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Adaptively Single-Key Secure Constrained PRFs for \(\mathrm {NC}^1\)

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Public-Key Cryptography – PKC 2019 (PKC 2019)

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Abstract

We present a construction of an adaptively single-key secure constrained PRF (CPRF) for \(\mathbf {NC}^1\) assuming the existence of indistinguishability obfuscation (IO) and the subgroup hiding assumption over a (pairing-free) composite order group. This is the first construction of such a CPRF in the standard model without relying on a complexity leveraging argument.

To achieve this, we first introduce the notion of partitionable CPRF, which is a CPRF accommodated with partitioning techniques and combine it with shadow copy techniques often used in the dual system encryption methodology. We present a construction of partitionable CPRF for \(\mathbf {NC}^1\) based on IO and the subgroup hiding assumption over a (pairing-free) group. We finally prove that an adaptively single-key secure CPRF for \(\mathbf {NC}^1\) can be obtained from a partitionable CPRF for \(\mathbf {NC}^1\) and IO.

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Notes

  1. 1.

    It is also known as delegatable PRF [36] and functional PRF [12].

  2. 2.

    We note that the role of the constraining function f is “reversed” from the definition by Boneh and Waters [10], in the sense that the evaluation by a constrained key \(\mathsf {sk}_f\) is possible for inputs x with \(f(x) = 1\) in their definition, while it is possible for inputs x for \(f(x) = 0\) in our paper. Our treatment is the same as Brakerski and Vaikuntanathan [14].

  3. 3.

    A CPRF is called collusion-resistant if it remains secure even if adversaries are given polynomially many constrained keys.

  4. 4.

    In previous works, both selective-challenge and selective-constraint security are simply called selective security. We use different names for them for clarity.

  5. 5.

    More precisely, they also generalized their construction to obtain a CPRF for t-puncturing functions, which puncture the input space on t points for a polynomial t (rather than a single point).

  6. 6.

    Actually, we use an extended notion called a balanced admissible hash function (Sect. 2.2).

  7. 7.

    It assumes that holds, where , , , and are groups of order N, p, and q, respectively, g, \(g_1\), and \(g_2\) are generators of , , and , respectively, and .

  8. 8.

    Note that being given both and does not lead to a trivial attack since we use “pairing-free” groups.

  9. 9.

    We note that even if the underlying partitionable CPRF only supports \(\mathbf {NC}^1\), we can naturally define a constrained key for a function outside \(\mathbf {NC}^1\) in the CPRF given in Sect. 4 because a function class supported by the partitionable CPRF matters only in the security proof and does not matter for the correctness.

  10. 10.

    The L-DDH assumption was called Assumption 2 by Hohenberger et al. [33].

  11. 11.

    In this paper, a “class of functions” is a set of “sets of functions”. Each \(\mathcal {F}_{\lambda ,k}\) in \(\mathcal {F}\) considered for a CPRF is a set of functions parameterized by a security parameter \(\lambda \) and an input-length k.

  12. 12.

    For clarity, we will define a CPRF as a primitive that has a public parameter. However, this treatment is compatible with the standard syntax in which there is no public parameter, because it can always be contained as part of a master secret key and constrained secret keys.

  13. 13.

    Selective-constraint no-evaluation security was simply called no-evaluation security in [2].

  14. 14.

    Though it is possible to define the adaptive security for PCPRFs in the similar way, we only define the selective-constraint no-evaluation security since we only need it.

  15. 15.

    The construction will be partition-hiding with respect to h. Looking ahead, we will show that PCPRF that is partition-hiding with respect to a balanced AHF is adaptively single-key secure in Sect. 4. There, we will set h to be a balanced AHF. However, in this section, h can be any efficiently computable function.

  16. 16.

    This can be done by sampling in \(\mathbb {Z}_N\); if it is not in \(\mathbb {Z}_N^*\), sampling again until it is. This will succeed with an overwhelming probability since N is a composite with two large prime factors.

  17. 17.

    If one relies on the technique of “exponential number of hybrids” (e.g., [18]), then we can prove the indistinguishability of these two cases without relying on subgroup hiding. However, the technique requires sub-exponentially secure \(\mathsf {iO}\), which we want to avoid.

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Acknowledgments

We would like to thank Yilei Chen for the valuable discussion about adaptive security of the LWE-based constraint-hiding CPRFs. The first, second, and fourth authors were supported by JST CREST Grant Number JPMJCR1688, Japan.

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Correspondence to Takashi Yamakawa .

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Attrapadung, N., Matsuda, T., Nishimaki, R., Yamada, S., Yamakawa, T. (2019). Adaptively Single-Key Secure Constrained PRFs for \(\mathrm {NC}^1\). In: Lin, D., Sako, K. (eds) Public-Key Cryptography – PKC 2019. PKC 2019. Lecture Notes in Computer Science(), vol 11443. Springer, Cham. https://doi.org/10.1007/978-3-030-17259-6_8

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