Isogenies and the Discrete Logarithm Problem in Jacobians of Genus 3 Hyperelliptic Curves
Résumé
We describe the use of explicit isogenies to reduce Discrete Logarithm Problems (DLPs) on Jacobians of hyperelliptic curves of genus three to Jacobians of non-hyperelliptic curves of genus three, which are vulnerable to faster index calculus attacks. We provide algorithms which compute an isogeny with kernel isomorphic to $(Z/2Z)^3$ for any hyperelliptic genus three curve. These algorithms provide a rational isogeny for a positive fraction of all hyperelliptic genus three curves defined over a finite field of characteristic p > 3. Subject to reasonable assumptions, our algorithms provide an explicit and efficient reduction from hyperelliptic DLPs to non-hyperelliptic DLPs for around $18.57\%$ of all hyperelliptic genus three curves over a given finite field.