An Extension of the Uniform Distribution to Infinite Groups from the Cryptographic Viewpoint
by
Eonkyung Lee
Sejong University, Seoul, Korea

Modern cryptography is deeply related to groups. Based on finite groups, there have been proposed various types of schemes and many of them are known to be secure. Compared to finite groups, in infinite groups there are only a few types of schemes (e.g. key agreement protocol or public key encryption) and none of them are not proved to be secure.

A natural question is how we can proceed one more step. An impediment to this seems to be connected with "probability". Indeed, the uniform distribution is popularly used in finite groups, but not available in infinite groups. Our motivation is that there is nothing discussed seriously for it in the literature on infinite-group-based cryptography.

As a first step for this line of research, we choose a particular probability-theoretic property, the so-called right-invariance, from finite groups, and formalize the notion in arbitrary groups. Next, we explore this property in infinite groups analyzing the structure of their s-algebra. Lastly, we discuss probability measures for right-invariance property both in ideal case and in practical case, and show applications.

Date received: March 8, 2005