| Project/Area Number |
26400025
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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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| Research Field |
Algebra
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| Research Institution | Tokyo Woman's Christian University |
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Principal Investigator |
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| Research Collaborator |
Akiyama Kenji
Suetake Chihiro
Taniguchi Hiroaki
Nakagawa Nobuo
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| Project Period (FY) |
2014-04-01 – 2019-03-31
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| Project Status |
Completed (Fiscal Year 2018)
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| Budget Amount *help |
¥2,990,000 (Direct Cost: ¥2,300,000、Indirect Cost: ¥690,000)
Fiscal Year 2017: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2016: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2015: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2014: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
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| Keywords | 高次元超卵型 / 生成空間 / 分裂性 / APN関数 / 高次元双対超卵型 / 非線形関数 / ＤＨＯ（双対超卵形） / APN関数の同値性 / 生成空間の次元 / bilinear DHO / ambient space / Huybrechts DHO / Buratti-Del Fra DHO / CCZ-同値 / 単項関数 / DHO(高次元双対超卵型) / quadratic 関数 / EA-同値 / DHO / APN 関数 |
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| 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.
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| 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.
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