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Exploration for mathematical attacks against isogeny-based cryptography and their complexity analysis

Research Project

Project/Area Number 19K22847
Research Category

Grant-in-Aid for Challenging Research (Exploratory)

Allocation TypeMulti-year Fund
Review Section Medium-sized Section 60:Information science, computer engineering, and related fields
Research InstitutionRikkyo University (2020-2022)
Kyushu University (2019)

Principal Investigator

Yasuda Masaya  立教大学, 理学部, 教授 (30536313)

Co-Investigator(Kenkyū-buntansha) 高島 克幸  早稲田大学, 教育・総合科学学術院, 教授 (70723964)
Project Period (FY) 2019-06-28 – 2023-03-31
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥6,500,000 (Direct Cost: ¥5,000,000、Indirect Cost: ¥1,500,000)
Fiscal Year 2021: ¥2,210,000 (Direct Cost: ¥1,700,000、Indirect Cost: ¥510,000)
Fiscal Year 2020: ¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000)
Fiscal Year 2019: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Keywords同種写像暗号 / 楕円曲線 / 同種写像問題 / Deuring対応 / 耐量子計算機暗号 / 超特異楕円曲線 / 自己準同型環 / 四元数環 / 同種問題 / Velu公式 / ポスト量子暗号 / グレブナー基底計算 / 同種写像パス探索問題 / 連立代数方程式 / 解読アルゴリズム
Outline of Research at the Start

楕円曲線上の同種写像を利用した同種写像暗号は量子計算機に耐性のあるポスト量子暗号として近年注目されている.同種写像暗号の安全性は同種写像計算問題と呼ばれる数学問題の計算困難性に依存するが,解読アルゴリズムの開発・解読実験を含めた安全性解析が不十分で,今後の実用化に向けた最重要課題となっている.本研究では,同種写像が持つ代数的性質を利用した新しい解読法の探求を目指す.さらに,解読実験による計算量評価を行い,同種写像暗号の安全パラメータ選択の指針を示す.本研究により,既存方法よりも高速な解読アルゴリズムの開発を行うと共に,安全パラメータ選択法などの安全性解析法の確立を目指す.

Outline of Final Research Achievements

Isogeny-based cryptography is one of the candidates for quantum-safe cryptography. In this research, we developed several mathematical attacks against isogeny problems that support the security of isogeny-based cryptography. We also analyzed their computational complexity based on implementation results. Specifically, we reduced the general isogeny problem to a system of algebraic equations, and solved the system using Groebner basis calculation algorithms. We also reported the running time for breaking SIKE by our method. Furthermore, we succeeded in speeding up the constructive Deuring correspondence calculation for supersingular elliptic curves over finite fields. We also developed and implemented an algorithm for computing the endomorphism ring of a supersingular elliptic curve, which is equivalent to solving the isogeny path-finding problem.

Academic Significance and Societal Importance of the Research Achievements

本研究では, 耐量子計算機暗号候補の1つである同種写像暗号の安全性を支える同種写像計算問題に対して, 代数的手法に基づく解読実験によって多角的に安全性解析を行った.今回得られた解読手法と解析結果は, 同種写像暗号における暗号方式として安全なパラメータの選択時に活用することができる.特に, 本研究による多角的な安全性解析は, 同種写像暗号がどの程度安全か評価するための学術的データを与えるため, 耐量子計算機暗号としての同種写像暗号の標準化活動への貢献が期待できる.

Report

(5 results)
  • 2022 Annual Research Report   Final Research Report ( PDF )
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • Research Products

    (28 results)

All 2023 2022 2021 2020 2019

All Journal Article (9 results) (of which Peer Reviewed: 8 results,  Open Access: 6 results) Presentation (18 results) (of which Int'l Joint Research: 5 results,  Invited: 1 results) Book (1 results)

  • [Journal Article] Introduction to algebraic approaches for solving isogeny path-finding problems2022

    • Author(s)
      Ryoya Fukasaku, Yasuhiko Ikematsu, Momonari Kudo, Masaya Yasuda, Kazuhiro Yokoyama
    • Journal Title

      RIMS講究録別冊

      Volume: B90

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Counting superspecial Richelot isogenies by reduced automorphism groups2022

    • Author(s)
      Katsuyuki Takashima
    • Journal Title

      RIMS講究録別冊

      Volume: B90

    • Related Report
      2022 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Solving the Constructive Deuring Correspondence via the Kohel-Lauter-Petit-Tignol Algorithm2022

    • Author(s)
      Kambe Yuta, Yasuda Masaya, Noro Masayuki, Yokoyama Kazuhiro, Aikawa Yusuke, Takashima Katsuyuki, Kudo Momonari
    • Journal Title

      Mathematical Cryptology

      Volume: 1 Pages: 10-24

    • Related Report
      2021 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Implementation Report of the Kohel-Lauter-Petit-Tignol Algorithm for the Constructive Deuring Correspondence2022

    • Author(s)
      Kambe Yuta, Aikawa Yusuke, Kudo Momonari, Yasuda Masaya, Takashima Katsuyuki, Yokoyama Kazuhiro
    • Journal Title

      Advances in Intelligent Systems and Computing, Springer

      Volume: 1412 Pages: 953-966

    • DOI

      10.1007/978-981-16-6890-6_72

    • ISBN
      9789811668890, 9789811668906
    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] Symbolic Computation of Isogenies of Elliptic Curves by Vélu’s Formula2020

    • Author(s)
      Masayuki NORO, Masaya YASUDA, and Kazuhiro YOKOYAMA
    • Journal Title

      Commentarii mathematici Universitatis Sancti Pauli = Rikkyo Daigaku sugaku zasshi

      Volume: 68 Pages: 93-130

    • DOI

      10.14992/00020348

    • NAID

      120006954959

    • URL

      http://id.nii.ac.jp/1062/00020348/

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Hybrid Meet-in-the-Middle Attacks for the Isogeny Path-Finding Problem2020

    • Author(s)
      Ikematsu Yasuhiko、Fukasaku Ryoya、Kudo Momonari、Yasuda Masaya、Takashima Katsuyuki、Yokoyama Kazuhiro
    • Journal Title

      APKC20: Proceedings of the 7-th ACM Workshop on ASIA Public-Key Cryptography

      Volume: -- Pages: 36-44

    • DOI

      10.1145/3384940.3388956

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] Complexity bounds on Semaev’s naive index calculus method for ECDLP2020

    • Author(s)
      Yokoyama Kazuhiro、Yasuda Masaya、Takahashi Yasushi、Kogure Jun
    • Journal Title

      Journal of Mathematical Cryptology

      Volume: 14 Issue: 1 Pages: 460-485

    • DOI

      10.1515/jmc-2019-0029

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Algebraic approaches for solving isogeny problems of prime power degrees2020

    • Author(s)
      Takahashi Yasushi、Kudo Momonari、Fukasaku Ryoya、Ikematsu Yasuhiko、Yasuda Masaya、Yokoyama Kazuhiro
    • Journal Title

      Journal of Mathematical Cryptology

      Volume: 15 Issue: 1 Pages: 31-44

    • DOI

      10.1515/jmc-2020-0072

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] ディジタル署名EdDSAで使われている曲線の安全性に関する調査及び評価2020

    • Author(s)
      安田雅哉
    • Journal Title

      CRYPTREC外部評価報告書

      Volume: CRYPTREC EX-3001-2020 Pages: 1-47

    • Related Report
      2020 Research-status Report
  • [Presentation] 有限体上の通常楕円曲線の自己準同型環の生成元計算2023

    • Author(s)
      片山瑛, 安田雅哉
    • Organizer
      日本応用数理学会第19回研究部会連合発表会 「数論アルゴリズムとその応用」
    • Related Report
      2022 Annual Research Report
  • [Presentation] 超特異楕円曲線の自己準同型環計算の実装報告2023

    • Author(s)
      神戸祐太, 片山瑛, 相川勇輔, 石原侑樹, 安田雅哉, 横山和弘
    • Organizer
      暗号と情報セキュリティシンポジウム(SCIS2023)
    • Related Report
      2022 Annual Research Report
  • [Presentation] On the feasibility of computing constructive Deuring correspondence2022

    • Author(s)
      Yuta Kambe, Yasushi Takahashi, Masaya Yasuda, Kazuhiro Yokoyama
    • Organizer
      Number-Theoretic Methods in Cryptology (NuTMiC 2021)
    • Related Report
      2022 Annual Research Report
    • Int'l Joint Research
  • [Presentation] 構成的Deuring対応の計算可能性について2022

    • Author(s)
      神戸祐太, 安田雅哉, 横山和弘
    • Organizer
      日本応用数理学会2022年度年会「数論アルゴリズムとその応用」(JANT)セッション
    • Related Report
      2022 Annual Research Report
  • [Presentation] ρ法による超特異同種写像グラフにおけるサイクル探索2022

    • Author(s)
      神戸祐太, 片山瑛, 相川勇輔, 安田雅哉, 横山和弘
    • Organizer
      日本応用数理学会2022年度年会「数論アルゴリズムとその応用」(JANT)セッション
    • Related Report
      2022 Annual Research Report
  • [Presentation] 有限体上の楕円曲線の積によるアーベル曲面のブラウアー群の位数計算2022

    • Author(s)
      片山瑛, 安田雅哉
    • Organizer
      日本応用数理学会2022年度年会「数論アルゴリズムとその応用」(JANT)セッション
    • Related Report
      2022 Annual Research Report
  • [Presentation] 同種写像グラフの数理と耐量子計算機暗号への応用2022

    • Author(s)
      高島克幸
    • Organizer
      早稲田大学・整数論セミナー
    • Related Report
      2022 Annual Research Report
  • [Presentation] 耐量子計算機暗号の進展2022

    • Author(s)
      高島克幸
    • Organizer
      東京大学大学院数理科学研究科・情報数学セミナー
    • Related Report
      2022 Annual Research Report
  • [Presentation] 適切な素数選択によるKLPTアルゴリズムを利用した同種写像構成計算2022

    • Author(s)
      高橋康, 神戸祐太, 安田雅哉, 横山和弘
    • Organizer
      2022年暗号と情報セキュリティシンポジウム(SCIS2022)
    • Related Report
      2021 Research-status Report
  • [Presentation] SIKEに対するvOW法の内部関数の新計算手法2022

    • Author(s)
      神戸祐太, 高橋康, 相川勇輔, 工藤桃成, 安田雅哉, 高島克幸, 横山和弘
    • Organizer
      2022年暗号と情報セキュリティシンポジウム(SCIS2022)
    • Related Report
      2021 Research-status Report
  • [Presentation] Selection of primes in the KLPT algorithm for construction of fast isogeny (poster)2021

    • Author(s)
      Takahashi Yasushi, Kambe Yuta, Yasuda Masaya, Yokoyama Kazuhiro
    • Organizer
      IWSEC2021
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research
  • [Presentation] Solving the Constructive Deuring Correspondence via the Kohel-Lauter-Petit-Tignol Algorithm2021

    • Author(s)
      Kambe Yuta, Yasuda Masaya, Noro Masayuki, Yokoyama Kazuhiro, Aikawa Yusuke, Takashima Katsuyuki, Kudo Momonari
    • Organizer
      MathCrypt2021
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research
  • [Presentation] Kohel-Lauter-Petit-Tignolアルゴリズムの構成的Deuring対応への適用2021

    • Author(s)
      神戸祐太、相川勇輔、工藤桃成、安田雅哉、高島克幸、横山和弘
    • Organizer
      2021年暗号と情報セキュリティシンポジウム(SCIS2021)
    • Related Report
      2020 Research-status Report
  • [Presentation] Introduction to algebraic approaches for solving isogeny path-finding problems2020

    • Author(s)
      Masaya YASUDA, Kazuhiro YOKOYAMA
    • Organizer
      RIMS Conference on Theory and Applications of Supersingular Curves and Supersingular Abelian Varieties
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Kohel-Lauter-Petit-Tignolアルゴリズムのsageにおける実装報告2020

    • Author(s)
      神戸祐太 , 安田雅哉 , 横山和弘
    • Organizer
      日本応用数理学会2020年度年会
    • Related Report
      2020 Research-status Report
  • [Presentation] 同種写像パス探索問題に対する中間一致攻撃のハイブリッド手法2020

    • Author(s)
      池松泰彦、深作亮也、工藤桃成、安田雅哉、高島克幸、横山和弘
    • Organizer
      2020年暗号と情報セキュリティシンポジウム(SCIS2020)
    • Related Report
      2019 Research-status Report
  • [Presentation] Algebraic approaches for solving isogeny problems of prime power degrees2019

    • Author(s)
      高橋康、工藤桃成、池松泰彦、安田雅哉、横山和弘
    • Organizer
      MathCrypt 2019
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research
  • [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年度年会:「数論アルゴリズムとその応用」(JANT)セッション
    • Related Report
      2019 Research-status Report
  • [Book] Theory and Applications of Supersingular Curves and Supersingular Abelian Varieties2022

    • Author(s)
      原下秀士, 工藤桃成, 高島克幸
    • Total Pages
      219
    • Publisher
      RIMS講究録別冊(B90)
    • Related Report
      2022 Annual Research Report

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Published: 2019-07-04   Modified: 2024-01-30  

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