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Published in:
29-09-2021 | Original Research
A new cryptosystem based on a twisted Hessian curve \(H^{4}_{a,d}\)
Published in: Journal of Applied Mathematics and Computing | Issue 4/2022
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Abstract
In this paper, we are going to study the twisted Hessian curves on the local ring \(\mathbb {F}_{q}[\epsilon ]\), \(\epsilon ^{4}=0\), with \(\mathbb {F}_{q}\) is a finite field of order \(q=p^{b}\), where p is a prime number \( \ge 5\) and \(b\in \mathbb {N}^{*}\). In a first time, we study the arithmetic of the ring \(\mathbb {F}_{q}[\epsilon ]\), \(\epsilon ^{4}=0\), which will be used in the remainder of this work. After, we define the twisted Hessian curves \(H^{4}_{a,d}\) over this ring and we give essential properties and the classification of these elements. In addition, we define the group extension \(H^{4}_{a,d}\) of \(H_{a_{0},d_{0}}\) by \(Ker \ \tilde{\pi }\). We finish this work by introducing a new public key cryptosystem which is a variant of Cramer-Shoup public key cryptosystem on a twisted Hessian curves and study its security and efficiency. Our future work will focus on the generalist these studies for all integers \(n>4\), \(\epsilon ^{n}=0\), which is beneficial and interesting in cryptography.