Variations on a Theme by Massey

TL;DR

This paper explores the relationship between discrete and differential entropy, establishing bounds that connect Massey's guessing entropy with continuous entropy measures, and applies these to derive bounds on guessing difficulty in cryptography.

Contribution

It links Massey's inequalities for entropy and guessing entropy within a unified framework relating discrete and continuous entropy measures, providing new bounds for cryptographic guessing metrics.

Findings

Discrete entropy can be upper bounded by differential entropy of a related continuous variable.

Lower bounds on guessing entropy are derived using entropy and R'enyi entropy measures.

Results apply to scenarios with and without side information in cryptographic contexts.

Abstract

In 1994, Jim Massey proposed the guessing entropy as a measure of the difficulty that an attacker has to guess a secret used in a cryptographic system, and established a well-known inequality between entropy and guessing entropy. Over 15 years before, in an unpublished work, he also established a well-known inequality for the entropy of an integer-valued random variable of given variance. In this paper, we establish a link between the two works by Massey in the more general framework of the relationship between discrete (absolute) entropy and continuous (differential) entropy. Two approaches are given in which the discrete entropy (or R\'enyi entropy) of an integer-valued variable can be upper bounded using the differential (R\'enyi) entropy of some suitably chosen continuous random variable. As an application, lower bounds on guessing entropy and guessing moments are derived in terms of entropy or R\'enyi entropy (without side information) and conditional entropy or Arimoto conditional entropy (when side information is available).