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Journal Articles Finite Fields and Their Applications Year : 2022

The crooked property

Abstract

Crooked permutations were introduced twenty years ago to cons- truct interesting objects in graph theory. These functions, over F2n with odd $n$, are such that their derivatives have as image set a com- plement 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 ex- ist. We extend the concept and propose to study the crooked property for any characteristic. A function $F$, from Fpn to itself, satisfies this property if all its derivatives have as image set an a ne 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 ap- proach 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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Dates and versions

hal-03091422 , version 1 (31-12-2020)
hal-03091422 , version 2 (16-11-2021)
hal-03091422 , version 3 (13-05-2022)

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Pascale Charpin. The crooked property. Finite Fields and Their Applications, 2022, ⟨10.1016/j.ffa.2022.102032⟩. ⟨hal-03091422v3⟩

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