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Limits on Low-Degree Pseudorandom Generators (Or: Sum-of-Squares Meets Program Obfuscation)

  • Boaz Barak
  • Zvika Brakerski
  • Ilan Komargodski
  • Pravesh K. Kothari
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10821)

Abstract

An m output pseudorandom generator \(\mathcal {G}:(\{\pm 1\}^b)^n \rightarrow \{\pm 1\}^m\) that takes input n blocks of b bits each is said to be \(\ell \)-block local if every output is a function of at most \(\ell \) blocks. We show that such \(\ell \)-block local pseudorandom generators can have output length at most \(\tilde{O}(2^{\ell b} n^{\lceil \ell /2 \rceil })\), by presenting a polynomial time algorithm that distinguishes inputs of the form \(\mathcal {G}(x)\) from inputs where each coordinate is sampled from the uniform distribution on m bits.

As a corollary, we refute some conjectures recently made in the context of constructing provably secure indistinguishability obfuscation (iO). This includes refuting the assumptions underlying Lin and Tessaro’s [47] recently proposed candidate iO from bilinear maps. Specifically, they assumed the existence of a secure pseudorandom generator \(\mathcal {G}:\{ \pm 1 \}^{nb} \rightarrow \{\pm 1\}^{2^{cb}n}\) as above for large enough \(c>3\) and \(\ell =2\). (Following this work, and an independent work of Lombardi and Vaikuntanthan [49], Lin and Tessaro retracted the bilinear maps based candidate from their manuscript.)

Our results actually hold for the much wider class of low-degree, non-binary valued pseudorandom generators: if every output of \(\mathcal {G}:\{\pm 1\}^n \rightarrow \mathbb R^m\) (\(\mathbb R\) = reals) is a polynomial (over \(\mathbb R\)) of degree at most d with at most s monomials and \(m \ge \tilde{\varOmega }(sn^{\lceil d/2 \rceil })\), then there is a polynomial time algorithm for distinguishing the output \(\mathcal {G}(x)\) from z where each coordinate \(z_i\) is sampled independently from the marginal distribution on \(\mathcal {G}_i\). Furthermore, our results continue to hold under arbitrary pre-processing of the seed. This implies that any such map \(\mathcal {G}\), with arbitrary seed pre-processing, cannot be a pseudorandom generator in the mild sense of fooling a product distribution on the output space. This allows us to rule out various natural modifications to the notion of generators suggested in other works that still allow obtaining indistinguishability obfuscation from bilinear maps.

Our algorithms are based on the Sum of Squares (SoS) paradigm, and in most cases can even be defined more simply using a canonical semidefinite program. We complement our algorithm by presenting a class of candidate generators with block-wise locality 3 and constant block size, that resists both Gaussian elimination and sum of squares (SOS) algorithms whenever \(m = n^{1.5-\varepsilon }\). This class is extremely easy to describe: Let \(\mathbb G\) be any simple non-abelian group with the group operation “\(*\)”, and interpret the blocks of x as elements in \(\mathbb G\). The description of the pseudorandom generator is a sequence of m triples of indices (ijk) chosen at random and each output of the generator is of the form \(x_i *x_j *x_k\).

Notes

Acknowledgements

We thank Prabhanjan Ananth, Dakshita Khurana and Amit Sahai for discussions regarding the class of generators needed for obfuscation. Thanks to Rachel Lin and Stefano Tessaro for discussing the parameters of their construction with us. We thank Avi Wigderson and Andrei Bulatov for references regarding Gaussian elimination in non-abelian groups.

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Copyright information

© International Association for Cryptologic Research 2018

Authors and Affiliations

  • Boaz Barak
    • 1
  • Zvika Brakerski
    • 2
  • Ilan Komargodski
    • 3
  • Pravesh K. Kothari
    • 4
    • 5
  1. 1.Harvard UniversityCambridgeUSA
  2. 2.Weizmann Institute of ScienceRehovotIsrael
  3. 3.Cornell TechNew YorkUSA
  4. 4.Princeton UniversityPrincetonUSA
  5. 5.IASPrincetonUSA

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