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arXiv · 2401.15255

Enumerating hyperelliptic curves over finite fields in quasilinear time

Everett W. Howe
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Abstract

We present an algorithm that, for every fixed genus $g$, will enumerate all hyperelliptic curves of genus $g$ over a finite field $k$ of odd characteristic in quasilinear time; that is, the time required for the algorithm is $\widetilde{O}(q^{2g-1})$, where $q=\#k$. Such an algorithm already exists in the case $g=2$, thanks to work of Mestre and Cardona and Quer, and in the case $g=3$, thanks to work of Lercier and Ritzenthaler. Experimentally, it appears that our new algorithm is about two orders of magnitude faster in practice than ones based on their work.

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Everett W. Howe. 2024-01-27. Enumerating hyperelliptic curves over finite fields in quasilinear time. https://arxiv.org/abs/2401.15255

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