The only crooked power functions are x2k+2lx2k+2l

A map f:F2n→F2nf:F2n→F2n is called crooked if the set {f(x+a)+f(x):x∈F2n}{f(x+a)+f(x):x∈F2n} is the complement of a hyperplane for every fixed a∈F2n∗ (where F2nF2n is considered as a vector space over F2F2). We prove that the only crooked power maps are the quadratic maps x2k+2lx2k+2l with gcd(n,k−l)=1.