Section: New Results

Improved complexity bounds for counting points on hyperelliptic curves

Participants : Simon Abelard, Pierrick Gaudry [contact] , Pierre-Jean Spaenlehauer [contact] .

In [3], we presented a probabilistic Las Vegas algorithm for computing the local zeta function of a hyperelliptic curve of genus $g$ defined over ${𝔽}_q$. It is based on the approaches by Schoof and Pila combined with a modeling of the $\ell$-torsion by structured polynomial systems. Our main result improves on previously known complexity bounds by showing that there exists a constant $c>0$ such that, for any fixed $g$, this algorithm has expected time and space complexity $O\left({(logq)}^{cg}\right)$ as $q$ grows and the characteristic is large enough.