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Succinct LWE Sampling, Random Polynomials, and Obfuscation

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Theory of Cryptography (TCC 2021)

Part of the book series: Lecture Notes in Computer Science ((LNSC,volume 13043))

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Abstract

We present a construction of indistinguishability obfuscation (iO) that relies on the learning with errors (LWE) assumption together with a new notion of succinctly sampling pseudorandom LWE samples. We then present a candidate LWE sampler whose security is related to the hardness of solving systems of polynomial equations. Our construction improves on the recent iO candidate of Wee and Wichs (Eurocrypt 2021) in two ways: first, we show that a much weaker and simpler notion of LWE sampling suffices for iO; and secondly, our candidate LWE sampler is secure based on a compactly specified and falsifiable assumption about random polynomials, with a simple error distribution that facilitates cryptanalysis.

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Notes

  1. 1.

    Our notion of succinct randomized encodings is weaker than prior works: indeed, [BGL+15] required the encoder to run in time sublinear in N, whereas we allow the encoder run-time to be polynomial in N.

  2. 2.

    In the WW terminology, this would be a candidate K-sim functional encoding for \(f_1,\ldots ,f_K : \{0,1\}^\ell \rightarrow \{0,1\}^M\).

  3. 3.

    It is simpler in terms of syntax, since we do not refer to LWE trapdoors for \(\mathbf {A}\), and in terms of the security requirement since we do not require a simulator, but instead have a simple indistinguishability criterion.

  4. 4.

    In general, we can use a different (small) distributions \(D_P\) and \(D_{P'}\) for \(\mathbf {P}\), \(\mathbf {P}'\). We only set \(D_P = D_P' = \chi \) to minimize the number of distributions and parameters.

  5. 5.

    The first constraint is redundant with the constraints of Corollary 1.

  6. 6.

    We prove that \(\mathrm {rank}\left( \overline{\mathbf {A}}^*\right) \le m^d - (m-w)^d\) in Sect. 4.5, paragraph Rank of \(\mathbf {A}^* \mathbf {S}^*\).

  7. 7.

    Writing \(m = m'\,+\,w\) where \(m'>0\), the difference \((m'\,+\,w)^d \,-\, (m'^d\,+\,dw(m'\,+\,w)^{d-1})\) is the sum of monomials in \(m',w\) with positive coefficients.

  8. 8.

    This is without loss of generality by defining for instance \(\chi ' = \chi \,+\, [-B,B]\) where \(B'\) is large enough to satisfy the previous constraint. A direct reduction ensures that if LWE holds with \(\chi \), then it holds with \(\chi '\).

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Acknowledgements

We thank Pravesh Kothari for his pointers to and conversations about the literature on SOS and low-degree polynomial attacks. LD and VV were supported by DARPA under Agreement No. HR00112020023, a grant from the MIT-IBM Watson AI, a grant from Analog Devices, a Microsoft Trustworthy AI grant, and a DARPA Young Faculty Award. WQ completed part of this work during an internship at NTT Research. DW was supported by NSF grant CNS-1750795, CNS-2055510, and the Alfred P. Sloan Research Fellowship.

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Authors and Affiliations

  1. Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    Lalita Devadas & Vinod Vaikuntanathan

  2. Northeastern University, Boston, MA, 02115, USA

    Willy Quach & Daniel Wichs

  3. NTT Research, Sunnyvale, CA, 94085, USA

    Hoeteck Wee & Daniel Wichs

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  1. Lalita Devadas
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Correspondence to Lalita Devadas .

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  1. Georgetown University, Washington, WA, USA

    Kobbi Nissim

  2. The University of Texas at Austin, Austin, TX, USA

    Brent Waters

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Devadas, L., Quach, W., Vaikuntanathan, V., Wee, H., Wichs, D. (2021). Succinct LWE Sampling, Random Polynomials, and Obfuscation. In: Nissim, K., Waters, B. (eds) Theory of Cryptography. TCC 2021. Lecture Notes in Computer Science(), vol 13043. Springer, Cham. https://doi.org/10.1007/978-3-030-90453-1_9

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