Index Calculus in Class Groups of Plane Curves of Small Degree

We present a novel index calculus algorithm for the discrete logarithm problem (DLP) in degree 0 class groups of curves over finite fields. A heuristic analysis of our algorithm indicates that asymptotically for varying q, “essentially all” instances of the DLP in degree 0 class groups of curves represented by plane models of a fixed degree d over Fq can be solved in an expected time of Õ(q ). A particular application is that heuristically, “essentially all” instances of the DLP in degree 0 class groups of non-hyperelliptic curves of genus 3 (represented by plane curves of degree 4) can be solved in an expected time of Õ(q). We also provide a method to represent “sufficiently general” (nonhyperelliptic) curves of genus g ≥ 3 by plane models of degree g+1. We conclude that on heuristic grounds the DLP in degree 0 class groups of “sufficiently general” curves of genus g ≥ 3 (represented initially by plane models of bounded degree) can be solved in an expected time of Õ(q).