| Project/Area Number |
18K03432
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Review Section |
Basic Section 12040:Applied mathematics and statistics-related
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| Research Institution | Rikkyo University |
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Principal Investigator |
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| Co-Investigator(Kenkyū-buntansha) |
野呂 正行 立教大学, 理学部, 教授 (50332755)
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)
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| Project Period (FY) |
2018-04-01 – 2021-03-31
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| Project Status |
Completed (Fiscal Year 2020)
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| Budget Amount *help |
¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000)
Fiscal Year 2020: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2019: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Fiscal Year 2018: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
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| Keywords | グレブナー基底 / 計算機代数 / F5アルゴリズム / F4アルゴリズム / 公開鍵暗号 / 計算可換環論 / 計算代数幾何 / F5アリゴリズム / F4 アルゴリズム / F5 アルゴリズム |
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| Outline of Final Research Achievements |
Groebner bases, bases of polynomial ideals, have useful computational properties and used in a various areas such as mathematics and engineering science. However, there still remains a problem on their computational efficiency. In this project, focusing an efficient technique named SBA, we succeeded in completing its theoretical correctness and termination, and an efficient its implementation on a
real computer. As a theoretical result, under a reasonable condition "compatibility" on monomial orders, the correctness and termination of our SBA are theoretically guaranteed. By its implementation on a real computer, we devised several algorithms based on SBA, which have certain superiority to existing algorithms. As application study, we applied Groebner basis computation to analyze the security level of public key cryptosystems.
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| Academic Significance and Societal Importance of the Research Achievements |
グレブナー基底は多項式イデアルのよい性質をもつ基底で、連立代数方程式の求解に留まらずに、解の代数的構造などが計算によりわかることから、純粋数学から情報（暗号理論等）や工学（制御・最適化等）への応用まで幅広く用いられている。本研究によるグレブナー基底計算の高速化により、その適用範囲が広がることで、さらなる数学研究の進展や情報・工学での活用が期待される。
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