Abstract
A pair of sources X, Y over {0,1}ⁿ are kindistinguishable 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 lowcomplexity 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 degreeO(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 degreed polynomial maps are εindistinguishable 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 εindistinguishable by AC^0 circuits, then the binary inner product function can have at most an εcorrelation with AC^0 ◦ ⊕ circuits.
Finally, we motivate the question and our results by presenting applications of positive results to lowcomplexity secret sharing and applications of negative results to leakageresilient cryptography.
BibTeX  Entry
@InProceedings{bogdanov_et_al:LIPIcs.ITCS.2022.26,
author = {Bogdanov, Andrej and Dinesh, Krishnamoorthy and Filmus, Yuval and Ishai, Yuval and Kaplan, Avi and Srinivasan, Akshayaram},
title = {{Bounded Indistinguishability for Simple Sources}},
booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
pages = {26:126:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772174},
ISSN = {18688969},
year = {2022},
volume = {215},
editor = {Braverman, Mark},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/15622},
URN = {urn:nbn:de:0030drops156223},
doi = {10.4230/LIPIcs.ITCS.2022.26},
annote = {Keywords: Pseudorandomness, bounded indistinguishability, complexity of sampling, constantdepth circuits, secret sharing, leakageresilient cryptography}
}
Keywords: 

Pseudorandomness, bounded indistinguishability, complexity of sampling, constantdepth circuits, secret sharing, leakageresilient cryptography 
Collection: 

13th Innovations in Theoretical Computer Science Conference (ITCS 2022) 
Issue Date: 

2022 
Date of publication: 

25.01.2022 