I wondered about the classes of invertible functions over $n$-dimensional Galois fields, $f: GF(q)^n \to GF(q)^n$ s.t. $f^{-1}$ exists.
I know that all linear mappings, e.g. of the form $f(x) = Gx$ with some matrix $G\in GF(q)^{n\times n}$, are invertible if the respective matrix is invertible. But are there any other, non-linear, invertible mappings?
Currently, I'm looking for examples of existence or arguments of non-existence. Ultimately I'd like to characterize and construct all invertible mappings $f: GF(q)^n \to GF(q)^n$.