Hybrid secure beamforming and vehicle selection using hierarchical agglomerative clustering for C-RAN-based vehicle-to-infrastructure communications in vehicular cyber-physical systems

Abstract

The architecture of C-RAN-based soft-defined vehicular networks can support increasing demands of various applications in the future vehicular cyber-physical systems, but lacks strong and practical security–enhancing mechanisms in the physical layer. In this article, we propose a hybrid beamforming and vehicle-selection framework for vehicle-to-infrastructure communications to broadcast high-speed confidential messages. A one-dimensional roadway scenario having multiple lanes is considered, and the concept of secure region and interference-selection region is, respectively, developed for each roadside unit. We formulate an optimization problem of maximizing secrecy sumrate with the constraint of orthogonality-based vehicle selection. The original problem is then divided into two subproblems subtly. In terms of the first subproblem, we, by exploiting the orthogonality assumption, obtain a corresponding secure beamformer for each vehicle located in the secure region to reduce the interference and confidential information leakage. Based on secure beamformers, the second subproblem is changed to be a vehicle-selection problem, which is then solved by adopting hierarchical agglomerative clustering method and implemented by two proposed novel algorithms blending vehicle selection with beamformer generation. Simulation results verify the effectiveness of the proposed algorithms in the respect of secrecy sumrate compared with the conventional zero-forcing beamforming using semi-orthogonal user selection algorithms. Furthermore, simulations show how our proposed algorithms resist against eavesdropping from collusive vehicles located in the same secure region by flexibly altering the size of secure region and interference-selection according to the traffic density.

Introduction

Cyber-physical systems (CPS), a new generation of systems with integrated computational and physical capabilities, can achieve human’s real-time and high-efficiency interactions with advanced systems through many new modality transformations from the nano-world to large-scale wide-area systems of systems.^1,2 In general, the interactions with various complex systems require co-working of control, computing, network, and physical subsystems.^3–5 By integrating those different subsystems, the potential benefits of the CPS can be developed in different networks such as cellular networks, ⁶ cognitive ad hoc networks, ⁷ and in wide application areas: intervention, precision, operation in dangerous or inaccessible environments and coordination, for example, vehicle scheduling and traffic control in vehicular networks. ⁸

With the aid of CPS, the combination of vehicular networks and CPS comes to be possible, considered as vehicular cyber-physical systems (VCPS). ⁹ By fully exploiting CPS, the next generation of VCPS is enabled to construct intelligence across different subsystems to improve security, efficiency, and energy conservation in transportation and further requires more innovative approaches in terms of system architectures that enable seamless integration of control, communication, and computation. ¹⁰ However, the integration of various subsystems, while keeping VCPS functional and operational, has been time-consuming and costly. Moreover, the increasing complexity of system components and the use of more advanced technologies for sensors, wireless communication, and processors exert much more pressure on the design of next-generation VCPS during its journey of enabling reliable integration of different critical system components. ¹¹ Traditionally, those vital components or subsystems include several specific vehicular networks with various communicating and computing abilities. It is the variety of networks and interfaces that make the joint and efficient use of resource across these different networks full of challenges. In terms of architecture designs for vehicular networks, powerful vehicular communication networks play a vital role in connecting different system components of VCPS and has been drawn great attention from both industry and academic due to its social and economic benefits expected from Intelligent Transport System (ITS). Thus, accompanying with the evolution and updating of VCPS, new innovative communication architectures of vehicular networks also need to be more integrated and compatible with future networks, which is vital and necessary to efficiently utilize vehicles and infrastructures for road services in VCPS.

Generally, in vehicular communication networks,^12–14 vehicles can flexibly communicate with each other and roadside unit (RSU) via vehicle-to-vehicle (V2V) and vehicle-to-infrastructure (V2I), respectively, and achieve information delivering and exchanging for various applications. However, with the scale of dynamic network topology and various quality-of-service requirements increasing, the architecture of conventional vehicular networks has a tendency to be more heterogeneous, such as dedicated short-range communication (DSRC) based on IEEE 802.11p for V2V communications,^15,16 co-existing with the fourth generation (4G) long-term evolution (LTE) exploited to support V2I communications. ¹⁷ Unfortunately, interfaces usually have been standardized between different protocols in conventional vehicular communication networks, thus constraining the resource utilization across different domains.

To achieve the resource sharing between different types of networks and interfaces, specialized developments for higher layer protocols need to be considered. Heterogeneous Vehicular NETworks (HetVNET), ¹⁸ by integrating different access networks including DSRC and LTE through a unified framework, can support a dynamic and instant composition of different networks and allow operators to utilize radio resources in an efficient and flexible way. Therefore, HetVNET is expected to be a practicable solution to meet various communication requirements for ITS services. In HetVNET, soft-defined network (SDN)-based cloud radio access network (Cloud-RAN) has been widely accepted to be a feasible solution for integrating various interfaces and other wireless resources efficiently, thus forming a Cloud-RAN-based HetVNET infrastructures. In general, in Cloud-RAN, ¹⁹ all RAN functionalities are achieved in the centralized central processor (CP) responsible for all of the baseband unit (BBU) processing tasks. BBUs are connected to remote radio heads (RRHs) via high bandwidth links on a wide geographical area. When performing large-scale and real-time dynamic resource scheduling in HetVNET, C-RAN is triggered and enabled to take advantages of technologies such as cloud computing, advanced RRH techniques, and SDN approaches with more open platform and more flexible operation.

However, though providing a rich set of available resource, the C-RAN-based HetVNET infrastructure is incurring significant security risks and threats from complex environment, including a formidable set of intentional overhearing on confidential message and malicious jamming on useful information possibly due to its heterogeneous features. ²⁰ Those security issues undoubtedly impact both the protocol design in the upper layer and information transmission in the physical layer, which calls for new consideration on the protocol and mechanism design. It is a critical challenge because uncertainty in the environment and possible security attacks often impose significant impact on system robustness and security, thus determining how to design a practical and robust system. Exploiting the physical nature of HetVNET by leveraging location-based, channel-based, and tag-based mechanisms is a feasible way of providing security solutions, especially for wireless vehicular communication networks.^21,22

In this article, we consider in a vehicular network the one-dimensional road scenario where RSUs connected by fiber are deployed as a C-RAN architecture to support confidential message transmission for V2I communications. Seen as a kind of hybrid location–based and channel-based scheme, a hybrid secure beamforming and vehicle-selection framework is proposed. First, we define a secure region restricted by accessing control and interference-selection region for each corresponding RSU, respectively, both of which is regulable and adjusted according to vehicle traffic, selection algorithms, and secrecy considerations. As a worst case, vehicles located in the secure region serve as the collusive eavesdroppers in a genie-aided way to overhear confidential messages transmitted by a target vehicle in the same secure region. By performing a novel general algorithm of vehicle selection, each RSU managing the corresponding secure region is enabled to perform secure beamforming for each vehicle located in this region with high speed of confidential message transmission. Our contributions can be summarized as follows:

1. The concept of secure region for C-RAN-based V2I architecture is first proposed, based on which a joint optimization problem that maximizes the secrecy sumrate with constraints of orthogonality-based vehicle selection is formulated.

2. By fully exploiting orthogonality between any two channels of vehicles, we transform the original optimization problem into two subproblems. The first subproblem gives a solution of how to determine optimal secure beamformers for vehicles with relative high orthogonality, and the other one describes how to conduct efficient vehicle-selection strategies to guarantee selected vehicles to have good channel orthogonality, thus improving the efficiency of proposed beamformer generation algorithms as described in subproblem 1.

3. In terms of the first subproblem, using generalized singular value decomposition (GSVD) method, we, for each vehicle, design a corresponding optimal beamformer, which is suboptimal for the original problem. To solve the second subproblem, we then adopt the hierarchical agglomerative clustering method and introduce two kinds of measures to determine the vehicle cluster in each process of clustering, based on which we first implement a vehicle-selection algorithm Suboptimal Algorithm for Maximum Secrecy Sumrate (SAMSS); then, when the concept of interference-selection region is defined to better tackle with the multi-user diversity and interference across different secure region with low complexity, new algorithm denoted as Suboptimal Algorithm for Maximum Secrecy Sumrate With Low Complexity (SAMSS-LC) is proposed to further implement the dynamic vehicle selection.

4. Simulation results prove that our proposed scheme outperforms the conventional Zero-forcing Semi-orthogonal User Selection (ZF-SUS) user selection algorithm. Interestingly, the secrecy sumrate of SAMSS-LC outperforms that of SAMSS in various scenarios. What’s more, our simulations show how our proposed algorithms are affected by the various traffic densities and then robustly demonstrate what characteristic the secrecy sumrate has in the presence of a high traffic density.

The remainder of this article is organized as follows: section “Related work” reviews the related works regarding vehicular networks in VCPS. In section “System model,” we present the system model overview. In section “Problem formulation and solution approach,” an optimization of joint maximizing secrecy sumrate and vehicle selection is proposed and solved. Simulation results are presented in section “Simulation results.” Finally, we conclude our work in section “Conclusion.”

Notation: Boldface is used for matrixes $\mathbf{A}$ and vectors $\mathbf{a}$ . ${\mathbf{A}}^H$ denotes the conjugate transpose of the matrix $\mathbf{A}$ . ||·|| denotes the Euclidean norm of a vector or a matrix. $\mathcal{A}$ represents a set and $\left|\mathcal{A}\right|$ denotes its cardinality. $[·{]}^{+}$ is the representation of the operation $max(0,·)$ . ⊗ is Kronecker matrix product and $\mathrm{E}\{·\}$ is the expectation operator.

Related work

The research in area of VCPS has formally commenced since National Science Foundation (NSF) CPS workshop held in 2006. The US project called the United States Council for Automotive Research (USCAR) is considering cyber-physical concepts for building a smart adaptive vehicle. One such autonomous vehicle concept is presented in Chen and Fraichard. ²³ From the academic point, research into the architecture design of vehicular network in VCPS has been extensively conducted in recent years. The multi-layer Cloud-RAN architecture ¹⁸ is proposed for implementing the new vehicular network, where the multi-domain resources can be exploited as needed for vehicle users. Authors in Zheng et al. ²⁴ develop a generic HetVNET framework focusing specifically on various services that can be provisioned by heterogeneous vehicular networks and flexible switch among various candidate networking techniques.

With the consideration of security and privacy, Papadimitratos et al. ²² propose a security architecture as a kind of location-based scheme that provides a comprehensive solution for securing vehicular communication in the respect of identity, credential, and key management in upper protocols. By assuming that a large number of certification authorities (CAs) are instantiated, served areas can be divided into many regions, in each of which one CA manages the identities and credentials of all vehicle nodes registered with it. Each vehicle located in a specific region managed by the corresponding CA cannot access its neighboring CA, while vehicles belonging to the same CA adopt secret-key based cryptographic encryption and decryption methods to keep pseudonymous with each other. However, secret-key distribution and management in the system is subject to dynamic and variable wireless environment, which incurs security vulnerabilities in wireless networks. Smart and harmful vehicles with powerful computation ability may be able to obtain the keys and eavesdrop confidential messages of vehicles in the same region.

As a type of channel-based mechanism and simultaneously a complement of cryptography techniques in the upper layer, physical layer techniques have been developed for further enhancing on the security of wireless systems in recent years.^25,26 In Wyner, ²⁷ a three-terminal network having a transceiver pair and an extra eavesdropper, known as the wiretap channel, was considered. To measure secrecy performance, secrecy capacity is first defined as the maximum achievable rate between the source and the destination while ensuring that no information can be inferred by the eavesdropper. It was shown that a non-zero secrecy capacity can be achieved for the source–destination pair if the desire receiver has better channel conditions than the eavesdropper(s).^28,29

Rather than those security analysis from the view of information theory, many coding and signal processing techniques have been developed to support and to further enhance security especially in some specific networks. ³⁰ Geraci et al. ³¹ studied the secrecy sumrate for multi-user multiple-input-multiple-output (MIMO) system with regularized channel inversion (RCI) precoding by considering a worst-case scene in which malicious users collude with each other and act as an eavesdropper to decode a certain user for obtaining its confidential message. The author obtains a deterministic approximation for the achievable secrecy sumrate with a fixed ratio of the user number to the transmit antenna number in the large-system regime. However, the method is based on the assumption that the system with a large number of antennas communicates with users simultaneously without user selection, which may be impossible in practical systems with limited transmit antenna number and the necessary user selection and scheduling mechanism for some business requirements such as the user fairness. To overcome this problem, user selection is necessary but need special design. In general, the user selection mechanism for the multi-user system has been widely studied in the past years. In MIMO systems with M transmit antennas and K ≥ M single-antenna users, the full multiplexing gain M can be achieved. Specifically, it was shown that when channel state information at the transmitter (CSIT) is known ideally, the system capacity grows like M log logK in a large-user regime K M due to multi-user diversity. ³² The scheme based on Zero-forcing Beamforming (ZFBF) or zero-forcing dirty-paper coding (ZF-DPC) has been proposed to reach this growth rate with low complexity. ³³

However, the multi-user diversity may also be exploited by malicious users which jointly act as an eavesdropper and decode the confidential messages for intended users. Worst of all, for an intended user, all the other users always tend to collude with each other for obtaining its message, whether the user selection is performed or not at the transmitter. Thus, the conventional user selection such as SUS will be limited in the respect of system secrecy sumrate. In another work, ³⁴ a joint secure beamforming and user selection (J-SBS) scheme is proposed to improve the secrecy sumrate in multi-user multiple-input-single output (MISO) broadcast channels with confidential messages (BCC). However, those mechanisms usually possibly do not apply when the topology of networks is huge and specific application scenarios are considered.

System model

In this section, we present the system model, including the C-RAN-based V2I communications architecture, scenario configuration, and confidential message transmission model. We present a more detailed framework of communications in V2I systems and propose some novel concepts accommodated to this framework. The main notations used are summarized in Table 1.

Table 1.

Summary of notations.

Notations	Description
$\mathcal{R}$	Set of RSUs in the networks
${\mathcal{L}}^i$	Set of vehicles located in the secure region
${\mathcal{S}}^i$	Set of selected vehicles from the ith secure region
${\mathcal{Q}}^{s,i}$	Set of selected vehicles in the ith secure region in the sth step of hierarchical agglomerative clustering
${\mathcal{S}}^{\mathrm{Nei},i}$	Set of vehicles served both by RSU i and RSU $i+1$ for vehicle selection
${\mathcal{B}}^{s,i}$	Set of vehicles served both by RSU i and RSU $i+1$ for vehicle selection and simultaneously located into the interference-selection region of their own adjacent RSUs in the sth process of clustering
R	Number of RSUs, $R=\left|\mathcal{R}\right|$
$L_i$	Number of vehicles in the secure region, $L_i=\left|{\mathcal{L}}^i\right|$
$S_i$	Number of selected vehicles for RSU i, $S_i=\left|{\mathcal{S}}^i\right|$
$S_{(\mathrm{num})}$	Intermediate variable indicating fixed number of selected vehicles
$S^{\mathbf{max}}$	Predefined maximum selected vehicles on the same time and frequency resource
S	Total number of selected vehicles
${\gamma }_S$	Radius of secure region
${\gamma }_L$	Radius of interference region
$N_T$	Number of antennas at each RSU
$x_{(i,j)}$	Confidential message of vehicle i
$n_{(i,j)}$	Noise at each vehicle
${\beta }_{(i,j)}$	Long term CSI between ith RSU and jth vehicle
${\mathbf{w}}_{(i,j)}$	Beamforming vector for jth vehicle at RSU i
${\mathbf{h}}_{(i,j)}$	Downlink channel vector for jth vehicle at RSU i
${\mathbf{H}}_{(i,j)}$	Downlink eavesdropping channel for jth vehicle at RSU i
$p_{i,j}$	Transmit power for vehicle j at RSU i
${\sigma }^2$	Noise power at vehicle j accessing RSU i
$\mathrm{SIN}{\mathrm{R}}_{(i,j)}$	Receiving SINR at vehicle j accessing RSU i
${\mathrm{SINR}}_{(i,j)}^a$	Receiving SINR at collusive vehicles j accessing RSU i

RSU: roadside unit; CSI: channel state information; SINR: signal to interference plus noise ratio.

C-RAN-based V2I communications architecture

This work considers the V2I communications in a C-RAN-based HetVNET ¹⁸ responsible for downlink confidential message providing to vehicles in one-dimensional roads, as shown in Figure 1. As a part of HetVNET, the architecture consisting of C-RAN based on specific wireless access techniques, for example, LTE, is adopted for providing more reliable and confidential messages. C-RAN-based networks have been researched extensively and also been deployed in many hotspots in cities of China for supporting high-speed communications service seamlessly. According to the work in Zheng et al., ¹⁸ using C-RAN techniques, Soft-defined HetVNET is enabled to integrate different access networks including DSRC and LTE to meet various communication requirements for ITS service. Thanks to its own Remote Cloud and Local Cloud, the C-RAN has ability to manage various applications such as confidential message transmission service through a large-scale centralized process which is beneficial for V2I communications. Furthermore, confidential data services, we assume, are solely provided through V2I communications, although they could be delivered among vehicles via V2V communications guaranteed by some specific protocols. Because many vehicles possibly act as untrusted nodes to track target vehicles for a long time and overhear the confidential messages, especially when traffic jam cause long-time waiting in the roads and provide sufficient time for eavesdropping vehicles to break down the upper protocol.

Figure 1.

C-RAN Based V2I communication systems under lots of one-dimensional roads.

Scenario configuration

We consider a one-dimensional road with multiple lanes, which is partitioned into segments. Without loss of generality, the segments are numbered from 1 to R, increasing in the car moving direction, as shown in Figure 2. There is one RSU placed in each segment and the RSU is placed at the center of the segment. In C-RAN-based V2I networks, each of RSUs can be seen as the RRH, usually used in C-RAN networks. In this article, we assume each of RSU has the same hardware configuration as the RRH, which usually consists of antenna system, power amplifier, low noise amplifier (LNA), and A/D converter. Since all the baseband processing functions have been concentrated into the BBUs, these modules have much less energy consumption and complexity which directly lowers their price, thus inducing easy and massive deployment along the road. Based on the above settings, there are more specific details to be considered.

Figure 2.

Secure region and interference-selection region in one-dimensional road scenario.

First, centralized control and processing in BBUs makes the centralized scheduling for large-scale vehicles possible. However, due to the limit of the antennas at RSUs, the number of selected vehicles on the same time and frequency resource is restricted. Therefore, to better provide high-speed transmission of confidential messages, vehicle selection is necessary and important for vehicle network.

Second, besides the goal of improving secrecy rate, the interference from adjacent RSUs is unavoidable during RSU’s transmitting messages to their own vehicles registered with it. Given the large-scale vehicles, how to precisely measure the interference level with low complexity is vital for high-speed message transmission. So, the measure range of interference level first needs to be defined. We assume that each service cell of RSUs can be overlapped due to the uncertainty of transmission power, channel condition, and vehicle traffic, as shown in Figure 2. This configuration also has the advantages of providing flexible multi-user diversity for vehicle selection. Now let’s introduce the details about the two types of proposed regions as follows.

Secure region

To reduce the confidential message leakage, we divide the segments served by RSUs into many small secure regions, as is shown in Figure 2. The range of secure region is predefined and each range of secure regions is identical for facilitating the modeling and analysis. Access control ³⁵ is adopted to manage the vehicles located in secure regions and tending to access RSUs for confidential message transmission request. Especially, vehicles outside the secure region served by one RSU cannot access this RSU absolutely. Nowadays, vehicles are usually equipped with global positioning system (GPS) and the movement of vehicles is constrained by the layout of roads. Thus, each RSU can know the location of all vehicles accurately by vehicles’ periodical reporting their GPS information. Therefore, location-based accessing control is practical for the current vehicle networks. Moreover, local cloud is responsible for the confidential message transmission to each vehicle going through a corresponding secure region and manages the accessing number of vehicles in this region by controlling the range of the secure region. However, although the confidentiality across different regions can be guaranteed by above location-based accessing control, confidential messages of vehicles within secure regions can be eavesdropped and obtained by some other deliberate vehicles within the same secure region due to its vulnerability of upper protocols, especially for vehicle communications usually with high vehicle density. Usually, those messages including safety beacons that report a vehicles location, which is possibly utilized by collusive vehicles to track the target vehicles location and transactions, infer confidential information about its driver and passengers. Moreover, one secure region should disjoint with its neighboring secure region, so that systems can reserve one spare region for handover. However, dealing with handover problems is not the focus of this article.

Interference-selection region

Also to better avoid co-channel interference among selected vehicles on the same time and frequency resource block, we define the interference-selection region as such a segment with its radius larger than that of the secure region belonging to the same RSU. The range of interference-selection is predefined and each range of regions is identical. Although a large number of vehicles in the network usually generate interference among each other, we only consider the interference from vehicles located in the interference-selection region when we perform interference estimation during the vehicle selection. Obviously, the range of interference-selection determines the number of vehicles deserving being considered to calculate the interference, thus influencing the accuracy of interference calculation methods. It is worth noting that reasonable setting of the subsequent simulation has verified the effectiveness of this type of region.

Confidential message transmission model

We consider the downlink information transmission process in which each RSU i and its serving vehicles can be seen as a downlink multi-user MISO BCC system with a transmitter having $N_T$ antennas and $S_i$ single-antenna-eq1 vehicles, as is shown in Figure 2. Each RSU i has its own serving vehicles constituting set ${\mathcal{L}}^i$ satisfying ${\mathcal{L}}^i\supset {\mathcal{S}}^i,i\in \mathcal{R}$ for secure region i and its own interference vehicles in ${\mathcal{I}}^i$ satisfying ${\mathcal{I}}^i\supset {\mathcal{L}}^i\supset {\mathcal{S}}^i,i\in \mathcal{R}$ for interference-selection region i. The ith RSU transmits $S_i$ independent confidential messages to $S_i$ vehicles only located in the secure region i over the downlink channel ${\mathbf{h}}_{(i,j)}=\sqrt{{\beta }_{(i,j)}}{\mathbf{g}}_{(i,j)}\in {\mathbb{C}}^{1\times N_T},i\in \mathcal{R},j\in {\mathcal{S}}^i$ and those vehicles always suffer from the interference from other RSUs. As is interpreted in the above section, the interference-selection regions are set in order to measure the interference and reduce computational complexity and may overlap with secure regions to provide more multi-user diversity: ${\mathcal{I}}^i\cap {\mathcal{L}}^i\ne \varnothing ,\forall i\in \mathcal{R}$ . Moreover, $S_i$ is usually predetermined by the transmitter in the previous slot or block according to a certain principle which is not our focus in this article. Therefore, given the confidential message $x_{(i,j)}~\mathcal{C}\mathcal{N}(0,1),i\in \mathcal{R},j\in {\mathcal{S}}^i$ and the noise $n_{(i,j)}~\mathcal{C}\mathcal{N}(0,{\sigma }^2)$ at the vehicle j, the received signal $y_{(i,j)}$ at vehicle j served by RSU i can be given by

$$\begin{array}{cc}\multicolumn{1}{c}{y_{(i,j)}=}& \sqrt{p_{i,j}}{\mathbf{h}}_{(i,j)}{\mathbf{w}}_{(i,j)}{x}_{(i,j)}+{\mathbf{h}}_{(i,j)}\sum _{m\in {\mathcal{S}}^i,m\ne j}\sqrt{p_{i,m}}{\mathbf{w}}_{(i,m)}{x}_{(i,m)}\hfill \\ \multicolumn{1}{c}{}& +\sum _{k\in \mathcal{R},k\ne i}\sum _{m\in {\mathcal{S}}^k}\sqrt{p_{k,m}}{\mathbf{h}}_{(k,j)}{\mathbf{w}}_{(k,m)}{x}_{(k,m)}+n_{(i,j)}\hfill \end{array}$$

(1)

We assume perfect channel state information (CSI) is known at the transmitter, based on which each beamforming vector ${\mathbf{w}}_{(i,j)}\in {\mathbb{C}}^{N_T\times 1},i\in \mathcal{R},j\in {\mathcal{L}}^i$ can be derived. Moreover, we assume a worst-case scenario where for each intended vehicle j satisfying $j\in {\mathcal{S}}^i,i\in \mathcal{R}$ the remaining $L_i-1,i\in \mathcal{R}$ vehicles can cooperate to jointly eavesdrop on the confidential message $x_{(i,j)},j\in {\mathcal{S}}^i,i\in \mathcal{R}$ . There are always $L_i-1$ users seen as an equivalent eavesdropper denoted by a to decode the message $x_{(i,j)},j\in {\mathcal{S}}^i,i\in \mathcal{R}$ , although the ith RSU in fact just select $S_i$ users to perform data transmission in one block fading channel realization. Thus, the signals observed by the user a can be given by

$$\begin{array}{cc}\multicolumn{1}{c}{{\mathbf{y}}_{(i,j)}^a=}& \sqrt{p_{i,j}}{\mathbf{H}}_{(i,j)}{\mathbf{w}}_{(i,j)}{x}_{(i,j)}\hfill \\ \multicolumn{1}{c}{}& +{\mathbf{H}}_{(i,j)}\sum _{m\in {\mathcal{S}}^i,m\ne j}\sqrt{p_{i,m}}{\mathbf{w}}_{(i,m)}{x}_{(i,m)}\hfill \\ \multicolumn{1}{c}{}& +\sum _{k\in \mathcal{R},k\ne i}\sum _{m\in {\mathcal{S}}^k}\sqrt{p_{k,m}}{\mathbf{H}}_{(k,j)}{\mathbf{w}}_{(k,m)}{x}_{(k,m)}+{\mathbf{n}}_{(i,j)}\hfill \end{array}$$

(2)

where the channel matrix ${\mathbf{H}}_{(i,j)}$ aggregated by the channel of eavesdropping vehicles is the channel of the equivalent eavesdropper a given by

$$H_{\left(i,j\right)}={\left[\begin{array}{cccccc}{h}_{\left(i,1\right)}^H& \cdots & h_{\left(i,j-1\right)}^H& h_{\left(i,j+1\right)}^H& \cdots & h_{\left(i,L_i\right)}^H\end{array}\right]}^H,i\in ℛ,j\in {ℒ}^i$$

(3)

Problem formulation and solution approach

In this section, we define the achievable secrecy sumrate in C-RAN-based V2I communication systems. Based on the secrecy sumrate, we formulate the vehicle selection and secure beamforming design as a joint optimization problem. To solve the optimization problem, we propose two suboptimal algorithms with different complexities.

Achievable secrecy sumrate

As is shown in Geraci et al., ³¹ if the transmitter uses independent codebooks for each user, where each codebook is a code for the scalar wiretap channel, an achievable secrecy sumrate can be derived as the sum of secrecy rate of each user. In this article, we adopt the assumption that each RSU $i,i\in \mathcal{R}$ transmit $S_i$ independent confidential messages to $S_i$ vehicles using independent codebooks simultaneously. Although vehicles served by a certain RSU cannot access any other neighboring RSUs due to the access control policy, the information leakage during downlink confidential message transmission is unavoidable. So information acquisition and interference canceling can be done by the collusive eavesdropper, especially based on some learning algorithms. As a worst case, collusive vehicles can perform cancellation of the interference not only from each other but also from the other vehicles served by other RSUs in a genie-aided way, and it does not see any undesired signal term apart from the received noise. In the end, according to the previous considerations, the sum of secrecy sumrate of the systems can be given by

$$R_s=\sum _{i\in \mathcal{R}}\sum _{j\in {\mathcal{S}}^i}{[{\log }_2(1+\mathrm{SIN}{\mathrm{R}}_{(i,j)})-{\log }_2(1+{\mathrm{SINR}}_{(i,j)}^a)]}^{+}$$

(4)

where $\mathrm{SIN}{\mathrm{R}}_{(i,j)}$ and ${\mathrm{SINR}}_{(i,j)}^a$ , respectively, denote the signal to interference plus noise ratios (SINRs) for message $x_{(i,j)}$ at the user j and the eavesdropper a, which are given by

$${\mathrm{SINR}}_{(i,j)}=\frac{\frac{p_{i,j}}{{\sigma }^2}{\left|h_{\left(i,j\right)}{w}_{\left(i,j\right)}\right|}^2}{1+\frac{p_{i,m}}{{\sigma }^2}{\displaystyle \sum _{m\in {\mathcal{S}}^i,m\ne j}{\left|h_{\left(i,j\right)}{w}_{\left(i,m\right)}\right|}^2}+\frac{p_{k,m}}{{\sigma }^2}{\displaystyle \sum _{k\in ℛ,k\ne i}{\displaystyle \sum _{m\in {\mathcal{S}}^k}{\left|h_{\left(k,j\right)}{w}_{\left(k,m\right)}\right|}^2}}}$$

(5)

and

$${\mathrm{SINR}}_{(i,j)}^a=\frac{p_{i,j}}{{\sigma }^2}{\left|{\mathbf{H}}_{(i,j)}{\mathbf{w}}_{(i,j)}\right|}^2$$

(6)

Existing ZFBF-SUS algorithm

To exploit the multi-user diversity sufficiently, the authors in Yoo and Goldsmith ³⁶ adopt a ZFBF strategy with SUS algorithm where the system can achieve the same asymptotic sum capacity as that dirty-paper coding (DPC), as the number of users goes to infinity. Using SUS algorithm, the transmitter selects user group ${\mathcal{S}}^i,i\in \mathcal{R}$ with semi-orthogonal channels and relatively large multi-user gain. For convenience, we review the SUS algorithm as follows:

1. Initialization: ${\mathcal{L}}_{i,m}={\mathcal{L}}_i,{\mathcal{S}}^i=\varnothing ,m=1$ for the ith secure region served by RSU $i,i\in \mathcal{R}$ .

2. The transmitter calculates ${\mathbf{r}}_{(i,j)}$ , which is the component of the beamforming vector ${\mathbf{h}}_{(i,j)}$ of each user j in ${\mathcal{L}}_{i,m}$ . The component is orthogonal to the subspace spanned by $\{{\mathbf{r}}_{(i,1)},\dots ,{\mathbf{r}}_{(i,m-1)}\}$

$${\mathbf{r}}_{(i,j)}={\mathbf{h}}_{(i,j)}-\sum _{k=1}^{m-1}\frac{{\mathbf{h}}_{(i,j)}{\mathbf{r}}_{(i,k)}^H}{{\Vert {\mathbf{r}}_{(i,k)}\Vert }^2}{\mathbf{r}}_{(i,k)}$$

(7)

if $m=1,{\mathbf{r}}_{(i,j)}={\mathbf{h}}_{(i,j)}$

• 3. In this step, we update the user group as follows

$$\begin{array}{c}\pi (i,m)=\arg \underset{j\in L_{(i,m)}}{max}\Vert {\mathbf{r}}_{(i,j)}\Vert \hfill \\ {\mathcal{S}}^i\leftarrow {\mathcal{S}}^i\cup \{\pi (i,m)\},{\mathbf{h}}_{(i,m)}={\mathbf{h}}_{(i,\pi (i,m))},{\mathbf{r}}_{(i,m)}={\mathbf{r}}_{(i,\pi (i,m))}\hfill \end{array}$$

(8)

• 4. If $\left|{\mathcal{S}}^i\right|\le S_i$ , we calculate ${\mathcal{L}}_{i,m+1}$ and go to step 2; otherwise, the algorithm is finished

$${ℒ}_{i,m+1}=\left\{j\in {ℒ}_{i,m},j\ne \pi (i,m),\frac{h_{(i,j)}{r}_{(i,m)}^H}{\Vert h_{(i,j)}\Vert \Vert r_{(i,m)}\Vert }<\alpha \right\},m=m+1$$

(9)

where $\alpha$ is a positive constant. When the user selection has been done, for selected user group ${\mathcal{S}}^i,i\in \mathcal{R}$ , each beamforming column vector ${\mathbf{w}}_{(i,j)}\in {\mathbb{C}}^{N_T\times 1},i\in \mathcal{R},j\in {\mathcal{L}}^i$ can be easily obtained.

However, when considering the scene where the eavesdropper a composed of multiple collusive vehicles decodes the confidential message $x_{(i,j)},j\in {\mathcal{S}}^i,i\in \mathcal{R}$ , the system secrecy sumrate will be limited even reduced. Although adopting ZFBF can reduce the confidential message eavesdropped by the other co-selection vehicle users, the transmitter does not adopt any technique to suppress the confidential message leakage caused by the unselected collusive vehicle users. Furthermore, ZF-SUS algorithm usually takes into consideration the interference in the same cell but not any matter of interference from RSUs around, which is not optimal for our proposed V2I communications network. In this article, we adopt ZF-SUS algorithm only as a reference, in which each RSU performs ZF-SUS-based vehicle selection for vehicles located in the corresponding secure region.

Novel optimization problem

As is shown in section “Existing ZFBF-SUS algorithm,” the vehicle selection and secure beamformer design should be considered jointly. This article considers the problem of joint secure beamforming design and vehicle selection to each RSU $i,i\in \mathcal{R}$ transmit $S_i$ . The objective of this problem is to determine the selection scheme of vehicles in each secure region and then how to conduct beamforming for those selected vehicles such that the achievable secrecy sumrate in the systems can be maximized. Mathematically, this problem can be formulated as follows:

1. The maximum number of selected vehicles for each RSU i is no more than $S^{\mathbf{max}}$ .

2. Each RSU has identical and fixed power budget. Among vehicles located in the secure region i, average power allocation is adopted based on the number of selected vehicles.

3. Each RSU only selects from the vehicles located in the secure region the desired vehicles, each of which has beamformer being orthogonal with each other. This restriction is imposed in order to reduce the interference caused by vehicles located in the same secure region.

Thus, we formulate the problem maximizing secrecy sumrate of the V2I communications as follows

$$\begin{array}{c}\underset{{\mathbf{w}}_{(i,j)},{\mathcal{S}}^i,i\in \mathcal{R},j\in {\mathcal{S}}^i}{\max }{R}_s\hfill \\ \mathrm{subject}\mathrm{to}:max\left|{\mathcal{S}}^i\right|\le S^{\mathbf{max}},i\in \mathcal{R}\hfill \\ \Vert {\mathbf{w}}_{(i,j)}{\Vert }^2=1,i\in \mathcal{R},j\in {\mathcal{S}}^i\hfill \\ {\left|{\mathbf{w}}_{(i,j)}^H{\mathbf{w}}_{(i,k)}\right|}^2\le \alpha ,i\in \mathcal{R},j\in {\mathcal{S}}^i,k\in {\mathcal{S}}^i,k\ne j\hfill \\ p_{i,j}=P_{max}/S_i,i\in \mathcal{R},j\in {\mathcal{S}}^i\hfill \\ {\mathcal{S}}^i\subset {\mathcal{L}}^i\hfill \end{array}$$

(10)

Obviously, the problem is non-convex and difficult to obtain its global optimal solution, accompanying with extremely high computational complexity. However, we propose suboptimal algorithms to efficiently obtain the suboptimal solutions with low complexity. We divide the original optimization problem into two subproblems. The subproblem 1 aims to perform beamforming for each of vehicles in secure region under the assumption that each RSU always select fixed number of vehicles in secure regions but do not know which vehicle deserve being selected. Thus, subproblem 1 creates a situation that the number of alternative beamformers in each secure region is relatively large, providing much more multi-user diversity for the system. And then the next step is to design suitable algorithms able to select vehicles inducing maximum secrecy sumrate. Therefore, subproblem 2 is adopted aiming to select vehicles which satisfies orthogonal properties and brings maximum secrecy maximum sumrate. The mathematical expression of two subproblems is given as follows:

• 1. Subproblem 1 (Beamformer Design). Optimization of ${\mathbf{w}}_{(i,j)},i\in \mathcal{R},j\in {\mathcal{L}}^i$ under fixed $S_{(\mathrm{num})}$

$$\begin{array}{c}\mathrm{P}\left(\left|{\mathcal{S}}^i\right|\right):\underset{{\mathbf{w}}_{(i,j)},{\mathcal{S}}^i,i\in \mathcal{R},j\in {\mathcal{S}}^i}{\max }{R}_s(\left|{\mathcal{S}}^i\right|,{\mathbf{w}}_{(i,j)})\hfill \\ \mathrm{subject\; to}:\left|{\mathcal{S}}^i\right|=S_{\left(\mathrm{num}\right)},i\in \mathcal{R}\hfill \\ \Vert {\mathbf{w}}_{(i,j)}{\Vert }^2=1,i\in \mathcal{R},j\in {\mathcal{L}}^i\hfill \\ {\left|{\mathbf{w}}_{(i,j)}^H{\mathbf{w}}_{(i,k)}\right|}^2\le \alpha ,i\in \mathcal{R},j\in {\mathcal{S}}^i,k\in {\mathcal{S}}^i,k\ne j\hfill \\ p_{i,j}=P_{max}/S_{\left(\mathrm{num}\right)},i\in \mathcal{R},j\in {\mathcal{L}}^i\hfill \end{array}$$

(11)

• 2. Subproblem 2 (Vehicle Selection). Optimization of ${\mathcal{S}}_i$ under fixed ${\mathbf{w}}_{(i,j)}^{\#},i\in \mathcal{R},j\in {\mathcal{L}}^i$ which is the solution of subproblem 1

$$\begin{array}{c}\mathcal{T}\left(\left|{\mathcal{S}}^i\right|\right):\underset{{\mathcal{S}}^i,i\in \mathcal{R}}{\max }{R}_s\left({\mathbf{w}}_{(i,j)}^{\#}\right)\hfill \\ \mathrm{subject\; to}:\left|{\mathcal{S}}^i\right|=S_{\left(\mathrm{num}\right)},i\in \mathcal{R}\hfill \\ 1\le S_{\left(\mathrm{num}\right)}\le S^{max}\hfill \\ {\left|{\mathbf{w}}_{(i,j)}^H{\mathbf{w}}_{(i,k)}\right|}^2\le \alpha ,i\in \mathcal{R},j\in {\mathcal{S}}^i,k\in {\mathcal{S}}^i,k\ne j\hfill \end{array}$$

(12)

Solution approach

To implement valid algorithms solving above two subproblems, we propose two corresponding methods, such as secrecy beamforming design and novel vehicle-selection algorithm. Specifically, secrecy beamforming design method can induce an optimal solution of the subproblem 1 when $\alpha =0$ , while using novel vehicle-selection algorithms we can obtain two suboptimal solutions of subproblem 2:

1. Secrecy beamforming design. In terms of the subproblem 1, the optimal beamformer is difficult to obtain. The interference to vehicle $(i,j)$ from neighboring RSU is small if we adopt a reasonable beamformer.

The interference to vehicle $(i,j)$ from other RSU $l,l\ne i,l\ne i-1,l\ne i+1$ is usually very small due to the low transmit power and long transmission distance of these RSUs. Therefore, we can approximately transform the problem as

$$R_{s,\left(i,j\right)}={\left[\underset{w_{\left(i,j\right)}:{\Vert w_{\left(i,j\right)}\Vert }^2=1}{\max }{\log }_2\left(\frac{1+\frac{\frac{p_{i,j}}{{\sigma }^2}{h}_{\left(i,j\right)}{w}_{\left(i,j\right)}{w}_{\left(i,j\right)}^H{h}_{\left(i,j\right)}^H}{1+\frac{p_{i,j}}{{\sigma }^2}{h}_{\left(i,j\right)}{\displaystyle \sum _{m\in {\mathcal{S}}^i,m\ne j}{w}_{\left(i,j\right)}{w}_{\left(i,j\right)}^H}{h}_{\left(i,j\right)}^H}}{1+\frac{p_{i,j}}{{\sigma }^2}{w}_{\left(i,j\right)}^H{H}_{\left(i,j\right)}^H{H}_{\left(i,j\right)}{w}_{\left(i,j\right)}}\right)\right]}^{+},i\in ℛ,j\in {ℒ}^i$$

(13)

From the above, the optimal beamforming ${\mathbf{w}}_{(i,j)}^{\#}$ can be derived by maximizing the ratio inside the logarithm. Obviously, we need to obtain the beamforming vectors of the interference. However, since the transmit beamformers employed by RSUs are unknown due to the uncertainty of vehicle selection, a feasible solution is that each vehicle employs an expected covariance matrix of the possible interference to overcome this type of selection uncertainty.

Lemma ³⁷

For a given user j served by i RSU with channel ${\mathbf{h}}_{(i,j)}$ , if $\alpha$ is zero, the expected covariance matrix of the interference whose streams are distributed in the null space of ${\mathbf{w}}_{(i,j)}$ can be computed as

$$\begin{array}{l}E\left[h_{\left(i,j\right)}{\displaystyle \sum _{m\in {\mathcal{S}}^i,m\ne j}{w}_{\left(i,j\right)}{w}_{\left(i,j\right)}^H}{h}_{\left(i,j\right)}^H\right]\\ =\frac{S_i-1}{N_T-1}{h}_{\left(i,j\right)}{W}_{\left(i,j\right)}^{\perp }{h}_{\left(i,j\right)}^H,i\in ℛ,j\in {ℒ}^i\end{array}$$

(14)

where ${\mathbf{W}}_{(i,j)}^{\perp }$ satisfies ${\mathbf{W}}_{(i,j)}^{\perp }={\mathbf{I}}_{N_T}-{\mathbf{w}}_{(i,j)}{\mathbf{w}}_{(i,j)}^H$ . Thus, we obtain a lower bound on the SINR for the vehicle j served by RSU i

$${\mathrm{SINR}}_{(i,j)}^{\mathrm{Lower}}=\frac{\frac{p_{i,j}}{{\sigma }^2}{\mathbf{h}}_{(i,j)}{\mathbf{w}}_{(i,j)}{\mathbf{w}}_{(i,j)}^H{\mathbf{h}}_{(i,j)}^H}{1+\frac{p_{i,j}}{{\sigma }^2}\frac{S_i-1}{N_T-1}{\mathbf{h}}_{(i,j)}({\mathbf{I}}_M-{\mathbf{w}}_{(i,j)}{\mathbf{w}}_{(i,j)}^H){\mathbf{h}}_{(i,j)}^H}$$

(15)

According to the above principle, the optimization problem (8) can be transformed into maximizing the lower bound of secrecy rate for each vehicle. Thus, the achievable secrecy rate for each vehicle can be updated as maximizing the lower bound of secrecy rate, which is shown as follows

$$R_{s,\left(i,j\right)}={\left[\underset{w_{\left(i,j\right)}:{\Vert w_{\left(i,j\right)}\Vert }^2=1}{\max }{\log }_2\left(\frac{1+\frac{\frac{p_{i,j}}{{\sigma }^2}{h}_{\left(i,j\right)}{w}_{\left(i,j\right)}{w}_{\left(i,j\right)}^H{h}_{\left(i,j\right)}^H}{1+\frac{p_{i,j}}{{\sigma }^2}\frac{S_i-1}{N_T-1}{h}_{\left(i,j\right)}\left(I_M-w_{\left(i,j\right)}{w}_{\left(i,j\right)}^H\right)h_{\left(i,j\right)}^H}}{1+\frac{p_{i,j}}{{\sigma }^2}{w}_{\left(i,j\right)}^H{H}_{\left(i,j\right)}^H{H}_{\left(i,j\right)}{w}_{\left(i,j\right)}}\right)\right]}^{+}$$

(16)

$$={\left[\underset{w_{\left(i,j\right)}:{\Vert w_{\left(i,j\right)}\Vert }^2=1}{\max }{\log }_2\left(\frac{w_{\left(i,j\right)}^H{A}_{\left(i,j\right)}{w}_{\left(i,j\right)}}{w_{\left(i,j\right)}^H{B}_{\left(i,j\right)}{w}_{\left(i,j\right)}}\right)\right]}^{+},i\in ℛ,j\in {ℒ}^i$$

(17)

where

$$A_{\left(i,j\right)}=\left(1+\frac{p_{i,j}}{{\sigma }^2}\frac{S_i-1}{N_T-1}{h}_{\left(i,j\right)}{h}_{\left(i,j\right)}^H\right)\otimes I_{N_T}+\frac{p_{i,j}}{{\sigma }^2}\frac{N_T-S_i}{N_T-1}{h}_{\left(i,j\right)}^H{h}_{\left(i,j\right)},i\in ℛ,j\in {ℒ}^i$$

(18)

and

$$B_{\left(i,j\right)}=\left(\left(1+\frac{p_{i,j}}{{\sigma }^2}\frac{S_i-1}{N_T-1}{h}_{\left(i,j\right)}{h}_{\left(i,j\right)}^H\right)\otimes I_{N_T}-\frac{p_{i,j}}{{\sigma }^2}\frac{S_i-1}{N_T-1}{h}_{\left(i,j\right)}^H{h}_{\left(i,j\right)}\right)\left(I_{N_T}+\frac{p_{i,j}}{{\sigma }^2}{H}_{\left(i,j\right)}^H{H}_{\left(i,j\right)}\right),i\in ℛ,j\in {ℒ}^i$$

(19)

This definition of achievable secrecy rate can be seen as maximizing the lower bound of genuine secrecy rate at each vehicle, which is reasonable. To solve the above optimization, we maximize the ratio inside the logarithm by applying the GSVD method. ³⁸

Theorem 1

Given that the selected users in group ${\mathcal{S}}^i,i\in \mathcal{R}$ has mutually orthogonal beamformers, the maximum secrecy rate achievable for the j vehicle belonging to ${\mathcal{S}}^i,i\in \mathcal{R}$ served by RSU i can be given by

$$R_{s,(i,j)}={\left[\mathrm{lo}{\mathrm{g}}_2{\lambda }_{max}({\mathbf{A}}_{(i,j)},{\mathbf{B}}_{(i,j)})\right]}^{+},i\in \mathcal{R},j\in {\mathcal{L}}^i$$

(20)

where ${\mathbf{A}}_{(i,j)}$ and ${\mathbf{B}}_{(i,j)}$ can be obtained from equations (18) and (19), respectively. ${\lambda }_{max}({\mathbf{A}}_{(i,j)},{\mathbf{B}}_{(i,j)})$ is the maximum generalized eigenvalue of the matrix pair $({\mathbf{A}}_{(i,j)},{\mathbf{B}}_{(i,j)})$ . The corresponding optimal precoding matrix is given by

$${\mathbf{w}}_{(i,j)}^{\#}={\psi }_{max}({\mathbf{A}}_{(i,j)},{\mathbf{B}}_{(i,j)}),i\in \mathcal{R},j\in {\mathcal{L}}^i$$

(21)

where ${\psi }_{max}({\mathbf{A}}_{(i,j)},{\mathbf{B}}_{(i,j)})$ is the generalized eigenvector corresponding to the maximum generalized eigenvalue ${\lambda }_{max}(\mathbf{A},\mathbf{B})$ .

When deriving the secure beamformers above, we now calculate the true secrecy sumrate obtained by each vehicle as follows

$$R_{s,\left(i,j\right)}^{\mathrm{True}}{=\log }_2{\left[\frac{1+\frac{\frac{p_{i,j}}{{\sigma }^2}{h}_{\left(i,j\right)}{w}_{\left(i,j\right)}{w}_{\left(i,j\right)}^H{h}_{\left(i,j\right)}^H}{1+\frac{p_{i,j}}{{\sigma }^2}{h}_{\left(i,j\right)}{\displaystyle \sum _{m\in {\mathcal{S}}^i,m\ne j}{w}_{\left(i,j\right)}{w}_{\left(i,j\right)}^H{h}_{\left(i,j\right)}^H}+\frac{p_{k,m}}{{\sigma }^2}{\displaystyle \sum _{k\in ℛ,k\ne i}{\displaystyle \sum _{m\in {\mathcal{S}}^k}{\left|h_{\left(k,j\right)}{w}_{\left(k,m\right)}\right|}^2}}}}{1+\frac{p_{i,j}}{{\sigma }^2}{w}_{\left(i,j\right)}^H{H}_{\left(i,j\right)}^H{H}_{\left(i,j\right)}{w}_{\left(i,j\right)}}\right]}_{w=w^{\#}}^{+},i\in ℛ,j\in {ℒ}^i$$

(22)

Once the above secrecy sumrate is obtained, we can perform the following vehicle-selection algorithms based on clustering analysis to solve subproblem 2.

• 2. Novel vehicle-selection algorithms. In this subsection, we first describe the cluster analysis method, ³⁹ based on which we propose two algorithms to select vehicles: SAMSS and SAMSS-LC, respectively.

Cluster analysis distinguishes and groups users (observations, events) based on the information found in the data describing the users or their relationships. The goal is that the users in a group will be similar to one another and different from the objects in other groups. The greater the similarity within a group, and the greater the difference between groups, the better or more distinct the clustering. How to define similarity and difference and adopt which clustering method is important for clustering analysis.

Hierarchical method: agglomerative clustering

As one technique of cluster analysis, hierarchical clustering usually starts with one large cluster and split it, which is denoted as Division. Or it starts with sub-clusters each of which containing a point or multiple points, and then merges them, which is defined as Agglomeration.

We, in this article, adopt the agglomerative clustering method as is shown in Figure 3 and start with sub-clusters each of which is obtained according to orthogonality-based vehicle pairing (OBVP) measure. Each sub-cluster has at most R vehicles. Then, we, at each step, merge or combine the closest pair of clusters according to the sum of neighboring secrecy sumrate (SNSS) measure until the decision criterion for the algorithm is broken, as is shown in Figure 3. Both of two measures will be interpreted in the following.

Figure 3.

An illustration of the process in the hierarchical agglomerative clustering method using traditional nested set (left) and dendrogram (right).

Objective functions

Many clustering techniques are based on trying to minimize or maximize a global objective function, which, in theory, can be solved by enumerating all possible ways of dividing the points into clusters and evaluating each potential set of clusters according to the objective function. However, this approach is computationally infeasible (NP complete) and as a result, many practical techniques for optimizing a global objective function have been developed. One approach is to rely on algorithms, which find solutions that are often good, but not optimal.

Hierarchical clustering procedures proceed by making local decisions at each step of the clustering process. These “local” decisions are also based on an objective function or a specified measure, though the overall or global result is not easily interpreted in terms of a global objective function. Different from the common proximity measures such as distance measures, ordinal measures, and cosine measure, we in this article define a two-limit measure model including OBVP measure for vehicles in each secure region and SNSS measure for vehicles across different secure regions, to help systems make local decisions. Obviously, SNSS measure determines the element belonging to each sub-cluster in each step, and OBVP measure determines how to merge different sub-clusters across different steps.

OBVP

To determine each sub-cluster jointly with the design mechanism of beamformer generation, we, from each secure region, select outstanding vehicles among which the orthogonality measure should be satisfied. This measure is derived by calculating the orthogonality of channels between vehicles in the secure region

$$O_{(i,j,k)}={\left|{\left({\mathbf{w}}_{(i,j)}^{\#}\right)}^H{\mathbf{w}}_{(i,k)}^{\#}\right|}^2,i\in \mathcal{R},k\in {\mathcal{S}}^i,k\ne j$$

(23)

Using this measure, we obtain the vehicle set as follows

$${\mathcal{Q}}^{s,i}=\{\begin{array}{c}\bigcup _{j\in {\mathcal{S}}^i,k\in {\mathcal{Q}}^{s-1,i}}\{j|O_{(i,j,k)}=0,i\in \mathcal{R}\},s\ne 1\hfill \\ \multicolumn{1}{c}{\{j|\underset{j}{max}{\left|{\mathbf{h}}_{(i,j)}{\mathbf{w}}_{(i,j)}^{\#}\right|}^2,i\in \mathcal{R},j\in {\mathcal{S}}^i\},s=1}\end{array}$$

(24)

which constituting the sub-cluster s. The operational sth sub-cluster in each step s can then be defined as $\bigcup _{i=1,\dots ,R}{\mathcal{Q}}^{s,i}$ satisfying the condition

$$\bigcup _{s=1,\dots ,S_{\left(\mathrm{num}\right)}}{\mathcal{Q}}^{s,i}={\mathcal{S}}^i,1\le j\le R$$

(25)

As is shown above, there possibly exists many alternative sub-cluster sets. It is vital to know how to confirm which sub-cluster is optimal according to some measure method. In the following, we give the specific definition of SNSS.

SNSS

We first calculate a simplified message transmission rate $R_{s,(i,j)}^{\mathrm{Sim}},j\in {\mathcal{Q}}^{s,i}$ , which is derived by calculating the secrecy sumrate of pre-selection vehicles in set ${\mathcal{Q}}^{s,i},i\in \mathcal{R}$ with only consideration of interference from its adjacent RSU $i+1,i\in \mathcal{R}$ . Similarly, $R_{s,(i,j)}^{\mathrm{Sim}},j\in {\mathcal{Q}}^{s,i+1}$ , is derived by calculating the secrecy sumrate of pre-selection vehicles in set ${\mathcal{Q}}^{s,i+1},i\in \mathcal{R}$ with only consideration of interference from its adjacent RSU $i,i\in \mathcal{R}$ .

Then, we define the neighboring secrecy sumrate (NSS) as the intermediate measure which is obtained as: $\sum _{j\in {\mathcal{Q}}^{s,i}}{R}_{s,(i,j)}^{\mathrm{Sim}}+\sum _{j\in {\mathcal{Q}}^{s,i+1}}{R}_{s,(i+1,j)}^{\mathrm{Sim}}$ . SNSS measure can then be obtained by summing up the NSS with i from 1 to $R-1$ . In the end, average SNSS serving as the objective function represents the ultimate measure of agglomerative clustering method, which can be formulated as follows

$$M\left(\bigcup _{i\in \mathcal{R}}{\mathcal{Q}}^{s,i}\right)=\frac{1}{S}\sum _{i=1}^{R-1}(\sum _{j\in {\mathcal{Q}}^{s,i}}{R}_{s,(i,j)}^{\mathrm{Sim}}+\sum _{j\in {\mathcal{Q}}^{s,i+1}}{R}_{s,(i+1,j)}^{\mathrm{Sim}})$$

(26)

where $R_{s,(i,j)}^{\mathrm{Sim}}$ satisfies

$$R_{s,\left(i,j\right)}^{\mathrm{True}}{=\log }_2{\left[\frac{1+\frac{\frac{p_{i,j}}{{\sigma }^2}{h}_{\left(i,j\right)}{w}_{\left(i,j\right)}{w}_{\left(i,j\right)}^H{h}_{\left(i,j\right)}^H}{1+\frac{p_{i,j}}{{\sigma }^2}{h}_{\left(i,j\right)}{\displaystyle \sum _{m\in {\mathcal{Q}}^{s,i},m\ne j}{w}_{\left(i,j\right)}{w}_{\left(i,j\right)}^H{h}_{\left(i,j\right)}^H}+\frac{p_{k,m}}{{\sigma }^2}{\displaystyle \sum _{m\in {\mathcal{S}}^{\mathrm{Nei,}i}\backslash {\mathcal{Q}}^{s,i+1}}{\left|h_{\left(k,j\right)}{w}_{\left(k,m\right)}\right|}^2}}}{1+\frac{p_{i,j}}{{\sigma }^2}{w}_{\left(i,j\right)}^H{H}_{\left(i,j\right)}^H{H}_{\left(i,j\right)}{w}_{\left(i,j\right)}}\right]}_{w=w^{\#}}^{+},i\in ℛ,j\in {ℒ}^i$$

(27)

and ${\mathcal{S}}^{\mathrm{Nei},i}={\mathcal{Q}}^{s,i}\cup {\mathcal{Q}}^{s,i+1}$ .

Based on the two measures above, SAMSS algorithm is given as shown in Algorithm 1. However, due to the specificity of vehicle network, the number of vehicles in the road at a certain time may be very large, even though the traffic density is not high. Therefore, when we calculate the interference to a certain RSU i caused by its adjacent RSU $i+1$ , a large amount of computational complexity will be encountered even in low traffic density.

As a consequence, we propose SAMSS-LC algorithm, shown in Algorithm 2, to reduce the computational complexity with relatively desirable secrecy performance. Different from SAMSS algorithm, we ignore the interference to a certain vehicle $(i,j)$ caused by RSU $i+1$ , if the vehicle $(i,j)$ locates outside the interference-selection region of the RSU $i+1$ . Because in this situation, RSU $i+1$ usually impose small interference to vehicle $(i,j)$ due to its low transmit power and long transmission distance. Therefore, the sub-cluster set is changed under the impact of interference-selection region. Specifically, in SAMSS-LC algorithm, ${\mathcal{Q}}^{s,i}$ is changed to ${\mathcal{Q}}^{s,i}\cap {\mathcal{I}}^{i+1}$ , while ${\mathcal{Q}}^{s,i+1}$ is changed to ${\mathcal{Q}}^{s,i+1}\cap {\mathcal{I}}^i$ . Furthermore, ${\mathcal{B}}^{s,i}$ is introduced as follows and vehicle set ${\mathcal{S}}^{\mathrm{Nei},i}$ served both by RSU i and RSU $i+1$ for vehicle selection in above SAMSS algorithm is replaced by ${\mathcal{B}}^{s,i}$

$${\mathcal{B}}^{s,i}=({\mathcal{Q}}^{s,i}\cap {\mathcal{I}}^{i+1})\cup ({\mathcal{Q}}^{s,i+1}\cap {\mathcal{I}}^i)$$

(28)

The details of SAMSS-LC are given in Table 1.

Simulation results

In this section, we evaluate the performance of our proposed joint secure beamforming and vehicle-selection strategy via simulations. The road topology is shown in Figure 2, which consists of a one-dimensional road with eight lanes. The region for each road measures 3750 m × 30 m, and the RSUs are with equal interval deployed in the middle of the road, such that they can serve vehicles on both roads. The mobility of vehicles is generated by cellular automata (CA) model, ⁴⁰ in which each vehicle occupy a space measuring 7.5 m × 3.5 m. ${\rho }_{\mathrm{jam}}(\mathrm{veh}/\mathrm{km}/\mathrm{lane})$ is the maximum vehicle density at traffic jam and $\rho (\mathrm{veh}/\mathrm{km}/\mathrm{lane})$ is the current vehicle density. We adopt the normalized vehicle density ${\rho }_{\mathrm{nor}}$ , satisfying ${\rho }_{\mathrm{nor}}=\rho /{\rho }_{\mathrm{jam}}$ to implement the simulations. The cellular automata approach has been proved to be quite useful in the field of vehicular traffic flow modeling to emulate realistic vehicular environments. There are different numbers of vehicles moving at different time on the roads. Vehicles on the road move from left to right. Each simulation is a snapshot, based on which we perform the corresponding mechanism design.

Algorithm 1. SAMSS
Input: $S,S_{(\mathrm{num})},\mathrm{Num}=0,{\mathcal{S}}^i=\varnothing ,{\mathcal{L}}^i,i\in \mathcal{R}$
Output: ${\mathcal{S}}^i,{\mathbf{w}}_{(i,j)}^{\#},i\in \mathcal{R},j\in {\mathcal{S}}^i$
1.for all S such that $1\le S\le S^{\mathbf{max}}$ do
2. $S=1,S_{(\mathrm{num})}=1$
3.while $max\left|{\mathcal{S}}^i\right|\le S,i\in \mathcal{R}$
4. Determine the vehicle set ${\mathcal{S}}^i$ , satisfying $\left|{\mathcal{S}}^i\right|\le S_{(\mathrm{num})}$ ;
5. for all i such that $1\le i\le R$ do
6.   $p_{i,j}=\sqrt{P_{max}/S_{(\mathrm{num})},j\in {\mathcal{L}}^i}$ ;
7.  Generate beamformer ${\mathbf{w}}_{(i,j)}^{\#}$ for each vehicle $(i,j),i\in \mathcal{R},j\in {\mathcal{L}}^i$ according to equation (21);
8.  Calculate the ${\mathcal{Q}}^{s,i}$ according to equation (24);
9. end for
10. if $s=1$ then
11.   $\mathcal{S}=\bigcup _{i=,\dots ,R}{\mathcal{Q}}^{s,i}$
12. else
13.   $\mathcal{S}=\bigcup _{i=,\dots ,R}({\mathcal{Q}}^{s-1,i}\cup {\mathcal{Q}}^{s,i})$
14. end if
15.  $S_{(\mathrm{num})}=S_{(\mathrm{num})}+1$ ;
16.   $S=S+1$ ;
17. end while
18.  $\mathrm{Num}=\mathrm{Num}+1$ ;
19.  ${\mathcal{S}}_{\mathrm{Num}}=\mathcal{S}$ ;
20. end for
21. Determine the vehicle cluster ${\mathcal{S}}^{\mathbf{index}}$ with maximum secrecy sumrate by comparing the SNSS measure of each vehicle cluster ${\mathcal{S}}_s,1\le s\le \mathrm{Num}$ ;
22. Determine the ultimate vehicle set ${\mathcal{S}}^i$ using for communications according to ${\mathcal{S}}^{\mathbf{index}}$
23. Select the corresponding beamformer ${\mathbf{w}}_{(i,j)}^{\#},i\in \mathcal{R},j\in {\mathcal{S}}^i$ .

We adopt location-based channel access control policy and the free space model for channel propagation ${\mathbf{g}}_{(i,j)}$ . The ultimate channel can be specified as

$$\begin{array}{l}{h}_{\left(i,j\right)}=\sqrt{{\beta }_{\left(i,j\right)}}{g}_{\left(i,j\right)}={10}^{-L\left(d_{\left(i,j\right)}\right)/20}\\ \sqrt{{\psi }_{\left(i,j\right)}{s}_{\left(i,j\right)}}{g}_{\left(i,j\right)},i\in ℛ,j\in {ℒ}^i\end{array}$$

(29)

The common radius of transmission range for each RSU is ${\gamma }_S\le 250$ and the common interference-selection radius is ${\gamma }_L\ge 250$ in our simulations. Each vehicle can transmit to the RSU in its forward or backward driving direction, using the shortest path strategy to choose which RSU to access the secure region. The details about the simulation configuration are summarized in Table 2.

Table 2.

Simulation parameters and values.

Simulation parameters	Values
Path-loss model $L(d_{(i,j)})$ , m 41	$30.18+26\log 10(d_{(i,j)})$
Standard deviation of log-norm shadowing ${\sigma }_{s_{(i,j)}}$	4 (dB)
Noise power ${\sigma }^2$ (10 MHZ bandwidth)	−102 (dBm)
Maximum transmit power of RRH $P_i,i\in \mathcal{R}$	15 (dBm)
Transmit antenna power gain ${\psi }_{(i,j)}$	9 (dBi)
The number of each RRH transmit antennas $N_T$	4
RSU height	15 m

RRH: remote radio head; RSU: roadside unit.

The impact of secure region and interference-selection region

The key indicator of the performance offered by the proposed algorithms in the vehicular scenario is the secrecy sumrate in the respect of different radius of secure region and interference-selection region, respectively. In general, the radius of those two regions affects the number of vehicles to be scheduled. Due to the adopted model of vehicle mobility, the vehicle traffic in fact also imposes an impact on the scheduling conducted by RSUs.

Algorithm 2. SAMSS-LC.
Input: $S,S_{(\mathrm{num})},{\mathcal{S}}^i=\varnothing ,\mathrm{Num}=0,{\mathcal{L}}^i,i\in \mathcal{R}$
Output: ${\mathcal{S}}^i,{\mathbf{w}}_{(i,j)}^{\#},i\in \mathcal{R},j\in {\mathcal{S}}^i$
1.for all S such that $1\le S\le S^{\mathbf{max}}$ do
2. $S=1,S_{(\mathrm{num})}=1$
3.while $max\left|{\mathcal{S}}^i\right|\le S,i\in \mathcal{R}$
4. i. Determine the vehicle set ${\mathcal{Q}}^{s,i}$ , satisfying $\left|{\mathcal{Q}}^{s,i}\right|\le S_{(\mathrm{num})}$ ;
5.  for all i such that $1\le i\le R$ do
6.   $p_{i,j}=\sqrt{P_{max}/S_{(\mathrm{num})},j\in {\mathcal{L}}^i}$ ;
7.  Generate beamformer ${\mathbf{w}}_{(i,j)}^{\#}$ for each vehicle $(i,j),i\in \mathcal{R},j\in {\mathcal{L}}^i$ according to equation (21);
8.  Calculate the ${\mathcal{Q}}^{s,i}$ according to equation (24);
9. end for
10. ii. Across the adjacent secure region, determine the set ${\mathcal{B}}^i$ , ${\mathcal{B}}^i\subset {\mathcal{Q}}^{s,i},i\in \mathcal{R}$ . Each set includes two neighboring vehicles, which are able to locate in the adjacent interference region:
11. for all i such that $1\le i\le R-1$ do
12.  for all j such that $j\in {\mathcal{Q}}^{s,i+1}$ do
13.  if $(i+1,j)\in {\mathcal{I}}^i,j\in {\mathcal{Q}}^{s,i+1}$ , then
14.   Add $(i+1,j)$ to ${\mathcal{B}}^{s,i}$ ;
15.  end if
16.  if $(i,j)\in {\mathcal{I}}^{i+1},j\in {\mathcal{Q}}^{s,i}$ , then
17.   Add $(i,j)$ to ${\mathcal{B}}^{s,i}$ ;
18.  end if
19.  end for
20. end for
21. if $s=1$ then
22.   $\mathcal{S}=\bigcup _{i=,\dots ,R}{\mathcal{B}}^{s,i}$
23. else
24.   $\mathcal{S}=\bigcup _{i=,\dots ,R}({\mathcal{B}}^{s-1,i}\cup {\mathcal{B}}^{s,i})$
25. end if
26.   $S_{(\mathrm{num})}=S_{(\mathrm{num})}+1$ ;
27.   $S=S+1$ ;
28. end while
29.  $\mathrm{Num}=\mathrm{Num}+1$ ;
30.  ${\mathcal{S}}_{\mathrm{Num}}=\mathcal{S}$ ;
31. end for
32. Determine the vehicle cluster ${\mathcal{S}}^{\mathbf{index}}$ with maximum secrecy sumrate by comparing the SNSS measure of each vehicle cluster ${\mathcal{S}}_s,1\le s\le \mathrm{Num}$ ;
33. Determine the ultimate vehicle set ${\mathcal{S}}^i$ using for communications according to ${\mathcal{S}}^{\mathbf{index}}$
34. Select the corresponding beamformer ${\mathbf{w}}_{(i,j)}^{\#},i\in \mathcal{R},j\in {\mathcal{S}}^i$ .

Based on this more practical model, we fix the traffic density ${\rho }_{\mathrm{nor}}=0.5$ , also with $N_T=4,S=4$ to research the relationship between secrecy sumrate and different regions.

Figure 4 shows that when increasing ${\gamma }_S$ to predefined maximum radius of secure region, the secrecy sumrate increases to a maximum which is about 53.37 bit/s/Hz at ${\gamma }_S=10$ and then decreases to zero, which is almost irrelevant to the size of interference-selection region. The above phenomenon can be attributed to the increased number of eavesdropping vehicles in the secure region, which equivalently improves the antennas of collusive vehicles and fundamentally reduces secrecy sumrate. And then it is also noted that the maximum radius of secure region with non-zero secrecy sumrate decreases with increasing the radius of interference-selection region. The corresponding reason may lies in the fact that the increased size of interference-selection region hinders the network from gaining more multi-user diversity, for more redundant inter-RSU interference is considered thus reducing the number of orthogonal vehicles. To better compare the performance, Figure 5 shows the secrecy sumrate under the conventional ZF-SUS user selection algorithm which has the same configuration as Figure 4. In can be concluded that the secrecy sumrate in ZF-SUS has lower secrecy sumrate with smaller non-zero secure region, which is contributed to the fact that ZF-SUS is less efficient in dealing with the eavesdropping behaviors.

Figure 4.

Secrecy sumrate versus ${\gamma }_S$ , and ${\gamma }_L$ under SAMSS-LC, with the traffic density ${\rho }_{\mathrm{nor}}=0.5$ and the transmit power $P=5\mathrm{dBm}$ .

Figure 5.

Secrecy sumrate versus ${\gamma }_S$ , and ${\gamma }_L$ under ZF-SUS, with the traffic density ${\rho }_{\mathrm{nor}}=0.5$ and the transmit power $P=5\mathrm{dBm}$ .

The performance of proposed algorithms

To explore the characteristic of our proposed two algorithms, we generate several vehicle traffics such as ${\rho }_{\mathrm{nor}}=0.1$ and ${\rho }_{\mathrm{nor}}=0.5$ which are enough to demonstrate the characteristic of the proposed algorithms in the following simulations. There are more fluctuations in the respect of secrecy sumrate using two algorithms. Figures 6 and 7 show the secrecy sumrate versus transmit power under SAMSS and SAMSS-LC algorithm, respectively, with ${\rho }_{\mathrm{nor}}=0.1$ , ${\rho }_{\mathrm{nor}}=0.5$ . It can be observed that when the traffic density is relatively low and the radius of secure region is fixed, the secrecy sumrate using SAMSS-LC is always higher than that using SAMSS with ${\gamma }_L$ from 250 to 650. This can be attributed to the fact that SAMSS-LC provides more multi-user diversity by considering partial interference between two adjacent RSUs based on their respective interference-selection region, while SAMSS usually considers interference induced by all the possible selected vehicles which actually constrained the multi-user diversity and improved the computational complexity. However, the difference between two figures is that the secrecy sumrate under ${\rho }_{\mathrm{nor}}=0.5$ is zero for the SAMSS algorithms but nearly 28 bit/s/Hz for the SAMSS-LC algorithm with ${\gamma }_L=650$ . This phenomenon verifies that SAMSS greatly limits the secrecy performance when the traffic density is relatively high and SAMSS-LC can flexibly control the secrecy sumrate by altering the radius the interference-selection thus bearing higher traffic density.

Figure 6.

Secrecy sumrate versus different transmit power P under algorithm SAMSS and algorithm SAMSS-LC, respectively, ${\rho }_{\mathrm{nor}}=0.1$ , ${\gamma }_S=120$ .

Figure 7.

Secrecy sumrate versus different transmit power P under algorithm SAMSS and algorithm SAMSS-LC, respectively, ${\rho }_{\mathrm{nor}}=0.5$ , ${\gamma }_S=120$ .

The impact of interference-selection region and transmit power

To better analyze the performance of proposed algorithm SAMSS-LC, we explore the secrecy sumrate of SAMSS-LC versus transmit power P under low traffic density and high traffic density such as ${\rho }_{\mathrm{nor}}=0.2$ and ${\rho }_{\mathrm{nor}}=0.8$ . In Figure 8, it is shown that when the traffic density is low, the secrecy sumrate is relatively unchanged with increasing the transmit power P. Although high transmit power means high interference, our proposed SAMSS-LC algorithm can reduce the influence by choosing suitable interference-selection region as is shown in the figure. However, when the interference-selection region is enlarged, the system’s ability of controlling the interference during vehicle selection is gradually deprived and vehicle selection by RSU is more easily influenced by the enhanced interference caused by increasing the transmitting power. Specifically, when ${\gamma }_L\ge 400$ , the secrecy sumrate increases with the transmit power S slowly. Especially, the secrecy sumrate turns to be almost zero at $p=-5\mathrm{dBm},{\gamma }_L\ge 650$ .

Figure 8.

Secrecy sumrate versus different transmit power P and ${\gamma }_L$ , with ${\rho }_{\mathrm{nor}}=0.2$ , ${\gamma }_S=100$ .

Compared with Figure 8, high traffic density in Figure 9 induces several changes in secrecy sumrate, one of which is that when ${\gamma }_L\ge 500$ , the secrecy sumrate is zero. The other change can be that secrecy sumrate is decreasing with rather than increasing with the increasing transmit power when ${\gamma }_L\le 500$ . The big difference between two figures means that traffic has a significant impact on the secrecy sumrate and even determines how to transmit reasonable power to improve the secrecy performance of the network.

Figure 9.

Secrecy sumrate versus different transmit power P and ${\gamma }_L$ , with, ${\rho }_{\mathrm{nor}}=0.8$ , ${\gamma }_S=100$ .

The impact of secure region and transmit power

To further explore the characteristic of SAMSS-LC, we fix the radius of interference-selection region but alter the radius of the secure region, accompanying with different transmit power. In general, the radius of secure region and traffic density jointly determines the number of vehicles located in one secure region. Therefore, we illustrate three different configurations of traffic density, under which we conduct the simulation of secrecy sumrate versus transmit power. In Figure 10, ${\rho }_{\mathrm{nor}}$ set to 0.1 represents a situation of low traffic density. It is shown that the secrecy sumrate increases with the transmit power under different radius of secure region. When we keep high transmit power at RSUs, the secrecy sumrate decreases with the increasing radius of secure region. However, when the transmit power is altered to be low, the secrecy sumrate decreases first with the increasing radius of secure region from 10 to 160 and then increase with the increasing radius of secure region from 160 to 250. The reason may lie in the fact that high transmit power makes the number of vehicles having better channel condition high, which provides higher power gain to RSU for making up the secrecy loss due to eavesdropping. Low transmit power incurs low power gain and then makes eavesdropping stealing more useful confidential messages. Therefore, there exists a point, at which the secrecy sumrate is lowest such as ${\gamma }_S=160,P=-5\mathrm{dBm}$ with the value of 43 bit/s/Hz.

Figure 10.

Secrecy sumrate versus different transmit power P and ${\gamma }_S$ with ${\rho }_{\mathrm{nor}}=0.1$ , ${\gamma }_L=500$ .

However, as is shown in Figure 11, when the traffic density increases to be medium such as ${\rho }_{\mathrm{nor}}=0.5$ , the region of lowest secrecy sumrate is more enlarged compared with Figure 12. Interestingly, there exists a partial secure region having zero secrecy rate and a partial secure region having non-zero secrecy rate in which the secrecy rate increases with the transmit power. The reason that the region of low secrecy sumrate increases is that increased secure region makes the number of possibly collusive vehicles increase thus indirectly enhancing the ability of eavesdropping. Specifically, secure region increases to a certain threshold, less than which power increasing can reduce secrecy sumrate gain but not force secrecy sumrate to zero. However, when secure region increases beyond a certain threshold, the secrecy sumrate turns to be zero, which means that more power cannot bring about secrecy sumrate gain any more for the unsurpassable ability of eavesdropping. Therefore, how to control the size of secure region is vital for the system scheduling.

Figure 11.

Secrecy sumrate versus different transmit power P and ${\gamma }_S$ with ${\rho }_{\mathrm{nor}}=0.5$ , ${\gamma }_L=500$ .

Figure 12.

Secrecy sumrate versus different transmit power P with ${\rho }_{\mathrm{nor}}=0.9$ , ${\gamma }_L=500$ .

As the traffic density get to ${\rho }_{\mathrm{nor}}=0.9$ , there exists suitable secure regions in which non-zero secrecy sumrate can be obtained. Specifically, when ${\gamma }_S=10$ , the maximum secrecy sumrate is achieved at $P=15\mathrm{dBm}$ and denoted as 45 bit/s/Hz. When ${\gamma }_S=40$ , the maximum secrecy sumrate is achieved at $P=-5\mathrm{dBm}$ and denoted as 41 bit/s/Hz. In general, the high traffic density usually compels the secure region to be very small. However, when ${\gamma }_S=100,160$ , the maximum secrecy sumrate is achieved at $P=5\mathrm{dBm}$ and denoted as, respectively, 14 and 9 bit/s/Hz. This is attributed to the fact that the increasing traffic density provides more vehicles having channels of better condition though the number of collusive vehicles also increases, thus inducing more multi-user diversity gain which outperforms and makes up the sacrifice in the respect of secrecy sumrate loss due to eavesdropping.

Conclusion

In this article, we have studied the hybrid secure beamforming and vehicle selection for a C-RAN-based V2I communications in VCPS. We propose the concept of secure region and interference-selection region, based on which a secure beamforming strategy is generated, together with two vehicle-selection algorithms, denoted as SAMSS and SAMSS-LC, to further improve the secrecy sumrate of the whole network. Specifically, we verify that SAMSS-LC algorithm has the lower complexity and better performance than SAMSS. SAMSS-LC outperforms the ZF-SUS algorithm in the respect of secrecy sumrate and has wider secure region and interference-selection region in which non-zero secrecy sumrate can be achieved. Furthermore, different size of secure region and interference-selection region determines what the secrecy sumrate can reach. When the radius of the secure region is relatively small, high secrecy sumrate can be reached if we control the interference-selection region to be a suitable extent. Otherwise, unsuitable configurations of secure region and interference-selection region can induce undesirable secrecy sumrate. What’s more, we show that the transmit power, respectively, together with the secure region and interference-selection region can impact the secrecy sumrate jointly. In the end, we found that the region of non-zero secrecy sumrate can change with different traffic densities. Therefore, reasonable configurations of transmit power, secure region, interference region, and traffic density jointly are necessary procedure for desirable secrecy sumrate, which will be our next work in the future.