New trends in the research of dimensional dual hyperovals

2018 Fiscal Year Final Research Report

• Project/Area Number: 26400025
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Tokyo Woman's Christian University
• Principal Investigator: Yoshiara Satoshi 東京女子大学, 現代教養学部, 教授 (10230674)
• Research Collaborator: Akiyama Kenji ; Suetake Chihiro ; Taniguchi Hiroaki ; Nakagawa Nobuo
• Project Period (FY): 2014-04-01 – 2019-03-31
• Keywords: 高次元超卵型 / 生成空間 / 分裂性 / APN関数

Outline of Final Research Achievements

This research has the following three objects: (a) to establish the sharpest upper bound for the dimension of the ambient space, (c) to show the splitness, of a dimensional dual hyperoval (DHO), which is a natural geometric object generalizing conics in a projective plane, and, as an application, (b) to solve the equivalence problem among known APN functions.
I obtained the following results. As for (a), I showed the exact bound for bilinear dual hyperovals and classified those attaining the upper bound. As for (b), I solved the problem with known monomial APN functions. As for (c), I developed several methods to show the splitness of a DHO with a concrete model, based on a certain function which can be calculated from the model. As an application, all known DHO are verified to split.

Free Research Field

代数学

Academic Significance and Societal Importance of the Research Achievements

本研究では, 高次元双対超卵型と呼ばれる平面上の二次曲線の一般化である幾何学的な対象を扱う. 生成空間の次元問題は, この対象がどの程度大きくなりうるかを追究し, また分裂性の問題は, この対象がどの程度非線形な関数と関連するかを調べている. 本研究の成果を通じて, 高次元双対超卵型という概念が数学的に自然なものであるのみならず, 深く追及すべき数理科学的具体例があることが示された. 後者は非線形関数と深く関連しており, それは対称暗号を実装する際に役立つことが知られている. 本研究は暗号理論において知られている具体的な非線形関数を統合する数学的理論が高次元超卵型論に内在することを示唆する.