Pell hyperbolas in DLP-based cryptosystems

TL;DR

This paper explores the use of Pell hyperbolas in cryptography, introducing a new parameterization linked to Rédé rational functions, leading to more efficient ElGamal-based cryptosystems over Pell conics.

Contribution

It introduces a novel algebraic and geometric parameterization of Pell hyperbolas, enabling the development of more efficient cryptosystems based on these structures.

Findings

Proposed a new parameterization connected to Rédé rational functions

Developed three public key cryptosystems based on Pell hyperbolas

Achieved improved efficiency over classical finite field schemes

Abstract

We present a study on the use of Pell hyperbolas in cryptosystems with security based on the discrete logarithm problem. Specifically, after introducing the group's structure over generalized Pell conics (and also giving the explicit isomorphisms with the classical Pell hyperbolas), we provide a parameterization with both an algebraic and a geometrical approach. The particular parameterization that we propose appears to be useful from a cryptographic point of view because the product that arises over the set of parameters is connected to the R\'edei rational functions, which can be evaluated in a fast way. Thus, we exploit these constructions for defining three different public key cryptosystems based on the ElGamal scheme. We show that the use of our parameterization allows to obtain schemes more efficient than the classical ones based on finite fields.