Pallier Cryptosystem is probabilistic in nature and IND-CPA secure. By design given two ciphertexts one cannot distinguish whether decrypting those two ciphertexts will result in same or different plaintexts.
But in certain situations like Onion-layered encryption in CryptDB. There are other onions like DET and OPE that reveal whether two ciphertexts map to same underlying plaintext or whether the plaintexts are greater or less when compared to each other.
For example , in CryptDB the plaintext is encrypted using multiple schemes and stored separately. So two different plaintexts $m_1,m_2$, assuming both are numbers, would look in the tables as below
- HOM Onion : Pallier_Enc($m_n,k_1$) = $c^h_n$ , $h$ denotes homomorphic layer
- DET Onion : AES_Enc($m_n, k_2$) = $c^d_n$, $d$ denotes deterministic layer
- OPE Onion : OPE_ENC($m_n, k_3$) = $c^o_n$, $o$ denotes order preserving layer
Now although $c^h_1 , c^h_2$ do not reveal whether $m_1, m_2$ are equal because pallier scheme is probabilistic . There are other onion layers revealing whether the $m_1,m_2$ are same or less or greater to each other.
My hunch is these inferences from other layers weaken the pallier cryptosystem's security guarantees but I could not come up with a way to weaken. So how can we take advantage of other onions leaking inferences (about equality, comparison) to break Pallier system in some way ?