International Association
for Cryptologic Research

An obfuscator $\\O$ is Virtual Grey Box (VGB) for a class $\\C$ of circuits if, for any $C\\in\\C$ and any predicate $\\pi$, deducing $\\pi(C)$ given $\\O(C)$ is tantamount to deducing $\\pi(C)$ 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 {\\em strong IO} (SIO). Essentially, $\\O$ is SIO for class $\\C$ if $\\O(C)\\approx\\O(C\')$ whenever the pair $(C,C\')$ is taken from a distribution over $\\C$ where, for all $x$, $C(x)\\neq C\'(x)$ 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.

{\\em 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 {\\em necessary} for showing VGB obfuscation.