Point Counting on Genus 3 Non Hyperelliptic Curves

We propose an algorithm to compute the Frobenius polynomial of an ordinary non hyperelliptic curve of genus 3 over F2N . The method is a generalization of Mestre’s AGM-algorithm for hyperelliptic curves and leads to a quasi quadratic time algorithm for point counting. The current methods for point counting on curves over finite fields of small characteristic rely essentially on a p-adic approach. They split up in three classes : those based on cohomology (Kedlaya), those based on deformation theory (Lauder) and those based on the canonical lift, proposed initially by Satoh. The AGM-algorithm developed by Mestre [?] for elliptic curves belongs to this last category. It is an elegant and natural variant in characteristic 2 of Satoh’s one using, in analogy with the complex field, the machinery of theta functions. Mestre generalized it later to the hyperelliptic case and Lercier-Lubicz’s implementations of these algorithms are the current records for point counting in characteristic 2 [?]. In this paper, we propose a generalization of the AGM-algorithm to the genus 3 non hyperelliptic case, as described in the author’s PhD thesis [?]. Fast algorithms for point counting on non hyperelliptic curves exist in the literature. However, as far as we know, the present one is the first which is fast enough for problems of cryptographic size in characteristic 2. The aim of the paper is to give all the details for an implementation. With this end in view, we shall give an algorithm based on Weber’s work for the computation of the initial theta constants. This computation, which is fairly easy for hyperelliptic curves, requires some work for this case. The other issue is to find a good lift in order that the computations take place in the field of definition of the lift. It is the main achievement of this paper that it shows how find good lifts. Once this is achieved, we are able to use the optimized iteration process of [?]. We shall illustrate the method with an example over F2100 . ? C. Ritzenthaler acknowledges the financial support provided through the European Community’s Human Potential Programme under contract HPRN-CT-2000-00114, GTEM