International Association for Cryptologic Research

International Association
for Cryptologic Research

CryptoDB

Jan Denef

Publications

Year
Venue
Title
2006
JOFC
2006
EPRINT
Computing Zeta Functions of Nondegenerate Curves
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.
2002
EPRINT
An Extension of Kedlaya's Algorithm to Hyperelliptic Curves in Characteristic 2
We present an algorithm for computing the zeta function of an arbitrary hyperelliptic curve over a finite field $\FF_q$ of characteristic 2, thereby extending the algorithm of Kedlaya for odd characteristic. For a genus $g$ hyperelliptic curve defined over $\FF_{2^n}$, the average-case time complexity is $O(g^{4 + \varepsilon} n^{3 + \varepsilon})$ and the average-case space complexity is $O(g^{3} n^{3})$, whereas the worst-case time and space complexities are $O(g^{5 + \varepsilon} n^{3 + \varepsilon})$ and $O(g^{4} n^{3})$ respectively.