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Unbounded Quadratic Functional Encryption and More from Pairings

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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

We propose the first unbounded functional encryption (FE) scheme for quadratic functions and its extension, in which the sizes of messages to be encrypted are not a priori bounded. Prior to our work, all FE schemes for quadratic functions are bounded, meaning that the message length is fixed at the setup. In the first scheme, encryption takes \(\{x_{i}\}_{i \in S_{c}}\), key generation takes \(\{c_{i,j}\}_{i,j \in S_{k}}\), and decryption outputs \(\sum _{i,j \in S_{k}} c_{i,j}x_{i}x_{j}\) if and only if \(S_{k} \subseteq S_{c}\), where the sizes of \(S_{c}\) and \(S_{k}\) can be arbitrary. Our second scheme is the extension of the first scheme to partially-hiding FE that computes an arithmetic branching program on a public input and a quadratic function on a private input. Concretely, encryption takes a public input \(\textbf{u}\) in addition to \(\{x_{i}\}_{i \in S_{c}}\), a secret key is associated with arithmetic branching programs \(\{f_{i,j}\}_{i,j \in S_{k}}\), and decryption yields \(\sum _{i,j \in S_{k}} f_{i,j}(\textbf{u})x_{i}x_{j}\) if and only if \(S_{k} \subseteq S_{c}\). Both our schemes are based on pairings and secure in the simulation-based model under the standard MDDH assumption.

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Notes

  1. 1.

    Concretely, p is an order of bilinear groups that the scheme based on.

  2. 2.

    Note that ABPs are a stronger computational model than NC1 circuits.

  3. 3.

    This does not mean that our results imply the listed schemes since we ignore the security requirement here and focus on only functionalities.

  4. 4.

    We require only indistinguishability-based security for unbounded slotted IPFE to prove simulation-based security of unbounded quadratic FE schemes. Note that the slotted property with indistinguishability-based security basically implies simulation-based security, and thus our approach essentially follows previous quadratic FE schemes with simulation-based security [19, 21, 33].

  5. 5.

    The second property is required for our unbounded quadratic FE from MDDH\(_{k}\) for \(k >1\) and FE for \(\text {ABP}\circ \text {UQF}\).

  6. 6.

    It is not hard to see that the security of the partially garbling scheme implies that \( \langle \textbf{d}_{f,\textbf{u}}, \textsf{pgb}^{*}(f, \textbf{u}, \alpha ; \textbf{t}) \rangle =\alpha \) for all \(\alpha \in \mathbb {Z}_p\).

  7. 7.

    We consider only selective (or semi-adaptive more precisely) security in this paper.

  8. 8.

    This condition implies selective security (or semi-adaptive security more precisely).

  9. 9.

    In general, this condition is necessary since the adversary can publicly encrypt (x, e) for all \(x \in \mathcal {X}_{\textsf{pub}} \times \mathcal {X}_{\textsf{priv1}}\) and decrypt the ciphertexts with its own secret keys. In this paper, however, we handle only function classes where this condition is always satisfied as long as the public parts of \(f^{\ell ,0}\) and \(f^{\ell ,1}\) are the same. Thus, we can ignore this condition in this paper.

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Authors and Affiliations

  1. NTT Social Informatics Laboratories, Tokyo, Japan

    Junichi Tomida

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  1. Junichi Tomida
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Correspondence to Junichi Tomida .

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  1. Bar-Ilan University, Ramat Gan, Israel

    Carmit Hazay

  2. Simula UiB, Bergen, Norway

    Martijn Stam

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Tomida, J. (2023). Unbounded Quadratic Functional Encryption and More from Pairings. 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_18

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  • DOI: https://doi.org/10.1007/978-3-031-30620-4_18

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