Efficient Search for Superspecial Hyperelliptic Curves of Genus Four with Automorphism Group Containing $${\textbf{C}}_6$$

Abstract

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 $$g \ge 4$$ g ≥ 4 , it is not even known whether there exists such a curve of genus g in general characteristic $$p > 0$$ p > 0 , and in the case of $$g=4$$ 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 as a subgroup. In this paper, we propose an algorithm with complexity $${\tilde{O}}(p^4)$$ O ~ ( p 4 ) in theory but $${\tilde{O}}(p^3)$$ 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 $$p \equiv 2 \pmod {3}$$ p ≡ 2 ( mod 3 ) .

Fast Enumeration of Superspecial Hyperelliptic Curves of Genus 4 with Automorphism Group $$V_4$$

“…We recall a classification by Kudo-Nakagawa-Takagi [14] (a preliminary version of [15]) of hyperelliptic curves of genus 4 by means of their automorphism groups. For a hyperelliptic curve H over k, we denote by ι H its hyperelliptic involution.…”

“…On the other hand, some polynomial-time algorithms have been proposed in recent years, for restricted cases where curves have non-trivial automorphism groups: the paper [13] for the non-hyperelliptic case where Aut ⊃ V 4 , and [22] (resp. [15]) for the hyperelliptic case where Aut ⊃ V 4 (resp. Aut ⊃ C 6 ), where V 4 and C n respectively denote the Klein 4-group and the cyclic group of order n. In particular, Ohashi-Kudo-Harashita [22] proposed an algorithm to enumerate superspecial hyperelliptic curves H of genus 4 such that Aut(H) ⊃ V 4 , and its complexity is estimated as O(p 4 ) operations in F p 4 .…”