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Privately Puncturing PRFs from Lattices: Adaptive Security and Collusion Resistant Pseudorandomness

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Advances in Cryptology – EUROCRYPT 2023 (EUROCRYPT 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14006))

Abstract

A private puncturable pseudorandom function (PRF) enables one to create a constrained version of a PRF key, which can be used to evaluate the PRF at all but some punctured points. In addition, the constrained key reveals no information about the punctured points and the PRF values on them. Existing constructions of private puncturable PRFs are only proven to be secure against a restricted adversary that must commit to the punctured points before viewing any information. It is an open problem to achieve the more natural adaptive security, where the adversary can make all its choices on-the-fly.

In this work, we solve the problem by constructing an adaptively secure private puncturable PRF from standard lattice assumptions. To achieve this goal, we present a new primitive called explainable hash, which allows one to reprogram the hash function on a given input. The new primitive may find further applications in constructing more cryptographic schemes with adaptive security. Besides, our construction has collusion resistant pseudorandomness, which requires that even given multiple constrained keys, no one could learn the values of the PRF at the punctured points. Private puncturable PRFs with collusion resistant pseudorandomness were only known from multilinear maps or indistinguishability obfuscations in previous works, and we provide the first solution from standard lattice assumptions.

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Notes

  1. 1.

    For example, if we use an adaptively secure private puncturable PRF in the construction of restricted searchable encryption given in [BLW17], the scheme will additionally achieve adaptive security, which allows the database owner to issue restricted search keys on restrictions determined after the system has been put in use.

  2. 2.

    A 1-puncturable PRF punctures each PRF key on only one input.

  3. 3.

    The general construction also works for larger puncture sets if we use a stronger building block in the construction. Looking ahead, this needs an explainable hash that can reprogram the outputs on multiple inputs simultaneously, which is much more difficult to construct (compared to the standard explainable hash constructed in this work).

  4. 4.

    In the formal definition of explainable hash, the simulator may fail and abort with a non-negligible probability. In this overview, we assume that the simulator always succeeds for simplicity.

  5. 5.

    Here, the adversary cannot view the hash key before submitting \(x^*\), and this allows the simulator to choose a suitable hash key after receiving \(x^*\).

  6. 6.

    This also relies on the fact that \(\boldsymbol{s}^{\intercal } \cdot \boldsymbol{A}_x \cdot \boldsymbol{G}^{-1}(\boldsymbol{v}_1)\) is not close to the borders (i.e., 0 and \(\frac{q}{2}\)), which can be guaranteed by adding an additional random element to it.

  7. 7.

    Note that if \(x\not \in \mathcal {P}_1 \cap \mathcal {P}_2\), i.e., \(x\not \in \mathcal {P}_1\) or \(x\not \in \mathcal {P}_2\), then the PRF value at x can be trivially learned from one of the constrained keys.

  8. 8.

    This also relies on the fact that \((\boldsymbol{s}^{\intercal } \cdot \boldsymbol{A}_x)[1]\) is not close to the “rounding border”, which can be ensured either by the 1D-SIS assumption [Reg04, BV15, BKM17] or via adding an additional random element to it. In this work, we use the latter method.

  9. 9.

    A similar idea is also employed in [BKM17] to achieve \(\tau \)-puncture PRF from 1-puncture PRF. However, as discussed below, we cannot achieve collusion resistance merely from this approach.

  10. 10.

    As a byproduct, this also leads to puncturable PRFs for puncture sets of unbounded sizes.

  11. 11.

    The key-homomorphism property requires that \(\textsf{PRF}_0(\boldsymbol{t}_{1},x) + \textsf{PRF}_0(\boldsymbol{t}_{2},x)=\textsf{PRF}_0(\boldsymbol{t}_{1}+\boldsymbol{t}_{2},x)\). Actually, due to the rounding operation, \(\textsf{PRF}_0\) is only “almost key-homomorphic”, i.e., there may exist a small difference between \(\textsf{PRF}_0(\boldsymbol{t}_{1},x) + \textsf{PRF}_0(\boldsymbol{t}_{2},x)\) and \(\textsf{PRF}_0(\boldsymbol{t}_{1}+\boldsymbol{t}_{2},x)\). We close the gap by summing the variables before rounding and then rounding the result to \(\mathbb {Z}_p\).

  12. 12.

    Recall that both \(\boldsymbol{s}\) and \(\boldsymbol{t}_{x}\) are PRF keys of \(\textsf{PRF}_0\).

  13. 13.

    We allow \(x_i=x_j\) for some distinct \(i,j\in [1,Q]\).

  14. 14.

    We implicitly assume that a set \(\mathcal {P}\) is described by listing all elements in \(\mathcal {P}\), thus, the puncture set is always of polynomial-size in this paper.

  15. 15.

    In this setting, \(\mathcal {A}_1\) can still make queries to the evaluation oracle, and \(\mathcal {A}_2\) can still query the evaluation oracle and the constrain oracle adaptively for a priori unbounded number of times.

  16. 16.

    Note that there are \(2^{n+\lambda }\) possible subsets of \(\mathcal {U}\).

  17. 17.

    We can use \((n+1)\)-bit strings to represent all strings with length not larger than n.

  18. 18.

    Since the sizes of the puncture sets are a priori bounded, the restriction described by Eq. (6) is not needed.

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We appreciate the anonymous reviewers for their valuable comments.

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  1. Institute of Cybersecurity and Cryptology, School of Computing and Information Technology, University of Wollongong, Wollongong, NSW, Australia

    Rupeng Yang

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Yang, R. (2023). Privately Puncturing PRFs from Lattices: Adaptive Security and Collusion Resistant Pseudorandomness. In: Hazay, C., Stam, M. (eds) Advances in Cryptology – EUROCRYPT 2023. EUROCRYPT 2023. Lecture Notes in Computer Science, vol 14006. Springer, Cham. https://doi.org/10.1007/978-3-031-30620-4_6

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