Design of Ciphers based on the Geometric Structure of the Laguerre and Minkowski Planes

TL;DR

This paper explores the design of cryptographic ciphers based on the geometric structures of Laguerre and Minkowski planes, demonstrating their potential for perfectness and completeness in encryption schemes.

Contribution

It introduces novel encryption schemes utilizing Laguerre and Minkowski geometries, extending geometric cryptography beyond previously studied projective and M"obius planes.

Findings

Laguerre cipher achieves Shannon's perfectness exactly.

Minkowski cipher approximates Shannon's perfectness.

Laguerre cipher satisfies Kam and Davida's completeness.

Abstract

Till now geometric structures don't play a major role in cryptography. Gilbert, MacWilliams and Sloane introduced an authentication scheme in the projective plane and showed its perfectness in the sense of Shannon. In arXiv:2102.10321 we introduced an encryption scheme in the M\"obius plane and showed that it fulfills Shannon's requirement of perfectness in first approximation and also the requirement of completeness according to Kam and Davida. In this paper we will apply a similar approach to define encryption schemes in the geometries of the Laguerre plande and the Minkowski plane. We will show that the encryption scheme in the Laguerre geometry meets Shannon's requirement of perfectness sharp and that the encryption scheme in the Minkowski geometry meets this requirement in first approximation. The Laguerre cipher also fulfills the requirement of completeness according to Kam and Davida.