Theory of Cryptography Conference

TCC 2016: Theory of Cryptography pp 269-296

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

  • Shashank Agrawal
  • Manoj Prabhakaran
  • Ching-Hua Yu
Conference paper

DOI: 10.1007/978-3-662-53644-5_11

Volume 9986 of the book series Lecture Notes in Computer Science (LNCS)
Cite this paper as:
Agrawal S., Prabhakaran M., Yu CH. (2016) Virtual Grey-Boxes Beyond Obfuscation: A Statistical Security Notion for Cryptographic Agents. In: Hirt M., Smith A. (eds) Theory of Cryptography. TCC 2016. Lecture Notes in Computer Science, vol 9986. Springer, Berlin, Heidelberg


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).

Copyright information

© International Association for Cryptologic Research 2016

Authors and Affiliations

  • Shashank Agrawal
    • 1
  • Manoj Prabhakaran
    • 2
  • Ching-Hua Yu
    • 2
  1. 1.University of Texas at AustinAustinUSA
  2. 2.University of Illinois at Urbana-ChampaignChampaignUSA