Remarks on the zeta function of some diagonal hyperelliptic curves
This paper studies some diagonal hyperelliptic curves defined over \(\mathbb{F}_p\), in particular the ones given by \(y^2 = \gamma x^5 + \delta\) where \(\gamma \delta \not= 0\) and \(p \equiv 1 \pmod{10}\). The zeta functions are computed using Jacobi sums and results from [\textit{B. C. Berndt}, \textit{R. J. Evans} and \textit{K. S. Williams}, Gauss and Jacobi sums (Series of Monographs and Advanced Texts, Canadian Mathematical Society, Vol. 32, Wiley-Interscience Publications, New York) (1997; Zbl 0906.11001)]. Using results from \textit{T. Honda} [Osaka J. Math. 3, 189--194 (1966; Zbl 0145.17702)] and a computation of Hasse-Witt matrices, it is shown that the Jacobians of these curves are simple and ordinary. The use of these curves for cryptosystems is also discussed, in particular their resistance to some known classical attacks.
scientific article; zbMATH DE number 2093448
scientific article