# Virtual Grey-Boxes Beyond Obfuscation: A Statistical Security Notion for Cryptographic Agents

## Abstract

We extend the simulation-based definition of Virtual Grey Box (VGB) security – originally proposed for obfuscation (Bitansky and Canetti 2010) – to a broad class of cryptographic primitives. These include functional encryption, graded encoding schemes, bi-linear maps (with über assumptions), as well as unexplored ones like homomorphic functional encryption.

Our main result is a characterization of VGB security, in all these cases, in terms of an *indistinguishability-preserving* notion of security, called \(\Gamma ^*\)-\(\textit{s-}{\textsf {IND}}\text{- }\!{\textsf {PRE}} \) security, formulated using an extension of the recently proposed *Cryptographic Agents* framework (Agrawal et al. 2015). We further show that this definition is equivalent to an indistinguishability based security definition that is restricted to “concentrated” distributions (wherein the outcome of any computation on encrypted data is essentially known ahead of the computation).

A result of Bitansky et al. (2014), who showed that VGB obfuscation is equivalent to strong indistinguishability obfuscation (SIO), is obtained by specializing our result to obfuscation. Our proof, while sharing various elements from the proof of Bitansky et al., is simpler and significantly more general, as it uses \(\Gamma ^*\)-\(\textit{s-}{\textsf {IND}}\text{- }\!{\textsf {PRE}} \) security as an intermediate notion. Our characterization also shows that the semantic security for graded encoding schemes (Pass et al. 2014), is in fact an instance of this same definition.

We also present a composition theorem for \(\Gamma ^*\)-\(\textit{s-}{\textsf {IND}}\text{- }\!{\textsf {PRE}} \) security. We can then recover the result of Bitansky et al. (2014) regarding the existence of VGB obfuscation for all \({\textsf {NC}}^{1}\) circuits, simply by instantiating this composition theorem with a reduction from obfuscation of \({\textsf {NC}}^{1}\) circuits to graded encoding schemas (Barak et al. 2014) and the assumption that there exists an \(\Gamma ^*\)-\(\textit{s-}{\textsf {IND}}\text{- }\!{\textsf {PRE}} \) secure scheme for the graded encoding schema (Pass et al. 2014).

## Notes

### Acknowledgments

This work was supported in part by NSF grant 12-28856. Part of this work was carried out while the authors were visiting the Simons Institute for Theoretical Computer Science, supported by the Simons Foundation and by the DIMACS/Simons Collaboration in Cryptography through NSF grant CNS 15-23467.

Part of this work was done when the first author was at the University of Illinois at Urbana-Champaign. At University of Texas at Austin, he is supported by NSF CNS-1228599, CNS-1414082 and DARPA SafeWare.

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