Bounded Indistinguishability for Simple Sources

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

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


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 AC0 function distinguish between two such sources when k is big, say k = n0.1? 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(log10(1/ϵ)). - Extending (weaker versions of) the above negative results to AC0 distinguishers would require settling a conjecture of Servedio and Viola (ECCC 2012). Concretely, if every pair of n0.9indistinguishable X, Y that are samplable by linear maps is ϵ-indistinguishable by AC0 circuits, then the binary inner product function can have at most an ϵ-correlation with AC0 ◦ ⨁ 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.

Original languageEnglish
Title of host publication13th Innovations in Theoretical Computer Science Conference, ITCS 2022
EditorsMark Braverman
ISBN (Electronic)9783959772174
StatePublished - 1 Jan 2022
Event13th Innovations in Theoretical Computer Science Conference, ITCS 2022 - Berkeley, United States
Duration: 31 Jan 20223 Feb 2022

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs


Conference13th Innovations in Theoretical Computer Science Conference, ITCS 2022
Country/TerritoryUnited States


  • Bounded indistinguishability
  • Complexity of sampling
  • Constant-depth circuits
  • Leakage-resilient cryptography
  • Pseudorandomness
  • Secret sharing

All Science Journal Classification (ASJC) codes

  • Software


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