DLP–based cryptosystems with Pell cubics

The classical Pell equation x 2−dy 2 = 1 can be extended to the cubic case considering the points (x, y, z) ∈ F 3 such that, for fixed r ∈ F, x 3 + ry 3 + r 2 z 3 − 3rxyz = 1. The set of solutions over a finite field Fq equipped with a generalized Brahmagupta product is a cyclic group for some choices of q and 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.