An Index Calculus Algorithm for Plane Curves of Small Degree

  • Claus Diem
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4076)


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

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 \(\tilde{O}(q^{2 -2/(g-1)})\).


Class Group Random Graph Plane Model Plane Curf Hyperelliptic Curve 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Claus Diem
    • 1
  1. 1.Mathematisches InstitutUniversität LeipzigLeipzigGermany

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