Cryptography Over the Twisted Hessian Curve $$H^{3}_{a,d}$$

In this paper, we will give some properties of the twisted Hessian curve over the ring \(\mathbb {F}_{q}[\epsilon ]\) denoted by \(H^{3}_{a,d}\), with \(\mathbb {F}_{q}\) is a finite field of order \(q=p^{b}\), where p is a prime number \( \ge 5\) and \(b \in \mathbb {N}^{*}\), and we prove that when p doesn’t divide \(\# (H_{a_{0},d_{0}})\), then \(H^{3}_{a,d}\) is a direct sum of \( H_{a_{0},d_{0}}\) and \(\mathbb {F}_{q}^2\), where \( H_{a_{0},d_{0}}\) is the twisted Hessian curve over \(\mathbb {F}_{q}\). Other results are deduced from, we cite the equivalence of the discrete logarithm problem on the twisted Hessian curves \(H^{3}_{a,d}\) and \( H_{a_{0},d_{0}}\), which is beneficial for cryptography and cryptanalysis as well, and we give an application in cryptography.