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**2005/119**

**Title:**

Index Calculus in Class Groups of Plane Curves of Small Degree

**Authors:**
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)}).

**ePrint:**
https://eprint.iacr.org/2005/119

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