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2022 | OriginalPaper | Buchkapitel

Indistinguishability Obfuscation from LPN over \(\mathbb {F}_p\), DLIN, and PRGs in NC\(^0\)

verfasst von : Aayush Jain, Huijia Lin, Amit Sahai

Erschienen in: Advances in Cryptology – EUROCRYPT 2022

Verlag: Springer International Publishing

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Abstract

In this work, we study what minimal sets of assumptions suffice for constructing indistinguishability obfuscation (\(i\mathcal {O}\)). We prove:

Theorem(Informal): Assume sub-exponential security of the following assumptions:

– the Learning Parity with Noise (\(\mathsf {LPN}\)) assumption over general prime fields \(\mathbb {F}_p\) with polynomially many \(\mathsf {LPN}\) samples and error rate \(1/k^\delta \), where k is the dimension of the \(\mathsf {LPN}\) secret, and \(\delta >0\) is any constant;

– the existence of a Boolean Pseudo-Random Generator (\(\mathsf {PRG}\)) in \(\mathsf {NC}^0\) with stretch \(n^{1+\tau }\), where n is the length of the \(\mathsf {PRG}\) seed, and \(\tau >0\) is any constant;

– the Decision Linear (\(\mathsf {DLIN}\)) assumption on symmetric bilinear groups of prime order.

Then, (subexponentially secure) indistinguishability obfuscation for all polynomial-size circuits exists. Further, assuming only polynomial security of the aforementioned assumptions, there exists collusion resistant public-key functional encryption for all polynomial-size circuits.

This removes the reliance on the Learning With Errors (LWE) assumption from the recent work of [Jain, Lin, Sahai STOC’21]. As a consequence, we obtain the first fully homomorphic encryption scheme that does not rely on any lattice-based hardness assumption.

Our techniques feature a new notion of randomized encoding called Preprocessing Randomized Encoding (PRE), that essentially can be computed in the exponent of pairing groups. When combined with other new techniques, PRE gives a much more streamlined construction of \(i\mathcal {O}\) while still maintaining reliance only on well-studied assumptions.

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Vorheriges Kapitel Cryptanalysis of Candidate Obfuscators for Affine Determinant Programs

Nächstes Kapitel Incompressible Cryptography

Besides [20], there are other works that contain FE security amplification techniques [4, 5, 26]. However, it has been recently acknowledged that there is a common issue in these techniques due to an incorrect application of the leakage simulation lemma. The work of [20] circumvents the use leakage simulation lemma, but achieves only weaker security amplification, which nevertheless is still sufficient for constructing \(i\mathcal {O}\).

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Titel: Indistinguishability Obfuscation from LPN over , DLIN, and PRGs in NC
verfasst von: Aayush Jain
Huijia Lin
Amit Sahai
Verlag: Springer International Publishing
Buch: Advances in Cryptology – EUROCRYPT 2022
Print ISBN: 978-3-031-06943-7

Electronic ISBN: 978-3-031-06944-4

Copyright-Jahr: 2022
DOI: https://doi.org/10.1007/978-3-031-06944-4_23

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