Study on special algebraic curves over fields of positive characteristic via computer algebra

2019 Fiscal Year Final Research Report

• Project/Area Number: 19K21026
• Project/Area Number (Other): 18H05836 (2018)
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Multi-year Fund (2019); Single-year Grants (2018)
• Review Section: 0201:Algebra, geometry, analysis, applied mathematics,and related fields
• Research Institution: Kobe City College of Technology
• Principal Investigator: Kudo Momonari 神戸市立工業高等専門学校, その他部局等, 講師 (10824708)
• Project Period (FY): 2018-08-24 – 2020-03-31
• Keywords: 代数幾何学 / 計算代数幾何 / 代数曲線 / 超特別曲線 / 最大曲線 / 正標数 / Hasse-Witt行列 / フロベニウス射

Outline of Final Research Achievements

Algebraic curves are central objects studied in algebraic geometry, number theory and related areas, and they are expected to be applied to information technology, in particular, cryptography and coding theory.
In this study, we focused on algebraic curves said to be superspecial or maximal, and studied them by the combined approach of the theory of algebraic geometry and computer algebra. As a result, we have succeeded in solving several problems on the (non-)existence of superspecial curves and maximal curves, and problems on the enumeration of these curves.
Furthermore, as an application to cryptography, we proposed efficient algorithms to compute isogenies between superspecial (supersingular) elliptic curves. Developing these algorithms shall contribute to evaluate the security of the state-of-art isogeny-based cryptosystems.

Free Research Field

代数学

Academic Significance and Societal Importance of the Research Achievements

本研究では，超特別曲線・最大曲線と呼ばれる代数曲線について，種数と呼ばれるパラメータを固定したときに，上記曲線の存在・非存在性や数え上げに関する幾つかの問題の解決に取り組んだ．
本研究とその成果の学術的意義としては，先行研究では種数3以下の場合に多くの結果が得られていたのに対し，本研究では，これまで困難とされてきた種数4以上の場合を主に考察し結果が得られたという点で新規性が非常に高い．また，本研究では計算代数の手法を駆使している点で独自性が高い．
社会的意義としては，本研究で得られた曲線は暗号・符号理論において具体パラメータとして活用されうるという点で，情報通信分野などへの応用価値が期待される．