Abstract
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 \(\tilde{O}(q^{2 -2/(g-1)})\).
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Diem, C. (2006). An Index Calculus Algorithm for Plane Curves of Small Degree. In: Hess, F., Pauli, S., Pohst, M. (eds) Algorithmic Number Theory. ANTS 2006. Lecture Notes in Computer Science, vol 4076. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11792086_38
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DOI: https://doi.org/10.1007/11792086_38
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