Crooked maps in F22

Abstract 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 an affine 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 i + 2 j with gcd ( n , i − j ) = 1 . This is a consequence of the following result of independent interest: for any prime p and almost all exponents 0 ⩽ d ⩽ p n − 2 the set { x d + γ ( x + a ) d : x ∈ F p n } contains n linearly independent elements, where γ and a ≠ 0 are arbitrary elements from F p n .

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