Improved Complexity Bounds for Counting Points on Hyperelliptic Curves

We present 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 modelling of the ℓ -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((log q)^c g) as q grows and the characteristic is large enough.