Secure RIS-Assisted Hybrid Beamforming Design With Low-Resolution Phase Shifters

The low-resolution reality of the hardware elements associated with massive mmWave antenna or reflector arrays is associated with the performance degradation of the wireless link when it is not properly controlled. In particular, the unintended angular radiations of the transmission or reflection arrays (e.g., transmission in non-intended directions) would invalidate the usual assumptions of information secrecy, even with perfect channel state information (CSI) knowledge at the transmitter, in the presence of low-resolution hardware. In this paper, we study a hybrid beamforming design for reconfigurable intelligent surface (RIS)-assisted multi-user multiple-input multiple-output (MU-MIMO) downlink (DL) communication, from the prospect of information secrecy maximization, wherein the array element phase rotations belong to the known discrete space. To address the NP-hard and non-convex nature of the problem at hand, we propose an iterative procedure by re-structuring the obtained discrete-domain problem into a tractable form which solves the problem numerically and guarantees the convergence to a stationary point. Further, we confirm the accuracy of the proposed optimization algorithm by an exhaustive search method based on graphical simulations. The minimal performance disparity that exists between the proposed algorithm and the considered digital beamforming (DBF) scheme as the upper bound validates the hybrid beamforming design. Moreover, the proposed work highlights the superiority of discrete-aware design over various existing baseline schemes, demonstrating the significant gains attainable by adopting discrete space design from the outset. Additionally, the proposed solution discusses the improvement in secrecy system performance by deploying RIS with an increased number of reflecting elements and thereby restricting the effect of eavesdroppers on secure communication.


I. INTRODUCTION
O VER the past decade, the exponential rise of data traffic and the quick spread of wireless devices has necessitated the investigation of promising next-generation wireless communication frameworks [1].The massive increase in mobile traffic has greatly worsened spectrum congestion in the predefined frequency bands, which has sparked a new paradigm shift for spectral-efficient wireless communication architecture.Primarily, secure communication along with ultra-high data rates has become even more crucial due to the potential integration of various sensitive applications, such as autonomous driving, telemedicine, and critical infrastructure monitoring, which require high reliability and security [2].Securing wireless communication in 6G networks is a challenging task that requires innovative and advanced physical-layer design to ensure reliable and secure data transmission.Nevertheless, the upsurge in S. Pala, M. Katwe, K. Singh, and C.-P. Li are with the Institute of Communications Engineering, National Sun Yat-sen University, Kaohsiung 804, Taiwan (email: sony.pj12@gmail.com,mayurkatwe@gmail.com,keshav.singh@mail.nsysu.edu.tw,cpli@faculty.nsysu.edu.tw).
Anke Schmeink is with the Chair of Information Theory and Data Analytics at RWTH Aachen University, Germany (Email: anke.schmeink@inda.rwthaachen.de).mobile devices and dynamic channel conditions induce significant challenges in providing end-to-end security and mitigating the risks of eavesdropping.Moreover, conventional encryption and cryptography may not satisfy the unprecedented shift in security and rate requirements of communication services for worse channel conditions and massive user connectivity [3]- [5].Overall, securing wireless communication in 6G networks is an ongoing research topic that will continue to attract significant interest and offer exciting research opportunities in the future.
Apparently, multiple input multiple output (MIMO) has emerged as a crucial technology for massive wireless communication [6]- [8].In general, MIMO systems enhance the spectral efficiency and security of wireless communication in the presence of eavesdroppers via a beamforming design that enables the signal to be transmitted in a specific direction, thus reducing the chances of interception by an eavesdropper.Additionally, beamforming techniques can provide a directional and robust signal transmission that minimizes the impact of interference and fading.However, the implementation of conventional digital precoding renders impractical constraints as it necessitates as many RF chains as there as antennas, which demands increased power consumption and enormous cost.To tackle the limitations of conventional digital precoding, hybrid beamforming techniques have been proposed which splits beamforming into analog and digital domains, which significantly reduces power consumption and cost while maintaining the advantages of beamforming for secure communication [9], [10].The primary goal of the hybrid beamforming is to minimize the use of RF chains and digital-to-analog converters (DACs) to realize low dimensional baseband digital precoders followed by the high dimensional analog precoders which solely incorporate a larger number of phase shifters for providing the flexibility needed to carry out advanced multiplexing techniques [11].Hybrid beamforming offers the flexibility to implement advanced multiplexing techniques, making it a promising solution for achieving high spectral efficiency and secure communication in 6G networks.
Besides, cutting-edge technology called reconfigurable intelligent surfaces (RISs) have gained a lot of interest in the research community and academia due to their improved coverage capabilities and excellent system performance, especially in the multi-users worst channel conditions [12].In particular, RIS is a two-dimensional meta-surface comprised of a plethora of lowcost passive reflection elements each of which may individually cause the incoming signal to undergo a programmable phase shift, modifying the reflected signal's propagation direction [13]- [16].RISs have been put forth as a novel cost-effective and low-complexity technique to reconfigure the wireless propagation conditions in real-time, therefore considerably enhancing the performance of upcoming communication systems [17], [18].By adjusting the reflection phase of these elements, RIS can enhance the signal-to-noise ratio (SNR) of legitimate users and degrade the SNR of eavesdroppers, thus improving the security of wireless communication.Moreover, RIS also mitigates fading and interference and increases coverage and capacity in wireless networks which renders a secure and energy-efficient alternative to conventional relaying techniques.
A. Interplay of Hybrid Beamforming and RIS for Secure communication MIMO can compensate for extreme signal attenuation by producing narrow beams with significant beamforming gains, while RIS can offer additional aperture gains by adjustable signal reflection.Interestingly, the interplay of hybrid beamforming design and RIS has the potential to address several challenges associated with secure communication in 6G networks.
1) RIS can enhance signal strength and quality while hybrid beamforming design combines analog and digital beamforming (DBF) techniques, resulting in better user experience and reliable communication [19].
2) The interplay of hybrid beamforming and RIS can enhance spectral efficiency by mitigating interference and signal fading.Hybrid beamforming also enables stronger and more focused signal transmission for further improvement in spectral efficiency [20].
3) The use of hybrid beamforming techniques in combination with RIS technology can improve energy efficiency and reduce the number of required transmit antennas, which aligns with the demands of 6G networks for lower power consumption and reduced hardware complexity [21].Motivated by the aforementioned benefits of RIS-assisted communication and hybrid beamforming design, its detailed investigation for spectral-efficient secure communication is interesting and forms the prime motivation for this work.

B. Related Works and Motivations
Over the past years, several research works have been published on physical layer security and RIS-assisted communication networks with the prime motive of enhancing energy efficiency, spectral efficiency, coverage capability, and outage probability.Moreover, RIS-assisted communication systems have been studied for various system models [22]- [35].For instance, the authors in [22] proposed a joint active and passive beamforming design in order to minimize overall transmit power at the access point for a multi-user (MU)-multiple input single output (MISO) network.According to the authors of [23], who investigated the energy efficiency maximization problem, integrating RIS can enhance the energy efficiency of the system when compared to the system without RIS.Additionally, in a non-orthogonal multiple access (NOMA) downlink (DL) MISO cognitive radio network, a multi-objective optimization framework was developed in [24] to achieve a desired trade-off between spectral efficiency and energy efficiency.[25] investigates the ergodic capacity of RIS-assisted MU-MISO wireless systems with statistical CSI, proposing efficient solutions for power control and phase shift design.The investigation of robust transmission design with the influence of transceiver hardware impairments was carried out in a RIS-assisted MISO secure wireless communication system [26].Furthermore, there is a significant body of research focusing on RIS-assisted communication systems from the perspective of secure communication.For instance, authors in [27] investigated the optimization problem based on the Taylor series approximation for maximizing the secrecy rate and minimizing power in a single user/eavesdropper MIMO system.Later, in [28], the secrecy rate maximization problem was investigated for single-cell MISO networks under the constraint of minimum harvested energy.In addition, [29] followed an inexact block coordinate descent approach for solving the secrecy rate maximization problem in single-user MIMO simultaneous wireless information and power transfer systems.In [30], the secrecy throughput optimization problem based on the primal decomposition method is exploited for wireless-powered communication networks.Moreover, [31] conducts a comprehensive literature review on RIS-assisted physical layer security (PLS), discussing various applications, scenarios, and optimization methodologies.The authors in [32] focus on secure communication in RIS-aided MU massive MIMO systems, optimizing artificial noise power and RIS phase shifts.[33] proposes virtual partitioning of RIS elements to enhance physical layer security, optimizing secrecy capacity with rate constraints.Furthermore, [34] examines a RIS-assisted wireless secure communication system, optimizing active and passive beamforming for secrecy rate maximization.Furthermore, to maximize the secrecy rate and demonstrate the efficacy of system security via a RIS, authors in [35] have considered the optimization of active beamforming, passive beamforming, and the covariance matrix of artificial noise.
Interestingly, the authors in [21], proposed a two-stage algorithm based on manifold optimization for designing the RIS phase-shift matrix and hybrid beamforming by neglecting the direct channel between transmitter and receiver.For broadband MIMO systems with frequency selective channels, the authors in [36] developed a RIS phase-shift matrix and hybrid beamformer with an aim of minimizing the bit error rate (BER), which inevitably results in a reduction in spectral efficiency.Even though, the aforementioned works discussed the significant improvement in performance and reduction in complexity but failed to showcase the effect of the RIS phase-shift matrix on channels.But in order to effectively profit from such integration, it is required to jointly design the hybrid beamformer and RIS phase-shift matrix, which are prone to hardware limitations that must be carefully taken into consideration [17].Furthermore, it is worth noting that the considered hybrid precoder designs of the aforementioned works assume the infinite or high-resolution phase shifters for designing the analog precoders in order to achieve satisfactory performance.Assuming infinite or highresolution phase shifters would rapidly increase the complexity of the required hardware circuits and also energy consumption [37], [38].Since employing high-resolution phase shifters is impractical, low-resolution phase shifters will be used to construct analog beamformers.Further, to counteract the loss of beamforming precision caused by the low-resolution phase shifters, it is vital to explore signal-processing solutions for hybrid beamforming architectures.To the best of the author's knowledge, the methodology of optimizing hybrid active and passive beamforming with secrecy considerations remains an open problem, making our proposed approach of designing from the outset in the discrete space, specifically tailored for the secrecy rate expression, a novel and promising contribution to the field.

C. Contributions
In this work, we investigate the design and performance analysis of a secure joint hybrid transmission and reflection beamforming scheme in a multi-RIS-assisted MU-MIMO system.The primary contributions of this work can be summarized as follows: • Discrete-Aware Design for Information Secrecy: In our work, we propose a framework for inherently designing the phase rotations of RIS and radio frequency (RF) beamformers in the discrete space, tailored to RIS-assisted secrecy rate maximization problems.This approach systematically incorporates the inherent discreteness of phase shifter and reflector elements, enabling us to optimize the system parameters efficiently.Additionally, it is important to note that this approach differs from the studies of [19], [21] where the discrete variables are adjusted subsequently rather than in parallel.Moreover, the computational efficiency represents a significant advancement compared to existing approaches relying on serial/sequential updates of each phase rotation element.• Performance Comparison and Observations: The proposed algorithm is compared with two conventional schemes: the Continuous phase shifters [36] and the Discretized scheme [19], [38], where the design is done either entirely in continuous space or in continuous space and then discretized, respectively.Additionally, our proposed algorithm is compared with the conventional DBF scheme [34] to validate the effectiveness of the hybrid beamforming design.The results demonstrate that our proposed algorithm not only achieves adequate secrecy performance with fewer RF chains and reduced complexity compared to DBF but also outperforms the baseline Discretized scheme by providing substantial improvement in terms of secrecy performance.Furthermore, we also perform a comprehensive comparison between the proposed multi-RIS-assisted secure system and its non-secrecy counterpart [21] under various conditions, transmit power budget, RIS reflecting elements, RIS-free scheme [10], number of base station antennas, user distance from the base station and RISs, and varying numbers of eavesdroppers.Notably, even with the low-resolution assumption of phase rotations, deploying RISs with an increased number of reflecting elements significantly enhances secrecy performance and effectively mitigates the effect of eavesdropping.This dual evaluation provides valuable insights into the benefits of our proposed hybrid beamforming method for both secure and non-secure communication scenarios.Our work showcases the advantages of discrete-aware design compared to Discretized scheme and underscores the potential gains achieved when adopting discrete space design from the outset, making it a promising avenue for advancing RISbased physical layer security.

D. Structure of the Paper:
This paper is organized into five sections.Section II illustrates the system model.In Section III, we discuss problem formulation, outlining the key challenges and objectives.The proposed solution of the joint beamforming design with low-resolution phase shifters for a secure RIS-assisted MU-MIMO system is detailed in Section IV.The numerical evaluations are described in Section V. Finally, Section VI presents our conclusions and future works.

E. Mathematical Notations:
Throughout this paper, column vectors and matrices are denoted as lower-case and upper-case bold letters, respectively.The rank of a matrix, expectation, trace, transpose, conjugate, Hermitian transpose, determinant, and Euclidean norm are denoted by rank(•), respectively.Kronecker product is denoted by ⊗ and the identity matrix with dimension K is denoted as I  and vec(•) operator stacks the elements of a matrix into a vector, and (•) −1 represents the inverse of a matrix.The sets of real, real and positive, complex, natural, and the set {1 . . . } are respectively denoted by R, R + , C, N and F  .⌊A  ⌋  ∈F  denotes a tall matrix, obtained by stacking the matrices A  ,  ∈ F  .R  (X) returns the -th row of the matrix X. {  } denotes the set of   , ∀. [ ] denotes the -th row of the vector .The set of all positive semi-definite matrices is denoted by H . ⊥ represents statistical independence. ★ is the value of the variable  at optimality.

II. SYSTEM MODEL
We consider a multi-RIS-assisted DL communication system, where the BS is equipped with a hybrid analog-DBF scheme and communicates with multiple legitimate users in the presence of multiple illegitimate receivers (eavesdroppers).In particular, the BS is equipped with  BS transmit antennas and  C ≪  BS transmit RF chains, simultaneously serving  U DL users.Defining the set of RISs, users, and eavesdropper as where  R ,  U and  E represent the number of RIS segments, user nodes, as well as the eavesdroppers, respectively.Each RIS segment consists of an array of  R, ,  ∈ R reflecting elements which apply a tunable limited-resolution phase rotation on the reflecting electromagnetic wave.At the BS, each RF chain is connected to the transmit antenna array via limited resolution phase shifters following the hybrid fully-connected architecture [39] as shown in Fig. 1.The legitimate user terminals are denoted as UT  ,  ∈ U, the illegitimate/undesired receivers are denoted as Eve  ,  ∈ E. While the deployed RISs are represented as RIS  ,  ∈ R.

A. Channel Model
We assume a narrow-band block-fading propagation channel where the direct channel gain between the BS and -th user and BS and -th eavesdropper are denoted as h  ∈ C  BS and h E, ∈ C  BS , respectively.The channel gain between the BS and the th RIS segment is denoted as H R, ∈ C  R, ×  BS .The channel gain between the -th RIS segment and -th legitimate user and Channel gain between the BS and the -th user h E, ∈ C  BS Channel gain between the BS and the - Channel between the BS and the -th RIS segment g , ∈ C  R, Channel from the -th RIS segment to the -th legitimate user g E,, ∈ C  R, Channel from the -th RIS segment to the -th eavesdropper Baseband linear precoder for transmitting data symbol Transmit precoder for artificial noise -th eavesdropper are respectively denoted as g , ∈ C  R, and g E,, ∈ C  R, , respectively.Due to high free-space path loss, the mm-wave propagation environment is well characterized by a clustered channel model.We adopt the channel model in [40], and a general channel matrix L ∈ C   ×   with   receive and   transmit antennas is represented as where   is the number of scattering clusters,    is the number of rays per cluster.Here, the small-scale coefficient of the -th ray in the -th cluster would be given by    ∼ CN 0,   10 − PL 10 , where PL is the path loss and   is the fraction of power in -th cluster, which is given by , where  ′  is the power of the cluster .We assume that perfect knowledge of the channel estimation for all the links is available1 at the BS using the pilot signal transmission and at the RISs as the prior information [32], [41]- [46].
Moreover,    ∈ [0, 2) and    ∈ [0, 2) are the angle of arrivals and departures (AoAs/AoDs), respectively.The vectors a     ∈ C   and a     ∈ C   are the array response vectors at the receiver and the transmitter to the angular arrivals and departures, respectively, given as: where  is the wavelength and  is the antenna spacing, where we have assumed  = /2.

B. Signal Model
The transmit DL signal from the BS is expressed as where s = ⌊  ⌋  ∈ U ,   ∼ CN (0, 1) is the transmit data symbol associated with the -th user and denotes the base-band linear precoder.Similarly, the transmit artificial noise and the corresponding transmit precoder are denoted as z ∼ CN 0, I  C and F J ∈ C  C × C , respectively.The analog phase shifter arrays at BS is expressed as , D BS :=   ,  2 ,  3 , . . ., 2 ÑBS  , where  := 2/2 ÑBS and ÑBS represents the phase shifter resolution at the BS.So, the reflected signal from the -th RIS segment is expressed as where denotes the tunable finite-resolution phase rotation at RIS and D R := { ψ ,  2 ψ ,  3 ψ , . . .,  2 ÑR ψ } is the discrete phase-shift such that ψ := 2/2 ÑR and ÑR is the phase shifter resolution for the reflecting elements.Now, the received signal at the user and at the eavesdroppers are formulated as respectively, where   ∼ CN 0,  2  and  E, ∼ CN 0,  2 E are unavoidable receiver noise at the users and eavesdroppers.Note that the estimated data symbol at the receiver is obtained as ŝ =  *    , where   ∈ C is the linear receiver equalizer.

C. Achievable Secrecy Rate
In this work, in order to obtain worst-case guarantee with respect to the eavesdropper capabilities, we consider the pessimistic case where the multiple separately located but collaborative eavesdroppers 2 are capable of non-linear receiver processing, e.g., successive interference decoding and cancellation [47]- [49].The achievable secure information rate for the -th user can be hence formulated as (6), as shown at the top of the next page, where {•} + ≜ max(0, •),  (; ) indicates the mutual information, and h eq, () are the equivalent channel between the BS and the -th user and the eavesdroppers, respectively such that , y E :=  E,  ∈ E .In the above expressions, the vertically-stacked version of the channel matrices and RIS phase shifter arrays are used for notational convenience, where the index of the stacked dimension is dropped in each case.

III. PROBLEM FORMULATION
Primarily, we aim to solve the problem of sum secrecy rate maximization which can be formulated as max where  max is the maximum allowed transmit power from the BS, (9b) indicates the transmit power budget at BS, and (9c) is the beamforming design constraints for the BS and RIS.
In particular, the optimization problem in ( 9) is a nondeterministic polynomial-time hard (NP-hard) mixed-integer non-linear programming (MINLP) problem that is generally intractable.Specifically, the rate expression exhibits an implicit relationship with the continuous and discrete optimization variables which is hard to realize.The joint optimization of continuous and discrete beamforming variables with the discrete beamforming constraints aggravates the intractability of the optimization problem.In general, there exists no standard and systematic mathematical optimization scheme which can provide the optimal global solutions to these non-convex problems in polynomial time.Although the application of an exhaustive search can solve the design optimally, the associated computational complexity grows exponentially over the total number of variables which makes it practically impossible to implement.As a compromise, attaining a high-quality sub-optimal solution for the resource allocation problem in (9) at hand is more appealing.In the sequel, we propose an iterative procedure in order to solve (9) numerically.

IV. JOINT BEAMFORMING DESIGN WITH LOW-RESOLUTION PS
In this section, we first propose an equivalent form of the optimization problem (9), which shares the same feasible space as well as the optimum solution over the design variables F BB , F J , F RF ,  with the original problem (9).Afterward, we propose an iterative solution with guaranteed convergence to a stationary point.The following Lemmas establish the envisioned re-formulation.
Lemma 1: At the optimality of ( 9), the {•} + ≜ max(0, •) operator of the secrecy rate expressions for each user (6) can be neglected without loss of optimality.
Proof: Please refer to Appendix B.
Lemma 2: (WMMSE Lemma [50]- [52]) The rate expression   can be equivalently written as where is the mean squared-error of the estimated symbol.
Proof: Please refer to Appendix B. Proof: Please refer to Appendix B.
and U E ∈ C  E × C be the additionally introduced auxiliary variables.Then, the following identity holds Using Lemmas 1, 2, and 3, the equivalent form of the problem (9) can be constructed as min where F ∈ C  BS × U , F  ∈ C  BS × C are the auxiliary variables such that F and F  corresponds to the equivalent transmit precoder for the data and artificial noise signal, respectively.However, the above problem is still intractable due to the jointly non-convex objective, the discreteness of the feasible space, as well as the coupled inequality constraint.Nevertheless, when the discreteness constraints on , F RF are relaxed to the corresponding continuous domain, it complies with the PDD framework [53, Section II], [54], [55], due to the separate convexity of the objective with respect to the variable blocks as well as the separately affine nature of the coupled inequality constraints.However, we will refrain from delving into the detailed explanation of the employed PDD method for solving coupled equality-constraint non-convex optimization problems.For further details on this method, please refer to [56], where it is comprehensively discussed.In the section that follows, we will explore the intricacies of the PDD inner loop.

A. PDD Inner Loop
The inner loop of the PDD method is dedicated to the minimization of the Augmented Lagrangian (AL) function over the variable blocks.The corresponding AL-minimization problem for the problem in ( 16) is formulated as where F = f 1 , f 2 , . . ., f  U and the objective is reformulated by replacing F RF F J and F RF F BB respectively as F  and F in the expressions of   (.) and M  (.).Now, the optimization variables are updated as follows: The closed-form quadratic solution is obtained by taking the derivative of the quadratic objective (17a) with respect to the variable   and equating to zero i.e., After some mathematical simplification, the update of  ★  is given as Similarly, the closed-form quadratic solution of U E is obtained by taking the derivative of the quadratic objective (17a) with respect to the variable U E and equating to zero, By utilizing (13) and after mathematical simplification, the update of U ★ E is given as 2) Update of {  , S E , S ,1 } The auxiliary variables are updated by using Lemma 2 as The estimation of F ★ , F ★  can be done using a standard convex quadratic-constraint quadratic program (QCQP), with the help of efficient numerical solvers [57].

4) Update of F BB , F J
To update the variables F BB and F J , we obtain the closedform quadratic solution by taking the derivative of the quadratic objective (17a) with respect to the variables F BB and F J and equating them to zero, respectively as, After some mathematical simplification, the update of F BB and F J , respectively, can be obtained as ≈ 1 and ≈ 1

5) Update of F RF
The design of the low-resolution phase shifters is a critical issue to be solved.Using the identity tr {ABCD} = vec A   D  ⊗ B vec (C), the optimization over F RF can be written as min where Due to the discreteness of the variable space, associated prob- x ℓ ← solve ( 55)-( 57), ∀ℓ ∈ {1, • • • , }, until objective converges 8: x ← {x 1 , . . ., x  } 9: end for 10: return  (F RF ) ← x 11: end procedure lems are of combinatorial nature and thus exhibit NP-hard complexity.To this end, we propose a discrete-domain successive quadratic upper bound minimization method (D-SQUM) in order to efficiently solve (29a).In particular, the D-SQUM method utilizes the quadratic-convex nature of the objective function over the relaxed real continuous domain, as well as the separability of the discreteness constraint for each variable in order to achieve an efficient polynomial-time solution.For more details please see Appendix A and Algorithm 1.

6) Update of 𝜽
To solve , we reformulate the problem ( 16) by applying the identities tr (AB) = tr (BA), tr and  (ABC) = (C ⊗ A)(B).However, the problem ( 16) is dependent on  through the terms   (.), M  (.) and (16c).Thus, we reformulate   (.) and M  (.) by using the aforementioned mathematical identities.As defined in (13), tr S ,1 M ,1 and tr (S E M 2 ) are recast as (30) and (31), respectively, as shown at the top of the next page.Additionally, the expression in (30) can be equivalently expressed by applying the change of variables as tr S ,1 M ,1 =   J +  + 2 Re    , where tr S ,1 M ,1 = (S tr Similarly, (31) can also be expressed using the application of change of variables as tr (S  M 2 ) =   Z +  + 2 Re    , where Consequently,   (.) in ( 11) is reformulated as   (  , F BB , F J , F RF , ) ) Furthermore, the application of change of variables on   (.) transform the (40) as where Based on the derived expressions in (32), ( 36) and ( 41) and further simplification (by neglecting the terms that are independent on the ), the objective function in problem ( 16) can be reformulated as Algorithm 2 Successive Refinement algorithm 1: Initialize  (0) with random value and  = 0 2: Evaluate  (0) using  (0) .3: while |  () −  (−1) | ≥   do 4: for  = 1 :  do 5: Evaluate θ[] using (49) 6: end for 7: Update  () using θ[] , ∀ ∈ S 8: Calculate  () using  () Further, we propose a successive refinement algorithm [14] to address the non-convex nature of .Specifically, the phaseshift of each reflecting element in each RIS is optimized while fixing the phase-shift variables of other reflecting elements.This process is repeated until the optimum values of all the phaseshift variables in  are obtained.We consider   ℎ element of , i.e.,   =  [] such that  ∈ S ≜ {1, . . ., } and  =   =1  , and fix other elements i.e.,  [ ] ∀ ∈ S/{}.Thus, we can observe that the (45) can be reformulated into the linear form w.r.t.
where   and   are given as and respectively.By utilizing ( 47) and ( 48), the sub-optimal value of  [] can be evaluated as The discrete phase-shifts that are closest to the continuous phaseshift values are quantized in . Let  be the objective of problem in (16) i.e., The overall successive refinement algorithm is summarized in Algorithm 1 which can be formulated for discrete phase shifts.

7) Update of 𝜌, 𝝀, 𝝀 𝑜
The elimination of penalty violations in the standard penalty method may result in disproportionately high penalty coefficients, which hinder the numerical convergence and generate sub-problems with unfavorable conditions.This is due to the fact that as the violation gets smaller, the penalty imposed for violating the square of the constraint vanishes quadratically.As the violations get smaller, this necessitates the use of substantially larger penalty factors.The PDD technique uses a hybrid approach to address this problem i.e. when the violations are substantial, the penalty coefficient  is updated.However, when the violations are small, the dual variables ,   are updated [54], [55].Further, the update of variables in the outer loop is defined as Note that the updates in ( 52) and ( 53) is obtained by increasing the penalty parameters in the direction of the violation [54, see eq. ( 49)], [56].In the aforementioned equations,  stands for the number of algorithm iterations in the outer loop, and 0 <   < 1 is a constant that describes the growth of the penalty parameter.

B. Initialization of Optimization Variables:
We efficiently initialize several variables in our approach through closed-form expressions or quadratic convex optimizations based on other variables.For example, variables like F RF , F BB , and F J involves the random generation of values that fall within the prescribed constraint boundaries, ensuring that the initialization aligns with the defined power limitations in (9b).Using these initialized values, we calculate F and F  .Other variables such as   , U E ,   , S ,1 , and S E are also efficiently initialized based on closed-form expressions, eliminating the need for separate initialization steps.Please refer to the PDD inner loop subsection for more details.Additionally, we initialize   with a random vector drawn from a predefined set, promoting phase shift diversity during the iterative algorithm.

C. Convergence and Computational Complexity
In this section, we provide a comprehensive convergence analysis of the proposed algorithm based on the Penalty Dual Decomposition (PDD) method.The algorithm employs a dual loop structure, consisting of an outer loop and an inner loop, to solve the constraint optimization problem efficiently.
Proposition 1: The proposed algorithm based on PDD is assured to converge to a stationary point.
Proof: In the inner loop, the AL is updated at each iteration when the block of variables, whether in continuous or discrete space as defined in the problem, is updated.The convexity of the problem in continuous space ensures that the AL decreases monotonically with each iteration.Thus, we have  () ≤ ( − 1) ≤  ( − 2), where '' indicates the iteration number.This monotonic decrease guarantees continuous improvement in the AL value and, consequently, the objective function throughout the inner loop iterations.Furthermore, the AL is lower bounded by 0 by definition, ensuring that the AL value remains non-negative during the optimization process.The combination of monotonic decrease and lower boundedness guarantees the convergence of the algorithm using the PDD method in the inner loop.
In the outer loop, for the penalty methods for the constraint optimization problem, a large penalty parameter leads to a significantly slower algorithm convergence, as the direction of movement will be dominated by the terms representing constraints violation [53].In this regard, the PDD method employs a dual loop structure, wherein the inner loop of the AL is minimized over x, whereas the variables ,  are updated in the outer loop until a given stability criteria are met.In particular, the minimization over the variable blocks x  is done following the BSUM algorithm [58] when  (x) is a convex function over each block x  .At each outer iteration, depending on the constraint violation level, either the penalty parameter  (when the violation level is high) or the dual variables  are updated.The algorithm is proven to converge to a solution that satisfies KKT optimality conditions.For a detailed convergence proof and a summary of the general PDD framework, please see [53,Section III].
Moreover, the complexity analysis is particularly relevant for understanding the computational resources required for implementing the proposed framework that provides an optimal solution to the problem at hand.Notably, the computational complexity during the updates of each optimization variable in the inner loop mostly occurs due to the execution of Algorithm 1 and Algorithm 2, which are specifically designed to handle the discrete nature of F RF , .As a result, the complexity of these two update algorithms plays a significant role in determining the overall computational demands of our proposed solution.Based on this we define the worst-case computational complexity of the proposed algorithm as follows: In our proposed approach, Algorithm 1, known as D-SQUM, is utilized to calculate the F RF .This algorithm consists of two main loops: the first loop iterates over the number of Analog phase shifter arrays, which is  BS  C .The second loop within the algorithm is for convergence and is an internal loop.Considering the characteristics of the D-SQUM algorithm and its convergence behavior, we can estimate its computational complexity.Let us denote the convergence of the D-SQUM algorithm within a certain threshold as   .In accordance with the algorithm's structure, the complexity analysis can be expressed as        .By providing this complexity analysis, we aim to convey the computational resources required for implementing the D-SQUM algorithm.It takes into account the first loop over the number of Analog phase shifter arrays and the internal convergence loop.Moreover, we utilize a successive refinement approach, as described in Algorithm 2, to calculate the phases for the RIS segments.This algorithm consists of two main loops: the convergence loop and an internal loop for the calculation phase loop for each RIS segment.Based on the provided explanation and the algorithm's characteristics, we can estimate the computational complexity of the algorithm.Let us denote the convergence of the successive refinement algorithm within a maximum number of iterations as   .Considering this, the complexity analysis can be expressed as     , where  represents the sum of the number of RIS segments for all   RISs, i.e.,  =   =1  , .By evaluating the algorithm complexity as     , we aim to provide an indication of the computational resources required for the algorithm implementation.This complexity analysis considers the convergence loop and the calculation phase loop for each RIS segment.Lastly, we assume that the outer loop and inner loop converge within Ω 1 and Ω 2 iterations, respectively.Based on this assumption, we derive the overall computational complexity of the unified proposed solution as  Ω 1 ( Ω 2 (      +   )) .

V. NUMERICAL EVALUATION
This section examines the performance of the considered multi-RIS-assisted secure DL MU-MIMO system through extensive computer simulations.A total of 5000 Monte-Carlo simulations are utilized to average the simulation results.We evaluate the performance of the sum-secrecy rate subject to convergence, node density, transmit power, number of RIS surfaces and elements, number of eavesdroppers, etc.In order to analyze the blocking impacts of eavesdroppers on the transmission of sensitive information, all users are considered to be situated in a three-dimensional zone within a 100m radius around the BS.In particular, the BS is located at (20, 0, 2), while, the   RISs are uniformly deployed in the   plane at a height of 1m within the radius of 100m around the BS.Additionally, the channel state information (CSI) information is assumed to be available at the BS and the RIS where all the channels are statistically independent.The large-scale path-loss model parameters are set as  =  0 (   0 ) −  where  0 = 1 is the reference distance,  0 is the reference path-loss at  0 ,  is the distance and  is the path-loss exponent [59], while the small-scale fading for all channels are considered as Rician distributed [60].The path loss exponent for the channel between BS and RIS, the RIS and users, the RIS and eavesdroppers, the BS and users, and users and eavesdroppers are set as 2.8, 2.8, 2.8, 3.5, and 3.5, respectively.The reference path-loss for all the links is set as  0 = −30dB [59].Unless otherwise specified, the network parameters are set as   = 2,   = 2,   = 4,   = 2,  , = 10, ∀ ∈ R,   = 20 dBm,   = 10 −5 .For the PDD method, the initial penalty parameter  is set to 100/ BS and the control parameter   is set to 0.8 [55].Further, the noise variances of   and  E, are set to -100 dBm.
We label our proposed solution for the considered multi-RISassisted secure DL MU-MIMO system as "Proposed".Moreover, we compare our proposed solution with the following benchmark schemes: 1) Exhaustive search: We conduct a comparative analysis to assess the optimality of the proposed solution against the exhaustive search method.We consider the sample sizes for base-band precoders, low-resolution phase-shifters, and phase-shifters at RIS for exhaustive search method as   = 20,   = 20,   = 5, and   = 8, respectively [14].2) Continuous: We extend the continuous space design approach to the continuous phase shifters, considering both the transmitter and the RIS as capable of continuous design [36].3) Discretized: In this approach, we initially design the continuous space for the continuous phase shifters, and subsequently, at the final stage, we discretize the obtained continuous solutions [19], [38].4) Rate maximization: This scheme combines hybrid beamforming design at the BS along with passive beamforming at RIS, but it is specifically tailored for non-secrecy communication.In this case, we have not considered the effect of eavesdroppers, and the focus is on the system's performance without security considerations [21].5) DBF: For this baseline, we have utilized DBF design at BS and passive beamforming at the RIS to address secure communication.This scheme allows us to compare the performance of our proposed hybrid beamforming method against a more conventional digital beamforming approach in the context of secure communication [34].6) RIS-free: In this scheme, we consider the scenario without utilizing RIS, i.e., No RIS case, for secure communication.By comparing the proposed hybrid beamforming design against this case, we can highlight the benefits and improvements brought about by the deployment of RISs in the context of secure communication [10].7) Hybrid, Case 1: This scheme implements the proposed beamforming design at BS while considering random passive beamforming at RIS [38].8) Hybrid, Case 2: This scheme implements the proposed passive beamforming design at RIS while considering random hybrid beamforming design at BS [19].Fig. 2 describes the convergence behavior of the Proposed solution w.r.t. the varying number of iterations.In particular, we evaluate the convergence speed of the hybrid analog/DBF and passive beamforming design at the BS and each RIS, respectively, and the unified solution.Additionally, we analyze the optimality of the proposed solution in comparison with the Exhaustive search method.It is important to note that the performance of the proposed solution is close to the exhaustive method.Further, the results for Hybrid, Case 1 and Hybrid, Case 2 attain convergence within 4-5 iterations and 6-7 iterations, respectively.The convergence speed of Hybrid, Case 2 is quite slow as the number of optimization variables involved in designing a passive beamformer at the RIS are comparatively higher than the variables involved in designing a hybrid precoder at the BS.Interestingly, the hybrid beamforming design at BS recount for higher performance gain, when compared to the passive beamforming as the hybrid beamforming design efficiently, as it utilizes the imminent spatial resources available at the BS.Moreover, in another case, the RIS-free scheme exhibits lower performance than the proposed method and Hybrid, Case 1 and Hybrid, Case 2. This further demonstrates the significant performance improvement achieved by incorporating RIS in the system design.Furthermore, we compare the proposed algorithm with two conventional schemes: Continuous and Discretized schemes.Intuitively, the baseline Continuous scheme outperforms the proposed scheme due to the incorporation of a continuous space design approach.However, the results indicate that the proposed method achieves both adequate secrecy performance with fewer RF chains and substantial improvement in terms of secrecy performance compared to the Discretized scheme.This finding highlights the significant benefits of our proposed algorithm over conventional approaches, ensuring higher system performance and superior secrecy capabilities.Furthermore, we observe that the Rate maximization scheme, which represents non-secrecy communication, outperforms the proposed method in terms of achievable rate, as expected since it does not consider the impact of eavesdropping and security constraints.Overall, the proposed solution achieves convergence with higher system performance compared to the random setting of the beamforming matrices at the RISs and the BS, as well as the no RIS case, as defined in Hybrid, Case 1, Case 2, and the RIS-free, respectively, as validated in Fig. 2.
In Fig. 3, the comparative performance behavior of the proposed system w.r.t.varying transmit power budget at the BS is analyzed.We compare our Proposed solution with different numbers of RIS-reflecting elements and RIS-free scheme.For all the schemes under consideration, the sum secrecy rate is intuitively increased with an increase in maximum transmit power.However, the sum secrecy rate increases with maximum transmit power in the lower power regime and later the  performance gain becomes trivial at the higher power regime.
It is due to the fact that the excessive power at the BS results in more information leakage than the information rate at the intended users due to the presence of eavesdroppers in between the links.However, it is important to note that in scenarios where there are no eavesdroppers, the Rate maximization scheme, which is specifically tailored for non-secrecy communication, may demonstrate better performance than our Proposed method.However, the deployment of RIS significantly improves the secrecy system performance when compared to a system without RIS.Intuitively, RISs can offer additional paths and spatial degrees of freedom for sensitive information transfer, while also steering a significant amount of signal power in the direction of the intended users and reducing information leakage to the eavesdroppers.Additionally, an increase in the number of RIS reflecting elements from 16 to 32 by fixing the number of RISs also increase the system performance.Furthermore, our Proposed hybrid beamforming method offers adequate secrecy performance with a lesser number of RF chains and reduced complexity as compared to the conventional DBF scheme which can be validated from Fig. 3. Fig. 4 depicts the impact of varying the number of RISreflecting elements on the proposed system.Intuitively, the Proposed system performance will increase with an increase in the number of RIS-reflecting elements.It is due to the fact that the increase in the number of RIS elements increases the phase shifters for the incident signal which ultimately enhances the channel diversity.The performance gain achieved by the proposed beamforming design adheres to the power scaling law achieved by the optimal or near-optimal design of RIS phase shifters as discussed in [61].Nonetheless, the performance gain can be also improved by increasing the number of RIS deployments as validated in Fig. 4. Apparently, deploying a larger number of RIS scenarios (  = 2,   = 6,   = 8) uses fewer RIS reflecting elements ( , = 10) and provides better system performance when compared to only   = 1 RIS deployment with more RIS reflecting elements ( , = 50).
In particular, the multi-RIS deployment not only increases the phase-shifters for incident signals but also provides extra links that boost the overall signal quality.Moreover, we evaluate the performance of the Continuous, Discretized, and DBF schemes and juxtapose the results with our proposed hybrid beamforming approach.While it's worth noting that the Continuous and DBF schemes represent upper-performance limits for our proposed scheme, the latter has the benefits of requiring fewer RF chains, lower power consumption, and lower cost than the former, while still achieving good performance.
Fig. 5 indicates the impact of varying the maximum transmit power on the secrecy sum rate performance in the presence of eavesdropping attacks for different numbers of antennas at the BS.The increase in the number of antennas at the BS improves the system performance.An increase in power and consequently increasing the number of antenna sizes increases the degree of freedom (DoF) which leads to a high signal-tonoise ratio and better system performance as validated in Fig. 5.It is noteworthy that the higher number of antennas at BS (  = 10) that uses lower transmit power at the BS (  = 10 dBm) provides a better secrecy rate than the lesser number of antennas (  = 2,   = 4) with higher transmit power at the BS.Therefore, there exists a trade-off between the maximum available transmit power and the number of antennas at the BS.Thus, the judicious selection of the number of antennas at the BS is extremely important for minimal power consumption.Additionally, we consider the DBF scheme as an upper bound for comparison with the results obtained from our proposed hybrid beamforming design.Furthermore, we also conducted a comparative analysis with conventional methods, including Continuous and Discretized schemes for the cases of   = 4 and   = 10, respectively.While the continuous scheme outperforms our proposed method due to its incorporation of continuous space design, our proposed scheme demonstrates commendable performance in achieving adequate secrecy level performance compared to the Discretized scheme.Fig. 6 indicates the impact of the distance of the users from the BS and RIS on the sum secrecy rate of the system with different numbers of RIS reflecting elements and RIS-free scheme.The distance between RIS becomes particularly intriguing as it can influence how eavesdroppers strategically position themselves to intercept signals via the RIS.In this context, distance takes on a new dimension beyond traditional considerations of throughput and coverage; it becomes a critical factor in ensuring secrecy.For better understanding, we assume that the BS is located at (20, -10, 2), while, the   = 2 RISs are deployed in the   plane at (0, 10, 1) and (0, 11, 1), respectively.For this particular scenario, we assume that the considered   = 2 users are close to one  another and are moving along the  plane at certainly (slow) speed.With this setting, we can observe that when the users are closer to BS and RISs, there is a significant improvement in the system performance.It is obvious that the sum secrecy rate decreases with an increase in distance between the users and BS and also the users and RISs.This is due to the fact that as the distance increases, due to an increase in the path-loss there will be a significant amount of degradation in the sum secrecy rate as validated in Fig. 6.Furthermore, we assessed the performance of our proposed scheme by comparing it with the Continuous and Discretized schemes, focusing on the case of  , = 40.While the Continuous scheme acts as an upper bound, it is worth noting that our proposed scheme surpasses the performance of the Discretized scheme, showcasing its effectiveness in enhancing system security and secrecy performance.Furthermore, the increase in the RIS reflecting elements that render higher system performance when compared to the RISfree scheme in the system.Conclusively, the desired quality-ofservices (QoSs) depends in large part on the deployment of RISs between the BS and users over the area.Fig. 7 illustrates the impact of a varying number of eavesdroppers on the Proposed system.The results show that more confidential signals may be intuitively intercepted by multiple eavesdroppers than by single eavesdropper systems, resulting in a significant loss in the secrecy rate.It is because an increase in eavesdroppers will increase the likelihood of overhearing.In particular, the case of   = 6 in the system results in around 5.3 bits/s/Hz secrecy rate loss compared to the case of   = 1 when   = 2 and  , = 150 reflecting elements for each RIS are installed.Comparatively, an increase in   from 1 to 6 eventually provides the loss of 62.3% in the sum secrecy rate.This severe performance loss illustrates that eavesdroppers' blocking is as crucial to communication systems as their interception actions, and blocking awareness is one of the crucial factors to ascertain the security of wireless communication.However, Fig. 7 also validates that the system with   = 5 and  , = 150 reflecting elements achieve performance that is nearly comparable to the system with   = 3 and  , = 128 reflecting elements.Additionally, we conducted a comparative analysis of the proposed scheme against the Continuous and Discretized schemes under the specific condition of  , = 150.This analysis suggests that deploying RISs with a higher number of reflecting elements can effectively mitigate the impact of increased eavesdropping while maintaining a similar level of secrecy rate.

VI. CONCLUSIONS
This paper investigated the sum secrecy rate maximization problem for a secure RIS-assisted hybrid beamforming design with low-resolution phase-shifters at the BS and phase-shift matrix at each RIS for an MU-MIMO communication framework.In particular, we formulated a sum secrecy rate maximization problem subject to the joint optimization of hybrid beamforming design at the BS and phase-shift matrix at each RIS.To tackle the non-convex problem, we proposed a novel utilization of the WMMSE design framework to attain an expression in a quadratic form over the analog beamforming variables.The obtained expression is then reconstructed using the linked equality constraints and a well-known majorization matrix inequality as well as the PDD algorithm.Simulation results were presented to demonstrate the effectiveness of the proposed algorithm over various communication scenarios.Primarily, the proposed solution achieved convergence within 7-8 iterations.Moreover, the proposed work highlights the superiority of discrete-aware design over various baseline schemes, demonstrating the significant gains attainable by adopting discrete space design from the outset.Furthermore, it was shown that the deployment of RISs with the increased number of reflecting elements can restrict the effect of eavesdropping while maintaining the same secrecy rate.For instance, the proposed system undergoes performance loss of around 5.3 bits/s/Hz i.e., 62.3% loss in terms of sum secrecy rate for an increase in the number of eavesdroppers from   = 1 to   = 6.However, it is shown that the system with   = 6 and  , = 150 reflecting elements achieve performance equivalent to the system with   = 1 and  , = 128 reflecting elements.Overall, the results validated that the proposed hybrid beamforming approach and RIS-aided DL-MIMO can provide an effective solution for spectral-efficient secure networks.Nevertheless, a robust resource allocation design for the considered systems remains a future research direction for this work.where x, a ∈ C  A , A ⪰ 0, and D is a discrete set of elements in C where all elements are located on the unit-circle.The following lemma from [62] enables a mathematically tractable majorizerminimization approach for solving (54).Lemma 5: Let M be an  A ×  A Hermitian matrix such that M ⪰ A. Then for any point z ∈ C  A , the quadratic function x  Ax is tightly majorized by x  Mx + 2R x  (A − M) z + z  (M − A) z.
Employing Lemma 5 and by choosing M =  max (A) I  A , a majorizer of the objective function (54) at x (−1) , aiming to update x at the -th iteration is expressed as  (x () , x (−1) ) =  max (A) (x () )  x () + 2R{(x () )  ã() } +(x (−1) )   max A I  A −A (x (−1) ), (55) where ã() = A −  max (A) I  A x (−1) + a.As the last term is a constant, and the first term x ()  x () =  A is fixed, the resulting majorized problem can be expressed as: where P D : C → D is the projection into the discrete set D. It is worth mentioning that the obtained solution in (57) enjoys a linear complexity over the dimension of feasible solution space, i.e., |D| and  A .The iterations of discrete majorizationminimization ( 55)-( 57) will be continued until a stable solution is achieved.For the purpose of solving the original problem (29a), we can either choose x =  (F RF ), hence jointly updating the complete variable space of F RF , or setting x as a sub-block of  (F RF ) and updating the separated blocks iteratively.Please note that the employed majorizer in Lemma 5 leads to a joint and low-complexity calculation of the discrete variables over the complete variable block x.Nevertheless, it can be observed that while the employed majorizer coincides with the actual objective function when  A = 1, it leads to less accurate approximations and consequently a performance degradation as  A increases.In order to close this gap, the iterates of the block-variable updates will continue after convergence, by dividing the variable space into smaller blocks where the proposed majorizer in Lemma 5 enjoys a closer approximation to the actual objective (54).The detailed procedure is summarized in Algorithm 1.

1) Proof of Lemma 1:
The proof is obtained from contradiction and by observing the fact that a negative value of the relaxed rate expressions of (6) may never occur at the optimality of (9).Assume that for an optimum choice of the variables F BB , F J , F RF ,  the secrecy expression of (6) without the operator {•} + leads to a negative value for a user  = k.Then, by updating the assumed value of the  , k to 0, we obtain a zero  , , while not reducing the secrecy rate of the other users  ≠ k, which contradicts with the initial optimality assumption of the negative  , value at the optimality.Please note that the former statement follows from the fact that  , k only appears in the denominator of the   and does not impact  , for  ≠ k, and hence replacing of  , k by zero does not reduce secrecy rate of the other users.
2) Proof of Lemma 2: The correspondence of the inverse MMSE of a communication channel with the achievable rate is obtained by equalizing the derivative of the MSE of the estimated symbol over the receiver linear filter   to zero, and this derivation is elaborated in the section dealing with the inner loop updates.Therefore, the MSE of the estimated symbol is calculated as follows Please note that this is a known result also from references [50]- [52] and used in our paper in order to obtain a mathematically tractable form of the rate expression.
3) Proof of Lemma 3: The identity S ★ = X −1 is obtained by equating the derivative of the concave right-hand side of the identity with respect to S. The proof is concluded by replacing the obtained value in the identity [52].

Lemma 3 :
Let X ∈ C × ≻ 0. Then the optimization problems  X≻0 − log|X| and  X,S≻0 log|S| − tr X  X +  share the same X ★ as well as the optimum objective value, where S ★ = X −1 .

Fig. 6 :
Fig. 6: Impact of the distance of users from BS and RIS on the sum secrecy rate when   = 2,   = 2,   = 2 and   = 20 dBm.