Cooperative Secure Transmission in the Presence of Untrusted Relay

Abstract

Considering a nonregenerative untrusted relay network, we investigate three different secure transmission strategies, for example, noncooperative strategy, conventional amplify-and-forward (AF) strategy, and cooperative jamming (CJ) strategy. To thoroughly assess the secrecy performance achieved by the three strategies, we derive the closed-form expressions for the connection outage probability and secrecy outage probability regarding each strategy. Based on these expressions, the reliability and security tradeoff (RST) is examined to facilitate the design of the transmitting parameters. We then present the closed-form expression for the effective secrecy throughput (EST) and characterize the overall efficiency of these transmission strategies. Furthermore, we conduct the asymptotic analysis for the secrecy throughput, which enables us to determine the optimal transmission strategy under different scenarios. Our analytical and numerical results demonstrate that compared with the noncooperative and AF strategies, the CJ strategy possesses the best performance in terms of RST. Additionally, we also find that the EST performance of the AF strategy is nearly invariable when the quality of the second hop changes.

1. Introduction

Due to the broadcast nature of the open medium, wireless communication systems are particularly vulnerable to security attacks. Recently, the information-theoretic security, referred to as physical layer security, has been recognized as a promising method to develop low-complexity and effective wireless security mechanism [1, 2]. The initial work in this area goes back to Wyner [3], who introduced the wiretap channel and concluded that secure communication can be achieved without relying on private keys. In [4], the secrecy capacity of the Gaussian wiretap channel was investigated, which is the maximal achievable secrecy rate below which the eavesdropper is unable to decode any confidential message. The transmission of confidential message over broadcast channels was generalized in [5].

Some studies have focused on the physical layer security in wireless ad hoc networks [6] and cognitive radio networks [7]. Meanwhile, cooperative communication has been emerging as a promising technique, due to its significant spatial diversity and low hardware demands. Therefore, current studies concentrated on generalizing the physical layer security to the relay networks. The information security in the relay network is often susceptible to eavesdropping by unintended receivers. Thus, one possible scenario in which external eavesdropper(s) may overhear transmission in the relay network was investigated in [8, 9]. Considering the practical working condition, the design of secure transmission scheme under channel state information (CSI) uncertainty has already been studied by [10–12] with eavesdropper. However, from a robust perspective, relay can also be viewed as a potential eavesdropper [13–22]. For example, the relay might belong to a heterogeneous network without the same security clearance as the source and the destination nodes. The work in [13, 14] investigated the achievable secrecy rate with an untrusted relay and proved that using nonregenerative untrusted relay can help achieve a higher secrecy rate than simply treating it as an eavesdropper. Subsequently, [15] examined the outage performance in a three-node relay network using the adaptive transmission scheme. Moreover, the use of multi-input multioutput (MIMO) [16, 17], multiple untrusted relays [18, 19], opportunistic relay [20], and two-way relay [21] was investigated for the security of information transmission. The optimal power problem was examined in [21, 22]. However, the aforementioned contributions only focused on secrecy rate and outage performance without paying much attention to the reliability and security tradeoff (RST) and effective secrecy throughput (EST) performance.

This motivates us to explore the secrecy coding aided secure wireless communications to achieve the reliability and security tradeoff, where the reliability is quantified in terms of the connection outage probability, while the security is quantified in terms of the secrecy outage probability [23]. Although the notion of reliability and security tradeoff has been studied in [24, 25], this work only focused on the scenario where the eavesdropper is an external entity. By contrast, in our work we consider the untrusted relay network. That is, the relay, who helps forward the signal to the destination, is also a potential eavesdropper. We adopt the fixed-rate transmission scheme to meet the reliability and security requirements for three different transmission strategies, for example, noncooperative strategy, conventional amplify-and-forward (AF) strategy, and cooperative jamming (CJ) strategy. The main contributions of this work are summarized as follows. First, we characterize three key performance metrics, including the connection and secrecy outage performance, the reliability and security tradeoff, and the effective secrecy throughput. Second, the asymptotic analysis of the EST is conducted to reveal the optimal transmission strategy under different scenarios. Additionally, our numerical results highlight the advantage of the CJ strategy over the noncooperative and AF transmissions in terms of RST. We also find that the EST of the AF strategy is not sensitive to the channel coefficient of the second hop.

2. System Model

As shown in Figure 1, we consider a cooperative network composed of a source S, a nonregenerative untrusted relay R, and a destination D. We assume that all nodes are equipped with a single-antenna and operate in a half-duplex mode. The channel between i and j is quasistatic Rayleigh fading, denoted by $h_{ij}~\mathcal{C}\mathcal{N}(\mathrm{0},{\sigma }_{ij}^{\mathrm{2}})$ , with $i,j\in \left\{S,R,D\right\}$ . Besides, all nodes have the same power budget P, and additive white Gaussian noise (AWGN) with zero mean and covariance $N_{\mathrm{0}}$ encounter at each receiver, for example, R, D. Let ${\gamma }_{ij}=P{\left|h_{ij}\right|}^{\mathrm{2}}/N_{\mathrm{0}}$ denote the instantaneous signal-to-noise ratio (SNR) of the link from i to j and $\lambda =P/N_{\mathrm{0}}$ denote the transmit SNR. Therefore, ${\gamma }_{ij}$ is exponentially distributed with ${\overline{\gamma }}_{ij}=\lambda {\sigma }_{ij}^{\mathrm{2}}$ , that is, ${\gamma }_{ij}~\exp \left({\overline{\gamma }}_{ij}\right)$ .

Figure 1

System model.

In this paper, the constant-rate Wyner coding scheme [3] is employed for confidential message transmission to satisfy the physical layer security requirements. Specifically, there are two parameters, namely, the codeword transmission rate $R_{\mathrm{0}}$ and the confidential information rate $R_s$ . The extra redundancy $R_e=R_{\mathrm{0}}-R_s$ over the above $R_s$ is used to provide protection against eavesdropping. If the channel capacity of the legitimate user drops below $R_{\mathrm{0}}$ , the connection outage event occurs, which means that the destination is unable to recover the source message. When the channel capacity of the eavesdropper becomes higher than $R_e$ , the secrecy outage event occurs, which is regarded as the secrecy outage event.

2.1. Noncooperative Strategy

For the noncooperative strategy, the relay is treated as a pure eavesdropper, and the source does not cooperate with the relay. Information transmission is finished under the source broadcasting information in one time slot. Thus, the received signal at the relay is expressed as

$$\begin{array}{cc}& y_R=h_{\mathrm{S}\mathrm{R}}{x}_s+n_R,\end{array}$$

(1)

where $x_s$ is the transmitted signal emitted from the source with power P, $h_{\mathrm{S}\mathrm{R}}$ is the channel fading coefficient of the source-relay link, and $n_R$ is the AWGN at the relay. Meanwhile, the corresponding received signal at the destination is written as

$$\begin{array}{cc}& y_D=h_{\mathrm{S}\mathrm{D}}{x}_s+n_D,\end{array}$$

(2)

where $h_{\mathrm{S}\mathrm{D}}$ is the channel fading coefficient of the source-destination link and $n_D$ is the AWGN at the destination.

2.2. AF Strategy

When the untrusted AF relay is employed, the complete transmission can be divided into two slots. During the first slot, the source broadcasts information to the relay and the destination. During the second slot, the source keeps silent, and the relay adopts the nonregenerative protocol to process the received signal with a variable amplify coefficient ${\beta }_{\mathrm{1}}=\mathrm{1}/\sqrt{P{\left|h_{\mathrm{S}\mathrm{R}}\right|}^{\mathrm{2}}+N_{\mathrm{0}}}$ . Thus, we can express the received signal at the destination as

$$\begin{array}{cc}& {\mathbf{y}}_D=\left[\begin{array}{c}{h}_{\mathrm{S}\mathrm{D}}\\ \frac{\sqrt{P}{h}_{\mathrm{S}\mathrm{R}}{h}_{\mathrm{R}\mathrm{D}}}{\sqrt{P{\left|h_{\mathrm{S}\mathrm{R}}\right|}^{\mathrm{2}}+N_{\mathrm{0}}}}\end{array}\right]x_s+\left[\begin{array}{c}{n}_{D\mathrm{1}}\\ \frac{\sqrt{P}{h}_{\mathrm{R}\mathrm{D}}{n}_{R\mathrm{1}}}{\sqrt{P{\left|h_{\mathrm{S}\mathrm{R}}\right|}^{\mathrm{2}}+N_{\mathrm{0}}}}+n_{D\mathrm{2}}\end{array}\right],\end{array}$$

(3)

where $h_{\mathrm{R}\mathrm{D}}$ is the channel fading coefficient of the relay-destination link and the subscripts $\mathrm{1}$ and $\mathrm{2}$ represent the first and second transmission slot, respectively. Besides, we assume that $n_{D\mathrm{1}}$ and $n_{D\mathrm{2}}$ are uncorrelated Gaussian noise with variance $N_{\mathrm{0}}$ .

2.3. CJ Strategy

Under CJ strategy, in the first transmission slot the destination ignores the information from the source instead of transmitting a jamming signal. Thus, the received signal at the relay is

$$\begin{array}{cc}& y_R=h_{\mathrm{S}\mathrm{R}}{x}_s+h_{\mathrm{D}\mathrm{R}}{z}_D+n_R,\end{array}$$

(4)

where $z_D$ is the jamming signal sent by the destination with the power P. Similar to the AF strategy, during the second transmission slot, the relay forwards the received signal with a variable amplifying coefficient ${\beta }_{\mathrm{2}}=\mathrm{1}/\sqrt{P{\left|h_{\mathrm{S}\mathrm{R}}\right|}^{\mathrm{2}}+P{\left|h_{\mathrm{R}\mathrm{D}}\right|}^{\mathrm{2}}+N_{\mathrm{0}}}$ . Here, we assume that the channel between the relay and the destination is reciprocal, and the channel is constant within two time slots. Moreover, since the jamming signal is known to the destination, we assume that the jamming signal can be removed by self-interference cancellation technique. Thus, the received signal at the destination can be written as

$$\begin{array}{cc}& y_D=\sqrt{P}{\beta }_{\mathrm{2}}{h}_{\mathrm{R}\mathrm{D}}{h}_{\mathrm{S}\mathrm{R}}{x}_s+\sqrt{P}{\beta }_{\mathrm{2}}{h}_{\mathrm{R}\mathrm{D}}{n}_R+n_D.\end{array}$$

(5)

3. Reliability and Security Tradeoff

In this section, we investigate the reliability and security performance for the three transmission strategies. Then, we calculate the RST to help the legitimate users choose proper transmit parameters to meet the reliability and security requirements.

3.1. Noncooperative Strategy

For noncooperative strategy, the relay does not help the source forward information and is treated as a pure eavesdropper. Therefore, this model, simplified as a traditional wiretap model, can be regarded as a benchmark for comparison purposes. Let $I_D^{\text{Non-C}}$ and $I_R^{\text{Non-C}}$ represent the mutual information between S and D and between S and R, respectively. Then, the COP and SOP can be expressed as

$$\begin{array}{cc}& P_{\mathrm{c}\mathrm{o}}^{\text{Non-C}}=\mathrm{P}\mathrm{r}\left[I_D^{\text{Non-C}}<R_{\mathrm{0}}\right]=\mathrm{1}-e^{-\left({\mathrm{2}}^{R_{\mathrm{0}}}-\mathrm{1}\right)/{\overline{\gamma }}_{\mathrm{S}\mathrm{D}}},\\ P_{\mathrm{s}\mathrm{o}}^{\text{Non-C}}=\mathrm{P}\mathrm{r}\left[I_R^{\text{Non-C}}>R_e\right]=e^{-\left({\mathrm{2}}^{R_{\mathrm{0}}-R_s}-\mathrm{1}\right)/{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}}.\end{array}$$

(6)

Combining the COP and SOP expressions, the RST of the noncooperative strategy can be expressed as

$$\begin{array}{cc}& P_{\mathrm{c}\mathrm{o}}^{\text{Non-C}}=\mathrm{1}-e^{-\left({\mathrm{2}}^{R_s}\left(\mathrm{1}-{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}\ln \left(P_{\mathrm{s}\mathrm{o}}^{\text{Non-C}}\right)\right)-\mathrm{1}\right)/{\overline{\gamma }}_{\mathrm{S}\mathrm{D}}}.\end{array}$$

(7)

Note that (7) characterizes the tradeoff relationship between COP and SOP for the noncooperative strategy.

Remark 1.

According to (7), we can see that increasing the SOP reduces the COP and vice versa. This observation indicates that there exists a tradeoff relationship between reliability and security. As such, the security in the untrusted relay network can be improved at the cost of the reliability degradation.

3.2. AF Strategy

When the untrusted AF relay is employed for cooperative communication, the mutual information $I_D^{\mathrm{A}\mathrm{F}}$ and $I_R^{\mathrm{A}\mathrm{F}}$ can be, respectively, expressed as

$$\begin{array}{cc}& I_D^{\mathrm{A}\mathrm{F}}=\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{l}\mathrm{o}\mathrm{g}}_{\mathrm{2}}\left(\mathrm{1}+{\gamma }_{\mathrm{S}\mathrm{D}}+\frac{{\gamma }_{\mathrm{S}\mathrm{R}}{\gamma }_{\mathrm{R}\mathrm{D}}}{\mathrm{1}+{\gamma }_{\mathrm{S}\mathrm{R}}+{\gamma }_{\mathrm{R}\mathrm{D}}}\right),\\ I_R^{\mathrm{A}\mathrm{F}}=\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{l}\mathrm{o}\mathrm{g}}_{\mathrm{2}}\left(\mathrm{1}+{\gamma }_{\mathrm{S}\mathrm{R}}\right).\end{array}$$

(8)

Then, the COP of the AF strategy can be written as

$$\begin{array}{cc}& P_{\mathrm{c}\mathrm{o}}^{\mathrm{A}\mathrm{F}}=\mathrm{Pr}\left[I_D^{\mathrm{A}\mathrm{F}}<R_{\mathrm{0}}\right]=\mathrm{Pr}\left[{\gamma }_{\mathrm{S}\mathrm{D}}+\frac{{\gamma }_{\mathrm{S}\mathrm{R}}{\gamma }_{\mathrm{R}\mathrm{D}}}{\mathrm{1}+{\gamma }_{\mathrm{S}\mathrm{R}}+{\gamma }_{\mathrm{R}\mathrm{D}}}<t_{\mathrm{1}}\right]\approx \mathrm{Pr}\left[{\gamma }_{\mathrm{S}\mathrm{D}}+\mathrm{m}\mathrm{i}\mathrm{n}\left\{{\gamma }_{\mathrm{S}\mathrm{R}},{\gamma }_{\mathrm{R}\mathrm{D}}\right\}<t_{\mathrm{1}}\right]=\mathrm{1}-e^{-t_{\mathrm{1}}/{\overline{\gamma }}_{\mathrm{S}\mathrm{D}}}-\left(\mathrm{1}+u_{\mathrm{1}}\right)e^{-\left(\mathrm{1}/{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}+\mathrm{1}/{\overline{\gamma }}_{\mathrm{R}\mathrm{D}}\right)t_{\mathrm{1}}},\end{array}$$

(9)

where $t_{\mathrm{1}}={\mathrm{2}}^{\mathrm{2}{R}_{\mathrm{0}}}-\mathrm{1}$ and $u_{\mathrm{1}}=\left({\overline{\gamma }}_{\mathrm{S}\mathrm{D}}{\overline{\gamma }}_{\mathrm{R}\mathrm{D}}+{\overline{\gamma }}_{\mathrm{S}\mathrm{D}}{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}\right)/\left({\overline{\gamma }}_{\mathrm{R}\mathrm{D}}{\overline{\gamma }}_{\mathrm{S}\mathrm{D}}+{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}{\overline{\gamma }}_{\mathrm{S}\mathrm{D}}-{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}{\overline{\gamma }}_{\mathrm{R}\mathrm{D}}\right)$ . We can find from (9) that the COP of the AF will converge to zero for the high SNR. This indicates that, for the high transmit SNR, the reliability can be perfectly guaranteed.

The SOP of the AF strategy can be expressed as

$$\begin{array}{cc}& P_{\mathrm{s}\mathrm{o}}^{\mathrm{A}\mathrm{F}}=\mathrm{Pr}\left[I_R^{\mathrm{A}\mathrm{F}}>R_e\right]=\mathrm{Pr}\left[{\gamma }_{\mathrm{S}\mathrm{R}}>t_{\mathrm{2}}\right]=e^{-t_{\mathrm{2}}/{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}},\end{array}$$

(10)

where $t_{\mathrm{2}}={\mathrm{2}}^{\mathrm{2}\left(R_{\mathrm{0}}-R_s\right)}-\mathrm{1}$ . Similar to the noncooperative strategy, information will be not secure when the relay lies nearby the source. This is not surprising since the mutual information of the relay will approach infinite when ${\overline{\gamma }}_{\mathrm{S}\mathrm{R}}\to \infty$ .

Combining (9) and (10), the RST of the AF strategy can be formulated as

$$\begin{array}{cc}& P_{\mathrm{c}\mathrm{o}}^{\mathrm{A}\mathrm{F}}\approx \mathrm{1}-e^{-\left(t_{\mathrm{3}}-{\mathrm{2}}^{\mathrm{2}{R}_s}{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}\ln \left(P_{\mathrm{s}\mathrm{o}}^{\mathrm{A}\mathrm{F}}\right)\right)/{\overline{\gamma }}_{\mathrm{S}\mathrm{D}}}-\left(\mathrm{1}+u_{\mathrm{1}}\right)e^{-\left(\mathrm{1}/{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}+\mathrm{1}/{\overline{\gamma }}_{\mathrm{R}\mathrm{D}}\right)\left(t_{\mathrm{3}}-{\mathrm{2}}^{\mathrm{2}{R}_s}{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}\ln \left(P_{\mathrm{s}\mathrm{o}}^{\mathrm{A}\mathrm{F}}\right)\right)},\end{array}$$

(11)

where $t_{\mathrm{3}}={\mathrm{2}}^{\mathrm{2}{R}_s}-\mathrm{1}$ . The tradeoff relationship between COP and SOP of the AF strategy is characterized by (11).

Remark 2.

Without the second link of the noncooperative strategy, it is difficult to distinguish the RST performance between the noncooperative strategy and the AF strategy. In other words, given a required SOP, which strategy can achieve a better reliability performance is not determined. Conversely, the security performance which is better is also not estimated between the two strategies with a target COP requirement.

3.3. CJ Strategy

According to the signal model of the CJ strategy mentioned in Section 2, the mutual information $I_D^{\mathrm{C}\mathrm{J}}$ and $I_R^{\mathrm{C}\mathrm{J}}$ can be, respectively, expressed as

$$\begin{array}{cc}& I_D^{\mathrm{C}\mathrm{J}}=\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{l}\mathrm{o}\mathrm{g}}_{\mathrm{2}}\left(\mathrm{1}+\frac{{\gamma }_{\mathrm{S}\mathrm{R}}{\gamma }_{\mathrm{R}\mathrm{D}}}{\mathrm{1}+{\gamma }_{\mathrm{S}\mathrm{R}}+\mathrm{2}{\gamma }_{\mathrm{R}D}}\right),\\ I_R^{\mathrm{C}\mathrm{J}}=\frac{\mathrm{1}}{\mathrm{2}}{\mathrm{l}\mathrm{o}\mathrm{g}}_{\mathrm{2}}\left(\mathrm{1}+\frac{{\gamma }_{\mathrm{S}\mathrm{R}}}{\mathrm{1}+{\gamma }_{\mathrm{R}\mathrm{D}}}\right).\end{array}$$

(12)

Thus, the COP of the CJ strategy can be expressed as

$$\begin{array}{cc}& P_{\mathrm{c}\mathrm{o}}^{\mathrm{C}\mathrm{J}}=\mathrm{Pr}\left[I_D^{\mathrm{C}\mathrm{J}}<R_{\mathrm{0}}\right]=\mathrm{Pr}\left[\frac{{\gamma }_{\mathrm{S}\mathrm{R}}{\gamma }_{\mathrm{R}\mathrm{D}}}{\mathrm{1}+{\gamma }_{\mathrm{S}\mathrm{R}}+\mathrm{2}{\gamma }_{\mathrm{R}\mathrm{D}}}<t_{\mathrm{1}}\right]=\mathrm{1}-\sqrt{\frac{\mathrm{4}{t}_{\mathrm{1}}\left(\mathrm{2}{t}_{\mathrm{1}}+\mathrm{1}\right)}{{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}{\overline{\gamma }}_{\mathrm{R}\mathrm{D}}}}{e}^{-\left(\mathrm{2}/{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}+\mathrm{1}/{\overline{\gamma }}_{\mathrm{R}\mathrm{D}}\right)t_{\mathrm{1}}}{K}_{\mathrm{1}}\left(\sqrt{\frac{\mathrm{4}{t}_{\mathrm{1}}\left(\mathrm{2}{t}_{\mathrm{1}}+\mathrm{1}\right)}{{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}{\overline{\gamma }}_{\mathrm{R}\mathrm{D}}}}\right),\end{array}$$

(13)

where $K_{\mathrm{1}}\left(x\right)$ is the modified Bessel function of the second kind [26].

Then, the SOP of the CJ strategy is formulated as

$$\begin{array}{cc}& P_{\mathrm{s}\mathrm{o}}^{\mathrm{C}\mathrm{J}}=\mathrm{Pr}\left[I_R^{\mathrm{C}\mathrm{J}}>R_e\right]=\mathrm{Pr}\left[\frac{{\gamma }_{\mathrm{S}\mathrm{R}}}{\mathrm{1}+{\gamma }_{\mathrm{R}\mathrm{D}}}>t_{\mathrm{2}}\right]=\frac{{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}}{{\overline{\gamma }}_{\mathrm{R}\mathrm{D}}{t}_{\mathrm{2}}+{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}}{e}^{-t_{\mathrm{2}}/{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}}.\end{array}$$

(14)

In the medial and high SNR regime, (14) can be approximated as

$$\begin{array}{cc}& P_{\mathrm{s}\mathrm{o}}^{\mathrm{C}\mathrm{J}}=\mathrm{Pr}\left[\frac{{\gamma }_{\mathrm{S}\mathrm{R}}}{\mathrm{1}+{\gamma }_{\mathrm{R}\mathrm{D}}}>t_{\mathrm{2}}\right]\approx \mathrm{Pr}\left[\frac{{\left|h_{\mathrm{S}\mathrm{R}}\right|}^{\mathrm{2}}}{{\left|h_{\mathrm{R}\mathrm{D}}\right|}^{\mathrm{2}}}>t_{\mathrm{2}}\right],\end{array}$$

(15)

which is independent of the SNR and will not approach unity even as the transmit power increases. This suggests that, compared with the noncooperative and AF strategies, the CJ strategy processes better security performance in the medial and high SNR regime. This is not surprising since the jamming signal sent by the destination during the first phase makes the untrusted relay difficult to eavesdrop.

Then, combining (13) and (14), the RST of the CJ strategy can be written as

$$\begin{array}{cc}& P_{\mathrm{c}\mathrm{o}}^{\mathrm{C}\mathrm{J}}=\mathrm{1}-\sqrt{\frac{\mathrm{4}\left({\mathrm{2}}^{\mathrm{2}{R}_s+\mathrm{1}}{u}_{\mathrm{3}}-\mathrm{1}\right)\left({\mathrm{2}}^{\mathrm{2}{R}_s}{u}_{\mathrm{3}}-\mathrm{1}\right)}{{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}{\overline{\gamma }}_{\mathrm{R}\mathrm{D}}}}{e}^{-\left(\mathrm{2}/{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}+\mathrm{1}/{\overline{\gamma }}_{\mathrm{R}\mathrm{D}}\right)\left({\mathrm{2}}^{\mathrm{2}{R}_s}{u}_{\mathrm{3}}-\mathrm{1}\right)}{K}_{\mathrm{1}}\left(\sqrt{\frac{\mathrm{4}\left({\mathrm{2}}^{\mathrm{2}{R}_s+\mathrm{1}}{u}_{\mathrm{3}}-\mathrm{1}\right)\left({\mathrm{2}}^{\mathrm{2}{R}_s}{u}_{\mathrm{3}}-\mathrm{1}\right)}{{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}{\overline{\gamma }}_{\mathrm{R}\mathrm{D}}}}\right),\end{array}$$

(16)

where $u_{\mathrm{3}}=\mathrm{1}+{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}{W}_{\mathrm{0}}\left(e^{{\overline{\gamma }}_{\mathrm{R}\mathrm{D}}^{-1}}{\left({\overline{\gamma }}_{\mathrm{R}\mathrm{D}}{P}_{\mathrm{s}\mathrm{o}}^{\mathrm{C}\mathrm{J}}\right)}^{-\mathrm{1}}\right)-{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}/{\overline{\gamma }}_{\mathrm{S}\mathrm{D}}$ and $W_{\mathrm{0}}\left(·\right)$ is the real-valued principal branch of Lambert's W function. Legitimate users can use formula (16) to choose appropriate transmit parameters to balance reliability and security.

Remark 3.

Due to the jamming signal by the destination, the security performance is improved for the CJ strategy. In addition, the CJ strategy is more sensitive about the reliability without the direct link. Thus, the RST of CJ strategy is better than the cooperative and AF strategies which is consistent with the simulation result.

4. Effective Secrecy Throughput and Asymptotic Performance Analysis

Besides the fact that reliability and security need our attention, we are also concerned about the effective secrecy throughput of the transmission strategy, which can achieve both reliable and secure transmission. The EST proposed in [27, 28] is adopted here to measure the overall reliable and secure throughput given by

$$\begin{array}{cc}& \varsigma =R_s\mathrm{Pr}\left\{I_D>R_{\mathrm{0}},I_R<R_{\mathrm{0}}-R_s\right\}.\end{array}$$

(17)

Thus, in this section, the EST is derived to characterize the overall efficiency of the transmission strategy.

4.1. Noncooperative Strategy

Substituting the expression of $I_D^{\text{Non-C}}$ and $I_R^{\text{Non-C}}$ into (17), the EST of the noncooperative strategy can be written as

$$\begin{array}{cc}& {\varsigma }^{\text{Non-C}}=R_s\mathrm{Pr}\left[I_D^{\text{Non-C}}>R_{\mathrm{0}},I_R^{\text{Non-C}}<R_{\mathrm{0}}-R_s\right]=R_s\mathrm{Pr}\left[{\gamma }_{\mathrm{S}\mathrm{D}}>{\mathrm{2}}^{R_{\mathrm{0}}}-\mathrm{1},{\gamma }_{\mathrm{S}\mathrm{R}}<{\mathrm{2}}^{R_{\mathrm{0}}-R_s}-\mathrm{1}\right]=R_s{e}^{-\left({\mathrm{2}}^{R_{\mathrm{0}}}-\mathrm{1}\right)/{\overline{\gamma }}_{\mathrm{S}\mathrm{D}}}\left(\mathrm{1}-e^{-\left({\mathrm{2}}^{R_{\mathrm{0}}-R_s}-\mathrm{1}\right)/{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}}\right).\end{array}$$

(18)

Due to the independence of $I_D^{\text{Non-C}}$ and $I_R^{\text{Non-C}}$ , we can get the expression of ${\varsigma }^{\text{Non-C}}$ directly. From (18), we can see that the EST of the noncooperative strategy directly depends on the COP, the SOP, and the secrecy rate.

4.2. AF Strategy

According to (8) and (17), the EST of the AF strategy can be formulated as

$$\begin{array}{cc}& {\varsigma }^{\mathrm{A}\mathrm{F}}=R_s\mathrm{Pr}\left[I_D^{\mathrm{A}\mathrm{F}}>R_{\mathrm{0}},I_R^{\mathrm{A}\mathrm{F}}<R_{\mathrm{0}}-R_s\right]=R_s\mathrm{Pr}\left[{\gamma }_{\mathrm{S}\mathrm{D}}+\frac{{\gamma }_{\mathrm{S}\mathrm{R}}{\gamma }_{\mathrm{R}\mathrm{D}}}{\mathrm{1}+{\gamma }_{\mathrm{S}\mathrm{R}}+{\gamma }_{\mathrm{R}\mathrm{D}}}>t_{\mathrm{1}},{\gamma }_{\mathrm{S}\mathrm{R}}<t_{\mathrm{2}}\right],\end{array}$$

(19)

and the exact EST of the AF strategy is given by the following proposition because of the relativity of $I_D^{\mathrm{A}\mathrm{F}}$ and $I_R^{\mathrm{A}\mathrm{F}}$ .

Proposition 1.

The EST of the AF strategy can be expressed as

$$\begin{array}{cc}& {\varsigma }^{\mathrm{A}\mathrm{F}}=\frac{R_s}{{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}{\overline{\gamma }}_{\mathrm{R}\mathrm{D}}}{e}^{-t_{\mathrm{1}}/{\overline{\gamma }}_{\mathrm{S}\mathrm{D}}}{\displaystyle \sum }_{n=\mathrm{0}}^{\infty }\hspace{0.17em}{\displaystyle \sum }_{k=n-\mathrm{1}}^{\infty }\hspace{0.17em}{\displaystyle \sum }_{m=n-\mathrm{1}}^k\left(\begin{array}{c}k+\mathrm{1}-n\\ k-m\end{array}\right)\frac{{\left(-\mathrm{1}\right)}^{n+m}}{\left(k+\mathrm{1}\right)n!m!{\overline{\gamma }}_{\mathrm{S}\mathrm{D}}^n}\left[u_{\mathrm{2}}^{-\left(n+m+\mathrm{1}\right)}\gamma \left(n+m+\mathrm{1},u_{\mathrm{2}}{t}_{\mathrm{2}}\right)+u_{\mathrm{2}}^{-\left(n+m+\mathrm{2}\right)}\gamma \left(n+m+\mathrm{2},u_{\mathrm{2}}{t}_{\mathrm{2}}\right)\right],\end{array}$$

(20)

where $u_{\mathrm{2}}=\mathrm{1}/{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}-\mathrm{1}/{\overline{\gamma }}_{\mathrm{S}\mathrm{D}}$ and $\gamma \left(a,x\right)={\int }_{\mathrm{0}}^x{e}^{-t}{t}^{a-\mathrm{1}}dt$ .

Proof.

See Appendix A.

The performance of the AF strategy achieving both reliable and secure transmission is characterized by (20). From (20), we can choose appropriate λ, $R_{\mathrm{0}}$ , and $R_s$ to improve the EST performance.

4.3. CJ Strategy

Substituting the expression of $I_D^{\mathrm{C}\mathrm{J}}$ and $I_R^{\mathrm{C}\mathrm{J}}$ into (17), then the EST of the CJ strategy can be expressed as

$$\begin{array}{cc}& {\varsigma }^{\mathrm{C}\mathrm{J}}=R_s\mathrm{Pr}\left[I_D^{\mathrm{C}\mathrm{J}}>R_{\mathrm{0}},I_R^{\mathrm{C}\mathrm{J}}<R_{\mathrm{0}}-R_s\right]=R_s\mathrm{Pr}\left[\frac{{\gamma }_{\mathrm{S}\mathrm{R}}{\gamma }_{\mathrm{R}\mathrm{D}}}{\mathrm{1}+{\gamma }_{\mathrm{S}\mathrm{R}}+\mathrm{2}{\gamma }_{\mathrm{R}\mathrm{D}}}>t_{\mathrm{1}},\frac{{\gamma }_{\mathrm{S}\mathrm{R}}}{\mathrm{1}+{\gamma }_{\mathrm{R}\mathrm{D}}}<t_{\mathrm{2}}\right],\end{array}$$

(21)

and the exact EST of the CJ strategy is provided in the following proposition.

Proposition 2.

The EST of the CJ strategy can be expressed as

$$\begin{array}{cc}& {\varsigma }^{\mathrm{C}\mathrm{J}}=\frac{R_s{\overline{\gamma }}_{\mathrm{R}\mathrm{D}}{t}_{\mathrm{2}}}{{\overline{\gamma }}_{\mathrm{R}\mathrm{D}}{t}_{\mathrm{2}}+{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}}{e}^{\mathrm{1}/{\overline{\gamma }}_{\mathrm{R}\mathrm{D}}-u_{\mathrm{4}}/{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}-u_{\mathrm{4}}/{\overline{\gamma }}_{\mathrm{R}\mathrm{D}}{t}_{\mathrm{2}}}+\frac{R_s\left(u_{\mathrm{4}}-\mathrm{2}{t}_{\mathrm{1}}\right)}{{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}}{e}^{-t_{\mathrm{1}}/{\overline{\gamma }}_{\mathrm{R}\mathrm{D}}-\mathrm{2}{t}_{\mathrm{1}}/{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}}{\displaystyle \sum }_{n=\mathrm{0}}^{\infty }\frac{{\left(-\mathrm{1}\right)}^n{\left(u_{\mathrm{4}}-\mathrm{2}{t}_{\mathrm{1}}\right)}^n}{n!{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}^n}{E}_{n+\mathrm{2}}\left(\frac{t_{\mathrm{1}}\left(\mathrm{2}{t}_{\mathrm{1}}+\mathrm{1}\right)}{\left(u_{\mathrm{4}}-\mathrm{2}{t}_{\mathrm{1}}\right){\overline{\gamma }}_{\mathrm{R}\mathrm{D}}}\right),\end{array}$$

(22)

where $u_{\mathrm{4}}=\left(\mathrm{2}{t}_{\mathrm{1}}+t_{\mathrm{2}}+t_{\mathrm{1}}{t}_{\mathrm{2}}+\sqrt{{\left(\mathrm{2}{t}_{\mathrm{1}}+t_2+t_{\mathrm{1}}{t}_{\mathrm{2}}\right)}^{\mathrm{2}}-\mathrm{4}{t}_{\mathrm{1}}{t}_{\mathrm{2}}}\right)/\mathrm{2}$ and $E_{n+\mathrm{2}}\left(a\right)={\int }_{\mathrm{0}}^{\mathrm{1}}{x}^n{e}^{-a/x}dx$ .

Proof.

See Appendix B.

We see that the EST of the CJ strategy mainly depends on $R_s$ in the high SNR condition from (21) and (22). This is because the successful transmission approaches a constant at high SNR regime. Thus, increasing the confidential information rate is an effective method to promote the EST performance for the CJ strategy in the high SNR scenario.

4.4. Asymptotic Performance Analysis

Based on the above expressions of the EST for the three strategies, we investigate the asymptotic behavior for each to determine conditions, which allows us to determine the transmission strategy providing best efficiency.

4.4.1. Case of $\lambda =P/N_{\mathrm{0}}\to \infty$

From (18) and (19), we can know that

$$\begin{array}{cc}& \underset{\lambda \to \infty }{\mathrm{l}\mathrm{i}\mathrm{m}}{\varsigma }^{\text{Non-C}}=\mathrm{0},\\ \underset{\lambda \to \infty }{\mathrm{l}\mathrm{i}\mathrm{m}}{\varsigma }^{\mathrm{A}\mathrm{F}}=\mathrm{0}.\end{array}$$

(23)

The EST of the noncooperative and AF strategies will converge to zero as $\lambda \to \infty$ . However, for the CJ strategy, according to (21) we have

$$\begin{array}{cc}& \underset{\lambda \to \infty }{\mathrm{l}\mathrm{i}\mathrm{m}}{\varsigma }^{\mathrm{C}\mathrm{J}}=\frac{R_s}{{\sigma }_{\mathrm{R}\mathrm{D}}^{\mathrm{2}}}{{\displaystyle \int }}_{\mathrm{0}}^{\infty }\left(\mathrm{1}-e^{-t_{\mathrm{2}}y/{\sigma }_{\mathrm{S}\mathrm{R}}^{\mathrm{2}}}\right)e^{-y/{\sigma }_{\mathrm{R}\mathrm{D}}^{\mathrm{2}}}\hspace{0.17em}dy=\frac{R_s{t}_{\mathrm{2}}{\sigma }_{\mathrm{R}\mathrm{D}}^{\mathrm{2}}}{t_{\mathrm{2}}{\sigma }_{\mathrm{R}\mathrm{D}}^{\mathrm{2}}+{\sigma }_{\mathrm{S}\mathrm{R}}^{\mathrm{2}}}.\end{array}$$

(24)

Therefore, the EST of the CJ strategy converges to a constant when $\lambda \to \infty$ .

Remark 4.

It can be inferred from (23) that when transmit power is sufficiently large, noncooperative and AF strategies are obviously not applicable since ${\varsigma }^{\text{Non-C}}$ , ${\varsigma }^{\mathrm{A}\mathrm{F}}\to \mathrm{0}$ . On the other hand, based on (23) and (24), we can conclude that the CJ strategy is better than the noncooperative and AF strategies when transmit power is sufficiently large.

4.4.2. Case of ${\sigma }_{\mathrm{S}\mathrm{R}}^{\mathrm{2}}\to \mathrm{0}$

From (18), we have

$$\begin{array}{cc}& \underset{{\sigma }_{\mathrm{S}\mathrm{R}}^{\mathrm{2}}\to \mathrm{0}}{\mathrm{l}\mathrm{i}\mathrm{m}}{\varsigma }^{\text{Non-C}}=R_s{e}^{-\left({\mathrm{2}}^{R_{\mathrm{0}}}-\mathrm{1}\right)/{\overline{\gamma }}_{\mathrm{S}\mathrm{D}}},\end{array}$$

(25)

and according to (19), the EST of the AF strategy will converge to

$$\begin{array}{cc}& \underset{{\sigma }_{\mathrm{S}\mathrm{R}}^{\mathrm{2}}\to \mathrm{0}}{\mathrm{l}\mathrm{i}\mathrm{m}}{\varsigma }^{\mathrm{A}\mathrm{F}}=R_s{e}^{-t_{\mathrm{1}}/{\overline{\gamma }}_{\mathrm{S}\mathrm{D}}}.\end{array}$$

(26)

However, it is clearly seen that ${\varsigma }^{\mathrm{C}\mathrm{J}}\to \mathrm{0}$ as ${\sigma }_{\mathrm{S}\mathrm{R}}^{\mathrm{2}}\to \mathrm{0}$ based on (21), which shows that the CJ strategy is not suitable when the relay is far away from the source.

4.4.3. Case of ${\sigma }_{\mathrm{S}\mathrm{D}}^{\mathrm{2}}\to \infty$ or $\mathrm{0}$

From (18), it is easy to prove that

$$\begin{array}{cc}& \underset{{\sigma }_{\mathrm{S}\mathrm{D}}^{\mathrm{2}}\to \infty }{\mathrm{l}\mathrm{i}\mathrm{m}}{\varsigma }^{\text{Non-C}}=R_s\left(\mathrm{1}-e^{-\left({\mathrm{2}}^{R_{\mathrm{0}}-R_s}-\mathrm{1}\right)/{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}}\right),\end{array}$$

(27)

and according to (19), we have

$$\begin{array}{cc}& \underset{{\sigma }_{\mathrm{S}\mathrm{D}}^{\mathrm{2}}\to \infty }{\mathrm{l}\mathrm{i}\mathrm{m}}{\varsigma }^{\mathrm{A}\mathrm{F}}=\underset{{\sigma }_{\mathrm{S}\mathrm{D}}^{\mathrm{2}}\to \infty }{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{R_s}{{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}{\overline{\gamma }}_{\mathrm{R}\mathrm{D}}}{{\displaystyle \int }}_{\mathrm{0}}^{t_{\mathrm{2}}}{{\displaystyle \int }}_{\mathrm{0}}^{\infty }{e}^{-x/{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}-y/{\overline{\gamma }}_{\mathrm{R}\mathrm{D}}}\left(\mathrm{1}-\frac{t_{\mathrm{1}}}{{\overline{\gamma }}_{\mathrm{S}\mathrm{D}}}+\frac{xy}{\left(\mathrm{1}+x+y\right){\overline{\gamma }}_{\mathrm{S}\mathrm{D}}}\right)dx\hspace{0.17em}dy=R_s\left(\mathrm{1}-e^{-t_{\mathrm{2}}/{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}}\right).\end{array}$$

(28)

Since ${\varsigma }^{\mathrm{C}\mathrm{J}}$ does not depend on ${\sigma }_{\mathrm{S}\mathrm{D}}^{\mathrm{2}}$ , the EST of the CJ strategy is the same as in (22). Besides, we can know that ${\varsigma }^{\text{Non-C}}$ and ${\varsigma }^{\mathrm{A}\mathrm{F}}\to \mathrm{0}$ as ${\sigma }_{\mathrm{S}\mathrm{D}}^{\mathrm{2}}\to \mathrm{0}$ .

Remark 5.

It is shown that the AF strategy is better than the noncooperative strategy when the link between the source and the destination is strong, while the CJ strategy can achieve a better EST performance than the noncooperative and AF strategies when the link between the source and the destination is weak.

4.4.4. Case of ${\sigma }_{\mathrm{R}\mathrm{D}}^{\mathrm{2}}\to \infty$ or $\mathrm{0}$

Since ${\varsigma }^{\text{Non-C}}$ does not change with ${\sigma }_{\mathrm{R}\mathrm{D}}^{\mathrm{2}}$ , it will be the same as in (18). When ${\sigma }_{\mathrm{R}\mathrm{D}}^{\mathrm{2}}\to \infty$ , it is easy to verify that

$$\begin{array}{cc}& \underset{{\sigma }_{\mathrm{R}\mathrm{D}}^{\mathrm{2}}\to \infty }{\mathrm{l}\mathrm{i}\mathrm{m}}{\varsigma }^{\mathrm{A}\mathrm{F}}=\frac{R_s}{{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}{\overline{\gamma }}_{\mathrm{S}\mathrm{D}}}\left\{{{\displaystyle \int }}_{\mathrm{0}}^{t_{\mathrm{2}}}{{\displaystyle \int }}_{t_{\mathrm{1}}}^{\infty }{e}^{-x/{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}-y/{\overline{\gamma }}_{\mathrm{S}\mathrm{D}}}dx\hspace{0.17em}dy+\underset{{\sigma }_{\mathrm{R}\mathrm{D}}^{\mathrm{2}}\to \infty }{\mathrm{l}\mathrm{i}\mathrm{m}}{{\displaystyle \int }}_{\mathrm{0}}^{t_{\mathrm{2}}}{{\displaystyle \int }}_{t_{\mathrm{1}}-x}^{t_{\mathrm{1}}}\left[\mathrm{1}-\frac{\left(t_{\mathrm{1}}-y\right)\left(\mathrm{1}+x\right)}{{\overline{\gamma }}_{\mathrm{R}\mathrm{D}}\left(x+y-t_{\mathrm{1}}\right)}\right]e^{-x/{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}-y/{\overline{\gamma }}_{\mathrm{S}\mathrm{D}}}\hspace{0.17em}dx\hspace{0.17em}dy\right\}=\frac{R_s{\overline{\gamma }}_{\mathrm{S}\mathrm{D}}}{{\overline{\gamma }}_{\mathrm{S}\mathrm{D}}-{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}}{e}^{-t_{\mathrm{1}}/{\overline{\gamma }}_{\mathrm{S}\mathrm{D}}}\left(\mathrm{1}-e^{-\left(\mathrm{1}/{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}-\mathrm{1}/{\overline{\gamma }}_{\mathrm{S}\mathrm{D}}\right)t_{\mathrm{2}}}\right).\end{array}$$

(29)

For CJ, when ${\sigma }_{\mathrm{R}\mathrm{D}}^{\mathrm{2}}\to \infty$ , we have

$$\begin{array}{cc}& \underset{{\sigma }_{\mathrm{R}\mathrm{D}}^{\mathrm{2}}\to \infty }{\mathrm{l}\mathrm{i}\mathrm{m}}{\varsigma }^{\mathrm{C}\mathrm{J}}=\underset{{\sigma }_{\mathrm{R}\mathrm{D}}^{\mathrm{2}}\to \infty }{\mathrm{l}\mathrm{i}\mathrm{m}}\left\{R_s\left[{{\displaystyle \int }}_{\mathrm{2}{t}_{\mathrm{1}}}^{\infty }\frac{e^{-x/{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}}}{{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}}dx-{{\displaystyle \int }}_{u_{\mathrm{4}}}^{\infty }\left(\frac{x-t_{\mathrm{2}}}{{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}{\overline{\gamma }}_{\mathrm{R}\mathrm{D}}{t}_{\mathrm{2}}}\right)e^{-x/{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}}dx-{{\displaystyle \int }}_{\mathrm{2}{t}_{\mathrm{1}}}^{u_{\mathrm{4}}}\frac{e^{-x/{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}}}{{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}{\overline{\gamma }}_{\mathrm{R}\mathrm{D}}}\left(\frac{t_{\mathrm{1}}+t_{\mathrm{1}}x}{x-\mathrm{2}{t}_{\mathrm{1}}}\right)dx\right]\right\}=R_s{e}^{-\mathrm{2}{t}_{\mathrm{1}}/{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}}.\end{array}$$

(30)

We can see that the EST of the three strategies approaches different constants when ${\sigma }_{\mathrm{R}\mathrm{D}}^{\mathrm{2}}\to \infty$ .

When ${\sigma }_{\mathrm{R}\mathrm{D}}^{\mathrm{2}}\to \mathrm{0}$ , for the AF strategy, the EST will converge to

$$\begin{array}{cc}& \underset{{\sigma }_{\mathrm{R}\mathrm{D}}^{\mathrm{2}}\to \mathrm{0}}{\mathrm{l}\mathrm{i}\mathrm{m}}{\varsigma }^{\mathrm{A}\mathrm{F}}=R_s{e}^{-t_{\mathrm{1}}/{\overline{\gamma }}_{\mathrm{S}\mathrm{D}}}\left(\mathrm{1}-e^{-t_{\mathrm{2}}/{\overline{\gamma }}_{\mathrm{S}\mathrm{R}}}\right).\end{array}$$

(31)

However, it is clearly seen that ${\varsigma }^{\mathrm{C}\mathrm{J}}\to \mathrm{0}$ as ${\sigma }_{\mathrm{R}\mathrm{D}}^{\mathrm{2}}\to \mathrm{0}$ . Thus, the noncooperative and AF strategies outperform the CJ strategy in terms of their EST performance when the second hop is weak.

5. Numerical Results

In this section, we present some numerical simulations to illustrate the accuracy of our analytical results.

Figure 2 plots the COP and SOP as a function of the transmit SNR λ. In this figure, we first observe that the Monte Carlo simulation results precisely match the numerical analysis, which validates our analysis. We also observe that as the transmit power increases, for each strategy the COP is reduced accordingly, whereas the corresponding SOP increases. This demonstrates that the reliability and security tradeoff between COP and SOP exists in the untrusted relay network. Besides, the figure also shows that the CJ strategy strictly performs better than the noncooperative and AF strategies in terms of the SOP. Meanwhile, the COP of the noncooperative and AF strategies outperforms the CJ strategy. This is due to the fact that the jamming signal of the CJ strategy makes the relay eavesdrop difficult, while ignoring the information from the source during the first slot results in the reliability degradation. Therefore, this figure highlights that the security performance of the CJ strategy is upgraded at the cost of the reliability depression.

Figure 2

SOP and COP versus transmit SNR with ${\sigma }_{\mathrm{S}\mathrm{D}}^{\mathrm{2}}={\sigma }_{\mathrm{R}\mathrm{D}}^{\mathrm{2}}=\mathrm{10}\hspace{0.17em}\mathrm{d}\mathrm{B}$ , ${\sigma }_{\mathrm{S}\mathrm{R}}^{\mathrm{2}}=\mathrm{5}\hspace{0.17em}\mathrm{d}\mathrm{B}$ , $R_{\mathrm{0}}=\mathrm{2}$  bits/s, and $R_s=\mathrm{1}$  bit/s.

Figure 3 plots the COP of the three strategies versus the SOP. We first observe that the COP decreases as the SOP increases and vice versa. This observation explicitly demonstrates the tradeoff relationship between reliability and security in the untrusted relay network. Moreover, we can see that the RST of CJ strategy outperforms the noncooperative and AF strategies. Therefore, given a required SOP, CJ strategy can achieve a better reliability than the noncooperative and AF strategies. Conversely, with a target COP requirement, the SOP performance of CJ strategy would be lower than the noncooperative and AF strategies. Furthermore, it is also worth noting that there is an almost linear relationship between the COP and SOP of CJ strategy. Thus, we conclude that CJ strategy is more sensitive to reliability and security than noncooperative and AF strategies.

Figure 3

COP versus SOP with ${\sigma }_{\mathrm{S}\mathrm{D}}^{\mathrm{2}}=\mathrm{10}\hspace{0.17em}\mathrm{d}\mathrm{B}$ , ${\sigma }_{\mathrm{S}\mathrm{R}}^{\mathrm{2}}=\mathrm{5}\hspace{0.17em}\mathrm{d}\mathrm{B}$ , ${\sigma }_{\mathrm{R}\mathrm{D}}^{\mathrm{2}}=\mathrm{20}\hspace{0.17em}\mathrm{d}\mathrm{B}$ , $\lambda =\mathrm{10}\hspace{0.17em}\mathrm{d}\mathrm{B}$ , and $R_s=\mathrm{0.6}$  bits/s.

Figure 4 plots the EST of the three strategies versus the transmit power. It is seen from Figure 4 that there is an optimal transmit power maximizing the EST of the noncooperative and AF strategies. This is because not only the destination benefited from increasing the transmit power but also the untrusted relay benefited from it. Moreover, we observe that the EST of the CJ strategy first increases as transmit power increases and then approaches a constant as transmit power approaches large values. This can be explained by the fact that the jamming signal leads to the enhancement of the security. Therefore, we can conclude that increasing transmit power is an efficient method of enhancing the EST performance of the CJ strategy, while it is not appropriate for noncooperative and AF strategies.

Figure 4

EST versus transmit SNR with ${\sigma }_{\mathrm{S}\mathrm{D}}^{\mathrm{2}}=\mathrm{10}\hspace{0.17em}\mathrm{d}\mathrm{B}$ , ${\sigma }_{\mathrm{S}\mathrm{R}}^{\mathrm{2}}=\mathrm{15}\hspace{0.17em}\mathrm{d}\mathrm{B}$ , ${\sigma }_{\mathrm{R}\mathrm{D}}^{\mathrm{2}}=\mathrm{20}\hspace{0.17em}\mathrm{d}\mathrm{B}$ , $R_{\mathrm{0}}=\mathrm{1.5}$ bits/s, and $R_s=\mathrm{1}$  bit/s.

Figure 5 plots the EST against ${\sigma }_{\mathrm{S}\mathrm{R}}^{\mathrm{2}}$ . As shown in Figure 5, the security is critical to the overall performance when the direct link exists. For the CJ strategy, we observe that a unique value of ${\sigma }_{\mathrm{S}\mathrm{R}}^{\mathrm{2}}$ exists to maximize the EST. When ${\sigma }_{\mathrm{S}\mathrm{R}}^{\mathrm{2}}\to \infty$ , information is not likely to be absolutely secure for the untrusted relay. When ${\sigma }_{\mathrm{S}\mathrm{R}}^{\mathrm{2}}\to \mathrm{0}$ , it is impossible to receive information reliably for the destination without the direct link. Based on this tradeoff, there is an optimal ${\sigma }_{\mathrm{S}\mathrm{R}}^{\mathrm{2}}$ maximizing the EST for the CJ strategy.

Figure 5

EST versus ${\sigma }_{\mathrm{S}\mathrm{R}}^{\mathrm{2}}$ with ${\sigma }_{\mathrm{S}\mathrm{D}}^{\mathrm{2}}=\mathrm{6}\hspace{0.17em}\mathrm{d}\mathrm{B}$ , ${\sigma }_{\mathrm{R}\mathrm{D}}^{\mathrm{2}}=\mathrm{20}\hspace{0.17em}\mathrm{d}\mathrm{B}$ , $\lambda =\mathrm{10}\hspace{0.17em}\mathrm{d}\mathrm{B}$ , $R_{\mathrm{0}}=\mathrm{1.5}$  bits/s, and $R_s=\mathrm{1}$  bit/s.

Figure 6 plots the EST versus ${\sigma }_{\mathrm{S}\mathrm{D}}^{\mathrm{2}}$ . We first observe that the EST of noncooperative and AF strategies increases as ${\sigma }_{\mathrm{S}\mathrm{D}}^{\mathrm{2}}$ increases and approaches a constant when ${\sigma }_{\mathrm{S}\mathrm{D}}^{\mathrm{2}}$ is sufficiently large. When ${\sigma }_{\mathrm{S}\mathrm{D}}^{\mathrm{2}}\to \mathrm{0}$ , the noncooperative strategy cannot establish a reliability transmission, while the AF strategy cannot achieve a security transmission. Therefore, the EST of noncooperative and AF strategies approach zero when the direct link is nonexistent. When ${\sigma }_{\mathrm{S}\mathrm{D}}^{\mathrm{2}}\to \infty$ , the performance of noncooperative and AF strategies is subject to security. Besides, due to two time slots, AF strategy has better performance than noncooperative strategy when ${\sigma }_{\mathrm{S}\mathrm{D}}^{\mathrm{2}}\to \infty$ .

Figure 6

EST versus ${\sigma }_{\mathrm{S}\mathrm{D}}^{\mathrm{2}}$ with ${\sigma }_{\mathrm{S}\mathrm{R}}^{\mathrm{2}}=\mathrm{6}\hspace{0.17em}\mathrm{d}\mathrm{B}$ , ${\sigma }_{\mathrm{R}\mathrm{D}}^{\mathrm{2}}=\mathrm{5}\hspace{0.17em}\mathrm{d}\mathrm{B}$ , $\lambda =\mathrm{5}\hspace{0.17em}\mathrm{d}\mathrm{B}$ , $R_{\mathrm{0}}=\mathrm{1.5}$  bits/s, and $R_s=\mathrm{1}$  bit/s.

Figure 7 plots the EST versus ${\sigma }_{\mathrm{R}\mathrm{D}}^{\mathrm{2}}$ . We first observe that, with the increasing of ${\sigma }_{\mathrm{R}\mathrm{D}}^{\mathrm{2}}$ , the EST performance curve of the AF strategy has little change, which indicates that AF strategy is not sensitive to the second hop. However, comparing with the noncooperative strategy, the second hop is critical to the overall performance for the AF strategy. Moreover, we can also observe that the EST of the CJ strategy increases as ${\sigma }_{\mathrm{R}\mathrm{D}}^{\mathrm{2}}$ increases and approaches a constant when ${\sigma }_{\mathrm{R}\mathrm{D}}^{\mathrm{2}}\to \infty$ . When ${\sigma }_{\mathrm{R}\mathrm{D}}^{\mathrm{2}}$ is sufficiently large, the system performance of the CJ strategy mainly depends on the link between the source and the destination.

Figure 7

EST versus ${\sigma }_{\mathrm{R}\mathrm{D}}^{\mathrm{2}}$ with ${\sigma }_{\mathrm{S}\mathrm{R}}^{\mathrm{2}}=\mathrm{2}\hspace{0.17em}\mathrm{d}\mathrm{B}$ , ${\sigma }_{\mathrm{S}\mathrm{D}}^{\mathrm{2}}=\mathrm{5}\hspace{0.17em}\mathrm{d}\mathrm{B}$ , $\lambda =\mathrm{5}\hspace{0.17em}\mathrm{d}\mathrm{B}$ , $R_{\mathrm{0}}=\mathrm{1.5}$  bits/s, and $R_s=\mathrm{1}$  bit/s.

6. Conclusion

Considering a three-node nonregenerative untrusted relay network, in this paper we investigated the three secrecy transmission strategies, for example, noncooperative strategy, conventional AF strategy, and CJ strategy. We derived the closed-form expressions for the COP, the SOP, the RST, and the EST. Furthermore, the asymptotic analysis of the secrecy throughput was examined to determine the optimal transmission strategy under different scenarios. Results showed that the CJ strategy performs consistently better than the noncooperative and AF strategies in terms of RST performance, demonstrating the tradeoff advantage of the CJ strategy. Moreover, we found that there exists an optimal transmit power maximizing the EST for the noncooperative and AF strategies, and the EST performance of the AF strategy does not change significantly when the quality of the second hop varies.