Normal forms of hyperelliptic curves of genus 3
type: Item
instance of: scholarly article
title: Normal forms of hyperelliptic curves of genus 3
zbMATH Open document ID: 1345.14037
author: Ernst Kani
author: Gerhard Frey
published in: Designs, Codes and Cryptography
publication date: 2015-10-26T00:00:00Z
review text: Public Cryptography matters are usually handled via so-called discrete logarithm (DL) in \(\mathrm{Pic}^0\), the group of divisor classes of degree 0 of a carefully chosen curve defined over a finite field; see e.g. \textit{H. Cohen} et al. [Handbook of elliptic and hyperelliptic curve cryptography. Boca Raton, FL: Chapman \& Hall/CRC (2006; Zbl 1082.94001)], \textit{G. Frey} [Adv. Math. Commun. 4, No. 2, 281--305 (2010; Zbl 1225.11149)]. Up to logarithm factors, the use of index-calculus methods allow to compute a bound on the complexity of the corresponding DL, namely of order \(\mathcal O(q^{(2-2/g)}\), being \(q\) the order of the ground field and \(g\) the genus of the base curve [\textit{P. Gaudry} et al., Math. Comput. 76, No. 257, 475--492 (2007; Zbl 1179.94062)]. Then the curves to be considered are those of genus up to \(3\), as \(\# \mathrm{Pic}^0\) is approximately \(q^g\) for fixed \(g\). Curves of genus 1 or 2 are so far currently being used under certain precautions [loc. cit.]. In this paper, the authors consider curves of genus 3 in characteristic different from 2. For plane curves, up to logarithm factors, one can also bound the complexity of the DL in \(\mathrm{Pic}^0\) by \(k(d)\) which depends on the degree \(d\) of the plane model of the curve [\textit{C. Diem}, J. Théor. Nombres Bordx. 24, No. 3, 639--667 (2012; Zbl 1270.11128)]. It turns out that in the non-hyperelliptic case; i.e., \(d=4\), \(k(4)=\mathcal O(q)\); otherwise, \(k(d)=\mathcal O(q^a)\) with \(a>1\). Thus hyperelliptic curves of genus 3 are the candidates from which one could obtain better performance for the DL. Now each of such a hyperelliptic curve \(C\) is related to a degree 4 covering \(f:C\to\mathbb P^1\) whose monodromy group is the symmetric group \(S_4\), and whose branch locus consists exactly of 8 points such that 4 of them satisfy \(f^*(\alpha)=2P_1+P_2+P_3\) (type \((2,1,1)\)) and the others \(f^*(\beta)=2Q_1+1Q_2\) (type \((2,2)\)) with pairwise different points \…
reviewed by: Fernando Torres
Mathematics Subject Classification ID: 14H30
Mathematics Subject Classification ID: 14Q05
Mathematics Subject Classification ID: 14G50
Mathematics Subject Classification ID: 14H45
zbMATH DE Number: 6498071
zbMATH Keywords: hyperelliptic curves
zbMATH Keywords: normal curves
zbMATH Keywords: Hurwitz spaces attached to curves of genus 3
MaRDI profile type: Publication
full work available at URL: s10623-015-0122-2
OpenAlex ID: W1466287087
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DOI: 10.1007/S10623-015-0122-2
description: scientific article; zbMATH DE number 6498071