2014 Fiscal Year Final Research Report
Researches of functions on finite fields coming regular affine planes
| Project/Area Number |
23540035
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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 | Kinki University |
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Principal Investigator |
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| Project Period (FY) |
2011-04-28 – 2015-03-31
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| Keywords | APN functions / finite fields / permutation group / EA-equivalence / plnar functions / alternative product / cryptography / linear equations |
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| Outline of Final Research Achievements |
In the cryptography theory,it is important to construct functions over finite fields which have hight non-linearities. Almost perfect nonlinear(APN) functions are one of them. APN functions are classified in EA-equivalent classes. In the study,I consider some permutation group (G,S) where G is the linear group of degree n over GF(2) and S is a set of subspaces of the alternative product of a finite field F with some properties. I proved the number of G-orbits on S equals to the namber of EA equivalent classes of quadratic APN functions on F.I obtained some conditions that a special linear equation over a finite field has exactly two solutions and as a application of the results, I constructed three APN functions and decided subspaces corresponding to these functions up to EA-equivalence. Moreover I obtained an effective expression of APN functions which are equivalent to Gold functions which are most interesting ones among APN functions.
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| Free Research Field |
代数学
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