On Virtual Grey Box Obfuscation for General Circuits - Bitansky, Nir; Canetti, Ran; Kalai, Yael Tauman; Paneth, Omer (Algorithmica, 2017)
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 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 N C 1 circuits under assumptions that are almost the same as those used by Pass et al. to obtain IO for N C 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 .