Vectorial boolean functions on distance one from apn functions | Prikladnaya Diskretnaya Matematika - Applied Discrete Mathematics. 2014. № 7 (Приложение).

The metric properties of the class of vectorial Boolean functions are studied. A vectorial Boolean function F in n variables is called a differential 8-uniform function if the equation F(x) ф F(x ф а) = b has at most 8 solutions for any vectors а, b, where а = 0. In particular, if it is true for 8 = 2, then the function f is called APN. The distance between vectorial Boolean functions F and G is the cardinality of the set {x £ Zn : F(x) = G(x)}. It is proved that there are only differential 4-uniform functions which are on the distance 1 from an APN function.