2006 Fiscal Year Final Research Report Summary
Researches of finite geometries and functions over finite fields
| Project/Area Number |
16540129
|
|---|
| Research Category |
Grant-in-Aid for Scientific Research (C)
|
| Allocation Type | Single-year Grants |
|---|
| Section | 一般 |
|---|
| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
|
|---|
| Research Institution | Kinki University |
|---|
Principal Investigator |
NAKAGAWA Nobuo Kinki University, Faculty of Science and Technology, Associate professor, 理工学部, 助教授 (10088403)
|
|---|
| Co-Investigator(Kenkyū-buntansha) |
NAGAOKA Shouyu Kinki University, Faculty of Science and Technology, Professor, 理工学部, 教授 (20164402)
HIRAMINE Yutaka Kumamoto University, the department of education, Professor, 教育学部, 教授 (30116173)
|
|---|
| Project Period (FY) |
2004 – 2006
|
| Keywords | planar function / finite semifield / almost perfect nonlinear / bent function / trace mapping / finite fields / blocking semioval / projective plane |
|---|
| Research Abstract |
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.
|
