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 double-struck F signq, can be solved in an expected time of Õ(q2-2/ (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 Õ(q2-2/ (g-1)). © Springer-Verlag Berlin Heidelberg 2006.
CITATION STYLE
Diem, C. (2006). An index calculus algorithm for plane curves of small degree. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 4076 LNCS, pp. 543–557). Springer Verlag. https://doi.org/10.1007/11792086_38
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