Security-aware dual-hop communication for amplify-and-forward relay networks

Abstract

The authentication scheme for vehicular ad hoc networks aims to improve the security and integrity of message delivery. The base station manages a large number of vehicular nodes, so the security communications are non-trivial. In this article, we propose an amplify-and-forward strategy for a dual-hop cooperative network in order to improve secure communications for vehicular ad hoc networks. We assume that each vehicular node equipped with a single antenna and derive closed-form expressions for the secure communication rate calculation. Moreover, we propose a cooperative strategy by jointly considering average power scaling and instantaneous power scaling, which are proved to be able to achieve information security. The simulation result shows that the proposed scheme can achieve better performance in scenarios with different signal-to-noise ratio.

Introduction

Due to the nature of the broadcasting communication, wireless communications are susceptible to eavesdropping. The traditional security mechanisms mainly rely on cryptographic protocols. The strategies of the physical layer security exploit the randomness of wireless channels to significantly strengthen the security of wireless communications.^1–4 Several potential benefits of studying secure information from physical layers have been reported by Maurer. ¹ The basic principle of information-theoretic security has been widely used in the strictest secure communication because it guarantees that the messages cannot be easily decoded by a malicious eavesdropper.^1,3–5 Wyner ⁶ introduced a wiretap channel model to evaluate secure transmissions for the physical layer, where Alice transmits confidential data to Bob and Eve eavesdrops on the transmission. Csiszár and Körner ⁷ and Leung-Yan-Cheong and Hellman ⁸ generalized this model to the broadcasting and the Gaussian channels, respectively. Kang and Liu ⁹ studied secure communications over two-user semi-deterministic broadcasting channels. The secrecy capacity is defined as the difference between the main channel capacity (Alice to Bob) and the eavesdroppers channel capacity (Alice to Eve). ⁸ Barros et al.^5,10 generalized the Gaussian wiretap channel model to wireless quasi-static fading channels, while other authors studied secure properties in multiple-input multiple-output (MIMO) systems.^11,12

Motivated by emerging wireless applications, there is a growing interest in exploiting the benefits of relay and cooperative strategies to guarantee secure transmissions.^13,14 Lai et al. ¹³ presented that the secure communication can occur via an untrusted relay node by jamming an eavesdropper. Recently, the physical layer secures protocols based on decode-and-forward (DF) and amplify-and-forward (AF) strategies proposed in previous works^14,15 to employ trusty relay nodes. To maximize the secrecy capacity, several power allocation schemes among the source node and relay nodes were investigated in studies by Dong et al.^2,15 On the other side, in vehicular ad hoc networks (VANETs), vehicular nodes require to transmit data to base station (BS) via relays. In this scenario, we should ensure the security and privacy of dual-hop communication to guarantee the related services for users.^14,16 By leveraging the broadcasting security in VANETs, the on-board units (OBUs) communicate with each other and between the infrastructure points. There are three main communication modes: (1) in vehicle and vehicle (V2V) communication mode, vehicles use OBUs to exchange information within the line free communication range; (2) in vehicle and infrastructure (V2I) communication mode, the RSU (road side unit) in VANET communicates with the vehicle to exchange information; and (3) in the hybrid communication mode (V2V + V2I), the vehicle can communicate with other vehicles and infrastructure points. In wireless communication, security and privacy are important issues in avoiding network threats. Consequently, secure communication can be used in many application scenarios, including not only the self-security of terminals and platforms^17,18 but also the wireless communication security in networked communication. There are many studies by jointly considering the dual-hop secure communication and the technology of VANETs to achieve a secure vehicular network. Waghmode et al. ¹⁹ put forward a group-based V2V communication scheme, using the intra-group symmetric key to complete the V2V communication. By using the encryption operation, it can resist the attack, prevent the vehicle from being threatened and achieve the goals of security and privacy. Horng et al. ²⁰ proposed an identity-based batch verification scheme to ensure anonymity authentication, integrity, privacy and traceability of the message in the communication between V2V and V2I. The effectiveness and practicability of the proposed scheme was proved theoretically. Remyakrishnan and Tripti ²¹ proposed a user authentication security method based on biometrics. Based on the concept of biometric encryption, this security method can ensure the convenience of authentication and the security of the biometric template. This method was committed to improving the security of V2I in VANETs.

In this article, inspired by the previous work in the AF dual-hop relay for secure communications within Wyner’s wire-tap channel, such as Hasna and Alouini ²² and Suraweera et al., ²³ we investigate the secure communications in VANETs in terms of the bit error rate and the outage probability. Specifically, unlike using the fixed channel conditions,^2,15 we consider the impact of time-varying channel fading on the secrecy capacity of an AF relay network with two power constraints, that is, average power scaling (APS) and instantaneous power scaling (IPS).^22,24 These two constraints are employed to scale the output power of the relayed signals in two different ways. ²⁴ Based on an information-theoretic formulation of secure communications over wireless channels, ²⁵ we characterize the secrecy of an AF relay network in terms of the average secrecy rate and the outage probability. The closed-form expression of the secrecy rate is derived for IPS when source-relay channels experience Rayleigh fading without a source-to-eavesdropper (S-E) link. The approximate expressions also are derived both for APS and IPS. Our main contributions in this article are shown as follows.

• A close-form expression of the secrecy rate is derived for IPS with Rayleigh fading without an S-E link. The approximate expressions of average secrecy rate are derived both for APS and IPS.

• The cooperative strategies by combined APS and IPS are proved to achieve better performance via simulations.

We organize the remainder of this article as follows. The system model is described in section ‘System model.’ The ‘Secure communications without S-E link’ section investigates the performance of a dual-hop relay wiretap channel without an S-E link. A dual-hop relay wiretap channel with an S-E link is discussed in the ‘Secure communications with S-E link’ section. The security analysis and performance evaluation are presented in the ‘Simulation results’ section. This article is concluded in the ‘Conclusion’ section.

System model

Figure 1 shows a typical architecture of VANETs that consists of vehicular nodes, roadside installed infrastructure (including RSU) and BSs. The vehicle is equipped with a wireless communication terminal, that is, OBU. The OBU realizes communications through 802.11p controller and exchanges information with vehicles and roadside infrastructures. As a typical roadside installed infrastructure, RSU is usually installed on the roadside, which can communicate with OBU in the communication area.

Figure 1.

A fundamental architecture of VANETs.

According to the aforementioned architecture of VANETs, a simple dual-hop relay system with wire-tap channel model is presented in Figure 2. The source $S$ (BS) transmits confidential information to the destination $D$ (OBU) via the trusty relay $R$ (RSU). A third-party entity acts as an eavesdropper on the relay transmissions. The opposite communication scenario is similar. BS, RSU and Eve are assumed to be equipped with a single antenna.

Figure 2.

A simple dual-hop relay communication system.

In the first transmission phase, $S$ communicates with $R$ . The received signal at the relay is

$$Y_r=\sqrt{E_{\mathrm{sr}}}{h}_{\mathrm{sr}}X+N_r$$

(1)

During the next phase, $R$ transmits a scaled replica of $Y_r$ towards the destination. The $D$ observes the output of a discrete-time fading channel (the main channel). The output can be written as

$$Y_d=\sqrt{E_{\mathrm{rd}}}{h}_{\mathrm{rd}}G{Y}_r+N_d$$

(2)

During the transmission phase, Eve observes the output of an independent discrete-time fading channel (the eavesdropper’s channel) from $R$ and $S$ . The output is given by

$$\{\begin{array}{c}{Y}_{e1}=\sqrt{E_{\mathrm{se}}}{h}_{\mathrm{se}}{X}_r+N_e\hfill \\ \multicolumn{1}{c}{Y_{e2}=\sqrt{E_{\mathrm{re}}}{h}_{\mathrm{re}}G{Y}_r+N_e}\end{array}$$

(3)

In equations (1)–(3), $h_{\mathrm{sr}}$ , $h_{\mathrm{se}}$ , $h_{\mathrm{rd}}$ and $h_{\mathrm{re}}$ are complex Gaussian random variables (RVs) with zero-mean and unit-variance; $N_r$ , $N_d$ and $N_e$ denote zero-mean complex Gaussian noise RVs with unit variance. From the point of view of user location, the average reception power is $E_{\mathrm{ij}}=E/d_{\mathrm{ij}}^{\varphi }$ where $E$ corresponds to the transmit power, $d_{\mathrm{ij}}$ is the distance between node $i$ and node $j$ , and $\varphi$ is the path loss exponent ( $i\in \{s,r\}$ , $j\in \{r,d,e\}$ ). The power scaling factor is $G^2=1/(E_{\mathrm{sr}}+1)$ for APS and $G^2=1/(E_{\mathrm{sr}}|h_{\mathrm{sr}}{|}^2+1)$ for IPS. It is assumed that the channel fading coefficients $h$ and noise $N$ are all independent. We also assume that all nodes have perfect channel state information (CSI).

We utilize maximum ratio combining (MRC) at the receiving sides. Then, for APS, the instantaneous signal to noise ratio (SNR) is

$${\gamma }_d=\frac{E_{\mathrm{sr}}\mu \alpha }{1+E_{\mathrm{sr}}+\alpha }$$

(4a)

$${\gamma }_e=E_{\mathrm{se}}{\left|h_{\mathrm{se}}\right|}^2+\frac{E_{\mathrm{sr}}\mu \beta }{1+E_{\mathrm{sr}}+\beta }$$

(4b)

For IPS, the instantaneous SNR is

$${\gamma }_d=\frac{E_{\mathrm{sr}}\mu +\alpha }{1+E_{\mathrm{sr}}\mu +\alpha }$$

(5a)

$${\gamma }_e=E_{\mathrm{se}}{\left|h_{\mathrm{se}}\right|}^2+\frac{E_{\mathrm{sr}}\mu \beta }{1+E_{\mathrm{sr}}\mu +\beta }$$

(5b)

where ${\gamma }_d$ is the instantaneous SNR of the main channel and ${\gamma }_e$ denotes the instantaneous SNR of the eavesdropper’s channel. Denote $\mu ={\left|h_{\mathrm{sr}}\right|}^2$ , $\alpha =E_{\mathrm{rd}}{\left|h_{\mathrm{rd}}\right|}^2$ and $\beta =E_{\mathrm{re}}{\left|h_{\mathrm{re}}\right|}^2$ . Since the channel fading coefficients are zero-mean complex Gaussian RVs, $\mu$ , $\alpha$ and $\beta$ are exponentially distributed, specifically

$$f(\alpha )=\frac{1}{E_{\mathrm{rd}}}\exp (-\frac{\alpha }{E_{\mathrm{rd}}})$$

(6)

$$f(\beta )=\frac{1}{E_{\mathrm{re}}}\exp (-\frac{\beta }{E_{\mathrm{re}}})$$

(7)

In this article, the source $S$ is assumed to be able to obtain CSI both for the main channel and the eavesdropper’s channel. For example, Eve may not be a secret eavesdropper, but another user. The source $S$ wants to send some confidential data to $D$ . The $S$ does not want Eve to know. Therefore, $S$ can estimate the CSI of the eavesdropper’s channel.^5,10

We summarize all the symbols used in this article in Table 1.

Table 1.

Symbols and notations.

Symbol	Physical meaning
$S$	source (BS)
$R$	relay (RSU)
$D$	relay (OBU)
$h_{\mathrm{sr}}$ , $h_{\mathrm{se}}$ , $h_{\mathrm{rd}}$ , $h_{\mathrm{re}}$	complex Gaussian random variables
$N_r$ , $N_d$ , $N_e$	zero-mean complex Gaussian noise
$E$	transmit power
$d_{\mathrm{ij}}$	distance between node $i$ and node $j$
$\varphi$	path loss exponent
${\gamma }_d$	instantaneous SNR of the main channel
${\gamma }_e$	instantaneous SNR of the eavesdropper’s channel
$K_m(x)$	$m$ -order modified Bessel function of the second kind
$S_{\mu ,\nu }(·)$	Lommel functions
${\overline{R}}_{\mathrm{APS}}$	average secrecy rate of APS
${\overline{R}}_{\mathrm{IPS}}$	average secrecy rate of IPS
$G$	power scaling factor
${\overline{R}}_{\mathrm{IPS}}^{\mathrm{app}}$	average secrecy rate under high S-R SNRs for AF relay with IPS
${\overline{R}}_{\mathrm{APS}}^{\mathrm{app}}$	average secrecy rate under high S-R SNRs for AF relay with APS

BS: base station; RSU; road side unit; OBU: on-board units; SNR; signal to noise ratio; AF: amplify-and-forward; APS: average power scaling; IPS: instantaneous power scaling.

Secure communications without S-E link

This section characterizes the secrecy rate of a dual-hop relay wiretap channel without S-E link in terms of average secure communication rates. For example, Eve and $D$ are located in a same cluster that is far away from $S$ . First, we compute the secrecy rate for one realization of $h_{\mathrm{sr}}$ . Then, the average secrecy rate is derived. The approximate closed-form expressions are derived by assuming sufficiently high SNR for the S-R links, that is, $E_{\mathrm{sr}}>\max (E_{\mathrm{rd}},E_{\mathrm{re}})$ .

Start by deriving the secrecy rate for one realization of fading. Recalling the results in Bloch et al. ⁵ for the Rayleigh fading wiretap channel, the secrecy rate for one realization can be written as

$$R_s({\gamma }_d,{\gamma }_e)=\{\begin{array}{cc}\ln (1+{\gamma }_d)-\ln (1+{\gamma }_e),\hfill & \mathrm{if}{\gamma }_d>{\gamma }_e\hfill \\ \multicolumn{1}{c}{0,}& \mathrm{if}{\gamma }_d\le {\gamma }_e\hfill \end{array}$$

(8)

where $\ln (1+{\gamma }_d)$ is the rate of main channel and $\ln (1+{\gamma }_e)$ denotes the rate of eavesdropper’s channel. From equation (8), it follows that secrecy rate is positive when ${\gamma }_d>{\gamma }_e$ . In other words, a non-zero secrecy rate exists only when $\alpha >\beta$ .

In order to compute average secrecy rates, we need the following two integrals given in Gradshteyn and Ryzhik ²⁶

$$\begin{array}{cc}\multicolumn{1}{c}{\underset{0}{\overset{\infty }{\int }}v{e}^{-\mathrm{vx}}\log (1+\mathrm{ux})\mathrm{dx}}& =\exp \left(\frac{v}{u}\right)E_1\left(\frac{v}{u}\right)\hfill \\ \multicolumn{1}{c}{}& =F_e\left(\frac{v}{u}\right)\hfill \end{array}$$

(9a)

$$\underset{0}{\overset{\infty }{\int }}{x}^{m-1}\exp (-\mathrm{mx}-\frac{\nu }{x})\mathrm{dx}=2{\left(\frac{v}{m}\right)}^{\frac{m}{2}}{K}_m(2\sqrt{\mathrm{mv}})$$

(9b)

where $E_1(x)={\int }_1^{\infty }\frac{e^{-\mathrm{xt}}}{t}\mathrm{dt}$ is the exponential-integral function and $K_m\left(x\right)$ is the $m$ -order modified Bessel function of the second kind.

First, we derive the secrecy rate for the fixed realization of $h_{\mathrm{sr}}$ , which is the function of $h_{\mathrm{sr}}$ . The $\alpha$ and $\beta$ are independent and exponentially distributed. The secrecy rate of APS is expressed as

$$\begin{array}{cc}\multicolumn{1}{c}{{\overline{R}}_{\mathrm{APS}}\left(\mu \right)}& =\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}{R}_s({\gamma }_d,{\gamma }_e)f\left(\alpha \right)f\left(\beta \right)d\alpha d\beta \hfill \\ \multicolumn{1}{c}{}& =\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\alpha }{\int }}\log (1+\frac{E_{\mathrm{sr}}\alpha \mu }{1+E_{\mathrm{sr}}+\alpha })f\left(\alpha \right)f\left(\beta \right)d\alpha d\beta \hfill \\ \multicolumn{1}{c}{}& -\underset{0}{\overset{\infty }{\int }}\underset{\beta }{\overset{\infty }{\int }}\log (1+\frac{E_{\mathrm{sr}}\beta \mu }{1+E_{\mathrm{sr}}+\beta })f\left(\alpha \right)f\left(\beta \right)d\alpha d\beta \hfill \\ \multicolumn{1}{c}{}& =[F_e\left(\frac{1}{{\phi }_1{E}_{\mathrm{rd}}}\right)-F_e\left(\frac{1}{{\phi }_1{E}_{\mathrm{re}}{P}_0}\right)]\hfill \\ \multicolumn{1}{c}{}& -[F_e\left(\frac{\phi }{E_{\mathrm{rd}}}\right)-F_e\left(\frac{\phi }{E_{\mathrm{re}}{P}_0}\right)]\hfill \end{array}$$

(10)

where ${\phi }_1=(1+E_{\mathrm{sr}}\mu )/(1+E_{\mathrm{sr}})$ and $\phi =1+E_{\mathrm{sr}}$ . With similar calculations, the secrecy rate of IPS is

$$\begin{array}{cc}\multicolumn{1}{c}{{\overline{R}}_{\mathrm{IPS}}\left(\mu \right)=}& [F_e\left(\frac{1}{E_{\mathrm{rd}}}\right)-F_e\left(\frac{1}{E_{\mathrm{re}}{P}_0}\right)]\hfill \\ \multicolumn{1}{c}{}& -[F_e\left(\frac{1+E_{\mathrm{sr}}\mu }{E_{\mathrm{rd}}}\right)-F_e\left(\frac{1+E_{\mathrm{sr}}\mu }{E_{\mathrm{re}}{P}_0}\right)]\hfill \end{array}$$

(11)

Subsequently, by using equations (10) and (11), we can obtain the final average rate by taking expectations with respect to the S-R channel

$${\overline{R}}_{\mathrm{APS}}=\underset{0}{\overset{\infty }{\int }}{\overline{R}}_{\mathrm{APS}}(\mu )f(\mu )d\mu$$

(12a)

$${\overline{R}}_{\mathrm{IPS}}=\underset{0}{\overset{\infty }{\int }}{\overline{R}}_{\mathrm{IPS}}(\mu )f(\mu )d\mu$$

(12b)

The closed-form solution of IPS can be expressed as

$$\begin{array}{cc}\multicolumn{1}{c}{{\overline{R}}_{\mathrm{IPS}}}& =\frac{E_{\mathrm{sr}}}{E_{\mathrm{sr}}-E_{\mathrm{rd}}}{F}_e\left(\frac{1}{E_{\mathrm{rd}}}\right)\hfill \\ \multicolumn{1}{c}{}& -\frac{E_{\mathrm{sr}}}{E_{\mathrm{sr}}-E_{\mathrm{re}}{P}_0}{F}_e\left(\frac{1}{E_{\mathrm{re}}{P}_0}\right)\hfill \\ \multicolumn{1}{c}{}& +\frac{E_{\mathrm{sr}}(E_{\mathrm{re}}{P}_0-E_{\mathrm{rd}})}{(E_{\mathrm{sr}}-E_{\mathrm{rd}})(E_{\mathrm{sr}}-E_{\mathrm{re}}{P}_0)}{F}_e\left(\frac{1}{E_{\mathrm{sr}}}\right)\hfill \end{array}$$

(13)

when $E_{\mathrm{sr}}\ne E_{\mathrm{rd}}\ne E_{\mathrm{re}}{P}_0$ . After integration by parts, similar results can be derived when $E_{\mathrm{sr}}=E_{\mathrm{rd}}$ or $E_{\mathrm{sr}}=E_{\mathrm{re}}{P}_0$ .

However, we are unaware of closed-form analytical solutions to the integrals (equation (12b)), unless some assumptions are imposed on the SNRs. Relatively high SNR for the S-R links, that is, $E_{\mathrm{sr}}>\max (E_{\mathrm{rd}},E_{\mathrm{re}})$ , is assumed. This assumption has been used to analyse the BER (bit error rate) performance of AF relay in previous works.^16,27 For IPS, under this assumption, the instantaneous SNR is^16,27

$${\gamma }_d=E_{\mathrm{rd}}{\left|h_{\mathrm{rd}}\right|}^2$$

(14a)

$${\gamma }_e=E_{\mathrm{re}}{\left|h_{\mathrm{re}}\right|}^2$$

(14b)

For APS, we express the instantaneous SNR as [14][15]

$${\gamma }_d=E_{\mathrm{rd}}{\left|h_{\mathrm{sr}}\right|}^2{\left|h_{\mathrm{rd}}\right|}^2$$

(15a)

$${\gamma }_e=E_{\mathrm{re}}{\left|h_{\mathrm{sr}}\right|}^2{\left|h_{\mathrm{re}}\right|}^2$$

(15b)

Proof of equation (13): we rewrite $F_e\left(x\right)$ as

$$F_e\left(x\right)=\underset{0}{\overset{\infty }{\int }}\frac{\exp (-\mathrm{tx})}{t+1}\mathrm{dt}$$

(16)

Therefore,

$$\begin{array}{cc}\multicolumn{1}{c}{{\overline{F}}_e\left(\lambda \right)}& =\underset{0}{\overset{\infty }{\int }}{F}_e\left(\frac{1}{\lambda x}\right)f\left(x\right)\mathrm{dx}\hfill \\ \multicolumn{1}{c}{}& =\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}\exp (-x-\frac{t}{\lambda x})\frac{1}{t+1}\mathrm{dxdt}\hfill \end{array}$$

(17)

Using the the integrals given in equation (17), equation (17) is given by

$${\overline{F}}_e\left(\lambda \right)=\underset{0}{\overset{\infty }{\int }}\frac{2}{t+1}{\left(\frac{t}{\lambda 1}\right)}^{1/2}{K}_1\left(2\sqrt{\frac{t}{\lambda }}\right)\mathrm{dt}$$

(18)

The integral can be evaluated by using equation [17]

$$\begin{array}{c}\underset{0}{\overset{\infty }{\int }}{x}^{1+\nu }{(x^2+a^2)}^{\mu }{K}_v(\mathrm{bx})\mathrm{dx}\hfill \\ {=2}^v\Gamma (1+v)a^{\nu +\mu +1}{b}^{-1-\mu }{S}_{\mu -\nu ,\mu +\nu +1}(\mathrm{ab})\hfill \end{array}$$

(19)

After some manipulations, equation (18) can be reexpressed as

$$\begin{array}{cc}\multicolumn{1}{c}{{\overline{F}}_e\left(\lambda \right)}& =\frac{4}{2\Gamma (1)}\underset{0}{\overset{\infty }{\int }}\frac{x^2{K}_1(x)}{x^2+a^2}\mathrm{dx}\hfill \\ \multicolumn{1}{c}{}& =4a{S}_{-2,1}(a)\hfill \end{array}$$

(20)

where $a=2\sqrt{1/\lambda }$ .

By taking expectation with respect to $\mu$ , we evaluate equation (10b) as

$$\begin{array}{cc}\multicolumn{1}{c}{\underset{0}{\overset{\infty }{\int }}{F}_e(\alpha +\beta \mu )d\mu }& =\underset{0}{\overset{\infty }{\int }}\frac{e^{-\alpha t}}{(t+1)(t\beta +1)}\hfill \\ \multicolumn{1}{c}{}& =\frac{F_e\left(\frac{\alpha }{\beta }\right)-F_e(\alpha )}{\beta -1}\hfill \end{array}$$

(21)

If $\beta$ is equal to 1, we evaluate equation (10b) as

$$\underset{0}{\overset{\infty }{\int }}{F}_e(\alpha +\mu )d\mu =1-\alpha F_e(\alpha )$$

(22)

The average secrecy rate under high S-R SNRs for AF relay can be approximately given by

$${\overline{R}}_{\mathrm{IPS}}^{\mathrm{app}}=F_e\left(\frac{1}{E_{\mathrm{rd}}}\right)-F_e\left(\frac{1}{E_{\mathrm{re}}{P}_0}\right)$$

(23)

$$\begin{array}{cc}\multicolumn{1}{c}{{\overline{R}}_{\mathrm{APS}}^{\mathrm{app}}}& =\underset{0}{\overset{\infty }{\int }}[F_e\left(\frac{1}{E_{\mathrm{rd}}\mu }\right)-F_e\left(\frac{1}{E_{\mathrm{re}}{P}_0\mu }\right)]f(\mu )d\mu \hfill \\ \multicolumn{1}{c}{}& =4\left(\sqrt{a}\right)S_{-2,1}\left(\sqrt{a}\right)-4\left(\sqrt{b}\right)S_{-2,1}\left(\sqrt{b}\right)\hfill \end{array}$$

(24)

where $a=4/E_{\mathrm{rd}}$ , $b=4/E_{\mathrm{re}}{P}_0$ , and $S_{\mu ,\nu }(·)$ is the Lommel functions. ²⁵

For IPS, we can prove that the secrecy rate improves when the value of $E_{\mathrm{sr}}$ increases. So, equation (14) is the maximum value of the average secrecy rate, that is, ${\overline{R}}_{\mathrm{IPS}}$ . For APS, simulation results also show that the higher the $E_{\mathrm{sr}}$ , the larger the average secrecy rate ${\overline{R}}_{\mathrm{APS}}$ .

Proof of equation (14) is the maximum: for the fixed realization of $\mu$ , the secrecy rate of IPS is rewritten as

$$\begin{array}{cc}\multicolumn{1}{c}{{\overline{R}}_{\mathrm{IPS}}(E_{\mathrm{sr}})}& =[F_e\left(\frac{1}{E_{\mathrm{rd}}}\right)-F_e\left(\frac{1}{E_{\mathrm{re}}{P}_0}\right)]\hfill \\ \multicolumn{1}{c}{}& -[F_e\left(\frac{1+E_{\mathrm{sr}}\mu }{E_{\mathrm{rd}}}\right)-F_e\left(\frac{1+E_{\mathrm{sr}}\mu }{E_{\mathrm{re}}{P}_0}\right)]\hfill \\ \multicolumn{1}{c}{}& =R-R(E_{\mathrm{sr}})\hfill \end{array}$$

(25)

where $F_e\left(x\right)={\int }_0^{\infty }\frac{e^{-\mathrm{xt}}}{t+1}\mathrm{dt}$ . It can be shown that

$$F{\prime }_e(x)=-\underset{0}{\overset{\infty }{\int }}\frac{t{e}^{-\mathrm{xt}}}{t+1}\mathrm{dt}<0$$

(26a)

$${F_e}^{\prime \prime }(x)=\underset{0}{\overset{\infty }{\int }}\frac{t^2{e}^{-\mathrm{xt}}}{t+1}\mathrm{dt}>0$$

(26b)

So, function ${F_e}^{\prime }(x)$ is the increasing function about SNR $x$ .

We define $\lambda =E_{\mathrm{re}}{P}_0=\frac{E_{\mathrm{rd}}{E}_{\mathrm{re}}}{E_{\mathrm{re}}+E_{\mathrm{rd}}}$ $(\lambda <E_{\mathrm{rd}})$ and have

$$R\prime (E_{\mathrm{sr}})=\mu [\frac{1}{E_{\mathrm{rd}}}{F\prime }_e\left(\frac{1+E_{\mathrm{sr}}\mu }{E_{\mathrm{rd}}}\right)-\frac{1}{\lambda }{F\prime }_e\left(\frac{1+E_{\mathrm{sr}}\mu }{\lambda }\right)]$$

(27)

Since $F{\prime }_e\left(x\right)$ is the increasing function about $x$ , it is obviously

$$\frac{1}{E_{\mathrm{rd}}}F{\prime }_e\left(\frac{1+E_{\mathrm{sr}}\mu }{E_{\mathrm{rd}}}\right)<\frac{1}{\lambda }F{\prime }_e\left(\frac{1+E_{\mathrm{sr}}\mu }{\lambda }\right)$$

(28)

Then $R\prime (E_{\mathrm{sr}})<0$ . Therefore, $R(E_{\mathrm{sr}})$ is the decreasing function about SNR $E_{\mathrm{sr}}$ . Conversely, ${\overline{R}}_{\mathrm{IPS}}(E_{\mathrm{sr}})$ is the increasing function about the SNR of $E_{\mathrm{sr}}$ , and the maximum secrecy rate is denoted by equation (14).

Secure communications with S-E link

This section characterizes the secrecy rate of a dual-hop relay wiretap channel with S-E link in terms of average secure communication rates. Apparently, the S-E link deteriorate the performance of the dual-hop relay system. Because the closed-form analytical results for AF relay with S-E link is difficult, we also assume relatively high SNR for the S-R links, that is, $E_{\mathrm{sr}}>\max (E_{\mathrm{rd}},E_{\mathrm{re}})$ . So, the approximate results are shown in this section.

Under this assumption, for IPS, the instantaneous SNR is

$${\gamma }_d=E_{\mathrm{rd}}{\left|h_{\mathrm{rd}}\right|}^2=\alpha$$

(29a)

$${\gamma }_e=E_{\mathrm{se}}{\left|h_{\mathrm{se}}\right|}^2+E_{\mathrm{re}}{\left|h_{\mathrm{re}}\right|}^2=\nu +\beta$$

(29b)

For APS, the instantaneous SNR is

$${\gamma }_d=E_{\mathrm{rd}}{\left|h_{\mathrm{sr}}\right|}^2{\left|h_{\mathrm{rd}}\right|}^2=\mu \alpha$$

(30a)

$${\gamma }_e=E_{\mathrm{se}}{\left|h_{\mathrm{se}}\right|}^2+E_{\mathrm{re}}{\left|h_{\mathrm{sr}}\right|}^2{\left|h_{\mathrm{re}}\right|}^2=\nu +\mu \beta$$

(30b)

where $\nu$ is exponentially distributed. For IPS, the probability density function (PDF) of the instantaneous SNR ${\gamma }_e$ (equation (29b)) is

$$f({\gamma }_e)=\{\begin{array}{c}\frac{1}{E_{\mathrm{re}}-E_{\mathrm{se}}}(e^{-\frac{{\gamma }_e}{E_{\mathrm{re}}}}-e^{-\frac{{\gamma }_e}{E_{\mathrm{se}}}}),E_{\mathrm{re}}\ne E_{\mathrm{se}}\hfill \\ \frac{{\gamma }_e}{E_{\mathrm{re}}^2}{e}^{-\frac{{\gamma }_e}{E_{\mathrm{re}}}},E_{\mathrm{re}}=E_{\mathrm{se}}\hfill \end{array}$$

(31)

Similar to section of Secure communications without S-E link, $E_{\mathrm{re}}\ne E_{\mathrm{se}}$ , the secrecy rate averaged over all the realizations can be calculated by

$$\begin{array}{cc}\multicolumn{1}{c}{{\overline{R}}_{\mathrm{IPS}}^{\mathrm{app}}}& =\underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}{R}_s({\gamma }_d,{\gamma }_e)f({\gamma }_d)f({\gamma }_e)d{\gamma }_{d}d{\gamma }_e\hfill \\ \multicolumn{1}{c}{}& =\frac{E_{\mathrm{re}}}{E_{\mathrm{re}}-E_{\mathrm{se}}}[F_e\left(\frac{1}{E_{\mathrm{rd}}}\right)-F_e\left(\frac{E_{\mathrm{rd}}+E_{\mathrm{re}}}{E_{\mathrm{rd}}{E}_{\mathrm{re}}}\right)]\hfill \\ \multicolumn{1}{c}{}& -\frac{E_{\mathrm{se}}}{E_{\mathrm{re}}-E_{\mathrm{se}}}[F_e\left(\frac{1}{E_{\mathrm{rd}}}\right)-F_e\left(\frac{E_{\mathrm{rd}}+E_{\mathrm{se}}}{E_{\mathrm{rd}}{E}_{\mathrm{se}}}\right)]\hfill \end{array}$$

(32)

when $E_{\mathrm{re}}=E_{\mathrm{se}}$ , the average secrecy rate is

$$\begin{array}{cc}\multicolumn{1}{c}{{\overline{R}}_{\mathrm{IPS}}^{\mathrm{app}}}& =F_e\left(\frac{1}{E_{\mathrm{rd}}}\right)-\frac{E_{\mathrm{rd}}}{E_{\mathrm{rd}}+E_{\mathrm{re}}}\hfill \\ \multicolumn{1}{c}{}& -\frac{E_{\mathrm{re}}-1}{E_{\mathrm{re}}}{F}_e\left(\frac{E_{\mathrm{re}}+E_{\mathrm{rd}}}{E_{\mathrm{re}}{E}_{\mathrm{rd}}}\right)\hfill \end{array}$$

(33)

In order to obtain the average secrecy rate of APS, we first take integrations over (equation (35)) fixed channel realization $\mu$ . When $\mu \ne E_{\mathrm{se}}/E_{\mathrm{re}}$ , the secrecy rate of APS conditioning $\mu$ is given by

$$\begin{array}{cc}\multicolumn{1}{c}{{\overline{R}}_{\mathrm{APS}}^{\mathrm{app}}(\mu )}& =F_e\left(\frac{1}{\mu E_{\mathrm{rd}}}\right)\hfill \\ \multicolumn{1}{c}{}& -\frac{\mu E_{\mathrm{re}}}{\mu E_{\mathrm{re}}-E_{\mathrm{se}}}{F}_e\left(\frac{E_{\mathrm{rd}}+E_{\mathrm{re}}}{\mu E_{\mathrm{rd}}{E}_{\mathrm{re}}}\right)\hfill \\ \multicolumn{1}{c}{}& +\frac{E_{\mathrm{se}}}{\mu E_{\mathrm{re}}-E_{\mathrm{se}}}{F}_e\left(\frac{\mu E_{\mathrm{rd}}+E_{\mathrm{se}}}{\mu E_{\mathrm{rd}}{E}_{\mathrm{se}}}\right)\hfill \end{array}$$

(34)

when $\mu =E_{\mathrm{se}}/E_{\mathrm{re}}$ , it is

$$\begin{array}{cc}\multicolumn{1}{c}{{\overline{R}}_{\mathrm{APS}}^{\mathrm{app}}(\mu )}& =F_e\left(\frac{1}{\mu E_{\mathrm{rd}}}\right)-\frac{E_{\mathrm{rd}}}{E_{\mathrm{rd}}+E_{\mathrm{re}}}\hfill \\ \multicolumn{1}{c}{}& -\frac{E_{\mathrm{se}}-1}{E_{\mathrm{se}}}{F}_e\left(\frac{E_{\mathrm{re}}+E_{\mathrm{rd}}}{E_{\mathrm{se}}{E}_{\mathrm{rd}}}\right)\hfill \end{array}$$

(35)

The average secrecy rate of APS can be expressed as ${\overline{R}}_{\mathrm{APS}}^{\mathrm{app}}={\int }_0^{\infty }{\overline{R}}_{\mathrm{APS}}^{\mathrm{app}}(\mu )f(\mu )d\mu$ and computed numerically.

Simulation results

In this section, we evaluate the average secure communication rates of dual-hop AF relay under APS and IPS constraints.

Performance of a dual-hop relay wiretap channel without an S-E link

Figure 3 shows the average secrecy rate versus $E_{\mathrm{rd}}$ for different values of $E_{\mathrm{sr}}$ in Rayleigh fading. In contrast with the Gaussian wiretap channel, we observe that the average secrecy rate is positive and can be improved when the values of $E_{\mathrm{sr}}$ increase. For example, when $E_{\mathrm{sr}}$ is less than $E_{\mathrm{rd}}$ , $E_{\mathrm{sr}}=E_{\mathrm{rd}}-10\mathrm{dB}$ , the secrecy rates of AF-APS and AF-IPS are lower. Therefore, we can conclude that the higher transmitting power between the source and relay can improve the average secrecy rate. In the moderate values of $E_{\mathrm{sr}}$ as shown in Figure 3, the secrecy rates of AF-APS outperform the secrecy rates of AF-IPS. However, when the transmitting power between the source and relay is sufficiently high, for instance, $E_{\mathrm{sr}}=E_{\mathrm{rd}}+20\mathrm{dB}$ , the secrecy rates of AF-IPS are higher to the secrecy rates of AF-APS.

Figure 3.

The average secrecy rate versus $E_{\mathrm{rd}}$ for selected values of $E_{\mathrm{sr}}$ in Rayleigh/Rayleigh fading when $E_{\mathrm{rd}}=E_{\mathrm{re}}$ .

Figure 4 describes the average secrecy rate versus $E_{\mathrm{rd}}$ . We assume that the SNR for the S-R links is sufficiently high in Rayleigh/Rayleigh fading. This assumption is widely adopted to compute the BER performance of dual-hop AF relay network in the study by Wu et al. ¹⁶ From the approximate results, we also observe the similar trends as shown in Figure 3. The approximate results of IPS match well only when the SNR gap is quite high, for example, $E_{\mathrm{sr}}-E_{\mathrm{rd}}=30\mathrm{dB}$ . However, the approximate results of APS match well when the SNR gap is relative lower, for example, $E_{\mathrm{sr}}-E_{\mathrm{rd}}=10\mathrm{dB}$ . We can consider that the approximate results of APS are more robust.

Figure 4.

The approximate average secrecy rate versus $E_{\mathrm{rd}}$ in Rayleigh/Rayleigh fading.

Performance of a dual-hop relay wiretap channel with an S-E link

We show the approximate average secure communication rates when Eve can observe the output from $R$ and $S$ by assuming the sufficiently high SNR of the S-R links in this subsection. In general, the secrecy rate has the similar trends as shown in Figures 3 and 4. Furthermore, it is obvious that the secrecy rate is deteriorated when S-E link exits. Figure 5 shows the performance with S-E link versus the transmitting power between the relay and the destination assuming sufficiently high SNR for the S-R link in Rayleigh/Rayleigh fading. We also plot the performance curves without S-E link for comparison. It is obvious that the average secrecy rate and outage probability become worse when the value of $E_{\mathrm{se}}$ increases. Moreover, in the scenario of the high SNR for the S-R link, IPS can provide a better performance than APS with S-E link.

Figure 5.

The approximate average secrecy rate with S-E link versus $E_{\mathrm{rd}}$ in Rayleigh/Rayleigh fading.

Conclusion

In order to solve the key issue of authentication schemes for dual-hop communications, an AF relay for secure communication scheme is proposed in this article. The closed-formed expression of the secrecy rate is derived for IPS when wireless channels experience Rayleigh fading without an S-E link. The approximate expressions of average secrecy rate are also shown for APS and IPS. The cooperative strategies combined APS with IPS are proved to achieve information security. Due to the S-E link, the performance of the AF dual-hop relay becomes worse. Simulation results show that the performance of IPS is better than that of APS if the SNR of the S-R link is higher than that of R-D/E links. However, APS outperforms IPS in the moderate SNR regime for the S-R link.