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Large Intelligent Surface Aided Physical Layer Security Transmission

In this paper, we investigate a large intelligent surface-enhanced (LIS-enhanced) system, where a LIS is deployed to assist secure transmission. Our design aims to maximize the achievable secrecy rates in different channel models, i.e., Rician fading and (or) independent and identically distributed Gaussian fading for the legitimate and eavesdropper channels. In addition, we take into consideration an artificial noise-aided transmission structure for further improving system performance. The difficulties of tackling the aforementioned problems are the structure of the expected secrecy rate expressions and the non-convex phase shift constraint. To facilitate the design, we propose two frameworks, namely the sample average approximation based (SAA-based) algorithm and the hybrid stochastic projected gradient-convergent policy (hybrid SPG-CP) algorithm, to calculate the expectation terms in the secrecy rate expressions. Meanwhile, majorization minimization (MM) is adopted to address the non-convexity of the phase shift constraint. In addition, we give some analyses on two special scenarios by making full use of the expectation terms. Simulation results show that the proposed algorithms effectively optimize the secrecy communication rate for the considered setup, and the LIS-enhanced system greatly improves secrecy performance compared to conventional architectures without LIS.


I. INTRODUCTION
With the popularizing of user devices, a variety of wireless technologies have been proposed to improve both spectrum efficiency (SE) and energy efficiency (EE) of wireless networks. Recently, a novel concept of large intelligent surface (LIS) has been introduced as a promising technique due to its capability of achieving high SE and EE. It is convenient to control the beamforming design at the access point (AP) and phase shifts at the LIS dynamically according to the changes in the environment. Reference [1] summarizes four specific benefits of LIS-enhanced wireless communication systems as follows: (i) easy deployment and sustainable operations; (ii) flexible B. Feng, and Y. Wu  reconfiguration via passive beamforming; (iii) enhanced capacity and SE/EE performance; (iv) exploration of emerging wireless applications. Many works have emerged to generalize classical scenarios to LIS-enhanced systems, such as channel estimation [2]- [5] and unmanned aerial vehicles [6]- [7], and to verify the effectiveness of LIS-enhanced systems compared with conventional ones [8]- [11].
Related Works: Recently, LIS-enhanced systems have been introduced into the physical layer security design in wireless communications. Reference [12] first proposes the security issues in LIS-enhanced systems and solves the beamforming and phase shift problem efficiently with both block coordinate decent and majorization minimization (MM) techniques [24] under the conditions of multi-input, single-output, singleeavesdropper (MISOSE) and perfect channel state information (CSI) of both legitimate and eavesdropper channels. Reference [13] develops suboptimal solutions for both beamforming and phase shifts with semidefinite relaxation (SDR) and Gaussian randomization methods with direct links existing in the environment. Reference [14] simplifies the phase shift scheme with MM in the network model of one legitimate receiver and one eavesdropper and applies it to the multipleantenna eavesdropper case. The model of multiple legitimate receivers is investigated in [15], where manifold optimization is applied to handle the phase shifts and successive convex approximation (SCA) method is used for beamforming and artificial noise (AN) injection. Meanwhile, the model of multiple eavesdroppers is investigated in [16], where SDR method is applied for beamforming, AN and phase shifts. Reference [17] studies the maximization of the minimum secrecy rate among several legitimate users in the presence of multiple eavesdroppers, where the unit modulus constraints of the phase shifts are approximated by a set of convex constraints. References [18]- [19] investigate the system with multi-input, multi-output, multiple-antenna eavesdropper (MIMOME). In summary, references [12]- [19] focus on a series of schemes under the assumption of perfect CSI of all channels. Reference [20] takes into account the model of multiple singleantenna legitimate receivers, which do not have line-of-sight (LoS) communication links, in the presence of multiple multiantenna potential eavesdroppers whose CSI is not perfectly known. It is worth noting that reference [20] solves the phase shift constraint with norm-difference method instead of the aforementioned MM, SCA or SDR. Reference [21] tries to take into consideration the model of rank-one AP-LIS channel and statistical LIS-Receiver/Eavesdropper channels and eliminates the effect of phase shifts for statistical CSI arXiv:2009.00473v1 [cs.IT] 1 Sep 2020 channel coefficients.
Main Contributions: In contrast to the previous literature, this paper studies a LIS-enhanced MISO system with Rician channel in the AP-LIS link and independent and identically distributed (i.i.d.) Gaussian fading channel in the LISeavesdropper link due to the challenge of acquiring perfect CSI of the eavesdropping channels at the AP. The main contributions of this paper can be summarized as follows.
• To the best of our knowledge, this is the first work to explore the use of the LIS to enhance the physical layer secret communication rate under the condition of i.i.d. Gaussian fading channel in the LIS-eavesdropper link. We formulate four achievable secrecy rates under different assumptions and jointly optimize the AN-aided beamforming at the AP and phase shifts at the LIS. • The problem is quite challenging due to the following reasons. First, the non-convex phase shift constraint makes the problem essentially an NP-hard problem. Second, computing the expected secrecy rate expressions is computationally expensive. In view of these problems, we adopt the MM algorithm to tackle the phase shift constraint. Meanwhile, we propose two algorithms, a sample average approximation based (SAA-based) algorithm and a hybrid stochastic projected gradient-convergent policy (hybrid SPG-CP) algorithm, to convert the expectation operations to the determined structure at each iteration. Besides, we generalize the SPG-based algorithm and prove that the expectation of the projected gradient approaches 0 as the number of iterations approaches infinity. • For the case of Rician fading channel at legitimate receiver and i.i.d. Gaussian fading channel at eavesdropper, we develop an alternating optimization method to solve the problem more efficiently based on the exact calculation of the expectation function. For the case of i.i.d. Gaussian fading channels at both legitimate receiver and single-antenna eavesdropper sides, we verify that it is unnecessary to take the AN-aided structure to suppress the single-antenna eavesdropper. • Finally, simulation results validate the effectiveness of the proposed algorithms. It can be seen that the LIS dramatically improves the quality of the whole secrecy system. Furthermore, the larger number of LIS elements we use, the better performance the system achieves.
The rest of this paper is organized as follows. Section II introduces the secure transmission model, achievable secrecy rate and problem formulation. Section III develops two efficient algorithms for the stochastic optimization problem of secrecy rates. Section IV provides analysis on some specific cases. Section V shows some simulation results to evaluate the performances of the proposed algorithms. Section VI concludes the paper.
The notations used in this paper are as follows. Boldface lowercase and uppercase letters, such as a and A, are used to represent vectors and matrices, respectively. In denotes the n-by-n identity matrix. Superscripts T , * , and H stand for the transpose, conjugate, and conjugate transpose, respectively. ∇ X f denotes the gradient of f with respect to X and ∇ X * f = ∇ X * f * = (∇ X f ) * with a real-valued function f . a * denotes the * -norm of the complex vector a. a and A , respectively, denote the 2-norm of the complex vector a and the Frobenius norm of the complex matrix A. λmax (A) and γmax (A), respectively, denote the maximum eigenvalue of matrix A and its corresponding eigenvector. arg (v) denotes the phases of complex elements in the vector v. CN (µ, Σ) denotes a complex circular Gaussian distribution with mean µ and covariance Σ. R(a) denotes the real part of a complex value a. The inner product •, • : C n×n ×C n×n → R is defined as A, B = R tr A H B . diag (A) denotes a vector whose elements are extracted from the main diagonals of matrix A. a ⊥ b denotes a · b = 0. a represents the smallest integer greater than or equal to a.

A. Secure Transmission Model
As shown in Fig. 1, we consider a MISO wiretap channel model with a LIS-enhanced link. In this system, there are an AP with Nt antennas, a LIS with NI passive and lowcost reflecting elements, a single-antenna legitimate receiver and an eavesdropper equipped with Ne antennas. We assume that the direct signal paths between the AP and the legitimate receiver/eavesdropper are neglected due to unfavorable propagation conditions in our model. Compared with the traditional wiretap channel model, the LIS-enhanced system introduces a LIS device, which is an intelligent control system that can dynamically adjust the phase through passive beamforming according to the changes in the environment to upgrade the communication quality. The baseband equivalent channels from AP to LIS, from LIS to the legitimate receiver and from LIS to the eavesdropper are, respectively, denoted by G ∈ C N t ×N I , hr ∈ C N I ×1 and He ∈ C N I ×Ne . Then, the received complex baseband signals at the legitimate receiver and the eavesdropper are, respectively, given by where ρr and ρe denote the signal-noise-ratios (SNR) at the transmitter for the legitimate receiver and eavesdropper, respectively; Θ diag e jθ 1 , e jθ 2 , ..., e jθ N I denotes the phase shift matrix with θn being the phase shift introduced by the nth element of LIS, and nr, ne are additive white Gaussian noise with variance one. We apply the linear channel prefixing and Gaussian signaling as where s ∼ CN (0, Σs) and z ∼ CN (0, Σz) are independent vectors to convey the message and AN, respectively. Hence, the case of x = s and z = 0 implies the non-AN aided system.

B. Transmission Schemes and Secrecy Rates
In this paper, we will discuss transmission schemes from the following three aspects:

C. Problem Formulation
In this paper, our goal is to design an effective scheme to maximize the achievable secrecy rates by adjusting the beamforming, AN spatial covariance and phase shifts. The corresponding optimization problems in non-AN and ANcases are, respectively, formulated as: (7c) θn ∈ [−π, π), n = 1, . . . , NI . (7d) The problems (P 1) and (P 2) are non-convex due to the non-concave objective functions with respect to Σs, Σz and θn and the expectations in the objective functions. Unfortunately, there is no standard method for solving them.
Prior to solving the problems (P 1) and (P 2), we present the feasibility. For notational simplicity, we define three sets There exists a feasible point that Σs = 1 N t I and θn = 0, n = 1, . . . , NI satisfies the constraint X1 ∩ Y, hence the problem (P 1) is feasible. Similarly, the point that Σs = 1 N t I, Σz = 0 and θn = 0, n = 1, . . . , NI satisfies the constraint X2 ∩Y, which verifies the feasible of the problem (P 2) [25, §4.1.1]. In the sequel, we will develop some iterative algorithms to solve (P 1) and (P 2) efficiently.

III. TWO FRAMEWORKS FOR SECRECY RATES MAXIMIZATION
Overall, both of our proposed frameworks decompose the original problem into two kinds of problems, namely optimizing phase shifts with fixed AN-aided beamforming and optimizing AN-aided beamforming with fixed phase shifts. In this section, we first study the effect of phase shifts on legitimate receiver and eavesdropper with the statistics of channel coefficients. Then, we give the iterative structure of our frameworks and study their performance. Note that we mainly concentrate on the realization of C3(Σs, Σz, Θ) in the following algorithms in this section, since C3(Σs, Σz, Θ) contains all variables and C1(Σs, Θ), C2(Σs, Θ), C4(Σs, Σz, Θ) have similar optimization structures with C3(Σs, Σz, Θ) in our frameworks.

A. Phase Shifts Optimization
Proposition 1. If the channels between the transmitter and legitimate and eavesdropper are i.i.d. Gaussian fading, phase shifts have no contribution to the expectation terms.
Proof. Since each element in hr and He is distributed as CN (0, 1), the distributions of hr and He could be expressed as Hence, Θhr and ΘHe have the same distributions as hr and He, respectively [39, A.26]. The same distribution implies that phase shifts have no contribution to the expectation terms. Then we have For given Σs and Σz, we Then we have the following problem: This problem belongs to fractional programming, and we consider the corresponding parametric program: where µ > 0 is an introduced parameter. Many works have emerged to solve the optimization problem efficiently with unit modulus constraint such as MM [12]- [14], manifold optimization [15], SDR [13][16] and DRL [11]. Here, we adopt MM algorithm to solve (P 3.1) due to its closed-form solution at each iteration. At iteration n, for any feasible Then we have the following problem: Moreover, R v H β is maximized when the phases of vi and βi are equal, where vi and βi are the ith entry of v and β, respectively. Therefore, the optimal solution at iteration n is More details are provided in Algorithm 1.

B. SAA-based Algorithm
The basic idea of the SAA-based algorithm is to generate some independent random samples and approximate the expectation function by the corresponding sample average function. Then, the result of the sample average optimization problem is considered as an approximate solution to the original problem [23]. We define Set µ = (µmax + µmin)/2, and n = 0;

8:
until Convergence 9:  (5), and each C3(Σs, Σz, Θ, H i e ) is a random variable with mean s and variance Thus, we have the following problem: It is obvious that (P 4) is a non-convex programming with the concavity of fr (Σz) and f i e (Σs + Σz). To facilitate the application of MM [24], we derive their upper bounds. Besides, considering that a large amount of log-det type functions −f i e (Σz) (i = 1, · · · , K) seriously influence the complexity in each iteration, we use descent lemma [24,Lemma 12] to get the upper bound due to the fact that the gradients of them are Lipschitz continuous, which has been verified in Appendix A.
At iteration t, fr (Σz), f i e (Σs + Σz) and −f i e (Σz) satisfy the following inequalities for any feasible points Σ t s and Σ t z : with the derivative functions ∇fr (X) and ∇f i e (X) shown as follows where Li, i = 1, 2, ..., K are the Lipschitz constants of ∇f i e (Σz). Then, we minimize the surrogate function as follows: where As We have proved that for any Li e G H 2 is too large, then it will cause our algorithm to converge very slowly, which is similar to the effect of a too small step-size on the classical gradient descent algorithm. Hence, by replacing 1 Li with an adaptive Lipschitz constant Lt at iteration t, we have As (P 4.2) is a convex semidefinite program problem, it can be solved by existing convex optimization solvers such as CVX [26]. More details are provided in Algorithm 2.
A key feature of Algorithm 2 is the adaptive line search: it always tries to use a smaller Lipschitz constant at the beginning of each iteration by setting Lt+1 = Lt/4. The line search procedure starts with an estimated Lipschitz constant Lt, and increases its value by a factor of 2 until the construction rule of MM is satisfied [24,Eqn. (24)]. In other words, the stopping criteria that C3 Σ t+1 for line search is also satisfied [41] [42].

C. Hybrid SPG-CP Algorithm
In the hybrid SPG-CP algorithm, we take the SPG-based algorithm and MM algorithm to optimize (Σs, Σz) and Θ, respectively.
Here we first introduce some conclusions about the generalized projection associated with the proximal operator. Let PX 2 [(Σs, Σz), (Hs, Hz), r] denote the projection of (Hs, Hz) onto the set X2 at the point (Σs, Σz) ∈ X2 with the step size r > 0. We define  over H t e . The derivative functions ∇fr,t (X) and ∇f i e,t (X) are shown as follows Now we solve the following problem for any given Σ t s , Σ t z : The constant r ∈ (0, 2/L) and L is relevant to the gradient of C3(Σs, Σz, Θ), which has been given in Proposition 2. A common practice in stochastic optimization is to estimate L by using the stochastic gradients computed at a small number of trial points [33] [40].
Proposition 3. The optimal solution of problem (P 5) is given by where [X] + denotes the projection of X onto the positive semidefinite cone, and λ is the multiplier such that 0 ≤ λ ⊥ tr(Σ t+1 s + Σ t+1 z ) − 1 ≤ 0, which can be found by bisection.
Proof. In the following we solve (P 5) via the partial Lagrangian function. 2 We use the notation t α to denote the sample size because α is not necessarily an integer. As we will see in Appendix D, α > 1 is needed and also enough to prove that the expectation of the projected gradient of C3 (Σs, Σz, Θ) approaches 0 as the number of iterations approaches infinity in Algorithm 3.
The idea in step 2 is similar with [38, Algorithm 9]. As α increases, the sample size t α increases. Based on Lemma 1, the gradients Gs and Gz become more accurate in approximating the gradients of C3(Σs, Σz, Θ) with respect to Σs and Σz. However, for a larger α, computational complexity also increases in each iteration. Therefore, in practice, we need to make a trade-off and select a suitable α for our algorithm. Then the stationary point is given by
) ≥ 0, ∀i holds according to step 9 in Algorithm 2, we take the limit and obtain the inequality On the other hands, due to for all possible (Σs, Σz), we have Based on [24, Eqn. (6)], (33) implies that the limit point Σs, Σz, θ is a stationary point.

Hybrid SPG-CP Algorithm
For the performance of Algorithm 3, we have the following proposition. Proof. See Appendix D.

MM Algorithm for Phase Shifts
At the start of Algorithm 1, it is necessary to compute the maximum and the minimum eigenvalues of the matrices Y1 and Y2, whose complexity is O(N 3 I ). Suppose that the MM algorithm requires T1 iterations to converge in total. The complexity of each iteration mainly depends on the computation of β in step 5 of Algorithm 1 and the corresponding complexity is given by O(N 2 I ). Therefore, the complexity of evaluating v is approximated as O(N 3 I + T1N 2 I ).

SAA-based Algorithm
The complexity of updating Σs and Σz mainly depends on the optimization of problem (P 4.2) and the comparison with the previous solution. Firstly, the complexity of computing As, Az and optimizing (P 4.2) can be asymptotically estimated as K(NeNI Nt + NeN 2 respectively [35]. Secondly, we mainly need to compute the matrix multiplication and determinant in C3, so the corresponding complexities are O(NeNI Nt + NeN 2 t + N 2 e Nt) and O(N 3 e ), respectively. Suppose that the adaptive algorithm needs T2 iterations to search for a suitable L and Algorithm 2 requires T3 iterations to converge. Then the total complexity is approximated as T3(N 3

Hybrid SPG-CP Algorithm
Suppose that we need NL trial points to estimate L, whose complexity is estimated as O(N 2 [40]. The complexity of updating Σs and Σz mainly depends on the search of a suitable λ and eigenvalue decomposition. Suppose we need T4 iterations to get a suitable λ by bisection and Algorithm 3 needs N iterations. Then the total complexity is approximated as O(N 2

IV. ANALYSIS ON SOME SPECIAL CASES
One difficulty in dealing with the maximizing secrecy rate problem is the expectation terms existing in the objective functions. In this section, we utilize some exact result of the expectation and analyze some special cases. To begin with, we identify three important properties: (i) the expectation of log-like function; (ii) the smooth and convex property for a specific optimization problem; (iii) the rank of the optimal Σs for C1(Σs, Θ). Lemma 4. Suppose the eigenvalues of ρσ 2 Q are {0, 0..., 0,t,t, ...t}, where the number oft s is N1. For z ∼ CN (0, σ 2 I), the expectation of log 1 + z H ρQz and its first-and second-order derivatives with respect tot are Proof. See Appendix E.
Lemma 5. For given G and h, we define Then, φ(z) is a smooth function over [0,1] and φ(z) is a convex function. Please see Lemma 6 and Lemma 7 in [27] for details.
Proposition 5. The optimal covariance matrix of Σs for C1(Σs, Θ) is of rank one.
Proof. See Appendix F.

A. Alternating Optimization for C 1 (Σ s , Θ)
Since Σs has rank one, let us write Σs = ωω H with ω 2 = 1. Then, the ergodic secrecy rate is reduced to Proposition 6. For any given beamforming ω, the optimal phase shifts are given by where a is a vector with all same element.
Proof. Invoking (36) and Cauchy-Schwarz inequality, we have In the last inequality, the equality is achieved at For the propose of accelerating the optimization of ω, an auxiliary variable z is introduced. 1]. For any given v and z, the aim of maximizing C1(Σs, Θ) motivates us to write The above optimization implies that we can reduce the eavesdropper's rate while guaranteeing the legitimate receiver's performance, thereby improving secrecy rate. Then, the original beamforming and phase shift problem is reduced to the optimization with respect to v and z: Since φ(v, z) is a smooth function, any converge method (e.g. Newton-type method) can be used to solve it. Note the firstand second-order derivatives can be determined numerically [30, §25.3]. For any given v, a Newton-type method is taken to update z, thereby optimizing ω according to (39). Based on (37), (39) and (40), the overall algorithm proposed in this section is summarized in Algorithm 4.

Algorithm 4
Alternating Optimization for C 1 (Σ s , Θ) Input: v 0 , t = 1 1: repeat 2: Update z t and ω t for given v t−1 according to (39) and (40) with Newton-type method; 3: Update v t for given ω t according to (37); 4: t ← t + 1; In this subsection, we study the system with both i.i.d. Gaussian fading channels at receiver and eavesdropper and ANaided transmission signal. Now, we consider a special scenario where the eavesdropper is equipped with a single antenna, i.e., Ne = 1. Therefore, the secrecy rate of C4(Σs, Σz, Θ) reduces to: By randomly choosing a possible value of he, we have The monotonically decreasing function with respect to t (Σz) implies that the optimal Σz should be 0. Intuitively, neither legitimate receiver nor eavesdropper can directly distinguish noise from the overall received signal so that AN is not helpful to the secrecy rate. Hence, the best policy is that when the legitimate channel is better than the eavesdropper channel, all power should be allocated to the information-carrying signal, and if the channel condition is reversed, communication should break up.

A. Simulation Setup
In this section, we numerically evaluate the performance of the proposed algorithms. In Fig.2, we take the Cartesian coordinate system to describe positions of all components in the LIS-enhanced system model. In practical systems, one of the motivations for deploying LIS in secure wireless systems is to enhance favorable communication links by suppressing the undesired eavesdropper. AP and LIS are usually deployed in advance and we assume AP and LIS are located at (0, 0, 15)m and (0, 50, 15)m, respectively. The channel matrix G between AP and LIS channel is modeled as follows where the small-scale fading is assumed to be Rician fading. G LOS and G NLOS represent the LoS and NLoS components, respectively. The distance-dependent path loss model is given by where C0 = −30dB is the path loss at the reference distance D0 = 1 meter, d denotes the individual link distance, and ζ denotes the path loss exponent. Some parameters are given in Table I.
B. SAA-based vs. Hybrid SPG-CP vs. Alternating Optimization for C 1 (Σ s , Θ) In this subsection, we compare the performance and the convergence rates of the SAA-based algorithm, the hybrid SPG-CP algorithm and the proposed alternating optimization algorithm. Fig. 3(a) depicts that the achievable secrecy rate versus the transmit power which ranges from 10dBm to 30dBm. The system rates increase monotonically with more transmit power budget. It results from the improved SNR 10   Also, we observe that the average secrecy rate achieved by three algorithms is the same. In Fig. 3(b), we compare the convergence of three algorithms when the total transmit power is 10dBm. Fig. 3(b) implies that alternating optimization algorithm makes full use of expectation information and achieves a faster convergence rate than the other two algorithms.
C. SAA-based vs. Hybrid SPG-CP for C 1 (Σ s , Θ) and For fairness, Fig. 4 compares the non-AN system C1(Σs, Θ) and AN-aided system C3(Σs, Σz, Θ) under the same condition. Fig. 4(a) shows that the achievable secrecy rate C1(Σs, Θ) exceeds the x-axis baseline, indicating that a performance gain can be achieved by introducing LIS into the scenario where the direct link is severely blocked. Meanwhile, the achievable secrecy rate C3(Σs, Σz, Θ) exceeds the C1(Σs, Θ) baseline, indicating that the introduced AN-aided structure achieves even greater performance gain due to AN suppressing the channel gain of the eavesdropper. Therefore, the AN-aided structure significantly outperforms the non-AN structure. Fig.  4(b) sketches the number of iterations versus the secrecy rate by considering three cases with configurations given by: case (1) Σs = 0.5IN t , Σz = 0.5IN t , θi = 0, ∀i; case (2) Σs = IN t , Σz = 0, θi = π, ∀i; case (3) Σs, Σz, θ are random matrix. To verify the sensitivity of algorithms with respect to the selection of the initial point, we set the total transmit power P = 15dBm for SAA-based algorithm in the three cases when P = 25dBm for the hybrid SPG-CP algorithm. It can be seen that they always converge to the similar secrecy rates, the differences between them are almost negligible in all the considered cases. The simulation results illustrate that the proposed two algorithms are accurate to optimize the objective problems.
D. Hybrid SPG-CP Algorithm for C 2 (Σ s , Θ) and In this subsection, for the purpose of demonstrating that AN structure has no contribution to the system with both i.i.d. Gaussian channel models and Ne = 1, we compare the secrecy rates C2(Σs, Θ) and C4(Σs, Σz, Θ). Fig. 5 sketches the transmit power versus the secrecy rate by considering two cases with configurations given by: case (1) ζIR = 2.8; case (2) ζIR = 2.2. Simulation results show that the achievable secrecy rate is the same in both cases, which implies the power  allocated to AN is 0. Besides, we also compare secrecy rates by setting different path losses. As can be observed from Fig.  5, increasing transmit power results in an improved secrecy rate. Therefore, we should allocate all power to beamforming when both i.i.d. Gaussian channel models are considered.

E. Secrecy Rate vs. Number of LIS elements
In Fig. 6, we study the impact of the number of LIS elements under two cases: (1) transmit power P = 15dBm; (2) transmit power P = 25dBm. We assume that the number ranges from [8,40]. As clearly shown, both proposed algorithms yield very similar performance curves and the differences between them are almost negligible. Furthermore, it is observed that as the number of LIS elements increases, the secrecy rate increases accordingly because more LIS elements can reflect more energy. Therefore, the LIS-based system contributes to improving performance significantly.

VI. CONCLUSIONS
In this paper, we have considered the problem of improving the physical layer security of wireless communication networks by deploying LIS. Two efficient algorithms, i.e., the SAA-based algorithm and the hybrid SPG-CP algorithm, were proposed for joint optimization of the AN-aided beamforming at the transmitter and phase shifts at the LIS. The SAAbased algorithm has high complexity at each iteration due to 8  in the following general form: where H i = GH i e . Then, for the Lipschitz continuous of the gradient, we have is given as follows: where (c) follows from tr(AB) ≤ 1 2 (tr(A)+tr(B)) and nonnegative definite matrices A, B; (d) is due to the fact that the maximal eigenvalue of I + ρeH iH XH i −1 is no more than 1. For any Li and the probability density function (p.d.f.) of X satisfies In addition, we assume that X = x, then the upper bound of ∇f i e,t (Σs + Σz) defined in (20) The first-and second-order derivatives [31, §12.212] of F1 t , N1 with respective tot are given by: 3 Here, we think the singular matrix is also invertible and the inverse matrix has some ∞ as eigenvalues.