Intelligent Reflecting Surface: A Programmable Wireless Environment for Physical Layer Security

Jie Chen; Ying-Chang Liang; Yiyang Pei; Huayan Guo

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Intelligent Reflecting Surface: A Programmable Wireless Environment for Physical Layer Security

Jie Chen, Ying-Chang Liang, Yiyang Pei, Huayan Guo

arXiv:1905.03689v2May 9, 2019eess.SP

TL;DR

The paper addresses physical-layer security when transceiver beamforming cannot reliably distinguish legitimate receivers from eavesdroppers. It introduces IRS-assisted programmable propagation, jointly optimizes BS beamformers and IRS coefficients under continuous and discrete constraints, and reports convergent algorithms whose effectiveness is validated by simulations.

Problem
Highly correlated legitimate and eavesdropper channels can make secret communication intractable with transceiver beamforming alone.
Method
The paper jointly optimizes BS beamformers and IRS reflecting coefficients using alternating optimization with path-following, plus lower-complexity suboptimal algorithms.
Results
Simulations validate the advantages of IRS assistance and the effectiveness of the proposed algorithms across transmit-power, reflecting-element, and coefficient-resolution settings.
Takeaways & Limitations
IRS provides additional programmable links that increase legitimate-receiver SNR while suppressing eavesdropper SNR in the studied MISO broadcast system.

Abstract

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In this paper, we introduce an intelligent reflecting surface (IRS) to provide a programmable wireless environment for physical layer security. By adjusting the reflecting coefficients, the IRS can change the attenuation and scattering of the incident electromagnetic wave so that it can propagate in a desired way toward the intended receiver. Specifically, we consider a downlink multiple-input single-output (MISO) broadcast system where the base station (BS) transmits independent data streams to multiple legitimate receivers and keeps them secret from multiple eavesdroppers. By jointly optimizing the beamformers at the BS and reflecting coefficients at the IRS, we formulate a minimum-secrecy-rate maximization problem under various practical constraints on the reflecting coefficients. The constraints capture the scenarios of both continuous and discrete reflecting coefficients of the reflecting elements. Due to the non-convexity of the formulated problem, we propose an efficient algorithm based on the alternating optimization and the path-following algorithm to solve it in an iterative manner. Besides, we show that the proposed algorithm can converge to a local (global) optimum. Furthermore, we develop two suboptimal algorithms with some forms of closed-form solutions to reduce the computational complexity. Finally, the simulation results validate the advantages of the introduced IRS and the effectiveness of the proposed algorithms

I. INTRODUCTION

The paper introduces IRS-assisted physical layer security for a downlink MISO broadcast system, addressing cases where transceiver beamforming alone cannot separate legitimate receivers from eavesdroppers. It jointly designs BS beamformers and IRS reflecting coefficients, develops iterative and lower-complexity algorithms, and validates the approach through simulations.

Motivation: IRS reconfigures incident electromagnetic waves to create a programmable wireless environment that guides signals toward intended receivers.Its reflecting elements adjust phase or amplitude through a preprogrammed controller.
Problem: When legitimate receivers and eavesdroppers have highly correlated channels, beamforming alone may maximize both legitimate and eavesdropped SNRs.The paper therefore explores IRS-provided communication links to increase legitimate-receiver SNR while suppressing eavesdropper SNR.
System and objective: The study considers a downlink MISO broadcast system in which a BS sends independent confidential streams to multiple single-antenna legitimate receivers while eavesdroppers attempt to intercept them.The system includes one IRS and multiple eavesdroppers.
Contributions: The main optimization jointly designs BS beamformers and IRS reflecting coefficients to maximize the minimum secrecy rate under continuous and discrete coefficient constraints.The variables are coupled and the objective is non-concave, making the formulation non-convex.
Algorithms: An alternating-optimization path-following algorithm handles the coupled non-convex problem iteratively and is shown to converge to a local or global optimum.The converged solution is also shown to satisfy the KKT conditions.
Algorithms: Two suboptimal algorithms reduce computational complexity through closed-form or heuristic solutions for single-user/single-eavesdropper and multi-user/multi-eavesdropper cases.The multi-user case uses heuristic zero-forcing beamforming.
Results: Simulations validate the advantages of IRS assistance and the effectiveness of the proposed algorithms.The reported comparisons examine transmit power, reflecting-element count, and discrete coefficient resolution.

B. Reflecting Coefficient Model

The reflecting-coefficient model represents the IRS as a diagonal matrix whose entries are selected from a coefficient set. It covers optimized- or constant-amplitude continuous phase shifts and constant-amplitude discrete phase shifts, while defining the signal and secrecy-rate quantities used in the system model.

Reflecting coefficient representation: The IRS reflection matrix is Θ = diag(θ), where θ contains the L reflecting coefficients and each θ_l belongs to the set Φ.The set Φ determines the coefficient constraint imposed on the reflecting elements.
Continuous reflecting coefficients: Continuous coefficients include optimized-amplitude with continuous phase shift and constant-amplitude with continuous phase shift configurations.These configurations correspond to distinct feasible reflecting-coefficient sets.
Discrete reflecting coefficients: Discrete coefficients use constant amplitude and a discrete phase shift with Q available reflecting-coefficient values.Discrete control is described as more practical under hardware limitations than continuous control.
Signal model: The BS transmits independent CSCG confidential messages using downlink beamforming vectors, and the legitimate and eavesdropped received signals follow the corresponding beamformed channel model.The beamforming vector w_k is associated with confidential message s_k.
Secrecy-rate model: The secrecy rate for message s_k is defined by comparing its achievable rate at legitimate receiver B_k with its wiretapped rates across the eavesdroppers.Because each eavesdropper may intercept any confidential message, the resulting secrecy rate uses the minimum over the relevant receivers and eavesdroppers.

III. PROBLEM STATEMENT

The paper formulates joint beamformer and IRS-reflection optimization to maximize the minimum secrecy rate under continuous and discrete reflection constraints. The resulting problem remains non-convex because the objective is non-concave and the optimization variables are coupled.

The design jointly optimizes BS beamformers and IRS reflecting coefficients to maximize the minimum secrecy rate among legitimate receivers.
The original formulation is transformed into an equivalent problem using the augmented vector v = [θ; 1].
The transformed objective R(W, v) is not jointly concave in the beamformers and reflection variables, which remain coupled.
The path-following method constructs a concave lower-bound approximation, while alternating optimization addresses the coupling between variables.The approximated problems use lower and upper bounds for the relevant non-concave terms.
The approximated problem remains non-convex because the variables are coupled and some reflection-coefficient feasible sets are non-convex.For Φ = Φ1 the constraint set is convex; for Φ = Φ2 and Φ = Φ3 it is non-convex.

B. Alternating Optimization with Continuous and Discrete Reflecting Coefficients

For continuous reflecting coefficients, alternating optimization separates the coupled beamformer and IRS-variable updates into two convex subproblems. The resulting updates can be solved efficiently within the path-following iterations.

With Φ = Φ1, the constraint set is convex, so the remaining non-convexity comes from coupling between W and v.
The alternating procedure decomposes each approximated problem into separate subproblems for optimizing W and v.
Each subproblem is convex when the other variable is fixed and can therefore be solved optimally using CVX.
Algorithm 1 alternates between updating W with fixed v and updating v with fixed W until convergence.

2) The Solutions of (P2) with Φ = Φ2:

For Φ = Φ2, the paper handles the non-convex reflection constraint using either relaxation with Taylor approximation or direct projection. The projection-based method avoids the relaxation factor used by the first approach.

When Φ = Φ2, the reflection constraint is non-convex, so two methods are proposed to obtain feasible solutions.
The first method introduces a positive relaxation factor λ and approximates the resulting convex term using a first-order Taylor expansion.
The remaining coupling between W and v is handled by applying the same alternating optimization procedure used in Algorithm 1.
The relaxation-based method uses an approximation and lacks a predetermined choice of λ to accelerate convergence.
The second method directly projects the Φ1 solution into the Φ2 feasible set and can then iteratively update the projected variables.
After projection, the update retains the new v only when it does not decrease the objective value.

3) The Solution of (P2) with Φ = Φ3:

For Φ = Φ3, the reflection design becomes a combinatorial optimization problem that is generally NP-hard, so the paper uses heuristic projection. The proposed alternating path-following algorithm has monotonic objective improvement and converges to a KKT point, but its convex subproblems can be computationally costly.

With Φ = Φ3, the optimized problem is generally NP-hard, making exact optimal-solution computation intractable.
The Φ3 solution is obtained by directly projecting the Φ1 solution into the Φ3 constraint set while keeping W unchanged initially.
The objective value increases at every iteration of Algorithm 1, guaranteeing convergence to a local (global) optimum.
The bounded iterates converge to a solution that is ultimately a Karush-Kuhn-Tucker point.
The convergence guarantee requires solving convex subproblems with complexities O((KN +1)K^2M^2) and O((KN +L+1)(L+1)^2).

V. SUBOPTIMAL ALGORITHMS WITH LOW-COMPLEXITY

The paper develops two lower-complexity suboptimal algorithms for special cases of the secrecy-rate optimization problem. One uses alternating optimization with closed-form updates for one legitimate user and one eavesdropper; the other uses noniterative zero-forcing beamforming for multiple users and eavesdroppers.

Algorithm Overview: Two suboptimal algorithms reduce complexity: a closed-form alternating method for K = 1 and N = 1, and a noniterative ZF method for multiple users and eavesdroppers.The first method alternates between beamformer and reflecting-coefficient updates; the second is based on zero-forcing beamforming.
Single-User Case: For K = 1 and N = 1, alternating optimization decouples the coupled beamformer W and reflecting coefficients v.The resulting iterations admit closed-form solutions, yielding a low-complexity algorithm despite the original problem's non-convexity.
Single-User Case: The single-user method alternately solves an optimization problem for w1 with fixed v and one for v with fixed w1.The two subproblems define the alternating updates used in the low-complexity procedure.
Single-User Case: The reflecting-coefficient solution is adjusted after convergence to satisfy the original constraint by setting vL+1 to 1 and rotating the other coefficients accordingly.The transformation is v*l = vl / exp(j arg(vL+1)) for 1 ≤ l ≤ L.
Single-User Case: The beamformer subproblem is the downlink MISO basic wiretap-channel problem, whose optimal solution is characterized using the principal eigenvector of matrix Z.The cited formulation identifies q as the eigenvector corresponding to Z's largest eigenvalue.

2) The Solutions to (P5-B)

For the reflecting-coefficient subproblem, the paper reformulates the objective using an auxiliary-variable identity and then applies alternating optimization. The resulting convex subproblems are solved through dual methods, producing a rank-one solution and a convergent iterative algorithm.

Reformulation: Lemma 5.1 rewrites the logarithmic term through an auxiliary variable y, enabling an equivalent formulation of (P5-B).The identity uses −ln x = max y>0 (−xy + ln y + 1), with optimizer y* = 1/x.
Alternating Optimization: The reformulated problem remains non-convex jointly but is convex in v or y when the other variable is fixed.This structure motivates alternating optimization over y and V = vv^H.
Alternating Optimization: The V-subproblem is solved as a convex problem without an explicit rank-one constraint because its optimal V is proved to have rank one.The rank-one structure allows recovery of V = v*v*^H.
Dual Solution: Strong duality holds for the convex V-subproblem, so its optimum can be obtained by solving the Lagrangian dual problem.The dual variables χ are constrained to be nonnegative.
Complexity: Using semidefinite relaxation to solve (P5-B) would require complexity on the order of O((N + 1)^4.5).This motivates the lower-complexity iterative treatment developed in this subsection.
Dual Solution: The algorithm iteratively solves the primal variables and updates χ using subgradient methods such as the ellipsoid method.The stated procedure alternates these updates until convergence.
Convergence: The proposed alternating optimization algorithm is guaranteed to converge.The paper states that the convergence proof is analogous to that of Theorem 4.1.

B. Heuristic Algorithm for (P2) with K ≥1 and N ≥1

The heuristic algorithm addresses the multi-user, multi-eavesdropper case under asymptotically favorable channel conditions. It combines phase alignment, zero-forcing beamforming, and power allocation to simplify the secrecy-rate optimization.

Channel Assumptions: When L →∞, the direct BS-to-user signal can be ignored because reflected-path received powers dominate asymptotically.The heuristic also assumes Rician factors κF, κhr,k, and κgr,n tend to infinity.
Reflecting-Phase Design: The optimal reflecting phase θ* is selected to maximize the total received signal power.This phase choice follows from substituting the asymptotic channel expressions into the received-signal objective.
ZF Beamforming: ZF beamforming forces information leakage to each eavesdropper En to zero.The construction uses the effective eavesdropper channel matrix Ĝ and its null space.
ZF Beamforming: The ZF beamformer is constructed from zero-singular-value vectors of Ĝ and user-space vectors uk, with M − N available null-space dimensions.The beamformer also includes diagonal power allocation P = diag(p).
Power Allocation: After ZF removes leakage, the problem becomes a minimum-SINR maximization problem with total allocated power satisfying Σk=1 pk = P.The resulting beamformer structure uses positive virtual-dual-uplink power variables zk.
ZF Beamforming: ZF beamforming requires M ≥ N in the studied system.This dimensional condition is stated as necessary for the adopted ZF scheme.

VI. SIMULATION RESULTS

The simulations evaluate IRS-assisted secrecy-rate performance, algorithm convergence, and complexity across transmit power, IRS size, coefficient resolution, and user count. Results consistently support IRS benefits and show trade-offs between performance and computational complexity.

Transmit power: The ZF-based heuristic is worse than the “Rand” baseline when K = 2 but better when K = 1, indicating greater effectiveness for small K at lower complexity.The paper notes that the “Rand” baseline still has higher complexity than the ZF-based heuristic.
IRS size: Minimum secrecy rates for IRS-assisted methods increase with the number of reflecting elements L, whereas the no-IRS system remains constant.The paper attributes the improvement to higher array gain from larger IRSs.
Number of users: Minimum secrecy rate decreases as the number of legitimate users K increases because beamforming and array gains must be shared among more users.The low-complexity heuristic is especially effective relative to other suboptimal baselines when K = 1 or 2.
Convergence: All evaluated objective functions increase across iterations, while Algorithm 2 reaches convergence in fewer iterations than Algorithm 1.Algorithm 2 optimizes the original problem and provides a global optimum for each subproblem; Algorithm 1 with Φ = Φ2(1) converges more slowly than with Φ = Φ1.

APPENDIX

The appendix derives lower-bound approximations for the non-concave objective by exploiting convexity, concavity, and first-order expansions. These approximations support the iterative solution procedure.

Iterative approximation: The resulting approximated problem has an optimal value that can increase at every iteration toward a local or global optimum.The construction uses the current iterate as the expansion point for the next approximation.
Upper-bound construction: The appendix also derives an upper bound for f E_k,n(W, v) using a proof approach similar to prior work.The cited passage states that this upper-bound derivation is analogous to the proof in reference.
Lower-bound construction: The proof constructs a lower bound for f B_k(W, v) using convexity and a first-order Taylor expansion.It defines f(x, y) = −ln(1 − |x|^2/y) and expands it around (x̃, ỹ).
Lower-bound construction: The logarithm inequality ln(1 + z) ≤ ln(1 + z̄) + (z − z̄)/(1 + z̄) provides another bound used in the derivation.The inequality follows from concavity of ln(1 + z).

B. Proof of Theorem 4.2

The theorem proof shows that the sequence of iterates converges and that the limiting solution satisfies the KKT conditions. It also characterizes an optimal lifted matrix as rank one for recovering a vector solution.

Convergence and stationarity: The iterates (W(t), v(t)) converge to a limiting point (W*, v*) as t approaches infinity.The proof then evaluates the Lagrangian and corresponding KKT conditions at this limit.
Convergence and stationarity: The limiting solution (W*, v*) is a KKT point because it solves the relevant convex subproblems and satisfies all stated KKT conditions.The proof identifies W* and v* as optimal solutions of the two convex optimization problems.
Rank-one recovery: The proof rewrites intermediate equations to obtain the optimal vector and then determines the associated beamforming variable from a subsequent problem.These steps complete the theorem’s closed-form characterization.
Rank-one recovery: The optimal lifted matrix Ṽ has rank one and can therefore be represented as ṽṽ^H up to its scalar weight.The proof normalizes ṽ so that ||ṽ||^2 = 1 before recovering the vector solution.