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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
LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry, Inria Nancy - Grand Est
Abstract : 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.
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Submitted on : Thursday, June 7, 2018 - 9:55:49 AM
Last modification on : Monday, November 23, 2020 - 10:20:03 AM
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Simon Abelard, Pierrick Gaudry, Pierre-Jean Spaenlehauer. Improved Complexity Bounds for Counting Points on Hyperelliptic Curves. Foundations of Computational Mathematics, Springer Verlag, 2019, 19 (3), pp.591-621. ⟨10.1007/s10208-018-9392-1⟩. ⟨hal-01613530v2⟩



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