What hash functions can be (efficiently) computed over GF(2^m)?
Given an arithmetic circuit over a finite field of characteristic 2, what families of cryptographic hash functions can be computed with this circuit?


Answer: all of them.  The combination of addition and multiplication in $GF(2^m)$ is complete.  That is, for any arbitrary function that takes $n$ inputs in $GF(2^m)$ and an output in $GF(2^m)$, there exists a polynomial in $n$ variables that produces that function.

Note that this polynomial need not be simple; however it will exist.