Quasi-quadratic elliptic curve point counting using rigid cohomology
Journal of Symbolic Computation
Computing zeta functions in families of Ca,bcurves using deformation
ANTS-VIII'08 Proceedings of the 8th international conference on Algorithmic number theory
An extension of Kedlaya's algorithm for hyperelliptic curves
Journal of Symbolic Computation
Fast arithmetic in unramified p-adic fields
Finite Fields and Their Applications
Fast Polynomial Factorization and Modular Composition
SIAM Journal on Computing
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Let E Γ be a family of hyperelliptic curves defined by $Y^{2}=\bar {Q}(X,\ensuremath {\Gamma })$, where $\bar{Q}$ is defined over a small finite field of odd characteristic. Then with $\ensuremath {\bar {\gamma }}$ in an extension degree n field over this small field, we present a deterministic algorithm for computing the zeta function of the curve $E_{\ensuremath {\bar {\gamma }}}$ by using Dwork deformation in rigid cohomology. The time complexity of the algorithm is $\ensuremath {\mathcal {O}}(n^{2.667})$ and it needs $\ensuremath {\mathcal {O}}(n^{2.5})$ bits of memory. A slight adaptation requires only $\ensuremath {\mathcal {O}}(n^{2})$ space, but costs time $\ensuremath {\widetilde {\mathcal {O}}}(n^{3})$. An implementation of this last result turns out to be quite efficient for n big enough.