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I believe I know quite well OPE and ORE, but I'm unsure about what family to put them in. Can we consider them as a sub family of Functional Encryption, like Attribute Based Encryption or Inner Product Encryption or are they a new paradigm of their own ?

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No, order-preserving encryption is not a subclass of functional encryption as usually defined, that is with public-key encryption.

Argument: In such functional encryption, encryption uses a public key. But we can't have secure public-key order-preserving encryption, because knowledge of the public key allows adversaries to encrypt, and ability to encrypt combined with the order-preserving property allows to decipher by dichotomy.

As noted in comment, that argument does not apply to Secret Key Functional Encryption, a notion I'm unfamiliar with. So I reserve my opinion on if OPE can be obtained in the framework of SKFE.

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    $\begingroup$ Secret-key functional encryption is a thing. So while I agree with your conclusion, your argument doesn't quite track. $\endgroup$
    – Maeher
    Jun 27 at 13:08
  • $\begingroup$ @Maeher I was going to comment that... FE for inner product suffers from more or less the same problem fgrieu pointed out: given a functional key for a hidden vector $v$, if one is able to encrypt several vectors $u_i$ by themselves, then they can decrypt several inner products $u_i \cdot v$ and then recover $v$ by linear algebra... So this type of FE for inner product is also secret key FE. $\endgroup$ Jun 28 at 8:48

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