Secure Beamforming via Amplify-and-Forward Relays in Machine-to-Machine Communications

Abstract

We consider the collaborative use of amplify-and-forward relays to form a beamforming system and provide physical layer security for a wireless machine-to-machine (M2M) communication network. We investigate two objectives: (i) the achievable secrecy rate maximization subject to the relay power constraint and (ii) the relay transmit power minimization under a secrecy rate constraint. For the first objective, we propose a secrecy rate maximization (SRM) beamforming scheme. The secrecy rate maximization problem can be formed into a two-level optimization problem and we solve it using semidefinite relaxation (SDR) techniques. To reduce the complexity of the SRM beamforming scheme, a virtual eavesdropper-based SRM (VE-SRM) beamforming scheme is proposed, in which we hypothesize a virtual eavesdropper instead of all eavesdroppers and maximize the secrecy rate according to the virtual eavesdropper. In addition, for the second objective, we design a relay power minimization (RPM) beamforming scheme, in which an iterative algorithm combining the SDR technology and the gradient-based method is devised by studying the convexity of the RPM problem. By relaxing the constraints of the RPM beamforming scheme, we propose a virtual eavesdropper-based RPM (VERPM) beamforming scheme, which reduces the multivariate RPM problem to a problem of a single variable, and thus an analytical solution is obtained. Our proposed beamforming designs can work well even if the number of eavesdroppers is larger than that of relays, while the existing schemes, for example, the null-space beamforming schemes, cannot work under this condition. Simulation results are presented to demonstrate the significance of performance improvements with the SRM and RPM beamforming schemes. It is also shown that the virtual eavesdropper approaches significantly reduce the complexity with acceptable performance degradation.

1. Introduction

The demands of wireless personal communications drive the development of abundant advanced wireless technologies and systems such as the cognitive radio network (CRN) and Third Generation Partnership Project (3GPP) Long Term Evolution-Advanced (LTE-Advanced) [1]. In addition to human-to-human (H2H) communications, an emerging technology empowering full mechanical automation (e.g., the Internet of Things and the smart grid) that may change our living styles is vigorously being developed. Such communications among machine-type devices are known as M2M communications [2–4]. Due to the use of machine-type devices and the broadcast nature of wireless media, privacy and security issues play an important role in wireless M2M networks. Although security is typically addressed via cryptographic approaches, there have been several attempts at addressing security at the physical layer, following the pioneering work of Wyner [5]. Physical layer security can be considered to solve the security problem from the point of view of information theory. The basic principle of information-theoretic security calls for the combination of cryptographic schemes with channel coding techniques. Physical layer security exploits the randomness of communication channels to guarantee that the transmitted messages cannot be decoded by a third party maliciously eavesdropping on the wireless media. The following work by Leung-Yan-Cheong and Hellman characterized the secrecy capacity of scalar Gaussian wiretap channel [6]. Along the same lines, the Gaussian multiple-input-multiple-output (MIMO) wiretap channel was investigated and its secrecy capacity was established in terms of an optimization problem over all possible input covariance matrices [7–9]. There are some recent works focusing on secrecy rates based on partial channel state information (CSI) [10–12] and multiuser scenarios [13–15].

The secrecy rate based on single-antenna systems is hampered by channel conditions. When the channel condition between the source and the destination is worse than the source-to-eavesdropper channel, positive secrecy rate cannot be achieved unless multiple antennas are employed at the transmitter. Since the use of multiple antennas on a node involves a potentially significant hardware cost, node cooperation is an alternative low-cost scheme that enables single-antenna nodes to enjoy the benefits of multiple-antenna systems. In particular, node cooperation via relays can increase the achievable secrecy rate by exploiting/mitigating the channel effects [16–20]. There are mainly two relaying protocols for the cooperative secure transmission: decode-and-forward (DF) [16] and amplify-and-forward (AF) [17, 18]. Dong et al. [18] proposed a two-stage amplify-and-forward cooperative approach. In [18], the source and relays are located in the same cluster, while the destination and eavesdroppers are at far away locations outside this cluster. A system design was proposed to maximize the upper and lower bounds of secrecy capacity in the presence of a single eavesdropper. For the multiple eavesdroppers case, Dong et al. [19] proposed suboptimal and conservative null-space beamforming schemes for both the secrecy rate maximization problem subject to a power constraint and the relay power minimization problem subject to a secrecy rate constraint, which completely nulled out the signal leakage to all eavesdroppers in Stage 2. However, this kind of null-space beamforming scheme requires that the number of relays is larger than that of eavesdroppers. In addition, Yang et al. [20] obtained the secrecy rate maximization (SRM) weights design in the presence of multiple eavesdroppers, but they assumed that there are no direct links between the source and the destination, as well as the source and the eavesdroppers.

In this paper, we consider a more general case of multiple relays and multiple eavesdroppers, where there are possible direct links between the source and the destination as well as the source and the eavesdroppers. Global CSI is assumed to be available and even the eavesdroppers' channels are known. CSI on the eavesdroppers' channels can be obtained when the eavesdroppers are active in the network as their transmissions can be monitored. This is applicable particularly in wireless networks, in which terminals play dual roles as legitimate receivers for some signals and eavesdroppers for others [19]. We focus on the cooperative secure beamforming designs for both the secrecy rate maximization problem subject to a power constraint, and the relay power minimization problem subject to a secrecy rate constraint. The contributions of this paper can be summarized as follows.

• (i)

For the first optimization problem, we propose an SRM beamforming scheme. Using the semidefinite relaxation (SDR) technique, we perform the two-level optimization SRM problem to a one-dimensional search problem. Its performance approaches that of the optimal beamforming scheme.

(ii)

From a practical point of view, it is necessary to design a more simplified solution. Null-space beamforming has low complexity to be implemented, while it is out of operation when the eavesdroppers outnumber the relays. In order to overcome the drawback of null-space beamforming scheme, we propose a virtual eavesdropper-based SRM (VE-SRM) beamforming scheme by hypothesizing a virtual eavesdropper instead of all the eavesdroppers and maximize the secrecy rate according to the virtual eavesdropper.

(iii)

For the second optimization problem, we propose a relay power minimization (RPM) beamforming scheme to minimize the relay power subject to a secrecy rate constraint. To solve the multivariate problem, an iterative algorithm combining SDR and gradient-based method is devised.

(iv)

To reduce the complexity of the RPM problem, we relax the eavesdropper constraint and propose a virtual eavesdropper-based RPM (VE-RPM) beamforming scheme, which reduces the multivariate RPM problem to a problem of a single variable, and thus an analytical solution is obtained.

Moreover, the beamforming schemes we proposed can work well even if the number of eavesdroppers is larger than that of relays.

The rest of this paper is organized as follows. In Section 2, an AF-based cooperative network with multiple eavesdroppers scenario is described and the corresponding secure beamforming problems are formulated. In Section 3, we propose an SRM beamforming scheme and a practical VE-SRM beamforming scheme for the SRM problem subject to a power constraint. In Section 4, we propose an RPM beamforming scheme and an analytical VE-RPM beamforming scheme for the RPM problem subject to a secrecy rate constraint. Simulation results are provided in Section 5 and the conclusions are drawn in Section 6.

2. System Model and Problem Formulation

2.1. Channel Model

In this paper, we consider a communication unit of wireless M2M network which consists of one source (Alice), N ( $N\ge \mathrm{1}$ ) relay nodes, one destination node (Bob), and J ( $J>\mathrm{1}$ ) eavesdroppers (Eves). Define the set of the relays $ℛ\triangleq \{R_{\mathrm{1}},R_{\mathrm{2}},\dots ,R_N\}$ and the set of the Eves $ℰ\triangleq \{E_{\mathrm{1}},E_{\mathrm{2}},\dots ,E_J\}$ . All channels are assumed to be flat fading and quasistatic. In particular, denote by ${\mathbf{h}}_{SR}$ the channel vector between Alice and the relays in ℛ, by $h_{SD}$ the channel coefficient between Alice and Bob, by $h_{SE,j}$ the channel vector between Alice and the jth Eve, by ${\mathbf{h}}_{RD}$ the channel vector between the relays in ℛ and Bob, and by ${\mathbf{h}}_{RE,j}$ the channel vector between the relays in ℛ and jth Eve. The noise at each node is assumed to be zero-mean white complex Gaussian with variance ${\sigma }^{\mathrm{2}}$ . Each node is equipped with a single omnidirectional antenna. The configuration is illustrated in Figure 1.

Figure 1

AF relay assisted wireless M2M network in the presence of eavesdroppers.

There are two stages in the AF protocol. In Stage 1, Alice broadcasts its n encoded symbols to its relays. For the purpose of evaluating the achievable secrecy rate, it is assumed that the codewords used at Alice are Gaussian inputs. In a time unit, let x be the transmitted symbol, and the average power of an symbol is normalized; that is, $𝔼\{{|x|}^2\}=\mathrm{1}$ . $P_{\mathrm{0}}$ is the total power budget for transmitting one symbol in the two stages. When the source transmits the symbol x, the received signals at the N relays, stacked in vector ${\mathbf{y}}_R(N\times \mathrm{1})$ , become

$$\begin{array}{cc}& {\mathbf{y}}_R=\sqrt{P_S}{\mathbf{h}}_{SR}x+{\mathbf{n}}_R,\end{array}$$

(1)

where $P_S$ is the transmit power of Alice. It holds that $\mathrm{0}<P_S\le P_{\mathrm{0}}$ . ${\mathbf{n}}_R$ is the noise vector at the relays.

In Stage 2, the N relays forward the signals that they received during Stage 1 to the destination. More specifically, each relay transmits a weighted version of the noisy signal that they received during Stage 1. Thus, the transmitted signals of all the relays are $\mathbf{D}(\mathbf{w}){\mathbf{y}}_R$ , where $\mathbf{D}(\mathbf{a})$ denote the diagonal matrix with the vector a on its main diagonal and w is the weight vector and ${\mathbf{y}}_R$ is given in (1). The transmit power budget for Stage 2 is $P_R\triangleq P_{\mathrm{0}}-P_S$ .

The received signal at the destination is

$$\begin{array}{l}{y}_D=\sqrt{P_S}{\mathbf{h}}_{RD}^{†}\mathbf{D}\left(\mathbf{w}\right){\mathbf{h}}_{SR}x+{\mathbf{h}}_{RD}^{†}\mathbf{D}\left(\mathbf{w}\right){\mathbf{n}}_R+n_D\\ \hspace{1em}=\sqrt{P_S}{\mathbf{h}}_{RD}^{†}\mathbf{D}\left({\mathbf{h}}_{SR}\right)\mathbf{w}x+{\mathbf{n}}_R^T\mathbf{D}\left({\mathbf{h}}_{RD}^{†}\right)\mathbf{w}+n_D\\ \hspace{1em}={\mathbf{a}}^{†}\mathbf{w}x+{\mathbf{n}}_R^T\mathbf{D}\left({\mathbf{h}}_{RD}^{†}\right)\mathbf{w}+n_D,\end{array}$$

(2)

where ${(·)}^{†}$ denotes the conjugate transpose, ${(·)}^T$ denotes the transpose, $\mathbf{a}\triangleq \sqrt{P_S}\mathbf{D}({\mathbf{h}}_{SR}^{†}){\mathbf{h}}_{RD}$ , and $n_D$ is the noise at Bob.

The received signal at the jth Eve is

$$\begin{array}{l}{y}_E^j=\sqrt{P_S}{\mathbf{h}}_{RE,j}^{†}\mathbf{D}\left(\mathbf{w}\right){\mathbf{h}}_{SR}x+{\mathbf{h}}_{RE,j}^{†}\mathbf{D}\left(\mathbf{w}\right){\mathbf{n}}_R+n_{E,j}\\ \hspace{1em}=\sqrt{P_S}{\mathbf{h}}_{RE,j}^{†}\mathbf{D}\left({\mathbf{h}}_{SR}\right)\mathbf{w}x+{\mathbf{n}}_R^T\mathbf{D}\left({\mathbf{h}}_{RE,j}^{†}\right)\mathbf{w}+n_{E,j}\\ \hspace{1em}={\mathbf{b}}_j^{†}\mathbf{w}x+{\mathbf{n}}_R^T\mathbf{D}\left({\mathbf{h}}_{RE,j}^{†}\right)\mathbf{w}+n_{E,j},\end{array}$$

(3)

where ${\mathbf{b}}_j\triangleq \sqrt{P_S}\mathbf{D}({\mathbf{h}}_{SR}^{†}){\mathbf{h}}_{RE,j}$ and $n_{E,j}$ is the noise at the jth Eve.

2.2. Achievable Secrecy Rate

Recalling that $P_{\mathrm{0}}$ is the transmit power in Stage 1, and the achievable rate at Bob is

$$\begin{array}{cc}& R_D=\frac{\mathrm{1}}{\mathrm{2}}\text{lo}{\text{g}}_{\mathrm{2}}\left(\alpha +\frac{{\mathbf{w}}^{†}{\mathbf{R}}_a\mathbf{w}}{\left({\mathbf{w}}^{†}\mathbf{U}\mathbf{w}+\mathrm{1}\right){\sigma }^{\mathrm{2}}}\right),\end{array}$$

(4)

where $\alpha \triangleq \mathrm{1}+P_S|h_{SD}{|}^{\mathrm{2}}/{\sigma }^{\mathrm{2}}$ , ${\mathbf{R}}_a\triangleq \mathbf{a}{\mathbf{a}}^{†}$ , and $\mathbf{U}\triangleq \mathbf{D}({\mathbf{h}}_{RD})\mathbf{D}({\mathbf{h}}_{RD}^{†})$ . Note that $P_S{|h_{SD}|}^{\mathrm{2}}/{\sigma }^{\mathrm{2}}$ is the received signal-noise ratio (SNR) in Stage 1 at Bob. Similarly, the achievable rate at the jth Eve is

$$\begin{array}{cc}& R_{E,j}=\frac{\mathrm{1}}{\mathrm{2}}\text{lo}{\text{g}}_{\mathrm{2}}\left({\beta }_j+\frac{{\mathbf{w}}^{†}{\mathbf{R}}_{b,j}\mathbf{w}}{\left({\mathbf{w}}^{†}{\mathbf{V}}_j\mathbf{w}+\mathrm{1}\right){\sigma }^{\mathrm{2}}}\right),\end{array}$$

(5)

where ${\beta }_j\triangleq \mathrm{1}+P_S{|h_{SE,j}|}^{\mathrm{2}}/{\sigma }^{\mathrm{2}}$ , ${\mathbf{R}}_{b,j}\triangleq {\mathbf{b}}_j{\mathbf{b}}_j^{†}$ , and ${\mathbf{V}}_j\triangleq \mathbf{D}({\mathbf{h}}_{RE,j})\mathbf{D}({\mathbf{h}}_{RE,j}^{†})$ . Note that $P_S{|h_{SE,j}|}^{\mathrm{2}}/{\sigma }^{\mathrm{2}}$ is the SNR in Stage 1 at the jth Eve.

With the presence of J Eves, the achievable secrecy rate is given by [19]

$$\begin{array}{cc}& R_s={\left(R_D-\max \left(R_{E,\mathrm{1}},R_{E,\mathrm{2}},\dots ,R_{E,J}\right)\right)}_{+},\end{array}$$

(6)

where ${(x)}_{+}$ represents the maximal one between x and $\mathrm{0}$ . The achievable secrecy rate in (6) can be interpreted as a worst-case result; that is, the Eve with the best channel dominates the secrecy rate. For convenience, we drop the operator ${(·)}_{+}$ and the constant coefficient $\mathrm{1}/\mathrm{2}$ .

Our beamforming design is based on amplify-and-forward cooperative scheme, and the optimization criterion is the maximization of the achievable secrecy rate subject to a relay power constraint or the minimization of the relay transmit power subject to a secrecy rate constraint. The specific problems addressed in this paper are as follows.

2.2.1. Maximize the Secrecy Rate under a Relay Power Constraint

Given a source power $P_S$ and a relay power $P_R$ , design the cooperative beamforming vector w to maximize the secrecy rate; that is,

$$\begin{array}{l}\underset{\mathbf{w}}{\max }\hspace{0.17em}\underset{E_j\in ℰ}{\min }\hspace{1em}\{\text{lo}{\text{g}}_{\mathrm{2}}\left(\alpha +\frac{{\mathbf{w}}^{†}{\mathbf{R}}_a\mathbf{w}}{\left({\mathbf{w}}^{†}\mathbf{U}\mathbf{w}+\mathrm{1}\right){\sigma }^{\mathrm{2}}}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}\hspace{0.5em}-\text{lo}{\text{g}}_{\mathrm{2}}\left({\beta }_j+\frac{{\mathbf{w}}^{†}{\mathbf{R}}_{b,j}\mathbf{w}}{\left({\mathbf{w}}^{†}{\mathbf{V}}_j\mathbf{w}+\mathrm{1}\right){\sigma }^{\mathrm{2}}}\right)\}\\ \text{s}.\text{t}.\hspace{1em}{\mathbf{w}}^{†}\mathbf{T}\mathbf{w}\le P_R,\end{array}$$

(7)

where $\mathbf{T}\triangleq P_S\mathbf{D}\{{\mathbf{h}}_{SR}\}\mathbf{D}\{{\mathbf{h}}_{SR}^{†}\}+\mathbf{I}$ .

2.2.2. Minimize the Relay Transmit Power under a Secrecy Rate Constraint

Given a secrecy rate constraint ${\mathrm{\Gamma }}_{\mathrm{0}}$ , design the cooperative beamforming vector w to minimize the relay power; that is,

$$\begin{array}{l}\underset{\mathbf{w}}{\min }\hspace{1em}{\mathbf{w}}^{†}\mathbf{T}\mathbf{w}\\ \text{s}.\text{t}.\hspace{1em}{R}_s\ge {\mathrm{\Gamma }}_{\mathrm{0}},\end{array}$$

(8)

where $R_s$ is given by (6). Here we point out that the constraints $R_s\ge {\mathrm{\Gamma }}_{\mathrm{0}}$ and $R_s={\mathrm{\Gamma }}_{\mathrm{0}}$ are equivalent. The reason is that $R_s$ is a nondecreasing function of the relay power $P_R$ .

3. Secrecy Rate Maximization Beamforming Design under Relay Power Constraint

3.1. Existing Beamforming Scheme

In this subsection, we demonstrate the existing beamforming scheme under the relay power constraint. In [19], a simple null-space beamforming design was proposed when the number of relays is larger than that of eavesdroppers. Under the assumption of no signal leakage to all eavesdroppers in Stage 2, the associated SRM problem (7) can be simplified to

$$\begin{array}{l}\underset{\mathbf{w}}{\max }\hspace{1em}\left\{\text{lo}{\text{g}}_{\mathrm{2}}\left(\alpha +\frac{{\mathbf{w}}^{†}{\mathbf{R}}_a\mathbf{w}}{\left({\mathbf{w}}^{†}\mathbf{U}\mathbf{w}+\mathrm{1}\right){\sigma }^{\mathrm{2}}}\right)-\text{lo}{\text{g}}_{\mathrm{2}}\left({\beta }_{\max }\right)\right\}\\ \text{s}.\text{t}.\hspace{1em}{\mathbf{h}}_{RE,j}^{†}\mathbf{w}=\mathrm{0},\hspace{1em}\forall E_j\in ℰ\\ \hspace{1em}\hspace{1em}{\mathbf{w}}^{†}\mathbf{T}\mathbf{w}\le P_R,\end{array}$$

(9)

where ${\beta }_{\max }\triangleq \mathrm{1}+\underset{}{{\max }_{E_j\in ℰ}}(P_{\mathrm{0}}{|h_{SE,j}|}^2/{\sigma }^{\mathrm{2}})$ . Under the relay power constraint, the null-space SRM problem was studied in [19], where a closed-form solution was derived.

3.2. Secrecy Rate Maximization Beamforming

In order to maximize the secrecy rate under the total power constraint at relays, a SRM beamforming scheme is developed. By introducing a slack variable τ, problem (7) can be equivalently rewritten as

$$\begin{array}{l}\underset{\mathbf{w},\tau >\mathrm{0}}{\max }\hspace{1em}\left\{\text{lo}{\text{g}}_{\mathrm{2}}\left(\alpha +\frac{{\mathbf{w}}^{†}{\mathbf{R}}_a\mathbf{w}}{\left({\mathbf{w}}^{†}\mathbf{U}\mathbf{w}+\mathrm{1}\right){\sigma }^{\mathrm{2}}}\right)-\text{lo}{\text{g}}_{\mathrm{2}}\left(\frac{\mathrm{1}}{\tau }\right)\right\}\\ \text{s}.\text{t}.\hspace{1em}\text{lo}{\text{g}}_{\mathrm{2}}\left({\beta }_j+\frac{{\mathbf{w}}^{†}{\mathbf{R}}_{b,j}\mathbf{w}}{\left({\mathbf{w}}^{†}{\mathbf{V}}_j\mathbf{w}+\mathrm{1}\right){\sigma }^{\mathrm{2}}}\right)\le \text{lo}{\text{g}}_{\mathrm{2}}\left(\frac{\mathrm{1}}{\tau }\right),\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}\forall E_j\in ℰ\\ \hspace{1em}\hspace{1em}{\mathbf{w}}^{†}\mathbf{T}\mathbf{w}\le P_R.\end{array}$$

(10)

Since the logarithm function is monotonically increasing, we remove it and further obtain

$$\begin{array}{l}\underset{\mathbf{w},\tau >\mathrm{0}}{\min }\hspace{1em}\frac{\left({\mathbf{w}}^{†}\mathbf{U}\mathbf{w}+\mathrm{1}\right){\sigma }^{\mathrm{2}}}{\left({\mathbf{w}}^{†}\left(\alpha {\sigma }^{\mathrm{2}}\mathbf{U}+{\mathbf{R}}_a\right)\mathbf{w}+\alpha {\sigma }^{\mathrm{2}}\right)\tau }\\ \text{s}.\text{t}.\hspace{1em}\frac{\left({\mathbf{w}}^{†}{\mathbf{V}}_j\mathbf{w}+\mathrm{1}\right){\sigma }^{\mathrm{2}}}{{\mathbf{w}}^{†}\left({\beta }_j{\sigma }^{\mathrm{2}}{\mathbf{V}}_j+{\mathbf{R}}_{b,j}\right)\mathbf{w}+{\beta }_j{\sigma }^{\mathrm{2}}}\ge \tau ,\hspace{1em}\forall E_j\in ℰ\\ \hspace{1em}\hspace{1em}{\mathbf{w}}^{†}\mathbf{T}\mathbf{w}\le P_R.\end{array}$$

(11)

To solve problem (11), following the idea of SDR [21], define $\mathbf{W}\triangleq \mathbf{w}{\mathbf{w}}^{†}$ and drop the nonconvex rank-one constraint to get a relaxed problem of (11) as follows:

$$\begin{array}{l}\underset{\mathbf{W}\succcurlyeq \mathbf{0},\tau >\mathrm{0}}{\min }\hspace{1em}\frac{\left(\mathrm{Tr}\left(\mathbf{U}\mathbf{W}\right)+\mathrm{1}\right){\sigma }^{\mathrm{2}}}{\left(\mathrm{Tr}\left(\left(\alpha {\sigma }^{\mathrm{2}}\mathbf{U}+{\mathbf{R}}_a\right)\mathbf{W}\right)+\alpha {\sigma }^{\mathrm{2}}\right)\tau }\\ \text{s}.\text{t}.\hspace{1em}\frac{\left(\mathrm{Tr}\left({\mathbf{V}}_j\mathbf{W}\right)+\mathrm{1}\right){\sigma }^{\mathrm{2}}}{\mathrm{Tr}\left(\left({\beta }_j{\sigma }^{\mathrm{2}}{\mathbf{V}}_j+{\mathbf{R}}_{b,j}\right)\mathbf{W}\right)+{\beta }_j{\sigma }^{\mathrm{2}}}\ge \tau ,\hspace{1em}\forall E_j\in ℰ,\\ \hspace{1em}\hspace{1em}\mathrm{Tr}\left(\mathbf{T}\mathbf{W}\right)\le P_R.\end{array}$$

(12)

While problem (12) is still nonconvex due to the variable τ, one may check that, for a given τ, problem (12) changes into a quasiconvex problem [22]. By this observation, treat (12) as a two-level optimization problem—the inner level (problem (12) with a fixed τ) is a quasiconvex problem, and the outer level (optimization over τ) is a single-variable optimization problem.

Let us start with the inner-level optimization problem. Though this class of quasiconvex problems can be solved by using the bisection method [22], one usually needs to deal with many semidefinite programs (SDPs) during the bisection search [22]. Here, we adopt a more efficient approach by reformulating quasiconvex problem into a convex SDP through the Charnes-Cooper transformation [23]. For some $\mathbf{Z}\succcurlyeq \mathbf{0}$ and $\eta >\mathrm{0}$ , we let $\mathbf{W}=\mathbf{Z}/\eta$ and recast (12) as

$$\begin{array}{l}\underset{\mathbf{Z}\succcurlyeq \mathbf{0},\eta >\mathrm{0}}{\min }\hspace{1em}\left\{\mathrm{Tr}\left(\mathbf{U}\mathbf{Z}\right)+\eta \right\}\\ \text{s}.\text{t}.\hspace{1em}\tau \left(\mathrm{Tr}\left(\left(\alpha {\sigma }^{\mathrm{2}}\mathbf{U}+{\mathbf{R}}_a\right)\mathbf{Z}\right)+\alpha {\sigma }^{\mathrm{2}}\eta \right)=\mathrm{1}\\ \hspace{1em}\hspace{1em}\tau \left(\mathrm{Tr}\left(\left({\beta }_j{\sigma }^{\mathrm{2}}{\mathbf{V}}_j+{\mathbf{R}}_{b,j}\right)\mathbf{Z}\right)+{\beta }_j{\sigma }^{\mathrm{2}}\eta \right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\le \left(\mathrm{Tr}\left({\mathbf{V}}_j\mathbf{Z}\right)+\eta \right){\sigma }^{\mathrm{2}},\hspace{1em}\forall E_j\in ℰ,\\ \hspace{1em}\hspace{1em}\mathrm{Tr}\left(\mathbf{T}\mathbf{Z}\right)\le \eta P_R.\end{array}$$

(13)

Problem (13) is an SDP whose optimal solution can be efficiently obtained by available solvers, for example, CVX. In the following, we turn our attention to the outer-level part, which is a single-variable optimization problem as follows:

$$\begin{array}{l}\underset{\tau }{\min }\hspace{1em}\varphi \left(\tau \right),\\ \text{s}.\text{t}.\hspace{1em}{\tau }_{\text{lb}}\le \tau \le {\tau }_{\text{ub}},\end{array}$$

(14)

where $\varphi (\tau )$ denotes the optimal value of (13) and ${\tau }_{\text{lb}}$ and ${\tau }_{\text{ub}}$ are the lower and upper bound of τ hidden in (10)–(12), respectively. Upon carefully examining the feasibility of the constraints in (12), we have

$$\begin{array}{l}\tau \le \underset{E_j\in ℰ}{\min }\frac{\left(\mathrm{Tr}\left({\mathbf{V}}_j\mathbf{W}\right)+\mathrm{1}\right){\sigma }^{\mathrm{2}}}{\mathrm{Tr}\left(\left({\beta }_j{\sigma }^{\mathrm{2}}{\mathbf{V}}_j+{\mathbf{R}}_{b,j}\right)\mathbf{W}\right)+{\beta }_j{\sigma }^{\mathrm{2}}}\\ \hspace{0.17em}<\underset{E_j\in ℰ}{\min }\left(\frac{\mathrm{1}}{{\beta }_j}\right)\triangleq {\tau }_{\text{ub}}.\end{array}$$

(15)

The effective ${\tau }_{\text{lb}}$ arises from the requirement of nonnegative secrecy rate in (10); that is,

$$\begin{array}{l}\tau \ge \frac{\left({\mathbf{w}}^{†}\mathbf{U}\mathbf{w}+\mathrm{1}\right){\sigma }^{\mathrm{2}}}{{\mathbf{w}}^{†}\left(\alpha {\sigma }^{\mathrm{2}}\mathbf{U}+{\mathbf{R}}_a\right)\mathbf{w}+\alpha {\sigma }^{\mathrm{2}}}\\ \hspace{0.17em}\hspace{0.17em}=\frac{\mathrm{1}}{\alpha +{\mathbf{w}}^{†}{\mathbf{R}}_a\mathbf{w}/\left({\mathbf{w}}^{†}\mathbf{U}\mathbf{w}+\mathrm{1}\right){\sigma }^{\mathrm{2}}}\\ \hspace{0.17em}\hspace{0.17em}\ge \frac{\mathrm{1}}{\alpha +{\mathbf{w}}^{†}{\mathbf{R}}_a\mathbf{w}/{\mathbf{w}}^{†}\left(\mathbf{U}+\mathbf{T}/P_R\right)\mathbf{w}{\sigma }^{\mathrm{2}}}\\ \hspace{0.17em}\hspace{0.17em}\ge \frac{\mathrm{1}}{\alpha +{\lambda }_{\max }\left({\left(\mathbf{U}+\mathbf{T}/P_R\right)}^{-\mathrm{1}}{\mathbf{R}}_a\right)/{\sigma }^{\mathrm{2}}}\triangleq {\tau }_{\text{lb}},\end{array}$$

(16)

where (16) is a generalized Rayleigh Quotient (GRQ) problem [24] and ${\lambda }_{\max }(\mathbf{A})$ represents the maximal eigenvalue of the matrix A.

The lower bound implies that, if $\tau <{\tau }_{\text{lb}}$ , the secrecy rate will be negative and the design does not make sense. Since (13) is a single-variable problem ( ${\tau }_{\text{lb}}\le \tau \le {\tau }_{\text{ub}}$ ), one-dimensional optimization techniques can be used to search for the solution.

So far, we can cope with (12) through performing a one-dimensional search during which we solve a sequence of SDPs. We denote the solution of (12) by $({\mathbf{W}}^{*},{\tau }^{*})$ . As we have dropped the rank one constraint of W, the ${\mathbf{W}}^{*}$ can achieve a upper bound of the optimal secrecy rate. If the rank of ${\mathbf{W}}^{*}$ is equal to $\mathrm{1}$ , the feasible w of (10) can be obtained from ${\mathbf{W}}^{*}$ via eigenvalue decomposition. If the rank of ${\mathbf{W}}^{*}$ is larger than 1, we can extract an approximate solution w from ${\mathbf{W}}^{*}$ using the Gaussian Randomization procedure [21]. We will compare the achievable secrecy rate of the SRM beamforming scheme and the upper bound of the optimal secrecy rate in the simulation section.

3.3. Virtual Eavesdropper-Based Secrecy Rate Maximization Beamforming

As mentioned above, the SRM beamforming is still difficult to implement due to the existence of multiple SDPs. To further reduce the complexity of the implementation, we propose a VE-SRM beamforming scheme. The SRM problem (7) can be rewritten as

$$\begin{array}{l}\underset{\mathbf{w}}{\max }\hspace{1em}\{\text{lo}{\text{g}}_{\mathrm{2}}\left(\alpha +\frac{{\mathbf{w}}^{†}{\mathbf{R}}_a\mathbf{w}}{\left({\mathbf{w}}^{†}\mathbf{U}\mathbf{w}+\mathrm{1}\right){\sigma }^{\mathrm{2}}}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}\hspace{1em}-\underset{E_j\in ℰ}{\max }\hspace{0.17em}\text{lo}{\text{g}}_{\mathrm{2}}\left({\beta }_j+\frac{{\mathbf{w}}^{†}{\mathbf{R}}_{b,j}\mathbf{w}}{\left({\mathbf{w}}^{†}{\mathbf{V}}_j\mathbf{w}+\mathrm{1}\right){\sigma }^{\mathrm{2}}}\right)\}\\ \text{s}.\text{t}.\hspace{1em}{\mathbf{w}}^{†}\mathbf{T}\mathbf{w}\le P_R.\end{array}$$

(17)

Obviously, it is the two-level optimization that makes the SRM problem complicated to solve. The key to address problem (17) is to solve $\underset{}{{\max }_{E_j\in ℰ}}\text{lo}{\text{g}}_{\mathrm{2}}({\beta }_j+{\mathbf{w}}^{†}{\mathbf{R}}_{b,j}\mathbf{w}/({\mathbf{w}}^{†}{\mathbf{V}}_j\mathbf{w}+\mathrm{1}){\sigma }^{\mathrm{2}})$ . The eavesdropping rate at $E_j$ is related to its received SNR, and the eavesdropper that has the largest SNR is the one that has the maximal eavesdropping rate. Thus, the largest SNR in all Eves for a given w in problem (17) can be expressed as

$$\begin{array}{cc}& \text{SN}{\text{R}}_{\max }^{\mathbf{w}}=\underset{E_j\in ℰ}{\max }\left\{{\beta }_j-\mathrm{1}+\frac{{\mathbf{w}}^{†}{\mathbf{R}}_{b,j}\mathbf{w}}{\left({\mathbf{w}}^{†}{\mathbf{V}}_j\mathbf{w}+\mathrm{1}\right){\sigma }^{\mathrm{2}}}\right\}.\end{array}$$

(18)

In order to drop the max expression in (18), we need to find an upper bound of the largest SNR (corresponding to the largest eavesdropping rate). Note that $\text{SN}{\text{R}}_{\mathbf{w}}^{\max }\le {\beta }_{\max }-\mathrm{1}+{\sum }_{E_j\in ℰ}‍(\text{SN}{\text{R}}_{\mathrm{2},j}^{\mathbf{w}})$ is always valid, where $\text{SN}{\text{R}}_{\mathrm{2},j}^{\mathbf{w}}$ is the received SNR of $E_j$ in Stage 2. Problem (18) can be relaxed as

$$\begin{array}{cc}& \text{SN}{\text{R}}_{\max }^{\mathbf{w}}\le {\beta }_{\max }-\mathrm{1}+\underset{E_j\in ℰ}{\sum ‍}\frac{{\mathbf{w}}^{†}{\mathbf{R}}_{b,j}\mathbf{w}}{\left({\mathbf{w}}^{†}{\mathbf{V}}_j\mathbf{w}+\mathrm{1}\right){\sigma }^{\mathrm{2}}}.\end{array}$$

(19)

Notice that

$$\begin{array}{cc}& \frac{{\mathbf{w}}^{†}{\mathbf{V}}_j\mathbf{w}}{{\mathbf{w}}^{†}\mathbf{T}\mathbf{w}}\ge {\lambda }_{\min }\left({\mathbf{T}}^{-\mathrm{1}}{\mathbf{V}}_j\right),\hspace{1em}\forall E_j\in ℰ,\hspace{0.17em}\hspace{0.17em}{\mathbf{w}}^{†}\mathbf{T}\mathbf{w}\le P_R\end{array}$$

(20)

and the upper bound of ${\mathbf{w}}^{†}{\mathbf{V}}_j\mathbf{w}+\mathrm{1}$ can be expressed as

$$\begin{array}{cc}& {\mathbf{w}}^{†}{\mathbf{V}}_j\mathbf{w}+\mathrm{1}\ge {\mathbf{w}}^{†}\mathbf{T}\mathbf{w}\left({\lambda }_{\min }\left({\mathbf{T}}^{-\mathrm{1}}{\mathbf{V}}_j\right)+\frac{\mathrm{1}}{P_R}\right).\end{array}$$

(21)

According to the inequality in (21), the formula in (19) can be rewritten as

$$\begin{array}{cc}& \text{SN}{\text{R}}_{\max }^{\mathbf{w}}\le {\beta }_{\max }-\mathrm{1}+\frac{{\mathbf{w}}^{†}{\mathbf{R}}_g\mathbf{w}}{{\mathbf{w}}^{†}\mathbf{T}\mathbf{w}{\sigma }^{\mathrm{2}}},\end{array}$$

(22)

where ${\mathbf{R}}_g\triangleq {\sum }_{E_j\in ℰ}({\mathbf{R}}_{b,j}/{\Omega }_j)$ and ${\Omega }_j={\lambda }_{\min }({\mathbf{T}}^{-\mathrm{1}}{\mathbf{V}}_j)+\mathrm{1}/P_R,\text{for}\mathrm{\hspace{0.17em}\hspace{0.17em}}\text{all}\mathrm{\hspace{0.17em}\hspace{0.17em}}{E}_j\in ℰ$ . It seems that there is a virtual eavesdropper whose channel gain is ${\beta }_{\max }-\mathrm{1}$ in Stage 1 and instantaneous channel gain matrix is ${\mathbf{R}}_g$ in Stage 2. By using the virtual eavesdropper instead of all the Eves, the two-level optimization SRM problem becomes a one-dimension optimization problem. Substituting (22) into (17), we can get

$$\begin{array}{l}\underset{\mathbf{w}}{\max }\hspace{1em}\{\text{lo}{\text{g}}_{\mathrm{2}}\left(\alpha +\frac{{\mathbf{w}}^{†}{\mathbf{R}}_a\mathbf{w}}{\left({\mathbf{w}}^{†}\mathbf{U}\mathbf{w}+\mathrm{1}\right){\sigma }^{\mathrm{2}}}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.5em}-\text{lo}{\text{g}}_{\mathrm{2}}\left({\beta }_{\max }+\frac{{\mathbf{w}}^{†}{\mathbf{R}}_g\mathbf{w}}{{\mathbf{w}}^{†}\mathbf{T}\mathbf{w}{\sigma }^{\mathrm{2}}}\right)\}\\ \text{s}.\text{t}.\hspace{1em}{\mathbf{w}}^{†}\mathbf{T}\mathbf{w}\le P_R.\end{array}$$

(23)

The VE-SRM beamforming scheme relaxes the constraint of selecting the Eve with the highest eavesdropping rate, while the null-space beamforming scheme introduces a strong constraint that nulls out all signals at Eves. Thus, the VE-SRM beamforming design is a better solution than the null-space beamforming scheme. Since the logarithm function is monotonically increasing, remove it and obtain

$$\begin{array}{l}\underset{\mathbf{w}}{\max }\hspace{1em}\left\{\frac{\alpha {\sigma }^{\mathrm{2}}{\mathbf{w}}^{†}\mathbf{U}\mathbf{w}+\alpha {\sigma }^{\mathrm{2}}+{\mathbf{w}}^{†}{\mathbf{R}}_a\mathbf{w}}{{\mathbf{w}}^{†}\mathbf{U}\mathbf{w}+\mathrm{1}}\frac{{\mathbf{w}}^{†}\mathbf{T}\mathbf{w}}{{\beta }_{\max }{\mathbf{w}}^{†}\mathbf{T}\mathbf{w}{\sigma }^{\mathrm{2}}+{\mathbf{w}}^{†}{\mathbf{R}}_g\mathbf{w}}\right\}\\ \text{s}.\text{t}.\hspace{1em}{\mathbf{w}}^{†}\mathbf{T}\mathbf{w}\le P_R.\end{array}$$

(24)

Then problem (24) can be further simplified as

$$\begin{array}{cc}& \hspace{0.17em}\underset{\mathbf{w}}{\max }\hspace{1em}\left\{\frac{{\mathbf{w}}^{†}{\tilde{\mathbf{R}}}_h\mathbf{w}}{{\mathbf{w}}^{†}{\tilde{\mathbf{R}}}_g\mathbf{w}}\frac{{\mathbf{w}}^{†}{\mathbf{R}}_v\mathbf{w}}{{\mathbf{w}}^{†}{\mathbf{R}}_u\mathbf{w}}\right\}\end{array}$$

(25)

$$\begin{array}{cc}& \hspace{0.17em}\text{s}.\text{t}.\hspace{1em}{\mathbf{w}}^{†}\mathbf{T}\mathbf{w}\le P_R,\end{array}$$

(26)

where ${\tilde{\mathbf{R}}}_h\triangleq \alpha {\sigma }^{\mathrm{2}}\mathbf{U}+(\alpha {\sigma }^{\mathrm{2}}/P_R)\mathbf{T}+{\mathbf{R}}_a$ , ${\tilde{\mathbf{R}}}_g\triangleq {\beta }_{\max }{\sigma }^{\mathrm{2}}\mathbf{T}+{\mathbf{R}}_g$ , ${\mathbf{R}}_v\triangleq \mathbf{T}/P_R$ , and ${\mathbf{R}}_u\triangleq \mathbf{U}+\mathbf{T}/P_R$ . The objective function in (25) is a product of two GRQ problems, and it is a difficult problem in general. To simplify the analysis, a gradient-based method is used in order to obtain the updating equation for the beamforming vector. We denote the product of two GRQ problems in (25) by $\mathbf{J}(\mathbf{w})$ , and its gradient with respect to w is

$$\begin{array}{l}\nabla \mathbf{J}\left(\mathbf{w}\right)=\frac{\mathrm{2}}{{\mathbf{w}}^{†}{\tilde{\mathbf{R}}}_g\mathbf{w}}\left({\tilde{\mathbf{R}}}_h\mathbf{w}-{\tilde{\mathbf{R}}}_g\mathbf{w}\frac{{\mathbf{w}}^{†}{\tilde{\mathbf{R}}}_h\mathbf{w}}{{\mathbf{w}}^{†}{\tilde{\mathbf{R}}}_g\mathbf{w}}\right)\frac{{\mathbf{w}}^{†}{\mathbf{R}}_v\mathbf{w}}{{\mathbf{w}}^{†}{\mathbf{R}}_u\mathbf{w}}\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\frac{{\mathbf{w}}^{†}{\tilde{\mathbf{R}}}_h\mathbf{w}}{{\mathbf{w}}^{†}{\tilde{\mathbf{R}}}_g\mathbf{w}}\frac{\mathrm{2}}{{\mathbf{w}}^{†}{\mathbf{R}}_u\mathbf{w}}\left({\mathbf{R}}_v\mathbf{w}-{\mathbf{R}}_u\mathbf{w}\frac{{\mathbf{w}}^{†}{\mathbf{R}}_v\mathbf{w}}{{\mathbf{w}}^{†}{\mathbf{R}}_u\mathbf{w}}\right).\end{array}$$

(27)

Considering the simple steepest gradient method [25], the equation for updating the weight vector regarding the function $\mathbf{J}(\mathbf{w})$ is

$$\begin{array}{cc}& \mathbf{w}\left(k+\mathrm{1}\right)=\mathbf{w}\left(k\right)+{\mu }_k\nabla \mathbf{J}\left(\mathbf{w}\left(k\right)\right),\end{array}$$

(28)

where $\mathbf{w}(k)$ denotes the weight vector at the kth iteration and ${\mu }_k$ is a real scalar step size at the kth iteration which is determined for the convergence of the iterative procedure. (28) is the updating equation for finding the new weight vector $\mathbf{w}(k+\mathrm{1})$ . The convergence analysis of the gradient method has been proposed in [22].

As shown in Algorithm 1, we construct an iterative VE-SRM beamforming algorithm. The proposed algorithm requires an initial beamforming vector $\mathbf{w}(\mathrm{0})$ . The first GRQ in (26) is related to the received signals both in Bob and Eves, while the second GRQ is mainly corresponding to the noise amplification at the relays. Therefore, we let the initial weight vector be $\rho \mathbf{v}$ , where v is the unit-norm eigenvector of the matrix ${\tilde{\mathbf{R}}}_g^{-\mathrm{1}}{\tilde{\mathbf{R}}}_h$ corresponding to the largest eigenvalue of the matrix and ρ is a scalar chosen to satisfy the power constraint and it equals

$$\begin{array}{cc}& \rho =\sqrt{\frac{P_R}{{\mathbf{v}}^{†}\mathbf{T}\mathbf{v}}}.\end{array}$$

(29)

Algorithm 1: The proposed iterative VE-SRM beamforming algorithm.

(1) Initial State:

  Initialize ${\tilde{\mathbf{R}}}_h$ , ${\tilde{\mathbf{R}}}_g$ , ${\mathbf{R}}_v$ , ${\mathbf{R}}_u$ and the initial beamforming vector $\mathbf{w}\left(\mathrm{0}\right)$ .

(2) Iterative  VE-SRM  Beamforming   Algorithm:

 (i) Clock index $k=\mathrm{0}$ .

 (ii) Repeat

   (a) Linear search. Choose step size ${\mathrm{\mu }}_k$ via backtracking line

     search [22].

   (b) Update. $\mathbf{w}\left(k+\mathrm{1}\right)=\mathbf{w}\left(k\right)+{\mathrm{\mu }}_k\nabla \mathbf{J}\left(\mathbf{w}\left(k\right)\right)$ .

   (c) Calculate. $\mathbf{J}\left(\mathbf{w}\left(k+\mathrm{1}\right)\right)$ .

   (d) Increase. clock index $k=k+\mathrm{1}$ .

 (iii) Until   $\mathbf{J}\left(\mathbf{w}\left(k\right)\right)\le \mathbf{J}\left(\mathbf{w}\left(k-\mathrm{1}\right)\right)$ and ${\mathrm{\mu }}_k<{\mathrm{\mu }}_{\text{thresh}}$ .

(3) Secure Transmission:

  Distribute the beamforming weights to the relays, and begin the secure data transmission.

4. Relay Power Minimization Beamforming Design under Secrecy Rate Constraint

4.1. Existing Beamforming Scheme

In this subsection, we demonstrate the existing beamforming scheme under the secrecy rate constraint. In [19], a simple null-space RPM beamforming design was proposed when there were more relays than eavesdroppers. Under the assumption of no information leakage to all eavesdroppers in Stage 2, the associated RPM problem (8) can be simplified to

$$\begin{array}{l}\underset{\mathbf{w}}{\min }\hspace{1em}{\mathbf{w}}^{†}\mathbf{T}\mathbf{w}\\ \text{s}.\text{t}.\hspace{1em}{\left({\mathbf{h}}_{RE}^j\right)}^{†}\mathbf{w}=\mathrm{0},\hspace{1em}\forall E_j\in ℰ\\ \hspace{1em}\hspace{1em}\frac{{\mathbf{w}}^{†}{\mathbf{R}}_a\mathbf{w}}{{\mathbf{w}}^{†}\mathbf{U}\mathbf{w}+\mathrm{1}}\ge \left({\beta }_{\max }{\mathrm{4}}^{{\mathrm{\Gamma }}_{\mathrm{0}}}-\alpha \right){\sigma }^{\mathrm{2}}.\end{array}$$

(30)

Under the secrecy rate constraint, the null-space RPM problem was studied in [19], where a closed-form solution was derived.

4.2. Relay Power Minimization Beamforming

In order to minimize the transmit power at relays under the secrecy rate constraint, an RPM beamforming scheme is developed. Problem (8) can be reformulated into

$$\begin{array}{l}\underset{\mathbf{w}}{\min }\hspace{1em}{\mathbf{w}}^{†}\mathbf{T}\mathbf{w}\\ \text{s}.\text{t}.\hspace{1em}\frac{\alpha +{\mathbf{w}}^{†}{\mathbf{R}}_a\mathbf{w}/\left({\mathbf{w}}^{†}\mathbf{U}\mathbf{w}+\mathrm{1}\right){\sigma }^{\mathrm{2}}}{\underset{}{{\max }_{E_j\in ℰ}}\left\{{\beta }_j+{\mathbf{w}}^{†}{\mathbf{R}}_{b,j}\mathbf{w}/\left({\mathbf{w}}^{†}{\mathbf{V}}_j\mathbf{w}+\mathrm{1}\right){\sigma }^{\mathrm{2}}\right\}}\ge {\mathrm{4}}^{{\mathrm{\Gamma }}_{\mathrm{0}}}.\end{array}$$

(31)

Introducing a list of auxiliary slacking variables $\mathbf{r}\triangleq \{r_{\mathrm{1}},\dots ,r_J\}$ , which satisfies

$$\begin{array}{cc}& {\beta }_j+\frac{{\mathbf{w}}^{†}{\mathbf{R}}_{b,j}\mathbf{w}}{\left({\mathbf{w}}^{†}{\mathbf{V}}_j\mathbf{w}+\mathrm{1}\right){\sigma }^{\mathrm{2}}}\le r_j,\hspace{1em}j=\mathrm{1,2},\dots ,J.\end{array}$$

(32)

Then we can further rewrite problem (31) as

$$\begin{array}{l}\underset{\mathbf{w},\mathbf{r}}{\min }\hspace{1em}{\mathbf{w}}^{†}\mathbf{T}\mathbf{w}\\ \text{s}.\text{t}.\hspace{1em}\alpha +\frac{{\mathbf{w}}^{†}{\mathbf{R}}_a\mathbf{w}}{\left({\mathbf{w}}^{†}\mathbf{U}\mathbf{w}+\mathrm{1}\right){\sigma }^{\mathrm{2}}}\ge r_j{\mathrm{4}}^{{\mathrm{\Gamma }}_{\mathrm{0}}}\\ \hspace{1em}\hspace{1em}{\beta }_j+\frac{{\mathbf{w}}^{†}{\mathbf{R}}_{b,j}\mathbf{w}}{\left({\mathbf{w}}^{†}{\mathbf{V}}_j\mathbf{w}+\mathrm{1}\right){\sigma }^{\mathrm{2}}}\le r_j,\hspace{1em}\forall E_j\in ℰ.\end{array}$$

(33)

The equivalence of the problems (33) and (31) can be shown as follows. Let w be a feasible solution to (31), and we can get

$$\begin{array}{cc}& \frac{\alpha +{\mathbf{w}}^{†}{\mathbf{R}}_a\mathbf{w}/\left({\mathbf{w}}^{†}\mathbf{U}\mathbf{w}+\mathrm{1}\right){\sigma }^{\mathrm{2}}}{\underset{}{{\max }_{E_j\in ℰ}}\left\{{\beta }_j+{\mathbf{w}}^{†}{\mathbf{R}}_{b,j}\mathbf{w}/\left({\mathbf{w}}^{†}{\mathbf{V}}_j\mathbf{w}+\mathrm{1}\right){\sigma }^{\mathrm{2}}\right\}}\ge {\mathrm{4}}^{{\mathrm{\Gamma }}_{\mathrm{0}}}.\end{array}$$

(34)

If we select

$$\begin{array}{cc}& r_j\in \left[{\beta }_j+\frac{{\mathbf{w}}^{†}{\mathbf{R}}_{b,j}\mathbf{w}}{\left({\mathbf{w}}^{†}{\mathbf{V}}_j\mathbf{w}+\mathrm{1}\right){\sigma }^{\mathrm{2}}},\left(\alpha +\frac{{\mathbf{w}}^{†}{\mathbf{R}}_a\mathbf{w}}{\left({\mathbf{w}}^{†}\mathbf{U}\mathbf{w}+\mathrm{1}\right){\sigma }^{\mathrm{2}}}\right){\mathrm{4}}^{-{\mathrm{\Gamma }}_{\mathrm{0}}}\right].\end{array}$$

(35)

Then $\{\mathbf{w},r_{\mathrm{1}},\dots ,r_J\}$ is a feasible solution to (33). On the other hand, let $\{\mathbf{w},r_{\mathrm{1}},\dots ,r_J\}$ denote a feasible solution to (33), it is easy to certify that w is also feasible for (31). Therefore problems (33) and (31) are equivalent.

4.2.1. Optimization of w for Fixed r

Just as the SRM beamforming design, we introduce $\mathbf{W}\triangleq \mathbf{w}{\mathbf{w}}^{†}$ based on the SDR idea and drop the nonconvex constraint $\mathrm{rank}(\mathbf{W})=\mathrm{1}$ . We rewrite (33) as

$$\begin{array}{l}\underset{\mathbf{W}\succcurlyeq \mathbf{0}}{\min }\hspace{1em}\mathrm{Tr}\left(\mathbf{T}\mathbf{W}\right)\\ \text{s}.\text{t}.\hspace{1em}\mathrm{Tr}\left(\left(\alpha {\sigma }^{\mathrm{2}}\mathbf{U}+{\mathbf{R}}_a-r_j{\mathrm{4}}^{{\mathrm{\Gamma }}_{\mathrm{0}}}{\sigma }^{\mathrm{2}}\mathbf{U}\right)\mathbf{W}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\ge \left(r_j{\mathrm{4}}^{{\mathrm{\Gamma }}_{\mathrm{0}}}-\alpha \right){\sigma }^{\mathrm{2}}\\ \hspace{1em}\hspace{1em}\mathrm{Tr}\left(\left({\beta }_j{\sigma }^{\mathrm{2}}{\mathbf{V}}_j+{\mathbf{R}}_{b,j}-r_j{\sigma }^{\mathrm{2}}{\mathbf{V}}_j\right)\mathbf{W}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\le \left(r_j-{\beta }_j\right){\sigma }^{\mathrm{2}},\hspace{1em}\forall E_j\in ℰ.\end{array}$$

(36)

The advantage of (36) is that it is an SDP problem and can be optimally solved [22]. Note that the obtained matrix W might not be rank 1; therefore, (36) is a relaxation of (34) for fixed r. If the rank of the solution W is larger than 1, the Gaussian Randomization method could be employed to find an approximated solution.

4.2.2. Optimization Over r

Denote the objective value of (36) as a function of r by ${\mathbb{P}}_{\mathbf{r}}$ . It can be seen that solving the original problem (33) is equivalent to minimizing ${\mathbb{P}}_{\mathbf{r}}$ with respect to r. The optimization of W has been embedded in evaluating ${\mathbb{P}}_{\mathbf{r}}$ . Since $r_j$ is a nonnegative variable with a range shown in (35), the optimal solution can be found by global optimization algorithms [26]. However, we will study the analytical property of ${\mathbb{P}}_{\mathbf{r}}$ to devise more efficient algorithms. In the following we focus on finding the optimal r to minimize ${\mathbb{P}}_{\mathbf{r}}$ .

Theorem 1.

${\mathbb{P}}_{\mathbf{r}}$ is a convex function of r.

Proof.

The Lagrangian of problem (36) is

$$\begin{array}{l}L\left(\mathbf{W},\mathrm{\Phi },\mathbf{r},\mathbf{y},\mathbf{z}\right)\\ \hspace{1em}\hspace{1em}=\mathrm{Tr}\left(\mathbf{T}\mathbf{W}\right)-\mathrm{Tr}\left(\mathrm{\Phi }\mathbf{W}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}-\underset{E_j\in ℰ}{\sum ‍}{y}_j\{\mathrm{Tr}\left(\left(\alpha {\sigma }^{\mathrm{2}}\mathbf{U}+{\mathbf{R}}_a-r_j{\mathrm{4}}^{{\mathrm{\Gamma }}_{\mathrm{0}}}{\sigma }^{\mathrm{2}}\mathbf{U}\right)\mathbf{W}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\left(r_j{\mathrm{4}}^{{\mathrm{\Gamma }}_{\mathrm{0}}}-\alpha \right){\sigma }^{\mathrm{2}}\}\\ \hspace{1em}\hspace{1em}\hspace{1em}+\underset{E_j\in ℰ}{\sum ‍}{z}_j\{\mathrm{Tr}\left(\left({\beta }_j{\sigma }^{\mathrm{2}}{\mathbf{V}}_j+{\mathbf{R}}_{b,j}-r_j{\sigma }^{\mathrm{2}}{\mathbf{V}}_j\right)\mathbf{W}\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-\left(r_j-{\beta }_j\right){\sigma }^{\mathrm{2}}\},\end{array}$$

(37)

where $\mathrm{\Phi }\succcurlyeq \mathbf{0},\mathbf{y},\mathbf{z}\ge \mathrm{0}$ are dual variables associated with the constraints in (36).

The Lagrange dual problem can be written as

$$\begin{array}{l}\underset{\mathbf{y},\mathbf{z}}{\max }\hspace{1em}\sum _{E_j\in ℰ}‍\left\{y_j\left(r_j{\mathrm{4}}^{{\mathrm{\Gamma }}_{\mathrm{0}}}-\alpha \right){\sigma }^{\mathrm{2}}+z_j\left({\beta }_j-r_j\right){\sigma }^{\mathrm{2}}\right\}\\ \text{s}.\text{t}.\hspace{1em}\mathbf{T}+\sum _{E_j\in ℰ}‍\{z_j\left({\beta }_j{\sigma }^{\mathrm{2}}{\mathbf{V}}_j+{\mathbf{R}}_{b,j}-r_j{\sigma }^{\mathrm{2}}{\mathbf{V}}_j\right)\\ \hspace{1em}\hspace{1em}\hspace{0.5em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-y_j\left(\alpha {\sigma }^{\mathrm{2}}\mathbf{U}+{\mathbf{R}}_a-r_j{\mathrm{4}}^{{\mathrm{\Gamma }}_{\mathrm{0}}}{\sigma }^{\mathrm{2}}\mathbf{U}\right)\}\succcurlyeq \mathbf{0}.\end{array}$$

(38)

Due to the strong duality between the primary problem (36) and the dual problem (38), the objective value of (38) is also ${\mathbb{P}}_{\mathbf{r}}$ . For given $\mathbf{y},\mathbf{z}$ , the objective function of (38) is linear and convex about r; therefore, its maximum over $\{\mathbf{y},\mathbf{z}\}$ , ${\mathbb{P}}_{\mathbf{r}}$ , is convex about r [22]. This completes the proof.

Since ${\mathbb{P}}_{\mathbf{r}}$ is a convex function, a gradient-based algorithm can find the global optimum of ${\mathbb{P}}_{\mathbf{r}}$ , and thus greatly reduces the complexity. We have the following straightforward results to facilitate the design of gradient-based algorithm [27].

The derivative of ${\mathbb{P}}_{\mathbf{r}}$ with respect to r is given by

$$\begin{array}{l}\frac{\partial {\mathbb{P}}_{\mathbf{r}}}{\partial r_j}=\frac{\partial L}{\partial r_j}\\ \hspace{1em}\hspace{0.17em}=y_j{\mathrm{4}}^{{\mathrm{\Gamma }}_{\mathrm{0}}}{\sigma }^{\mathrm{2}}\left(\mathrm{Tr}\left(\mathbf{U}\mathbf{W}+\mathrm{1}\right)\right)-z_j{\sigma }^{\mathrm{2}}\left(\mathrm{Tr}\left({\mathbf{V}}_j\mathbf{W}\right)+\mathrm{1}\right).\end{array}$$

(39)

The main iterative algorithm can be summarized as follows. For given r, solve both the primary problem (36) and the Lagrange dual problem (38) to obtain the primary variable W and dual variables $\mathbf{y},\mathbf{z}$ . Then update r using the gradient-based algorithm in (39). The overall iterative algorithm to solve (31) is given in Algorithm 2.

Algorithm 2: The proposed iterative RPM beamforming algorithm.

(1)  Initial State:

   Initialize α, ${\mathbf{R}}_a$ , ${\mathrm{\beta }}_j$ , ${\mathbf{R}}_{b,j},\forall E_j\in \mathcal{E}$ .

(2) Iterative  RPM  Beamforming   Algorithm:

  (i) Clock index $k=\mathrm{0}$ .

  (ii) Repeat

    (a) Calculate. calculate W in (34) for a fixed ${\mathbf{r}}_k$

    (b) Update. update ${\mathbf{r}}_{k+\mathrm{1}}$ using the gradient-based algorithm in (39).

    (c) $Increase$ . clock index $k=k+\mathrm{1}$ .

  (iii) Until convergence.

(3) Secure Transmission:

  Distribute the beamforming weights to the relays, and begin the secure data transmission.

4.3. Virtual Eavesdropper-Based Relay Power Minimization Beamforming

Similarly to the VE-SRM beamforming design, we propose a practical VE-RPM beamforming scheme to simplify the complexity of the RPM beamforming scheme. The eavesdroppers constraint in (31) can be relaxed as

$$\begin{array}{l}\underset{E_j\in ℰ}{\max }\left\{{\beta }_j+\frac{{\mathbf{w}}^{†}{\mathbf{R}}_{b,j}\mathbf{w}}{\left({\mathbf{w}}^{†}{\mathbf{V}}_j\mathbf{w}+\mathrm{1}\right){\sigma }^{\mathrm{2}}}\right\}\\ \hspace{1em}\hspace{1em}\le {\beta }_{\max }+\underset{E_j\in ℰ}{\sum ‍}\frac{{\mathbf{w}}^{†}{\mathbf{R}}_{b,j}\mathbf{w}}{\left({\mathbf{w}}^{†}{\mathbf{V}}_j\mathbf{w}+\mathrm{1}\right){\sigma }^{\mathrm{2}}}.\end{array}$$

(40)

Similar to (20),

$$\begin{array}{cc}& \frac{{\mathbf{w}}^{†}{\mathbf{V}}_j\mathbf{w}}{{\mathbf{w}}^{†}\mathbf{U}\mathbf{w}}\ge {\lambda }_{\min }\left({\mathbf{U}}^{-\mathrm{1}}{\mathbf{V}}_j\right),\hspace{1em}\forall E_j\in ℰ.\end{array}$$

(41)

The formula in (40) can be rewritten as

$$\begin{array}{l}\left(\mathrm{40}\right)\\ \le {\beta }_{\max }+\underset{E_j\in ℰ}{\sum ‍}\frac{{\mathbf{w}}^{†}{\mathbf{R}}_{b,j}\mathbf{w}}{{\lambda }_{\min }\left({\mathbf{U}}^{-\mathrm{1}}{\mathbf{V}}_j\right)\left({\mathbf{w}}^{†}\mathbf{U}\mathbf{w}+\mathrm{1}/{\lambda }_{\min }\left({\mathbf{U}}^{-\mathrm{1}}{\mathbf{V}}_j\right)\right){\sigma }^{\mathrm{2}}}\\ \le {\beta }_{\max }+\frac{{\mathbf{w}}^{†}{\mathbf{R}}_g\mathbf{w}}{\left({\mathbf{w}}^{†}\mathbf{U}\mathbf{w}+\mathrm{1}\right){\sigma }^{\mathrm{2}}},\end{array}$$

(42)

where

$$\begin{array}{cc}& {\lambda }_j\triangleq \{\begin{array}{cc}{\lambda }_{\min }\left({\mathbf{U}}^{-\mathrm{1}}{\mathbf{V}}_j\right),\hfill & {\lambda }_{\min }\left({\mathbf{U}}^{-\mathrm{1}}{\mathbf{V}}_j\right)<\mathrm{1},\hfill \\ \mathrm{1},\hfill & {\lambda }_{\min }\left({\mathbf{U}}^{-\mathrm{1}}{\mathbf{V}}_j\right)\ge \mathrm{1},\hfill \end{array}\end{array}$$

(43)

and ${\mathbf{R}}_g\triangleq {\sum }_{E_j\in ℰ}‍({\mathbf{R}}_{b,j}/{\lambda }_j)$ . Through the relaxation, it seems that there was a virtual eavesdropper, whose received SNR is

$$\begin{array}{cc}& \text{SN}{\text{R}}_{\text{VE}}={\beta }_{\max }-\mathrm{1}+\frac{{\mathbf{w}}^{†}{\mathbf{R}}_g\mathbf{w}}{\left({\mathbf{w}}^{†}\mathbf{U}\mathbf{w}+\mathrm{1}\right){\sigma }^{\mathrm{2}}},\end{array}$$

(44)

and the relay power minimization problem (31) under the secrecy rate constraint can be expressed as

$$\begin{array}{l}\underset{\mathbf{w}}{\min }\hspace{1em}{\mathbf{w}}^{†}\mathbf{T}\mathbf{w}\\ \text{s}.\text{t}.\hspace{1em}\frac{\alpha +{\mathbf{w}}^{†}{\mathbf{R}}_a\mathbf{w}/\left({\mathbf{w}}^{†}\mathbf{U}\mathbf{w}+\mathrm{1}\right){\sigma }^{\mathrm{2}}}{{\beta }_{\max }+{\mathbf{w}}^{†}{\mathbf{R}}_g\mathbf{w}/\left({\mathbf{w}}^{†}\mathbf{U}\mathbf{w}+\mathrm{1}\right){\sigma }^{\mathrm{2}}}\ge {\mathrm{4}}^{{\mathrm{\Gamma }}_{\mathrm{0}}}.\end{array}$$

(45)

Note that the minimum relay power is achieved when the constraint in (45) is an equality; that is,

$$\begin{array}{l}\underset{\mathbf{w}}{\min }\hspace{1em}{\mathbf{w}}^{†}\mathbf{T}\mathbf{w}\\ \text{s}.\text{t}.\hspace{1em}\frac{{\mathbf{w}}^{†}\left({\mathbf{R}}_a+\alpha {\sigma }^{\mathrm{2}}\mathbf{U}\right)\mathbf{w}+\alpha {\sigma }^{\mathrm{2}}}{{\mathbf{w}}^{†}\left({\mathbf{R}}_g+{\beta }_{\max }{\sigma }^{\mathrm{2}}\mathbf{U}\right)\mathbf{w}+{\beta }_{\max }{\sigma }^{\mathrm{2}}}={\mathrm{4}}^{{\mathrm{\Gamma }}_{\mathrm{0}}}.\end{array}$$

(46)

Defining $\mathbf{w}={\mathbf{T}}^{-\mathrm{1}/\mathrm{2}}\tilde{\mathbf{w}}$ , the optimization problem in (46) is equivalent to

$$\begin{array}{l}\underset{\tilde{\mathbf{w}}}{\min }\hspace{1em}{\tilde{\mathbf{w}}}^{†}\tilde{\mathbf{w}}\\ \text{s}.\text{t}.\hspace{1em}{\tilde{\mathbf{w}}}^{†}\tilde{\mathbf{R}}\tilde{\mathbf{w}}=\varsigma ,\end{array}$$

(47)

where

$$\begin{array}{cc}& \tilde{\mathbf{R}}\triangleq {\mathbf{T}}^{-\mathrm{1}/\mathrm{2}}\left({\mathbf{R}}_a+\alpha {\sigma }^{\mathrm{2}}\mathbf{U}-{\mathrm{4}}^{{\mathrm{\Gamma }}_{\mathrm{0}}}{\mathbf{R}}_g-{\mathrm{4}}^{{\mathrm{\Gamma }}_{\mathrm{0}}}{\beta }_{\max }{\sigma }^{\mathrm{2}}\mathbf{U}\right){\mathbf{T}}^{-\mathrm{1}/\mathrm{2}},\\ \varsigma \triangleq \left({\beta }_{\max }{\mathrm{4}}^{{\mathrm{\Gamma }}_{\mathrm{0}}}-\alpha \right){\sigma }^{\mathrm{2}}.\end{array}$$

(48)

Note that if ${\mathrm{\Gamma }}_{\mathrm{0}}$ is chosen such that $\varsigma >\mathrm{0}$ but $\tilde{\mathbf{R}}$ is negative definite, then the optimization problem in (46) will be infeasible and the transmit power in Stage 2 is zero. In the simulation section we will discuss the infeasibility probabilities of different beamforming schemes. If $\varsigma \le \mathrm{0}$ , the secrecy rate constraint is satisfied in Stage 1, and there is no need to waste power in Stage 2. Above all, let us assume that $\varsigma >\mathrm{0}$ and $\tilde{\mathbf{R}}$ is positive definite. By using the method of Lagrange multipliers, we obtain

$$\begin{array}{cc}& \tilde{\mathbf{R}}\tilde{\mathbf{w}}=\frac{\mathrm{1}}{\lambda }\tilde{\mathbf{w}},\end{array}$$

(49)

where λ is the Lagrange multiplier. Multiplying both sides of (49) with $\lambda {\tilde{\mathbf{w}}}^{†}$ yields

$$\begin{array}{cc}& {\tilde{\mathbf{w}}}^{†}\tilde{\mathbf{w}}=\lambda {\tilde{\mathbf{w}}}^{†}\tilde{\mathbf{R}}\tilde{\mathbf{w}}=\lambda \varsigma .\end{array}$$

(50)

Then minimizing ${\tilde{\mathbf{w}}}^{†}\tilde{\mathbf{w}}$ is equivalent to minimizing λ, so $\mathrm{1}/\lambda$ corresponds to the largest eigenvalue of $\tilde{\mathbf{R}}$ , and thus $\tilde{\mathbf{w}}$ is related to the largest eigenvector of $\tilde{\mathbf{R}}$ . Finally, the solution of (46) is

$$\begin{array}{cc}& \mathbf{w}=\mu {\mathbf{T}}^{-\mathrm{1}/\mathrm{2}}{\mathbf{v}}_{\max },\end{array}$$

(51)

where ${\mathbf{v}}_{\max }$ is the largest unit-norm eigenvector of $\tilde{\mathbf{R}}$ and the scalar μ is

$$\begin{array}{cc}& \mu =\sqrt{\frac{\varsigma }{{\mathbf{v}}_{\max }^{†}\tilde{\mathbf{R}}{\mathbf{v}}_{\max }}}.\end{array}$$

(52)

5. Simulation Results

In this section, we carry out some simulations to evaluate the secrecy rate performance of the proposed designs. For simplicity and without loss of generality, we consider a cooperative M2M network with N relays located in a ( $\mathrm{20}\hspace{0.17em}\text{m}\times \mathrm{20}\hspace{0.17em}\text{m}$ ) square area whose center is ( $\mathrm{0}\hspace{0.17em}\text{m},\mathrm{0}\hspace{0.17em}\text{m}$ ). Alice and Bob are located at ( $-\mathrm{20}\hspace{0.17em}\text{m},\mathrm{0}\hspace{0.17em}\text{m}$ ) and ( $\mathrm{20}\hspace{0.17em}\text{m},\mathrm{0}\hspace{0.17em}\text{m}$ ), respectively. The eavesdroppers are located in a $\mathrm{10}\hspace{0.17em}\text{m}\times \mathrm{10}\hspace{0.17em}\text{m}$ square area surrounding Bob. The network scenario is shown in Figure 2. The other simulation parameters are set up as follows. The noise variance is normalized; that is, ${\sigma }^{\mathrm{2}}=\mathrm{1}$ . We also assume that the power transmitted in Stage 1 is 10 dB; that is, $P_S=\mathrm{10}\hspace{0.17em}\text{dB}$ . Rayleigh fading channel is assumed, where the channel gain consists of both the path loss (i.e., free-space loss) and the Rayleigh fading coefficient. The path loss factor is $\mathrm{3.5}$ [28]. The Monte Carlo simulations consisting of $\mathrm{1000}$ independent trails are performed to obtain the average results.

Figure 2

Network scenario for simulations.

In Figure 3, we show the secrecy rates of different schemes versus the total transmit power budget when the number of relays is $\mathrm{8}$ . The number of Eves J is $\mathrm{2}$ or $\mathrm{7}$ . The figure clearly demonstrates the advantages of the proposed schemes. It can be observed that the performance of SRM beamforming scheme is always best in the three schemes, and its performance is almost the same as the upper bound of the optimal beamforming scheme when the number of relays is larger than that of Eves. Due to the relaxation in the VE-SRM beamforming scheme, there is a performance degradation compared to the SRM beamforming design. The performance of VE-SRM beamforming scheme is still better than that of the null-space beamforming scheme, especially when the number of Eves approaches the number of relays. It is because that the degree of freedom (DoF) of the null-space beamforming design becomes small as the number of eavesdroppers increases. When the number of Eves is larger than that of relays, there will be no extra DoF for the null-space beamforming scheme.

Figure 3

Secrecy rate versus total power constraint.

Figure 4 illustrates the relationship between the secrecy rate and the number of Eves. The relay number N is $\mathrm{8}$ . The source power $P_{\mathrm{0}}$ is set to $\mathrm{15}$  dB or $\mathrm{20}$  dB. The proposed SRM beamforming scheme approaches the upper bound of optimal beamforming even if the number of Eves is larger than that of relays. Also, the proposed designs exhibit great robust performance compared to the null-space beamforming scheme. As shown in Figure 4, the null-space method suffers a serious secrecy rate degradation compared to the proposed designs. When the number of Eves approaches that of the relays, the secrecy rate of the null-space beamforming drops quickly, while the proposed beamforming schemes can still yield consistent secrecy rate even when the number of Eves exceeds that of the relays.

Figure 4

Secrecy rate versus number of Eves.

Figure 5 shows the results for the relay transmit power against the secrecy rate constraint. The relay number N is $\mathrm{8}$ . The number of Eves is set to $\mathrm{2}$ , $\mathrm{4}$ , or $\mathrm{6}$ . It is observed that the relays require more and more power for a fixed secrecy rate as the number of Eves increases. However, the performance of the proposed beamforming schemes, RPM and EV-RPM beamforming schemes, is more efficient than the performance of the null-space beamforming scheme. The gaps between the proposed designs and the null-space beamforming scheme grows as the number of Eves increases, because the DoF has a serious impact on the null-space approach but less impact on our beamforming methods.

Figure 5

Relay transmit power versus secrecy rate constraint.

In Figure 6, we illustrate the infeasibility probability against the number of Eves. We assume that there are N = $\mathrm{8}$ relays and the secrecy rate constraints are ${\mathrm{\Gamma }}_{\mathrm{0}}=\mathrm{0.5}$  bps/Hz or ${\mathrm{\Gamma }}_{\mathrm{0}}=\mathrm{1}$  bps/Hz. Here, the infeasibility probability is obtained by counting the failed number of trials among the 100000 channel realizations that each algorithm is able to find the solution to meet the secrecy rate constraint. It can be seen that all the schemes will not work sometimes when secrecy rate target ${\mathrm{\Gamma }}_{\mathrm{0}}$ is large or the number of Eves is large. It is also shown that the proposed beamforming schemes are feasible under the considered secrecy rate constraint. Here, the reason is that the proposed schemes are not forced to null out all the signal at all Eves.

Figure 6

Infeasibility probability versus number of Eves.

6. Conclusion

In this paper, we have considered an AF-based cooperative protocol to improve the performance of secure wireless M2M communications in the presence of multiple eavesdroppers. The optimization criterion is the maximization of the achievable secrecy rate subject to a transmit power constraint or the minimization of the transmit power subject to a secrecy rate constraint. In previous works, the null-space beamforming schemes were imposed, at the expense of optimality, in order to obtain a simple solution. In our works, we propose beamforming schemes that do not rely on the null-space conditions, and thus our proposed beamforming designs can work well even if the number of eavesdroppers is larger than that of relays. Moreover, in the cooperative M2M network we have assumed that the global CSI is available, and even the eavesdroppers channels are known. System design of cooperative schemes that uses partial CSI or channel statistics will be considered in the future work.