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 ...

In 1899, Hutchinson [Hut99] presented a way to obtain a threeparameter family of Hessians of cubic surfaces as blowups of Kummer surfaces. We show that this family consists of those Hessians containing an extra class of conic curves. Based on this, we find the invariant of a cubic surface C in pentahedral form that vanishes if its Hessian is in Hutchinson’s family, and we give an explicit map b...

This work considers the problem of fast and secure scalar multiplication using curves of genus one defined over a field of prime order. Previous work by Gaudry and Lubicz in 2009 had suggested the use of the associated Kummer line to speed up scalar multiplication. In this work, we explore this idea in detail. The first task is to obtain an elliptic curve in Legendre form which satisfies necess...

We study degenerations of Kummer surfaces associated to certain divisors in Nieto’s quintic threefold and show how they arise from boundary components of a suitable toroidal compactification of the corresponding Siegel modular threefold. The aim of this paper is to study the degenerations of an interesting class of Kummer surfaces in P in terms of degenerations of the corresponding abelian surf...

We give a general framework for uniform, constant-time oneand two-dimensional scalar multiplication algorithms for elliptic curves and Jacobians of genus 2 curves that operate by projecting to the xline or Kummer surface, where we can exploit faster and more uniform pseudomultiplication, before recovering the proper “signed” output back on the curve or Jacobian. This extends the work of López a...

The spinor Bethe-Salpeter equation describing bound states of a fermion-antifermion pair with massless-boson exchange reduces to a single (uncoupled) partial differential equation for special combinations of the fermion-boson couplings. For spinless bound states with positive or negative parity this equation is a generalization to nonvanishing bound-state masses of the equations studied by Kumm...

The expansion of Kummer’s hypergeometric function as a series of incomplete Gamma functions is discussed, for real values of the parameters and of the variable. The error performed approximating the Kummer function with a finite sum of Gammas is evaluated analytically. Bounds for it are derived, both pointwisely and uniformly in the variable; these characterize the convergence rate of the serie...

The Weierstrass semigroups and pure gaps can be helpful in constructing codes with better parameters. In this paper, we investigate explicitly the minimal generating set of the Weierstrass semigroups associated with several totally ramified places over arbitrary Kummer extensions. Applying the techniques provided by Matthews in her previous work, we extend the results of specific Kummer extensi...

In the preprint [1] one of the authors formulated some conjectures on monotonicity of ratios for exponential series sections. They lead to more general conjecture on monotonicity of ratios of Kummer hypergeometric functions and was not proved from 1993. In this paper we prove some conjectures from [1] for Kummer hypergeometric functions and its further generalizations for Gauss and generalized ...

Based on an explicit representation of the Artin map for Kummer extensions, we present a method to compute arbitrary class fields. As in the proofs of the existence theorem, the problem is first reduced to the case where the field contains sufficiently many roots of unity. Using Kummer theory and an explicit version of the Artin reciprocity law we show how to compute class fields in this case. ...