Secure and Energy-Efficient Configuration Strategies for UAV-RIS System with Uplink NOMA

Abstract

This paper investigated the configuration of the reflecting elements for uplink non-orthogonal multiple access (NOMA) unmanned aerial vehicle (UAV)–reconfigurable intelligent surface (RIS) systems. By analyzing the practical air-to-ground (A2G) channels and phase estimation errors, a closed-form expression for the range of reflecting elements has been formulated to enhance the reliability and security of the system. Considering the energy efficiency of the system, the number of reflecting elements is optimized, aiming to maximize the energy secrecy efficiency (ESE) index under the given constraints. The simulation results verified the correctness of the derivation, which offers theoretical guidance for configuring RISs in uplink NOMA UAV systems with heterogeneous service demands. The uplink NOMA UAV system outperforms traditional terrestrial systems. The results also show that when the number of eavesdroppers increases, the influence of the number of reflecting elements on the system’s ESE becomes more significant. This demonstrates the benefits of equipping UAVs with RISs for the security of multiple eavesdropping systems.

1. Introduction

The crucial role of unmanned aerial vehicles (UAVs) in future wireless networks is gaining recognition due to their autonomous nature, adaptability, and cost-effectiveness [1]. As aerial communication facilitators, UAVs can dynamically adjust their positions and communication links to meet changing demands, thereby improving the capacity and reliability of wireless networks [2,3]. UAVs enhance the flexibility and resilience of communication networks by supporting various applications, including mobile edge computing, data collection, and emergency communications [4,5,6].

The advancement of wireless communication and the growth of Internet of Things (IoT) devices have intensified the pressure on spectral resources. Non-orthogonal multiple access (NOMA) addresses this issue by allowing multiple users to simultaneously share the same spectrum through power domain multiplexing, thus improving the spectral efficiency [7,8,9]. NOMA efficiently supports numerous connections and provides equitable service to users with diverse channel conditions, aligning with 6G network requirements [10,11,12].

Incorporating NOMA into UAV communication shows promise in enhancing systems’ reliability [13]. In UAV communications, this technology enables simultaneous access for numerous users, effectively boosting the system’s overall capacity. The combination of NOMA technology and UAVs facilitates concurrent access for many users, meeting the high-capacity demands of future 6G networks [14,15]. Furthermore, integrating NOMA and UAVs not only improves the spectral efficiency and system capacity but also increases the network’s flexibility and reliability [16,17]. In [18], the researchers proposed a secure communication scheme for NOMA-based UAV mobile edge computing systems in the presence of a flying eavesdropper, utilizing successive convex approximation and block coordinate descent methods to address security concerns. The authors of [18] also suggested a dual-UAV-assisted IoT system using NOMA to enhance the IoT capacity and improve the UAVs’ energy efficiency. Similarly, ref. [19] proposes a dual-UAV-assisted IoT system using non-NOMA to achieve the same goals. This study formulated a non-convex optimization problem to maximize UAVs’ energy efficiency by jointly optimizing the communication scheduling, UAV transmission power, and motion parameters, solving it using an alternating iterative optimization algorithm. In [20], the researchers examined a multi-antenna UAV-assisted NOMA communication system where a multi-antenna base station communicated with indoor and outdoor users via a multi-antenna UAV. Their numerical and simulation results show that adjusting the UAV parameters and increasing the antenna numbers can significantly improve the system performance.

Despite these advancements, existing UAV-NOMA studies have mainly focused on downlink or edge computing scenarios, and the research on UAV-based uplink NOMA transmissions remains limited, particularly in systems with heterogeneous service requirements. UAV communication typically relies on line-of-sight (LoS) links, which are characterized by a high signal strength and low propagation loss. However, the inherent broadcast characteristics and LoS-dominated channels of UAV communications render these systems susceptible to security vulnerabilities [21]. In comparison to traditional terrestrial communication, UAV communication exhibits increased vulnerability to security threats, such as malicious eavesdropping and privacy leakages [22,23].

Furthermore, UAVs predominantly operate on battery power, which is limited in capacity. During flight and communication operations, UAVs consume substantial amounts of energy. In complex environments such as urban areas, buildings can obstruct LoS links, requiring UAVs to expend additional energy to maintain stable communication. The constraints of their limited energy capacity and inaccessibility for battery replacement significantly impact the operational longevity of UAVs [24]. Addressing security and energy limitations in a synergistic manner is imperative for the advancement and sustainability of UAV-based communication systems [25].

In light of these challenges, reconfigurable intelligent surfaces (RISs) are regarded as a revolutionary technology for forthcoming 6G wireless networks [26]. By integrating a large number of low-cost passive reflecting elements, RISs can intelligently reconfigure the wireless propagation environment, thereby significantly enhancing the performance of wireless communication networks. The utilization of a controlled propagation environment facilitated by RISs is anticipated to enhance the reliability of UAV communications [27]. With the assistance of UAVs, RISs can be deployed with greater flexibility at elevated locations, which will enable the easier establishment of LoS channel links. Moreover, incorporating a RIS can enhance the communication quality through phase adjustments to the reflection, thus mitigating the power consumption and equipment burden of UAV systems. RISs can be lightweight and seamlessly integrated into UAVs, thereby increasing the channel capacity [28,29,30]. Additionally, a RIS possesses the capability to reconfigure radio channels, enhancing the signal reception for authorized users while concurrently degrading the channels accessible to eavesdroppers [31,32]. Physical layer security (PLS) can complement or potentially supplant traditional cryptography by leveraging the randomness of wireless fading channels to secure UAV wireless communications from an information theoretic perspective [33]. Advancements in RISs are enabling a multitude of novel opportunities to enhance the security of UAV communications. Gu et al. developed an efficient algorithm based on an alternating optimization technique to jointly design a UAV’s trajectory, a RIS’s passive beamforming, and the transmit power [34]. Experimental validations by Li et al. [35] demonstrate the enhancements to the secrecy capacity of a terrestrial RIS made possible by the use of 3D beamforming optimization. The existing literature, such as [13,31], focuses on using NOMA technology to enhance UAV systems’ capacity or RISs to enhance PLS, respectively, but does not explore the potential for synergistic optimization of the two. For example, Gu et al. proposed a RIS-assisted UAV trajectory design, but their model was based on orthogonal multiple access (OMA) and did not take advantage of NOMA’s spectral efficiency [34]. Li et al. increased the secrecy capacity through 3D beamforming optimization but did not combine this with multi-user power domain multiplexing [35].

The current research on UAV-RIS systems predominantly concentrates on enhancing either their reliability [28,36] or security [31,37], yet the simultaneous achievement of both remains insufficiently explored. Furthermore, the number of RIS elements on UAVs is typically predetermined. While increasing the number of elements can enhance the reliability, it also facilitates the interception of signals by eavesdroppers. Additionally, a greater number of elements results in increased costs and heavier payloads for the UAVs, potentially diminishing their flight durations. Existing studies have attempted to balance security and energy efficiency in determining the number of RIS elements. However, no research has yet specifically addressed this issue for uplink NOMA UAV-RIS systems [38].

Motivated by the aforementioned challenges and issues, the range of reflecting elements possible within the constraints of both reliability and security is first determined for uplink NOMA UAV-RIS systems. Subsequently, the optimal number of reflecting elements is determined by optimizing the energy secrecy efficiency (ESE). The contributions of this paper can be summarized as follows:

• A practical elevation-based A2G LoS link subject to Nakagami-m fading is established. This study specifically evaluates the effects of the phase estimation errors that arise from imperfect conditions such as UAV jitter, providing insights into how these errors influence the overall communication performance.
• The range of reflecting elements for simultaneously reliable and secure transmissions from an uplink NOMA UAV-RIS system is derived. The derivation serves as a theoretical reference for configuring RISs in UAV systems with heterogeneous service requirements, ensuring that the system can achieve both reliability and security.
• To address the energy efficiency of the system, we define and utilize the indexed ESE metric to characterize the overall performance of the system. This metric comprehensively characterizes the system’s performance by balancing energy consumption and security. Under the constraints of reliability and security, the number of reflecting elements is optimized to maximize the ESE.
• Numerical results are presented to validate the accuracy of our theoretical analysis, revealing the impact of key system parameters on the system performance. These findings provide valuable guidance for the practical deployment and optimization of uplink NOMA UAV-RIS systems in future wireless communication networks.

2. System Model

As shown in Figure 1, we consider a ground–aerial uplink NOMA communication system. Specifically, two users $D_1$ and $D_2$ upload information to the base station (S) via a UAV-mounted RIS in the presence of K non-colluding eavesdroppers, denoted as $E_1,E_2,\dots ,E_K$. Specifically, we assume $D_1$ has been accorded a high-priority status with the highest quality-of-service requirements, while $D_2$ is a low-priority user. This makes sense in real-world communication application scenarios, where sensitive user data necessitate prioritized transmissions. However, non-critical key performance indicator data lack stringent real-time demands and can consequently be treated as a lower transmission priority. The RIS is composed of N reflecting elements, which are denoted as $R_1,R_2,\dots ,R_N$. The horizontal distance between the ground node i and te UAV U is expressed as $r_{Ui}$ ($i\in \left\{S,D_1,D_2,E_k\right\}$) (Although it is extremely difficult for legitimate systems to directly estimate the CSI in real-world scenarios, it can be obtained in the following ways. Initially, in certain scenarios, the system may possess prior knowledge of the network topology or the eavesdropper’s location, which can facilitate an estimation of the channel information. Subsequently, during the transmission process, the system can make a channel estimation by transmitting pilot signals or employing alternative estimation techniques. Furthermore, the system can estimate the eavesdropper’s channel when it actively attacks by detecting the signals sent during active eavesdropping or an attack.).

2.1. The Channel Model

A probabilistic LoS channel model, predicated on the elevation angle, is widely utilized to characterize A2G channels [39]. The probability of a LoS channel from the UAV to the ground node i can be expressed as

$$P_L^{Ui}={\displaystyle \frac{1}{1+\varphi \exp [-\omega ({\theta }_{Ui}-\varphi )]}},$$

(1)

where $\varphi$ and $\omega$ are influenced by environmental factors. The elevation angle ${\theta }_{Ui}$ is a function of the UAV’s flight altitude H, which is given by

$${\theta }_{Ui}={\displaystyle \frac{180°}{\pi }}arctan\left({\displaystyle \frac{H}{r_{Ui}}}\right),$$

(2)

The path loss exponent between U and i is represented as

$$\mathsf{\Lambda }\left({\theta }_{Ui}\right)=a_1{P}_L^{Ui}+b_1,$$

(3)

where $a_1\approx {\tau }_{{\displaystyle \frac{\pi }{2}}}-{\tau }_0$, $b_1={\tau }_0-{\tau }_1{P}_L^{Ui}\approx {\tau }_0$. ${\tau }_{{\displaystyle \frac{\pi }{2}}}$ and ${\tau }_0$ are environment-dependent constants. Accordingly, the average A2G channel gain $L_{Ui}$ can be written as

$$L_{Ui}=\left(P_L^{Ui}+\left(1-P_L^{Ui}\right)\eta \right){d_{Ui}}^{-\mathsf{\Lambda }\left({\theta }_{Ui}\right)},$$

(4)

where $\eta$ denotes the additional attenuation factor under NLoS conditions, and $d_{Ui}=\sqrt{{r_{Ui}}^2+H^2}$ represents the distance between U and i with the horizontal distance $r_{Ui}$.

In characterizing the small-scale fading within A2G channels, we employ the generalized Nakagami-m distribution. The channel vectors between the RIS and S, $D_1$, $D_2$, and $E_k$ are denoted as ${\mathbf{h}}_{RS}=\left[h_{R_{1}S},h_{R_{2}S},\dots ,h_{R_{N}S}\right]\in {\mathbb{C}}^{1\times N}$, ${\mathbf{h}}_{D_{1}R}={\left[h_{D_1{R}_1},h_{D_1{R}_2},\dots ,h_{D_1{R}_N}\right]}^T\in {\mathbb{C}}^{N\times 1}$, ${\mathbf{h}}_{D_{2}R}={\left[h_{D_2{R}_1},h_{D_2{R}_2},\dots ,h_{D_2{R}_N}\right]}^T\in {\mathbb{C}}^{N\times 1}$, and ${\mathbf{h}}_{R{E}_k}=\left[h_{R_1{E}_k},h_{R_2{E}_k},\dots ,h_{R_3{E}_k}\right]\in {\mathbb{C}}^{1\times N}$. Each entry of ${\mathbf{h}}_{RS}$, ${\mathbf{h}}_{D_{1}R}$, ${\mathbf{h}}_{D_{2}R}$, and ${\mathbf{h}}_{R{E}_k}$ is assumed to undergo independent and identically distributed (i.i.d.) Nakagami-m small-scale fading, i.e., $h_{R_{n}S}$∼$\mathrm{Nakagami}\left(m_{US},1\right)$, $h_{D_1{R}_n}$∼$\mathrm{Nakagami}\left(m_{U{D}_1},1\right)$, $h_{D_2{R}_n}$∼$\mathrm{Nakagami}\left(m_{U{D}_2},1\right)$, $h_{R_n{E}_k}$∼$\mathrm{Nakagami}\left(m_{U{E}_k},1\right)$. The parameter $m_{Ui}$ is a variable related to the elevation ${\theta }_{Ui}$, which can be expressed as

$$m_{Ui}={\displaystyle \frac{{\left(K\left({\theta }_{Ui}\right)+1\right)}^2}{2K\left({\theta }_{Ui}\right)+1}},$$

(5)

where $K\left({\theta }_{Ui}\right)=a_2\exp \left(b_2{\theta }_{Ui}\right)$ denotes the Rician factor with $a_2=K\left(0\right)$ and $b_2={\displaystyle \frac{2}{\pi }}\ln {\displaystyle \frac{K\left(\frac{\pi }{2}\right)}{K\left(0\right)}}$ being environment-dependent parameters.

2.2. The Phase Shift Design and SNR Distribution

To ensure reliability and security for high-priority users, the phase shifts in the RIS are adjusted to maximize the channel quality for $D_1$. Therefore, the channel quality for $D_1$ is superior to that fpr $D_2$. According to the uplink NOMA principle, S first decodes the signal of the stronger user $D_1$ from the received superimposed signals by treating the signal of the weaker user $D_2$ as noise. Subsequently, S employs successive interference cancellation (SIC) to mitigate the signal from $D_1$, followed by decoding $D_2$’s signal. The received signal-to-noise ratio (SNR) for decoding the signals of $D_1$ and $D_2$ can be expressed as

$${\gamma }_{D_1}={\displaystyle \frac{{\rho }_s{\left|{\mathbf{h}}_{RS}\mathbf{\Phi }{\mathbf{h}}_{D_{1}R}\right|}^2{L}_{US}{L}_{U{D}_1}}{{\rho }_s{\left|{\mathbf{h}}_{RS}\mathbf{\Phi }{\mathbf{h}}_{D_{2}R}\right|}^2{L}_{US}{L}_{U{D}_2}+1}},$$

(6)

$${\gamma }_{D_2}={\rho }_s{\left|{\mathbf{h}}_{RS}\mathbf{\Phi }{\mathbf{h}}_{D_{2}R}\right|}^2{L}_{US}{L}_{U{D}_2},$$

(7)

where ${\rho }_s=P_s/N_0$ denotes the transmission SNR of the users, $P_s$ is the transmision power of the users, $N_0$ is the variance in the additive white Gaussian noise, and $\mathbf{\Phi }=\mathrm{diag}\left[e^{j{\varphi }_1},e^{j{\varphi }_2},\dots ,e^{j{\varphi }_N}\right]$ is the reflection-coefficient matrix of the RIS. ${\left\{{\varphi }_n\right\}}_{n=1}^N$ denotes the reflection coefficient matrix of the RIS. Ideally, the optimal phase shift ${\varphi }_n^{opt}$ is designed to maximize the SNR of the signal of $D_1$, which can be expressed as

$${\varphi }_n^{opt}=-\left({\varphi }_{R_{n}S}+{\varphi }_{D_1{R}_n}\right),$$

(8)

where ${\varphi }_{R_{n}S}$ and ${\varphi }_{D_1{R}_n}$ are the phases of $h_{R_{n}S}$ and $h_{D_1{R}_n}$, respectively.

In alignment with prior studies [10,11,40], our analysis adopts a conservative security framework where eavesdroppers are assumed to possess the optimal multi-user detection techniques. This allows them to intercept and decode signals intended for either $D_1$ or $D_2$ by suppressing the interference from the other device. Under this adversarial assumption, the achievable SNRs for eavesdroppers to recover $D_1$ and $D_2$’s transmissions are, respectively, defined as

$${\gamma }_{E_k\to D_1}={\rho }_s{\left|{\mathbf{h}}_{R{E}_k}\mathbf{\Phi }{\mathbf{h}}_{D_{1}R}\right|}^2{L}_{U{D}_1}{L}_{U{E}_k},$$

(9)

$${\gamma }_{E_k\to D_2}={\rho }_s{\left|{\mathbf{h}}_{R{E}_k}\mathbf{\Phi }{\mathbf{h}}_{D_{2}R}\right|}^2{L}_{U{D}_2}{L}_{U{E}_k}.$$

(10)

In practical implementations, hardware imperfections (e.g., mechanical jitter) and environmental disturbances (e.g., airflow turbulence) constrain the ability to perfectly align reconfigurable surface elements with their theoretical phase states. This mismatch induces non-negligible deviations in the realized phase profile of the n-th reflecting unit.

$${\varphi }_n={\varphi }_n^{opt}+{\varphi }_n^{err}.$$

(11)

The phase estimation error ${\varphi }_n^{err}$ typically follows a von Mises distribution with the mean 0 and the concentration parameter $\kappa$. The probability density function (PDF) and eigenfunction of ${\varphi }_n^{err}$ are, respectively, given by

$$f_{{\varphi }_n^{err}}\left(x\right)={\displaystyle \frac{e^{\kappa cos\left(x\right)}}{2\pi I_0\left(\kappa \right)}},$$

(12)

$${\phi }_p=\mathbb{E}\left(e^{jp{\varphi }_n^{err}}\right)={\displaystyle \frac{I_p\left(\kappa \right)}{I_0\left(\kappa \right)}},$$

(13)

where $I_0\left(·\right)$ and $I_p\left(·\right)$ represent the zero-order and p-order modified Bessel function of the first kind, respectively. The concentration parameter $\kappa$ reflects the impact of jitter and exhibits an inverse relationship with the phase estimation error [41]. As the UAV experiences substantial jitter due to the effects of airflow, the phase estimation error increases, resulting in a decrease in $\kappa$. Conversely, a higher $\kappa$ indicates reduced variability in the phase errors, attributable to enhanced directional concentration. Notably, when $\kappa =0$, the phase error follows a uniform distribution.

Under phase estimation errors, the SNRs of $D_1$ and $D_2$ can be given by

$${\gamma }_{D_1}={\displaystyle \frac{{\rho }_s{\left|{\displaystyle \sum _{n=1}^N}\left|h_{R_{n}S}\right|\left|h_{D_1{R}_n}\right|e^{j{\varphi }_n^{err}}\right|}^2{L}_{US}{L}_{U{D}_1}}{{\rho }_s{\left|{\displaystyle \sum _{n=1}^N}\left|h_{R_{n}S}\right|\left|h_{D_2{R}_n}\right|e^{j{\varphi }_{D_2}}\right|}^2{L}_{US}{L}_{U{D}_2}+1}},$$

(14)

$${\gamma }_{D_2}={\rho }_s{\left|\sum _{n=1}^N\left|h_{R_{n}S}\right|\left|h_{D_2{R}_n}\right|e^{j{\varphi }_{D_2}}\right|}^2{L}_{US}{L}_{U{D}_2},$$

(15)

where ${\varphi }_{D_2}={\varphi }_{D_2{R}_n}-{\varphi }_{D_1{R}_n}+{\varphi }_n^{err}$.

The eavesdropping SNRs can, respectively, be expressed as

$${\gamma }_{E_k\to D_1}={\rho }_s{\left|\sum _{n=1}^N\left|h_{R_n{E}_k}\right|\left|h_{D_1{R}_n}\right|e^{j{\varphi }_{E_1}}\right|}^2{L}_{U{D}_1}{L}_{U{E}_k},$$

(16)

$${\gamma }_{E_k\to D_2}={\rho }_s{\left|\sum _{n=1}^N\left|h_{R_n{E}_k}\right|\left|h_{D_2{R}_n}\right|e^{j{\varphi }_{E_2}}\right|}^2{L}_{U{D}_2}{L}_{U{E}_k},$$

(17)

where ${\varphi }_{E_1}={\varphi }_{R_n{E}_k}-{\varphi }_{R_{n}S}+{\varphi }_n^{err}$, ${\varphi }_{E_2}={\varphi }_n^{err}-{\varphi }_{R_{n}S}+{\varphi }_{R_n{E}_k}+{\varphi }_{D_1{R}_n}+{\varphi }_{D_2{R}_n}$.

2.3. Statistical Characteristics of the Channels

To facilitate subsequent calculations, we present the following lemmas regarding the statistical properties of legitimate and eavesdropping channels, respectively.

Lemma 1.

By denoting $H_{D_1}={\left|{\displaystyle \sum _{n=1}^N}\left|h_{R_{n}S}\right|\left|h_{D_1{R}_n}\right|e^{j{\varphi }_n^{err}}\right|}^2$ and $H_{D_2}={\left|{\displaystyle \sum _{n=1}^N}\left|h_{R_{n}S}\right|\left|h_{D_2{R}_n}\right|e^{j{\varphi }_{D_2}}\right|}^2$, the distribution of $H_{D_1}$ and $H_{D_2}$ can be approximated as

$$H_{D_1}\sim \mathsf{\Gamma }\left(m_{D_1},{\mathsf{\Omega }}_{D_1}\right),$$

(18)

$$H_{D_2}\sim EXP\left(N\right),$$

(19)

where $m_{D_1}={\displaystyle \frac{\frac{N}{2}{\xi }_1^4{{\phi }_1}^2}{1+{\phi }_2-2{\xi }_1^4{{\phi }_1}^2}}$, ${\mathsf{\Omega }}_{D_1}=N^2{\xi }_1^4{{\phi }_1}^2$, ${\xi }_1=\sqrt{\mathbb{E}\left(\left|h_{R_{n}S}\right|\right)\mathbb{E}\left(\left|h_{D_1{R}_n}\right|\right)}$, $\mathbb{E}\left(\left|h_{R_{n}S}\right|\right)={\displaystyle \frac{\mathsf{\Gamma }(m_{US}+0.5)}{\mathsf{\Gamma }\left(m_{US}\right)\sqrt{m_{US}}}}$, $\mathbb{E}\left(\left|h_{D_1{R}_n}\right|\right)={\displaystyle \frac{\mathsf{\Gamma }(m_{U{D}_1}+0.5)}{\mathsf{\Gamma }\left(m_{U{D}_1}\right)\sqrt{m_{U{D}_1}}}}$, and ${\phi }_1$ and ${\phi }_2$ can be obtained in (13).

Proof. 

See Appendix A.    □

3. Analysis of the Configuration of Reflecting Elements and Their Optimization

This study establishes a framework for analysis of the reflecting elements in uplink NOMA systems, where the minimum number of elements N is analytically characterized to jointly satisfy reliability and secrecy constraints. Given $D_1$’s designation as a high-priority user, the feasible domain of N is determined through a rigorous analysis of its connection outage probability (COP) and secrecy outage probability (SOP). Subsequently, we formalize the outage performance metrics as follows: for any user $D_i$ ($i\in \left\{1,2\right\}$), the COP quantifies the likelihood of failed signal decoding at the legitimate receiver, while the SOP represents the probability of confidential information being intercepted by eavesdroppers.

$$P_{co{p}_1}=Pr\left\{{log}_2\left(1+{\gamma }_{D_1}\right)<R_1\right\},$$

(20)

where $R_1$ denotes the target transmission rate for $D_1$. For an uplink NOMA transmission, S first decodes the signal of $D_1$ and then removes it by using SIC to decode the signal of $D_2$. Hence, the COP for user $D_2$ can be derived as

$$P_{co{p}_2}=1-Pr\left\{{log}_2\left(1+{\gamma }_{D_1}\right)>R_1,{log}_2\left(1+{\gamma }_{D_2}\right)>R_2\right\}.$$

(21)

We consider a non-colluding scenario where eavesdroppers work independently without cooperation to decode legitimate users’ messages. In order to investigate the secrecy of transmissions from $D_i$, wiretap code is utilized to describe the secrecy performance [42]. A secrecy outage occurs when the capacity of the wiretap channel exceeds the redundancy rate of the wiretap code. Consequently, the secrecy outage probability (SOP) for $D_i$ is given by

$$P_{so{p}_i}=\mathrm{Pr}\left\{{\log }_2\left(1+{\gamma }_{E\to D_i}\right)>R_i-R_s\right\},$$

(22)

where $R_s$ denotes the target secrecy rate, and $R_i-R_s$ is the redundancy rate against eavesdropping for $D_i$.

To holistically characterize the performance of the system, we define and apply the energy secrecy efficiency (ESE) index to evaluate both the security and energy efficiency of the system, which is defined as the ratio of the effective secrecy throughput to the consumed power. The ESE of user $D_i$ ($i\in \left\{1,2\right\}$) can be expressed as

$$ES{E}_i={\displaystyle \frac{R_s\left(1-P_{co{p}_i}\right)\left(1-P_{so{p}_i}\right)}{N{P}_{RIS}+P_s+P_c}},$$

(23)

where $P_{RIS}$ and $P_c$ represent the power consumption for the RIS’s phase shift control and the other node circuitry, respectively.

Our objective is to select the optimal number of reflecting elements that maximizes the system’s ESE while ensuring reliable and secure transmissions for user $D_1$. Therefore, the problem can be formulated as

$$\begin{array}{cc}\hfill \mathrm{P}1:& \underset{N}{max}ES{E}_{sum},\hfill \end{array}$$

(24a)

$$\begin{array}{cc}\hfill \mathrm{s}.\mathrm{t}.& P_{co{p}_1}<v_c,\hfill \end{array}$$

(24b)

$$\begin{array}{cc}\hfill & P_{so{p}_1}<v_s,\hfill \end{array}$$

(24c)

where $ES{E}_{sum}=ES{E}_1+ES{E}_2$, $v_c$ and $v_s$ denote the pre-set thresholds for $P_{cop}$ and $P_{sop}$, respectively.

Given that $D_1$ is a high-priority user, the number of reflecting elements necessary to facilitate reliable and secure transmissions is calculated by constraining the COP and the SOP for $D_1$ to a preset threshold. In the following, the range of N, i.e., constraints (24b) and (24c), can be given in the following theorems.

Theorem 1.

To ensure reliability for $D_1$, the number of reflecting elements should satisfy

$$\begin{array}{c}\hfill N\ge N_c=⌈{\displaystyle \frac{-\ln v_c{\chi }_1{L}_{U{D}_2}+\sqrt{{\left(\ln v_c{\chi }_1{L}_{U{D}_2}\right)}^2+\frac{4{\xi }_1^4{{\phi }_1}^2{L}_{U{D}_1}{\chi }_1}{L_{US}{\rho }_s}}}{2{\xi }_1^4{{\phi }_1}^2{L}_{U{D}_1}}}⌉,\end{array}$$

(25)

where ${\chi }_1=2^{R_1}-1$, $v_c$ denotes the preset threshold for the COP, and $⌈·⌉$ represents the ceiling function.

Proof. 

See Appendix B.    □

Theorem 2.

To ensure security for $D_1$, the number of reflecting elements should satisfy

$$N\le N_s=⌊-{\displaystyle \frac{{\chi }_e}{\ln \left(1-\sqrt[K]{1-v_s}\right){\rho }_s{L}_{U{D}_1}{L}_{U{E}_k}}}⌋,$$

(26)

where ${\chi }_e=2^{R_1-R_s}-1$, $v_s$ denotes the preset threshold for the SOP, and $⌊·⌋$ denotes the floor function.

Proof. 

See Appendix C.    □

$$\begin{array}{cc}\hfill ES{E}_{sum}& ={\displaystyle \frac{R_s}{N{P}_{RIS}+P_s+P_c}}\left(\left(1-e^{{\displaystyle \frac{1}{N{\rho }_s{L}_{US}{L}_{U{D}_2}}}}{\left({\displaystyle \frac{L_{U{D}_1}{\mathsf{\Omega }}_{D_1}}{{\chi }_{1}N{L}_{U{D}_2}{m}_{D_1}}}+1\right)}^{-m_{D_1}}\right)\right.\hfill \\ \hfill & +\left(1-{\left(1-e^{-{\displaystyle \frac{{\chi }_e}{{\rho }_s{L}_{U{D}_1}{L}_{U{E}_k}N}}}\right)}^K\right)\left({\displaystyle \frac{1}{N}}\sum _{n=0}^{m_{D_1}-1}{e}^{-{\displaystyle \frac{{\chi }_1{m}_{D_1}}{{\rho }_s{\mathsf{\Omega }}_{D_1}{L}_{US}{L}_{U{D}_1}}}}{\left({\displaystyle \frac{{\chi }_1{m}_{D_1}}{{\rho }_s{\mathsf{\Omega }}_{D_1}{L}_{US}{L}_{U{D}_1}}}\right)}^n\right.\hfill \\ \hfill & \times {\displaystyle \frac{1}{n!}}\sum _{t=0}^n\left({\displaystyle \genfrac{}{}{0pt}{}{n}{t}}\right){\left({\rho }_s{L}_{US}{L}_{U{D}_2}\right)}^t{\left({\displaystyle \frac{{\chi }_1{L}_{U{D}_2}{m}_{D_1}}{{\mathsf{\Omega }}_{D_1}{L}_{U{D}_1}}}+{\displaystyle \frac{1}{N}}\right)}^{-t-1}\hfill \\ \hfill & \times \left.\gamma \left(m_{D_1}+1,\left({\displaystyle \frac{{\chi }_1{L}_{U{D}_2}{m}_{D_1}}{{\mathsf{\Omega }}_{D_1}{L}_{U{D}_1}}}+{\displaystyle \frac{1}{N}}\right){\displaystyle \frac{{\chi }_2}{{\rho }_s{L}_{US}{L}_{U{D}_2}}}\right)\right)\left(1-{\left(1-e^{-{\displaystyle \frac{{\chi }_e}{{\rho }_s{L}_{U{D}_2}{L}_{U{E}_k}N}}}\right)}^K\right)\hfill \end{array}$$

(27)

Theorem 3.

Based on the aforementioned analysis, the problem P1 can be further formulated as

$$\begin{array}{cc}\hfill \mathrm{P}2:& \underset{N}{max}ES{E}_{sum},\hfill \\ \hfill s.t.& N_c\le N\le N_s\hfill \end{array}$$

(28)

where ${\chi }_2=2^{R_2}-1$, and $ES{E}_{sum}$ is shown at the top of the next page. The feasible conditions for problem P2 require the preset thresholds to satisfy

$$\begin{array}{c}\hfill v_s\le 1-{\left(1-e^{-{\displaystyle \frac{{\chi }_e}{{\rho }_s{L}_{U{D}_1}{L}_{U{E}_k}⌈\frac{-\ln v_c{\chi }_1{L}_{U{D}_2}+\sqrt{{\left(\ln v_c{\chi }_1{L}_{U{D}_2}\right)}^2+\frac{4{\xi }_1^4{{\phi }_1}^2{L}_{U{D}_1}{\chi }_1}{L_{US}{\rho }_s}}}{2{\xi }_1^4{{\phi }_1}^2{L}_{U{D}_1}}⌉}}}\right)}^K.\end{array}$$

(29)

Proof. 

See Appendix D.    □

As indicated in Problem P2, deriving a closed-form expression for the optimal N presents significant challenges. Consequently, we propose an iterative approach, leveraging the golden section search method, to identify the optimal $N_{opt}$. This methodology is encapsulated in Algorithm 1, which is detailed below.

Algorithm 1: Golden search iterative algorithm for $N_{opt}$

Initialization: $a_1=N_c$, $a_2=N_s$, and the maximum tolerance $\epsilon ={10}^{-4}$;

Calculate $N_1=a_2-0.618\left(a_2-a_1\right)$ and $N_2=a_1+0.618\left(a_2-a_1\right)$;

While $\left|N_1-N_2\right|>\epsilon$ do;

$N=N_c$, calculate $ES{E}_{sum}^1$ using (27);

$N=N_s$, calculate $ES{E}_{sum}^2$ using (27) again;

if $ES{E}_{sum}^1\ge ES{E}_{sum}^2$ do;

$a_1=N_1$, $N_1=N_2$, $N_2=a_1+0.618\left(a_2-a_1\right)$;

else do

$a_2=N_2$, $N_2=N_1$, $N_1=a_2-0.618\left(a_2-a_1\right)$;

end if

end while

Obtain the optimal $N_{opt}=\left[{\displaystyle \frac{a_1+a_2}{2}}\right]$.

4. Simulation Results and Discussions

In this section, we evaluate the effect of key parameters on the system performance. The horizontal distance is set as $r_{US}=50$ m, $r_{U{D}_1}=50$ m, $r_{U{D}_2}=50$ m, and $r_{U{E}_k}=50$ m, respectively. Without the loss of generality, we set $H=50$ m, $P_{RIS}=-10$ dBm, $P_s=10$ dBm, $N_0=-60$ dBm, $R_2=2$, $R_2=1.5$, $v_c=v_s=0.1$, and $\kappa =20$. According to [43], the environmental parameters are set as $\varphi =4.88$, $w=0.43$, ${\tau }_0=3.5$, ${\tau }_{{\displaystyle \frac{\pi }{2}}}=2$, $K\left(0\right)=5$ dB, $K\left({\displaystyle \frac{\pi }{2}}\right)=15$ dB, and $\eta =0.2$.

As shown in Figure 2, Lemma 1 is verified via Monte Carlo simulations. The distribution of $H_{D_1}$ and $H_{D_2}$ closely matches the simulated values, thereby validating the correctness of the analysis. Additionally, Lemma 1 demonstrates that the distribution of $H_{D_2}$ is independent of the channel qualities of ${\mathbf{h}}_{RS}$ and ${\mathbf{h}}_{D_{1}R}$ and is solely dependent on the number of RIS elements. This indicates that the RIS possesses the capability to intelligently reconfigure the wireless propagation environment, thereby significantly enhancing the secrecy performance for wireless communication.

Figure 3 depicts the number of reflecting element boundaries $N_c$ under COP constraints $v_c$. A random phase shift scheme, continuous phase shift scheme [36], and OMA scheme [38] are designed as benchmark schemes for comparison. The results depicted in the figure demonstrate a strong correlation between the theoretical analysis and the simulation data, thereby validating the accuracy of the theoretical analysis. The necessary range for the number of reflecting elements can be determined using Theorems 1. It is observed that the minimum number of $N_c$ decreases with an increment in $v_c$. Additionally, a reduction in $\kappa$ leads to a decrease in $N_c$. When $\kappa$ reaches 20, the system’s performance is nearly indistinguishable from that of an ideal continuous phase shift system, implying that the phase shift errors become insignificant when $\kappa \ge 20$. As evidenced by the results in Figure 3, in contrast to the OMA scheme, NOMA achieves a substantial reduction in the number of reflecting elements required while maintaining an equivalent, reliable performance. Compared with the randomly distributed phase shift, the designed phase shift can reduce the number of reflecting elements required. This underscores the efficiency of the passive beamforming enabled by the RIS in enhancing the system’s reliability.

Figure 4 characterizes the relationship between the number of reflecting elements required $N_s$ versus $v_s$. As $v_s$ increases, the required $N_s$ increases. As the altitude of the UAV rises, $N_s$ also increases, indicating that the UAV should be equipped with more reflecting elements when it is deployed at higher altitudes. Compared with traditional terrestrial RIS schemes [44], the number of reflecting elements needed in terrestrial systems is usually much higher than that in UAV-based systems. This discrepancy arises because deploying UAVs at higher altitudes can improve the channel quality, despite the increased path loss. Additionally, the random distribution of the phases does not significantly affect $N_s$. This is because the RIS can dynamically adjust the phase, and the phase of the signal received by the eavesdropper is inherently random. Consequently, the overall requirement for reflecting elements remains unaffected.

Figure 5 illustrates the relationship between the ESE and the number of reflecting elements N. The simulation results validate the correctness of the derivation. It can be observed from the figure that the range of reflecting elements that ensures the system’s reliability and security narrows as the user transmission power $P_s$ increases. This trend is primarily attributed to the fact that with a higher transmission power, fewer reflecting elements are required to maintain security. Specifically, a reduced number of reflecting elements helps mitigate the risk of eavesdropping by limiting the potential exposure of signals to unauthorized parties. The addition of N enhances the RIS’s beamforming gain and improves the channel quality for legitimate users, thereby facilitating the secrecy performance. The power consumption of the RIS increases linearly with N, resulting in an increase in the denominator. These results also indicate that an optimal number of reflecting elements $N_{opt}$ exists that maximizes the ESE. The optimal ESE does not increase with an increment in $P_s$. This is due to the fact that as $P_s$ increases, the system’s energy consumption also rises, which in turn diminishes the ESE. The numerical optimal number of elements and the maximum ESE obtained from the proposed iterative algorithm are consistent, confirming the correctness of the algorithm. When the number of eavesdroppers K is small, the number of reflecting elements has a minor impact on the system’s ESE, but as K increases, the impact on the ESE becomes more significant. This demonstrates the benefits of equipping UAVs with RISs for security in systems with multiple eavesdroppers.

5. Conclusions

This paper has investigated the optimal configuration of reflecting elements in uplink NOMA UAV-RIS systems, focusing on optimizing the ESE of the system under reliability and security constraints. By modeling practical A2G channels and incorporating phase estimation errors, we derived a closed-form expression for the range of reflecting elements. This derivation provides a clear guideline for configuring RISs in such systems, ensuring a reliable and secure performance under heterogeneous service demands. The uplink NOMA UAV-RIS system is superior to conventional terrestrial systems. The results also show that the range of reflecting elements to ensure the system’s reliability and security narrows as the user transmission power $P_s$ increases. Moreover, the optimal ESE does not increase with an increase in $P_s$. The findings provide valuable guidance for the practical deployment and optimization of NOMA UAV-RIS systems in future wireless communication networks. This work assumes static, optimal UAV deployments, which may affect the systems’ adaptability. Future research will explore robust optimization in dynamic settings and architectures to boost scalability and resilience.