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Multi-key and Multi-input Predicate Encryption from Learning with Errors

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Advances in Cryptology – EUROCRYPT 2023 (EUROCRYPT 2023)

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

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Abstract

We put forward two natural generalizations of predicate encryption (PE), dubbed multi-key and multi-input PE. More in details, our contributions are threefold.

  • Definitions. We formalize security of multi-key PE and multi-input PE following the standard indistinguishability paradigm, and modeling security both against malicious senders (i.e., corruption of encryption keys) and malicious receivers (i.e., collusions).

  • Constructions. We construct adaptively secure multi-key and multi-input PE supporting the conjunction of poly-many arbitrary single-input predicates, assuming the sub-exponential hardness of the learning with errors (LWE) problem.

  • Applications. We show that multi-key and multi-input PE for expressive enough predicates suffices for interesting cryptographic applications, including non-interactive multi-party computation (NI-MPC) and matchmaking encryption (ME).

In particular, plugging in our constructions of multi-key and multi-input PE, under the sub-exponential LWE assumption, we obtain the first ME supporting arbitrary policies with unbounded collusions, as well as robust (resp. non-robust) NI-MPC for so-called all-or-nothing functions satisfying a non-trivial notion of reusability and supporting a constant (resp. polynomial) number of parties. Prior to our work, both of these applications required much heavier tools such as indistinguishability obfuscation or compact functional encryption.

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Notes

  1. 1.

    Sometimes, we also refer to x as the predicate input. Throughout the paper, we use the terms attribute and input interchangeably.

  2. 2.

    This is one of the differences between multi-key PE and multi-input PE: the former has a public-key encryption algorithm, whereas the latter could have a secret-key encryption algorithm.

  3. 3.

    Note that, in the setting with no corruptions, assuming the presence of a (single) wildcard \(x^\star _i\) for each \(\mathbb {P}_i\) does not affect the expressiveness and the security guarantees of multi-input PE. This is because the i-th sender can simply choose not to encrypt \(x^\star _i\), which will not permit the receiver to evaluate \(\mathbb {P}_i\) over \(x^\star _i\).

  4. 4.

    Note that, in case of no corruptions, our CPA-1-sided construction supports \(n=\textsf{poly}(\lambda )\). However, to achieve CPA-2-sided security we use complexity leveraging and this reduces n from \(\textsf{poly}(\lambda )\) to \(O(\log (\lambda ))\).

  5. 5.

    Observe that the decryption keys can be interleaved. For example, starting from \((\textsf{dk}_{v_1},\ldots ,\textsf{dk}_{v_i},\ldots \textsf{dk}_{v_{n}})\) representing the predicate \(\mathbb {P}_{v_1,\ldots ,v_{i},\ldots ,v_{n}}\), the adversary can ask for an additional i-th decryption key \(\textsf{dk}_{v'_i}\) and rearrange the decryption keys as \((\textsf{dk}_{v_1},\ldots ,\textsf{dk}_{v'_i},\ldots \textsf{dk}_{v_{n}})\) in order to obtain the tuple representing a different predicate \(\mathbb {P}_{v_1,\ldots , v'_i,\ldots , v_{n}} \ne \mathbb {P}_{v_1,\ldots , v_i,\ldots , v_{n}}\).

  6. 6.

    By leveraging hybrid encryption, we can transform any PE into one with \(\textsf{poly}(\lambda )+ |m|\) ciphertext expansion, i.e., \(\textsf{Enc}'(\textsf{mpk},x,m) = \textsf{Enc}(\textsf{mpk},x,s) || \textsf{PRG}(s) \oplus m\) where .

  7. 7.

    When we write CPA secure PE, without specifying 1-sided or 2-sided security, we refer to a PE scheme that guarantees only the secrecy of the message. CPA secure PE is the same as CPA secure ABE.

  8. 8.

    Indeed, as we discuss in Remark 1, CPA-1-sided (resp. CPA-2-sided) secure multi-input PE for arbitrary predicates implies CPA-1-sided (resp. CPA-2-sided) secure multi-key PE.

  9. 9.

    If this condition is not satisfied, the adversary has obtained through the encryption oracles a set of ciphertexts that can be interleaved with one (or more) parts of the challenge ciphertext in order to satisfy the predicate \(\mathbb {P}^*\).

  10. 10.

    As we discuss in the full version [25], our construction remains secure if we consider a weaker form of collusion in which the adversary can only obtain multiple decryption keys for predicates \(\mathbb {P}\) such that there is a unique j for all predicates (submitted to \(\textsf{KGen}\)) that satisfies the validity condition (ii).

  11. 11.

    The secret-key construction achieves this by linking multiple PE ciphertexts via SKE, and including the secret key \(\textsf{k}_{i+1}\) into the PE ciphertext.

  12. 12.

    Recall that wildcards must be efficiently computable.

  13. 13.

    Note that security of NI-MPC for general functions is formalized by an indistinguishability-based definition [14, 32]. This is because simulation-based NI-MPC implies virtual black-box (VBB) obfuscation that is known to be impossible for certain classes of functions [13].

  14. 14.

    Reusable NI-MPC remains secure even when the same setup is used over multiple rounds. On the other hand, non-reusable NI-MPC does not permit to reuse the same setup, i.e., after each round the setup algorithm needs to be executed.

  15. 15.

    Observe that \(\mathcal {Q}^0_i\) and \(\mathcal {Q}^1_i\) are identical except for the last element.

  16. 16.

    If we restrict the n-key PE’s encryption algorithm to be secret-key (i.e., \(\textsf{Enc}(\textsf{ek},\cdot ,\cdot )\) where \(\textsf{ek}\) is kept secret) then we can start from a secret-key \((n+1)\)-input PE, i.e., 0 corruptions.

  17. 17.

    This is also reflected by the results achieved in this paper. For example, our multi-key PE construction for conjunctions of arbitrary predicates tolerates unbounded collusions whereas our multi-input PE constructions (for the same class of predicates with wildcards) are significantly more complex and are secure only in the case of no collusions.

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Acknowledgements

The authors would like to thank the anonymous reviewers for useful feedback. The first author was supported by the Carlsberg Foundation under the Semper Ardens Research Project CF18-112 (BCM); the second and the fourth author were partially supported by project SERICS (PE00000014) under the NRRP MUR program funded by the EU - NGEU and by Sapienza University under the project SPECTRA; the third author was partially supported by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038 and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - EXC 2092 CASA - 390781972.

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  1. Aarhus University, Aarhus, Denmark

    Danilo Francati

  2. Sapienza University of Rome, Rome, Italy

    Daniele Friolo & Daniele Venturi

  3. Max Planck Institute for Security and Privacy, Bochum, Germany

    Giulio Malavolta

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Francati, D., Friolo, D., Malavolta, G., Venturi, D. (2023). Multi-key and Multi-input Predicate Encryption from Learning with Errors. 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_19

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