Design of Ciphers based on the Geometric Structure of the M\"obius Plane

TL;DR

This paper explores the use of geometric structures, specifically the M"obius plane, to design cryptographic ciphers, extending prior work on geometric authentication schemes and demonstrating their cryptographic properties.

Contribution

It introduces a novel encryption scheme based on the M"obius plane and analyzes its cryptographic properties, including completeness and perfectness, expanding geometric cryptography applications.

Findings

The scheme fulfills the completeness requirement.

It approximates Shannon's perfectness.

Extends geometric cryptography to circle geometries.

Abstract

Till now geometric structures don't play a major role in cryptography. Gilbert, MacWilliams and Sloane introduced in 1974 an authentication scheme in the projective plane and showed its perfectness in the sense of the definition of Shannon. In this paper we will show that this authentication scheme also fulfills the requirement of completeness according to Kam and Davida and we will extend the application of geometric structures in cryptography by introducing an encryption scheme in the M\"obius plane. We will further examine its properties, showing that it also fulfills the requirement of completeness and Shannon's requirement of perfectness in first approximation. The results of this paper can be used to define similar encryption schemes in the circle geometries of Laguerre and Minkowski.