Exceptional crooked functions (Q2081341)

In this paper, the authors investigate conditions on the shape of a polynomial to be crooked (it is already known that the only monomial and binomial crooked functions are quadratic). The notion of exceptional crooked is introduced, similarly to those of APN or PN exceptional functions. Via a connection with algebraic varieties over finite fields, the authors provide non-existence results of exceptional crooked functions (functions that are crooked over infinitely many extensions of \(\mathbb{F}_2\)). Their approach is based on an attaching of algebraic variety \(W_f\) defined by a given affine equation to a function \(f:\mathbb{F}_{2^n} \rightarrow \mathbb{F}_{2^n}\) with \(f(0) = 0\).