Limits of Reliable Communication with Low Probability of Detection on AWGN Channels (1202.6423v4)

Abstract: We present a square root limit on the amount of information transmitted reliably and with low probability of detection (LPD) over additive white Gaussian noise (AWGN) channels. Specifically, if the transmitter has AWGN channels to an intended receiver and a warden, both with non-zero noise power, we prove that $o(\sqrt{n})$ bits can be sent from the transmitter to the receiver in $n$ channel uses while lower-bounding $\alpha+\beta\geq1-\epsilon$ for any $\epsilon>0$, where $\alpha$ and $\beta$ respectively denote the warden's probabilities of a false alarm when the sender is not transmitting and a missed detection when the sender is transmitting. Moreover, in most practical scenarios, a lower bound on the noise power on the channel between the transmitter and the warden is known and $O(\sqrt{n})$ bits can be sent in $n$ LPD channel uses. Conversely, attempting to transmit more than $O(\sqrt{n})$ bits either results in detection by the warden with probability one or a non-zero probability of decoding error at the receiver as $n\rightarrow\infty$.

• The paper establishes the square root law, demonstrating that covert bits scale as O(√n) over n channel uses.
• It leverages Neyman–Pearson hypothesis testing to quantify the trade-off between detection probability and decoding error.
• The analysis offers practical guidelines for deploying covert communication strategies under realistic AWGN and power constraints.

Limits of Reliable Communication with Low Probability of Detection on AWGN Channels

The paper investigates the boundaries of transmitting information with low probability of detection (LPD) in additive white Gaussian noise (AWGN) channels. It introduces a concept termed the "square root limit", establishing that in scenarios where both the intended receiver and a potential eavesdropper (warden) experience non-zero noise, the amount of information reliably and covertly transmitted is limited to $o(\sqrt{n})$ bits over $n$ channel uses.

Key Contributions

1. Square Root Law: The paper proves that the maximum number of bits sent covertly scales with $\mathcal{O}(\sqrt{n})$, a significant restriction implied by asymptotic behavior. This limit is a mathematical boundary beyond which either detection by the warden occurs with high probability or the probability of decoding error at the receiver increases non-trivially.
2. Information-Theoretic Limits: Extending beyond the spread-spectrum techniques, this research fills a gap by analytically defining the bounds of LPD communication. It elaborates a scenario with both the receiver and warden channels being characterized by AWGN, framing conditions for covert communication.
3. Analysis Framework: Utilizing alternating hypotheses, the research employs statistical tools such as Neyman--Pearson hypothesis testing and total variation distance to delineate the conditions under which covert communication with low error probability is achievable.

Numerical and Theoretical Insights

• The results indicate that Alice (the transmitter) can maintain a combined detection error probability ($\alpha + \beta$) that sufficiently reaches or exceeds a threshold, given certain assumptions on noise power and shared randomness with Bob (the receiver).
• Detailed proofs show the achievability under realistic constraints such as peak power and average power limitations, reinforcing the robustness of the square root law.

Implications and Future Directions

Practical Impacts: The findings have relevance across fields requiring covert communication, such as military operations where avoiding detection is crucial. It underscores the limitations of encryption when detectability itself poses a threat.

Theoretical Extensions: By confirming the zero capacity for LPD communication, the work prompts further exploration into the characterization of multipath, fading scenarios, and alternative adversary models, such as active wardens.

Speculative Projections: The paper alludes to the potential for developing LPD techniques that rely on environmental noise; this could be revolutionary in network situations where traditional cryptographic approaches are infeasible.

In summary, the paper contributes to a deeper understanding of covert communications within noisy channels, setting the scene for practical applications and theoretical advancement in securing wireless communications from detection.