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Covert Communication in Intelligent Reflecting Surface-Assisted NOMA Systems: Design, Analysis, and Optimization

Lu Lv, Qingqing Wu, Zan Li, Zhiguo Ding, Naofal Al-Dhahir, Jian Chen

arXiv:2012.03244v1cs.IT

TL;DR

The paper proposes IRS-assisted downlink and uplink NOMA schemes for covert communication, exploiting IRS phase-shift uncertainty and Roy’s non-orthogonal transmission as cover. It evaluates Willie’s minimum average detection error probability and reports positive covert rates with improved performance over benchmark schemes.

  • Problem

    Existing approaches neglect useful uncertainty inherent in the considered system for protecting communication content.

  • Method

    The paper proposes IRS-assisted downlink and uplink NOMA schemes that exploit IRS phase-shift uncertainty and Roy’s non-orthogonal transmission as the cover medium.

  • Results

    The proposed schemes guarantee positive covert rates and significantly outperform other benchmark schemes.

  • Takeaways & Limitations

    Increasing Roy’s public-signal transmit power and the number of IRS reflecting elements benefits covert communication.

Abstract

from arXiv · show

In this paper, we investigate covert communication in an intelligent reflecting surface (IRS)-assisted non-orthogonal multiple access (NOMA) system, where a legitimate transmitter (Alice) applies NOMA for downlink and uplink transmissions with a covert user (Bob) and a public user (Roy) aided by an IRS. Specifically, we propose new IRS-assisted downlink and uplink NOMA schemes to hide the existence of Bob's covert transmission from a warden (Willie), which cost-effectively exploit the phase-shift uncertainty of the IRS and the non-orthogonal signal transmission of Roy as the cover medium without requiring additional uncertainty sources. Assuming the worst-case covert communication scenario where Willie can optimally adjust the detection threshold for his detector, we derive an analytical expression for the minimum average detection error probability of Willie achieved by each of the proposed schemes. To further enhance the covert communication performance, we propose to maximize the covert rates of Bob by jointly optimizing the transmit power and the IRS reflect beamforming, subject to given requirements on the covertness against Willie and the quality-of-service (QoS) at Roy. Simulation results demonstrate the covertness advantage of the proposed schemes and confirm the accuracy of the derived analytical results. Interestingly, it is found that covert communication is impossible without using IRS or NOMA for the considered setup while the proposed schemes can always guarantee positive covert rates.

I. INTRODUCTION

The paper motivates covert communication for hiding communication existence, then proposes IRS-assisted NOMA schemes that use IRS phase-shift uncertainty and public NOMA signaling as cover media. Analytical and numerical results indicate positive covert rates, improved covertness with stronger public transmission or more IRS elements, and impossibility without IRS or NOMA in the considered system.

  • Motivation: Physical-layer security can protect message content, but covert communication additionally hides whether communication is occurring.This need arises in settings such as tactical networks and monitored financial-institution networks.
  • Limitations of existing approaches: Existing covert strategies may degrade legitimate-user communication because of resource consumption and stringent covertness constraints.Some public-cover approaches also assume different codebooks or random transmit power that may be impractical.
  • Proposed approach: The paper proposes downlink and uplink IRS-assisted NOMA schemes using IRS phase-shift uncertainty and Roy’s non-orthogonal transmission as joint cover media.The schemes avoid requiring transmitter random power or warden noise uncertainty, making them simpler and more cost-effective than existing approaches.
  • Optimization: The schemes formulate joint transmit-power allocation and IRS reflect-beamforming optimization to improve covert communication performance.The optimization is applied to the proposed schemes while supporting their covertness design.
  • Results: The proposed schemes can always guarantee positive covert rates with non-zero transmit power, whereas covert communication is impossible without IRS or NOMA in the considered system.Increasing Roy’s transmit power and the IRS’s reflecting-element count helps degrade Willie’s detection performance.
  • Results: As Roy’s public-user transmit power grows large, Willie’s minimum average detection error probability approaches one, making detection resemble random guessing.The paper also reports that the analytical and numerical results provide useful insights into the proposed schemes.

II. SYSTEM MODEL

The system is an IRS-assisted NOMA network with Alice, covert user Bob, public user Roy, and warden Willie. It supports downlink and uplink transmission while using IRS phase shifts and signal design to improve reception and reduce Willie’s detection capability.

  • System architecture: The considered network contains Alice, Bob, Roy, Willie, and an IRS assisting downlink and uplink transmissions.Each node has one antenna and operates half-duplex; the IRS has N independently reconfigurable reflecting elements.
  • IRS operation: The IRS independently reflects phase-shifted incident signals to improve reception at Roy and Bob and create uncertainty at Willie.Its smart controller manages reflection reconfiguration and assists channel estimation for IRS-involved links.
  • Channel model: Channels follow quasi-static block fading with distance-based path loss, independent zero-mean unit-variance coefficients, and AWGN at each receiver.The model includes direct and IRS-reflected links among Alice, Bob, Roy, and Willie.
  • Transmission protocol: TDD is assumed for downlink and uplink, so channel reciprocity holds.The channel coefficients across the modeled links are assumed independent and identically distributed.
  • CSI assumptions: Alice knows instantaneous legitimate-link CSI but only statistical Willie-link CSI, whereas Willie is assumed to know the relevant instantaneous CSI.These assumptions represent a worst-case setting for covert communication design.
  • CSI assumptions: For uplink transmission, Roy and Bob know instantaneous CSI toward Alice but only statistical CSI for their Willie links, while Willie knows instantaneous CSI of their links.This asymmetric CSI availability supports covert design against an external warden.

III. COVERT COMMUNICATION IN IRS-ASSISTED DOWNLINK NOMA

The downlink scheme superimposes Roy’s public signal and Bob’s covert signal, using NOMA and IRS phase-shift uncertainty to make Bob’s transmission difficult for Willie to detect. It derives Willie’s detection model and shows why both NOMA and IRS are essential in the considered setup.

  • Scheme design: The proposed downlink scheme uses NOMA to transmit Roy’s public signal and Bob’s covert signal simultaneously.Bob sequentially decodes Roy before Bob’s signal, while Roy decodes its own signal by treating Bob’s signal as noise.
  • Scheme design: Roy’s higher allocated power, Pr ≥ Pb, supports successful SIC and better protects Bob’s covert communication.Willie uses energy detection rather than decoding, so the SIC order does not affect his detector.
  • Willie’s detector: Willie performs a Neyman–Pearson binary test using average received power and an optimally selected detection threshold.The hypotheses represent absence and presence of Alice’s covert transmission to Bob.
  • Covertness mechanism: IRS phase-shift uncertainty and Roy’s public transmission obscure whether Willie’s received-power change comes from Bob’s covert signal.The uncertainty is created without extra random transmit power or uncertain warden noise.
  • Necessity of NOMA: Without NOMA in an IRS-assisted OMA setting, Bob’s covert communication cannot be achieved because Willie can measure Bob’s non-zero received power.Without instantaneous Willie CSI, Alice cannot completely neutralize the signal received by Willie through IRS design.
  • Necessity of IRS: Replacing the IRS with a constant-power full-duplex relay also prevents covert communication because Willie can raise an alarm when Bob’s additional power is received.These comparisons identify IRS and NOMA as necessary components for the considered setup.

B. Detection Error Probability of Willie

The paper derives Willie’s minimum average detection error probability for the IRS-assisted downlink scheme under an optimally chosen threshold. The derivation uses a large-IRS approximation for the reflected-channel distribution and reveals how public-signal power affects covertness.

  • Distribution analysis: The phase value ψn is independently and uniformly distributed on [0, 2π), supporting the reflected-channel distribution analysis.This characterization is used before deriving the distribution of δN.
  • Distribution analysis: The analysis approximates the reflected-channel random variable δN by a complex Gaussian distribution when the IRS has sufficiently many elements.The exact distribution is difficult because δN sums complex-valued random variables with correlated real and imaginary components.
  • Detection-error derivation: The derived detection-error expression applies to an arbitrary threshold, after which Willie’s threshold is optimized to minimize detection error.The minimum average metric averages over the unknown direct Alice–Willie channel realization.
  • Detection-error derivation: Theorem 2 gives a closed-form expression for Willie’s minimum average detection error probability under the proposed IRS-assisted downlink scheme.The result is obtained by applying the total probability theorem and evaluating the resulting integral.
  • Covertness insight: As Pr approaches infinity with finite Pb, the minimum average detection error probability approaches 1.This asymptotic result indicates that sufficiently strong public transmission can make Willie’s decision equivalent to random guessing.

C. Joint Power and Beamforming Optimization

The paper maximizes Bob’s covert rate by jointly optimizing Alice’s transmit powers and IRS passive beamforming under power, Roy-QoS, SIC, and covertness constraints. An alternating optimization method combines power optimization with semidefinite relaxation and Gaussian randomization for beamforming.

  • Problem formulation: The optimization maximizes Bob’s covert rate by jointly selecting transmit powers and IRS reflect beamforming under power, QoS, SIC, and covertness constraints.Roy’s minimum-rate requirement and Willie’s minimum average detection-error requirement are explicitly included.
  • Problem formulation: The original problem is difficult because transmit powers and IRS phases are coupled, while the constraints and detection-error expression are non-convex or complicated.These properties prevent direct solution of the original formulation.
  • Power optimization: For fixed IRS phases, the method reduces the problem to transmit-power optimization and uses structural monotonicity results to simplify power allocation.The total power constraint is active at the optimum, with Pr + Pb = P_a^max.
  • Beamforming optimization: For fixed powers, the method formulates passive beamforming as a semidefinite problem by lifting the IRS phase vector into matrix U.The lifted formulation preserves unit-modulus constraints but retains a rank-one constraint.
  • Beamforming optimization: Semidefinite relaxation removes the rank-one constraint, and Gaussian randomization extracts an approximate rank-one beamforming solution when needed.The relaxed problem provides an upper bound when its solution has rank greater than one.
  • Algorithm: The alternating optimization algorithm repeatedly updates transmit powers and beamforming until the objective increase falls below a threshold, and it is guaranteed to converge.The power and beamforming subproblems are solved in alternating steps.

IV. COVERT COMMUNICATION IN IRS-ASSISTED UPLINK NOMA

The section introduces an IRS-assisted uplink NOMA scheme for covert communication and derives its worst-case detection performance and optimization procedure.

  • The proposed uplink NOMA scheme uses IRS assistance to achieve covert wireless communication.
  • Under Willie’s optimally selected detection threshold, the scheme’s minimum average detection probability is derived in closed form.
  • The covert rate is optimized through joint transmit-power and passive-beamforming optimization.

A. IRS-Assisted Uplink NOMA Scheme

The uplink scheme superposes Roy’s public and Bob’s covert signals, uses SIC at Alice, and exploits IRS phase-shift uncertainty and Roy’s transmission as cover.

  • Roy transmits a public signal while Bob simultaneously transmits a covert signal to Alice using uplink NOMA.
  • Roy’s public transmission is exploited as a cover for Bob’s covert signal transmission.
  • Users are ordered by composite-channel strength, with |g_ra|^2 ≥ |g_ba|^2, so Bob is decoded at the last SIC stage.This ordering helps Bob avoid severe inter-user interference.
  • Bob’s covert rate increases as his average transmit power increases.
  • The IRS phase shifts are designed from CSI of the direct and IRS-assisted Roy/Bob-to-Alice links to create phase-shift uncertainty at Willie.

B. Detection Error Probability of Willie

The analysis derives Willie’s optimal threshold and the resulting minimum average detection error probability for the IRS-assisted uplink NOMA scheme.

  • The analysis models relevant IRS-assisted channel sums using complex Gaussian approximations with zero mean and variance N.
  • The resulting false-alarm and miss-detection probabilities are used to characterize the scheme’s detection error probability.
  • Willie’s optimal detection threshold minimizes the detection error probability of the IRS-assisted uplink NOMA scheme.
  • The minimum average detection error probability is expressed in closed form for the proposed uplink scheme.
  • The derived observations about the minimum detection error probability are summarized in Remark 5.

C. Joint Power and Beamforming Optimization

The optimization jointly controls uplink transmit powers and IRS passive beamforming under power, Roy-QoS, and Willie-covertness constraints using alternating optimization.

  • The objective maximizes Bob’s covert rate through joint transmit-power and passive-beamforming optimization.
  • The optimization imposes individual transmit-power constraints, Roy’s QoS requirement, and a minimum detection-error constraint at Willie.
  • Because P̃_r, P̃_b, and Φ are highly coupled, the problem is decomposed into power and passive-beamforming subproblems solved by alternating optimization.
  • With fixed Φ, the transmit-power subproblem is solved using the stated optimal allocation coefficients.
  • With fixed transmit powers, the passive-beamforming problem uses unit-modulus IRS variables, semidefinite relaxation, CVX, and Gaussian randomization.
  • The downlink detection probability is evaluated against average transmit power and the number of IRS reflecting elements.

V. NUMERICAL STUDIES

Numerical studies evaluate the proposed IRS-assisted downlink and uplink NOMA schemes under stated channel and simulation settings. The results show improved covertness and covert rates, with joint power and IRS beamforming outperforming benchmarks.

  • Covertness evaluation: Increasing Roy’s transmit power improves covertness, whereas increasing Bob’s transmit power makes covert communication easier for Willie to detect.Roy’s public transmission acts as the cover, and stronger public power creates greater uncertainty at Willie.
  • Validation: The analytical detection-error results closely match simulations, confirming the accuracy of the derived expressions.This agreement is reported for the downlink and uplink evaluations.
  • Covertness evaluation: The minimum average detection error probability increases with IRS reflecting elements because larger arrays create greater phase-shift uncertainty for Willie.The same trend is reported for the downlink and uplink settings.
  • Covert-rate evaluation: The proposed scheme significantly outperforms random-phase, fixed-power, IRS-assisted OMA, and NOMA-without-IRS benchmarks in covert rate.The benchmark schemes IRS-assisted OMA and NOMA without IRS achieve zero covert rate in the reported downlink comparisons.
  • Covert-rate evaluation: Joint power and IRS beamforming design dynamically satisfies Roy’s QoS and Willie’s covertness constraints while maximizing Bob’s covert rate.The covert rate decreases as the covertness requirement becomes stricter or Roy’s minimum QoS rate increases.

B. IRS-Assisted Uplink NOMA Scheme

The uplink scheme exploits IRS phase-shift uncertainty and Roy’s public transmission to conceal Bob’s signal. Analytical detection-error expressions and joint optimization yield positive covert rates that outperform the benchmarks.

  • Covertness performance: The minimum average detection error probability increases with IRS size and with Roy’s power, but decreases with Bob’s power.These trends are attributed to stronger IRS phase-shift uncertainty and a stronger public cover signal.
  • Covert-rate performance: The proposed uplink scheme achieves a remarkable covert-rate gain over random-phase, fixed-power, IRS-assisted OMA, and NOMA-without-IRS benchmarks.The comparisons vary the common transmit-power budget and the number of reflecting elements.
  • Scheme and rationale: The uplink scheme uses IRS phase-shift uncertainty and Roy’s non-orthogonal public transmission as the cover medium for Bob’s covert communication.The design is intended to avoid requiring additional uncertainty sources.
  • Optimization: Joint optimization maximizes Bob’s covert rate subject to Willie’s minimum covertness requirement and Roy’s QoS requirement.The optimization jointly selects transmit power and IRS passive beamforming.
  • Conclusion: The proposed schemes guarantee positive covert rates and significantly outperform other benchmark schemes in the reported simulations.The conclusion identifies positive covert rates as a central advantage of the proposed designs.
  • Conclusion: Increasing Roy’s public-signal power and the number of IRS reflecting elements benefits covert communication performance.This conclusion summarizes the reported covertness trends.

APPENDIX A: PROOF OF LEMMA 1

The appendix proves the phase variables’ distributional properties and analyzes detection-threshold optimization and algorithm convergence. It establishes uniformity, independence, worst-case detection behavior, and non-decreasing objective values.

  • Threshold optimization: The downlink detection-error objective is optimized piecewise across three regions of Willie’s detection threshold τ_dl.Its monotonicity changes across the threshold regions, determining the optimal threshold where one exists.
  • Threshold optimization: In one channel-distance regime, Willie’s minimum average detection error probability is always 1, so an optimal detection threshold does not exist.This is identified as Willie’s worst-case detection performance in that regime.
  • Optimization properties: The transmit-power constraint is active at the optimum because increasing Bob’s power increases the covert-rate objective when unused budget remains.The proof scales Bob’s power until Pr + Pb = Pmax.
  • Algorithm convergence: Algorithm 1 has a non-decreasing objective sequence and converges because the objective is bounded above.The alternating updates optimize power variables and IRS beamforming in successive steps.
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