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

Research Project

Project/Area Number 19K21026
Project/Area Number (Other) 18H05836 (2018)
Research Category

Grant-in-Aid for Research Activity Start-up

Allocation TypeMulti-year Fund (2019)
Single-year Grants (2018)
Review Section 0201:Algebra, geometry, analysis, applied mathematics,and related fields
Research InstitutionKobe 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
Project Status Completed (Fiscal Year 2019)
Budget Amount *help
¥2,990,000 (Direct Cost: ¥2,300,000、Indirect Cost: ¥690,000)
Fiscal Year 2019: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2018: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
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 / フロベニウス写像 / コホモロジー群 / 超特異曲線 / 同種写像 / 計算代数 / 数式処理
Outline of Research at the Start

数学とその応用分野において,曲線は古くから研究されてきた重要な研究対象であり,その中でも特に代数曲線は,主に代数幾何学・整数論と呼ばれる分野で研究されている図形である.
本研究では,特殊な代数曲線の探索・決定を目的としており,理論と計算の融合的なアプローチによって解決を目指している.
本研究で得られる特殊な代数曲線は,暗号・符号理論への応用可能性を併せ持ち,将来的には情報通信などへの応用が期待される.

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.

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.

Report

(3 results)
  • 2019 Annual Research Report   Final Research Report ( PDF )
  • 2018 Annual Research Report
  • Research Products

    (18 results)

All 2020 2019 2018 Other

All Journal Article (5 results) (of which Peer Reviewed: 5 results) Presentation (9 results) (of which Int'l Joint Research: 4 results,  Invited: 2 results) Remarks (4 results)

  • [Journal Article] Automorphism groups of superspecial curves of genus 4 over F112020

    • Author(s)
      Kudo Momonari、Harashita Shushi、Senda Hayato
    • Journal Title

      Journal of Pure and Applied Algebra

      Volume: 224 Issue: 9 Pages: 106339-106339

    • DOI

      10.1016/j.jpaa.2020.106339

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Computational approach to enumerate non-hyperelliptic superspecial curves of genus 42020

    • Author(s)
      Kudo Momonari、Harashita Shushi
    • Journal Title

      Tokyo Journal of Mathematics

      Volume: -

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Superspecial trigonal curves of genus 52020

    • Author(s)
      Kudo Momonari、Harashita Shushi
    • Journal Title

      Experimental Mathematics

      Volume: -

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On the existence of superspecial and maximal nonhyperelliptic curves of genera four and five2019

    • Author(s)
      Kudo Momonari
    • Journal Title

      Communications in Algebra

      Volume: 47 Issue: 12 Pages: 5020-5038

    • DOI

      10.1080/00927872.2019.1609013

    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Superspecial Hyperelliptic Curves of Genus 4 over Small Finite Fields2018

    • Author(s)
      Kudo Momonari、Harashita Shushi
    • Journal Title

      L. Budaghyan, F. Rodriguez-Henriquez (eds), Arithmetic of Finite Fields, WAIFI 2018, Lecture Notes in Computer Science

      Volume: 11321 Pages: 58-73

    • DOI

      10.1007/978-3-030-05153-2_3

    • ISBN
      9783030051525, 9783030051532
    • Related Report
      2018 Annual Research Report
    • Peer Reviewed
  • [Presentation] Computing representation matrices for the Frobenius on cohomology groups2019

    • Author(s)
      Kudo Momonari
    • Organizer
      Effective Methods in Algebraic Geometry (MEGA2019)
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Algebraic approaches for solving isogeny problems of prime power degrees2019

    • Author(s)
      Yasushi Takahashi, Momonari Kudo, Yasuhiko Ikematsu, Masaya Yasuda, Kazuhiro Yokoyama
    • Organizer
      MathCrypt 2019
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Computational approaches to superspecial curves of genera 4 and 5 over finite fields2019

    • Author(s)
      Kudo Momonari
    • Organizer
      Supersingular Abelian Varieties and Related Arithmetic
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 代数多様体のコホモロジー群へのフロベニウス作用を計算するアルゴリズム2019

    • Author(s)
      工藤桃成
    • Organizer
      日本数式処理学会第28回大会
    • Related Report
      2019 Annual Research Report
  • [Presentation] 同種写像問題に対する代数的求解法の解析と計算量評価2019

    • Author(s)
      高橋康、工藤桃成、池松泰彦、安田雅哉、横山和弘
    • Organizer
      日本応用数理学会2019年年会「数論アルゴリズムとその応用」
    • Related Report
      2019 Annual Research Report
  • [Presentation] Superspecial curves of genera four and five2019

    • Author(s)
      工藤桃成
    • Organizer
      第6回代数幾何学研究集会-宇部-
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Superspecial trigonal curves of genus five2019

    • Author(s)
      工藤桃成、原下秀士
    • Organizer
      日本数学会2019年年会
    • Related Report
      2018 Annual Research Report
  • [Presentation] Superspecial Hyperelliptic Curves of Genus 4 over Small Finite Fields2018

    • Author(s)
      Kudo Momonari、Harashita Shushi
    • Organizer
      International Workshop on the Arithmetic of Finite Fields (WAIFI2018)
    • Related Report
      2018 Annual Research Report
    • Int'l Joint Research
  • [Presentation] Superspecial Trigonal Curves of Genus 52018

    • Author(s)
      工藤桃成、原下秀士
    • Organizer
      日本応用数理学会2018年年会 「数論アルゴリズムとその応用」(JANT) セッション
    • Related Report
      2018 Annual Research Report
  • [Remarks] Automorphisms of superspecial curves

    • URL

      https://sites.google.com/view/m-kudo-official-website/english/code/kudo-harashita-senda-comp

    • Related Report
      2019 Annual Research Report
  • [Remarks] Computation of the Frobenius on cohomology groups

    • URL

      https://sites.google.com/view/m-kudo-official-website/english/code/comp_frobenius

    • Related Report
      2019 Annual Research Report
  • [Remarks] Data base of superspecial curves of genus 4

    • URL

      http://www2.math.kyushu-u.ac.jp/~m-kudo/Ssp-curves-genus-4.html

    • Related Report
      2018 Annual Research Report
  • [Remarks] Computation for superspecial trigonal curves

    • URL

      http://www2.math.kyushu-u.ac.jp/~m-kudo/kudo-harasita-comp-trigonal.html

    • Related Report
      2018 Annual Research Report

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Published: 2018-08-27   Modified: 2024-03-26  

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