The only crooked power functions are

A map f : F 2 n → F 2 n is called crooked if the set { f ( x + a ) + f ( x ) : x ∈ F 2 n } is the complement of a hyperplane for every fixed a ∈ F 2 n ∗ (where F 2 n is considered as a vector space over F 2 ). We prove that the only crooked power maps are the quadratic maps x 2 k + 2 l with gcd ( n , k − l ) = 1 .