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Superspecial Hyperelliptic Curves of Genus 4 over Small Finite Fields

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Arithmetic of Finite Fields (WAIFI 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11321))

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Abstract

In this paper, we enumerate superspecial hyperelliptic curves of genus 4 over finite fields \(\mathbb {F}_q\) for small q. This complements our preceding results in the non-hyperelliptic case. We give a feasible algorithm to enumerate superspecial hyperelliptic curves of genus g over \(\mathbb {F}_q\) in the case that q and \(2g+2\) are coprime and \(q>2g+1\). We executed the algorithm for \((g,q)= (4,11^2)\), \((4,13^2)\), \((4,17^2)\) and (4, 19) with our implementation on a computer algebra system Magma. Moreover, we found many maximal hyperelliptic curves and some minimal hyperelliptic curves over \(\mathbb {F}_{q}\) from among enumerated superspecial curves.

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Authors and Affiliations

  1. Kobe City College of Technology, 8-3, Gakuen-Higashimachi, Nishi-ku, Kobe, 651-2194, Japan

    Momonari Kudo

  2. Graduate School of Environment and Information Sciences, Yokohama National University, 79-7, Tokiwadai, Hodogaya-ku, Yokohama, 240-8501, Japan

    Shushi Harashita

Authors
  1. Momonari Kudo
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  2. Shushi Harashita
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Correspondence to Momonari Kudo .

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Editors and Affiliations

  1. Department of Informatics, University of Bergen, Bergen, Norway

    Lilya Budaghyan

  2. Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional, Mexico, Mexico

    Francisco Rodríguez-Henríquez

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Kudo, M., Harashita, S. (2018). Superspecial Hyperelliptic Curves of Genus 4 over Small Finite Fields. In: Budaghyan, L., Rodríguez-Henríquez, F. (eds) Arithmetic of Finite Fields. WAIFI 2018. Lecture Notes in Computer Science(), vol 11321. Springer, Cham. https://doi.org/10.1007/978-3-030-05153-2_3

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  • DOI: https://doi.org/10.1007/978-3-030-05153-2_3

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-05152-5

  • Online ISBN: 978-3-030-05153-2

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