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.
DLP–based cryptosystems with Pell cubics / Dutto, Simone. - 126:(2023), pp. 123-136. (Intervento presentato al convegno Number-Theoretic Methods in Cryptology NuTMiC 2022) [10.4064/bc126-8].
DLP–based cryptosystems with Pell cubics
Simone Dutto
2023
Abstract
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.File | Dimensione | Formato | |
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https://hdl.handle.net/11583/2986145