Correspondences on Hyperelliptic Curves and Applications to the Discrete Logarithm

Abstract

The discrete logarithm is an important crypto primitive for public key cryptography. The main source for suitable groups are divisor class groups of carefully chosen curves over finite fields. Because of index-calculus algorithms one has to avoid curves of genus ≥ 4 and non-hyperelliptic curves of genus 3. An important observation of Smith [17] is that for “many” hyperelliptic curves of genus 3 there is an explicit isogeny of their Jacobian variety to the Jacobian of a non-hyperelliptic curve. Hence divisor class groups of these hyperelliptic curves are mapped in polynomial time to divisor class groups of non-hyperelliptic curves. Behind his construction are results of Donagi, Recillas and Livné using classical algebraic geometry. In this paper we only use the theory of curves to study Hurwitz spaces with monodromy group S ₄ and to get correspondences for hyperelliptic curves. For hyperelliptic curves of genus 3 we find Smith’s results now valid for ground fields with odd characteristic, and for fields with characteristic 2 one can apply the methods of this paper to get analogous results at least for curves with ordinary Jacobian.