On the Construction of Polar Codes for Achieving the Capacity of Marginal Channels

TL;DR

This paper introduces a polar coding scheme for secure communication over non-symmetric binary-input channels, achieving secrecy capacity and satisfying security and reliability conditions in degraded wiretap scenarios.

Contribution

It extends polar coding techniques to non-symmetric channels for secure communication, demonstrating the existence of frozen vectors that meet security and reliability criteria.

Findings

Achieves secrecy capacity in non-symmetric channels.

Ensures security and reliability with a specific coding scheme.

Empirically attains the upper bound of equivocation rate.

Abstract

Achieving security against adversaries with unlimited computational power is of great interest in a communication scenario. Since polar codes are capacity achieving codes with low encoding-decoding complexity and they can approach perfect secrecy rates for binary-input degraded wiretap channels in symmetric settings, they are investigated extensively in the literature recently. In this paper, a polar coding scheme to achieve secrecy capacity in non-symmetric binary input channels is proposed. The proposed scheme satisfies security and reliability conditions. The wiretap channel is assumed to be stochastically degraded with respect to the legitimate channel and message distribution is uniform. The information set is sent over channels that are good for Bob and bad for Eve. Random bits are sent over channels that are good for both Bob and Eve. A frozen vector is chosen randomly and is sent over channels bad for both. We prove that there exists a frozen vector for which the coding scheme satisfies reliability and security conditions and approaches the secrecy capacity. We further empirically show that in the proposed scheme for non-symmetric binary-input discrete memoryless channels, the equivocation rate achieves its upper bound in the whole capacity-equivocation region.