A New Fast Algorithm for Computation of \({log_{\omega } \left( {2} \right)}\) on Finite Fermat Fields

The quantity of data generated by the Internet of Things creates new challenges for the pillars of security (confidentiality, integrity and availability). These challenges are due to the diversity of characteristics of connected objects: size, computing power, storage capacity and autonomy. Based on this observation, the criticality of this diversity requires us to define new security algorithms capable of processing all objects in all circumstances. To overcome the majority of these new challenges, the use of cryptographic elliptic curves (ECC) is essential for any development in the field. The difficulty of research in this area is linked to the complexity of arithmetic in a finite field. The latter is a central axis to be optimized in the coming years to make ECCs more adapted to all the constraints of connected objects. The aim of this work is to propose a fast algorithm allowing to calculate the discrete logarithm of number two in a finite Fermat field \({\mathbb{F}}_{{\text{p}}}\) . Therefore, this problem consists to find an integer k ∈ {1,…, p − 1} such that \(\omega^{k} = 2\) to gather ω is a primitive element of a Fermat field \({\mathbb{F}}_{{\text{p}}}\) .