Secure Backscatter Communications Through RIS: Modeling and Performance

1 Secure Backscatter Communications Through RIS: Modeling and Performance Masoud Kaveh, Member, IEEE, Farshad Rostami Ghadi, Member, IEEE, Zhao Li, Member, IEEE, Zheng Yan, Fellow, IEEE, and Riku J¨ antti, Senior Member, IEEE Abstract—Backscatter communication (BC) has emerged as a pivotal wireless communication paradigm owing to its low-power and cost-effective characteristics. However, BC faces various challenges from its low signal detection rate to its security vulnerabilities. Recently, reconfigurable intelligent surfaces (RIS) have surfaced as a transformative technology addressing power and communication performance issues in BC. However, the potential of RIS in addressing the security challenges of BC remains uncharted. This paper investigates the secrecy performance of RIS-aided BC, where all channels are distributed according to the Fisher-Snedecor Fdistribution. Specifically, we consider a RIS with Nreflecting elements to help a backscatter device (BD) establish a smart environment and enhance the secrecy performance in BC. Due to the nature of BC systems, our analysis considers two possible scenarios (i) in the absence of direct links and (ii) in the presence of direct links. In both cases, we first derive compact analytical expressions of the probability density function (PDF) and cumulative distribution function (CDF) for the received signal-to-noise ratio (SNR) at both a legitimate receiver and an eavesdropper. Then, to analyze the secrecy performance, we further derive analytical expressions of the average secrecy capacity (ASC) and secrecy outage probability (SOP) for both mentioned scenarios. In addition, regarding the importance of system behavior in a high SNR regime, we provide an asymptotic analysis of the SOP and ASC. Eventually, the Monte-Carlo simulation is used to validate the analytical results, revealing that utilizing RIS can greatly improve the secrecy performance of the BC system relative to traditional BC setups that do not incorporate RIS. Index Terms—Backscatter communication, reconfigurable intelligent surfaces, Fisher-Snedecor Ffading, physical layer security, secrecy outage probability, average secrecy capacity. I. INTRODUCTION BACKSCATTER communication (BC) is a wireless communication technique that enables low-power and lowdata-rate devices to transmit data by modulating and reflecting existing radio frequency (RF) signals in the environment [1]. This technology holds exceptional promise, particularly in the The work of M. Kaveh and R. J¨ antti has received funding from the Smart Networks and Services Joint Undertaking under the European Union’s Horizon Europe research and innovation programme under Grant Agreement No 101192113. The work of F. R. Ghadi is supported in part by the European Union’s Horizon 2022 Research and Innovation Programme under Marie Skłodowska-Curie Grant No. 101107993. The work of Z. Yan is supported in part by the Academy of Finland under Grants 345072 and 350464. M. Kaveh and R. J¨ antti are with the Department of Information and Communication Engineering, Aalto University, Espoo, Finland. (e-mail: [email protected],[email protected]) F. R. Ghadi is with the Department of Signal Theory, Networking and Communications, University of Granada, 18071, Granada, Spain. (e-mail: [email protected]). Z. Li and Z. Yan are with the School of Cyber Engineering, Xidian University, Xi’an, China, (e-mail: [email protected],[email protected]) context of the Internet of Things (IoT), due to its inherent energy efficiency, cost-effectiveness, and simplicity [2]. Backscatter devices (BDs) are designed to consume minimal energy as they do not need to generate their own signals; instead, they efficiently utilize the ambient RF signals already present in the environment. Therefore, BC allows the batteryless IoT devices operate for extended periods without requiring frequent battery replacements [3]. While BC offers several advantages for IoT, there are some challenges that have yet to be addressed to ensure its reliable operation and widespread adoption. These challenges, inherently arising due to BC’s low-power operation nature, unlicensed spectrum usage, and reliance on ambient signals, mainly include low signal detection rate, limited transmission range, low data rate and throughput, and concerns related to energy harvesting and consumption [4]. Recently, reconfigurable intelligent surfaces (RIS) [5] has shown promising potentials to improve the communication performance in BC systems [6], [7]. RIS is essentially a metasurface composed of a large number of passive elements, which takes the advantage of meta-materials to dynamically control and shape the reflection and scattering of RF signals and improve the signal quality in wireless propagation environment [8], [9]. RIS has demonstrated remarkable efficacy in tackling significant challenges in BC including improving the channel conditions [10], [11], transmission range [12], system throughput [13]–[15], energy efficiency [16], [17], detection performance [18]–[20], and energy harvesting [21], [22] (see II-C). These enhancements been observed across various BC systems, including monostatic BC, bistatic BC, and ambient BC [6]. Moreover, RIS has shown to be harnessed in BC through different approaches i.e., serving as a helper element or a dedicated BD [7]. Furthermore, BC signals are typically passive and operate in an open broadcast fashion, rendering them susceptible to interception and eavesdropping by malicious entities. As BDs often operate with stringent resource constraints, including limited processing power and memory, their ability to employ sophisticated security protocols and cryptographic primitives is further restricted. Therefore, physical layer security (PLS) stands as a compelling approach for establishing secure BC by offering lower complexity and better security features than cryptographic schemes [23]–[25]. In this regard, examining PLS performance is essential for establishing resilient and efficient secure communication within wireless networks. In this context, two critical PLS performance metrics, namely average secrecy capacity (ASC) and secrecy outage probability (SOP), have been assessed across various scenarios within BC through recent years [26]–[32] (see II-A). This article has been accepted for publication in IEEE Transactions on Vehicular Technology. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TVT.2025.3612485 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ 2 A. Research Gaps and Motivations While there have been several efforts to enhance PLS performance in BC during recent years [26]–[32], achieving optimal PLS performance in BC systems has been a challenging task for the related works, primarily due to constraints inherent in conventional BC paradigms, such as restricted signal strength, interference, limited channel knowledge, and resource constraints of BDs [25]. Despite of RIS great enhancement to BC [10]–[22], it has been remained unexplored whether RIS can also enhance the PLS performance of BC systems. Motivated by the suboptimal PLS performance in BC, and the notable benefits that RIS can offer to BC, we are propelled to leverage the great potential of RIS to improve the PLS performance in BC. As an initial step in this process, developing analytical expressions for secrecy metrics enables a quantitative evaluation of how various system parameters influence PLS performance. Specifically, our analysis focuses on assessing ASC and SOP as crucial PLS metrics, which to the best of our knowledge, have not been studied in previous works in RIS-aided BC. On the other hand, since the impact of multipath fading and shadowing on the received signal strength in BC has not yet been thoroughly explored, despite its substantial significance in evaluating the performance of BC [14], [33], employing a flexible composite fading channel model, such as FisherSnedecor F[34], can provide a more precise assessment of secrecy performance. Moreover, given the inherently weak direct links 1in BC, it becomes imperative to encompass both potential scenarios in the secrecy performance evaluation process. This includes assessing the performance of RIS-aided BC systems with and without direct links. The inclusion of both scenarios allows for a comprehensive evaluation of BC’s effectiveness and the potential enhancements offered by RIS technology [35]. B. Contributions In this paper, we investigate the impact of RIS in enhancing the performance of secure BC under Fisher-Snedecor Ffading channels. Our work addresses critical gaps in the understanding of secure BC systems by encompassing a range of important aspects. The main contributions of this paper can be summarized as follows. • For the first time in this paper, we assess the secrecy performance of the RIS-aided BC system, considering how different system parameters influence its effectiveness. In addition, due to the relatively weaker direct link in practical scenarios for BC, our analysis includes both possible cases, one with RIS-aided links alone and the other with a combination of RIS-aided and direct links. • We employ the Fisher-Snedecor Fdistribution [34] to model the fading channels, allowing us to accurately characterize the simultaneous occurrence of multi-path fading and shadowing in BC, and consequently, to reach a more accu1In this context, direct link refers to the links between BDs and either backscatter or eavesdropping receivers. As for the links between the RF source and BDs, it is referred to as the source link. rate secrecy performance analysis compared to other fading distributions. • For providing the necessary mathematical foundation for analyzing the secrecy performance metrics, we introduce compact analytical expressions for the probability density function (PDF) and cumulative distribution function (CDF) of the received SNR at the legitimate receiver and the eavesdropper in the RIS-aided BC system for both scenarios (i.e., with and without direct links) with assumption of a perfect channel state information (CSI) of the legitimate links and imperfect CSI of eavesdropping links at the RIS. Then, by using the derived PDFs and CDFs, we are able to derive accurate analytical expressions of the ASC and SOP based on the bivariate and multivariate Fox’s H-function for analyzing the system’s secrecy performance. • We provide an asymptotic analysis of the obtained ASC and SOP by employing the residue method [37], for studying the RIS-aided BC system’s behavior in high SNR regime and gaining insights into the fundamental performance trends and capabilities of BC systems under optimal conditions. • We validate the analytical results using Monte-Carlo simulation. Our findings indicate that RIS can significantly improve the PLS performance in BC systems across diverse system configurations. Furthermore, utilizing the Fisher-Snedecor F distribution can accurately model channel and offer a more precise secrecy performance analysis in comparison to other fading distributions in BC. C. Organization and Notations The rest of this paper is organized as follows. Section II delves into the existing literature on the subject. Section III presents the RIS-aided BC system and channel models. Section IV demonstrates the SNR distributions at the legitimate receiver and eavesdropper. We analyze the secrecy performance of RIS-aided BC by deriving the compact analytical expressions of ASC and SOP in Section V. Section VI presents the asymptotic analysis of the secrecy metrics. The simulation results are discussed in Section VII, the practical applications and constraints discussed in Section VIII, and finally a conclusion is drawn in the last section. Notations:Γ(.)is the complete Gamma function [68, Eq. 8.31], B(., .)is the Beta function [68, Eq. 8.38], Gm,n p,q (.)is the Meijer’s G-function [65, Eq. 8.2.1.1], Hm,n:m1,n1;...;mr,nr p,q:p1,q1;...;pr,qr(.) is the multivariate Fox’s H-function [67], j=√−1, and XT is the transpose of X. II. RELATED WORKS In this section, we present a review of the literature concerning performance analysis frameworks for PLS in BC, strategies for enhancing PLS performance through RIS, and the utilization of RIS in BC systems. Table I shows the differences between previous studies and our research, highlighting the distinctive aspects of our analysis in this paper. A. PLS Performance Analysis for BC Systems In recent years, there has been a notable body of work dedicated to assessing the secrecy performance of BC systems This article has been accepted for publication in IEEE Transactions on Vehicular Technology. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TVT.2025.3612485 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ 3 TABLE I COMPARISON OF RELATED WORKS: RIS-AIDED BC SYSTEMS AND PLS PERFORMANCE ANALYSIS IN BC VERSUS OUR WORK Works RISIBC PLSPE EMFS AA Evaluation Metrics [10], [11] ✓× × × Channel Condition [12] ✓× × × Transmission Range [13]–[15] ✓× × × System Throughput [16], [17] ✓× × × Energy Efficiency [18]–[20] ✓× × × Bit Error Rate [21], [22] ✓× × × Energy Harvesting [26]–[28] ×✓× × SOP [29]–[32] ×✓×✓SOP, ASC [44] ×✓×✓SOP Ours ✓ ✓ ✓ ✓ SOP, ASC RISIBC: RIS integration into BC, PLSPE: PLS performance evaluation, EMFS: Considering effect of multipath fading and shadowing, AA: Asymptotic analysis, ✓: Item is supported, ×: Item is not supported. [26]–[32]. In [26], researchers conducted an examination of the performance of wireless backscatter systems, specifically focusing on the evaluation of SOP. Additionally, [27] delved into the analysis of SOP within the context of a multi-tag BC system, considering the presence of an eavesdropper. Furthermore, in [28], the authors improved SOP by introducing an optimal tag selection scheme for passive BC systems characterized by multiple tags and a single eavesdropper. In [29], a novel tag selection scheme was proposed with the aim of enhancing both the ASC and SOP of a multi-tag selfpowered BC system in scenarios involving an eavesdropper. The authors also derived an analytical expression for SOP to facilitate their analysis. In [30], the investigation centered around the influence of eavesdroppers and the motion of readers on the secrecy performance of ambient BC systems, particularly in cases where channel estimation is imperfect. In [31], researchers introduced an overlay cognitive ambient BC non-orthogonal multiple access (NOMA) system tailored for intelligent transportation systems. Within this context, they scrutinized the secrecy performance of their proposed system model in the presence of an eavesdropping vehicle by deriving the SOP. Additionally, in [32], an exploration was conducted into secure multi-antenna transmission within ambient BC-based intelligent transportation systems. This investigation was carried out in the presence of a passive eavesdropper with jamming, where a cooperative jammer was strategically positioned within the system to disrupt the eavesdropper without affecting the reader. To assess the performance of this proposed scheme, a new closed-form expression for SOP was derived. While a notable body of work has been dedicated to assessing the secrecy performance of BC systems under various conditions, attaining optimal PLS performance in BC has proven to be a formidable challenge, primarily due to the inherent limitations of traditional BC systems such as restricted signal strength in direct link and resource constraints of BDs. These factors motivate the integration of RIS into BC to potentially overcome the suboptimal PLS performance by enhancing signal strength and providing improved control over channel characteristics in BC. Furthermore, another critical gap in previous studies is the overlooking of the simultaneous effects of multipath fading and shadowing in PLS performance analysis, which are prevalent in real-world BC environments [14], [33]. This gap motivates the need for a general and comprehensive channel fading model such as the FisherSnedecor Fdistribution, to provide a more precise and robust analysis of secrecy performance in RIS-aided BC systems. B. Enhancing PLS Performance Using RIS Through recent years, RIS technology has been offering a promising avenue to enhance the PLS performance in various wireless communication systems. These include but are not limited to smart grid communications [38], vehicular networks [39], device-to-device communications [40], IoT networks [41], integrated satellite-vehicle networks [42], and networks involving unmanned aerial vehicles (UAVs) [43]. In addition, by assuming an ambient BC system, the authors in [44] derived SOP for a RIS-aided set-up; however, they did not consider the source link (i.e., product channels) in their performance analysis. Tang et al. introduced an innovative RIS design aimed at enhancing PLS for RIS-aided NOMA networks, showcasing the potential of RIS to improve security measures dynamically [45]. Similarly, Zhang et al. explored the general benefits of RIS in enhancing PLS, providing robust strategies against eavesdroppers in diverse network conditions [46]. Gu et al. addressed the challenge of uncertain eavesdropper locations by employing RIS in their security strategies, demonstrating the flexibility of RIS in adapting to varying security requirements and enhancing overall network resilience [47]. In a similar vein, Zhang et al. focused specifically on the integration of RIS into 6G networks using NOMA, illustrating significant improvements in secrecy performance due to the smart deployment of RIS elements [48]. Extending the application of RIS, Wang et al. investigated the uplink secrecy performance of RIS-based radio frequency/free space optics (RF/FSO) three-dimensional heterogeneous networks, highlighting the versatility of RIS in different transmission mediums and its effectiveness in securing communications across multiple layers [49]. Elhoushy et al. utilized RIS to limit information leakage in a cell-free massive multiple input multiple output (MIMO) setting, facing active eavesdroppers. This study particularly noted the efficacy of RIS in environments where traditional security measures fall short [50]. Although recent advancements have highlighted RIS as a promising tool to improve PLS in various wireless communication systems, its impact on the secrecy performance of BC systems remains largely unexplored. Our proposed secure RISaided BC model introduces two main distinctions from prior work: First, in analytical complexity, BDs in BC rely on an RF signal from a source, creating additional links among the source, RIS, BD, reader, and eavesdropper, which complicates deriving the PDF and CDF of the SNR for secrecy analysis. Second, BC’s unique properties, such as passive operation and dependence on incident RF signals, limit power and range, which substantially affect secrecy performance and require tailored simulation settings. These constraints make RIS-aided enhancements necessary, yet complex, making BC systems different from traditional wireless systems. This article has been accepted for publication in IEEE Transactions on Vehicular Technology. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TVT.2025.3612485 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ 4 C. RIS-Aided BC Systems Quite recently, integrating RIS into BC has led to significant enhancements in different aspects. The authors in [10], [11] demonstrated that RIS is able to address channel condition challenges observed in conventional BC systems. These challenges arise from various propagation effects, resulting in the arrival of multiple out-of-phase signals at the BD and reader, ultimately leading to a decline in system performance. However, through intelligent signal reflection, RIS effectively mitigates the adverse impacts of electromagnetic radiation and enhances the overall channel gain. RIS can also tackle the limited transmission range issue in BC systems by introducing efficient supplementary paths. This capability facilitates the widespread deployment of BC and prevents its utilization from being restricted to short-range applications [12]. Due to the shared transmission medium among BDs in BC, system throughput can be constrained by interference and collisions between different devices. To address this challenge, researchers in [13]–[15] leveraged RIS in various BC scenarios, aiming to improve the quality of service (QoS) and weighted sum rate. This was achieved through optimization of the RIS phase-shift matrix and beamforming vectors at the transmitter. In addition, RIS’s passive reflections offer power gains, allowing RIS-aided BC to leverage this advantage for achieving higher performance gains while requiring less transmit power. As a result, RIS significantly enhances the energy efficiency of BC systems [16], [17]. In BC systems, the reader’s signal detection capability is often hindered by factors like direct-link interference. The authors in [18]–[20] utilized the RIS capabilities to control signal direction, effectively mitigating direct-link interference and enhancing signal detection performance with reduced complexity compared to conventional schemes. Furthermore, the authors in [21], [22] demonstrated that employing RIS for the coherent combination of reflected signals results in a substantial increase in the total received power within BC systems. This enhancement enables BDs to harvest energy from both direct and RIS-reflected signals, significantly boosting the total harvested energy and supporting long-term IoT network operation. Although prior studies demonstrate the significant enhancements that RIS can bring to BC systems, such as improved channel conditions, extended transmission range, increased system throughput, enhanced energy efficiency, reduced bit error rate, and optimized energy harvesting, the impact of RIS on the secrecy performance of BC has yet to be explored. Addressing this gap is critical, as the unique characteristics of BC systems require tailored approaches to secure communication, particularly in the presence of eavesdroppers. Based on limitations and gaps in previous works, our paper fills this gap by comprehensively evaluating the secrecy performance of RIS-aided BC, as summarized in Table I, where we account for realistic fading conditions, analyze critical secrecy metrics, and provide a robust framework to assess the potential of RIS to improve secure backscatter communication in low-power communication applications. Tag RIS Eve 𝒉𝑻𝑬 𝒉𝚯𝑬 𝒉𝚯𝑹 𝒉𝑻𝚯 𝒉𝑻𝑹 Source 𝒉𝑺𝑻 Reader 𝒉𝑻𝚯 𝒉𝑻𝚯 Obsrtruction RIS0 Controller Source0Link RIS-aided0Link Direct0Link Control0Link Fig. 1. The system model of RIS-aided BC. III. SYSTEM MODEL Fig. 1 shows the system model of RIS-aided BC. In this model, we consider a BD (like a tag) as a semi-passive device which is powered up through a continuous wave carrier signal transmitted by a source. The tag aims to send its confidential messages to a legitimate receiver (like a reader) within both direct and cascade links. We consider a RIS with N reflecting elements positioned between the tag and the reader 2 , configured to enhance communication by optimizing its phase shifts through a microcontroller, which is securely managed by a trusted entity within the network. We assume that there is also a passive eavesdropper named Eve that tries to decode the confidential message sent by the tag over both tag-toreader and RIS-to-reader links with unlimited computational power. For simplicity and without loss of generality, since the RF source is sending unmodulated carriers, both reader and Eve can utilize cancellation methods to mitigate the impact of interference due to the source’s link [2], [51], [52]. We assume that the CSI of cascade links are known by RIS, so RIS can adjust the phase shifting coefficients of its elements for maximizing the SNR at the reader. This process can be executed at the reader, which involves transmitting pilot symbols from the tag to the reader through different phase configurations of the RIS. This allows for a linear estimation of the cascaded channel by aggregating the received signals that correspond to each configuration [53]. Then, the estimated channel coefficients are communicated back to the RIS controller (see [54] and reference therein). We also assume that the tag, reader, and Eve are equipped with a single antenna for simplicity. Therefore, the received signal at the tag can be given as follows yT=pPshST +nT,(1) where Psrepresents the transmit power of the source, hST is the channel coefficient between source and tag, and nT denotes the additive white Gaussian noise (AWGN) at the tag. Since the noise power caused by the tag’s antenna is considerably smaller than the received signal from the source [28], we neglect it in the rest of this paper. 2While an alternative configuration could position the RIS between the source and the tag to enhance energy harvesting, this setup primarily benefits power transfer rather than secrecy [19], [22]. Since our study focuses on secrecy enhancement, we consider the direct source-to-tag transmission sufficient for energy supply, ensuring a practical system design without unnecessary complexity. This article has been accepted for publication in IEEE Transactions on Vehicular Technology. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TVT.2025.3612485 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ 5 In most practical scenarios, the direct link between the BD and reader may not be feasible due to distance, obstructions, or channel conditions in BC. Additionally, the use of lowpower BDs further contributes to the limited signal strength and reduces the direct link’s reliability. In such cases, RIS can act as the primary means of communication between the tag and reader [35]. In other cases where a reliable link exists between the BD and reader, RIS can serve as an enhancer, further optimizing signal strength and improving overall system capacity. Therefore, we divide our analysis into two scenarios: RIS-aided BC with and without direct links. A. Without Direct Links Assuming the direct links between tag-to-reader and tag-toEve is not feasible, the received signals at the reader and Eve can be expressed as yR=pPshST HTΘΘHT ΘRS(t) + nR,(2) yE=pPshST HTΘΘHT ΘES(t) + nE,(3) in which S(t)is the information signal backscattered from the tag with a unit power, nRand nErepresent the AWGN at the reader and Eve with zero mean and variances σ2 R and σ2 E, respectively, and Θis the adjustable phase matrix induced by the reflection of RIS elements, which is defined as Θ=diag ejθ1, ejθ2, ..., ejθN. The vectors HTΘ,HΘR, and HΘEcontain the Nchannel coefficients from the tag to the RIS, and from RIS to the reader and Eve, respectively. The above channel vectors are given as HTΘ=d−χ TΘ.hTΘ1e−jα1, hTΘ2e−jα2, ..., hTΘNe−jαN, HΘR=d−χ ΘR.hΘR1e−jβ1, hΘR2e−jβ2, ..., hΘRNe−jβN, and HΘE=d−χ ΘE.hΘE1e−jϵ1, hΘE2e−jϵ2, ..., hΘENe−jϵN, where dTΘdenotes the distance between the tag and the RIS, dΘRis the distance between the RIS and the reader, and dΘE defines the distance between the RIS and Eve, respectively. The term χindicates the path-loss exponent. Furthermore, the terms hTΘn,hΘRn, and hΘEn, for n∈ {1,2, ..., N}, are the amplitudes of the corresponding channel coefficients, and e−jαn,e−jβn, and e−jϵndenote the phase of the respective links. In order to precisely capture the coexistence of multipath fading and shadowing in BC and achieve a more precise evaluation of secrecy performance, we utilize the FisherSnedecor Fdistribution [34] as a means to accurately model and characterize the system behavior in our analysis. B. With Direct Links Suppose the direct links between tag-to-reader and tag-toEve are existed, thereby, the received signals at the reader and Eve can be given by yR=pPsS(t)hST hT R +hST HTΘΘHT ΘR+nR,(4) yE=pPsS(t)hST hT E +hST HTΘΘHT ΘE+nE,(5) where hT R and hT E denote the tag-to-reader and tag-to-Eve channel coefficients, respectively. IV. SNR DISTRIBUTION In this section, an analysis is conducted on the SNR at the reader and Eve by considering both without and with direct link cases. The compact analytical expressions of PDF and CDF are then further derived based on the received SNR. A. Without Direct Links 1) Legitimate link: From (2), the instantaneous SNR at the reader can be determined as γR=√PshST HTΘΘHΘR) 2 nR (6) = Ps|hST |2PN n=1 hTΘnhΘRnej(θn−αn−βn) 2 dχ ST dχ TΘdχ ΘRσ2 R (7) (a) = ¯γR|hST |2 N X n=1 hTΘnhΘRn 2 ,(8) where (a)is obtained by enabling ideal phase shifting for RIS [22], [55], [56], and ¯γRis the average SNR at the reader due to the RIS-aided link. By defining X1=|hST |2and Y1= |PN n=1 hTΘnhΘRn|2, where all the channels follow FisherSnedecor Ffading model, we will have fX1(x1)[57] and fY1(y1)[58] as fX1(x1) = CG1,1 1,1λ1x1−mSST mST −1,(9) fY1(y1) = y1 c−1 2e−√y1 ¯y 1 d 1 2 ¯y1 c+1 2Γ(c+ 1) dc+1 ,(10) where mij and mSij indicate the fading severity parameter and the amount of shadowing of the root-mean-square (rms) signal power parameters, respectively, λ1=mST σ2 T mSST PS, C=λ1 Γ(mST )Γ(mSST ),c=(N+1)B′2−A′C′ A′C′−B′2,d=D′(A′C′−B′2) B′C′, A′=B(mθR + 1, mSθR −1) B(mT θ + 1, mST θ −1), B′=BmθR +1 2, mSθR −1 2BmT θ +1 2, mST θ −1 2, C′=B(mθR, mSθR )B(mT θ, mST θ ),D′= q(mSθR −1)(mST θ −1)ΩθRΩT θ mθRmT θ , and Ωij is the mean power. Theorem 1. Assuming all channels follow the FisherSnedecor Ffading distribution, the PDF and CDF of γR without direct links are given by fγR(γR) = GG1,3 3,14λ1γR ¯y 2 d 1 2−c 2,3−c 2,2−mST 1 + mSST ,(11) FγR(γR)=GγRG1,4 4,24λ1γR ¯y 2 d 1 0,2−c 2,3−c 2,2−mST 1 + mSST ,−1,(12) where G=22c−3 2¯y c−1 d 1C √2π¯γR¯y c+1 2 1dc+1Γ(c+1) and ¯y1=PS dχ TΘdχ ΘRσ2 R . Proof. The proof is elaborated in Appendix A. This article has been accepted for publication in IEEE Transactions on Vehicular Technology. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TVT.2025.3612485 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ 6 2) Eavesdropper link: From (3), the instantaneous SNR at Eve can be determined as γE=√PshST HTΘΘHΘE) 2 nE (13) = Ps|hST |2PN n=1 hTΘnhΘEnej(θn−αn−ϵn) 2 dχ ST dχ TΘdχ ΘEσ2 E (14) = ¯γE|hST |2 N X n=1 hTΘnhΘEnej(θn−αn−ϵn) 2 ,(15) where ¯γEis the average SNR at Eve due to the RIS-aided link. When the phase shifts of RIS elements are optimally designed based on the legitimate link’s conditions solely, the resulting phase distributions for each of the Eve’s links (HTΘΘHT ΘE) become effectively random and uniformly distributed across the range [−π,π) due to the lack of Eve’s CSI at RIS. Consequently, the channel coefficient at Eve follows a circularly-symmetric complex normal distribution, resulting in a channel that exhibits characteristics similar to Rayleigh fading. This Rayleigh-like behavior at Eve is supported by previous studies [59]–[61], and it accurately models the degraded coherence of the reflected signal that arises from the phase randomness induced by optimizing the RIS configuration solely for the legitimate link. Therefore, by assuming Y2=PN n=1 hTΘnhΘEnej(θn−αn−ϵn) 2 ,fY2(y2) can be shown as fY2(y2) = 1 ae−y2 a,(16) where a=NPS dχ TΘdχ ΘEσ2 E . Now, by considering (9) and (16), the marginal distributions of γEcan be obtained as the following theorem. Theorem 2. Assuming all channels follow the FisherSnedecor Ffading distribution, the PDF and CDF of γE without direct links are given by fγE(γE) = C a¯γE G1,2 2,1aλ1γE 1,2−mST 1 + mSST ,(17) FγE(γE) = C a¯γE γEG1,3 3,2aλ1γE 0,1,2−mST 1 + mSST ,−1.(18) Proof. The proof is elaborated in Appendix B. B. With Direct Links 1) Legitimate Link: According to (4), the instantaneous SNR at the reader can be determined as γR=√Ps(hST hT R +hST HTΘΘHΘR) 2 nk ≈Ps|hST |2|hT R|2 dχ ST dχ T Rσ2 R + Ps|hST |2PN n=1hTΘn hΘRn ej(θn−αn−βn) 2 dχ ST dχ TΘdχ ΘRσ2 R (19) (a) = ¯γR1|hST |2|hT R|2+ ¯γR2|hST |2 N X n=1 hTΘnhΘRn 2 ,(20) where ¯γR1and ¯γR2are the average SNR at the reader due to the direct and the RIS-aided links, respectively. By re-writing (20) as γR=γR1+γR2,fγR1(γR1)can be given by [62] fγR1(γR1) = η1 γR1 G2,2 2,2 δ1γR1 ¯γR1 1−msST ,1−msT R mST , mT R !,(21) where η1=1 Γ(mST )Γ(mSST )Γ(mT R)Γ(mST R )and δ1= mST mT R (mSST −1)(mST R −1) . According to Thm. 1, fγR2(γR2)can be obtained as fγR2(γR2) = GG1,3 3,14λ1¯y−2 d 1γR2 2−c 2,3−c 2,2−mST mSST + 1 .(22) Now, since γR=γR1+γR2, we exploit the MomentGenerating function (MGF) of γR1and γR2to obtain the PDF and CDF of γRas fγR(γR) = L−1MγR1(s)MγR2(s),(23) FγR(γR) = L−11 sMγR1(s)MγR2(s),(24) where L−1shows the Laplace inverse transform and Mγ(t) = Mγ(−s)denotes the MGF of γ. Theorem 3. Assuming all channels follow Fisher-Snedecor F fading distribution, the PDF and CDF of γRwith direct links can be obtained as (20) and (21), respectively. Proof. The proof is elaborated in Appendix C. 2) Eavesdropper link: According to (5), the instantaneous SNR at Eve can be determined as γE=√Ps(hST hT E +hST HTΘΘHΘE) 2 nk =Ps|hST |2|hT E |2 dχ ST dχ T Eσ2 R + Ps|hST |2PN n=1hTΘn hΘEn ej(θn−αn−ϵn) 2 dχ ST dχ TΘdχ ΘRσ2 E (22) =¯γE1|hST |2|hT R|2+¯γE2|hST |2 N X n=1 hTΘn hΘRnej(θn−αn−ϵn)  2 ,(23) where ¯γE1and ¯γE2are the average SNR at Eve due to the direct and the RIS-aided links, respectively. By re-writing (23) as γE=γE1+γE2,fγE1(γE1)can be shown as follows [62]. fγE1(γE1) = η2 γE1 G2,2 2,2 δ2γE1 ¯γE1 1−msST ,1−msT E mST , mT E !,(24) where η2=1 Γ(mST )Γ(mSST )Γ(mT E )Γ(mST E )and δ2= mST mT E (mSST −1)(mST E −1) . As mentioned before, since Eve is a passive eavesdropper and the knowledge about its CSI is imperfect, and the equivalent channel reflected by RIS exhibits similarities to Rayleigh fading [59]–[61]. Thus, we can use the same analysis provided in Thm. 2 to obtain fγE2(γE2)as fγE2(γE2) = C a¯γE2 G1,2 2,1aλ1γE2 1,2−mST 1 + mSST .(25) Now, by using the MGF of γE1and γE2and considering (23) and (24), we can derive fγE(γE)as the following theorem. This article has been accepted for publication in IEEE Transactions on Vehicular Technology. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TVT.2025.3612485 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ 7 fγR(γR)=Gη1γR−2H0,0:2,3;1,4 1,0:3,2;4,1  δ1 ¯γR1γR 4λ1¯y−2 d 1 γR (−1; 1,1) : (1,1),(1 −mSST ,1),(1 −mST R ,1); (0,1),(2−c 2,1),(3−c 2,1),(2 −mST ,1) –––– : (mST ,1),(mT R,1); (1 + mSST ,1)  .(20) FγR(γR)=Gη1γR−3H0,0:2,3;1,4 1,0:3,2;4,1  δ1 ¯γR1γR 4λ1¯y−2 d 1 γR (−2; 1,1) : (1,1),(1 −mSST ,1),(1 −mST R ,1); (0,1),(2−c 2,1),(3−c 2,1),(2 −mST ,1) –––– : (mST ,1),(mT R,1); (1 + mSST ,1)  .(21) fγE(γE)= η2C ¯γE2aγE−2H0,0:2,3;1,3 1,0:3,2;3,1 δ2 ¯γE1γE aλ1 γE (−1; 1,1) : (1,1),(1 −mSST ,1),(1 −mST E ,1); (2,1),(1,1),(2 −mST ,1) –––– : (mST ,1),(mT E ,1); (1 + mSST ,1) !.(26) FγE(γE)= η2C ¯γE2aγE−3H0,0:2,3;1,3 1,0:3,2;3,1 δ2 ¯γE1γE aλ1 γE (−2; 1,1) : (1,1),(1 −mSST ,1),(1 −mST E ,1); (2,1),(1,1),(2 −mST ,1) –––– : (mST ,1),(mT E ,1); (1 + mSST ,1) !.(27) Theorem 4. Assuming all channels follow the FisherSnedecor Ffading distribution, the PDF and CDF of γEwith direct links can be obtained as (26) and (27), respectively. Proof. The proof is elaborated in Appendix D. V. SECRECY PERFORMANCE ANALYSIS In this section, we derive the analytical expressions of ASC and SOP for RIS-aided BC, exploiting the distributions obtained in the previous section. Secrecy capacity (SC) refers to the highest possible transmission rate at which information can be sent over the BC channels while ensuring that the transmitted information remains confidential. Thus, SC can be expressed as Cs(γR, γE) = hCR−CEi+ ,(28) where CR= log2(1 + γR)and CE= log2(1 + γE)denote the wireless channel capacity between the tag and reader, and the tag and Eve, respectively. A. Without Direct Links 1) ASC Analysis: The randomness of the SNRs, caused by varying BC channel conditions such as fading and shadowing, directly impacts the SC, making it inherently stochastic. To comprehensively evaluate the system performance, it is essential to compute the expectation of the achievable secrecy rate over all possible channel realizations. The ASC quantifies this average SC across diverse channel conditions and serves as a fundamental metric for assessing PLS performance. Referring to (28), for a complex AWGN wiretap channel, the SC is defined as the difference between the capacities of the main channel and the eavesdropper channel, particularly when Eve’s channel is degraded by higher noise levels compared to the main channel. Utilizing this definition, along with the derived PDFs and CDFs in Thms. 1 and 2, the ASC can be mathematically expressed as ¯ Cs ∆ =Z∞ 0Z∞ 0 Cs(γR, γE)fγR(γR)fγE(γE)dγRdγE.(29) Therefore, the ASC for the considered RIS-aided BC is derived in the following theorem. Theorem 5. The ASC for the considered RIS-aided BC system without direct links under Fisher-Snedecor Ffading channels is given by (30), where A=GC a¯γEln(2) . Proof. The proof is elaborated in Appendix E. 2) SOP Analysis: SOP is an important analytical measure used to assess the performance of PLS, which quantifies the probability that SC falls below a specific positive secrecy rate threshold, say Rs>0, i.e., Psop = Pr (Cs≤Rs).(31) Now, by inserting (28) into SOP definition we have Psop = Pr ln 1 + γR 1 + γE≤Rs(32) =Z∞ 0 FγR(γt)fγE(γE)dγE,(33) in which γt= (1+γE)eRs−1 = γEeRs+eRs−1 = γERt+ R′tis the SNR threshold. Theorem 6. The SOP for the considered RIS-aided BC system without direct links under Fisher-Snedecor Ffading channels is given by (34). Proof. The proof is elaborated in Appendix F. B. With Direct Links 1) ASC Analysis: By utilizing the ASC definition in (28) and the obtained PDFs and CDFs in Thms. 3 and 4, the ASC for RIS-aided BC with considering the direct links can be derived as the following theorem. Theorem 7. The ASC for RIS-aided BC with direct links under Fisher-Snedecor Ffading channels is given by (35), where A′=η1η2GC a¯γE2ln(2) ,λ1= (−4; 1,1,1,1),(−3; 1,1,1,1), λ2= (1,1),(1 −mSST ,1),(1 −mST R,1), λ3= (0,1),(2−c 2,1),(3−c 2,1),(2 −mST ,1),λ4= This article has been accepted for publication in IEEE Transactions on Vehicular Technology. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TVT.2025.3612485 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ 8 ¯ Cs=AH2,1:1,3;1,3 2,2:3,2;3,2  aλ1 4λ1 ¯y 2 d 1 (−2; 1,1),(−1; 1,1) : (0,1),(1,1),(2 −mST ,1); (2−c 2,1),(3−c 2,1),(2 −mST ,1) (−2; 1,1),(−2; 1,1) : (1 + mSST ,1),(−1,1); (1 + mSST ,1)   +AH2,1:1,4;1,2 2,2:4,2;2,1  4λ1 ¯y 2 d 1 aλ1 (−2; 1,1),(−1; 1,1) : (0,1),(2−c 2,1),(3−c 2,1),(2 −mST ,1); (1,1),(2 −mST ,1) (−2; 1,1),(−2; 1,1) : (1 + mSST ,1),(−1,1); (1 + mSST ,1)   −A GG3,3 4,3aλ1 1,2−mST ,−1,0 1 + mSST ,−1,−1.(30) Psop =GR′2 tC a¯γE2Rt H0,1:4,1;3,1 1,0:2,5;1,3  ¯y 2 d 1 4λ1R′ t Rt aλ1R′ t (3; 1,1) : (1,1),(−mSST ,1),(2,1); (−mSST ,1) –––– : (1,1),(c 2,1),(c−1 2,1),(mST ,1),(2,1); (0,1),(mST −1,1),(1,1) .(34) ¯ Cs=A′H0,1:2,3;1,4;2,3;1,3 2,2:3,2;4,1;3,2;3,1 δ1 ¯γR1 ,4λ1¯y−2 d 1,δ2 ¯γE1 , aλ1 Λ1: Λ2; Λ3; Λ4; Λ5 Π1: Π2; Π3; Π4; Π5! +A′H0,2:2,3;1,3;2,3;1,4 3,1:3,2;3,1;3,2;4,1 δ2 ¯γE1 , aλ1,δ1 ¯γR1 ,4λ1¯y−2 d 1 Λ′ 1: Λ4; Λ5; Λ2; Λ3 Π′ 1: Π4; Π5; Π2; Π3!+A′′H0,1:2,3;1,3 3,2:3,2;3,1 δ2 ¯γE1 , aλ1 τ1:τ2;τ3 ν1:ν2;ν3!.(35) Psop =PH0,0:2,3;1,4;2,3;1,3 0,1:3,2;4,1;3,2;3,1 δ1 ¯γR1R′ t,4λ1 ¯y 2 d 1R′ t ,δ2Rt ¯γE1 , aλ1Rt –– : Ξ1; Ξ2; Ξ3; Ξ4 Υ1: Υ2; Υ3; Υ4; Υ5!.(36) (0,1),(1 −mSST ,1),(1 −mST E ,1),λ5= (0,1),(1,1),(2 − mST ,1),Π1= (−3; 1,1,1,1),(−3; 1,1,1,1), Π2= (mST ,1),(mT R,1),Π3= (1 + mSST ,1), Π4= (mST ,1),(mT E ,1),Π5= (1 + mSST ,1), λ′ 1= (−3; 1,1,1,1),(−3; 1,1,1,1),(−3; 1,1,1,1), Π′ 1= (−4; 1,1,1,1),A′′ =η2C a¯γE2ln(2) ,τ1= (−1; 1,1),(−1; 1,1),(0; 1,1),τ2= (1,1),(1 − mSST ,1),(1 −mST E ,1),τ3= (2,1),(1,1),(2 −mST ,1), ν1= (0; 1,1),(0; 1,1),ν2= (mST ,1),(mT E ,1), and ν3= (1 + mSST ,1). Proof. The proof is elaborated in Appendix G. 2) SOP Analysis: By utilizing the SOP definition in (33) and the obtained PDFs and CDFs in Thms. 3 and 4, the SOP can be obtained as the following theorem. Theorem 8. The SOP for the considered RIS-aided BC system with direct links under Fisher-Snedecor Ffading channels is given by (36), where P=η1η2GCRt a¯γE2R′4 t ,Ξ1= (1,1),(1 − mSST ,1),(1 −mST R,1),Ξ2= (0,1),(2−c 2,1),(3−c 2,1),(2 − mST ,1),Ξ3= (1,1),(1 −mSST ,1),(1 −mST E ,1),Ξ4= (0,1),(1,1),(2−mST ,1),(1+mSST ,1),Υ1= (−3; 1,1,1,1), Υ2= (mST ,1),(mT R,1),Υ3= (1 + mSST ,1),Υ4= (mST ,1),(mT E ,1), and Υ5= (1 + mSST ,1). Proof. The proof is elaborated in Appendix H. Remark 1. The ASC and SOP behavior of RIS-aided BC systems under Fisher-Snedecor Ffading channels are accurately modeled as shown in Thms. 5–8. The intricacies of these theorems reveal how different parameters intricately influence the system’s secrecy performance. Thms. 5 and 6 demonstrate that the ASC and SOP in the absence of direct links are primarily governed by the parameters within the Fox’s Hfunction expressions, offering a nuanced understanding of how they modulate secrecy performance. These parameters include the number of RIS reflecting elements, the distance of every entity from the RIS, average SNR at Eve (¯γE2), and the fading and shadowing parameters (mST , and mSST ), which dictate system susceptibility to channel variability. A higher average SNR at Eve or more severe fading conditions tend to degrade the secrecy performance, while a larger number Nor more favorable links at the reader lead to reduced SOP and increased ASC, as evidenced by the derived expressions. The inclusion of direct links in Thms. 7 and 8 adds another layer of complexity. While the direct links provide an additional pathway for signal transmission, our analysis indicates that their contribution to enhancing secrecy performance is relatively minor compared to the significant role played by the RIS. This is particularly evident in the Fox’s H-function expressions of Thms. 7 and 8, where the direct link parameters subtly alter the overall secrecy metrics. The mathematical transition from Thms. 5 and 6 to Thms. 7 and 8, though marked by an increase in complexity, underscore the dominant influence of RIS in shaping the secrecy performance of BC systems. VI. ASYMPTOTIC ANALYSIS OF SECRECY METRICS In this section, given the importance of the secrecy metrics performance in the high SNR regime, we evaluate the asymptotic behaviour of both SOP and ASC by exploiting the residue approach [37]. This article has been accepted for publication in IEEE Transactions on Vehicular Technology. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TVT.2025.3612485 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ 9 A. Asymptotic ASC Since the exact analytical expression of ASC in first and second trms of (30) is in terms of the bivariate Fox’ Hfunction, we can derive the asymptotic behavior of the ASC at the high SNR regime (i.e., ¯γR→ ∞) by using the expansion of the bivariate Fox’s H-function. To do this, we need to evaluate the residue of the corresponding integrands at the closest poles to the contour, namely, the minimum pole on the right for large Fox’s H-function arguments and the maximum pole on the left for small ones. Hence, the asymptotic ASC can be determined according to the following proposition. Proposition 1. The asymptotic ASC (i.e., ¯γR→ ∞) for the considered RIS-aided BC system under Fisher-Snedecor F fading channels is given by ¯ Casy s=G1G4,4 4,5 ¯y 2 d 1a 4 0,1,2−mST ,−2−mSST 1 + mSST ,c−4 2,c−5 2,−3 + mST ,−1! +G2G3,5 5,4 4 ¯y 2 d 1a −2−mSST ,0,2−c 2,3−c 2,2−mST 1 + mSST ,−2,−3 + mST ,−1! −A GG3,3 4,3 aλ1 1,2−mST ,−1,0 1 + mSST ,−1,−1!,(37) where G1=GC¯y 4 d 1 16aλ2 1¯γEand G2=GC a3λ2 1¯γE. Proof. The proof is elaborated in Appendix I. B. Asymptotic SOP With same strategy, we can derive the asymptotic behavior of the SOP at the high SNR regime (i.e., ¯γR→ ∞) by using the expansion of the bivariate Fox’s H-function. Hence, the asymptotic SOP can be determined as follows. Proposition 2. The asymptotic SOP (i.e., ¯γR→ ∞) for the considered RIS-aided BC system under Fisher-Snedecor F fading channels is given by Pasy sop =GCRt a3λ2 1¯γE2 G5,2 3,5¯y 2 d 1a 4Rt−mSST , mST + 4,2 c 2,c−1 2, mST ,1, mSST + 3.(38) Proof. The proof is elaborated in Appendix J. Remark 2. The asymptotic analysis of ASC and SOP in high SNR regimes in Props. 1 and 2 provides essential insights into the secrecy performance of RIS-aided BC systems under Fisher-Snedecor Ffading channels. By leveraging the residue approach to expand the bivariate Fox’s H-function, we accurately capture the behavior of ASC and SOP as the system approaches high SNR limits. The derived expressions in (37) and (38) demonstrate the role of the RIS elements (N) and environmental factors on secrecy metrics in highSNR scenarios. Specifically, the presence of terms involving the average SNRs at the reader and Eve, G1, and G2in (37) for ASC, and the threshold secrecy rate, G, and Cin (38) for SOP, reflect how different system parameters like Nand various distances enhance the signal reflection towards the reader while suppressing leakage toward Eve, validating the Eve (47,(30) (53,(60) X Y Reader (72,(54) 🔵🔵 🔵 🖲🖲 (0,0) 🔲 Tag Source (–15,(0) Fig. 2. The simulation setup. scalability of RIS-aided systems in high-SNR regimes. Furthermore, the presence of mSST and mST within the Meijer’s G-function arguments show how channel conditions impact secrecy, particularly under severe fading environments. These asymptotic results emphasize the importance of accounting for multipath fading and shadowing in real-world RIS-aided BC deployments to maintain optimal secrecy performance. VII. SIMULATION RESULTS In this section, we validate the theoretical expressions of the derived ASC and SOP for RIS-aided BC through Monte-Carlo simulations. We conduct simulations for various BC scenarios, including cases with only a direct link (BC without RIS), only RIS-aided links, and both direct and RIS-aided links. We also evaluate the PLS performance of RIS-aided BC based on different system parameters. A. Simulation Setup The simulation setup is visually illustrated in Fig. 2. In this setup, the tag remains stationary at the coordinates (0,0), transmitting its confidential information to the reader by modulating and reflecting the RF signal emitted from the source, which is coordinated at (−15,0). The RIS is placed at coordinates (53,60) to enhance the received SNR at the reader, thereby improving the PLS performance of the BC system. Eve and the reader are located at coordinates (47,30) and (72,54), respectively. Notably, in order to represent a worst-case scenario for security analysis and have a rigorous evaluation of the system’s secrecy performance, the reader is positioned farther away from the tag compared to Eve. Unless otherwise specified, the simulation adopts fixed inter-node distances as follows: dST = 15 m,dT R = 90 m,dTΘ= 80 m, dΘR= 20 m,dT E = 60 m, and dΘE= 30 m. These values are selected to reflect a realistic deployment scenario in low-power long-range (LoRa) BC systems [63]. Additionally, the simulation assumes the secrecy rate threshold Rs= 1 bps/Hz, noise powers σ2 R=−70 dBm and σ2 E=−60 dBm, source transmit power Ps= 10 dBm, and path-loss exponent χ= 2.5. Monte Carlo simulations are carried out under these conditions with 106channel realizations per scenario to ensure statistical accuracy. The simulation results validate This article has been accepted for publication in IEEE Transactions on Vehicular Technology. This is the author's version which has not been fully edited and content may change prior to final publication. Citation information: DOI 10.1109/TVT.2025.3612485 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/