Two-cover descent on hyperelliptic curves

Weil descent attack for Kummer extensions

In this paper, we show how the Weil descent attack of Gaudry, Hess and Smart can be adapted to work for some hyperelliptic curves defined over fields of odd characteristic. This attack applies to a family of hyperelliptic and superelliptic curves over quadratic field extensions, as well as two families of hyperelliptic curves defined over cubic extensions. We also show that those are the only f...

An Extension of GHS Weil Descent Attack

The Weil descent attack, suggested by Frey, has been implemented by Gaudry, Hess and Smart (the so-called GHS attack), on elliptic curves over finite fields of characteristic two of composite degrees. The GHS attack has been extended by Galbraith to hyperelliptic curves of characteristic two. Recently, Diem presented a general treatment of GHS attack to hyperelliptic curves over finite fields o...

On a Problem of Hajdu and Tengely

We answer a question asked by Hajdu and Tengely: The only arithmetic progression in coprime integers of the form (a, b, c, d) is (1, 1, 1, 1). For the proof, we first reduce the problem to that of determining the sets of rational points on three specific hyperelliptic curves of genus 4. A 2-cover descent computation shows that there are no rational points on two of these curves. We find generat...

Elliptic curves with weak coverings over cubic extensions of finite fields with odd characteristic

In this paper, we present a classification of elliptic curves defined over a cubic extension of a finite field with odd characteristic which have coverings over the finite field therefore subjected to the GHS attack. The densities of these weak curves, with hyperelliptic and non-hyperelliptic coverings, are then analyzed respectively. In particular, we show, for elliptic curves defined by Legen...

2-Descent on the Jacobians of Hyperelliptic Curves