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What would happen if you added two inputs, one known and one unknown, both encrypted with a fully homomorphic encryption scheme, and then checked the result by dividing it by two? Wouldn’t this tell you if the data is identical?

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    $\begingroup$ How are you "checking" the result? Do you have the decryption key? If so, why not just decrypt the unknown one? $\endgroup$ – mikeazo Aug 23 '17 at 15:17
  • $\begingroup$ If the sum of the two inputs is revealed to you it should be trivial to compute the unknown input given the known (just subtract). This is a very simplistic example of a general problem with secure computation. Typically, the security properties of the system simply guarantees that nothing is revealed except for the intended result. However, wether the result it self leaks unacceptable information is not considered. $\endgroup$ – Guut Boy Aug 24 '17 at 18:28
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Maybe you are looking for something like MPC, where parties want to compute a shared result, but need to keep their data private. Well, as you observed, the result of the computation can reveal more or less information depending on the security model and the number of parties involved. However, you can achieve secure multi-party computation using homomorphic encryption, as indicated in this paper On-the-fly Multiparty Computation on the Cloud via Multipkey Homomorphic Encryption. Unfortunately, the scheme described in the paper is no longer secure for practical computation as recent work by Ducas et.al proved.

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