New Results on the DMC Capacity and Renyi's Divergence

Abstract

This work is part of a project "Walsh Spectrum Analysis and the Cryptographic Applications". The project initiates the study of finding the largest (and/or significantly large) Walsh coefficients as well as the index positions of an unknown distribution by random sampling. This proposed problem has great significance in cryptography and communications. In early 2015, Yi JANET Lu first constructed novel imaginary channel transition matrices and introduced Shannon's channel coding problem to statistical cryptanalysis. For the first time, the channel capacity results of well-chosen transition matrices, which might be impossible to calculate traditionally, become of hottest research focus. For a few Discrete Memoryless Channels (DMCs), it is known that the capacity can be computed analytically; in general, there is no closed-form solution. This work is concerned with analytical results of channel capacity in the new setting. We study both the Blahut-Arimoto algorithm (which gave the first numerical solution historically) and the most recent results [Sutter et al'2014] for the transition matrix of \(N\times M\). For an \(\epsilon\)-approximation (i.e., the desired absolute accuracy of the approximate solution) of the capacity, the former has the computational complexity \( O(MN^2 \log N/\epsilon) \), while the latter has the complexity \( O(M^2N\sqrt{\log N}/\epsilon) \). We also study the relation of Renyi's divergence of degree \(1/2\) and the generalized channel capacity of degree \(1/2\).