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

[math]\displaystyle{ \{D_aF(x) \colon x\in\mathbb{F}_2^n\} }[/math]

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

[math]\displaystyle{ W_{D_aF}(0,v)\neq 0 }[/math]

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

[math]\displaystyle{ W_{D_aF}(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) }[/math]

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