Researches of finite geometries and functions over finite fields

有限域上的有限几何和函数研究

• 批准号: 16540129
• 负责人: NAKAGAWA Nobuo
• 金额: $ 1.15万
• 依托单位: Kinki University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2004
• 资助国家: 日本
• 起止时间: 2004 至 2006
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-16540129/
• 关键词: planar function; finite semifield; almost perfect nonlinear; bent function; trace mapping; finite fields; blocking semioval; projective plane; planar functions; finite semifields; finite projective planes; differentially 4-uniform; bent functi

I obtains the following results under the granted science research subsidy. Firstly it is shown that functions over finite fields of characteristic 2 coming from cubic functions of finite commutative semifields of characteristic 2 has high nonlinearity and they are applied to cryptography and coding theory well.We constructed these explicit functions, and then we found an important key lemma, that is some functions from cubic functions of semifields naturally are two to one on a suitable sub domain of finite fields of characteristic 2.Moreover we gave a conjecture concerning two to one property of generalized functions in the key lemma above. This paper is a joint work with Satoshi Yoshiara and will be published in a Lecture Note of Computer Science.Secondly we proved that planes corresponding to quadratic planar functions over finite fields of characteristic p (p is an odd prime) are semifild planes, especially some of them are Desargusian planes. Moreover we proved square functions of finite commutative semifields of characteristic an odd prime and we calculated that functions from square functions of almost all known commutative semifields are quadratic functions. This paper is submitted as a joint work with Kaori Minami.Thirdly we classified blocking semiovals of all projective planes of order 9 which have 8 intersection points with a line. We sued a compute at a latter step of the proof. This paper was published in a Hokkaido Mathematical Journal 2006 as a joint work with Chihiro Suetake.

I obtains the following results under the granted science research subsidy. Firstly it is shown that functions over finite fields of characteristic 2 coming from cubic functions of finite commutative semifields of characteristic 2 has high nonlinearity and they are applied to cryptography and coding theory well.We constructed these explicit functions, and then we found an important key lemma, that is some functions from cubic functions of semifields naturally are two to one on a suitable sub domain of finite fields of characteristic 2.Moreover we gave a conjecture concerning two to one property of generalized functions in the key lemma above. This paper is a joint work with Satoshi Yoshiara and will be published in a Lecture Note of Computer Science.Secondly we proved that planes corresponding to quadratic planar functions over finite fields of characteristic p (p is an odd prime) are semifild planes, especially some of them are Desargusian planes. Moreover we proved square functions of finite commutative semifields of characteristic an odd prime and we calculated that functions from square functions of almost all known commutative semifields are quadratic functions. This paper is submitted as a joint work with Kaori Minami.Thirdly we classified blocking semiovals of all projective planes of order 9 which have 8 intersection points with a line. We sued a compute at a latter step of the proof. This paper was published in a Hokkaido Mathematical Journal 2006 as a joint work with Chihiro Suetake.

项目成果

A constructions of differentially 4-uniform functions from commutative semifields of characteristic 2

由特征 2 的交换半域构造微分 4-一致函数

• 期刊: To appear in Lecture Note of Computer Science
• 作者: Nobuo Nakagawa;Satoshi Yoshiara
• 通讯作者: Satoshi Yoshiara

On Hilbert modular forms modulo p

关于对 p 取模的希尔伯特模形式

• 发表时间: 2006
• 期刊: Revista Matematica Iberoamericana 22・No. 1
• 作者: Shoyu Nagaoka

Note onp-adic Hermitian Eisenstein Series

注意 onp-adic 埃尔米蒂安·爱森斯坦系列

• DOI: 10.1007/bf02960867
• 发表时间: 2006
• 期刊: Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg
• 作者: T. Munemoto;S. Nagaoka
• 通讯作者: S. Nagaoka

On the 2-dimensional dual hyperoval of polar type in PG(5,4) and its automorphism group

关于PG(5,4)中极型的二维对偶超椭圆及其自同构群

• 发表时间: 2005
• 期刊: 第21回代数的組合せ論シンポジュウム報告集
• 作者: Nobuo Nakagawa;Chihiro Suetake;Yutaka Hiramine;Shoyu Nagaoka;中川 暢夫
• 通讯作者: 中川 暢夫

On blocking semiovals with 8-secone in projective planes of order 9

关于在 9 阶射影平面上用 8 秒锥块半椭圆

• 发表时间: 2006
• 期刊: Hokkaido Mathematical Journal Vol. 35, No. 2
• 作者: Nobuo Nakagawa;Chihiro Suetake
• 通讯作者: Chihiro Suetake