Let $ \mathbb{Z}_2^n$ be the field of bitvectors of length $ n$ and define the xor operator $ \oplus$ and the addition operator $ +$ over this field, with $ +$ having the usual overflow semantics (take addition modulo $ 2^n$ ).

Is it possible to express any mapping $ f: \mathbb{Z}_2^n \rightarrow \mathbb{Z}_2^n$ entirely in terms of $ \oplus$ and $ +$ ? It seems like it might be possible due to the nonlinearity of $ +$ .