarios, respectively. We also compared the utility rate results with the baseline Stackelberg and the power-control-based algorithms in [46,48]. Here, we can see that the per user utility rate increases with increasing available $CJs$ in the case of the baseline algorithms, but the overall utility rate will be higher in the proposed scheme, as we have employed multiple $CJs$, while the baseline scheme chooses a better $CJ$ among the available $CJs$.

5. Conclusions

We have proposed and investigated a cooperative jamming-assisted untrusted relaying scheme, where the Stackelberg game was invoked between the PU and the CJs, while a power control game was employed by the CJs. It has been noted that when an untrusted relay is present, the UR-MCJST benchmark scheme suffers from a reduced secrecy rate, when the number of CJs are increased. Moreover, the proposed UR-MCJTDT scenario can maintain the same utility rate as the benchmark scheme while allowing the CJs to use a lower transmission power. The PU’s secrecy rate of the UR-MCJTDT scheme is significantly better than that of the UR-MCJST scheme, while maintaining the same utility rates for the CJs. However, UR-MCJTDT can sometimes be inefficient in terms of resource utilization because the data are divided into discrete time slots, and each slot is allocated to a specific user for transmission. This means that even if a particular user has only a small amount of data to transmit, it still occupies a full time slot. In such cases, UR-MCJST performs well initially, but as the number of users increases, interference becomes a challenge. This creates a trade-off between time-division transmission, which ensures isolated time slots for each user but may use more resources, and simultaneous transmission, which maximizes resource utilization but risks increased interference with a higher user count. Furthermore, the secrecy rate of the benchmark scheme depends on the locations of the relay node and the CJs. However, the secrecy rate of the proposed UR-MCJTDT scheme is mainly influenced by the location of the relay node. The simulation results indicated that three CJs are sufficient to support a secure transmission when one untrusted relay is employed, as investigated in the proposed scheme. The optimum number of relaying nodes and CJs for a given system is worth further investigation in future work.