[00110] Computation on Supersingular and Superspecial Curves and its Applications
Session Time & Room : 5D (Aug.25, 15:30-17:10) @G302
Type : Proposal of Minisymposium
Abstract : Supersingular and superspecial algebraic curves have been studied in coding theory and cryptography for the last few decades. The applications are based on explicit constructions and computational aspects of such algebraic curves, which give novel and fascinating mathematical challenges. Interestingly, we have different kinds of problems depending on the genus of curves. The supersingular genus 1 curves, i.e., elliptic curves, are a central ingredient in quantum-resistant isogeny-based cryptography. A series of recent research shows that the security of the cryptosystems is closely related to arithmetic on superspecial curves of higher genera, whose study is the main topic in this minisymposium.
[04024] Decomposed Richelot isogenies of Jacobian varieties of hyperelliptic curves and generalized Howe curves
Format : Talk at Waseda University
Author(s) :
Toshiyuki Katsura (University of Tokyo)
Katsuyuki Takashima (Waseda University)
Abstract : We advance previous studies on decomposed Richelot isogenies (Katsura–
Takashima (ANTS 2020) and Katsura (J. Algebra)) which are useful for analysing
superspecial Richelot isogeny graphs in cryptography. We first give a characterization
of decomposed Richelot isogenies between Jacobian varieties of hyperelliptic
curves of any genus. We then define generalized Howe curves, and present two
theorems on their relationships with decomposed Richelot isogenies. We also give
new examples including a non-hyperelliptic (resp. hyperelliptic) generalized Howe
curve of genus 5 (resp. of genus 4).
[04668] Some explicit arithmetics on curves of genus three and their applications
Format : Online Talk on Zoom
Author(s) :
Tomoki Moriya (University of Birmingham)
Momonari Kudo (Fukuoka Institute of Technology)
Abstract : A Richelot isogeny between Jacobian varieties is an isogeny whose kernel is included in the 2-torsion subgroup of the domain. In particular, a Richelot isogeny whose codomain is the product of two or more principally polarized abelian varieties is called a decomposed Richelot isogeny. In this talk, I provide some explicit arithmetics on curves of genus 3, including algorithms to compute the codomain of a decomposed Richelot isogeny. I also provide
explicit formulae of defining equations for Howe curves of genus 3 as solutions to computing the domain of a decomposed Richelot isogeny. Finally, I give a construction of an algorithm with complexity O~(p^3) (resp. O~(p^4)) to enumerate all hyperelliptic (resp. non-hyperelliptic) superspecial Howe curves of genus 3.
[05345] Construction of superspecial curves of higher genera with extra automorphisms
Format : Talk at Waseda University
Author(s) :
Momonari Kudo (Fukuoka Institute of Technology)
Abstract : Superspecial curves are one of the most important objects in algebraic geometry over fields of positive characteristic, with applications to coding theory and cryptography, but explicit constructions and enumerations of such curves of higher genera are known to be quite difficult in general. In this talk, we develop several algorithms to construct superspecial curves of higher genera, restricting ourselves to the case of curves with non-trivial automorphisms, e.g., genus-4 hyperelliptic curves with order-6 automorphisms.
[04048] Several examples of curves whose superspeciality imply maximality or minimality
Format : Talk at Waseda University
Author(s) :
Ryo Ohashi (University of Tokyo)
Shushi Harashita (Yokohama National University)
Abstract : A curve over the finite field $\mathbb{F}_q$ of characteristic $p > 0$ is called maximal (resp. minimal) if the number of its $\mathbb{F}_q$-rational points attains the Hasse-Weil upper (resp. lower) bound. Maximal curves have been investigated for their applications to cryptography and coding theory. It is known that maximal or minimal curves over $\mathbb{F}_{p^2}$ are all superspecial, while superspecial curves over $\mathbb{F}_{p^2}$ are not necessary maximal nor minimal. However in this lecture, we will present several examples of curves $C$ which have the following property: If $C$ is superspecial, then $C$ is maximal or minimal over $\mathbb{F}_{p^2}$.
