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

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.