On Virtual Grey Box Obfuscation for General Circuits

n 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_0)≈𝒪(C_1) whenever the pair (C_0,C_1) is taken from a distribution over 𝒞 where, for all x , C_0(x) C_1(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 observe that VGB obfuscation for all polynomial-size circuits implies the existence of semantically-secure graded encoding schemes with limited functionality known as jigsaw puzzles .