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For a (possibly complicated) function $f: X \rightarrow Y$, can we design a set of encryption scheme $Enc, Func\_Enc$ such that, for $\forall x \in X$,

$$Func\_Enc(f) = g \text{ where } g \text{ is another function} $$ $$ g(Enc(x)) = f(x) \quad \forall x \in X $$

Also, $g$ shouldn't leak information of $f$, and $g(Enc(x))$ shouldn't leak information of $f(x')$ for any $x' \neq x$.

I realize that the definition is kind of similar to functional encryption, but there are subtle differences between them. Also, if the problem is too ideal to be solved, any relaxation on security (e.g. information leakage can be tolerated within a small probability or limited amount, additional assumptions on $f$ or security introduced) is fine, but it's preferred if a concrete and practically efficient protocol can be used on this topic.

Any help, ideas, or impossibility results are appreciated.

Thanks a lot!

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    $\begingroup$ It sounds like you want to obfuscate the function $A(c) = f(\textsf{Dec}(k,c))$, where $f$ and $k$ are things to be hidden by the obfuscation. $\endgroup$
    – Mikero
    Feb 14 at 21:58
  • $\begingroup$ Yeah obfuscation is clearly a possible solution. If we have an obfuscation scheme, we can separate the encryption and function value calculation part, and use obfuscation on the latter one. However, the obfuscation schemes are really slow in practice nowadays, and I'm wondering, whether or not we can design the encrypted function g which works on encrypted text Enc(x) without first decoding it (just for a possibly inaccurate analogy, it is something like translating questions into a new language first, and do all the logical thinking and communication in the new language, then translate back). $\endgroup$
    – Marc_12
    Feb 15 at 4:03

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