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2020 | OriginalPaper | Buchkapitel

Adaptively Secure Constrained Pseudorandom Functions in the Standard Model

verfasst von : Alex Davidson, Shuichi Katsumata, Ryo Nishimaki, Shota Yamada, Takashi Yamakawa

Erschienen in: Advances in Cryptology – CRYPTO 2020

Verlag: Springer International Publishing

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Abstract

Constrained pseudorandom functions (CPRFs) allow learning “constrained” PRF keys that can evaluate the PRF on a subset of the input space, or based on some predicate. First introduced by Boneh and Waters [AC’13], Kiayias et al. [CCS’13] and Boyle et al. [PKC’14], they have shown to be a useful cryptographic primitive with many applications. These applications often require CPRFs to be adaptively secure, which allows the adversary to learn PRF values and constrained keys in an arbitrary order. However, there is no known construction of adaptively secure CPRFs based on a standard assumption in the standard model for any non-trivial class of predicates. Moreover, even if we rely on strong tools such as indistinguishability obfuscation (IO), the state-of-the-art construction of adaptively secure CPRFs in the standard model only supports the limited class of \(\mathbf{NC} ^1\) predicates.

In this work, we develop new adaptively secure CPRFs for various predicates from different types of assumptions in the standard model. Our results are summarized below.

We construct adaptively secure and O(1)-collusion-resistant CPRFs for t-conjunctive normal form (t-CNF) predicates from one-way functions (OWFs) where t is a constant. Here, O(1)-collusion-resistance means that we can allow the adversary to obtain a constant number of constrained keys. Note that t-CNF includes bit-fixing predicates as a special case.
We construct adaptively secure and single-key CPRFs for inner-product predicates from the learning with errors (LWE) assumption. Here, single-key security means that we only allow the adversary to learn one constrained key. Note that inner-product predicates include t-CNF predicates for a constant t as a special case. Thus, this construction supports more expressive class of predicates than that supported by the first construction though it loses the collusion-resistance and relies on a stronger assumption.
We construct adaptively secure and O(1)-collusion-resistant CPRFs for all circuits from the LWE assumption and indistinguishability obfuscation (IO).

The first and second constructions are the first CPRFs for any non-trivial predicates to achieve adaptive security outside of the random oracle model or relying on strong cryptographic assumptions. Moreover, the first construction is also the first to achieve any notion of collusion-resistance in this setting. Besides, we prove that the first and second constructions satisfy weak 1-key privacy, which roughly means that a constrained key does not reveal the corresponding constraint. The third construction is an improvement over previous adaptively secure CPRFs for less expressive predicates based on IO in the standard model.

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Vorheriges Kapitel New Constructions of Hinting PRGs, OWFs with Encryption, and More

Nächstes Kapitel Collusion Resistant Watermarkable PRFs from Standard Assumptions

Throughout the introduction, \(\mathsf {poly}\) will denote an arbitrary polynomial in the security parameter.

In general, we can upgrade selective security to adaptive security by complexity leveraging. However, we want to avoid this since complexity leveraging needs subexponentially hard assumptions.

Note that a lot of subtlety on parameter selections and technicalities regarding rounding are swept under the rug. However, we believe the rough details are enough to convey the intuition.

Recall that we assume that an adversary does not make an evaluation query and that the challenge query is made at the end of the game.

We can show that \(\mathsf {SHSF}.\mathsf {SEval}(\mathsf {sk}^{\mathsf {sim}},x)\) is uniformly distributed in \(\mathcal {Y}\) over the choice of \(\mathsf {sk}^{\mathsf {sim}}\) for any fixed x.

In the original definition of [39], there is an additional setup algorithm that generates a public parameter. We omit this algorithm since in general we can always include the public parameter in the secret key.

Note that the vector \(\mathbf {r}\) is sampled in the key generation, and not relevant to the vector \(\mathbf {v}\) that appeared above.

In the actual scheme, only \(\sigma _1\) will appear and \(\sigma _2,...,\sigma _{Q_1+1}\) are only used in the security proof.

Although there may be many ways to describe the circuit \(C_{\mathsf {eq}}[x^*, \mathbf {r}]\), we consider the most obvious and standard one.

In our scheme, a public parameter is just the security parameter. So we omit the setup algorithm \(\mathsf {CPRF.}\mathsf {Setup}\).

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Titel: Adaptively Secure Constrained Pseudorandom Functions in the Standard Model
verfasst von: Alex Davidson
Shuichi Katsumata
Ryo Nishimaki
Shota Yamada
Takashi Yamakawa
Verlag: Springer International Publishing
Buch: Advances in Cryptology – CRYPTO 2020
Print ISBN: 978-3-030-56783-5

Electronic ISBN: 978-3-030-56784-2

Copyright-Jahr: 2020
DOI: https://doi.org/10.1007/978-3-030-56784-2_19

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