breaking fully homomorphic encryption schemes
Even though all the operations you described can be performed homomorphically, the result remains encrypted, i.e., the attacker cannot "see" it.  So homomorphic computation is not useful (on its own) as an attack, because the results remain unknown to the attacker.

For example, given two ciphertexts $c, c'$, an attacker can homomorphically compute whether they encrypt the same message.  The result is a ciphertext $c''$ that is an encryption of either 1 ("same message") or 0 ("different messages").  But the attacker cannot decrypt $c''$, so the 0/1 answer remains hidden to him.