An (n, n)-function F is called crooked if, for every nonzero a, the set
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
This characterization can be expressed by means of the Walsh transform of F since
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