Properties of hash functions

[Aviel David Rubin]

Does anyone know the following, or can someone point me to the
proper references?

What properties are preserved by hash functions? For example, take
x = MD5(Y). Now, say that Y is a quadradic residue mod N, is it
the case that x is also a quadradic residue mod N? Are there any
interesting properties that are preserved by frequently used hash
functions?

Thanks a lot for any help,
Avi Rubin