Cubic Curves, Finite Geometry and Cryptography

TL;DR

This paper surveys the geometry of non-singular cubic curves over finite fields, their group structures, and explores new cryptographic security enhancements and combinatorial point set constructions in finite geometry.

Contribution

It introduces a potential method to strengthen elliptic curve cryptography security using shared secrets and presents novel geometric constructions in finite planes.

Findings

Classification of cubic curves by inflexion points

Proposal of a shared secret to enhance cryptographic security

Construction of point sets with special properties in finite planes

Abstract

Some geometry on non-singular cubic curves, mainly over finite fields, is surveyed. Such a curve has 9,3,1 or 0 points of inflexion, and cubic curves are classified accordingly. The group structure and the possible numbers of rational points are also surveyed. A possible strengthening of the security of elliptic curve cryptography is proposed using a `shared secret' related to the group law. Cubic curves are also used in a new way to construct sets of points having various combinatorial and geometric properties that are of particular interest in finite Desarguesian planes.