Enhancing Physical-Layer Security in UAV-Assisted Communications: A UAV-Mounted Reconfigurable Intelligent Surface Scheme for Secrecy Rate Optimization

Abstract

With the wide application of unmanned aerial vehicles (UAVs) in the military and civilian fields, the physical layer security of UAV-assisted communication has attracted more and more attention in recent years. Reconfigurable intelligent surface (RIS) is a revolutionizing and promising technology that can improve spectrum efficiency through intelligent reconfiguration of the propagation environment. In this paper, we investigate the physical layer security of RIS and UAV-assisted communication systems. Specifically, we consider the scenario of multiple eavesdroppers wiretapping the communication between the base station and the legitimate user and propose a secure mechanism that deploys the RIS on a dynamic UAV for security assistance. In order to maximize the average secrecy rate of the system, we propose a joint optimization problem of joint UAV flight trajectory, RIS transmit phase shift, and base station transmit power. Since the proposed problem is non-convex, it is difficult to solve it directly, so we propose a joint optimization algorithm based on block coordinate descent and successive convex optimization techniques. Simulation results verify the effectiveness of our proposed design in improving the secrecy performance of the considered system.

1. Introduction

With the characteristics of small size, strong flexibility, and ease of deployment, unmanned aerial vehicles (UAVs) have been widely used in various fields such as cargo transport, environmental monitoring, infrastructure inspection and maintenance, and urban planning [1]. One of the most important applications, UAV-assisted communication has attracted significant attention. Compared to traditional ground-based communication, UAV-assisted communication offers several distinct advantages [2]. First, it can provide temporary and emergency communication services, offering network coverage in areas without fixed communication infrastructure. UAVs can be rapidly deployed to ensure uninterrupted critical communication. Second, UAV-assisted communication can enhance network coverage and capacity, dynamically adjusting flight positions and communication capabilities in the air to provide real-time, flexible coverage [3]. Additionally, UAV-assisted communication can improve mobile communication services in scenarios that require high-speed networks and large-scale device connections, ensuring high-speed data transmission. For example, a UAV is deployed to collect data from low-power ground nodes in a multi-obstacle environment [4]. The completion time can be minimized while satisfying the communication link requirements of each device and ensuring obstacle avoidance and data transmission. In addition, UAV-assisted communication can be easily extended to serve more complex wireless networks, such as the energy harvesting systems [5] and data collection tasks in maritime scenarios [6].

The advantages make UAV-assisted communication a promising technology for future wireless communication networks. UAVs operate at low altitudes, which allows them to establish line-of-sight (LoS) communication links with ground users and can significantly reduce path loss. However, due to the LoS characteristics of UAV-assisted communication, the signal is more easily wiretapped by eavesdroppers. Therefore, security issues in UAV-assisted communication have become increasingly important. The traditional methods guaranteeing the confidentiality of the communication system are mainly based on the cryptography in wireless OSI layers, such as HTTP and FTP in the application layer [7] and TCP and UDP in the transport layer [8]. Although these methods based on cryptography can meliorate communication confidentiality, they require high computational complexity and lead to longer delays. Different from the traditional encryption methods, physical layer security (PLS) can enhance eavesdropping resistance and improve communication privacy by leveraging the physical characteristics of the channel itself [9]. With the advantages of not depending on computational complexity and being capable of achieving perfect secrecy, physical layer security has been widely studied with various promising technologies. In the UAV-assisted communication system, the flight path of UAVs is usually not fixed, which leads to dynamic variations in the communication channel [1,10]. Physical layer security just utilizes the characteristics of the channel to provide flexible and dynamic security protection for UAV-assisted communication [11]. Therefore, leveraging physical layer security to improve the secure performance of the UAV-assisted communication system has attracted significant attention in recent years.

The secure challenge of UAV-assisted communication by applying physical layer security is studied considering both downlink and uplink communications with a potential eavesdropper on the ground [12]. The utilization of the mobile UAV reduces the possibility of information exposure to eavesdroppers. In Ref. [13], a secure communication system was considered, where the UAV plays the role of a mobile relay where no direct links exist between a source node and a legitimate node. The authors developed an iterative algorithm and demonstrate that mobile UAV relay is superior to static relay in terms of secrecy enhancement. The authors in [14] addressed physical layer security in a network of untrusted UAVs communicating in a full-duplex way and propose a source-based jamming scheme to enhance the secure performance of the system. In Ref. [15], the authors considered the PLS in a UAV-assisted backscatter scenario and propose the artificial noise (AN) injection design to weaken the eavesdropper link. An iterative algorithm is proposed to meliorate the secrecy performance considering the UAV’s location, the reflection coefficient, and the power allocation. In Ref. [16], the authors studied a UAV-assisted energy harvesting system where the UAV is employed as a cooperative jammer to improve the physical layer security and meliorate the secrecy rate.

Recently, reconfigurable intelligent surface (RIS), as an emerging technology, has received widespread attention and can provide a new solution for enhancing physical layer security [17]. RIS can flexibly adjust the configuration of the reflective surface and intelligently adjust the propagation path of electromagnetic waves, effectively enhancing the directionality of the signal and reducing the probability of illegal receivers (such as eavesdroppers) capturing the signal [18]. The incorporation of RIS into physical layer security presents a multitude of significant advantages. Specifically, the passive beamforming can be effectively implemented through the use of passive elements of RIS, thereby enabling the manipulation of signal propagation direction and leading to notable enhancements in system secrecy performance [19]. Moreover, RIS can obtain signal coverage where there is a barrier in the direct link [20], which implies the RIS can be deployed around legitimate users in the scenario that obstacles are blocking its channel link. Meanwhile, RIS can be readily deployed on existing infrastructures, while avoiding the additional deployment expenditure [21]. The authors in [22] studied the PLS of an RIS-aided multi-antenna system and maximize the secrecy rate constrained by the transmit power and the phase shift of the RIS. The authors in [23] further proposed a secure power-efficient design of the considered system. In Ref. [24], the authors investigated the cooperation of RIS and artificial noise (AN) to meliorate the PLS. Besides, the PLS of the Device-to-Device communication and relay system are considered to study the assistance of RIS in improving the secrecy performance [25,26]. In summary, the unique advantages of RIS significantly enhance physical layer security in wireless communications.

Although UAV-assisted secure communication systems have shown significant advantages in providing flexible communication coverage, UAVs still face challenges such as energy consumption, signal interference, and coverage limitations in high-density or complex environments [27]. RIS, with its low power consumption, environmental friendliness, and superior service quality, can be integrated to further enhance the PLS in UAV-assisted communication systems [28]. The PLS of the UAV communication system is investigated where an eavesdropper wiretap the secure communication between the UAV and the ground user [29]. Given the UAV’s limited capacity, RIS is considered to meliorate the secrecy performance. Taking the imperfect channel state information (CSI) into consideration, the secrecy rate is maximized by jointly optimizing the dynamic UAV trajectory, the phase shift of the static RIS, and the power consumption. The authors in [30] further considered a malicious active jammer in the scenario and a secure scheme is proposed to improve the secure performance of the uplink communication. In Ref. [31], the authors studied PLS of the scenario without a direct link between the UAV and the base station and static RIS is deployed to assist the secure communication. Their proposed secure design utilizes the cascaded channel, while the authors further investigated a quantizer to quantify the channel gain fluctuations associated with the channel characteristics. In the RIS-assisted multi-UAV secure system, the authors explored the secure performance of three modes that the RIS utilizes to enhance the information transmission, energy transfer, and the two functions simultaneously [32].

Based on the exploration of deploying the independent RIS in the UAV-assisted communication scenario, researchers further investigated the impact of directly integrating the RIS and the UAV together, which can combine the advantages of two techniques and meliorate the communication performance [33,34,35,36]. The authors in [33] presented a stochastic analysis of UAV-mounted RIS-assisted wireless systems under composite fading and spatially correlated shadowing, deriving closed-form expressions for the outage probability, ergodic capacity, and energy efficiency, and demonstrating superior performance over decode-and-forward relaying in composite fading environments. However, this study focuses on different channel conditions, without considering the security issue and leveraging the mobility of the UAV. Considering the vulnerabilities of integrating RIS and UAV, the authors in [34] proposed a secure framework based on deep machine learning to robustly address security issues and ensure reliable communication by countering malicious threats. The authors in [35] investigated the PLS issue of a secure communication system where the communication links are established by mounting RIS on UAV. Specifically, a deep Q-network is designed to attain reliable and secure transmission under imperfect CSI and behaviors of mixed attacks. By jointly designing the UAV-RIS location, the transmit beamforming of the base station, the passive beamforming of RIS, and artificial noise, the worst-case secrecy rate can be maximized. In [36], the authors further investigated leveraging the UAV equipped with RIS relaying source signals against eavesdropping, interference, and jamming attacks. The robust transmit beamforming design of the base station is improved to counter three security threats and a joint optimization algorithm is proposed to obtain the reflect phase shift of RIS. Although [34,35,36] considered integrating RIS and UAV to improve the security performance of communication systems and defend against security attacks, none of these studies took into account the simultaneous use of the UAV’s flexible mobility and the reconfigurable channel characteristics of the RIS to enhance the physical layer security performance of the system.

Inspired by the above discussions, in this paper, we investigate the integration of RIS and dynamic UAV to improve the secrecy performance of the communication system. Different from the current studies introducing static RIS in UAV-assisted secure communications, we consider mounting the RIS on the dynamic UAV to enhance the secure communication of the base station and the legitimate user by adjusting the phase shift of the RIS and fully exploiting the mobility of the UAV. In order to maximize the average achievable secrecy rate, we formulate a joint optimization problem of the phase shifter of the RIS, the dynamic UAV trajectory, and the transmit power of the base station, and propose an iterative algorithm for the optimization problem. Simulation results demonstrate the efficiency of our proposed secure design in meliorating the secrecy performance. The contributions of the innovation points and the work of the simulation part in this paper can be outlined as follows:

• In this paper, we propose a flexible UAV-mounted reconfigurable intelligent surface secure design and attempt to investigate the assistance of the UAV-mounted reconfigurable intelligent surface in physical layer security. Specifically, we study a secure communication system consisting of a base station, a legitimate ground user, several passive eavesdroppers, and the dynamic UAV-mounted RIS. We jointly optimize the UAV trajectory, the RIS phase shifter, and transmit power, aiming to improve the average achievable secrecy rate in the multiple-eavesdroppers scenario.
• In order to maximize the average achievable secrecy rate, we formulate a joint RIS phase shifter, UAV trajectories, and base station transmit power optimization problem. To tackle the non-convexity of the formulated problem, we propose an iterative algorithm based on block coordinate descent and successive convex optimization techniques. The algorithm enables us to find an efficient solution to the optimization problem.
• To better present the superiority of our proposed secure design, we consider the conditions without the secure design, the proposed design in a static mode, and heuristic mode. Simulation results demonstrate that our proposed dynamic secure design outperforms other schemes in different channel and deployed height conditions in meliorating the secrecy rate. Besides, the numerical results also confirm the potential of using the proposed dynamic UAV-mounted RIS secure design to improve secrecy performance and save base station power.

The remaining parts of this paper are organized as follows. Section 2 presents the system model and the secure design. The formulated problem is discussed in Section 3 and the proposed optimization algorithm is discussed in Section 4. Section 5 depicts the numerical results. We conclude the paper in Section 6.

2. System Model and Secure Design

In this section, we consider a secure communication system where the base station communicates with the legitimate user wiretapped by the eavesdroppers around the legitimate user, which is a common scenario of practical application; for instance, in military operational contexts requiring UAV-enabled data acquisition or in urban deployment scenarios where the inherent complexity of built environments gives rise to multi-Eve eavesdropping scenarios. In order to improve the secrecy performance of the system, we propose a secure design that deploys the RIS on the dynamic UAV for the considered secure scenario.

2.1. System Model

The secure transmission system consists of a base station (Alice), a legitimate ground user (Bob), a UAV-mounted RIS, and K passive eavesdroppers (Eves). As shown in Figure 1, Alice, Bob, and Eves are all located on the ground. The base station communicates with the legitimate user wiretapped by the eavesdroppers around the legitimate user, which is a common scenario of practical application. Considering the complexity of practical deployment scenarios where base-station-to-user and base-station-to-Eves connections may potentially lack direct propagation paths due to physical obstructions or environmental factors, these particular links are modeled using Rayleigh fading channels.

We denote $\mathcal{K}=\{1,2,\cdots ,K\}$ as the set of eavesdroppers that are arbitrarily distributed on the ground. The signals are transmitted from Alice to Bob straightly and through RIS to Bob reflectively, while K Eves intend to intercept. Moreover, Alice, Bob, and K Eves are all with a single omnidirectional antenna. The scenario adopts a three-dimensional (3D) Cartesian coordinate system. We denote ${\mathit{c}}_A={[x_A, y_A]}^T, {\mathit{c}}_B={[x_B, y_B]}^T$, and ${\mathit{c}}_{Ek}={[x_{Ek}, y_{Ek}]}^T, k\in \mathcal{K}$ denote the horizontal coordinates of Alice, Bob, and the kth Eve, respectively. Besides, Alice’s height is denoted as $H_A$.

The channel coefficients of Alice–Bob link $g_{AB}$ and Alice–kth Eve $g_{AEk}$, $k\in \mathcal{K}$, can be, respectively, expressed as

$$\begin{array}{c}{g}_{AB}=\sqrt{g_0{d}_{AB}^{-\beta }}{h}_{AB},\end{array}$$

(1)

$$\begin{array}{c}{g}_{AEk}=\sqrt{g_0{d}_{AEk}^{-\beta }}{h}_{AEk},\end{array}$$

(2)

where $g_0$ represents the channel gain at $D_0=1$ m and $d_{AB}=\sqrt{\left|\right|{\mathit{c}}_A-{\mathit{c}}_B{\left|\right|}^2+H_A^2}$ and $d_{AEk}=\sqrt{\left|\right|{\mathit{c}}_A-{\mathit{c}}_{Ek}{\left|\right|}^2+H_A^2}$, $k\in \mathcal{K}$, denote the distance from Alice to Bob and the kth Eve. $\beta$ represents the path loss exponent of AB/AE links. Besides, $h_{AB}$ and $h_{AEk}$ denote the independent and random scattering components, modeled as circularly symmetric complex Gaussian (CSCG) random variables with zero mean and unit variance.

2.2. Dynamic UAV-RIS Secure Design

In order to meliorate the secrecy performance of the system, we propose a UAV-mounted RIS-aided (UAV-RIS) secure design for the considered system. The UAV-RIS can construct additional channel links for the secure system from different locations. Specifically, the UAV flies at a fixed height $H_0$ from the predetermined SP ${\mathit{c}}_0$ to the TP ${\mathit{c}}_f$ within flight period T. We assume that the RIS and UAV has the same coordinate. Its horizontal coordinate at t, $0\le t\le T$, is denoted as $\mathit{q}\left(t\right)={[x\left(t\right),y\left(t\right)]}^T$. According to the limit of the UAV flight speed in practice, the maximum speed of the UAV is represented as $V_{\max }$, which gives the flight constraints of the UAV as follows

$$\begin{array}{cc}\hfill \mathit{q}\left(0\right)& ={\mathit{c}}_0,\hfill \end{array}$$

(3a)

$$\begin{array}{cc}\hfill {\displaystyle \frac{d\left(\mathit{q}\right(t)-\mathit{q}(t-1\left)\right)}{dt}}& \le V_{\max },\hfill \end{array}$$

(3b)

$$\begin{array}{cc}\hfill \mathit{q}\left(T\right)& ={\mathit{c}}_f.\hfill \end{array}$$

(3c)

For ease of description, we divide the flight period T into N equal time slots, i.e., $T=N{\delta }_0$, where ${\delta }_0$ is small enough that the location change of the UAV can be regarded as constant. Besides, the set of the time slot is denoted as $\mathcal{N}=\{1,\cdots ,N\}$. Therefore, the UAV time-change horizontal coordinate can be given as $\mathit{q}[n]={[x[n],y[n]]}^T,n\in \mathcal{N}=\{1,\cdots ,N\}$, and the flight constraints can be expressed as

$$\begin{array}{cc}\hfill \parallel \mathit{q}[1]-{\mathit{c}}_0\parallel & \le V_{\max }{\delta }_0,\hfill \end{array}$$

(4a)

$$\begin{array}{cc}\hfill \parallel \mathit{q}[n+1]-\mathit{q}[n]\parallel & \le V_{\max }{\delta }_0,n=1,\cdots ,N-1,\hfill \end{array}$$

(4b)

$$\begin{array}{cc}\hfill \mathit{q}[N]& ={\mathit{c}}_f,\hfill \end{array}$$

(4c)

where $\parallel ·\parallel$ denotes the Euclidean norm and $V_{\max }{\delta }_0$ denotes the UAV maximum horizontal distance change during one time slot.

The UAV-mounted RIS utilizes a uniform linear array (ULA) with M passive elements, and the phase shifter matrix of the RIS is denoted by a diagonal matrix $\Theta [n]=\mathrm{diag}\{e^{j{\theta }_1[n]},\cdots ,e^{j{\theta }_M[n]}\}\in {\mathbb{C}}^{M\times M}$, where ${\theta }_m[n]\in [0,2\pi ),m=1,\cdots ,M$. Given the blockages of common communication scenarios, the channel fading of the Alice–Bob (AB) and Alice–Eves (AE) links are assumed following Rayleigh fading channel. The communication links between Alice–RIS–Bob (AIB)/Alice–RIS–Eve (AIE) exhibit dominant LoS A2G wireless channel characteristics, while the impact of the Doppler effect caused by the UAV’s mobility can be fully mitigated [37]. We assume that Alice and RIS perfectly possess the channel state information (CSI) of the AIB link and Alice knows the location of RIS, Bob, and Eves [38]. Moreover, we consider that all Eves are passive eavesdroppers, and obtaining small-scale fading between Alice and Eves is difficult, which is a worst-case scenario for consideration.

In the nth time slot, channel coefficients of LoS A2G channel Alice–UAV/RIS link ${\mathit{g}}_A[n]\in {\mathbb{C}}^{M\times 1}$, UAV/RIS–Bob ${\mathit{g}}_B[n]\in {\mathbb{C}}^{M\times 1}$, and UAV/RIS–kth Eve ${\mathit{g}}_{Ek}[n]\in {\mathbb{C}}^{M\times 1}$, $k\in \mathcal{K}$, are as follows

$${\mathit{g}}_A[n]=\sqrt{g_0{d}_A^{-\alpha }[n]}{[1,\cdots ,e^{-j{\displaystyle \frac{2\pi }{{\lambda }_0}}(M-1)d_e{\varphi }_A[n]}]}^T,$$

(5)

$${\mathit{g}}_B[n]=\sqrt{g_0{d}_B^{-\alpha }[n]}{[1,\cdots ,e^{-j{\displaystyle \frac{2\pi }{{\lambda }_0}}(M-1)d_e{\varphi }_B[n]}]}^T,$$

(6)

$${\mathit{g}}_{Ek}[n]=\sqrt{g_0{d}_{Ek}^{-\alpha }[n]}{[1,\cdots ,e^{-j{\displaystyle \frac{2\pi }{{\lambda }_0}}(M-1)d_e{\varphi }_{Ek}[n]}]}^T,$$

(7)

where $d_A[n]=\sqrt{\left|\right|\mathit{q}[n]-{\mathit{c}}_A{\left|\right|}^2+{(H_0-H_A)}^2}$ denotes the distance from the UAV to Alice and $d_B[n]=\sqrt{\left|\right|\mathit{q}[n]-{\mathit{c}}_B{\left|\right|}^2+H_0^2}$ and $d_{Ek}[n]=\sqrt{\left|\right|\mathit{q}[n]-{\mathit{c}}_{Ek}{\left|\right|}^2+H_0^2}$, $k\in \mathcal{K}$ denote the distance from the UAV to Bob/kth Eve. Moreover, $\alpha \ge 2$ denotes the pathloss exponent between RIS and Alice/Bob/Eves. ${\lambda }_0$ and $d_e$ denote the carrier wavelength and element separation distance at the RIS, respectively. Besides, ${\varphi }_A[n]={\displaystyle \frac{x[n]-x_A}{d_A[n]}}$ is the cosine of the AoA of the signal from Alice to RIS, ${\varphi }_B[n]={\displaystyle \frac{x_B-x[n]}{d_B[n]}}$ and ${\varphi }_{Ek}[n]={\displaystyle \frac{x_{Ek}-x[n]}{d_{Ek}[n]}}$ are the cosine of the AoD of the signal from RIS to Bob and the kth Eve.

With Equations (1)–(7), the signals received at Bob $y_B[n]$ and the kth Eve $y_{Ek}[n]$ are as follows

$$y_B[n]=\sqrt{P[n]}(g_{AB}+{\mathit{g}}_B{[n]}^H\Theta [n]{\mathit{g}}_A[n])s[n]+n_B,$$

(8)

$$y_{Ek}[n]=\sqrt{P[n]}(g_{AEk}+{\mathit{g}}_{Ek}{[n]}^H\Theta [n]{\mathit{g}}_A[n])s[n]+n_{Ek},$$

(9)

where $P[n]$ and $s[n]$ denote the transmit power and the transmit signal at Alice in the nth time slot and $n_B$ and $n_{Ek}$ represent the additive white Gaussian noise (AWGN) with zero mean and variance ${\sigma }^2$ at Bob and the kth Eve. Besides, the transmit power constraints at Alice can be given as

$$\begin{array}{cc}\hfill & \sum _{n=1}^{N}P[n]\le N{P}_{\mathrm{ave}},\hfill \end{array}$$

(10a)

$$\begin{array}{cc}\hfill & 0\le P[n]\le P_{\mathrm{p}},\hfill \end{array}$$

(10b)

where $P_{\mathrm{ave}}$ and $P_{\mathrm{p}}$ represent the average and peak transmit power. Equation (10a) indicates that the total transmit power during the transmission should be less than the budget and Equation (10b) indicates the peak power constraint.

3. Problem Formulation

In this paper, we aim to meliorate the average achievable secrecy rate of the considered multiple-eavesdroppers scenario. According to the channel coefficients and the coordinates, the transmission rate at the legitimate user Bob in the nth time slot can be expressed as

$$R_B[n]={\log }_2\left(1+{\displaystyle \frac{P[n]}{{\sigma }^2}}{\left|{\displaystyle \frac{\sqrt{g_0}{h}_{AB}}{d_{AB}^{\beta /2}}}+{\displaystyle \frac{g_0{f}_B[n]}{d_B^{\alpha /2}[n]d_A^{\alpha /2}[n]}}\right|}^2\right),$$

(11)

where

$$f_B[n]=\sum _{m=1}^M{e}^{j[{\displaystyle \frac{2\pi }{{\lambda }_0}}(m-1)d_e({\varphi }_B[n]-{\varphi }_A[n])+{\theta }_m[n]]}.$$

(12)

Similarly, the transmission rates at the kth Eve in the nth time slot can be given as

$$\begin{array}{c}\hfill R_{Ek}[n]=\mathbb{E}\left[{\log }_2\left(1+{\displaystyle \frac{P[n]}{{\sigma }^2}}{\left|{\displaystyle \frac{\sqrt{g_0}{h}_{AEk}}{d_{AEk}^{\beta /2}}}+{\displaystyle \frac{g_0{f}_{Ek}[n]}{d_{Ek}^{\alpha /2}[n]d_A^{\alpha /2}[n]}}\right|}^2\right)\right],\end{array}$$

(13)

where $\mathbb{E}[·]$ denotes the expectation operator and

$$f_{Ek}=\sum _{m=1}^M{e}^{j[{\displaystyle \frac{2\pi }{{\lambda }_0}}(m-1)d_e({\varphi }_{Ek}[n]-{\varphi }_A[n])+{\theta }_m[n]]}.$$

(14)

Considering that the small-scale fading between Alice and Eves is difficult to obtain, we use Jensen’s inequality to get an upper bound of $R_{Ek}[n]$, expressed as ${\widehat{R}}_{Ek}[n]$

$$\begin{array}{cc}\hfill R_{Ek}[n]& \le {\log }_2\left(1+{\displaystyle \frac{P[n]}{{\sigma }^2}}\mathbb{E}\left[{\left|{\displaystyle \frac{\sqrt{g_0}{h}_{AEk}}{d_{AEk}^{\beta /2}}}+{\displaystyle \frac{g_0{f}_{Ek}[n]}{d_{Ek}^{\alpha /2}[n]d_A^{\alpha /2}[n]}}\right|}^2\right]\right)\hfill \\ \hfill & ={\log }_2\left(1+{\displaystyle \frac{P[n]}{{\sigma }^2}}\left({\displaystyle \frac{g_0}{d_{AEk}^{\beta }}}+{\displaystyle \frac{g_0^2{f}_{Ek}^2[n]}{d_{Ek}^{\alpha }[n]d_A^{\alpha }[n]}}\right)\right)\hfill \\ \hfill & \triangleq {\widehat{R}}_{Ek}[n].\hfill \end{array}$$

(15)

The objective of the paper is to maximize the average achievable secrecy rate during the N time slots constrained by the UAV mobility constraints in Equation (4) and power constraint in Equation (10), where there are three variables: UAV’s trajectory $\mathit{Q}=\{\mathit{q}[n],n=1,2,\dots ,N\}$, the phase shifter of the RIS $\mathsf{\Phi }=\{\Theta [\mathit{n}],n=1,2,\dots ,N\}$, and transmit power at Alice $\mathit{P}=\{P[n],n=1,2,\dots ,N\}$. Thus, the average secrecy rate maximization problem in the proposed UAV-mounted RIS-aided dynamic communication system can be formulated as

$$\begin{array}{cc}\hfill \left(\mathbf{P}1\right):\underset{\mathsf{\Phi },\mathit{Q},\mathit{P}}{\max }& {\displaystyle \frac{1}{N}}\sum _{n=1}^N{\left[R_B[n]-\underset{k\in \mathcal{K}}{\max }{\widehat{R}}_{Ek}[n]\right]}^{+}\hfill \end{array}$$

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& \parallel \mathit{q}[1]-{\mathit{c}}_0\parallel \le V_{\max }{\delta }_0,\hfill \end{array}$$

(16a)

$$\begin{array}{cc}\hfill & \parallel \mathit{q}[n+1]-\mathit{q}[n]\parallel \le V_{\max }{\delta }_0,n=1,\cdots ,N-1,\hfill \end{array}$$

(16b)

$$\begin{array}{cc}\hfill & \mathit{q}[N]={\mathit{c}}_f,\hfill \end{array}$$

(16c)

$$\begin{array}{cc}\hfill & \sum _{n=1}^{N}P[n]\le N{P}_{\mathrm{ave}},\hfill \end{array}$$

(16d)

$$\begin{array}{cc}\hfill & 0\le P[n]\le P_{\mathrm{p}},\hfill \end{array}$$

(16e)

where ${\displaystyle \frac{1}{N}}{\sum }_{n=1}^N{\left[R_B[n]-\underset{k\in \mathcal{K}}{\max }{\widehat{R}}_{Ek}[n]\right]}^{+}$ is the average achievable secrecy rate, Equations (16a)–(16c) represent the mobility constraints of the UAV, and Equations (16d) and (16e) represent the average and peak transmit power of Alice. Due to the operator ${[Y]}^{+}=\max \{Y,0\}$, the objective function of (P1) is non-smooth. Moreover, the optimization problem (P1) is characterized by a non-convex objective function, which presents a considerable challenge for direct resolution. In order to overcome this difficulty, we propose an efficient algorithm that addresses the formulated problem by optimizing the RIS phase shifter $\mathsf{\Phi }$, the dynamic UAV trajectory $\mathit{Q}$, and transit power at Alice $\mathit{P}$ in an iterative way.

4. Proposed Algorithm

As for the non-smooth problem of the formulated problem, the optimization of transmit power in (P1) can invariably result in a non-negative secrecy rate within each time slot according to Lemma 1 in [12], which allows ${[]}^{+}$ operator to be omitted. Thus, (P1) can be rewritten as

$$\begin{array}{cc}\hfill \left(\mathbf{P}2\right):& \underset{\mathsf{\Phi },\mathit{Q},\mathit{P}}{\max }{\displaystyle \frac{1}{N}}\sum _{n=1}^N\left[R_B[n]-\underset{k\in \mathcal{K}}{\max }{\widehat{R}}_{Ek}[n]\right]\hfill \\ \hfill & \begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& \hfill \left(16\mathrm{a}\right),\left(16\mathrm{b}\right),\left(16\mathrm{c}\right),\left(16\mathrm{d}\right),\left(16\mathrm{e}\right).\end{array}\hfill \end{array}$$

(P2) is a problem that involves multiple variables and is challenging to solve directly. Considering the non-convexity and intractability of the multi-variable problem, block coordinate descent (BCD) is utilized to tackle the above optimization problem in an iterative way. Specifically, the optimization is performed in one direction at a time [39]. When we optimize the specific variable, the others will be treated as constant. The details will be presented in the following.

4.1. Phase Shifter Optimization

With the given UAV trajectory $\mathit{Q}$ and transmit power $\mathit{P}$, optimizing the phase shifter $\mathsf{\Phi }$ of the RIS can improve the value of secrecy rate, which means the secure performance enhanced. Because the small scale fading of the AE links cannot be obtained, the optimized phase shifter maximizes Bob’s received rate. Considering that ${\displaystyle \frac{\sqrt{g_0}{h}_{AB}}{d_{AB}^{\beta /2}}}$ and $f_B[n]={\displaystyle \sum _{m=1}^M}{e}^{j[{\displaystyle \frac{2\pi }{{\lambda }_0}}(m-1)d_e({\varphi }_B[n]-{\varphi }_A[n])+{\theta }_m[n]]}$ are complex values, $|{\displaystyle \frac{\sqrt{g_0}{h}_{AB}}{d_{AB}^{\beta /2}}}+{\displaystyle \frac{g_0{f}_B[n]}{d_B^{\alpha /2}[n]d_A^{\alpha /2}[n]}}|$ can be taken to the maximum when the phase of each element at the RIS is equal to the phase of $h_{AB}$. The optimal phase shifter of the mth element in the nth time slot can be expressed as

$${\theta }_m^{*}[n]={\displaystyle \frac{2\pi }{{\lambda }_0}}(m-1)d_e({\varphi }_A[n]-{\varphi }_B[n])+arg\left(h_{AB}\right).$$

(17)

With the optimal phase shifter, the terms of Equations (11) and (15) can be rewritten as

$$\begin{array}{c}\hfill R_B^{*}[n]={\log }_2\left(1+{\displaystyle \frac{P[n]}{{\sigma }^2}}{\left|{\displaystyle \frac{\sqrt{g_0}\left|h_{AB}\right|}{d_{AB}^{\beta /2}}}+{\displaystyle \frac{g_{0}M}{d_B^{\alpha /2}[n]d_A^{\alpha /2}[n]}}\right|}^2\right),\end{array}$$

(18)

$$\begin{array}{c}\hfill R_{Ek}^{*}[n]={\log }_2\left(1+{\displaystyle \frac{P[n]}{{\sigma }^2}}\left({\displaystyle \frac{g_0}{d_{AEk}^{\beta }}}+{\displaystyle \frac{g_0^2{\left(f_{Ek}^{*}[n]\right)}^2}{d_{Ek}^{\alpha }[n]d_A^{\alpha }[n]}}\right)\right),\end{array}$$

(19)

where

$$f_{Ek}^{*}[n]=\sum _{m=1}^M{e}^{j[{\displaystyle \frac{2\pi }{{\lambda }_0}}(m-1)d({\varphi }_{Ek}[n]-{\varphi }_B[n])+arg\left(h_{AB}\right)]}.$$

(20)

However, ${\varphi }_{Ek}[n]$ and ${\varphi }_B[n]$ still depend on the coordinate of UAV, which is difficult to handle. To circumvent the intractability, the upper bound of achievable rate in the kth Eve, denoted as $R_{Ek}^{†}[n]$, is used, shown as follows:

$$\begin{array}{cc}\hfill R_{Ek}^{*}[n]& \le {\log }_2\left(1+{\displaystyle \frac{P[n]}{{\sigma }^2}}\left({\displaystyle \frac{g_0}{d_{AEk}^{\beta }}}+{\displaystyle \frac{g_0^2{M}^2}{d_{Ek}^{\alpha }[n]d_A^{\alpha }[n]}}\right)\right)\hfill \\ & \triangleq R_{Ek}^{†}[n].\hfill \end{array}$$

(21)

4.2. Trajectory Optimization

Based on the optimized RIS phase shifter $\mathsf{\Phi }$ and the given transmit power $\mathit{P}$, we further study the subproblem of optimizing the UAV trajectory and ($\mathbf{P}2$), given as

$$\begin{array}{cc}\hfill \left(\mathbf{P}3\right):& \underset{\mathit{Q}}{\max }{\displaystyle \frac{1}{N}}\sum _{n=1}^N\left[R_B^{*}[n]-\underset{k\in \mathcal{K}}{\max }{R}_{Ek}^{†}[n]\right]\hfill \\ \hfill & \begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& \hfill \left(16\mathrm{a}\right),\left(16\mathrm{b}\right),\left(16\mathrm{c}\right).\end{array}\hfill \end{array}$$

Through the optimization of this subproblem, the UAV can optimize a better path to avoid multiple eavesdroppers or close to the legitimate user to achieve a larger channel rate difference between the legitimate channel and wiretap channel, which means the confidentiality of the system improved. However, ($\mathbf{P}3$) remains a non-convex problem with respect to $\mathit{Q}$. First of all, for convenience of marking, we define

$$A={\displaystyle \frac{g_0{\left|h_{AB}\right|}^2}{d_{AB}^{\beta }}},$$

(22)

$$B={\displaystyle \frac{2g_0^{\frac{3}{2}}\left|h_{AB}\right|M}{d_{AB}^{\beta /2}}},$$

(23)

$$C=g_0^2{M}^2,$$

(24)

$$D_k={\displaystyle \frac{g_0}{d_{AEk}^{\beta }}},k\in \mathcal{K}.$$

(25)

Then, $R_B^{*}[n]$ and $R_{Ek}^{†}[n]$ can be expressed as

$$\begin{array}{c}\hfill R_B^{*}[n]={\log }_2\left(1+{\displaystyle \frac{P[n]}{{\sigma }^2}}\left(A+{\displaystyle \frac{B}{d_B^{\frac{\alpha }{2}}[n]d_A^{\frac{\alpha }{2}}[n]}}+\right.\right.\left.\left.{\displaystyle \frac{C}{d_B^{\alpha }[n]d_A^{\alpha }[n]}}\right)\right),\end{array}$$

(26)

$$\begin{array}{c}\hfill R_{Ek}^{†}[n]={\log }_2\left(1+{\displaystyle \frac{P[n]}{{\sigma }^2}}\left(D_k+{\displaystyle \frac{C}{d_{Ek}^{\alpha }[n]d_A^{\alpha }[n]}}\right)\right).\end{array}$$

(27)

Some variables, $u[n]$, $v[n]$, $w[n]$, $z_k[n]$, ${\tau }_k[n],k\in \mathcal{K}$, are introduced to solve the non-convexity of ($\mathbf{P}3$). Therefore, $R_B^{*}[n]$ and $R_{Ek}^{†}[n]$, $k\in \mathcal{K}$, can be, respectively, formulated as

$$\begin{array}{c}\hfill R_B^{†}[n]={\log }_2\left(1+{\displaystyle \frac{P[n]}{{\sigma }^2}}\left(A+{\displaystyle \frac{B}{u^{\frac{1}{2}}[n]v^{\frac{1}{2}}[n]}}+{\displaystyle \frac{C}{u[n]v[n]}}\right)\right),\end{array}$$

(28)

and ${\tau }_k[n]$, $k\in \mathcal{K}$. With new related constraints added, we can rewrite ($\mathbf{P}3$) as

$$\begin{array}{cc}\hfill \underset{{\displaystyle \genfrac{}{}{0pt}{}{\mathit{Q},u[n],v[n],}{w[n],z_k[n],{\tau }_k[n]}}}{\max }& {\displaystyle \frac{1}{N}}\sum _{n=1}^N\left[R_B^{†}[n]-\underset{k\in K}{\max }{\tau }_k[n]]\right]\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \left(16\mathrm{a}\right),\left(16\mathrm{b}\right),\left(16\mathrm{c}\right),\hfill \end{array}$$

(29a)

$$\begin{array}{cc}\hfill & u^{{\displaystyle \frac{2}{\alpha }}}[n]\ge d_B^2[n],\hfill \end{array}$$

(29b)

$$\begin{array}{cc}\hfill & v^{{\displaystyle \frac{2}{\alpha }}}[n]\ge d_A^2[n],\hfill \end{array}$$

(29c)

$$\begin{array}{cc}\hfill {\tau }_k[n]\ge {\log }_2(& 1+{\displaystyle \frac{P[n]}{{\sigma }^2}}(D_k+C{e}^{w[n]+z_k[n]})),\forall k\in \mathcal{K},\hfill \end{array}$$

(29d)

$$\begin{array}{cc}\hfill & e^{-{\displaystyle \frac{2}{\alpha }}w[n]}\le d_A^2[n],\hfill \end{array}$$

(29e)

$$\begin{array}{cc}\hfill & e^{-{\displaystyle \frac{2}{\alpha }}{z}_k[n]}\le d_{Ek}^2[n],\forall k\in \mathcal{K}.\hfill \end{array}$$

(29f)

However, Equation (29) remains non-convex, which makes it difficult to solve. Note that $R_B^{†}[n]$ is jointly convex with respect to $u[n]$ and $v[n]$, and the first-order Taylor approximation of a convex function is a global underestimator. Therefore, we consider achieving an approximate optimal solution based on the successive convex optimization technique. It successively maximizes a concave lower bound of the objective function of Equation (29) within a convex feasible region until it converges. Due to the fact that the algorithm works iteratively, for simplicity, we consider a specific step in the iteration, denoted as the lth iteration. $u_l[n]$ and $v_l[n]$ are expressed as the given points in the lth iteration. It is clear that $R_B^{†}[n]$ is jointly convex with respect to $u[n]$ and $v[n]$. In accordance with the properties of the first-order Taylor expansion, it is a global underestimator. Thus, we apply the first-order Taylor expansion of $R_B^{†}[n]$ at the given points $u_l[n]$ and $v_l[n]$ as the lower-bound, expressed as $R_B^{\mathrm{lb}}[n]$

$$\begin{array}{cc}\hfill R_B^{†}[n]& \ge {\log }_2\left(A_l[n]\right)+{\displaystyle \frac{B_l[n]}{A_l[n]\ln 2}}(u[n]-u_l[n])+{\displaystyle \frac{C_l[n]}{A_l[n]\ln 2}}(v[n]-v_l[n])\hfill \\ \hfill & \triangleq R_B^{\mathrm{lb}}[n],\hfill \end{array}$$

(30)

where

$$\begin{array}{cc}\hfill A_l[n]& =1+{\displaystyle \frac{P[n]}{{\sigma }^2}}\left(A+{\displaystyle \frac{B}{{\left(u_l[n]v_l[n]\right)}^{\frac{1}{2}}}}+{\displaystyle \frac{C}{u_l[n]v_l[n]}}\right),\hfill \\ \hfill B_l[n]& =-{\displaystyle \frac{P[n]}{{\sigma }^2}}\left({\displaystyle \frac{B}{2u_l^{\frac{3}{2}}[n]v_l^{\frac{1}{2}}[n]}}+{\displaystyle \frac{C}{u_l^2[n]v_l[n]}}\right),\hfill \\ \hfill C_l[n]& =-{\displaystyle \frac{P[n]}{{\sigma }^2}}\left({\displaystyle \frac{B}{2u_l^{\frac{1}{2}}[n]v_l^{\frac{3}{2}}[n]}}+{\displaystyle \frac{C}{u_l[n]v_l^2[n]}}\right).\hfill \end{array}$$

(31)

Similarly, because the right-hand terms in Equations (29e) and (29f) are convex concerning $u[n]$, $v[n]$, the successive convex optimization technique can be exploited to get the lower-bounds of $d_A^2[n]$ and $d_{Ek}^2[n]$ in the lth iteration as

$$\begin{array}{cc}\hfill d_A^2[n]& \ge \left|\right|{\mathit{c}}_l[n]-{\mathit{c}}_A{\left|\right|}^2+{(H_0-H_A)}^2+2{({\mathit{c}}_l[n]-{\mathit{c}}_A)}^T(\mathit{c}[n]-{\mathit{c}}_l[n])\hfill \\ \hfill & \triangleq {\left(d_A^{\mathrm{lb}}[n]\right)}^2,\hfill \\ \hfill d_{Ek}^2[n]& \ge \left|\right|{\mathit{c}}_l[n]-{\mathit{c}}_{Ek}{\left|\right|}^2+H_0^2+2{({\mathit{c}}_l[n]-{\mathit{c}}_{Ek})}^T(\mathit{c}[n]-{\mathit{c}}_l[n])\hfill \\ \hfill & \triangleq {\left(d_{Ek}^{\mathrm{lb}}[n]\right)}^2.\hfill \end{array}$$

By replacing the terms $d_A^2[n]$ and $d_{Ek}^2[n]$ in Equations (29e) and (29f) with their lower bounds, problem Equation (29) can be transformed into its approximate form

$$\begin{array}{cc}\hfill \left(\mathbf{P}4\right):\underset{{\displaystyle \genfrac{}{}{0pt}{}{\mathit{Q},u[n],v[n],}{w[n],z_k[n],{\tau }_k[n]}}}{\max }& {\displaystyle \frac{1}{N}}\sum _{n=1}^N\left[R_B^{\mathrm{lb}}[n]-\underset{k\in \mathcal{K}}{\max }{\tau }_k[n]\right]\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \left(16\mathrm{a}\right),\left(16\mathrm{b}\right),\left(16\mathrm{c}\right),\left(29\mathrm{b}\right),\left(29\mathrm{c}\right),\left(29\mathrm{d}\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill & e^{-{\displaystyle \frac{2}{\alpha }}w[n]}\le {\left(d_A^{\mathrm{lb}}[n]\right)}^2,\hfill \end{array}$$

(32a)

$$\begin{array}{cc}\hfill & e^{-{\displaystyle \frac{2}{\alpha }}{z}_k[n]}\le {\left(d_{Ek}^{\mathrm{lb}}[n]\right)}^2,\forall k\in \mathcal{K},\hfill \end{array}$$

(32b)

which is a convex problem and can be solved by CVX now.

4.3. Transmit Power Optimization

With the given UAV trajectory $\mathit{Q}$ and RIS phase shifter $\mathsf{\Phi }$, this section studies the subproblem of transmit power optimization. The transmit power optimization problem can be expressed as

$$\begin{array}{cc}\hfill (\mathbf{P}5):\underset{\mathit{P}}{\max }& {\displaystyle \frac{1}{N}}\sum _{n=1}^N{\log }_2(1+P[n]\gamma [n])-{\log }_2(1+P[n]\xi [n])\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \left(16\mathrm{d}\right),\left(16\mathrm{e}\right),\hfill \end{array}$$

where

$$\gamma [n]={\displaystyle \frac{1}{{\sigma }^2}}(A+{\displaystyle \frac{B}{d_B^{\frac{\alpha }{2}}[n]d_A^{\frac{\alpha }{2}}[n]}}+{\displaystyle \frac{C}{d_B^{\alpha }[n]d_A^{\alpha }[n]}})),$$

(33)

$$\xi [n]=\underset{k\in \mathcal{K}}{\max }{\displaystyle \frac{1}{{\sigma }^2}}(D_k+{\displaystyle \frac{C}{d_{Ek}^{\alpha }[n]d_A^{\alpha }[n]}}).$$

(34)

By optimizing $\mathit{P}$, the base station can reasonably distribute the power so that the average transmission rate difference between the two channels is maximized depending on the UAV’s different positions and RIS’s reflecting phases.

According to [40], the optimal solution of the transmit power optimization problem can be expressed as

$$P^{*}[n]=\left\{\begin{array}{cc}min\left({\left[P_0[n]\right]}^{+},P_{\mathrm{p}}\right)\gamma [n]>\xi [n],\hfill & \\ 0\gamma [n]\le \xi [n],\hfill \end{array}\right.$$

(35)

where

$$\begin{array}{c}\hfill P_0[n]=\sqrt{{\left({\displaystyle \frac{1}{2\xi [n]}}-{\displaystyle \frac{1}{2\gamma [n]}}\right)}^2+{\displaystyle \frac{1}{\zeta \ln 2}}\left({\displaystyle \frac{1}{\xi [n]}}-{\displaystyle \frac{1}{\gamma [n]}}\right)}-{\displaystyle \frac{1}{2\xi [n]}}-{\displaystyle \frac{1}{2\gamma [n]}}.\end{array}$$

(36)

In Equation (36), $\zeta >0$ is a constant, which can make sure the constraints in Equation (10) are guaranteed when attaining the optimal solution. It can be obtained by the bisection search with ${\displaystyle \frac{1}{N}}\sum _{\mathcal{N}}{P}^{*}[n]=P_{\mathrm{ave}}$. However, before conducting the bisection search, it is necessary to verify the validity of ${\displaystyle \frac{1}{N}}\sum _{\mathcal{N}}{P}_{\mathrm{p}}<P_{\mathrm{ave}}$. Otherwise, $\zeta$ can be zero, and ${\displaystyle \frac{1}{N}}\sum _{\mathcal{N}}{P}^{*}[n]=P_{\mathrm{ave}}$ cannot be met [12].

4.4. Overall Algorithm

Algorithm 1 summarizes the whole algorithm which can solve ($\mathbf{P}$2). Specifically, we iteratively solve each subproblem of RIS phase shifter, UAV trajectory optimization, and transmit power at the base station to attain a suboptimal solution of ($\mathbf{P}$2). Since the target value of ($\mathbf{P}2$) after each iteration is monotonically finite and non-decreasing, Algorithm 1 guarantees convergence by $ϵ$, which controls the accuracy convergence. Moreover, the complexity of step 2 and step 4 are $\mathcal{O}\left(N_{\mathrm{ite}}{\left(KN\right)}^{3.5}\right)$ and $\mathcal{O}\left(N_{\mathrm{ite}}KN\log \left(K_{\mathrm{bis}}\right)\right)$, respectively, where $K_{\mathrm{bis}}$ and $N_{\mathrm{ite}}$ denote the number of values for $\zeta$ in the process of bisection search and the iteration number. The overall complexity of Algorithm 1 is $\mathcal{O}\left(N_{\mathrm{ite}}{\left(KN\right)}^{3.5}\right)$.

The proposed algorithm first calculates the initial ${\mathsf{\Phi }}^{\left(0\right)}$ with the given trajectory ${\mathit{Q}}^{\left(0\right)}$. Then, with the given transmit power ${\mathit{P}}^{(l-1)}$, the subproblem of the UAV trajectory can be optimized. It is reasonable to optimize the auxiliary position of the UAV in order to increase the disparity between the transmission rates of Bob and Eves. With the optimized UAV trajectory ${\mathit{Q}}^{\left(l\right)}$, the optimal ${\mathsf{\Phi }}^{\left(l\right)}$ of the RIS can be obtained by Equation (17). After that, by rationally distributing the transmit power ${\mathit{P}}^{\left(l\right)}$, the difference between the transmission rates of Bob and Eves can be further increased, which reveals that the secrecy performance can be enhanced. After being iteratively optimized, a suboptimal solution of the non-convex problem ($\mathbf{P}$2) can be obtained within the tolerance.

Algorithm 1 Proposed Algorithm for Solving ($\mathbf{P}2$)

Initialize: Set tolerance $ϵ$. Set initial variables ${\mathit{P}}^{\left(0\right)}$, ${\mathit{Q}}^{\left(0\right)}$, ${\mathit{u}}_0$, ${\mathit{v}}_0$, ${\mathit{w}}_0$, ${\mathit{z}}_{k0}$, ${\mathsf{\tau }}_{k0}$, and iteration number $l=0$. Use Equation (17) to get ${\mathsf{\Phi }}^{\left(0\right)}$ with ${\mathit{Q}}^{\left(0\right)}$. Solve ($\mathit{P}$2) to get ${\mathit{R}}^{\left(0\right)}$. Repeat: 1. Set $l=l+1$. 2. Update ${\mathit{Q}}^{\left(l\right)}$ with given ${\mathit{P}}^{(l-1)}$ through solving ($\mathbf{P}4$). 3. With given ${\mathit{Q}}^{\left(l\right)}$, obtain ${\mathsf{\Phi }}^{\left(l\right)}$ by using Equation (17). 4. Update ${\mathit{P}}^{\left(l\right)}$ through solving ($\mathbf{P}5$) with given ${\mathit{Q}}^{\left(l\right)}$. 5. Calculate ${\mathit{R}}^{\left(l\right)}$ with given ${\mathsf{\Phi }}^{\left(l\right)}$, ${\mathit{Q}}^{\left(l\right)}$, ${\mathbf{P}}^{\left(l\right)}$ by solving ($\mathbf{P}2$). Until:${\displaystyle \frac{{\mathit{R}}^{\left(l\right)}-{\mathit{R}}^{(l-1)}}{{\mathit{R}}^{\left(l\right)}}}<ϵ.$

5. Numerical Results

In this section, we present simulation results to illustrate the improvement in secrecy achieved by our proposed algorithm as compared to the scenario without a UAV-mounted RIS. The horizontal coordinates of Alice, Bob, and Eves are set as ${\mathit{c}}_A=[0,0]$ m, ${\mathit{c}}_B=[200,0]$ m, ${\mathit{c}}_{E1}=[300,300]$ m, ${\mathit{c}}_{E2}=[400,0]$ m, ${\mathit{c}}_{E3}=[0,-350]$ m, ${\mathit{c}}_{E4}=[300,-300]$ m. The UAV-mounted RIS flies from the starting point ${\mathit{c}}_0=[400,400]$ m to the terminal point ${\mathit{c}}_f=[400,-400]$ m. It should be noted that motivated by the accelerated development of wireless communication technologies, we have empirically configured the maximum element number of the RIS to 1024. This parameterization ensures a comprehensive evaluation of the performance boundaries inherent to our proposed system under technologically progressive conditions. The remaining parameters are summarized in

Table 1. Simulation parameters.

Parameters	Values
Height of the UAV $H_0$	40 m
Height of the Alice $H_A$	20 m
Average transmit power $P_{\mathrm{ave}}$	10 mW
Peak transmit power $P_{\mathrm{p}}$	40 mW
Tolerance $ϵ$	${10}^{-4}$
Reference channel gain ${\mathrm{g}}_0$	−20 dB
AWGN power ${\sigma }^2$	−80 dBm
Defaule UAV flight period T	150 s
Element separation distance at the RIS $d_e$	${\displaystyle \frac{{\lambda }_0}{2}}$
Pathloss exponent of RIS and Alice/Bob/Eves $\alpha$	2.5
Path loss exponent of AB and AE $\beta$	3

.

5.1. UAV Trajectories with Different RIS Elements

Figure 2 shows the UAV trajectories with different elements $M=64,256,1024$ of the RIS. The trends of the UAV trajectories are consistent with the geographical location of Bob and Alice and are appropriately far away from the eavesdroppers, revealing the advantage of the UAV-mounted RIS dynamically choosing the best trajectory. To be specific, the UAV always flies to Bob’s location from the starting point ${\mathit{c}}_0$, hovers for a while, then flies to Alice, and at the end, flies through Bob’s location to the terminal point ${\mathit{c}}_f$. We can observe that the majority of the flight time is concentrated between Alice and Bob, where the RIS can achieve secrecy enhancements significantly. Moreover, with the number of elements M increasing, the UAV obviously changes its trajectories between the starting/terminal point and Bob in order to evade eavesdroppers. Precisely, when the element at the RIS is set as $M=1024$, UAV distinctly adjusts its trajectories far away from Eve 1. This is because when M increases, the assistance role of the UAV-mounted RIS is remarkably enhanced, and the proposed algorithm enables the UAV to avoid eavesdroppers further.

5.2. Convergence of the Proposed Algorithm

Figure 3 illustrates the convergence of our proposed algorithm. We consider $M=0$ as the benchmark comparison, which means the considered system without the assistance of RIS. The curves of $M=64,256,1024$ all converge gradually, and as the number of M increases, the convergence speed slows down and can also converge within a finite round. Moreover, we can observe the considerable improvement achieved by our proposed scheme. To be specific, when $M=64$, the average secrecy rate increases obviously compared to the benchmark, which reveals the assistance performance of UAV-mounted RIS in multiple-eavesdroppers scenarios. Besides, the performance gaps between $M=64,256$, and 1024 illustrate the great potential of our proposed secure assistance scheme when it needs to meet a rigorous requirement of average secrecy rate.

5.3. The Transmit Power P Throughout the Optimization Process

Figure 4 illustrates the change of the base station transmit power P throughout the optimization process. It is evident from the results that with an increasing number of RIS elements M, the transmit power exhibits distinct variations over time. Specifically, when $M=1024$, P has a remarkable decline in the first 25 s. In contrast, when $M=256$, the change is slight. While for $M=64$, the transmit power does not change obviously compared with $P_{\mathrm{ave}}$. This is because when there are fewer reflective elements at the RIS, the transmit power does not change significantly when far away from Bob. However, when there are more reflective elements, Alice needs to provide greater power to ensure the average transmission rate when far away from Bob. Then, three curves maintain the average transmit power to transmit signals for a long period near Bob and Alice. However, after 130 s, when $M=64$, the transmit power does not change distinctly; when $M=256$, $P_{\mathrm{ave}}$ drops slightly; and when $M=1024$, $P_{\mathrm{ave}}$ drops visibly. This is because the total power is limited during the total flight period.

5.4. Average Secrecy Rate with Different Average Transmit Power $P_{\mathrm{ave}}$

Figure 5 depicts the average secrecy rate obtained by our proposed UAV-mounted RIS-aided scheme with different average transmit power $P_{\mathrm{ave}}$. We have the following crucial observations. To begin with, as expected, the average secrecy rates of different M ascend with $P_{\mathrm{ave}}$ increasing. Second, for the UAV-mounted RIS-aided multiple-eavesdroppers network, with $P_{\mathrm{ave}}$ fixed, the average secrecy rate gradually increases as M increases from 0 to 64, 256, and 1024. Moreover, we can observe that the performance gap between $M=256$ and $M=1024$ at $P_{\mathrm{ave}}=10$ mW is greater than the corresponding gap at $P_{\mathrm{ave}}=30$ mW. However, the performance gap between $M=64$ and benchmark at $P_{\mathrm{ave}}=30$ mW is greater than $P_{\mathrm{ave}}=10$ mW. This is reasonable because compared with the benchmark, when M is small, the increase of $P_{\mathrm{ave}}$ can effectively enhance the secrecy performance. When M becomes sufficiently large, the number of M plays a dominant role in improving the secrecy rate than the change of $P_{\mathrm{ave}}$. Third, the two cases where $P_{\mathrm{ave}}=15$ mW, $M=256$ and $P_{\mathrm{ave}}=25$ mW, $M=64$ obtain nearly the same average secrecy rate, which illustrates that with the assistance of a UAV-mounted RIS, our algorithm can save the energy consumption of the base station for a specific requirement. Additionally, the benchmark can be interpreted as scenarios where the current RIS is excluded from the optimization framework. The proposed algorithm achieves superior performance compared to cases involving solely the optimization of transmit power and UAV trajectory. This further validates the effectiveness of the proposed algorithm.

5.5. Transmission Rate with Different Number of RIS Elements

Figure 6 illustrates the impact of varying RIS element numbers on the transmission rates of a legitimate user and multiple eavesdroppers under a fixed average transmit power (10 mW), with benchmark curves representing scenarios without RIS deployment. As the RIS element number increases, both the user’s and Eves’ transmission rates exhibit enhancement trends, though the user rate improvement significantly outweighs that of Eves. This asymmetric performance stems from the RIS’s prioritized constructive signal superposition at the intended user, while residual electromagnetic reflections inadvertently create minor parasitic enhancements for Eves. The observed behavior aligns with our aforementioned analysis, confirming that the RIS-aided system inherently strengthens legitimate communication more effectively than unintended leakage, thereby preserving a favorable security ability.

5.6. Comparisons of UAV-Mounted Mobile RIS and Static RIS

Figure 7 compares the average secrecy rate achieved by our proposed UAV-mounted mobile RIS assistance scheme and a static RIS assistance scheme at $P_{\mathrm{ave}}=10$ mW. When the RIS is fixed, it indicates that the UAV trajectory is not optimized. The static RIS is deployed at the locations in a line between SP and TP at the same height $H_0$, and the distance in Figure 7 represents the distance to SP in the line. The three curves $M=64,256$, and 1024 are the optimized value by our proposed algorithm when $P_{\mathrm{ave}}$ = 10 mW in Figure 3. For the fixed RIS scheme, we can observe that when its location is close to the midpoint of the line, the value of the average secrecy rate increases significantly, although it also near Eve 2. This is because when RIS is positioned near the midpoint, the improvement in data rate at Bob is greater than the eavesdropping rate. Specifically, as expected, the average secrecy rate of the fixed RIS becomes greater with increasing of M. Under the same M, the average secrecy rate of our proposed scheme is greater than the static RIS. Moreover, it is evident that our proposed scheme outperforms the static RIS scheme with $M=1024$ in terms of average secrecy rate when $M=64$ is employed, which shows that our proposed scheme, fully exploiting the mobility of the UAV, can effectively improve the average secrecy rate.

5.7. The Influence of Pathloss Exponent and UAV Deployment Height

Figure 8 evaluates the influence of pathloss exponent $\alpha$ with different schemes for comparison. HEUR-TR is the heuristic trajectory scheme where the UAV-mounted RIS flies to a position above Bob, hovers long enough, and finally flies back to the destination with transmit power and phase shifts optimization during the process. Except for hovering, the UAV is always at maximum speed in HEUR-TR. TRIV-TR is the trivial trajectory scheme that optimizes the RIS phase shift and the transmit power with the UAV flying from the starting point and terminal point at a uniform speed in a straight line. PROP-TR is our proposed scheme. It can be observed that our proposed joint optimization always achieves a higher average secrecy rate than the heuristic trajectory scheme and the trivial trajectory scheme. With the increment of $\alpha$, the average secrecy rates all decrease because of the descending property of AIB/AIEs links concerning $\alpha$. The higher the value of $\alpha$, the easier the channel is to be wiretapped on, leading to a decrease in secrecy performance. Moreover, compared with HEUR-TR and TRIV-TR schemes, our proposed algorithm jointly optimizing $\mathsf{\Phi }$, $\mathit{Q}$, and $\mathit{P}$ can still achieve a better secrecy performance with the growing $\alpha$.

Figure 9 analyzes the influence of the UAV deployment height $H_0$ with different schemes for comparison. It can be clearly observed that there is a decrease in the average secrecy rate as $H_0$ increases in all three scenarios, which aligns with the practical implications of the scenario. Compared with the trivial trajectory TRIV-TR from the starting point and terminal point that only optimizes the RIS phase shifter and transmit power, the secrecy rates of heuristic trajectory HEUR-TR and our proposed design PROP-TR are more stable because their trajectories would require more time around the legitimate user. From the comparison between Figure 8 and Figure 9, we can find that $\alpha$ has a greater impact on the system secrecy performance than $H_0$.

6. Conclusions

In this paper, we investigated the assistance performance of a UAV-mounted dynamic RIS in secure communication with several eavesdroppers. Specifically, we considered the problem of maximizing the average secrecy rate by jointly optimizing the RIS phase shifter, the UAV trajectory, and the base station transmit power. To deal with the non-convexity of the problem, we proposed an iterative algorithm with a convergence guarantee based on the block coordinate descent and successive convex optimization techniques. Numerical results reveal that our proposed UAV-mounted RIS-aided scheme achieves a significant performance gain over the cases with a static RIS or without RIS for secrecy enhancement. Moreover, it also shows the inherent advantage of reducing energy consumption for a specific secrecy rate requirement. In the future, we will investigate the energy efficiency of the UAV-mounted RIS-aided scheme in the multiple-eavesdroppers scenario and the effectiveness of the proposed algorithm considering the mobility of the eavesdroppers. Besides, given the rapid evolutionary trajectory of artificial intelligence technologies, our forward-looking research framework proposes synthesizing the presented algorithm with deep learning architectures. This integration aims to systematically enhance the efficiency and adaptability of methods for handling complex challenges, further exploring the performance of the proposed secure design in practical scenarios.