In the public-key setting, known constructions of functionprivate functional encryption (FPFE) were limited to very restricted classes of functionalities like inner-product [Agrawal et al. - PKC 2015]. Moreover, its power has not been well investigated. In this paper, we construct FPFE for general functions and explore its powerful applications, both for general and specific functionalities.As warmup, we construct from FPFE a natural generalization of a signature scheme endowed with functional properties, that we call functional anonymous signature (FAS) scheme. In a FAS, Alice can sign a circuit C chosen from some distribution D to get a signature s and can publish a verification key that allows anybody holding a message m to verify that (1) sigma is a valid signature of Alice for some (possibly unknown to him) circuit C and (2) C(m) = 1. Beyond unforgeability the security of FAS guarantees that the signature s hide as much information as possible about C except what can be inferred from knowledge of D.Then, we show that FPFE can be used to construct in a black-box way functional encryption schemes for randomized functionalities (RFE).As further application, we show that specific instantiations of FPFE can be used to achieve adaptively-secure CNF/DNF encryption for bounded degree formulae (BoolEnc). Though it was known how to implement BoolEnc from inner-product encryption (IPE) [Katz et al. - EUROCRYPT 2008], as already observed by Katz et al. this reduction only works for selective security and completely breaks down for adaptive security; however, we show that the reduction works if the IPE scheme is function-private.Finally, we present a general picture of the relations among all these related primitives. One key observation is that Attribute-based Encryption with function privacy implies FE, a notable fact that sheds light on the importance of the function privacy property for FE.
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