Crooked Functions

An (n, n)-function F is called crooked if, for every nonzero a, the set

$\{D_{a}F(x)\colon x\in \mathbb {F} _{2}^{n}\}$

is an affine hyperplane (i.e. a linear hyperplane or its complement).

Conversely, crooked functions are strongly plateaued and APN.

The component functions of a crooked function are all partially-bent.

CCZ equivalence does not preserve crookedness

Characterization of Crooked Functions

For n odd, F is crooked if and only if F is almost bent (AB).

F is crooked if and only if, for every nonzero a, there exists a unique nonzero v such that

$W_{D_{a}F}(0,v)\neq 0$

This characterization can be expressed by means of the Walsh transform of F since

$W_{D_{a}F}(0,v)=\Delta _{v\cdot F}(a)=2^{-n}\sum _{u\in \mathbb {F} _{2}^{n}}(-1)^{u\cdot a}W_{F}^{2}(u,v)$

Known Crooked Functions

All quadratic APN functions are crooked

If a monomial is crooked, then it is quadratic

If a binomial is crooked, then it is quadratic

An open problem is to find a crooked function that is not quadratic