This paper presents an algorithm to construct cryptographically strong genus 2 curves and their Kummer surfaces via Rosenhain invariants and related Kummer parameters. The most common version of the complex multiplication (CM) algorithm for constructing cryptographic curves in genus 2 relies on the well-studied Igusa invariants and Mestre’s algorithm for reconstructing the curve. On the other h...

Let A be a skew field of degree 3 over a field containing the 3 roots of unity. We prove a sort of chain equivalence for Kummer elements in A. As a consequence one obtains a common slot lemma for presentations of A as a cyclic algebra. Châınes d’éléments de Kummer en degré 3 Résumé. Soit k un corps contenant les racines cubiques de l’unité, et soit A un corps gauche de centre k, avec [A : k] = ...

In this paper we study Seidel’s mirror map for abelian and Kummer surfaces. We find that mirror symmetry leads in a very natural way to the classical parametrization of Kummer surfaces in P. Moreover, we describe a family of embeddings of a given abelian surface into noncommutative projective spaces.

We generalize various classical results on specialization of fundamental groups to log schemes in the sense of Fontaine and Illusie. The key technical result relates the category of finite Kummer étale covers of a fs log scheme over a complete noetherian local ring to the Kummer étale coverings of its reduction, generalizing earlier results of Hoshi.

is the closure of the locus parametrizing H22-invariant quartics with 16 skew lines. The smooth surfaces of this type are parametrized by a non-empty open set N s of N . These surfaces are Kummer surfaces associated to abelian surfaces with a (1,3)–polarization. The action of the Heisenberg group on the Kummer surface corresponds to a level-2 structure on the abelian surface. If A is an abelian...

Let E be an elliptic curve, K1 its Kummer curve E/{±1}, E its square product, and K2 the split Kummer surface E /{±1}. The addition law on E gives a large endomorphism ring, which induce endomorphisms of K2. With a view to the practical applications to scalar multiplication on K1, we study the explicit arithmetic of K2.

The univariate Kummer-beta family of distributions has been proposed and studied recently by K. W. Ng and Samuel Kotz. This distribution is an univariate extension of the beta distribution. In this article, we derive the distributions of the product and the quotient of independent Kummer-beta variables.

The theory of Kummer surfaces is a classical topic in algebraic geometry.1 A generalization to higher dimensions has been given by W. Wirtinger.2 The varieties introduced by Wirtinger may be called Kummer varieties and have algebraic dimension n with 22n ordinary double points. If these double points are desingularized in a standard manner by means of "dilatations," one obtains nonsingular vari...

We show that a Magma implementation of Joux’s new L[1/4] algorithm can be used to compute discrete logarithms in the 1303-bit finite field F36·137 and the 1551-bit finite field F36·163 with very modest computational resources. Our F36·137 implementation was the first to illustrate the effectiveness of Joux’s algorithm for computing discrete logarithms in small-characteristic finite fields that ...