The crooked property |
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Institution: | 1. School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471023, China;2. Faculty of Mathematics and Statistics, Hubei University, Wuhan 430062, China;3. Department of Informatics, University of Bergen, Bergen N-5020, Norway;4. State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing 100093, China;5. Department of Electrical Engineering and Computer Science, University of Stavanger, 4036, Stavanger, Norway;1. Key Laboratory of Intelligent Computing Signal Processing, Ministry of Education, School of Mathematical Sciences, Anhui University, Hefei, Anhui, 230601, China;2. School of Mathematical Sciences, Anhui University, Hefei, 230601, China;3. Department of Mathematics and Institute of Applied Mathematics, Middle East Technical University, Ankara, Turkey;4. I2M, Aix Marseille Univ., Centrale Marseille, CNRS, Marseille, France |
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Abstract: | Crooked permutations were introduced twenty years ago to construct interesting objects in graph theory. These functions, over with odd n, are such that their derivatives have as image set a complement of a hyperplane. The field of applications was extended later, in particular to cryptography. However binary crooked functions are rare. It is still unknown if non quadratic crooked functions do exist. We extend the concept and propose to study the crooked property for any characteristic. A function F, from to itself, satisfies this property if all its derivatives have as image set an affine subspace. We show that the partially-bent vectorial functions and the functions satisfying the crooked property are strongly related. We later focus on the components of these functions, establishing that the existence of linear structures is here decisive. We then propose a symbolic approach to identify the linear structures. We claim that this problem consists in solving a system of linear equations, and can often be seen as a combinatorial problem. |
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Keywords: | Affine subspace Vectorial function Boolean function Linear structure Planar function Differential set Plateaued function Partially-bent function Bent function APN function |
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