Physical Layer Security Enhancement With Reconfigurable Intelligent Surface-Aided Networks

Let us consider the DL MIMO-RIS system and assume that a BS equipped with $M$ transmit antennas (TAs) communicates with $N$ users each of which is equipped with $K$ receive antennas (RAs). a RIS is installed between the BS and the users for assisting end-to-end communication. By jointly optimizing the transmit precoding at the BS and the reflect phase shifts at the RIS according to the wireless channel conditions, the reflected signals can be added constructively at the desired receiver to enhance the received signal power, which can potentially yield a secure transmission [22]. In addition, a malicious eavesdropper who is equipped with $K$ RAs aims to eavesdrop the information of the desired user. We assume that the number of reflecting elements on the RIS is $L$. The RIS is equipped with a controller to coordinate between the BS and the RIS for both channel acquisition and data transmission [3]. As such, the signal received from the BS can be manipulated by the RIS via adjusting the phase shifts and amplitude coefficients of the RIS elements.

On the other hand, various practical approaches have been proposed in the literature for the channel estimation of RIS-aided links [23, 24, 25]. Thus, we can assume that the global channel state information (CSI) of users is perfectly known for joint design of beamforming [26]. The CSI of the eavesdropper is typically unavailable as the eavesdropper is passive to hide its existence. Therefore, the passive eavesdropping scenario is considered [27]. To quantify the performance of such a RIS-aided communication system, we adopt SOP, PNSC, and ASR as performance metrics of interested.

In particularly, we assume that the users are located on a disc with a radius $r_{2}$ according to homogeneous Poisson point processes (HPPP) [28] with density $\lambda$ and the eavesdropper will try to choose a position that is close to the legitimate user.

In practice, the direct transmission link between the BS and the users may be blocked by trees or buildings. Such assumption is applicable for 5G and mmWave communication systems that are known to suffer from high path and penetration losses resulting in signal blockages. In order to unveil the benefits in adopting the RIS, we consider the following two practice of communication scenarios:

• •

  As shown in Fig. 1 (a), for the first scenario, RIS is not adopted for enhancing the PLS and the quality of wireless communication. Specifically, both LoS and NLoS links exist in our system. Without loss of generality, we focus our attention on user $n$. We use $d_{\upsilon,\sigma}$ to denote the distance between the BS and user $n$ $(n=1,\cdots,N)$. According to Table I, we have $\upsilon=u_{n}$, $\sigma={\rm L}$ for LoS links and $\upsilon=u_{n}$, $\sigma={\rm NL}$ for NLoS links.

• •

  For the second scenario, a RIS is deployed to leverage the LoS components with respect to both the BS and the users to assist their end-to-end communication of Fig. 1 (b). Thus, the NLoS users in scenario 1 now can communicate with BS through two LoS links with the help of RIS. Moreover, the locations of the BS and the RIS are fixed, hence we assume that the distance between the BS and the RIS is known and denoted by $d_{u_{R},{\rm L}}$. In addition, the distance between the RIS and user $n$ is denoted by $d_{u_{R}-u_{n},{\rm L}}$.

A blockage model was proposed in [29], which can be regarded as an accurate approximation of the statistical blockage model [30] and incorporates the LoS ball model proposed in [31] as a special case. In the considered system model, we adopt the blockage model to divide the users process in the spherical region around the BS into two independent HPPPs: LoS users process and NLoS users process. In particular, we define ${q_{L}}(r)$ as the probability that a link of length $r$ is LoS. Each access link of separation $r$ is assumed to be LoS with probability $B_{1}$ if $r\leq r_{1}$ and $0$ otherwise:

where $0\leq B_{1}\leq 1$. The parameter $B_{1}$ can be interpreted as the average LoS area in a circular ball with a radius of $r_{1}$ around the BS.

Let us assume that the users are located according to a HPPP within the disc shown as Fig. 1. The PDF of the user locations have been derived as [32, eq. (30)]. As a result, the CDF of the user locations is given by

where $r_{0}$ is the minimum distance which is used to avoid encountering a singularity [33]. Thus, the probability of the distance between the user and the BS is less than $r_{1}$ can be expressed as

For the first communication scenario without a RIS, different path loss equations are applied to model the LoS and NLoS links as [29, 31]

where ${\alpha_{1}}$ and ${\alpha_{2}}$ are the LoS and NLoS path loss exponents, respectively. Typical values of ${\alpha_{1}}$ and ${\alpha_{2}}$ are defined in [34, Table 1], while ${\alpha_{1}}<{\alpha_{2}}$ hold in general.

For the second communication scenario with the assistance by the RIS, recently, the free-space path loss models of RIS-aided wireless communications are developed for different situations in [35, Proposition 1]. Authors in [35] proposed that the free-space path loss of RIS-aided communications is proportional to ${(d_{u_{R},{\rm L}}d_{u_{R}-u_{n},{\rm L}})}^{2}$ in the far field case. Thus, we obtain

where $C_{L_{1}}$ and $C_{L_{2}}$ denote the path loss intercepts of BS-RIS and RIS-user links, respectively.

To unify the system performance with different channel environment, generalized fading distributions have been proposed that include the most common fading distributions as special cases [36, 37, 8]. It has been shown recently that the Fisher-Snedecor $\mathcal{F}$ composite fading model can provide a more comprehensive modeling and characterization of the simultaneous occurrence of multipath fading and shadowing, which is generally more accurate than most other generalized fading models [38]. Furthermore, the Fisher-Snedecor $\mathcal{F}$ model is more general and includes several fading distributions as special cases, while being more mathematically tractable. For example, authors in [32] conceived a RIS-aided MIMO framework using Nakagami-$m$ distribution. Note that Fisher-Snedecor $\mathcal{F}$ distribution includes the case of Nakagami-$m$ distribution for $m_{s}\to\infty$ as one of its subsequent special cases, such as Rayleigh $(m=1)$ and one-sided Gaussian $(m=1/2)$. Motivated by this, the Fisher-Snedecor $\mathcal{F}$ distribution has been adopted to analyze the RIS-aided systems, i.e., RIS-aided Internet-of-Things networks [39].

To conceive a practical RIS framework, we assume that the small-scale fading of each link follows Fisher-Snedecor $\mathcal{F}$ fading distributions [39]. The channel correlation between RIS elements may exist because the electrical size of RIS’s reflecting elements is between $\lambda/8$ and $\lambda/4$ in principle, where $\lambda$ is a wavelength of the signal [40]. However, it is hard to model the correlation in RIS because a tractable model for capturing such unique characteristics has not been reported in the literature yet. Therefore, we consider a scenario where the correlation is weak enough to be ignored, i.e., the electrical size of RIS’s reflecting elements is larger than $\lambda/4$ because the frequency of signals is high [41], [42]. Thus, we can assume that the small scale fading components are independent of each other.

For the first communication scenario, in order to characterize the LoS and NLoS links between the BS and users (eavesdropper), the small-scale fading matrices are defined as

where ${{{\bf Q}}_{\upsilon,\sigma}}$ $(\sigma={\rm L},{\rm NL})$ is a $K\times M$ matrix. Letting $\upsilon=u_{n}$, we obtain the small-scale fading matrix between the BS and user $n$. Using [43, Eq. (5)], the PDF of the elements of (6) can be expressed as

where $\Omega=\mathbb{E}\left[r^{2}\right]$ is the mean power, $m$ and $m_{s}$ are physical parameters which represent the fading severity and shadowing parameters, respectively.

For the second case, the small-scale fading matrices for the LoS links between the BS and users (eavesdropper) are same as the first case. In addition, in order to model the LoS links between the BS and the RIS, the small-scale fading matrix is defined as

where ${{{\bf Q}}_{u_{R},{\rm L}}}$ is a $L\times M$ matrix.

In order to model the LoS links between the RIS and users (eavesdropper), the small-scale fading matrices are defined as

where ${{{\bf Q}}_{\upsilon,{\rm L}}}$ is a $K\times L$ matrix. Letting $\upsilon=u_{R}-u_{n}$, we get the small-scale fading matrix between the RIS and user $n$.

Highly directional beamforming antenna arrays are deployed at the BS to compensate the significant path-loss in the considered system. For mathematical tractability and similar to [29, 30], the antenna pattern of users can be approximated by a sectored antenna model in [44] which is given by

where ${\theta}$, distributed in $\left[{0,2\pi}\right]$, is the angle between the BS and the user, $\theta_{c}$ denotes the beamwidth of the main lobe, for each TA, ${{{G}_{\upsilon,\kappa}}}$ and ${{{g}_{\upsilon,\kappa}}}$ are respectively the array gains of main and sidelobes. In practice, the BS can adjust their antennas according to the CSI. In the following, we denote the boresight direction of the antennas as ${0^{\circ}}$. For simplifying the performance analysis, different antennas of the authorized users and the malicious eavesdropper are assumed to have same array gains. Thus, without loss of generality, we assume that ${\cal G}_{\upsilon,\kappa}={\cal G}_{\upsilon}$, ${G}_{\upsilon,\kappa}={G}_{\upsilon}$ and ${g}_{\upsilon,\kappa}={g}_{\upsilon}$.

Let us consider a composite channel model of large-scale and small-scale fading. It is assumed that the distance ${d_{\upsilon,\sigma}}$, $\upsilon=u_{1},\cdots,u_{N},u_{e},$ and $\sigma={\rm L},{\rm NL}$ are independent but not identically distributed (i.n.i.d.) and the large-scale fading is represented by the path loss. In the DL transmission, the complex baseband transmitted signal at the BS can be then expressed as

where ${s_{u_{n}}}$ $(n=1,\cdots,N)$ are i.i.d. random variables (RVs) with zero mean and unit variance, denoting the information-bearing symbols of users, and ${\bf{p}}_{u_{n}}\in{\mathbb{C}}^{M\times 1}$ is the corresponding beamforming vector. The transmit power consumed at the BS can be expressed as

For the first communication scenario, e.g., Fig. 1 (a), there are both LoS and NLoS links. The signal received by user $n$ or the eavesdropper from the BS can be expressed as

where $\upsilon=u_{n},u_{e}$, $\sigma={\rm L},{\rm NL}$, ${\bf{v}}_{u_{n}}\in{\mathbb{C}}^{1\times K}$ is the detection vector of user $n$, and ${\bf N}_{0}\in{\mathbb{C}}^{K\times 1}$ denotes the additive white Gaussian noise, which is modeled as a realization of a zero-mean complex circularly symmetric Gaussian variable with variance $\sigma_{N}^{2}$.

For the second communication scenario, e.g., Fig. 1 (b), there are LoS and RIS-aided links. For LoS links, the signal received by the user or the eavesdropper can be obtained by letting $\sigma={\rm L}$ in (II-G). For RIS-aided links, the signal can be expressed as

where $\upsilon=u_{n},u_{e}$, ${\bf{\Phi}}\buildrel\Delta\over{=}{\mathop{\rm diag}\nolimits}\left[{{\beta_{1}}{\phi_{1}},{\beta_{2}}{\phi_{2}},\cdots,{\beta_{L}}{\phi_{L}}}\right]$ is a diagonal matrix accounting for the effective phase shift introduced by all the elements of the RIS ¹¹1We assume that the phase shifts can be continuously varied in $[0,2\pi)$ to characterize the fundamental performance of RIS. Although using the RIS with discrete phase shifts causes performance loss [45], our results serve as performance upper bounds for the considered system which are still useful for characterizing the fundamental performance of the RIS-aided system., ${\beta_{\ell}}\in(0,1]$ represents the amplitude reflection coefficient, ${\phi_{\ell}}=\exp\left({j{\theta_{\ell}}}\right)$, $n=1,\cdots,L$, ${\theta_{\ell}}\in[0,2\pi),$ denotes the phase shift.

In the literature, numerous methods have been proposed to obtain the effective phase shift introduced by all the elements of the RIS, e.g., [3, 46, 47, 48, 32]. To mitigate the interference at the $n_{\rm th}$ user and perform well both in RIS and non-RIS scenarios, we adopt the design of passive beamforming process and the downlink user-detection vectors proposed in [32]. The passive beamforming matrix at RIS can be obtained as [32, eq. (13)]

where $\beta_{\max}$ is obtained by finding the maximum amplitude coefficient, ${\bf{X}}$ is the signal sent by BS, and ${{{\bf{\tilde{H}}}}}$ can be formulated from ${{\bf{Q}}_{u_{R}-u_{1},{\rm L}}}{\bf{\Phi}}{{\bf{Q}}_{u_{R},{\rm L}}}$, $\cdots$, ${{\bf{Q}}_{u_{R}-u_{N},{\rm L}}}{\bf{\Phi}}{{\bf{Q}}_{u_{R},{\rm L}}}$ as [32, eq. (9)].

The detection vector of the $n_{\rm th}$ user can be written as [32, eq. (19)]

where ${{\bf{T}}_{{u_{n}}}}$ is derived by the left singular vectors of ${{{\bf{\tilde{H}}}}}$. With the aid of the classic maximal ratio combining technique, ${{\bf{x}}_{{u_{n}}}}$ can be expressed as

where ${{\bf{h}}_{{u_{n}}}}$ is the ${u_{n}}_{\rm th}$ column of the effective channel matrix ${{\bf{Q}}_{u_{R}-u_{n},{\rm L}}}{\bf{\Phi}}{{\bf{Q}}_{u_{R},{\rm L}}}$. Thus, the SINRs involved in both scenarios can be obtained.

For the first case, the received SINR at user $n$ and the eavesdropper are denoted as ${\rm SINR}_{\upsilon,\sigma}$, and it can be expressed as²²2To facilitate effective eavesdropping, the eavesdropper should try to close to BS because the SINR of the eavesdropper increases as $d_{u_{e},\sigma}$ decreases.

where $p_{u_{n}}$ denotes the transmit power of the BS for user, $Q\triangleq K-M-1$ is the effective antenna gain, and

is a $Q\times M$ matrix which denotes the channel gain of user $n$. In addition, we can see that the eavesdropper’s SINR is affected by the random directivity gain ${G_{{u_{e}}}(\theta)}$.

For the second case, the SINR of LoS path can be obtained by letting $\sigma={\rm L}$ in (18). For the RIS-aided path, with the help of the design of passive beamforming process [32], the SINR with the optimal phase shift design of RIS’s reflector array can be derived as [32, eq. (29)]

where $\upsilon=u_{R}-u_{n},u_{R}-u_{e}$, and

is a $Q\times L$ matrix.

With the help of (18) and (20), we derive the exact PDF and CDF expressions for each scenario’s SINR in terms of multivariate Fox’s $H$-function [49, eq. (A-1)], which are summarized in the following Theorems.

Then, we focus on the derivation for the performance metric in the second scenario.

The OP is defined as the probability that the instantaneous SINR is less than $Z_{\rm th}$, where $Z_{\rm th}$ is the determined SNR threshold. In the first communication scenario, the OP can be directly calculated as

which can be evaluated directly with the help of (22).

The secrecy outage probability (SOP), is defined as the probability that the instantaneous secrecy capacity falls below a target secrecy rate threshold. In the first communication scenario, the SOP can be written as