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Indistinguishability Obfuscation Without Multilinear Maps: New Paradigms via Low Degree Weak Pseudorandomness and Security Amplification

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Advances in Cryptology – CRYPTO 2019 (CRYPTO 2019)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 11694))

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Abstract

The existence of secure indistinguishability obfuscators (\(i\mathcal {O}\)) has far-reaching implications, significantly expanding the scope of problems amenable to cryptographic study. All known approaches to constructing \(i\mathcal {O}\) rely on d-linear maps. While secure bilinear maps are well established in cryptographic literature, the security of candidates for \(d>2\) is poorly understood.

We propose a new approach to constructing \(i\mathcal {O}\) for general circuits. Unlike all previously known realizations of \(i\mathcal {O}\), we avoid the use of d-linear maps of degree \(d \ge 3\).

At the heart of our approach is the assumption that a new weak pseudorandom object exists. We consider two related variants of these objects, which we call perturbation resilient generator (\(\varDelta \)RG) and pseudo flawed-smudging generator (\(\mathrm {PFG}\)), respectively. At a high level, both objects are polynomially expanding functions whose outputs partially hide (or smudge) small noise vectors when added to them. We further require that they are computable by a family of degree-3 polynomials over \(\mathbb {Z}\). We show how they can be used to construct functional encryption schemes with weak security guarantees. Finally, we use novel amplification techniques to obtain full security.

As a result, we obtain \(i\mathcal {O}\) for general circuits assuming:

  • Subexponentially secure LWE

  • Bilinear Maps

  • \(\mathrm {poly}(\lambda )\)-secure 3-block-local PRGs

  • \(\varDelta \)RGs or \(\mathrm {PFG}\)s

This paper is a merge of two independent works, one by Ananth, Jain, and Sahai [AJS18], and the other by Lin and Matt [LM18].

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Notes

  1. 1.

    Note that this does not necessarily mean that the resulting \(i\mathcal {O}\) constructions are insecure; in particular, there have been efforts (e.g., [GMM+16a]) in constructing \(i\mathcal {O}\) in more complex security models for multilinear maps (e.g. [MSZ16]) that have resisted polynomial-time attacks. There have also been several other \(i\mathcal {O}\) candidates proposed which are not known to polynomial-time broken (e.g. [CVW18, BGMZ18]).

  2. 2.

    A function has locality \(\ell \) if every output element depends on at most \(\ell \) input elements.

  3. 3.

    The attacks actually leave open a very small window of expansion. Nevertheless, they have weakened our confidence on the security of PRGs with block-locality 2.

  4. 4.

    n, m, p are parameterized by the security parameter \(\lambda \).

  5. 5.

    There is be a tradeoff between how much AJS can weaken the indistinguishability requirements of the \(\varDelta \)RG and the 3-block-local PRG.

  6. 6.

    In an earlier version of this paper, this overview focused on constructing degree-2 \(\varDelta \)RGs. However, as we describe now, our technical approach is more general, and we describe it in greater generality here.

  7. 7.

    The schemes in [AR17, Agr18a] has more complicated decryption equation, where the decryption noise is of form \(\sum _i p_i {{\mathbf {e}}}_i\) where \(\{p_i\}\) is a set of increasing moduli. Here we omit this complexity.

  8. 8.

    Thanks Yuval Ishai and Elette Boyal for pointing this out.

  9. 9.

    If not, one can always use garbled circuits to turn g into another circuit where every output bit is computable in size \(\mathrm {poly}(\lambda )\), at the cost of increasing the size, input length, and output length of the circuit by a multiplicative \(\mathrm {poly}(\lambda )\) factor.

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Acknowledgments

Huijia Lin and Christian Matt thank Shweta Agrawal for sharing an early version of [Agr18a], and Stefano Tessaro for many general discussions. This work was done in part when both of these two authors were at the University of California, Santa Barbara.

Prabhanjan Ananth, Aayush Jain and Amit Sahai thank Dakshita Khurana for collaboration in the initial stages of this research, for contributing to the writeup, and for countless discussions and comments supporting this work and improving the write up. Eventually, the current set of authors had to reluctantly agree to Dakshita’s repeated requests to not be listed in the set of authors, and hence she is in these acknowledgements instead. We thank Boaz Barak, Sam Hopkins and Pravesh Kothari for insights and extremely helpful suggestions about how attacks based on the Sum of Squares paradigm could impact our new assumptions on perturbation-resilient generators.

Huijia Lin and Christian Matt were supported by NSF grants CNS-1528178, CNS-1514526, CNS-1652849 (CAREER), a Hellman Fellowship, the Defense Advanced Research Projects Agency (DARPA) and Army Research Office (ARO) under Contract No. W911NF-15-C-0236, and a subcontract No. 2017-002 through Galois. The views expressed are those of the authors and do not reflect the official policy or position of the Department of Defense, the National Science Foundation, or the U.S. Government.

Prabhanjan Ananth, Aayush Jain and Amit Sahai were supported in part from a DARPA/ARL SAFEWARE award, NSF Frontier Award 1413955, and NSF grant 1619348, BSF grant 2012378, a Xerox Faculty Research Award, a Google Faculty Research Award, an equipment grant from Intel, and an Okawa Foundation Research Grant. Aayush Jain is also supported by a Google PhD fellowship award in Privacy and Security. This material is based upon work supported by the Defense Advanced Research Projects Agency through the ARL under Contract W911NF-15-C- 0205. The views expressed are those of the authors and do not reflect the official policy or position of the Department of Defense, the National Science Foundation, the U.S. Government or Google.

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Authors and Affiliations

  1. MIT, Cambridge, USA

    Prabhanjan Ananth

  2. UCLA, Los Angeles, USA

    Aayush Jain & Amit Sahai

  3. University of Washington, Seattle, USA

    Huijia Lin

  4. Concordium, Zurich, Switzerland

    Christian Matt

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  1. Prabhanjan Ananth
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Correspondence to Prabhanjan Ananth or Aayush Jain .

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  1. Georgia Institute of Technology, Atlanta, GA, USA

    Alexandra Boldyreva

  2. University of California at San Diego, La Jolla, CA, USA

    Daniele Micciancio

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Ananth, P., Jain, A., Lin, H., Matt, C., Sahai, A. (2019). Indistinguishability Obfuscation Without Multilinear Maps: New Paradigms via Low Degree Weak Pseudorandomness and Security Amplification. In: Boldyreva, A., Micciancio, D. (eds) Advances in Cryptology – CRYPTO 2019. CRYPTO 2019. Lecture Notes in Computer Science(), vol 11694. Springer, Cham. https://doi.org/10.1007/978-3-030-26954-8_10

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