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The arrival of indistinguishability obfuscation (iO) has transformed the cryptographic landscape by enabling several security goals that were previously beyond our reach. Consequently, one of the pressing goals currently is to construct iO from well-studied standard cryptographic assumptions.In this work, we make progress in this direction by presenting a reduction from iO to a natural form of public-key functional encryption (FE). Specifically, we construct iO for general functions from any single-key FE scheme for NC 1 that achieves selective, indistinguishability security against sub-exponential time adversaries. Further, the FE scheme should be compact, namely, the running time of the encryption algorithm must only be a polynomial in the security parameter and the input message length (and not in the function description size or its output length).We achieve this result by developing a novel arity amplification technique to transform FE for single-ary functions into FE for multi-ary functions (aka multi-input FE). Instantiating our approach with known, non-compact FE schemes, we obtain the first constructions of multi-input FE for constant-ary functions based on standard assumptions.Finally, as a result of independent interest, we construct a compact FE scheme from randomized encodings for Turing machines and learning with errors assumption.

In a functional encryption (FE) scheme, the owner of the secret key can generate restricted decryption keys that allow users to learn specific functions of the encrypted messages and nothing else. In many known constructions of FE schemes, security is guaranteed only for messages that are fixed ahead of time (i.e., before the adversary even interacts with the system). This so-called selective security is too restrictive for many realistic applications. Achieving adaptive security (also called full security), where security is guaranteed even for messages that are adaptively chosen at any point in time, seems significantly more challenging. The handful of known adaptively-secure schemes are based on specifically tailored techniques that rely on strong assumptions (such as obfuscation or multilinear maps assumptions).We show that any sufficiently-expressive selectively-secure FE scheme can be transformed into an adaptively-secure one without introducing any additional assumptions. We present a black-box transformation, for both public-key and private-key schemes, making novel use of hybrid encryption, a classical technique that was originally introduced for improving the efficiency of encryption schemes. We adapt the hybrid encryption approach to the setting of functional encryption via a technique for embedding a "hidden execution thread" in the decryption keys of the underlying scheme, which will only be activated within the proof of security of the resulting scheme. As an additional application of this technique, we show how to construct functional encryption schemes for arbitrary circuits starting from ones for shallow circuits (NC1 or even TC0).

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