The only crooked power functions are \(x^{2^k+2^l}\)
A map \(f\colon\mathbb F_ {2^ {n}}\rightarrow\mathbb F_ {2^ n}\) is called crooked if the set \(\{f(x+a)+f(x)\mid x\in\mathbb F_ {2^ {n}}\}\) is the complement of a hyperplane for every fixed \(a\in\mathbb F^ *_ {2^ n}\) (where \(\mathbb F_ {2^ {n}}\) is considered as a vector space over \(\mathbb F_ {2}\)). In Theorem 1 the author proves that the only crooked power maps are the quadratic maps \(x^{2^k+2^l}\) with \(\gcd(n,k-l)=1\). In Theorem 3 a geometrical characterization of crooked maps for almost bent permutations is given, and some connections with almost perfect nonlinear maps are revealed.
scientific article; zbMATH DE number 5141679
scientific article