Characterization of the Arithmetic Complexity of the Secrecy Capacity of Fast-Fading Gaussian Channels

TL;DR

This paper investigates whether the secrecy capacity of fast-fading Gaussian channels can be computed algorithmically, revealing that for some distributions it is non-computable and not classifiable within the arithmetical hierarchy.

Contribution

It introduces an algorithmic perspective to the secrecy capacity problem, showing non-computability results and classifying the problem within the arithmetical hierarchy.

Findings

Secrecy capacity can be non-computable for certain fading distributions.

No computable bounds exist for the achievability and converse of secrecy capacity.

The problem's classification within the arithmetical hierarchy is established.

Abstract

This paper studies the computability of the secrecy capacity of fast-fading wiretap channels from an algorithmic perspective, examining whether it can be computed algorithmically or not. To address this question, the concept of Turing machines is used, which establishes fundamental performance limits of digital computers. It is shown that certain computable continuous fading probability distribution functions yield secrecy capacities that are non-computable numbers. Additionally, we assess the secrecy capacity's classification within the arithmetical hierarchy, revealing the absence of computable achievability and converse bounds.