An Index Calculus Algorithm for Plane Curves of Small Degree

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