Cayley Integers (long version)

Abstract

We present here some results of applying the Cayley-Dickson process to certain alternative algebras (notably built upon Galois fields and congruence rings), in a manner which might yield new building blocks for cryptographic systems. We focus on enumeration properties rather than the classification and comparison questions which are extensively studied elsewhere, at least for algebras over fields. The results presented here are technical but not inherently difficult.