论文标题
有效地搜索属于$ \ mathbb {z} _6 $的自动形态组的属四属超椭圆形曲线
Efficient search for superspecial hyperelliptic curves of genus four with automorphism group containing $\mathbb{Z}_6$
论文作者
论文摘要
在算术和代数几何形状中，SuperSpecial（S.Sp. \ for Shtt）曲线是要研究的最重要的对象之一，应用于密码学和编码理论。如果$ g \ geq 4 $，甚至还不知道是否存在这种曲线的属属$ g $的一般特征$ p> 0 $，并且在$ g = 4 $的情况下，已经提出了几种搜索这些曲线的计算方法。在属-4 $ $ 4 $的过度外壳中，Kudo-Harashita提出了一种通用算法来枚举所有S.Sp. \ curves，最近Ohashi-Kudo-Harashita提出了一种针对自动形态组包含Klein 4 Group的案例的算法。在本文中，我们提出了一种具有复杂性$ \ tilde {o}（p^4）$的算法，但在实践中，$ \ tilde {o}（p^3）$枚举S.Sp. \ s.sp. \属属4具有自动形态组的属于订单组的属的超ellirtictic曲线。通过对岩浆执行算法，我们以$ P $ $ 1000 $的价格列举了这些曲线。我们还成功地找到了S.Sp. \ S.Sp. \属属$ 4 $在每$ p $中，带有$ p \ equiv 2 \ pmod {3} $。作为理论上的结果，我们根据附录中的自动形态组对属$ 4 $的过性曲线进行了分类。
In arithmetic and algebraic geometry, superspecial (s.sp.\ for short) curves are one of the most important objects to be studied, with applications to cryptography and coding theory. If $g \geq 4$, it is not even known whether there exists such a curve of genus $g$ in general characteristic $p > 0$, and in the case of $g=4$, several computational approaches to search for those curves have been proposed. In the genus-$4$ hyperelliptic case, Kudo-Harashita proposed a generic algorithm to enumerate all s.sp.\ curves, and recently Ohashi-Kudo-Harashita presented an algorithm specific to the case where automorphism group contains the Klein 4-group. In this paper, we propose an algorithm with complexity $\tilde{O}(p^4)$ in theory but $\tilde{O}(p^3)$ in practice to enumerate s.sp.\ hyperelliptic curves of genus 4 with automorphism group containing the cyclic group of order $6$. By executing the algorithm over Magma, we enumerate those curves for $p$ up to $1000$. We also succeeded in finding a s.sp.\ hyperelliptic curve of genus $4$ in every $p$ with $p \equiv 2 \pmod{3}$. As a theoretical result, we classify hyperelliptic curves of genus $4$ in terms of automorphism groups in the appendix.
