Fundamental Limits of Communication with Low Probability of Detection (1506.03236v3)

Abstract: This paper considers the problem of communication over a discrete memoryless channel (DMC) or an additive white Gaussian noise (AWGN) channel subject to the constraint that the probability that an adversary who observes the channel outputs can detect the communication is low. Specifically, the relative entropy between the output distributions when a codeword is transmitted and when no input is provided to the channel must be sufficiently small. For a DMC whose output distribution induced by the "off" input symbol is not a mixture of the output distributions induced by other input symbols, it is shown that the maximum amount of information that can be transmitted under this criterion scales like the square root of the blocklength. The same is true for the AWGN channel. Exact expressions for the scaling constant are also derived.

• The paper demonstrates that the maximum covert information rate scales with the square-root of blocklength over various channel models.
• The study employs Fisher Information and relative entropy to precisely characterize scaling behaviors and constants under low detection constraints.
• The authors analyze covert communication without relying on wiretap assumptions, providing insights critical for secure and undetectable data transmission.

Fundamental Limits of Communication with Low Probability of Detection

The paper "Fundamental Limits of Communication with Low Probability of Detection" by Ligong Wang, Gregory W. Wornell, and Lizhong Zheng investigates the theoretical constraints on covert communication over discrete memoryless channels (DMC) and additive white Gaussian noise (AWGN) channels. The novelty of the work lies in characterizing the maximum information rate that achieves covert communication, defined by ensuring that an adversary cannot detect whether communication is occurring.

Key Contributions

1. Communication with Low Probability of Detection (LPD): The paper analyzes communication protocols where the probability of detection by an adversary must be minimal. This involves ensuring that the output distribution when transmitting does not deviate significantly from the distribution when transmission is absent.
2. Square-Root Law: The authors demonstrate that the maximum information rate under the LPD constraint scales with the square root of the blocklength for a large class of DMCs and AWGN channels. This finding confirms and generalizes the square-root law reported in previous studies for specific channels.
3. Exact Characterization: Precise characterizations of the scaling behaviors and constants associated with the maximum information rate are provided, which goes beyond prior work that established the existence of the square-root law without offering exact formulations.
4. Information-Theoretic Perspective Without the Wiretap Assumption: Unlike some earlier works that assumed either a noisier channel for the eavesdropper or the use of a secret key, this paper assumes no such constraints, focusing on the pure LPD problem where both the legitimate receiver and the adversary observe the same outputs.
5. Channels and Techniques: The paper discusses results for both binary symmetric channels and Gaussian channels, employing tools like Fisher Information and relative entropy as metrics for probability of detection, providing a more tractable theoretical framework.

Theoretical and Practical Implications

The results are significant for applications involving covert communications, such as in military or secure communication settings where the mere detection of communication is a concern. The square-root scaling law implies that achieving truly covert communication requires exceedingly low rates as the blocklength increases, which may influence how systems are designed in practice for scenarios where covert capabilities are needed.

On a theoretical level, the paper enriches the understanding of how covert objectives constrain communication rates differently than traditional secrecy goals. This challenges the design strategies in scenarios where communication must not only remain secret but also undetected.

Speculations on Future Directions

Future research may explore extending these results in several directions. One possibility includes analyzing the effect of memory in the channel, which the authors suggest could potentially allow positive rates even under stringent LPD constraints. Another avenue could involve extending the framework to more complex channel models or incorporating realistic constraints such as power limitations. Additionally, the boundary conditions of the square-root law across a wider array of channel types, and exploring practical code constructions resonating with these theoretical findings, are potential areas for advancement.

Overall, the paper provides a robust foundation for understanding the covert facets of communication theory, with both immediate applications and numerous points of departure for future inquiry in this nuanced subfield.