Improved Complexity Bounds for Counting Points on Hyperelliptic Curves

Simon Abelard (1) , Pierrick Gaudry (1) , Pierre-Jean Spaenlehauer (1)

1 CARAMBA - Cryptology, arithmetic : algebraic methods for better algorithms

We present a probabilistic Las Vegas algorithm for computing the local zeta function of a hyperelliptic curve of genus $g$ defined over $\mathbb{F}_q$. 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 $O((\log q)^{cg})$ as $q$ grows and the characteristic is large enough.

Domaines: Théorie des nombres [math.NT]; Géométrie algébrique [math.AG]; Calcul formel [cs.SC]

Simon Abelard, Pierrick Gaudry, Pierre-Jean Spaenlehauer. Improved Complexity Bounds for Counting Points on Hyperelliptic Curves. Foundations of Computational Mathematics, 2019, 19 (3), pp.591-621. ⟨10.1007/s10208-018-9392-1⟩. ⟨hal-01613530v2⟩