# Crooked Functions

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