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Bounded Indistinguishability for Simple Sources

Authors Andrej Bogdanov, Krishnamoorthy Dinesh, Yuval Filmus, Yuval Ishai, Avi Kaplan, Akshayaram Srinivasan

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Author Details

Andrej Bogdanov
  • Department of Computer Science and Engineering and Institute of Theoretical Computer Science and Communications, The Chinese University of Hong Kong, Hong Kong
Krishnamoorthy Dinesh
  • Institute of Theoretical Computer Science and Communications, The Chinese University of Hong Kong, Hong Kong
Yuval Filmus
  • The Henry and Marylin Taub Faculty of Computer Science, Technion, Haifa, Israel
Yuval Ishai
  • The Henry and Marylin Taub Faculty of Computer Science, Technion, Haifa, Israel
Avi Kaplan
  • The Henry and Marylin Taub Faculty of Computer Science, Technion, Haifa, Israel
Akshayaram Srinivasan
  • School of Technology and Computer Science, Tata Institute of Fundamental Research, Mumbai, India

Funding

  • Bogdanov, Andrej: Supported by Hong Kong Research Grants Council GRF CUHK 14207618 and CUHK 14209419.
  • Filmus, Yuval: Supported by the European Union’s Horizon 2020 research and innovation programme under grant agreement No 802020-ERC-HARMONIC.
  • Ishai, Yuval: Supported by ERC Project NTSC (742754), NSF-BSF grant 2015782, BSF grant 2018393, and a grant from the Ministry of Science and Technology, Israel and Department of Science and Technology, Government of India.
  • Kaplan, Avi: Supported by the European Union’s Horizon 2020 research and innovation programme under grant agreement No 802020-ERC-HARMONIC, and ERC Project NTSC (742754).

Acknowledgements

We thank Chin Ho Lee, Igor Oliveira, Rahul Santhanam, Amir Shpilka, Justin Thaler, and Emanuele Viola for useful feedback. Special thanks go to Chin Ho Lee, who suggested [Andrej Bogdanov et al., 2021], and Justin Thaler, who suggested the construction in [Andrej Bogdanov et al., 2021].

Cite AsGet BibTex

Andrej Bogdanov, Krishnamoorthy Dinesh, Yuval Filmus, Yuval Ishai, Avi Kaplan, and Akshayaram Srinivasan. Bounded Indistinguishability for Simple Sources. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 26:1-26:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.ITCS.2022.26

Abstract

A pair of sources X, Y over {0,1}ⁿ are k-indistinguishable if their projections to any k coordinates are identically distributed. Can some AC^0 function distinguish between two such sources when k is big, say k = n^{0.1}? Braverman’s theorem (Commun. ACM 2011) implies a negative answer when X is uniform, whereas Bogdanov et al. (Crypto 2016) observe that this is not the case in general. We initiate a systematic study of this question for natural classes of low-complexity sources, including ones that arise in cryptographic applications, obtaining positive results, negative results, and barriers. In particular: - There exist Ω(√n)-indistinguishable X, Y, samplable by degree-O(log n) polynomial maps (over F₂) and by poly(n)-size decision trees, that are Ω(1)-distinguishable by OR. - There exists a function f such that all f(d, ε)-indistinguishable X, Y that are samplable by degree-d polynomial maps are ε-indistinguishable by OR for all sufficiently large n. Moreover, f(1, ε) = ⌈log(1/ε)⌉ + 1 and f(2, ε) = O(log^{10}(1/ε)). - Extending (weaker versions of) the above negative results to AC^0 distinguishers would require settling a conjecture of Servedio and Viola (ECCC 2012). Concretely, if every pair of n^{0.9}-indistinguishable X, Y that are samplable by linear maps is ε-indistinguishable by AC^0 circuits, then the binary inner product function can have at most an ε-correlation with AC^0 ◦ ⊕ circuits. Finally, we motivate the question and our results by presenting applications of positive results to low-complexity secret sharing and applications of negative results to leakage-resilient cryptography.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
  • Theory of computation → Circuit complexity
Keywords
  • Pseudorandomness
  • bounded indistinguishability
  • complexity of sampling
  • constant-depth circuits
  • secret sharing
  • leakage-resilient cryptography

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