Paper 2005/119
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 $\tilde{O}(q^{2 -2/(g-1)})$.
Metadata
- Available format(s)
-
PDF PS
- Category
- Public-key cryptography
- Publication info
- Published elsewhere. Unknown where it was published
- Keywords
- discrete logarithm problem
- Contact author(s)
- diem @ iem uni-due de
- History
- 2005-04-21: received
- Short URL
- https://ia.cr/2005/119
- License
-
CC BY
BibTeX
@misc{cryptoeprint:2005/119, author = {Claus Diem}, title = {Index Calculus in Class Groups of Plane Curves of Small Degree}, howpublished = {Cryptology {ePrint} Archive, Paper 2005/119}, year = {2005}, url = {https://eprint.iacr.org/2005/119} }