Experiments with Clifford Geometric Algebra Applied to Cryptography

Abstract

The combination of flexibility, simplicity, elegance, and power that is found in Clifford Geometric Algebra (GA) is probably one of the main reasons for growing interest from those willing to explore new algebraic structures for producing many applications in physics, engineering, and computer science. As a result, to the surprise of many, existing applications, when modified to run as implementations of Clifford GA algorithms, demonstrate higher performance, better maintenance, less complexity, and a friendlier learning curve for those new to GA. Nevertheless, Clifford GA is virtually unexplored in cryptography, an area of science that can be greatly benefited by a mathematical tool equipped with the aforementioned potential. In this work, we introduce preliminary experiments of cryptographic solutions based on Clifford geometric algebra, including a key exchange protocol, a hash algorithm, and a private-key encryption scheme, with the hope of providing appealing pieces of evidence that this powerful mathematical resource is worth investigating as a strong candidate for broad adoption in cryptography. As proof of concept, we provide access to a library written in Ruby language in which we implement the ideas discussed in this work and where the reader will be able to find some predefined numerical examples.