# Universality of $(+, \oplus)$ over $\mathbb{Z}_2^n$

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 $$+$$.