A group law on the projective plane with applications in Public Key Cryptography

TL;DR

This paper introduces a novel group law on a subset of the projective plane over any field, with applications in cryptography, and analyzes its security based on the discrete logarithm problem.

Contribution

It proposes a new group law on the projective plane and explores its cryptographic applications, including a variant over rings for enhanced security.

Findings

Security is equivalent to the discrete logarithm problem in a cubic extension

A variant over rings increases security but requires longer keys

The group law enables Diffie-Hellman-like key exchange

Abstract

We present a new group law defined on a subset of the projective plane $\mathbb{F}P^2$ over an arbitrary field $\mathbb{F}$, which lends itself to applications in Public Key Cryptography, in particular to a Diffie-Hellman-like key agreement protocol. We analyze the computational difficulty of solving the mathematical problem underlying the proposed Abelian group law and we prove that the security of our proposal is equivalent to the discrete logarithm problem in the multiplicative group of the cubic extension of the finite field considered. Finally, we present a variant of the proposed group law but over the ring $\mathbb{Z}/pq\mathbb{Z}$, and explain how the security becomes enhanced, though at the cost of a longer key length.