Abstract
We construct the first multi-input functional encryption (MIFE) scheme for quadratic functions from pairings. Our construction supports polynomial number of users, where user i, for \(i \in [n]\), encrypts input \(\mathbf{x}_i \in \mathbb {Z}^m\) to obtain ciphertext \(\mathsf {CT}_i\), the key generator provides a key \(\mathsf {SK}_\mathbf{c}\) for vector \(\mathbf{c} \in \mathbb {Z}^{({mn})^2}\) and decryption, given \(\mathsf {CT}_1,\ldots ,\mathsf {CT}_n\) and \(\mathsf {SK}_\mathbf{c}\), recovers \(\langle \mathbf{c}, \mathbf{x} \otimes \mathbf{x} \rangle \) and nothing else. We achieve indistinguishability-based (selective) security against unbounded collusions under the standard bilateral matrix Diffie-Hellman assumption. All previous MIFE schemes either support only inner products (linear functions) or rely on strong cryptographic assumptions such as indistinguishability obfuscation or multi-linear maps.
S. Agrawal—Research supported by the DST “Swarnajayanti” fellowship, an Indo-French CEFIPRA project and the CCD Centre of Excellence. Part of the research corresponding to this work was conducted while visiting the Simons Institute for the Theory of Computing.
R. Goyal—Research supported in part by NSF CNS Award #1718161, an IBM-MIT grant, and by the Defense Advanced Research Projects Agency (DARPA) under Contract No. HR00112020023. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the United States Government or DARPA. Work done in part while at the Simons Institute for the Theory of Computing, supported by Simons-Berkeley research fellowship.
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Notes
- 1.
Note that FE for quadratic functions are trivially constructible from FE for inner products (IPFE) by linearizing and encrypting all quadratic monomials. However, FE for quadratic functions requires that the ciphertext size be linear in input length.
- 2.
In an exciting recent work, iO has been constructed from sub-exponential hardness of four well-founded assumptions [24]. However, this construction relies on a series of intricate, lossy reductions and is primarily a feasibility result. We will focus on the polynomial hardness of a well-founded problem in this work.
- 3.
Recall that public-key MIFE does not imply secret-key MIFE. Roughly speaking, a user who has \(\mathsf {CT}_{1}\) for \(x_{1}\) and \(\mathsf {SK}\) for f of a public-key scheme is allowed to learn \(f(x_{1},x_{2} , \ldots ,x_{n})\) for all \((x_{2} , \ldots ,x_{n})\), since this is inherent leakage, while it is not the case in secret-key MIFE.
- 4.
In more detail, this follows since the scheme remains correct and secure even if input vectors for \(\mathsf {Enc}\) and \(\mathsf {KeyGen}\) consist of group elements, and \(\mathsf {Dec}\) first obtains decryption values on the exponent of a target-group generator and then computes its discrete log.
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Agrawal, S., Goyal, R., Tomida, J. (2021). Multi-input Quadratic Functional Encryption from Pairings. In: Malkin, T., Peikert, C. (eds) Advances in Cryptology – CRYPTO 2021. CRYPTO 2021. Lecture Notes in Computer Science(), vol 12828. Springer, Cham. https://doi.org/10.1007/978-3-030-84259-8_8
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