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.