Achieving the Secrecy Capacity of Wiretap Channels Using Polar Codes (1007.3568v2)

Abstract: 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 the mutual information between the message and Eve's observations. Wyner showed that the situation is characterized by a single constant C_s, called the secrecy capacity, which has the following meaning: for all $\epsilon > 0$, there exist coding schemes of rate $R \ge C_s - \epsilon$ 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 of general wiretap channels. Our construction works for any instantiation of the wiretap channel model, as originally defined by Wyner, as long as both C_1 and C_2 are symmetric and binary-input. Moreover, we show how to modify our construction in order to achieve strong security, as 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.

• The paper introduces a polar coding scheme that achieves the secrecy capacity for symmetric binary-input wiretap channels.
• It employs low-complexity encoding and decoding with O(n log n) performance to ensure reliable communication and minimal information leakage.
• Extensive mathematical proofs validate that the scheme meets strong security conditions while operating near the theoretical secrecy capacity.

Achieving the Secrecy Capacity of Wiretap Channels Using Polar Codes

The research by Mahdavifar and Vardy explores the application of polar codes to achieve the secrecy capacity of wiretap channels, expanding upon Wyner's foundational wiretap channel model from 1975. This work innovatively employs polar codes to provide both reliable and secure communication over wiretap channels, explicitly tackling the construction of coding schemes that can achieve secrecy capacity for symmetric binary-input discrete memoryless channels (DMCs).

Background and Motivation

Wyner introduced the wiretap channel as a communication model where a sender (Alice) transmits messages to a legitimate receiver (Bob) with an eavesdropper (Eve) possibly intercepting these communications. The model's goal is to achieve both reliability, in terms of minimizing Bob’s decoding errors, and security, by minimizing information leakage to Eve. Wyner established the concept of secrecy capacity—the maximum rate at which secure communication can occur.

Existing approaches often rely on nonconstructive arguments, making practical implementation challenging, especially for cases where Eve’s channel is not a binary erasure channel. This paper fills this notable gap by leveraging polar codes, which are known for their capacity-achieving properties and low complexity, to construct practical codes that approach secrecy capacity under specific conditions.

Main Contributions

The authors propose a coding scheme utilizing polar codes that achieve the secrecy capacity for symmetric binary-input DMCs, where the eavesdropper's channel is degraded with respect to the main channel. Key points include:

1. Construction and Complexity: The schemes employ polar codes, whose encoding/decoding complexities are $\mathcal{O}(n \log n)$, making them practical for implementation.
2. Security Enhancement: Through intricate construction adjustments, the research introduces modifications to meet strong security conditions, as defined by Maurer, while maintaining rates close to secrecy capacity.
3. Extensive Analysis: Mathematical proofs validate that the scheme satisfies reliability and weak security conditions at rates approaching the secrecy capacity. Theoretical bounds on the mutual information further support the security claims.

Implications and Future Directions

The practical implications of these findings are significant, particularly in fields requiring secure communications such as cybersecurity and data privacy. By providing an explicit construction method for achieving secrecy capacity, the paper offers a feasible pathway for implementing secure communication strategies in real-world systems.

From a theoretical perspective, this work opens avenues for further explorations into non-symmetric and non-binary input channels, with the aim of broadening the applicability of polar codes for achieving secrecy capacity under more generalized conditions.

Future research could focus on refining the practical decoding strategies when the main channel is not noiseless, potentially leveraging methods like decision feedback or alternate polar decoding mechanisms. Additionally, expanding techniques to handle channels where the standard degradation condition does not hold poses an intriguing challenge.

In conclusion, Mahdavifar and Vardy successfully bridge the gap between theoretical and practical aspects of secure communication over wiretap channels, offering a robust framework that could reshape approaches to data security using polar coding techniques.