Equivocations, Exponents and Second-Order Coding Rates under Various R\'enyi Information Measures

TL;DR

This paper analyzes the asymptotic behavior of equivocations, their decay rates, and second-order coding rates using Rényi information measures, extending security analysis beyond Shannon measures.

Contribution

It introduces a framework using Rényi information measures for security analysis, providing tight asymptotic results and bounds on second-order rates, generalizing previous Shannon-based approaches.

Findings

Proves tight asymptotic results for equivocation and decay exponents.

Establishes bounds on second-order asymptotics that match for large rates.

Develops non-asymptotic bounds evaluated via probabilistic limit theorems.

Abstract

We evaluate the asymptotics of equivocations, their exponents as well as their second-order coding rates under various R\'{e}nyi information measures. Specifically, we consider the effect of applying a hash function on a source and we quantify the level of non-uniformity and dependence of the compressed source from another correlated source when the number of copies of the sources is large. Unlike previous works that use Shannon information measures to quantify randomness, information or uniformity, we define our security measures in terms of a more general class of information measures--the R\'{e}nyi information measures and their Gallager-type counterparts. A special case of these R\'{e}nyi information measure is the class of Shannon information measures. We prove tight asymptotic results for the security measures and their exponential rates of decay. We also prove bounds on the second-order asymptotics and show that these bounds match when the magnitudes of the second-order coding rates are large. We do so by establishing new classes non-asymptotic bounds on the equivocation and evaluating these bounds using various probabilistic limit theorems asymptotically.