DLP-based cryptosystems with Pell cubics

The classical Pell equation $x^2 - \mathrm{D}y^2 = 1$ can be extended to the cubic case considering the points $(x, y, z) \in \mathbb{F}^3$ such that, for fixed $\mathrm{R}\in \mathbb{F}$, $$ x^3 + \mathrm{R}y^3 + \mathrm{R}^2 z^3 - 3 \mathrm{R}x y z = 1. $$ The set of solutions over a finite field $\mathbb{F}_q$ equipped with a generalized Brahmagupta product is a cyclic group for some choices of $q$ and $\mathrm{R}$. In these cases, novel cryptosystems can be built exploiting the discrete logarithm problem over this group. This paper focuses on the study of ElGamal-based cryptosystems as well as digital signature schemes with the Pell cubic. Finally, a comparison in terms of security, data-size and performance among these cryptosystems and the classical versions with finite fields, elliptic curves and also with Pell conics is provided.