Study on algebraic curves in positive characteristic via computational algebraic geometry and its application to cryptography

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K14301
• Research Category: Grant-in-Aid for Early-Career Scientists
• Allocation Type: Multi-year Fund
• Review Section: Basic Section 11010:Algebra-related
• Research Institution: Fukuoka Institute of Technology (2023); The University of Tokyo (2020-2022)
• Principal Investigator: Kudo Momonari 福岡工業大学, 情報工学部, 助教 (10824708)
• Project Period (FY): 2020-04-01 – 2024-03-31
• Keywords: 計算代数幾何学 / 代数曲線 / 正標数 / 超特異曲線 / 超特別曲線 / Jacobi多様体 / 有理点 / 同種写像暗号

Outline of Final Research Achievements

We studied algebraic curves in positive characteristic, for example, proving the (non-)existence of curves with prescribed invariants such as p-rank and a-number and enumerating such curves over finite field. We also developed related algorithms in algebraic geometry in positive characteristic, and analyzed the hardness of computational problems related to isogenies, which are security base in isogeny-based cryptography. In our study of algebraic curves in positive characteristic, we focused on curves birational to fiber products of hyperelliptic curves (such a curve is called a generalized Howe curve): We wrote down their explicit equations, and constructed algorithms to compute isogenies between their Jacobian varieties. As an application, we presented practical algorithms to find or enumerate superspecial curves, in each case of genus 3, 4, and 5. As a result, we wrote 14 journal papers and gave 27 conference talks in total, throughout the research period supported by this grant.

Free Research Field

代数幾何学

Academic Significance and Societal Importance of the Research Achievements

本研究で開発した計算代数幾何学のアルゴリズムは，代数曲線の研究のみならず他の代数学分野においてもツールとしての利活用が期待されるだけでなく，同種写像暗号や多変数多項式暗号などの量子計算機の解読にも耐性をもつ暗号（耐量子計算機暗号）の安全性解析に応用される．また，本研究で得られた超特別曲線は，同種写像暗号において安全なパラメータとしての利用が期待されるなど，情報セキュリティ分野への貢献にも繋がる．