**On Virtual Grey Box Obfuscation for General Circuits**

*Nir Bitansky and Ran Canetti and Yael Tauman-Kalai and Omer Paneth*

**Abstract: **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.

**Category / Keywords: **Obfuscation, Multilinear Graded Encodings, Semantic Security, VBB, VGB

**Original Publication**** (with major differences): **IACR-CRYPTO-2014

**Date: **received 15 Jul 2014, last revised 5 Aug 2014

**Contact author: **nirbitan at tau ac il

**Available format(s): **PDF | BibTeX Citation

**Note: **Noted that a preliminary version appears in the proceedings of crypto.

**Version: **20140805:181558 (All versions of this report)

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