# Polar Coding for Achieving the Capacity of Marginal Channels in Nonbinary-Input Setting

## TL;DR

This paper extends polar coding techniques to achieve secrecy capacity in asymmetric nonbinary-input channels, ensuring reliable and secure communication even when the wiretap channel is stochastically degraded.

## Contribution

It introduces a novel coding scheme for nonbinary channels that guarantees secrecy capacity under asymmetric conditions, expanding the applicability of polar codes in secure communications.

## Key findings

- Polar codes can achieve secrecy capacity in nonbinary asymmetric channels.
- A coding scheme is proposed that ensures both reliability and security.
- Uniform message distribution is necessary for secrecy capacity achievement.

## Abstract

Achieving information-theoretic security using explicit coding scheme in which unlimited computational power for eavesdropper is assumed, is one of the main topics is security consideration. It is shown that polar codes are capacity achieving codes and have a low complexity in encoding and decoding. It has been proven that polar codes reach to secrecy capacity in the binary-input wiretap channels in symmetric settings for which the wiretapper's channel is degraded with respect to the main channel. The first task of this paper is to propose a coding scheme to achieve secrecy capacity in asymmetric nonbinary-input channels while keeping reliability and security conditions satisfied. Our assumption is that the wiretap channel is stochastically degraded with respect to the main channel and message distribution is unspecified. The main idea is to send information set over good channels for Bob and bad channels for Eve and send random symbols for channels that are good for both. In this scheme the frozen vector is defined over all possible choices using polar codes ensemble concept. We proved that there exists a frozen vector for which the coding scheme satisfies reliability and security conditions. It is further shown that uniform distribution of the message is the necessary condition for achieving secrecy capacity.