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2016 | OriginalPaper | Buchkapitel

Secure Obfuscation in a Weak Multilinear Map Model

verfasst von : Sanjam Garg, Eric Miles, Pratyay Mukherjee, Amit Sahai, Akshayaram Srinivasan, Mark Zhandry

Erschienen in: Theory of Cryptography

Verlag: Springer Berlin Heidelberg

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Abstract

All known candidate indistinguishability obfuscation (iO) schemes rely on candidate multilinear maps. Until recently, the strongest proofs of security available for iO candidates were in a generic model that only allows “honest” use of the multilinear map. Most notably, in this model the zero-test procedure only reveals whether an encoded element is 0, and nothing more.

However, this model is inadequate: there have been several attacks on multilinear maps that exploit extra information revealed by the zero-test procedure. In particular, Miles, Sahai and Zhandry (Crypto’16) recently gave a polynomial-time attack on several iO candidates when instantiated with the multilinear maps of Garg, Gentry, and Halevi (Eurocrypt’13), and also proposed a new “weak multilinear map model” that captures all known polynomial-time attacks on GGH13.

In this work, we give a new iO candidate which can be seen as a small modification or generalization of the original candidate of Garg, Gentry, Halevi, Raykova, Sahai, and Waters (FOCS’13). We prove its security in the weak multilinear map model, thus giving the first iO candidate that is provably secure against all known polynomial-time attacks on GGH13. The proof of security relies on a new assumption about the hardness of computing annihilating polynomials, and we show that this assumption is implied by the existence of pseudorandom functions in NC\(^1\).

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Vorheriges Kapitel Multi-key FHE from LWE, Revisited

Nächstes Kapitel Virtual Grey-Boxes Beyond Obfuscation: A Statistical Security Notion for Cryptographic Agents

Nur mit Berechtigung zugänglich

For simplicity we abuse notations of branching programs, in that it outputs a ring element instead of a bit. It is straightforward to embed multiple branching programs into one to achieve this effect.

Here and for the remainder of the paper, we use \(z_i\) rather than \(r_i\) to denote the randomization values in GGH13 encodings, to avoid conflicting with the random matrices R chosen by the obfuscator. We will not need to work with the GGH13 level denominators, which were previously denoted by \(z_i\).

Note that this corresponds to finding a non-trivial element in the ideal \(\langle g \rangle \).

These can be constructed from any NC\(^1\) formula with \(m = \mathsf {poly}(n)\) and \(w = 5\) by Barrington’s theorem [Bar86]. Obfuscating NC\(^1\) formulas is sufficient to obfuscate all polynomial-size circuits [GGH+13b, BR14, App14].

For the analysis that we borrow from [BGK+14, BMSZ16], namely Lemma 1, we will not need the strong straddling set systems due to [MSW14].

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Titel: Secure Obfuscation in a Weak Multilinear Map Model
verfasst von: Sanjam Garg
Eric Miles
Pratyay Mukherjee
Amit Sahai
Akshayaram Srinivasan
Mark Zhandry
Verlag: Springer Berlin Heidelberg
Buch: Theory of Cryptography
Print ISBN: 978-3-662-53643-8

Electronic ISBN: 978-3-662-53644-5

Copyright-Jahr: 2016
DOI: https://doi.org/10.1007/978-3-662-53644-5_10

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