My end goal is to have an encryption function $e$ and a hash function $H$ such that for all m we have: $$H(e(m)) = e(H(m))$$
This would work if we use RSA encryption along with RSA "hash", using a different public key. I don't mind the fact that there would be one specific person (who created the private/public key for the hash) that would be able to reverse the "hash".
So my question is can I use RSA as a hash function? I still need all the other properties of hash functions, I just don't mind if a specific person can reverse it.
Or, would there be another hash/cipher pair that would satisfy $H(e(m)) = e(H(m))$?
I have looked into fully homomorphic encryption but it does not seem to help.