Covert Communication over Noisy Channels: A Resolvability Perspective (1503.08778v3)

Abstract: We consider the situation in which a transmitter attempts to communicate reliably over a discrete memoryless channel while simultaneously ensuring covertness (low probability of detection) with respect to a warden, who observes the signals through another discrete memoryless channel. We develop a coding scheme based on the principle of channel resolvability, which generalizes and extends prior work in several directions. First, it shows that, irrespective of the quality of the channels, it is possible to communicate on the order of $\sqrt{n}$ reliable and covert bits over $n$ channel uses if the transmitter and the receiver share on the order of $\sqrt{n}$ key bits; this improves upon earlier results requiring on the order of $\sqrt{n}\log n$ key bits. Second, it proves that, if the receiver's channel is "better" than the warden's channel in a sense that we make precise, it is possible to communicate on the order of $\sqrt{n}$ reliable and covert bits over $n$ channel uses without a secret key; this generalizes earlier results established for binary symmetric channels. We also identify the fundamental limits of covert and secret communications in terms of the optimal asymptotic scaling of the message size and key size, and we extend the analysis to Gaussian channels. The main technical problem that we address is how to develop concentration inequalities for "low-weight" sequences; the crux of our approach is to define suitably modified typical sets that are amenable to concentration inequalities.

• The paper introduces a novel framework using channel resolvability to achieve reliable and covert communication over noisy channels.
• It operationalizes conditions for achieving covert communication without secret keys and determines asymptotic scaling laws for message and key sizes.
• The proposed coding schemes leverage both source-resolvability and channel-resolvability, extending prior results to general discrete memoryless channels.

An Analysis of Covert Communication from a Resolvability Perspective

The paper by Matthieu R. Bloch titled "Covert Communication over Noisy Channels: A Resolvability Perspective" explores the intersection of reliable communication and low probability of detection in noisy discrete memoryless channels (DMCs). Central to the analysis is an exploration of covert communication, where the sender transmits messages not only reliably but also covertly, meaning without being detected by a warden observing through another DMC. This is an extension of previous works by utilizing channel resolvability as a principal framework.

Contributions and Results

The paper presents significant advances in understanding how covert communication can be efficiently achieved without heavy reliance on secret keys. The primary contributions can be summarized as follows:

1. Generalized Covert Communication Scheme: The paper extends prior results to demonstrate that, regardless of the channel conditions, reliable and covert communication is feasible using approximately $\sqrt{n} \log n$ secret key bits.
2. Keyless Communication Achievements: It operationalizes the condition under which communication can occur without any secret key, provided the receiver's channel qualitatively supersedes that of the warden. This is a marked generalization of existing results confined to binary symmetric channels (BSCs).
3. Asymptotic Scalings and Fundamental Limits: The work determines the asymptotic scaling laws for both message and key sizes, showing optimal covert communication limits in various parameter regimes, including continuous cases such as Gaussian channels.
4. Channel Resolvability Framework: A novel coding scheme utilizing channel resolvability is proposed to portray covert processes as "low-weight" sequences that traditional concentration inequalities do not typically cover.

Technical Insights and Methodologies

The framework leverages resolvability, traditionally applied to source coding, to develop robust communication schemes where covert messages remain statistically hidden amongst channel noise. By defining "modular typical sets," Bloch solves the problem of achieving concentration inequalities suitable for the non-standard low-weight scenarios. This approach aligns with addressing the dual goal of covertness and reliability, laying new pathways where secret-key requirements are minimized.

The proposed coding schemes underscore two primary cases:

• Source-Resolvability-Based Scheme: This scheme corresponds to revisiting previous covert communication strategies (like the AWGN case) but with nuanced adjustments to accommodate non-binary channels.
• Channel-Resolvability-Based Scheme: In contrast, this method focuses on leveraging diversity in channel quality, allowing for communication with minimal or even no secret-key bits under favorable channel conditions.

Implications and Future Directions

Theoretical implications of these findings gesture towards a unified understanding of covert communication constrained by channel resolvability. Practically, this means more secure and efficient communication systems that mitigate unnecessary key exchange. The results bear potential applications in dynamic spectrum access, secure wireless networks, and regulatory-compliant communication strategies.

Future research could explore extending such resolvability frameworks to address time-varying channels and adaptive communication environments. Another avenue could examine quantum channels, continuing Bloch's previous ventures into quantum-secure covert communication, aiming to cover a broader spectrum of practical scenarios.

Overall, this paper advances the state of the art in both the theory and application of covert communication systems by embedding covertness into the core design of communication schemes, leveraging the statistical nature of DMCs effectively. This adds an important dimension to the secure communication landscape, emphasizing the dual potential of resolvability techniques in both message and channel domains.

This work by Matthieu R. Bloch is a detailed exploration of resolvability in covert communication, expanding the boundaries of how we understand and apply communication secrecy principles in modern networks.