Homomorphic Encryption: how does the equality test on ciphertexts work?
No, what you want to do is not possible, because encryption is randomized: if you were to encrypt the same message many times, you'd get many different ciphertexts.  Therefore, Alice can't just compare two ciphertexts to see if they are the same; the two ciphertexts will be different even if they decrypt to the same thing.