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Properties of hash functions



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