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arXiv:2407.01401 · IEEE International Symposium on Information Theory (ISIT)

Finite-Length Analysis of Polar Secrecy Codes for Wiretap Channels

Hessam Mahdavifar, Fariba Abbasi

Published Jul 7, 2024• 1 citation
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Abstract

In a classical wiretap channel setting, Alice communicates with Bob through a main communication channel, while her transmission also reaches an eavesdropper Eve through a wiretap channel. In this paper, we consider a general class of polar secrecy codes for wiretap channels and study their finite-length performance. In particular, bounds on the normalized mutual information security (MIS) leakage, a fundamental measure of secrecy in information-theoretic security frameworks, are presented for polar secrecy codes. The bounds are utilized to characterize the finite-length scaling behavior of polar secrecy codes, where scaling here refers to the non-asymptotic behavior of both the gap to the secrecy capacity as well as the MIS leakage. Furthermore, the bounds are shown to facilitate characterizing numerical bounds on the secrecy guarantees of polar secrecy codes in finite block lengths of practical relevance, where directly calculating the MIS leakage is in general infeasible.

DOI
10.1109/isit57864.2024.10619177

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In a classical wiretap channel setting, Alice communicates with Bob through a main communication channel, while her transmission also reaches an eavesdropper Eve through a wiretap channel. In this paper, we consider a general class of polar secrecy codes for wiretap channels and study their finite-length performance. In particular, bounds on the normalized mutual information security (MIS) leakage, a fundamental measure of secrecy in information-theoretic security frameworks, are presented for polar secrecy codes. The bounds are utilized to characterize the finite-length scaling behavior of polar secrecy codes, where scaling here refers to the non-asymptotic behavior of both the gap to the secrecy capacity as well as the MIS leakage. Furthermore, the bounds are shown to facilitate characterizing numerical bounds on the secrecy guarantees of polar secrecy codes in finite block lengths of practical relevance, where directly calculating the MIS leakage is in general infeasible.
Suppose Alice wishes to send messages to Bob through a communication channel C_1, but her transmissions also reach an eavesdropper Eve through another channel C_2. The goal is to design a coding scheme that makes it possible for Alice to communicate both reliably and securely. Reliability is measured in terms of Bob's probability of error in recovering the message, while security is measured in terms of Eve's equivocation ratio. Wyner showed that the situation is characterized by a single constant C_s, called the secrecy capacity, which has the following meaning: for all ε>0ε> 0ε>0, there exist coding schemes of rate R≥Cs−εR \ge C_s - εR≥Cs​−ε that asymptotically achieve both the reliability and the security objectives. However, his proof of this result is based upon a nonconstructive random-coding argument. To date, despite a considerable research effort, the only case where we know how to construct coding schemes that achieve secrecy capacity is when Eve's channel C_2 is an erasure channel, or a combinatorial variation thereof. Polar codes were recently invented by Arikan; they approach the capacity of symmetric binary-input discrete memoryless channels with low encoding and decoding complexity. Herein, we use polar codes to construct a coding scheme that achieves the secrecy capacity for a wide range of wiretap channels. Our construction works for any instantiation of the wiretap channel model, as long as both C_1 and C_2 are symmetric and binary-input, and C_2 is degraded with respect to C_1. Moreover, we show how to modify our construction in order to provide strong security, in the sense defined by Maurer, while still operating at a rate that approaches the secrecy capacity. In this case, we cannot guarantee that the reliability condition will be satisfied unless the main channel C_1 is noiseless, although we believe it can be always satisfied in practice.
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.
This letter discusses a secrecy transmission scheme based on 2-D polar coding over block fading wiretap channels. Unlike the previous approaches using 1-D polar coding, we propose to encode the secret bits in two dimensions to adapt to variation of the instantaneous secrecy capacity. In the proposed scheme, intra-block coding in one dimension is used to guarantee reliability, and cross-block coding in the other dimension is used to combat eavesdropping. Theoretical analysis demonstrates that our scheme is able to achieve the maximum perfect secrecy rate asymptotically. Simulation results show that the equivocation rate of eavesdropper is close to the secrecy capacity of the system with finite codelength.
Wyner's work on wiretap channels and the recent works on information theoretic security are based on random codes. Achieving information theoretical security with practical coding schemes is of definite interest. In this note, the attempt is to overcome this elusive task by employing the polar coding technique of Arıkan. It is shown that polar codes achieve non-trivial perfect secrecy rates for binary-input degraded wiretap channels while enjoying their low encoding-decoding complexity. In the special case of symmetric main and eavesdropper channels, this coding technique achieves the secrecy capacity. Next, fading erasure wiretap channels are considered and a secret key agreement scheme is proposed, which requires only the statistical knowledge of the eavesdropper channel state information (CSI). The enabling factor is the creation of advantage over Eve, by blindly using the proposed scheme over each fading block, which is then exploited with privacy amplification techniques to generate secret keys.
This paper studies secrecy capacity in a semi-deterministic setting, in which the channel between legitimate users (called Alice and Bob) is deterministic, while that between Alice and the eavesdropper (called Eve) is a discrete memoryless channel. Such a model is particularly relevant when a pre-existing error correcting code tailored to the legitimate channel is in use on top of which secret information is to be shared. First, a point-to-point setting is considered with a single wiretapper, a situation in which the secrecy capacity has an elegant characterization. Next, a generalized multiple access setting with confidential messages is considered in which each user wishes to communicate secret information to a common destination without the other determining its message. In this latter situation, outer bounds on the secrecy capacity are obtained.
Practical codes for the Gaussian wiretap channel are designed aiming at satisfying information-theoretic metrics to ensure security against a passive eavesdropper (Eve). Specifically, a design criterion is introduced for the coset coding scheme in order to satisfy a strong secrecy condition described with the mutual information between the secret message and Eve's observation. In addition, mutual information neural estimation (MINE) powered from deep learning tools is applied in order to directly compute the information-theoretic security constraint, and verify the proposed solutions. It is shown that finite-length coset codes can indeed ensure secure transmission from an information-theoretic perspective.