Normal forms of hyperelliptic curves of genus 3
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 \(P_1,P_2,P_3,Q_1,Q_2\in C\) [the authors, Lect. Notes Comput. Sci. 7053, 1--19 (2012; Zbl 1345.14036)]. Thus one is lead to study the Hurwitz space \(\mathbf H\) which classifies classes of morphism \(f\) as above. By using certain normalization form which is different from the usual Weierstrass normal form, it is shown that \(\mathbf H\) is unirational, irreducible algebraic variety of dimension 4 whenever the ground field has characteristic bigger than 5. In particular, the Donagi-Livné-Smith locus has dimension four giving a formal prove for an heuristic claim made in [\textit{B. Smith}, Lect. Notes Comput. Sci. 4965, 163--180 (2008; Zbl 1149.94329)].
Hurwitz spaces attached to curves of genus 3
scientific article; zbMATH DE number 6498071
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