On bent functions associated to AB functions

In 1998, the second author, Charpin and Zinoviev characterized APN and AB (n, n)-functions by means of associated 2n-variable Boolean functions. In particular, they proved that a function F is AB if and only if the associated Boolean function γ_F is bent. This observation leads to potentially new bent functions associated to the known AB functions, or at least gives new insight on known bent functions. However, up to now, representations of γ_F are known only for Gold AB power functions and determining γ_F for the rest of AB functions is an open problem. In the present paper we determine γ_F for most of the known families of APN and AB functions.