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arXiv:1706.03866 · IEEE Transactions on Information Theory

Wiretap Channels: Nonasymptotic Fundamental Limits

Wei Qing Yang, Rafael F. Schaefer, H. Vincent Poor

Published Mar 12, 2019• 141 citations
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Abstract

This paper investigates the maximal secret communication rate over a wiretap channel subject to reliability and secrecy constraints at a given blocklength. New achievability and converse bounds are derived, which are uniformly tighter than existing bounds, and lead to the tightest bounds on the second-order coding rate for discrete memoryless and Gaussian wiretap channels. The exact second-order coding rate is established for semi-deterministic wiretap channels, which characterizes the optimal tradeoff between reliability and secrecy in the finite-blocklength regime. Underlying our achievability bounds are two new privacy amplification results, which not only refine the classic privacy amplification results, but also achieve secrecy under the stronger semantic-security metric.

DOI
10.1109/tit.2019.2904500

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This paper investigates the maximal secret communication rate over a wiretap channel subject to reliability and secrecy constraints at a given blocklength. New achievability and converse bounds are derived, which are uniformly tighter than existing bounds, and lead to the tightest bounds on the second-order coding rate for discrete memoryless and Gaussian wiretap channels. The exact second-order coding rate is established for semi-deterministic wiretap channels, which characterizes the optimal tradeoff between reliability and secrecy in the finite-blocklength regime. Underlying our achievability bounds are two new privacy amplification results, which not only refine the classic privacy amplification results, but also achieve secrecy under the stronger semantic-security metric.
This paper studies the maximum secrecy rate for a semi-deterministic wiretap channel, in which the channel between the transmitter and the legitimate receiver is deterministic, while that between the transmitter and the eavesdropper is a discrete memoryless channel. For a given decoding error probability and information leakage (measured by the total variation distance), the optimal second-order secrecy rate is derived. Unlike the secrecy capacity, the second-order secrecy rate characterizes the optimal tradeoff between secrecy and reliability at finite blocklength.
We derive lower bounds to the second-order coding rates for the wiretap channel. The decoding error probability and the information leakage measured in terms of the variational distance secrecy metric are fixed at some constants εrand εsrespectively. We leverage on the connection between wiretap channel coding and channel resolvability to derive tighter secrecy bounds than those available in the literature. We then use central limit theorem-style analysis to evaluate these bounds for the discrete memoryless wiretap channel with cost constraints and the Gaussian wiretap channel.
We develop three new results regarding the second-order asymptotics of secure communication over wiretap channels. We first establish the optimal second-order asymptotics for a class of degraded wiretap channels without feedback under an effective secrecy criterion. We then derive the optimal second-order asymptotics for degraded wiretap channels with feedback. We finally develop a new converse bound for channel resolvability with non-capacity achieving distributions, which we use to develop useful converse bounds for asymmetric degraded wiretap channels. Our results are illustrated with several numerical examples, and suggest that known coding techniques already achieve their best performance.
This work discusses the bounds for the wiretap channel (WTC) in the finite blocklength regime, characterizing the behavior of the performance trade-off between information delivery to the legitimate receiver and information leakage to the eavesdropper as a function of blocklength. Since the error probability at the legitimate receiver is the same as that of a conventional point-to-point channel, the characterization of the former builds on the finite blocklength analysis for that case. The information leakage bound relies on the analysis of the corresponding leakage density. This paper extends these concepts by incorporating the use of the Berry-Esseen theorem in assessing the convergence rates of leakage and secrecy metrics to their asymptotic values, which provides a refined analytical tool to evaluate the performance of secrecy systems under finite blocklength constraints. Furthermore, this approach facilitates a quantification of the trade-offs involved, integrating concepts such as the aggregate leakage rate and empirical average leakage, thus enabling a more comprehensive understanding of how system parameters affect the overall secrecy performance. This allows for a more detailed exploration of the dependencies between blocklength, secrecy, and reliability trade-offs required to balance effective communication against security imperatives. These analyses also highlight the critical balance necessary between ensuring low probability of error for the legitimate receiver while maintaining stringent secrecy requirements against an eavesdropper. Furthermore, the study introduces metrics such as aggregate leakage rate and empirical average leakage, providing new insights into how these factors interplay to influence the overall performance of secure communication systems.
For physical-layer secrecy performance, existing studies only focus on coding schemes that can achieve the secrecy capacity. However, finite blocklength penalty could be fatal for security of private message. In this paper, we explore the impact of finite blocklength on secrecy for the wiretap channel model. We propose secrecy performance matrices for finite blocklength analysis and provide analytical expressions to evaluate the secrecy performance. We demonstrate trade-off between secrecy and reliability, and also provide an upper bound of secret information rate in finite blocklength case. The results are also applied to indicate asymptotic performance.