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2017 | OriginalPaper | Buchkapitel

Functional Encryption for Bounded Collusions, Revisited

verfasst von : Shweta Agrawal, Alon Rosen

Erschienen in: Theory of Cryptography

Verlag: Springer International Publishing

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Abstract

We provide a new construction of functional encryption (FE) for circuits in the bounded collusion model. In this model, security of the scheme is guaranteed as long as the number of colluding adversaries can be a-priori bounded by some polynomial Q. Our construction supports arithmetic circuits in contrast to all prior work which support Boolean circuits. The ciphertext of our scheme is sublinear in the circuit size for the circuit class \(\mathsf{NC}_1\); this implies the first construction of arithmetic reusable garbled circuits for \(\mathsf{NC}_1\).

Additionally, our construction achieves several desirable features:

Our construction for reusable garbled circuits for \(\mathsf{NC}_1\) achieves the optimal “full” simulation based security.
When generalised to handle Q queries for any fixed polynomial Q, our ciphertext size grows additively with \(Q^2\). In contrast, previous works that achieve full security [5, 39] suffered a multiplicative growth of \(Q^4\).
The ciphertext of our scheme can be divided into a succinct data dependent component and a non-succinct data independent component. This makes it well suited for optimization in an online-offline model that allows a majority of the computation to be performed in an offline phase, before the data becomes available.

Security of our reusable garbled circuits construction for \(\mathsf{NC_1}\) is based on the Ring Learning With Errors assumption (RLWE), while the bounded collusion construction (with non-succinct ciphertext) may also be based on the standard Learning with Errors (LWE) assumption. To achieve our result, we provide new public key and ciphertext evaluation algorithms. These algorithms are general, and may find application elsewhere.

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Vorheriges Kapitel Decomposable Obfuscation: A Framework for Building Applications of Obfuscation from Polynomial Hardness

Nächstes Kapitel Attribute-Hiding Predicate Encryption in Bilinear Groups, Revisited

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We emphasise that we do not rely on FHE in a black box way, but rather adapt techniques developed in this domain to our setting.

We note that the FE scheme by Abdalla et al. [1] also supports linear functions but only over \(\mathbb {Z}\), while the bounded collusion FE of [5] requires an FE scheme that supports \(\mathbb {Z}_p\). Also note the difference from Inner Product orthogonality testing schemes [4, 45] which test whether \(\langle \mathbf {x}, \mathbf {v}\rangle = 0 \mod p\) or not.

As in FHE, approximate recovery is enough since the noise can be modded out.

Here \(\mu _{f(\mathbf {x})}\) is clubbed with the message \(f(\mathbf {x})\) rather than the \(\mathsf {RLWE}\) noise \(p_{d-1} \cdot \eta _{f}^{d-1}\) since \(\mu _{f(\mathbf {x})} + f(\mathbf {x})\) is what will be recovered after decryption of \(\mathsf {CT}_{f(\mathbf {x})}\), whereas \(p_{d-1} \cdot \eta _{f}^{d-1}\) will be removed by the decryption procedure. This is merely a matter of notation.

We note that the multiplication by s does not impact the nested message degree, number of nestings or growth of the expression as we proceed down the circuit.

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Titel: Functional Encryption for Bounded Collusions, Revisited
verfasst von: Shweta Agrawal
Alon Rosen
Verlag: Springer International Publishing
Buch: Theory of Cryptography
Print ISBN: 978-3-319-70499-9

Electronic ISBN: 978-3-319-70500-2

Copyright-Jahr: 2017
DOI: https://doi.org/10.1007/978-3-319-70500-2_7

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