Right-Invariance: A Property for Probabilistic Analysis of Cryptography Based on Infinite Groups

Infinite groups have been used for cryptography since about twenty years ago. However, it has not been so fruitful as using finite groups. An important reason seems the lack of research on building a solid mathematical foundation for the use of infinite groups in cryptography. As a first step for this line of research, this paper pays attention to a property, the so-called right-invariance, which makes finite groups so convenient in cryptography, and gives a mathematical framework for correct, appropriate use of it in infinite groups.