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Journal of Cryptology

Erschienen in:

23.11.2015

Obfuscating Conjunctions

verfasst von: Zvika Brakerski, Guy N. Rothblum

Erschienen in: Journal of Cryptology | Ausgabe 1/2017

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Abstract

We show how to securely obfuscate the class of conjunction functions (functions like \(f(x_1, \ldots , x_n) = x_1 \wedge \lnot x_4 \wedge \lnot x_6 \wedge \cdots \wedge x_{n-2}\)). Given any function in the class, we produce an obfuscated program which preserves the input–output functionality of the given function, but reveals nothing else. Our construction is based on multilinear maps, and can be instantiated using the recent candidates proposed by Garg, Gentry and Halevi (EUROCRYPT 2013) and by Coron et al. (CRYPTO 2013). We show that the construction is secure when the conjunction is drawn from a distribution, under mild conditions on the distribution. Security follows from multilinear entropic variants of the Diffie–Hellman assumption. We conjecture that our construction is secure for any conjunction, regardless of the distribution from which it is drawn. We offer supporting evidence for this conjecture, proving that our obfuscator is secure for any conjunction against generic adversaries.

Vorheriger Artikel An Algebraic Framework for Diffie–Hellman Assumptions

Nächster Artikel Efficient One-Sided Adaptively Secure Computation

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We use the asymmetric variant of the encoding scheme, where there are several distinct “source groups.”

We remark that in this case nothing at all can be learned from black box access to the function since it is infeasible to find an accepting input. We also remark that, for example, the conjunctions used for the k-resource application above naturally satisfy this condition, because of the entropy in each password.

This assumption is actually known to be false for the construction and formulation of [21]. However, we show a more careful definition of the scheme and the assumption for which no attack is known. Also, no attack is known for the recent construction of Coron et al. [14].

For the candidate of [21], the encodings are randomized, but there is a procedure for testing equality between encoded elements in the target group.

Wee [35] showed that these types of assumptions (hardness given only super-logarithmic entropy) are essential even for obfuscating point functions.

The “zero testing” parameter \({\mathbf {pzt}}\) defined in [21] is a part of \({ evparams }\).

This functionality is not explicitly provided by Garg et al. [21]; however, it can be obtained by combining their encoding and re-randomization procedures.

Recall that, as discussed in Remark 5.1, for security proofs in the random graded encoding scheme model, we assume that the encoding of each indexed ring element is unique.

The only dependence is that if for \(i \in [n]\) any of the \(\vec {\alpha }_{i,\vec {x}[i]}\) variables are 0, then \(\alpha _{n+1}\) is also 0.

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Titel: Obfuscating Conjunctions
verfasst von: Zvika Brakerski
Guy N. Rothblum
Publikationsdatum: 23.11.2015
Verlag: Springer US
Erschienen in: Journal of Cryptology / Ausgabe 1/2017
Print ISSN: 0933-2790
Elektronische ISSN: 1432-1378
DOI: https://doi.org/10.1007/s00145-015-9221-5

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