Constructing hyperelliptic curves of genus 2 suitable for cryptography

In this article we show how to generalize the CM-method for elliptic curves to genus two. We describe the algorithm in detail and discuss the results of our implementation.

     

Fields of moduli and definition of hyperelliptic curves of odd genus

Mestre has shown that if a hyperelliptic curve C of even genus is defined over a subfield k ì \mathbbC{k \subset \mathbb{C}} then C can be hyperelliptically defined over the same field k. In this paper, for all genera g > 1, g o 1{g\equiv1} mod 4, hence odd, we construct an explicit hyperelliptic curve defined over \mathbbQ{\mathbb{Q}} which can not be hyperelliptically defined over \mathbbQ{\mathbb{Q}}.     

2-Weierstrass points of genus 3 hyperelliptic curves with extra involutions

We consider families of curves with extra automorphisms in ?₃, the moduli space of smooth hyperelliptic curves of genus g = 3. Such families of curves are explicitly determined in terms of the absolute invariants of binary octavics. For each family of positive dimension where |Aut (C)|>4, we determine the possible distributions of weights of 2-Weierstrass points.     

Tropical hyperelliptic curves

We study the locus of tropical hyperelliptic curves inside the moduli space of tropical curves of genus g. We define a harmonic morphism of metric graphs and prove that a metric graph is hyperelliptic if and only if it admits a harmonic morphism of degree 2 to a metric tree. This generalizes the work of Baker and Norine on combinatorial graphs to the metric case. We then prove that the locus of 2-edge-connected genus g tropical hyperelliptic curves is a (2g?1)-dimensional stacky polyhedral fan whose maximal cells are in bijection with trees on g?1 vertices with maximum valence 3. Finally, we show that the Berkovich skeleton of a classical hyperelliptic plane curve satisfying a certain tropical smoothness condition is a standard ladder of genus g.     

Counting hyperelliptic curves

We find a closed formula for the number hyp(g) of hyperelliptic curves of genus g over a finite field k=Fq of odd characteristic. These numbers hyp(g) are expressed as a polynomial in q with integer coefficients that depend on g and the set of divisors of q−1 and q+1. As a by-product we obtain a closed formula for the number of self-dual curves of genus g. A hyperelliptic curve is defined to be self-dual if it is k-isomorphic to its own hyperelliptic twist.     

Singular hyperelliptic curves

Singular curves with a morphism of degree two onto a projective line should be classified into two types according as the equipped morphism is separable or not; we call a curve with a separable one a hyperelliptic curve of separable type, and the other a hyperelliptic curve of inseparable type. We give concrete expressions of a hyperelliptic curve of separable type by means of its global “equation” and a hyperelliptic curve of inseparable type by means of its local rings. Furthermore, we discuss about Weierstrass points of a hyperelliptic curve of inseparable type. Received: 26 March 1997 / Revised version: 21 May 1998     

A new method for decomposition in the Jacobian of small genus hyperelliptic curves

Decomposing a divisor over a suitable factor basis in the Jacobian of a hyperelliptic curve is a crucial step in an index calculus algorithm for the discrete log problem in the Jacobian. For small genus curves, in the year 2000, Gaudry had proposed a suitable factor basis and a decomposition method. In this work, we provide a new method for decomposition over the same factor basis. The advantage of the new method is that it admits a sieving technique which removes smoothness checking of polynomials required in Gaudry’s method. Also, the total number of additions in the Jacobian required by the new method is less than that required by Gaudry’s method. The new method itself is quite simple and we present some example decompositions and timing results of our implementation of the method using Magma.     

Maximal rank space curves of high genus are projectively normal

Riassunto Se il genere è alto, le curve di rango massimo sono proiettivamente normali. La nozione di ?genere alto? è precisata fornendo un limite explicito, dipendente solo dal grado.

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Summary Maximal rank space curves of high genus are projectively normal. We precise the meaning of ?high genus?, by giving an explicit bound, depending only on the degree.

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Dal 14 giugno 1990: Dip. di Matematica ?U. Dini?, viale Morgagni 67/A, 50134 Firenze.     

Elliptic subcovers of hyperelliptic curves

The main result of this paper states that if C is a hyperelliptic curve of even genus over an arbitrary field K , then there is a natural bijection between the set of equivalence classes of elliptic subcovers of and the set of elliptic subgroups of its Jacobian .     

Compactifying moduli of hyperelliptic curves

We construct a new compactification of the moduli spaceH _g of smooth hyperelliptic curves of genusg. We compare our compactification with other well-known remarkable compactifications ofH _g. The author was partially supported byCNP _q, Proc. 151610/2005-3, and by Faperj, Proc. E-26/152-629/2005.