From the previous properties, we can deduce two more, which we will use. As a matter of fact, can be defined by The second property which is a consequence of the previous ones, is the existence of a way to prove that two ciphertexts represent the encryption of the same integer without revealing this integer. Let and consider and . Then, there exists such that . In order to prove that , one can simply reveal .
Verification is performed by computing both and and checking their equality.