Efficient methods for Groebner basis computation, verification and thier applications

• Project/Area Number: 15K05008
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Rikkyo University
• Principal Investigator: NORO Masayuki 立教大学, 理学部, 教授 (50332755)
• Co-Investigator(Kenkyū-buntansha): 横山 和弘 立教大学, 理学部, 教授 (30333454) ; 篠原 直行 国立研究開発法人情報通信研究機構, サイバーセキュリティ研究所セキュリティ基盤研究室, 主任研究員 (70565986)
• Research Collaborator: AOYAMA Toru 神戸大学, 理学研究科
• Project Period (FY): 2015-04-01 – 2018-03-31
• Project Status: Completed (Fiscal Year 2017)
• Budget Amount: ¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000); Fiscal Year 2017: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2016: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000); Fiscal Year 2015: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
• Keywords: 応用数学 / 計算代数 / グレブナー基底 / モジュラー計算 / Ｆ５アルゴリズム / 楕円曲線 / ホロノミック勾配法 / 行列変数1F1 / 有限体 / 暗号 / Wishart分布 / signature / F5 / 同種写像

Outline of Final Research Achievements

We published a paper describing various modular methods for efficient Groebner basis computation. We published two papers concerned with the signature based
algorithm (SBA). In these papers we proposed several variants of SBA and showed their correctness and termination. F4 and SBA are main methods for attacking elliptic curve cryptography and post-quantum cryptography. We analyzed the complexity when we applied these methods to cryptanalysis. We studied the system of partial differential equations (PDE) satisfied by the cumulative distribution function (CDF) of Wishart matrices and proposed an efficient method for deriving a system of PDE satisfied by the restriction of the CDF on diagonal regions. This is an application of Groebner basis theory to statistics that is a very active research area.

Research Products

• [Journal Article] Usage of Modular Techniques for Efficient Computation of Ideal Operations (2018)
  • Author(s): Noro, M. & Yokoyama, K.
  • Journal Title: Mathematics in Computer Science Volume: 12,1 Issue: 1 Pages: 1-32
  • DOI: 10.1007/s11786-017-0325-1
  • Peer Reviewed
• [Journal Article] Modular Algorithms for Computing Minimal Associated Primes and radicals of Polynomial Ideals (2018)
  • Author(s): Aoyama, T. & Noro, M.
  • Journal Title: Proceedings of ISSAC 2018 Volume: -
  • Peer Reviewed
• [Journal Article] On Affine Tropical F5 Algorithms (2018)
  • Author(s): Tristan, V. & Yokoyama, K.
  • Journal Title: Proceedings of ISSAC2018 Volume: -
  • Peer Reviewed
• [Journal Article] 楕円曲線上の離散対数問題に関する指数計算法 (2017)
  • Author(s): 篠原直行, 野呂正行, 横山和弘
  • Journal Title: CRYPTREC Report 2016 Volume: - Pages: 71-100
  • Peer Reviewed
• [Journal Article] An Algorithm for Computing Minimal Associated Primes of Binomial Ideals without Producing Redundant Components (2017)
  • Author(s): Aoyama, Toru
  • Journal Title: Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation (ISSAC 2017) Volume: 印刷中
  • Peer Reviewed / Acknowledgement Compliant
• [Journal Article] System of Partial Differential Equations for the Hypergeometric Function 1F1 of a Matrix Argument on Diagonal Regions (2016)
  • Author(s): M. Noro
  • Journal Title: Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation (ISSAC 2016) Volume: 2016 Pages: 381-388
  • DOI: 10.1145/2930889.2930905
  • Peer Reviewed / Open Access
• [Presentation] 有理関数体上の最小多項式の計算について (2018)
  • Author(s): 野呂正行
  • Organizer: Risa/Asir Conference 2018
• [Presentation] 楕円曲線の同種写像の公式作成における計算機代数の利用 (2018)
  • Author(s): 横山和弘
  • Organizer: Risa/Asir Conference 2018
• [Presentation] Risa/Asir 2016-2017 (2017)
  • Author(s): 野呂正行
  • Organizer: Risa/Asir conference 2017
  • Place of Presentation: 金沢大学自然科学５号館第講義室（石川県金沢市）
  • Year and Date: 2017-03-29
• [Presentation] 楕円曲線上の離散対数問題へのグレブナー基底の応用 (2017)
  • Author(s): 篠原直行, 野呂正行, 横山和弘
  • Organizer: Risa/Asir conference 2017
  • Place of Presentation: 金沢大学自然科学５号館第講義室(石川県金沢市）
  • Year and Date: 2017-03-29
• [Presentation] 楕円曲線の同種写像の公式生成について (2017)
  • Author(s): 横山和弘
  • Organizer: 九州代数的整数論2017
  • Place of Presentation: 九州大学ＭＩ研究所（福岡県福岡市）
  • Year and Date: 2017-03-10
  • Invited
• [Presentation] 楕円離散対数問題に対する指数計算法 (2017)
  • Author(s): 篠原直行
  • Organizer: 代数幾何学と暗号数理の展開
  • Place of Presentation: 九州大学西新プラザ（福岡県福岡市）
  • Year and Date: 2017-02-06
  • Int'l Joint Research
• [Presentation] 耐量子計算機暗号の最新動向とNICTの取組み (2017)
  • Author(s): 篠原直行
  • Organizer: SecurityDay 2017
  • Invited
• [Presentation] 近年の数式処理と将来の展望 (2016)
  • Author(s): 横山和弘
  • Organizer: RIMS 研究集会「数式処理の新たな発展」
  • Place of Presentation: RIMS(京都府京都市）
  • Year and Date: 2016-09-08
  • Invited
• [Presentation] System of Partial Differential Equations for the Hypergeometric Function 1F1 of a Matrix Argument on Diagonal Regions (2016)
  • Author(s): Noro, Masayuki
  • Organizer: ISSAC2016
  • Place of Presentation: Waterloo (Canada)
  • Year and Date: 2016-07-21
  • Int'l Joint Research
• [Presentation] matrix 1F1 が対角領域上で満たす微分方程式系を用いた分布関数の値の計算 (2016)
  • Author(s): 野呂正行
  • Organizer: Risa/Asir Conference 2016
  • Place of Presentation: 金沢大学自然科学5号館大講義室(石川県・金沢市)
  • Year and Date: 2016-03-28
• [Presentation] 小標数の有限体上の離散対数問題の解法 (2015)
  • Author(s): 篠原直行
  • Organizer: 第 11 回「代数学と計算」研究集会 (AC2015)
  • Place of Presentation: 首都大学東京(東京都・八王子市)
  • Year and Date: 2015-12-14
  • Invited
• [Presentation] 行列変数1F1の対角領域への制限が満たす微分方程式系の計算 (2015)
  • Author(s): 野呂正行
  • Organizer: RIMS研究集会「数式処理とその周辺分野の研究 ― Computer Algebra and Related Topics」
  • Place of Presentation: 都大学北部総合教育研究棟「益川ホール」(京都府・京都市）
  • Year and Date: 2015-12-01
• [Presentation] Stability of Parametric Decomposition (2015)
  • Author(s): Kazuhiro Yokoyama
  • Organizer: Dagstuhl Seminar 15471 Symbolic Computation and Satisfiability
  • Place of Presentation: Schloss Dagstuhl, Wadern(Germany)
  • Year and Date: 2015-11-15
  • Int'l Joint Research / Invited
• [Presentation] グレブナー基底計算の効率化 (2015)
  • Author(s): 横山和弘
  • Organizer: RIMS研究集会「数式処理研究の新たな発展」
  • Place of Presentation: 京都大学数理解析研究所(京都府・京都市)
  • Year and Date: 2015-08-19
  • Invited
• [Presentation] Modular Techiniques for Efficient Computation of Ideal Operation (2015)
  • Author(s): Kazuhiro Yokoyama, Masayuki Noro
  • Organizer: ICIAM 2015
  • Place of Presentation: China National Convention Center, Beijing(China)
  • Year and Date: 2015-08-11
  • Int'l Joint Research / Invited