**Index Calculus in Class Groups of Plane Curves of Small Degree**

*Claus Diem*

**Abstract: **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 $\mathbb{F}_q$ can be solved in an expected time of $\tilde{O}(q^{2 -2/(d-2)})$.

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 $\tilde{O}(q)$.

We also provide a method to represent ``sufficiently general'' (non-hyperelliptic) curves of genus $g \geq 3$ 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 $g \geq 3$ (represented initially by plane models of bounded degree) can be solved in an expected time of $\tilde{O}(q^{2 -2/(g-1)})$.

**Category / Keywords: **public-key cryptography / discrete logarithm problem

**Date: **received 18 Apr 2005

**Contact author: **diem at iem uni-due de

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**Version: **20050421:174335 (All versions of this report)

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