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56
Constructing hyperelliptic curves of genus 2 suitable for cryptography
 Math. Comp
, 2003
"... Abstract. In this article we show how to generalize the CMmethod for elliptic curves to genus two. We describe the algorithm in detail and discuss the results of our implementation. 1. ..."
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Abstract. In this article we show how to generalize the CMmethod for elliptic curves to genus two. We describe the algorithm in detail and discuss the results of our implementation. 1.
On the performance of hyperelliptic cryptosystems
, 1999
"... In this paper we discuss various aspects of cryptosystems based on hyperelliptic curves. In particular we cover the implementation of the group law on such curves and how to generate suitable curves for use in cryptography. This paper presents a practical comparison between the performance of ellip ..."
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In this paper we discuss various aspects of cryptosystems based on hyperelliptic curves. In particular we cover the implementation of the group law on such curves and how to generate suitable curves for use in cryptography. This paper presents a practical comparison between the performance of elliptic curve based digital signature schemes and schemes based on hyperelliptic curves. We conclude that, at present, hyperelliptic curves offer no performance advantage over elliptic curves.
Examples of genus two CM curves defined over the rationals
 Math. Comp
, 1999
"... Abstract. We present the results of a systematic numerical search for genus two curves defined over the rationals such that their Jacobians are simple and have endomorphism ring equal to the ring of integers of a quartic CM field. Including the wellknown example y 2 = x 5 − 1 we find 19 nonisomorp ..."
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Cited by 23 (1 self)
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Abstract. We present the results of a systematic numerical search for genus two curves defined over the rationals such that their Jacobians are simple and have endomorphism ring equal to the ring of integers of a quartic CM field. Including the wellknown example y 2 = x 5 − 1 we find 19 nonisomorphic such curves. We believe that these are the only such curves. 1.
A CRT algorithm for constructing genus 2 curves over finite fields
, 2007
"... Abstract. — We present a new method for constructing genus 2 curves over a finite field Fn with a given number of points on its Jacobian. This method has important applications in cryptography, where groups of prime order are used as the basis for discretelog based cryptosystems. Our algorithm prov ..."
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Cited by 21 (8 self)
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Abstract. — We present a new method for constructing genus 2 curves over a finite field Fn with a given number of points on its Jacobian. This method has important applications in cryptography, where groups of prime order are used as the basis for discretelog based cryptosystems. Our algorithm provides an alternative to the traditional CM method for constructing genus 2 curves. For a quartic CM field K with primitive CM type, we compute the Igusa class polynomials modulo p for certain small primes p and then use the Chinese remainder theorem (CRT) and a bound on the denominators to construct the class polynomials. We also provide an algorithm for determining endomorphism rings of ordinary Jacobians of genus 2 curves over finite fields, generalizing the work of Kohel for elliptic curves. Résumé (Un algorithme fondé sur le théorème chinois pour construire des courbes de genre 2 sur des corps finis) Nous présentons une nouvelle méthode pour construire des courbes de genre 2 sur un corps fini Fn avec un nombre donné de points sur sa jacobienne. Cette méthode a des applications importantes en cryptographie, où des groupes d’ordre premier sont employés pour former des cryptosystèmes fondés sur le logarithme discret. Notre algorithme fournit une alternative à la méthode traditionnelle de multiplication complexe pour construire des courbes de genre 2. Pour un corps quartique K à multiplication complexe de type primitif, nous calculons les polynômes de classe d’Igusa modulo p pour certain petit premiers p et employons le théorème chinois et une borne sur les dénominateurs pour construire les polynômes de classe. Nous fournissons également un algorithme pour déterminer les anneaux d’endomorphismes des jacobiennes de courbes ordinaires de genre 2 sur des corps finis, généralisant le travail de Kohel pour les courbes elliptiques.
CalabiYau threefolds and moduli of abelian surfaces
"... be the moduli space of polarized abelian surfaces with canonical level structure. Both are (possibly singular) quasiprojective threefolds, and Alev d is a finite cover of Ad. We will also denote by Ãd and Ãlev d nonsingular models of suitable compactifications of these moduli spaces. We will use in ..."
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be the moduli space of polarized abelian surfaces with canonical level structure. Both are (possibly singular) quasiprojective threefolds, and Alev d is a finite cover of Ad. We will also denote by Ãd and Ãlev d nonsingular models of suitable compactifications of these moduli spaces. We will use in the sequel definitions and notation as in [GP1], [GP2]; see also [Mu1], [LB] and [HKW] for basic facts concerning abelian varieties and their moduli. Throughout the paper the base field will be C. Let Ad denote the moduli space of polarized abelian surfaces of type (1,d), and let A lev d The main goal of this paper, which is a continuation of [GP1] and [GP2], is to describe birational models for moduli spaces of these types for small values of d. Since the Kodaira dimension is a birational invariant, thus independent of the chosen compactification, we can decide the uniruledness, unirationality or rationality of nonsingular models of (any) compactifications of these moduli spaces.
Signatures of foliated surface bundles and the symplectomorphism groups of surfaces
, 2003
"... For any closed oriented surface Σg of genus g ≥ 3, we prove the existence of foliated Σgbundles over surfaces such that the signatures of the total spaces are nonzero. We can arrange that the total holonomy of the horizontal foliations preserve a prescribed symplectic form ω on the fiber. We relat ..."
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For any closed oriented surface Σg of genus g ≥ 3, we prove the existence of foliated Σgbundles over surfaces such that the signatures of the total spaces are nonzero. We can arrange that the total holonomy of the horizontal foliations preserve a prescribed symplectic form ω on the fiber. We relate the cohomology class represented by the transverse symplectic form to a crossed homomorphism ˜Flux: Symp Σg→H 1 (Σg; R) which is an extension of the flux homomorphism Flux: Symp 0 Σg→H 1 (Σg; R) from the identity component Symp 0 Σg to the whole group Symp Σg of symplectomorphisms of Σg with respect to the symplectic form ω.
Classical and minimal models of the moduli space of curves of genus two
, 2004
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