On information leakage during secure verification of compatibility between signals

We consider a secure verification problem in which Alice wants to verify whether her signal Xⁿ is compatible with Bob´s signal Yⁿ, where Xⁿ and Yⁿ are drawn i.i.d. according to a joint distribution p(x, y). The notion of compatibility is defined as the requirement that p(x, y) belongs to a certain set A of allowable joint distributions. For privacy, Alice jointly encrypts and encodes Xⁿ and transmits the result over a public channel to Bob. Using the information leaked by the encryption algorithm, Bob verifies the compatiblility of Xⁿ with Yⁿ. We characterize the minimum information that Alice´s encryption and coding algorithm must leak in order to guarantee reliable verification results. Further, we determine the maximum information that Bob can hope to extract about Xⁿ if he is curious. It is shown that a source/channel separation theorem holds for this scenario.