Counting points on Cab curves using Monsky--Washnitzer cohomology

Authors:
Jan Denef;Frederik Vercauteren
Affiliations:
Department of Mathematics, University of Leuven, Celestijnenlaan 200B, B-3001 Leuven-Heverlee, Belgium;Computer Science Department -- Cryptography Group, University of Bristol, Woodland Road, Bristol BS8 1UB, UK and Department of Electrical Engineering -- ESAT/COSIC, University of Leuven, Kasteelpa ...
Venue:
Finite Fields and Their Applications
Year:
2006

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Abstract

We describe an algorithm to compute the zeta function of any C"a"b curve over any finite field F"p"^"n. The algorithm computes a p-adic approximation of the characteristic polynomial of Frobenius by computing in the Monsky-Washnitzer cohomology of the curve and thus generalizes Kedlaya's algorithm for hyperelliptic curves. For fixed p the asymptotic running time for a C"a"b curve of genus g over F"p"^"n is O(g^5^+^@?n^3^+^@?) and the space complexity is O(g^3n^3).