An obfuscator O is Virtual Grey Box (VGB) for a class C of circuits if, for any C ∈ C and any predicate π, deducing π(C) given O(C) is tantamount to deducing π(C) given unbounded computational resources and polynomially many oracle queries to C. VGB obfuscation is often significantly more meaningful than indistinguishability obfuscation (IO). In fact, for some circuit families of interest VGB is equivalent to full-fledged Virtual Black Box obfuscation.We investigate the feasibility of obtaining VGB obfuscation for general circuits. We first formulate a natural strengthening of IO, called strong IO (SIO). Essentially, O is SIO for class C if O(C) ≈ O(C ) whenever the pair (C, C ) is taken from a distribution over C where, for all x, C(x) = C (x) only with negligible probability.We then show that an obfuscator is VGB for a class C if and only if it is SIO for C. This result is unconditional and holds for any C. We also show that, for some circuit collections, SIO implies virtual black-box obfuscation.Finally, we formulate a slightly stronger variant of the semantic security property of graded encoding schemes [Pass-Seth-Telang Crypto 14], and show that existing obfuscators, such as the obfuscator of Barak et al. [Eurocrypt 14], are SIO for all circuits in NC 1 , assuming that the underlying graded encoding scheme satisfies our variant of semantic security.Put together, we obtain VGB obfuscation for all NC 1 circuits under assumptions that are almost the same as those used by Pass et al. to obtain IO for NC 1 circuits. We also show that semantic security is in essence necessary for showing VGB obfuscation. * A preliminary version of this work appears in the proceedings of CRYPTO 2014.

We construct a 1-round delegation scheme (i.e., argument-system) for every language computable in time t = t ( n ), where the running time of the prover is poly ( t ) and the running time of the verifier is n · polylog ( t ). In particular, for every language in P we obtain a delegation scheme with almost linear time verification. Our construction relies on the existence of a computational sub-exponentially secure private information retrieval ( PIR ) scheme. The proof exploits a curious connection between the problem of computation delegation and the model of multi-prover interactive proofs that are sound against no-signaling (cheating) strategies , a model that was studied in the context of multi-prover interactive proofs with provers that share quantum entanglement, and is motivated by the physical principle that information cannot travel faster than light. For any language computable in time t = t ( n ), we construct a multi-prover interactive proof ( MIP ), that is, sound against no-signaling strategies, where the running time of the provers is poly ( t ), the number of provers is polylog ( t ), and the running time of the verifier is n · polylog ( t ). In particular, this shows that the class of languages that have polynomial-time MIP s that are sound against no-signaling strategies, is exactly EXP . Previously, this class was only known to contain PSPACE . To convert our MIP into a 1-round delegation scheme, we use the method suggested by Aiello et al. (ICALP, 2000), which makes use of a PIR scheme. This method lacked a proof of security. We prove that this method is secure assuming the underlying MIP is secure against no-signaling provers.