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Covert Communication Over Additive-Noise Channels

Cécile Bouette, Laura Luzzi, Ligong Wang

TL;DR

This paper resolves the covert communication problem over general memoryless additive-noise channels by establishing a universal upper bound on the square-root scaling constant $L$, namely $L\le\sqrt{2}\sqrt{\mathrm{Var}[\ln(p_Z(Z))]}$, and showing tightness under additional regularity assumptions on the noise PDF. It provides concrete expressions or bounds for $L$ in key noise models—exponential, generalized Gaussian, and generalized gamma—thereby extending the square-root law beyond AWGN. The authors also derive finite-key bounds, showing that a finite secret-key length $|\mathcal{K}|$ suffices to achieve the optimal scaling, with tighter requirements in Gaussian and exponential cases. The approach combines converse arguments based on entropy and KL-divergence with achievability via perturbations of the noise and channel-resolvability techniques, offering practical insights into when covert communication is feasible under non-Gaussian noise.

Abstract

We study the fundamental limits of covert communications over general memoryless additive-noise channels. We assume that the legitimate receiver and the eavesdropper share the same channel and therefore see the same outputs. Under mild integrability assumptions, we find a general upper bound on the square-root scaling constant, which only involves the variance of the logarithm of the probability density function of the noise. Furthermore, we show that, under some additional assumptions, this upper bound is tight. We also provide upper bounds on the length of the secret key required to achieve the optimal scaling.

Covert Communication Over Additive-Noise Channels

TL;DR

This paper resolves the covert communication problem over general memoryless additive-noise channels by establishing a universal upper bound on the square-root scaling constant , namely , and showing tightness under additional regularity assumptions on the noise PDF. It provides concrete expressions or bounds for in key noise models—exponential, generalized Gaussian, and generalized gamma—thereby extending the square-root law beyond AWGN. The authors also derive finite-key bounds, showing that a finite secret-key length suffices to achieve the optimal scaling, with tighter requirements in Gaussian and exponential cases. The approach combines converse arguments based on entropy and KL-divergence with achievability via perturbations of the noise and channel-resolvability techniques, offering practical insights into when covert communication is feasible under non-Gaussian noise.

Abstract

We study the fundamental limits of covert communications over general memoryless additive-noise channels. We assume that the legitimate receiver and the eavesdropper share the same channel and therefore see the same outputs. Under mild integrability assumptions, we find a general upper bound on the square-root scaling constant, which only involves the variance of the logarithm of the probability density function of the noise. Furthermore, we show that, under some additional assumptions, this upper bound is tight. We also provide upper bounds on the length of the secret key required to achieve the optimal scaling.
Paper Structure (15 sections, 8 theorems, 48 equations, 1 figure)

This paper contains 15 sections, 8 theorems, 48 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. A General Upper Bound on $L$
  4. Tightness of the Upper Bound
  5. Examples
  6. Exponential noise
  7. Generalized Gaussian noise
  8. Generalized gamma noise
  9. Bounds on the Key Length
  10. Concluding Remarks
  11. Uniform Noise
  12. Proofs of Propositions \ref{['proposition:minimum_key_length_general_case']} and \ref{['proposition:minimum_key_length_gaussian_exponential']}
  13. Gaussian noise
  14. Exponential noise
  15. Derivation of \ref{['L_for_ggamma_noise']}

Key Result

Lemma 1

If $p_Z$ satisfies eq:integrable0--eq:integrable2, then, for every $k\in\{0,1,2,3,4\}$, the integrals are uniformly bounded over $\nu\colon \mathbb{R}\to[\zeta,1]$, $z\mapsto \nu(z)$.

Figures (1)

  • Figure 1: General setup for covert communication.

Theorems & Definitions (14)

  • Lemma 1
  • Theorem 1
  • Lemma 2
  • Remark 1
  • Remark 2
  • Theorem 2
  • Definition 1: Characteristic function williams_probability_2018
  • Definition 2: Weak convergence williams_probability_2018
  • Theorem 3: Lévy's convergence theorem williams_probability_2018
  • Lemma 3
  • ...and 4 more