On the discrete logarithm problem for plane curves

In this article the discrete logarithm problem in degree 0 class groups of curves over finite fields given by plane models is studied. It is proven that the discrete logarithm problem for non-hyperelliptic curves of genus 3 (given by plane models of degree 4) can be solved in an expected time of O(q), where q is the cardinality of the ground field. Moreover, it is proven that for every fixed natural number d ≥ 4 the following holds: We consider the discrete logarithm problem for curves given by plane models of degree d for which there exists a line which defines a divisor which splits completely into distinct Fq-rational points. Then this problem can be solved in an expected time of O(q 2 d−2 ). This holds in particular for curves given by reflexive plane models.

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