Efficient methods for Groebner basis computation, verification and thier applications

2017 Fiscal Year Final Research Report

• 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
• Keywords: 応用数学 / 計算代数 / グレブナー基底 / モジュラー計算

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.

Free Research Field

計算代数