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(2005 / vol.8 / no.1)
A Property for Cryptography based on Infinite Groups
Eonkyung Lee
Pages. 155-166     



Cryptography using infinite groups has been studied since about
twenty years ago. However, it has not been so fruitful as using
finite groups.
An important reason is the absence of research on
probability in this area. Indeed, a number of cryptographic tools
concerning probability are playing significant roles in analyses
in the case of finite groups.

Our purpose is twofold---to deal with not a particular finite
subset (as before) of an infinite group but the whole group
itself, and to make cryptographic tools developed in finite groups
still useful in infinite groups. As a first step to serve this
purpose, we study a probability-theoretic property, the so-called
{em right-invariance}, that has been widely used in cryptography.
Like the uniform distribution over finite sets, right-invariance
property simplifies many complex situations.
However, it can be unused or misused since it is not known when this property can be
used.

We propose a method of deciding whether or not we can use this property in a given situation,
and prove that there is no right-invariant probability distribution on most infinite groups
which can be universally used.
Therefore, we discuss weaker, yet practical alternatives with concrete examples.



1. Introduction
2. Preliminaries
3. Role of Right-Invariance
4. Right-Invariant Probability Space
5. Universally Right-Invariant Probability Measure and Alternatives
6. Conclusions



Probabilistic cryptanalysis, Infinite group,Right-closedness, Right-invariance