Remarks on the zeta function of some diagonal hyperelliptic curves
type: Item
instance of: scholarly article
title: Remarks on the zeta function of some diagonal hyperelliptic curves
zbMATH Open document ID: 1090.11041
published in: Journal of Pure and Applied Algebra
publication date: 2004-08-19T00:00:00Z
review text: 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.
reviewed by: Imin Chen
Mathematics Subject Classification ID: 11G20
Mathematics Subject Classification ID: 14G50
Mathematics Subject Classification ID: 11T24
Mathematics Subject Classification ID: 14G10
Mathematics Subject Classification ID: 11T71
zbMATH DE Number: 2093448
zbMATH Keywords: hyperelliptic
zbMATH Keywords: zeta functions
zbMATH Keywords: Jacobi sums
zbMATH Keywords: cryptosystems
author: Jean-Pierre Cherdieu
MaRDI profile type: Publication
full work available at URL: j.jpaa.2003.10.017
OpenAlex ID: W2005736842
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DOI: 10.1016/J.JPAA.2003.10.017
description: scientific article; zbMATH DE number 2093448