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How to Use (Plain) Witness Encryption: Registered ABE, Flexible Broadcast, and More

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Advances in Cryptology – CRYPTO 2023 (CRYPTO 2023)

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Abstract

Witness encryption is a generalization of public-key encryption where the public key can be any \(\textsf{NP}\) statement x and the associated decryption key is any witness w for x. While early constructions of witness encryption relied on multilinear maps and indistinguishability obfuscation (\(i\mathcal {O}\)), recent works have provided direct constructions of witness encryption that are more efficient than \(i\mathcal {O}\) (and also seem unlikely to yield \(i\mathcal {O}\)). Motivated by this progress, we revisit the possibility of using witness encryption to realize advanced cryptographic primitives previously known only in “obfustopia.”

In this work, we give new constructions of trustless encryption systems from plain witness encryption (in conjunction with the learning-with-errors assumption): (1) flexible broadcast encryption (a broadcast encryption scheme where users choose their own secret keys and users can encrypt to an arbitrary set of public keys); and (2) registered attribute-based encryption (a system where users choose their own keys and then register their public key together with a set of attributes with a deterministic and transparent key curator). Both primitives were previously only known from \(i\mathcal {O}\). We also show how to use our techniques to obtain an optimal broadcast encryption scheme in the random oracle model.

Underlying our constructions is a novel technique for using witness encryption based on a new primitive which we call function-binding hash functions. Whereas a somewhere statistically binding hash function statistically binds a digest to a few bits of the input, a function-binding hash function statistically binds a digest to the output of a function of the inputs. As we demonstrate in this work, function-binding hash functions provide us new ways to leverage the power of plain witness encryption and use it as the foundation of advanced cryptographic primitives. Finally, we show how to build function-binding hash functions for the class of disjunctions of block functions from leveled homomorphic encryption; this in combination with witness encryption yields our main results.

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Notes

  1. 1.

    Note that we can generically obtain a flexible broadcast encryption scheme from a distributed broadcast encryption scheme by having users sample their index randomly. To support a maximum of \(\ell \) users, we would instantiate the scheme with \(n = \varOmega (\ell ^2)\) indices. This is sufficient if the number of users is a priori bounded, though it may incur a quadratic blowup in the size of some scheme parameters. If the underlying distributed broadcast encryption scheme supports a super-polynomial number of users (i.e., \(n = \lambda ^{\omega (1)}\)), then it directly implies a flexible broadcast encryption for an arbitrary polynomial number of users.

  2. 2.

    Note that set of public keys could just be a set of indices if we include a mapping between indices and user public keys (i.e., a public-key directory) as part of the public parameters of the flexible broadcast encryption scheme. This is the setting considered in [13]. More generally, the set of users could admit a succinct description (e.g., “all computer science students”) and the decrypter would look up the public keys associated with the members of the set.

  3. 3.

    Specifically, if \(c_i\) is the public key of an honest user, then \(\textsf{PKE}.\textsf{Dec} (\textsf{sk}, c_i) = 0\) whereas if \(x_i\) belongs to a corrupted user, then \(\mathcal {P}(x_i) = 0\) by the admissibility restriction on the registered ABE adversary. In either case, \(g(c_i, x_i) = 0\) for all \(i \in [n]\).

  4. 4.

    The transformed scheme is still policy-selective; however, this can be removed generically via complexity leveraging and assuming subexponential hardness. Note that one cannot use complexity leveraging to handle corruption queries without having the ciphertext size grow with the number of registered users.

  5. 5.

    Note that the direct approach of encoding the broadcast set as part of the policy in the ABE scheme does not yield a flexible broadcast encryption scheme with the required efficiency. Namely, in existing registered ABE schemes (including the one in this work), the size of the ciphertext scales with the size of the policy. Using this to implement broadcast encryption yields a flexible broadcast encryption scheme where the size of the ciphertext scales linearly with the number of users.

  6. 6.

    Here, we assume an ordered list of public keys for simplicity. However, we could have alternatively encrypted to an (unordered) set of public keys by first ordering the public keys in lexicographic order.

  7. 7.

    The definition from [46] also includes an algorithm \(\textsf{IsValid}(\textsf{pp}, i, \textsf{pk}_i) \rightarrow \{0,1\} \) that checks if a given public key is valid. Our construction does not require this check. However, to match their syntax, we could define it to simply output 1 on any public key of the correct length with respect to the public parameters \(\textsf{pp}\).

  8. 8.

    Note that because \(\textsf{Aggregate}\) is deterministic and can be run by A itself, there is no need to additionally provide \((\textsf{mpk}, \textsf{hsk}_1, \ldots , \textsf{hsk}_L)\) to A. Similarly, there is no advantage to allowing the adversary to select the challenge policy and messages after seeing the aggregated key.

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Acknowledgments

We thank Dan Boneh and Hoeteck Wee for helpful pointers on broadcast encryption. Cody Freitag’s work was done while at Cornell Tech, and he is supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-2139899, DARPA Award HR00110C0086, AFOSR Award FA9550-18-1-0267, NSF CNS-2128519, and DARPA under Agreement No. HR00112020023. Brent Waters is supported by NSF CNS-1908611, a Simons Investigator award, and the Packard Foundation Fellowship. David J. Wu is supported by NSF CNS-2151131, CNS-2140975, a Microsoft Research Faculty Fellowship, and a Google Research Scholar award.

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  1. Boston University, Boston, USA

    Cody Freitag

  2. UT Austin, Austin, USA

    Brent Waters & David J. Wu

  3. NTT Research, Sunnyvale, USA

    Brent Waters

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    Helena Handschuh

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    Anna Lysyanskaya

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Freitag, C., Waters, B., Wu, D.J. (2023). How to Use (Plain) Witness Encryption: Registered ABE, Flexible Broadcast, and More. In: Handschuh, H., Lysyanskaya, A. (eds) Advances in Cryptology – CRYPTO 2023. CRYPTO 2023. Lecture Notes in Computer Science, vol 14084. Springer, Cham. https://doi.org/10.1007/978-3-031-38551-3_16

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