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Perfectly Covert Communication Assisted by an Intelligent Reflecting Surface

Or Elimelech, Asaf Cohen

TL;DR

The necessary and sufficient conditions for perfect covertness are characterized, defined as zero received energy at the warden (Willie), and a complete analytical solution for the two-element case is provided.

Abstract

This work investigates the fundamental limits of \emph{perfect} covert communication assisted by an Intelligent Reflecting Surface (IRS). We first characterize the necessary and sufficient conditions for perfect covertness, defined as zero received energy at the warden (Willie), and provide a complete analytical solution for the two-element case. For general array sizes, we prove that the probability of achieving perfect covertness converges to $1$ almost surely as the number of reflecting elements increases. To enable practical implementation, we formulate the phase design problem as an interference-minimization problem. A key contribution is proving that the objective satisfies the \emph{strict-saddle} property. Consequently, we establish that gradient descent with random initialization converges almost surely to a global minimizer (zero interference), thereby identifying a feasible, perfectly covert configuration whenever one exists. Finally, to address physical limitations, such as imperfect CSI, we introduce the notion of \emph{operational perfect covertness}. In this setting, we assume a bounded CSI error model and an analogous constraint on Willie. We derive robust conditions guaranteeing that the warden's detection capability remains effectively nullified even when exact signal cancellation is infeasible.

Perfectly Covert Communication Assisted by an Intelligent Reflecting Surface

TL;DR

The necessary and sufficient conditions for perfect covertness are characterized, defined as zero received energy at the warden (Willie), and a complete analytical solution for the two-element case is provided.

Abstract

This work investigates the fundamental limits of \emph{perfect} covert communication assisted by an Intelligent Reflecting Surface (IRS). We first characterize the necessary and sufficient conditions for perfect covertness, defined as zero received energy at the warden (Willie), and provide a complete analytical solution for the two-element case. For general array sizes, we prove that the probability of achieving perfect covertness converges to almost surely as the number of reflecting elements increases. To enable practical implementation, we formulate the phase design problem as an interference-minimization problem. A key contribution is proving that the objective satisfies the \emph{strict-saddle} property. Consequently, we establish that gradient descent with random initialization converges almost surely to a global minimizer (zero interference), thereby identifying a feasible, perfectly covert configuration whenever one exists. Finally, to address physical limitations, such as imperfect CSI, we introduce the notion of \emph{operational perfect covertness}. In this setting, we assume a bounded CSI error model and an analogous constraint on Willie. We derive robust conditions guaranteeing that the warden's detection capability remains effectively nullified even when exact signal cancellation is infeasible.
Paper Structure (16 sections, 12 theorems, 59 equations, 4 figures, 1 algorithm)

This paper contains 16 sections, 12 theorems, 59 equations, 4 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. System Model and Problem Statement
  3. System Model
  4. Perfect Covertness
  5. Perfect Covertness with $N=2$ IRS Elements
  6. With Instantaneous CSI of Willie's Link
  7. Perfect Covertness with $N>2$ IRS Elements and With Instantaneous CSI of Willie's Link
  8. Probability Analysis
  9. A Perfectly Covert IRS Phase-Design Scheme
  10. Proof of Convergence
  11. Imperfect CSI and Detection Resolution
  12. Detector Resolution and Operational Covertness
  13. Bounded CSI Uncertainty Model
  14. Conclusion
  15. Proof of Lipschitz Gradient
  16. ...and 1 more sections

Key Result

Lemma 1

$SNR_w^1 = 0$ iff $\mathcal{D}(P_{y_w}^0||P_{y_w}^1) = 0$.

Figures (4)

  • Figure 1: IRS-Assisted Covert Communication.
  • Figure 2: Geometrical visualization of the Perfect Covertness solution for the case of N=2 IRS elements.
  • Figure 3: Probability of solution existence vs. $\sigma_y$ for different $\sigma_{X_1}=\sigma_{X_2}$, where $N=2$ IRS elements.
  • Figure 4: (a) Solution probability vs. The number of IRS elements, where ${\sigma_{a,s}=\sigma_{s,w}=1}$. (b) Simulation results for $\underset{\boldsymbol{\phi}}{\min}\left|{\mathbf{g}_{s,w}^T \mathbf{\Theta} \mathbf{h}_{a,s}}\right|$, $\underset{\boldsymbol{\phi}}{\max}\left|{\mathbf{g}_{s,w}^T \mathbf{\Theta} \mathbf{h}_{a,s}}\right|$ and $\left|h_{a,w}\right|$ vs. the number of IRS elements, where ${\sigma_{a,w}=\sigma_{a,s}=\sigma_{s,w}=1}$.

Theorems & Definitions (18)

  • Definition 1: Perfect Covertness
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4: Feasibility Probability
  • Theorem 1
  • Lemma 5
  • Remark 1
  • Theorem 2
  • Definition 2: Strict Saddle
  • ...and 8 more