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**2006/240**

**Title:**

Computing Zeta Functions of Nondegenerate Curves

**Authors:**
W. Castryck, J. Denef, F. Vercauteren

**Abstract:**

In this paper we present a p-adic algorithm to compute the zeta function of a nondegenerate curve over a finite field using Monsky-Washnitzer cohomology. The paper vastly generalizes previous work since all known cases, e.g. hyperelliptic, superelliptic and C_{ab} curves, can be transformed to fit the nondegenerate case. For curves with a fixed Newton polytope, the property of being nondegenerate is generic, so that the algorithm works for almost all curves with given Newton polytope. For a genus g curve over \FF_{p^n}, the expected running time is \widetilde{O}(n^3 g^6 + n^2 g^{6.5}), whereas the space complexity amounts to \widetilde{O}(n^3 g^4), assuming p is fixed.

**ePrint:**
https://eprint.iacr.org/2006/240

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