Exploiting Cascaded Channel Signature for PHY-Layer Authentication in RIS-Enabled UAV Communication Systems

Abstract

Reconfigurable Intelligent Surface (RIS)-assisted Unmanned Aerial Vehicle (UAV) communications face a critical security threat from impersonation attacks, where adversaries impersonate legitimate entities to infiltrate networks to obtain private data or unauthorized access. To combat such security threats, this paper proposes a novel physical layer (PHY-layer) authentication scheme for validating UAV identity in RIS-enabled UAV wireless networks. Considering that most existing works focus on traditional communication systems such as IoT and millimeter wave multiple-input multiple-output (MIMO) systems, there is currently no mature PHY-layer authentication scheme to serve RIS-UAV communication systems. To this end, our scheme leverages the unique characteristics of cascaded channels related to RIS to verify the legitimacy of UAV transmitting signals to the base station (BS). To be more precise, we first use the least squares estimate method and coordinate a descent-based algorithm to extract the cascaded channel feature. Next, we explore a quantizer to quantize the fluctuations of the channel gain that are related to the extracted channel feature. The 1-bit quantizer’s output findings are exploited to generate the authentication decision criteria, which are then tested using a binary hypothesis. The statistical signal processing technique is utilized to obtain the analytical formulations for detection and false alarm probabilities. We also conduct a computational complexity analysis of the proposed scheme. Finally, the numerical results validate the effectiveness of the proposed performance metric models and show that our detection performance can reach over 90% accuracy at a low signal-to-noise ratio (e.g., −8 dB), with a 10% improvement in detection accuracy compared with existing schemes.

1. Introduction

1.1. Background

The introduction of reconfigurable intelligent surfaces (RISs) is highly important for enhancing the performance of wireless communication systems, which provides significant benefits, such as extensive coverage, boosting capacity, and spectral efficiency enhancement [1,2]. Moreover, RIS is compatible with one or more of the most common standards and hardware architectures in existing wireless networks [3] and is easily deployed into wireless communication environments, making it extremely important for sixth generation (6G) wireless communications in the future [4,5]. RIS, serving as a promising technology, can be incorporated into the infrastructure of wireless communications to enhance connectivity for various applications, including smart grids, autonomous vehicles, the Internet of Things (IoT), etc. [6]. The emergence of UAV networks acting as aerial wireless relays for transferring information is becoming a feasible and cost-effective solution to enhance energy efficiency and provide additional capacity [7]. On the one hand, owing to their flexibility, maneuverability, and low manufacturing cost, UAVs provide traditional wireless communication with broader communication coverage and potentially overcome propagation constraints due to terrain characteristics, especially applicable in rural or disaster areas lacking adequate ground infrastructure [8]. On the other hand, UAVs are usually deployed at high altitudes. This results in a more dominant line-of-sight (LoS) channel with the ground nodes, which can be exploited to provide more reliable communication links compared with terrestrial wireless channels that generally suffer from severe path loss [9,10]. Hence, based on the advantages of the UAVs mentioned above, the integration of RIS with unmanned aerial vehicle (UAV) communication systems presents a promising avenue for enhancing the performance and capabilities of systems. We also note that the strategic placement of RIS in RIS-UAV systems is pivotal for optimizing performance by providing focused signal enhancement, which is crucial for maintaining a stable and high-quality link in the dynamic aerial environment [11,12]. By intelligently reflecting signals towards the UAV, RIS can mitigate signal attenuation and blockages caused by the terrain or other obstacles, ensuring a more consistent and reliable communication link even as the UAV navigates through varied landscapes. Consequently, the RIS-enabled UAV communication system leads to an extended coverage area and less movement of UAVs and improves communication performance in wireless networks. However, incorporating RIS into UAV communication systems also introduces new security considerations, such as impersonation attacks.

Ensuring authentication is crucial for validating the legitimacy of the source of transmitted signals in secure RIS-enabled UAV wireless networks [13,14]. For RIS-enabled wireless networks, the provision of a trustworthy and adaptable transmitter identity is becoming a pressing need. First, because of the open nature of the transmission media, RIS-enabled UAV communication systems are very vulnerable to impersonation attacks. Second, numerous heterogeneous wireless devices join in or leave wireless networks randomly, leading to high demands on the security and confidentiality of RIS-enabled communication systems. Therefore, it is significantly crucial to design a robust and effective authentication scheme to guarantee the secure operation of the considered RIS-UAV networks.

1.2. Related Work

The implementation of physical layer (PHY-layer) authentication offers a cost-effective approach to bolster and supplement traditional higher-layer authentication, and has garnered significant interest. The main methods of PHY-layer authentication are broadly divided into two categories: (1) radio frequency (RF) fingerprinting-based authentication and (2) channel fingerprinting-based authentication. RF fingerprinting-based approaches authenticate the legitimacy of the device identity by exploiting hardware features hidden in transmitted signals. In [15], the authors introduce a novel Radio Frequency Fingerprinting Identification (RFFI) protocol that leverages the unique and intrinsic hardware impairments of devices, such as oscillator imperfections, in-phase and quadrature (IQ) imbalances, and power amplifier (PA) nonlinearities, for device authentication in the Internet of Things. Utilizing signal-to-noise ratio (SNR) trace features, an effective PHY-layer authentication method is designed to detect impersonation attacks in millimeter wave (mmWave) communication systems [16]. In [17], it is proved that polarization fingerprints (PF) offer high identification accuracy and robustness against typical attacks by leveraging the unique spatial and frequency characteristics of wireless signals in the polarization domain. Extensive experiment results are provided to show the overall identification and authentication accuracy approach at $99\%$. In [18], a new beam fingerprinting referred to as SNR trace is exploited to resist the spoofing attack, and verification accuracy can reach 99% even for 200 training samples. It is important to note that RF fingerprinting-based authentication carries a greater computational complexity and always necessitates the employment of high-precision hardware equipment for hardware feature extraction.

Channel fingerprinting-based authentication primarily leverages the spatial variability of uncorrelated wireless channels across distinct geographical areas [19]. The authors in [20] exploit multi-path delay characteristics extracted from channel impulse response to develop an authentication scheme based on a two-dimensional quantization algorithm. In [21], the researchers examine the channel gain specific to the location and the phase noise specific to the transmitter, aiming to establish a framework for leveraging various PHY-layer attributes for authentication purposes. Following this line, the authors in [22] explore the use of multi-channel features to design a PHY-layer authentication scheme and also examine the performance loss caused by quantization errors caused by quantizers. A ResNet-based deep learning approach is introduced in [23] for physical layer authentication in the industrial Internet of Things (IIoT), demonstrating superior accuracy and robust performance through channel state information feature extraction and advanced training strategies, culminating in a 99.64% authentication accuracy that outperforms existing methods by 32.68%. In [24], the authors attempt to utilize a sparse mmWave channel for authenticating devices. Utilizing channel polarization response, a robust authentication technique for physical layer security is introduced in [25] to ensure high authentication accuracy across diverse scattering conditions and in low SNR environments through the distinctive polarization signatures of hardware devices. The existing methods above show that the uniqueness reflected by the channel enables it to be used as a device identity fingerprint to resist impersonation attacks. For a communication system, a channel is generally required to accurately estimate for ensuring the quality of service of message transmission in advance. Channel-based authentication can almost be conducted in an arbitrary communication system without additional feature extraction operations, and thus keeps lower signal processing cost and overhead. We also observe that schemes based on hardware impairments might not be able to authenticate transmitters because hardware feature extraction is difficult under strict error tolerance limitations and the extensive usage of passive components using an identical reflecting architecture. Therefore, we focus on the channel fingerprint-based authentication in this paper.

Table 1. Summary of the related works.

References	Authentication Parameters	Application Scenario
[15]	CFO, IQ imbalance, and PA nonlinearities	IoT systems
[16]	SNR trace feature	mmWave systems
[17]	polarization fingerprint	LoRaWAN
[19]	channel frequency response	MIMO systems
[20]	channel impulse response	mobile MIMO scenario
[21]	channel gain and phase noise	MIMO systems
[26]	channel sparsity	mmWave systems
Our scheme	cascaded channel gain	RIS-UAV communication systems

summarizes the related works and their adopted feature parameters and the application scenario.

Efficient channel feature extraction methods are vital for the channel-based authentication scheme. In recent years, the use of tensor decomposition technology for high-mobility channel estimation has attracted extensive attention. In [27], Wu et al. propose a low-complexity channel estimation algorithm based on a tensor, which is specially used for millimeter-wave large-scale MIMO-OTFS systems. This method takes advantage of the sparsity of the channel in the delay-Doppler-angle domain, and transforms the channel estimation problem into a sparse signal recovery problem by designing a new pilot structure. Zhang et al. propose a unified tensor method in [28] for channel and target parameter estimation in large-scale MIMO-ISAC systems. By parameterizing the high-dimensional communication channel, this method associates the channel state information with the angle, delay, and Doppler parameters of the target, and proposes a shared training mode, which makes the channel estimation and target parameter estimation unified into a tensor decomposition problem, and significantly improves the estimation accuracy and training overhead efficiency. Despite their excellent estimation performance, these methods mainly focus on the mmWave MIMO channel rather than the RIS-UAV scenario. Hence, it is imperative to design a feasible cascaded channel estimation technique for the RIS-UAV system.

1.3. Motivation and Contribution

Although current channel fingerprint-based authentication solutions mark a significant advancement in the design of PHY-layer authentication approaches, they typically detect the correction between the current channel associated with the transmitter and the previous channel validated as being associated with the legitimate transmitter in order to determine the legitimacy of an identity. These methods cannot be directly applied to wireless communication systems with RIS assistance because of the complexity and dynamic nature of the cascaded channels. In addition, the partial visibility of RIS’s effect on the data implies that the overall structure of the channel becomes indeterminate when considering RIS. It becomes exceedingly challenging to identify the specific components of the channel that are influenced by RIS. Moreover, the presence of a substantial number of passive RIS elements may complicate the acquisition of channel state information.

Some works have been devoted to the estimation of cascaded (also named composite) channels for RIS-enabled wireless communication systems. The authors in [29] investigate a passive frequency-guided channel estimation method, which can effectively reduce the pilot overhead. In [30], a minimum mean squared error-discrete Fourier transform-based channel estimation protocol is designed to separately estimate two-stage cascaded channels. A channel estimation framework referred to as two-stage RIS-enabled channel estimation is provided for RIS mmWave communication systems, which has a comparatively low training overhead and computational complexity [31]. In view of the severe threat of eavesdropping and/or impersonation attacks on RIS-enabled wireless communication systems, many methods have been proposed for secret key generation and secure transmission based on a PHY-layer security technique. The authors in [32] examine secret key generation using RIS-enabled wireless communication networks and the minimum achievable secret key capacity using a RIS beamformer in the presence of multiple eavesdroppers and solve an optimization problem on reflecting coefficients with the secret key capacity’s lower bound. Further, they also exhibit an encrypted data transmission scheme utilizing RIS to generate secret keys and provide the method for maximizing the secure transmission rate [33].

To the best of our knowledge, the issue of verifying transmitter identity in RIS-enabled UAV networks needs to be explored to ensure security. Based on the above research works, this paper attempts to develop an effective PHY-layer authentication method, considering the complexity brought by the cascade channels for RIS-enabled UAV wireless communication systems. Our proposed approach differs from previous channel-based approaches in that it uses RIS-related cascaded/composite channel fingerprinting to jointly authenticate transmitter identity and thwart impersonation attacks. Previous approaches conducted authentication based on a single-channel coefficient similarity. Additionally, the analytic expressions of statistical performances are provided for the RIS-related cascaded/composite channel-based authentication scheme in RIS-enabled UAV communication systems.

The main contributions are as follows:

• By leveraging the location-specific cascaded channel features’ user equipment (UE) RIS and RIS base station (BS), we propose a novel physical layer authentication scheme for RIS-enabled UAV wireless communication systems.
• Using the first-order Gauss–Markov process, we model the time-varying complicated cascaded channels involved in RIS. With the help of the coordinated descent-based estimation algorithm and least squares estimation algorithm, we fulfill the estimation of time-varying complicated cascaded channels.
• Based on a 1-bit quantizer, we exploit the extracted channel features to establish authentication verification under the framework of the hypothesis testing framework. The performance of our proposed authentication scheme is analytically evaluated by deriving the typical performance metrics, including false alarm and detection probabilities, and establishing the statistical performance analytically.
• Through extensive simulations, we verify the correctness of the theoretical models of the two probabilities. Simulation results further show how the system parameters can affect the statistical authentication performance.

The remainder of this paper is structured as follows: Section 2 provides a detailed introduction to the system model and problem formulation. The proposed PHY-layer authentication scheme is outlined in Section 3. In Section 4, the modeling of false alarm and detection probabilities is presented. Section 5 contains the numerical results. Lastly, Section 6 wraps up this paper. Important symbols are summarized in

Table 2. Summary of important symbols.

Symbols	Description
${(·)}^{*}$/${(·)}^H$/${(·)}^T$	Conjugate/conjugate transpose/transpose operators.
⊙	Hadamard product.
$|·|$	Absolute value operator.
Pr(·)	Probability operator.
≜	Definition.
$\mathrm{diag}\left(\mathbf{a}\right)$	A diagonal matrix with vector $\mathbf{a}$.
$\mathrm{vec}\left(\mathbf{G}\right)$	The vectorization of the matrix $\mathbf{G}$.
$\mathbf{f}\left(k\right)$	The RIS-UAV channel with $f_i\left(k\right)$ being the ith channel component.
$\mathbf{G}\left(k\right)$	The channel for BS-RIS.
$\varphi$	The coefficient of the reflecting elements on RIS.
$f_d$	The normalized maximum Doppler frequency.
v	The mobile speed of UAV.
$f_c$/c/$T_s$	Carrier frequency/speed of light/sampling interval.
$J_0(·)$	Zero-order Bessel function of the first kind.
$\alpha$	Correlation coefficient for the UAV(Alice/Eve)-RIS channel.
$P_{BS}$	Signal power.
${\tilde{\varphi }}_t$	The reflection coefficient vector of RIS in the t-th subframe.
A/E	Alice/Eve.
$\gamma$	A predefined threshold.
$O_{ij}$	The output of the quantizer of the subcascaded channel.
$H_{0/1}$	The current signal originates from Alice/Eve.
Z	The judgment threshold in the hypothesis testing.
${\wedge }_{H_q}$	The square of the difference between the current cascaded channel at time k and the previous one at time $k-1$.
$\exp (·)$	Exponential function.
$P_f{/}_d$	The false alarm probabilities/the detection probabilities.

.

2. System Model and Problem Formulation

In Figure 1, a RIS-enabled UAV wireless communication system is depicted, featuring a base station (Bob) with M antennas, a legitimate single-antenna UAV user equipment (UE, named Alice), a RIS with N reflecting elements, and a single-antenna attacker (Eve). The BS is assumed to operate in full-duplex mode, and uses the BS-RIS channel estimation method described in [34]. In this setup, UAV Alice transmits signals to the intended recipient, Bob, who needs to differentiate between legitimate and illegitimate data frames at time k. Meanwhile, Eve seeks to transmit maliciously modified data frames or inject aggressive signals by posing as Alice. Since the focus of this paper is on implementing the UAV identity authentication for RIS-UAV systems, the optimal placement determination of RIS can refer to [13,14] for more details.

In line with the assumption in previous works [35,36], our primary focus is on exploring the potential of RIS technology for authentication in scenarios where a direct link between the BS and UE is obstructed or unreliable. This could be due to factors like dense urban environments, indoor deployments, or non-line-of-sight communication. By excluding direct links, we can more accurately evaluate the impact of RIS deployment on system performance without interference from traditional direct-link communication channels.

The secure transmission of confidential information, such as the authentication scheme used by Alice and Bob, presents significant challenges for unauthorized parties like Eve. Nevertheless, through the analysis of transmitted signals from Alice, Eve can gain access to certain commonly repurposed information, such as pilot symbols and training sequences, owing to the inherent broadcast nature of the communication medium.

This study suggests a straightforward and adaptable physical layer authentication technique to enable Bob, the receiver, to easily determine the current signal’s source. For the two consecutive data frames, we assume that the first, received by Bob at time $k-1$, was validated by Alice by exploiting a cryptographic authentication scheme at the upper application layer and Bob measured and stored the cascaded channel parameters as authentication fingerprinting for subsequent certification. When receiving the second data frame at time k, the aim of Bob is to determine whether the current data frame is still from Alice based on the stored cascaded channel fingerprinting. The detailed authentication process is also illustrated in Figure 2.

The messages are exchanged between the communicating devices via the channels for BS-RIS and RIS-UAV (Alice/Eve). Similar to the work in [37], if we use RIS to reflect a signal in the direction $\theta$, then the path loss $PL$ at the distance r is expressed by

$$\begin{array}{c}\hfill PL={\displaystyle \frac{G_t{G}_r}{{\left(4\pi \right)}^2}}{\left({\displaystyle \frac{s}{dr}}\right)}^2{cos}^2\left(\theta \right),\end{array}$$

(1)

where $G_t$ and $G_r$ denote the antenna gain at the transceiver end. s denotes the size of RIS, and d is the distance between the signal source and RIS. Let $\mathbf{f}\left(k\right)={[f_1\left(k\right),\dots ,f_N\left(k\right)]}^T\in {\mathbb{C}}^{N\times 1}$ denote the RIS-UAV channel with $f_i\left(k\right)$ being the ith channel component, and all channel components are independent and identically distributed complex Gaussian random variables with zero mean and variance ${\sigma }_f^2$. Then, we have $\mathbf{f}\left(k\right)\sim \mathcal{CN}\left(0,{\sigma }_f^2{\mathbf{I}}_N\right)$. Denote $\mathbf{G}\left(k\right)\in {\mathbb{C}}^{M\times N}$ as the channel for BS-RIS and the channel component is i.i.d. Then the vectorization of the matrix $\mathbf{G}\left(k\right)$ is written as $\mathrm{vec}\left(\mathbf{G}\left(k\right)\right)\sim \mathcal{CN}\left(0,{\sigma }_G^2{\mathbf{I}}_{MN}\right)$. Let $\varphi ={[{\varphi }_1,\dots ,{\varphi }_N]}^T\in {\mathbb{C}}^{N\times 1}$ be the coefficient of the reflecting elements on RIS. Moreover, the hardware limitations of RIS make the phases at RIS elements controlled by quantization, i.e., ${\varphi }_i\in \left\{-1,+1\right\}$, for $i=1,2,\dots ,N$. The reflection coefficient is known to us. The received signal $\mathbf{y}\left(k\right)\in {\mathbb{C}}^{M\times 1}$ at BS is written as

$$\mathbf{y}\left(k\right)=\mathbf{G}\left(k\right)\left(\varphi \odot \mathbf{f}\left(k\right)\right)·x\left(k\right)+\mathbf{n}\left(k\right),$$

(2)

where $x\left(k\right)$ is the transmitted symbol from UAV (e.g., Alice) and $\mathbf{n}\left(k\right)\in {\mathbb{C}}^{M\times 1}$ is the additive white noise, i.e., $\mathbf{n}\left(k\right)\sim \mathcal{CN}\left(0,{\sigma }_n^2{\mathbf{I}}_M\right)$. The equivalent cascaded channel (i.e, BS-RIS-UAV channel) is given by the following equation:

$$\mathbf{H}\left(k\right)\triangleq \mathbf{G}\left(k\right)\mathrm{diag}\left(\mathbf{f}\left(k\right)\right).$$

(3)

From (3), one can see that the cascaded channel matrix $\mathbf{H}\left(k\right)\in {\mathbb{C}}^{M\times N}$ is a complex Gaussian random variable conditioned on a given $\mathbf{G}\left(k\right)$.

Due to the mobility of the user, the channels are spatially uncorrelated and changing continuously in time (i.e., time-varying). Temporal channel variations are caused by the Doppler rate. Let $f_d$ be the normalized maximum Doppler frequency. Then, we have a relationship between $f_d$ and the mobile speed of UAV v as $f_d={\displaystyle \frac{v{f}_c}{c}}{T}_s$. $f_c$, c, and $T_s$ denote carrier frequency, speed of light, and sampling interval, respectively. Using the Jakes model [38], the temporal variation of $\mathbf{f}\left(k\right)$ for an arbitrary time $k_s$ is given by

$$\begin{array}{cc}\hfill \mathbf{\Psi }\left(k_s\right)& =\mathbb{E}\left\{\mathbf{f}\left(k\right){\mathbf{f}}^{*}(k+k_s)\right\}\hfill \\ \hfill & =J_0\left(2\pi f_d{k}_s\right){\sigma }_f^2{\mathbf{I}}_N,\hfill \end{array}$$

(4)

where $J_0(·)$ is a zero-order Bessel function of the first kind. A first-order Gauss–Markov process is used to describe the fluctuation of channels [21]. Applying [21], the correlation coefficient of $\mathbf{f}\left(k\right)$ is computed as ${\displaystyle \frac{1}{{\sigma }_f^2}}\mathbf{\Psi }\left(k_s\right){\mathbf{I}}_N$. Thus, $\mathbf{f}\left(k\right)$ is expressed by

$$\begin{array}{cc}\hfill \mathbf{f}\left(k\right)& =\alpha \mathbf{f}(k-1)+\sqrt{1-{\alpha }^2}\mathbf{v}\left(k\right),\hfill \end{array}$$

(5)

where $\alpha$ denotes the correlation coefficient for the UAV (Alice /Eve)-RIS channel and $\mathbf{v}\left(k\right)\in {\mathbb{C}}^{N\times 1}\sim \mathcal{CN}\left(\mathbf{0},{\sigma }_f^2{\mathbf{I}}_N\right)$ is independent of $\mathbf{f}\left(k\right)$.

It is worth noting that when both RIS and BS are geographically deployed in fixed locations and scatterers and reflectors in the environments do not quickly change, the channel for BS-RIS can be regarded as a constant during the two successive times $k-1$ and k as follows:

$$\mathbf{G}\left(k-1\right)=\mathbf{G}\left(k\right)=\mathbf{G}.$$

(6)

By using the Gauss–Markov model, the time-varying cascaded channel $\mathbf{H}\left(k\right)$ relevant to UAV (Alice /Eve) is given by

$$\begin{array}{cc}\hfill \mathbf{H}\left(k\right)& =\alpha \mathbf{H}(k-1)+\sqrt{1-{\alpha }^2}\mathbf{G}\mathrm{diag}\left(\mathbf{v}(k-1)\right)\hfill \\ \hfill & =\alpha \mathbf{G}\mathrm{diag}\left(\mathbf{f}(k-1)\right)+\sqrt{1-{\alpha }^2}\mathbf{G}\mathrm{diag}\left(\mathbf{v}\left(k\right)\right).\hfill \end{array}$$

(7)

Since there is always inevitable noise and/or interference, the estimated value of the cascaded channel is expressed as the sum of the true value of the channel and the estimation error, as follows:

$$\widehat{\mathbf{H}}\left(k\right)=\mathbf{H}\left(k\right)+\mathbf{W}\left(k\right),$$

(8)

where $\mathrm{vec}\left(\mathbf{W}\left(k\right)\right)\in {\mathbb{C}}^{MN\times 1}\sim \mathcal{CN}\left(0,{\sigma }_w^2{\mathbf{I}}_{MN}\right).$

3. Proposed PHY-Layer Authentication Scheme

3.1. Estimation of the Cascaded Channel

Because of the fixed positions of RIS and BS, the BS-RIS channel $\mathbf{G}$ is generally regarded as quasi-static, meaning that its value does not change over time. As such, it can be estimated once on a large timescale, and the estimated value is expressed as the channel value during this time. However, this research takes the stance that since UAV UEs are movable, the channel RIS-UAV varies over time. It is essential to estimate the channel on a smaller timescale, that is, in a brief time interval, in order to produce a more accurate channel estimation of the channel RIS-UAV.

Since RIS is assumed to have only passive reflecting elements in this paper, it can neither transmit nor receive pilots, making it difficult to estimate the channel with conventional methods. Fortunately, a dual-link pilot transmission method can overcome this difficult problem. Based on [34], the BS works at full-duplex mode to facilitate the estimation of the BS-RIS channel. Since there may be severe self-interference in the full-duplex system, we use self-interference mitigation techniques [39] to reduce the impact of self-interference. Specific implementation details can be found in [34].

To estimate the BS-RIS channel, the dual-link pilot transmission approach [34] is employed in this study. Next, a least squares estimation technique using the traditional pilot transmission strategy is developed to achieve the estimate of RIS-UAV because of the low dimensionality of the RIS-UAV channel. The cascaded channel estimation framework is illustrated in Figure 3.

In an attempt to simplify the computation, here, we consider one single-antenna UAV UE, and thus, Equation (2) can be equivalently expressed in a compact form (the time index is omitted here for convenient notation) as

$$\begin{array}{cc}\hfill \mathbf{y}& =\mathbf{G}(\varphi \odot \mathbf{f})·x+\mathbf{n}\hfill \\ \hfill & =\mathbf{H}\varphi ·x+\mathbf{n}.\hfill \end{array}$$

(9)

Equation (3) can be rewritten (the time index is omitted here for convenient notation) as

$$\begin{array}{cc}\hfill \mathbf{H}& =\left(\mathbf{G}\left[\begin{array}{ccc}{p}_1& & \\ & \ddots & \\ & & p_N\end{array}\right]\right)\hfill \\ \hfill & \times \left(\left[\begin{array}{ccc}{p}_1^{-1}& & \\ & \ddots & \\ & & p_N^{-1}\end{array}\right]\mathrm{diag}\left(\mathbf{f}\right)\right)\hfill \\ \hfill & ={\mathbf{G}}^{\prime }\mathrm{diag}\left({\mathbf{f}}^{\prime }\right),\hfill \end{array}$$

(10)

where $p_i\in \mathbb{C}$ is non-zero, ${\mathbf{G}}^{\prime }=\mathbf{G}\mathrm{diag}\left(\mathbf{p}\right)\in {\mathbb{C}}^{M\times N},\mathbf{p}\triangleq {\left[p_1,p_2,\cdots ,p_N\right]}^T\in {\mathbb{C}}^{N\times 1}$, and ${\mathbf{f}}^{\prime }=\mathbf{f}\odot \mathbf{p}\in {\mathbb{C}}^{N\times 1}$.

In the transmission scheme, a pilot frame consisting of $\left(N+1\right)$ subframes is considered. A subframe includes L time slots. BS transmits a non-zero pilot signal in the t-th subframe $\left(t=1,2,\cdots ,N+1\right)$$s_{m_1,t}$ with the $m_1$-th antenna (the rest of the BS antennas transmit zeros) through the downlink to RIS, which then returns the pilot signal to BS with the corresponding reflection coefficient vector ${\overline{\varphi }}_t\in {\mathbb{C}}^{N\times 1}$. The signal at the $m_2$-th antenna for BS is then expressed by

$$\begin{array}{cc}\hfill y_{m_1,m_2,t}^d& =\left[{\mathbf{g}}_{m_2}^T\mathrm{diag}\left({\overline{\varphi }}_t\right){\mathbf{g}}_{m_1}+e_{m_1,m_2}\right]s_{m_1,t}\hfill \\ \hfill & +{\overline{q}}_{m_1,m_2,t}+{\overline{n}}_{m_1,m_2,t}\hfill \\ \hfill & =\left[{\left({\mathbf{g}}_{m_1}\odot {\mathbf{g}}_{m_2}\right)}^T{\overline{\varphi }}_t+e_{m_1,m_2}\right]s_{m_1,t}\hfill \\ \hfill & +{\overline{q}}_{m_1,m_2,t}+{\overline{n}}_{m_1,m_2,t},\hfill \\ \hfill & m_2=1,2,\cdots ,M,m_2\ne m_1,\hfill \end{array}$$

(11)

where $m_1$ and $m_2$ denote the $m_1$-th antenna (i.e., transmitting pilot signal) and the $m_2$-th BS antenna (i.e., receiving pilot signal), respectively. ${\mathbf{g}}_{m_2}^T\triangleq \mathbf{G}\left(m_2,:\right)\in {\mathbb{C}}^{1\times N}$, ${\mathbf{g}}_{m_1}\triangleq \mathbf{G}{\left(m_1,:\right)}^T\in {\mathbb{C}}^{N\times 1}$, and ${\overline{q}}_{m_1,m_2,t}$ denotes the interference. Let $e_{m_1,m_2}$ and ${\overline{n}}_{m_1,m_2,t}$ denote the propagation noise and the receiving noise at the $m_2$-th BS antenna, respectively. We consider ${\overline{n}}_{m_1,m_2,t}\sim \mathcal{CN}\left(0,{\sigma }_n^2\right)$.

After receiving the signals from the $m_1$-th transmitting antenna and the $m_2$-th receiving antenna across all $\left(N+1\right)$ subframes, we write (11) in vector form given by

$$\begin{array}{cc}\hfill {\left({{\mathbf{y}}^d}_{m_1,m_2}\right)}^T& =\left\{{\left({\mathbf{g}}_{m_1}\odot {\mathbf{g}}_{m_2}\right)}^T\left[{\overline{\varphi }}_1,{\overline{\varphi }}_2,\cdots ,{\overline{\varphi }}_{N+1}\right]\right.\hfill \\ \hfill & \left.+e_{m_1,m_2}{\mathbf{1}}_{1\times (N+1)}\right\}\hfill \\ \hfill & \times \sqrt{P_{BS}}+{\overline{\mathbf{q}}}_{m_1,m_2}^T+{\overline{\mathbf{n}}}_{m_1,m_2}^T\hfill \\ \hfill & =\sqrt{P_{BS}}{\mathbf{u}}_{m_1,m_2}^T\left[\begin{array}{c}{\mathbf{1}}_{1\times (N+1)}\\ \overline{\varphi }\end{array}\right]\hfill \\ \hfill & +{\overline{\mathbf{q}}}_{m_1,m_2}^T+{\overline{\mathbf{n}}}_{m_1,m_2}^T,\hfill \end{array}$$

(12)

where $\overline{\varphi }=[{\overline{\varphi }}_1,{\overline{\varphi }}_2,\cdots ,{\overline{\varphi }}_{N+1}]\in {\mathbb{C}}^{N\times (N+1)}$, ${\mathbf{u}}_{m_1,m_2}\triangleq {[e_{m_1,m_2},{({\mathbf{g}}_{m_1}\odot {\mathbf{g}}_{m_2})}^T]}^T\in {\mathbb{C}}^{(1+N)\times 1}$, and we assume that $\sqrt{P_{BS}}$ represents the pilot signal without loss of generality (i.e., $s_{m_1,t}=\sqrt{P_{BS}}$). $P_{BS}$ is the signal power. ${\overline{\mathbf{q}}}_{m_1,m_2}={[q_{m_1,m_2,1},q_{m_1,m_2,2},\cdots ,q_{m_1,m_2,N+1}]}^T\in {\mathbb{C}}^{(1+N)\times 1}$,${\overline{\mathbf{n}}}_{m_1,m_2}={[n_{m_1,m_2,1},n_{m_1,m_2,2},\cdots ,n_{m_1,m_2,N+1}]}^T\in {\mathbb{C}}^{(1+N)\times 1}$. From (12), the estimation of ${\mathbf{u}}_{m_1,m_2}^T$ denoted by ${\widehat{\mathbf{u}}}_{m_1,m_2}^T$ can be easily achieved by

$$\begin{array}{cc}\hfill {\widehat{\mathbf{u}}}_{m_1,m_2}^T& \triangleq \left[{\widehat{e}}_{m_1,m_2}{u}_{m_1,m_2,1}{u}_{m_1,m_2,2}\cdots u_{m_1,m_2,N}\right]\hfill \\ \hfill & ={\displaystyle \frac{1}{\sqrt{P_{BS}}}}{\left({{\mathbf{y}}^d}_{m_1,m_2}\right)}^T{\mathbf{F}}^{-1},\hfill \end{array}$$

(13)

where $u_{m_1,m_2,n}$ is the estimation of $g_{m_1,n}{g}_{m_2,n}$ and $\mathbf{F}\triangleq \left[\begin{array}{c}{\mathbf{1}}_{1\times (N+1)}\\ \overline{\varphi }\end{array}\right]\in {\mathbb{C}}^{(1+N)\times (1+N)}$. After combining (12) and (13), we obtain

$$\begin{array}{cc}\hfill {\widehat{\mathbf{u}}}_{m_1,m_2}^T& =\left[e_{m_1,m_2}{\left({\mathbf{g}}_{m_1}\odot {\mathbf{g}}_{m_2}\right)}^T\right]\hfill \\ \hfill & +{\displaystyle \frac{\left({\overline{\mathbf{q}}}_{m_1,m_2+}^T{\overline{\mathbf{n}}}_{m_1,m_2}^T\right){\mathbf{F}}^{-1}}{\sqrt{P_{BS}}}}.\hfill \end{array}$$

(14)

The independence of $a_{m_1,m_2,n}$ from the channel coefficients of other RIS elements allows us to partition the task of estimating the BS-RIS channel into N distinct subproblems. Each subproblem focuses on estimating the channel coefficients associated with a specific RIS element, indexed by n.

The n-th subproblem can be defined by Equation (15):

$${\widehat{g}}_{1,n},{\widehat{g}}_{2,n},\cdots ,{\widehat{g}}_{M,n}=\arg \underset{g_{1,n},g_{2,n},\cdots ,g_{M,n}}{min}Jn\left(g_{1,n},g_{2,n},\cdots ,g_{M,n}\right),$$

(15)

where

$$Jn\left(g_{1,n},g_{2,n},\cdots ,g_{M,n}\right)\triangleq \sum _{\left(m_1,m_2\right)\in S}{\left|a_{m_1,m_2,n}-g_{m_1,n}{g}_{m_2,n}\right|}^2.$$

(16)

$${\widehat{g}}_{m,n}^{\left(i\right)}={\displaystyle \frac{{\sum }_{\left({m,m}^{\prime }\right)\in S}{a}_{{m,m}^{\prime },n}{\left(g_{m^{\prime },n}^{\left(i,m\right)}\right)}^{*}+{\sum }_{\left(m^{\prime },m\right)\in S}{a}_{m^{\prime },m,n}{\left(g_{m^{\prime },n}^{\left(i,m\right)}\right)}^{*}}{{\sum }_{\left(m,m^{\prime }\right)\in S}{\left|g_{m^{\prime },n}^{\left(i,m\right)}\right|}^2+{\sum }_{\left(m^{\prime },m\right)\in S}{\left|g_{m^{\prime },n}^{\left(i,m\right)}\right|}^2}},$$

(17)

where

$$g_{m^{\prime },n}^{\left(i.m\right)}\triangleq \left\{\begin{array}{c}{\widehat{g}}_{m^{\prime },n}^{\left(i\right)},\hfill \\ {\widehat{g}}_{m^{\prime },n}^{\left(i-1\right)},\hfill \end{array}\right.\begin{array}{c}{m}^{\prime }<m\\ m^{\prime }>m\end{array}.$$

(18)

Define $S\triangleq \{\left(m_1,m_2\right)\left|1⩽m_1⩽\right.L,$$1⩽m_2⩽M,m_1\ne m_2\}$. In this step, we first calculate the initial channel estimates ${\widehat{g}}_{1,n}^{\left(0\right)}$,${\widehat{g}}_{2,n}^{\left(0\right)}$,⋯,${\widehat{g}}_{M,n}^{\left(0\right)}$ to help the following iterative refinement converge faster, which are given by

$${\widehat{g}}_{m_1,n}^{\left(0\right)}\leftarrow \sqrt{{\displaystyle \frac{a_{m_1,m_2,n}{a}_{m_1,m_3,n}}{a_{m_2,m_3,n}}}},$$

(19)

$${\widehat{g}}_{m^{\prime },n}^{\left(0\right)}\leftarrow {\displaystyle \frac{a_{m_1,m^{\prime },n}}{{\widehat{g}}_{m_1,n}^{\left(0\right)}}},1⩽m^{\prime }⩽M,m^{\prime }\ne m_1.$$

(20)

Then, we formulate the problem of estimation of BS-RIS channels as (15), shown at the top of the next page. The details can be further seen in [34]. Finally, we derive the expression for ${\widehat{g}}_{m,n}^{\left(i\right)}$ by making ${\displaystyle \frac{\partial Jn}{\partial g_{m,n}}}={\displaystyle \frac{\partial Jn}{\partial g_{m,n}^{*}}}=0$, and ${\widehat{g}}_{m,n}^{\left(i\right)}$ is given by (17).

A least squares method is proposed to address the problem of estimating a RIS-UAV channel. Similar to the work in [40], the transmission frame of the uplink pilot contains R subframes. The uplink’s received signal corresponding to the t-th subframe is written as

$$\begin{array}{cc}{\mathbf{y}}_t^u\hfill & =\left[\mathbf{G}\mathrm{diag}\left({\tilde{\varphi }}_t\right)\mathbf{f}\right]+{\tilde{\mathbf{n}}}_t\hfill \\ \hfill & ={\mathbf{A}}_t\mathbf{f}+{\tilde{\mathbf{n}}}_t,t=1,2,\cdots ,R,\hfill \end{array}$$

(21)

where ${\tilde{\varphi }}_t\in {\mathbb{C}}^{N\times 1}$ denotes the reflection coefficient vector of RIS in the t-th subframe, ${\tilde{\mathbf{n}}}_t\in {\mathbb{C}}^{M\times 1}$ is the noise, and we have

$${\mathbf{A}}_t\triangleq \begin{array}{c}\mathbf{G}\mathrm{diag}\left({\tilde{\varphi }}_t\right)\end{array}.$$

(22)

Due to the small-scale fading resulting from possible environment variations around the BS-RIS channel, we attempt to replace $\mathbf{G}$ with a $\widehat{\mathbf{G}}$ ($\widehat{\mathbf{G}}(i,j)={\widehat{g}}_{ij}$, $1⩽i⩽M$, $1⩽j⩽N$.) estimated BS-RIS channel to achieve a more practical authentication performance. As a result, the estimated model of a cascaded channel given in (8) (using a time index) is expressed by

$$\widehat{\mathbf{H}}\left(k\right)=\widehat{\mathbf{G}}\left(k\right)\mathrm{diag}\left(\mathbf{f}\left(k\right)\right)+\mathbf{W}\left(k\right).$$

(23)

Without loss of generality, the estimated value of the matrix $\mathbf{A}\triangleq {\left[{{\mathbf{A}}_1}^T,{{\mathbf{A}}_2}^T,\dots ,{\mathbf{A}}_R^T\right]}^T\in {\mathbb{C}}^{MR\times N}$ can be written as

$$\widehat{\mathbf{A}}\left(k\right)\triangleq \left[\begin{array}{c}\widehat{\mathbf{G}}\left(k\right)\mathrm{diag}\left({\tilde{\varphi }}_1\right)\\ ⋮\\ \widehat{\mathbf{G}}\left(k\right)\mathrm{diag}\left({\tilde{\varphi }}_R\right)\end{array}\right].$$

(24)

Finally, combing Equations (21) and (24), the RIS-UAV channel estimation is given by

$$\widehat{\mathbf{f}}\left(k\right)={\left({\widehat{\mathbf{A}}}^H\left(k\right)\widehat{\mathbf{A}}\left(k\right)\right)}^{-1}{\widehat{\mathbf{A}\left(k\right)}}^H{\mathbf{y}}^u\left(k\right),$$

(25)

where ${\mathbf{y}}^u\left(k\right)$ is the equivalent received signal corresponding to R collected subframes (i.e., ${\mathbf{y}}^u\left(k\right)\triangleq {\left[{{\mathbf{y}}_1^u}^T\left(k\right),{{\mathbf{y}}_2^u}^T\left(k\right),\cdots ,{{\mathbf{y}}_R^u}^T\left(k\right)\right]}^T\in {\mathbf{C}}^{MR\times 1}$).

3.2. Channel Quantization

After obtaining the cascaded channel estimates (BS-RIS channel estimation denoted by $\widehat{\mathbf{G}}\left(k\right)$ and RIS-UAV channel estimation denoted by $\widehat{\mathbf{f}}\left(k\right)$), based on the quantizer’s output, we view the method of authentication as a problem of testing binary hypotheses. Particularly, we utilize a cascaded channel gain quantizer introduced in [20]. This quantizer assesses the square of the disparity between the current and the preceding cascaded channel gain, calculated at consecutive time intervals, against a predefined threshold $\gamma$. For simplification, let $X\in \{A,E\}$ be the origin of the current signal with A and E being Alice and Eve, respectively. If the difference is larger than the given threshold, this means that the two channels are not correlated, which indicates that the transmitted signal may be from Eve ($X=E$); then the value of the quantizer is 1. Conversely, it is assumed that the source of the current signal is Alice ($X=A$), and the quantizer output is 0.

Consider that Bob has M antennas and RIS equipped with N antennas receives signals from a single UAV UE with only one antenna, and the total number of channels in the whole communication system is $L=M\times N$. For the quantization of cascaded channels, the result is the sum of the quantization results corresponding to each sub-cascaded channel (i.e., $h_{X,ij}$, $1⩽i⩽M$, $1⩽j⩽N$). Therefore, the quantizer $Q_h$ for an estimated sub-cascaded channel gain ${\widehat{h}}_{X,ij}$ ($X\in \{E,A\}$) at time $k-1$ and k can be given by

$$\begin{array}{cc}\hfill O_{ij}& \triangleq Q_h\left[{\left|{\widehat{h}}_{X,ij}\left(k\right)-{\widehat{h}}_{A,ij}\left(k-1\right)\right|}^2\right]\hfill \\ \hfill & =\left\{\begin{array}{cc}0,{\left|{\widehat{h}}_{X,ij}\left(k\right)-{\widehat{h}}_{A,ij}\left(k-1\right)\right|}^2⩽\gamma ,\hfill & \\ 1,\mathrm{otherwise}.\hfill & \end{array}\right.\hfill \end{array}$$

(26)

where $O_{ij}$ denotes the output of the quantizer of the subcascaded channel and ${\widehat{h}}_{ij}\left(k\right)$ is the $(i,j)$th (for $i,j\in \{1,2,\dots N\}$) channel complement of $\widehat{\mathbf{H}}\left(k\right)$ (i.e., ${\left[\widehat{\mathbf{H}}\left(k\right)\right]}_{ij}={\widehat{h}}_{ij}\left(k\right)$).

Let D be the output sum of channel gain quantizer and D is given by

$$D\triangleq \sum _{i=1}^M\sum _{j=1}^N{O}_{ij}.$$

(27)

Mathematically, D is a random variable being the sum of the output of the 1-bit channel quantizers.

Please note that 1-bit quantizers inevitably cause a slight decrease in authentication performance. In future work, we will theoretically model its quantization error and design strategies to optimize its impact on authentication systems. However, our simulation results also indicate that the proposed authentication scheme can still meet the basic requirements of authentication even under performance loss from quantization error.

3.3. Authentication Decision

Applying the adopted one-dimensional quantization scheme, an authentication decision criterion is formulated as a simple hypothesis testing. Using the quantified results of the cascade channel gain, the identity legitimacy of the transmitter is then determined. The authentication decision criterion is expressed by

$$\begin{array}{ccc}{H}_0:& D⩽Z,& \\ H_1:& D>Z.& \end{array}$$

(28)

where $H_0$ represents the hypothesis that the current signal originates from Alice, while $H_1$ denotes the hypothesis that it originates from Eve. The judgment threshold in the hypothesis testing is denoted as Z. If the quantization result of the cascaded channel gain is less than or equal to the judgment threshold, i.e., the hypothesis $H_0$ is satisfied, then the current signal transmitter can be considered as the legitimate UAV UE Alice, and vice versa, i.e., the hypothesis $H_1$ is satisfied and the current transmitter is considered as Eve. In fact, the authentication decision depends on the relation between the preset threshold and the sum of $MN$ outputs of the 1-bit quantizer $Q_h[·]$, and thus, D can be also considered a detector.

We employ a coordinate descent-based algorithm to iteratively optimize quasi-static BS-RIS channel estimation. This process includes partitioning the problem into N independent subproblems for estimating the channel between the BS and a single RIS element, calculating initial estimates for each subproblem, and iteratively refining the BS-RIS channel estimates using the coordinate descent algorithm. Following this, the LS-based algorithm is used to estimate the channels for the mobile but low-dimensional BS-UAV and RIS-UAV. The channel estimation methods utilized in this research are uncomplicated and economical. Moreover, the coordinate descent-based algorithm proposed here shows swift and consistent convergence. It is crucial to highlight that the estimation capabilities of composite channels based on these two approaches meet the authentication criteria (as illustrated in Section 5).

The primary challenge of the proposed approach is the complexity of estimating the high-dimensional BS-RIS channel due to the passive nature of RIS, which prevents the transmission or reception of pilots. The initial estimation of the BS-RIS channel using a full-duplex technique enables the subsequent estimation of the low-dimensional RIS-UAV and BS-UAV channels using a standard uplink pilot transmission scheme and a least squares-based algorithm. Consequently, the channel estimation for RIS-UAV and BS-UAV mirrors the process used in massive MIMO setups. It is noteworthy that while the total channel gains in massive MIMO setups increase linearly with the number of antennas/elements (N) in the far field, the gains in the RIS setup grow quadratically as $N^2$. The analytical results for performance metrics such as false alarm and detection probabilities will be derived in Section 4, where differences between the massive MIMO setup and the RIS setup will be discussed.

Based on Lemma 1 in [41], the BS-RIS channel cannot be exactly estimated when the considered system does not provide full-duplex technique support at the BS. Hence, investigating the use of unreliable channel estimation for authentication presents an intriguing area for future research, aiming to delve deeper into the effective utilization of channel characteristics to enhance the security of RIS-enabled communication systems. We also notice that the approach proposed in [29] without requiring full-duplex technique support poses a good estimation performance, but the channel estimation approach proposed in [41] with a similar protocol has a better channel estimation performance. Hence, in our future work, we will examine the authentication performance of the channel-based scheme with inaccurate channel estimation based on [29,41].

4. Modeling of Performance Metrics

4.1. Probability of Quantizer Output 1 under Each Hypothesis

Let ${\wedge }_{H_q},q\in \{0,1\}$ denote the square of the difference between the current cascaded channel at time k and the previous one at time $k-1$.

$${\wedge }_{H_q}\triangleq {\left|{\widehat{h}}_{X,ij}\left(k\right)-{\widehat{h}}_{A,ij}\left(k-1\right)\right|}^2.$$

(29)

On $H_0$, the current signal is from Alice, and the difference $△h_{A,A}$ can be written as

$$\begin{array}{cc}\hfill △h_{A,A}& \triangleq {\widehat{h}}_{A,ij}\left(k\right)-{\widehat{h}}_{A,ij}\left(k-1\right)\hfill \\ \hfill & =\left(\alpha -1\right){\widehat{g}}_{ij}\left(k-1\right)f_{A,j}\left(k-1\right)\hfill \\ \hfill & +\sqrt{\left(1-{\alpha }^2\right)}{\widehat{g}}_{ij}\left(k-1\right)v_{A,j}\left(k-1\right)\hfill \\ \hfill & +w_{ij}\left(k\right)-w_{ij}\left(k-1\right),\hfill \end{array}$$

(30)

where $f_{A,j}(k-1)$ and $v_{A,j}(k-1)$ and $w_{ij}(k-1)$ are elements in $\mathbf{f}(k-1)$, $\mathbf{v}(k-1)$, and $\mathbf{W}(k-1)$ at time $k-1$, respectively.

We know that $f_{A,j}(k-1)$, $v_{A,j}(k-1)$, $w\left(k\right)$, and $w(k-1)$ are zero-mean Gaussian distributed variables with variances ${\sigma }_{f_A}^2$ and ${\sigma }_w^2$, respectively. Hence, for a given ${\widehat{g}}_{ij}\left(k-1\right)$, $△h_{A,A}$ is a zero-mean complex Gaussian random variable with variance $2\left(1-\alpha \right){\left|{\widehat{g}}_{ij}\left(k-1\right)\right|}^2{\sigma }_{f_A}^2+2{\sigma }_w^2$. To sum up, ${\wedge }_{H_0}$ obeys exponential distribution, that is,

$${\wedge }_{H_0}\sim exp\left({\displaystyle \frac{1}{2\left(1-\alpha \right){\left|{\widehat{g}}_{ij}\left(k-1\right)\right|}^2{\sigma }_{f_A}^2+2{\sigma }_w^2}}\right).$$

(31)

For a given threshold $\gamma$, the probability of the output of each quantizer under null hypothesis $H_0$ can be evaluated as

$$\begin{array}{cc}\hfill P_{H_0}& \triangleq \mathrm{Pr}\left(Q_h\left[{\left|{\widehat{h}}_{X,ij}\left(k\right)-{\widehat{h}}_{A,ij}\left(k-1\right)\right|}^2\right]\right.\hfill \\ \hfill & \left.=1\left|H_0\right.\right)\hfill \\ \hfill & =\mathrm{Pr}\left({\wedge }_{H_0}>\gamma \right)\hfill \\ \hfill & =exp\left(-{\displaystyle \frac{\gamma }{2\left(1-\alpha \right){\left|{\widehat{g}}_{ij}\left(k-1\right)\right|}^2{\sigma }_{f_A}^2+2{\sigma }_w^2}}\right).\hfill \end{array}$$

(32)

On $H_1$, the current transmitter is Eve, and $△h_{E,A}$ is given by

$$\begin{array}{cc}\hfill △h_{E,A}& \triangleq {\widehat{h}}_{E,ij}\left(k\right)-{\widehat{h}}_{A,ij}\left(k-1\right)\hfill \\ \hfill & =\alpha {\widehat{g}}_{ij}\left(k-1\right)f_{E,j}\left(k-1\right)\hfill \\ \hfill & +\sqrt{\left(1-{\alpha }^2\right)}{\widehat{g}}_{ij}\left(k-1\right)v_{E,j}\left(k-1\right)\hfill \\ \hfill & -{\widehat{g}}_{ij}\left(k-1\right)f_{A,j}\left(k-1\right)\hfill \\ \hfill & +w_{ij}\left(k\right)-w_{ij}\left(k-1\right).\hfill \end{array}$$

(33)

Similarly, we can obtain that $△h_{E,A}$ is a zero-mean complex Gaussian random variable with variance $|{\widehat{g}}_{ij}\left(k-1\right){|}^2\left({\sigma }_{f_E}^2+{\sigma }_{f_A}^2\right)+2{\sigma }_{\omega }^2$. ${\wedge }_{H_1}$ obeys exponential distribution, which can be written as

$${\wedge }_{H_1}\sim exp\left({\displaystyle \frac{1}{|{\widehat{g}}_{ij}\left(k-1\right){|}^2\left({\sigma }_{f_E}^2+{\sigma }_{f_A}^2\right)+2{\sigma }_{\omega }^2}}\right).$$

(34)

For a given threshold $\gamma$, the probability of the output of each quantizer under hypothesis $H_1$ can be evaluated as

$$\begin{array}{cc}\hfill P_{H_1}& \triangleq \mathrm{Pr}\left(Q_h\left[{\left|{\widehat{h}}_{X,ij}\left(k\right)-{\widehat{h}}_{A,ij}\left(k-1\right)\right|}^2\right]\right.\hfill \\ \hfill & \left.=1\left|H_1\right.\right)\hfill \\ \hfill & =\mathrm{Pr}\left({\wedge }_{H_1}>\gamma \right)\hfill \\ \hfill & =exp\left(-{\displaystyle \frac{\gamma }{|{\widehat{g}}_{ij}\left(k-1\right){|}^2\left({\sigma }_{f_E}^2+{\sigma }_{f_A}^2\right)+2{\sigma }_{\omega }^2}}\right).\hfill \end{array}$$

(35)

4.2. False Alarm and Detection Probabilities

We use $P_f$ and $P_d$ to denote the false alarm and detection probabilities, respectively. For a given $P_{H_0}$ and $P_{H_1}$, $P_f$ and $P_d$ can be analytically evaluated. When employing location-specific cascaded channel features in the proposed scheme to authenticate the origin of the current signal, the evaluation of $P_f$ and $P_d$ is carried out as follows.

$$\begin{array}{cc}{P}_f\hfill & =\mathrm{Pr}\left(D=Z+1\mathrm{or}D=Z+2\mathrm{or}\cdots \mathrm{or}\right.\hfill \\ \hfill & \left.D=L\left|H_0\right.\right)\hfill \\ \hfill & =\mathrm{Pr}\left(D=Z+1\left|H_0\right.\right)+\mathrm{Pr}\left(D=Z+2\left|H_0\right.\right)\hfill \\ \hfill & +\cdots +\mathrm{Pr}\left(D=L\left|H_0\right.\right)\hfill \\ \hfill & =\sum _{z=Z+1}^L\mathrm{Pr}\left(D=z\left|H_0\right.\right)\hfill \\ \hfill & =\sum _{z=Z+1}^L\mathrm{Pr}\left(D=z\left|H_0\right.\right)\hfill \\ \hfill & =\sum _{z=Z+1}^L\left(\begin{array}{c}L\\ z\end{array}\right)P_{H_0}^z{\left(1-P_{H_0}\right)}^{L-z},\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill P_d& =1-\mathrm{Pr}\left(D=0orD=1or\cdots orD=Z\left|H_1\right.\right)\hfill \\ \hfill & =1-[\mathrm{Pr}\left(D=0\left|H_1\right.\right)+\mathrm{Pr}\left(D=1\left|H_1\right.\right)\hfill \\ \hfill & +\cdots +\mathrm{Pr}\left(D=Z\left|H_1\right.\right)]\hfill \\ \hfill & =1-\sum _{z=0}^Z\mathrm{Pr}\left(D=z\left|H_1\right.\right)\hfill \\ \hfill & =1-\sum _{z=0}^Z\left(\begin{array}{c}L\\ z\end{array}\right)P_{H_1}^z{\left(1-P_{H_1}\right)}^{L-z}.\hfill \end{array}$$

(37)

The derivation for $P_f$ and $P_d$ relies on the Gaussian assumption regarding the channels $\mathbf{G}\left(k\right)$, $\mathbf{f}\left(k\right)$, and $\mathbf{H}\left(k\right)$. Both the BS-RIS and RIS-UAV paths follow Rayleigh fading distribution due to the isotropic scattering assumption, which is a common assumption in all related works [34,42]. This Gaussian assumption provides an insight characterization for the typical authentication performance metrics (false alarm and detection probabilities) and is helpful in establishing statistically theoretical results.

4.3. Computational Complexity

The proposed authentication scheme uses channel gains, and hence, the computational complexity of the scheme mainly results from the channel estimation, channel quantization, and authentication decision. In particular, the computational complexity of the coordinate descent-based algorithm for the quasi-static BS-RIS channel estimation is $\mathcal{O}(N^3+ML{N}^2+MNLC)$ while taking into account the number of iterations, where C is the maximal number of outer iterations and L is the number of time slots for one subframe; the computational complexity of the LS-based algorithm for the mobile but low-dimensional BS-UE and RIS-UAV channels is $\mathcal{O}({(N+M)}^3+{(N+M)}^2)$; and the computational complexity of channel quantization and authentication decision is $\mathcal{O}\left(2MN\right)$. As a result, the whole complexity of the proposed scheme is $\mathcal{O}(N^3+ML{N}^2+MNLC+2MN)$. For the scheme in [21], the computational complexity is $\mathcal{O}\left(N^3{M}^5\right)$. It is evident for the massive MIMO systems ($N=M=64$) that our proposed scheme offers a substantial benefit in terms of reduced computational complexity. Given the powerful computational capabilities of smart devices in RIS-enabled communication systems, the additional computational complexity is also generally acceptable in engineering practices.

5. Evaluation

Two sets of 200 symbols are present in the data frames. The initial frame, which is authored by Alice, undergoes validation by Bob through the use of higher-layer authentication methods. Subsequently, the second frame is received by Bob with the assistance of RIS, 300 $\mu$s after the first one. If the second frame is also sent by Alice, we calculate time-varying channel coefficients that cover both frames, using specific parameters for each iteration. If the frame is sent by Eve, we generate independent channel matrices instead. The effectiveness of the suggested method’s authentication is closely tied to the average SNR at Bob. Extensive simulation experiments have been undertaken to validate the theoretical findings in relation to $P_f$ and $P_d$ under the specified configurations ($M=4$, $N=2$, ${\sigma }_{f_A}^2=1\times {10}^{-4}$, ${\sigma }_{f_E}^2=1\times {10}^{-5}$, $\alpha =0.9$).

In Figure 4, we investigate how SNR affects the detection probability $P_d$ with the settings of $Z\in \{1,2,3,4\}$. We can see from Figure 4 that, for a given threshold Z, $P_d$ improves monotonously as SNR increases in a relatively lower regime. Specifically, in a low SNR regime (e.g., SNR is smaller than −3 dB), $P_d$ quickly grows, and when SNR is larger than −2 dB, $P_d$ tends to be 1. It reveals that our proposed scheme has a good performance benefit from SNR. The decrease in estimation error is attributed to the increase in signal-to-noise ratio (SNR), resulting in a more precise estimation. Another notable finding is that, under a constant SNR, a smaller threshold, Z, is associated with a higher value for $P_d$, whereas a larger threshold, Z, is linked to a lower value for $P_d$. Given the energy constraint, it is commonly anticipated that the power level of wireless networks will be below a specific threshold. As a result, it becomes crucial to accurately establish the threshold parameter (e.g., Z) in order to attain the desired authentication performance within a specified power constraint. In addition, we note that multiple lines converge as SNR is in a high regime. The convergence observed in Figure 4 can be attributed to the fact that when the signal-to-noise ratio (SNR) reaches a certain threshold, such as −2 dB, the accuracy of the estimation algorithm reaches a bottleneck. At this point, the influence of SNR on channel fingerprint extraction becomes minimal. As a result, the detection rate of the authentication scheme converges.

As seen from Figure 5, $P_f$ decreases sharply when SNR increases for a given $\gamma$ in a relatively low SNR regime, while in a relatively high regime, $P_f$ gradually tends towards 0. This tells us that using a larger power to transmit data signals is helpful in reducing false alarm property $P_f$. We also observe that, given a fixed SNR, the false alarm probability $P_f$ decreases as the threshold $\gamma$ increases. Similarly, it needs to appropriately set the threshold parameters (e.g., Z and $\gamma$) to satisfy the performance requirements under a given power limitation.

Receiver operating characteristic (ROC) curves are employed to provide an additional illustration of the performance of the proposed scheme. In Figure 6, we show how SNR affects the authentication performance ($P_f$, $P_d$) for ($M=4$, $N=2$, ${\sigma }_{f_A}^2=1\times {10}^{-4}$, ${\sigma }_{f_E}^2=1\times {10}^{-5}$, $\alpha =0.9$). One can note from Figure 6 that the simulation and theoretical results are close to each other, verifying the correctness of the proposed theoretical performance models. In other words, we can use these models to precisely capture the authentication performance and conduct theoretical analysis. Moreover, one can also note from Figure 6 that increasing SNR can obtain a large $P_d$ and a low $P_f$. Even for a small SNR (e.g., SNR = −16 dB), we can see that when $P_f<0.02$, $P_d$ can exceed 0.95. It indicates that the scheme is effective for RIS-enabled wireless communication systems to authenticate transmitters and can be a complement to traditional cryptography-based authentication approaches.

The impact of a channel correlation coefficient for the cascaded channel on performance is also investigated in Figure 7. We plot in Figure 7 ROC curves under different channel correlation coefficients and the settings of $Z=1$, $\gamma \in \left[0.01\sim 0.02\right]$, and SNR $=-1$ dB. In Figure 7, it is evident that the proposed scheme demonstrates its optimal authentication performance when $\alpha =1$ and its weakest performance when $\alpha =0.5$. This indicates that the proposed scheme is better suited for slow-fading RIS-enabled wireless communication. It is noteworthy that the ROC curves for different $\alpha$ values are closely aligned, with all four curves yielding $P_d$ values exceeding 0.95 while maintaining very low $P_f$. Consequently, our proposed authentication scheme holds great promise for applications in high-speed environments.

To further validate the performance of the proposed scheme, we illustrate in Figure 8 the confidence interval and actual theoretical value of $P_f$ under different SNRs with the settings of $Z=1$, $M=4$, $N=2$, and $\alpha =0.9$. As observed from Figure 8, we can note that the actual value of $P_f$ is within the 95% confidence interval, indicating the reliability and correctness of the adopted false alarm performance model. In other words, the derived performance expression can effectively characterize the authentication performance of the proposed scheme.

In Figure 9, we present a comparison of the performances of the existing scheme without RIS (one-hop channel) described in [19,20] and our proposed scheme utilizing RIS-enabled UAV communication (cascaded channel with RIS). Here, we consider a low SNR regime mainly to explore the illegal device detection performance of the proposed scheme in RIS systems under poor signal environment conditions and to investigate the application potential in disaster scenarios or remote locations with weak signal strength. For the benchmark in [19,20], we consider that there is only a direct link between the transceiver ends and utilize the classic LMMSE algorithm for direct channel estimation. Then, these channel estimates are quantified to determine whether the current transmitter is a legitimate device based on the quantization results. As shown in Figure 9, we note that our scheme is superior to [19], while it is slightly poorer than [20]. Specifically, under an SNR = −8 dB, the $P_d$ of [19], our scheme, and [20] are 0.8, 0.9, and 0.92, respectively. The reason is that the method in [19] depends on the strong channel temporal correlation; such a method may encounter performance loss when considering the high-speed UAV mobility (i.e., small $\alpha$). In addition, the cascaded channel resulting from the introduction of RIS entails the multiplication of two individual channel components. As such, this channel is highly dynamic and complex, leading to reduced accuracy in cascaded channel estimation and degraded authentication performance compared with [20]. However, although the performance under cascaded channels is not as good as that of the existing scheme without RIS [20], it keeps a relatively high detection probability even in a low SNR regime. This indicates that we can achieve a trade-off between security and communication service: our proposed scheme is expected to achieve high authentication performance and guarantee the identification security of legitimate users while exploiting RIS to improve communication coverage to provide more reliable services for users.

6. Conclusions

In this study, we established a novel PHY-layer authentication using a cascade channel signature within the RIS-enabled UAV wireless communication system for verifying the transmitter’s identity. With the least squares estimate and a coordinate descent-based algorithm, we first effectively extracted the RIS-cascaded channel features. Then, a quantizer was introduced to quantify channel gain fluctuations, which are critical for generating authentication decision criteria under a binary hypothesis test. By leveraging hypothesis testing and statistical signal processing, we further analytically derived closed-form expressions for both false alarm and detection probabilities. These theoretical findings were then substantiated by extensive simulation results, affirming their accuracy. The outcomes of this investigation show that our scheme can achieve an improvement of 10% in detection accuracy compared with existing works and offers valuable insights for devising cost-effective authentication approaches centered on RIS-induced channel characteristics to fortify RIS-enabled wireless networks.