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Public-Key Function-Private Hidden Vector Encryption (and More)

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Advances in Cryptology – ASIACRYPT 2019 (ASIACRYPT 2019)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11923))

Abstract

We construct public-key function-private predicate encryption for the “small superset functionality,” recently introduced by Beullens and Wee (PKC 2019). This functionality captures several important classes of predicates:

  • Point functions. For point function predicates, our construction is equivalent to public-key function-private anonymous identity-based encryption.

  • Conjunctions. If the predicate computes a conjunction, our construction is a public-key function-private hidden vector encryption scheme. This addresses an open problem posed by Boneh, Raghunathan, and Segev (ASIACRYPT 2013).

  • d-CNFs and read-once conjunctions of d-disjunctions for constant-size d.

Our construction extends the group-based obfuscation schemes of Bishop et al. (CRYPTO 2018), Beullens and Wee (PKC 2019), and Bartusek et al. (EUROCRYPT 2019) to the setting of public-key function-private predicate encryption. We achieve an average-case notion of function privacy, which guarantees that a decryption key \(\mathsf {sk} _f\) reveals nothing about f as long as f is drawn from a distribution with sufficient entropy. We formalize this security notion as a generalization of the (enhanced) real-or-random function privacy definition of Boneh, Raghunathan, and Segev (CRYPTO 2013). Our construction relies on bilinear groups, and we prove security in the generic bilinear group model.

Research conducted at Princeton University.

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Notes

  1. 1.

    Note that we would require a public-key predicate encryption scheme for this scenario, with the assumption that an email client would encrypt any email to the user under the user’s public key.

  2. 2.

    Function privacy had been studied before the work of BRS [BRS13a], albeit in the private key setting [SWP00, OS07, BSW09, SSW09].

  3. 3.

    We remark that [BKM+18, BLMZ19] framed their results as obfuscation for conjunctions. Beullens and Wee [BW19] were the first to notice that these techniques are in fact obfuscating a more general class of “big subset” predicates, which in particular encompass conjunctions.

  4. 4.

    While our definitions are new, we are not the first to observe the connection between program obfuscation and function-privacy. See also [AAB+15, ITZ16, ABF16].

  5. 5.

    We note that we are not the first to give a public-key function-private definition that is agnostic to the predicate class. In particular, this is also achieved by the definition of [ITZ16]. However, their definition does not extend to enhanced function privacy, and furthermore they do not give any constructions achieving their definition except under a strengthening of indistinguishability obfuscation due to [BCKP14].

  6. 6.

    The “big subset” function of Beullens and Wee [BW19] is also parameterized by the same n, t, X, but it outputs 1 if and only if \(Y \subseteq X\) and \(|Y| \ge t\). The functionalities are seen to be equivalent by associating each input set Y with its complement \([n] \setminus Y\).

  7. 7.

    In more detail, an attacker trying to distinguish between an encryption of \(y_0\) and an encryption of \(y_1\) (for \(y_0,y_1\) of their choice) is free to request decryption keys corresponding to any function \(I_x\) provided that \(I_x\) does not trivially allow the attacker to distinguish between \(y_0\) and \(y_1\). The attacker can therefore request \(g_1^{rx}\) for any x that does not equal \(y_0\) or \(y_1\). Given challenge \(g_2^{\alpha r^{-1}y_b^{-1}},g_T^{\alpha }\) and decryption key \(g_1^{rx}\), the attacker can use the fact that they know \(x,y_0,y_1\) in the clear to determine b as follows. The attacker raise \(g_T^{\alpha }\) to the exponent \(xy_0^{-1}\) to obtain \(g_T^{\alpha x y_0^{-1}}\), and then computes \(e(g_1^{rx},g_2^{\alpha r^{-1}y_b^{-1}})\). If \(b = 0\), these quantities match, and otherwise they do not.

  8. 8.

    We use the shorthand \(g^{\mathbf {V}}\) where \(\mathbf {V} = (v_{i,j})_{i\in [k],j\in [\ell ]}\) to denote the matrix of group elements \((g^{v_{i,j}})_{i\in [k],j\in [\ell ]}\).

  9. 9.

    Indeed, m must be \(\omega (\log k)\) in order to make the function family evasive.

  10. 10.

    In [BW07], it is noted that this restriction on the size of the message space can be avoided in practice by essentially setting the payload to be the key of a symmetric key encryption scheme, and releasing an encryption of the actual message under this key (along with a consistency check). This technique can easily be applied in our setting.

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Acknowledgements

This research was supported in part by ARO and DARPA Safeware under contracts W911NF-15-C-0227, W911NF-15-C-0236, W911NF-16-1-0389, W911NF-15-C-0213, and by NSF grants CNS-1633282, 1562888, 1565208, and 1814919. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the ARO and DARPA.

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Authors and Affiliations

  1. UC Berkeley, Berkeley, USA

    James Bartusek

  2. Galois, Portland, USA

    Brent Carmer & Alex J. Malozemoff

  3. Johns Hopkins University, Baltimore, USA

    Abhishek Jain & Zhengzhong Jin

  4. Google, Mountain View, USA

    Tancrède Lepoint & Mariana Raykova

  5. Princeton University, Princeton, USA

    Fermi Ma

  6. Columbia University, New York, USA

    Tal Malkin

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  1. University of Auckland, Auckland, New Zealand

    Steven D. Galbraith

  2. Security Fundamentals Lab, NICT, Tokyo, Japan

    Shiho Moriai

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Bartusek, J. et al. (2019). Public-Key Function-Private Hidden Vector Encryption (and More). In: Galbraith, S., Moriai, S. (eds) Advances in Cryptology – ASIACRYPT 2019. ASIACRYPT 2019. Lecture Notes in Computer Science(), vol 11923. Springer, Cham. https://doi.org/10.1007/978-3-030-34618-8_17

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