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Erschienen in:
Stronger Security for Reusable Garbled Circuits, General Definitions and Attacks Kapitel lesen Erstes Kapitel lesen

2017 | OriginalPaper | Buchkapitel

Indistinguishability Obfuscation from SXDH on 5-Linear Maps and Locality-5 PRGs

verfasst von : Huijia Lin

Erschienen in: Advances in Cryptology – CRYPTO 2017

Verlag: Springer International Publishing

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Abstract

Two recent works [Lin, EUROCRYPT 2016, Lin and Vaikuntanathan, FOCS 2016] showed how to construct Indistinguishability Obfuscation (IO) from constant degree multilinear maps. However, the concrete degrees of multilinear maps used in their constructions exceed 30. In this work, we reduce the degree of multilinear maps needed to 5, by giving a new construction of IO from asymmetric L-linear maps and a pseudo-random generator (PRG) with output locality L and polynomial stretch. When plugging in a candidate PRG with locality-5 (e.g., [Goldreich, ECCC 2010, Mossel, Shpilka, and Trevisan, FOCS 2013, O’Donnald and Wither, CCC 2014]), we obtain a construction of IO from 5-linear maps.
Our construction improves the state-of-the-art at two other fronts: First, it relies on “classical” multilinear maps, instead of their powerful generalization of graded encodings. Second, it comes with a security reduction to (i) the SXDH assumption on algebraic multilinear maps [Boneh and Silverberg, Contemporary Mathematics, Rothblum, TCC 2013], (ii) the security of PRG, and (iii) sub-exponential LWE, all with sub-exponential hardness. The SXDH assumption is weaker and/or simpler than assumptions on multilinear maps underlying previous IO constructions. When noisy multilinear maps [Garg et al., EUROCRYPT 2013] are used instead, security is based on a family of more complex assumptions that hold in the generic model.

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Vorheriges Kapitel The First Collision for Full SHA-1
Nächstes Kapitel Indistinguishability Obfuscation from Trilinear Maps and Block-Wise Local PRGs
Fußnoten
1
The operation is according to some well-defined addition operation over the labels; for example, if labels are integers, \(+\) is integer addition, and if labels are sets, \(+\) is set union.
 
2
For instance, if labels are sets, then two groups are pairable, if their label-sets \(l_1, l_2\) are disjoint.
 
3
Note that in both distributions, the same exponents, a, b, r, are used in all groups in S.
 
4
Some bootstrapping theorems additionally assume LWE [27, 41] or the existence of public key encryption [13]).
 
5
Such IPEs should be contrasted with functional encryption for testing the orthogonality of two vectors (see, e.g., [38, 40] and many others), which reveals only whether the inner product is zero and nothing else. In particular, they do not compute the inner product in the exponent in a way that allows for further computation, which is needed for our quadratic FE construction.
 
6
Non-black-box security reduction may get around this difficulty, but is unclear how one can design a non-black-box reduction here.
 
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Metadaten
Titel
Indistinguishability Obfuscation from SXDH on 5-Linear Maps and Locality-5 PRGs
verfasst von
Huijia Lin
Copyright-Jahr
2017
Buch
Advances in Cryptology – CRYPTO 2017

Print ISBN: 978-3-319-63687-0

Electronic ISBN: 978-3-319-63688-7

Copyright-Jahr: 2017

DOI
https://doi.org/10.1007/978-3-319-63688-7_20