2026-07-22T18:37:01Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:15869 2024-03-06T10:56:24Z ddc:004 open_access

Sharp Indistinguishability Bounds from Non-Uniform Approximations Williamson, Christopher bounded indistinguishability randomness secret sharing polynomial approximation We study the basic problem of distinguishing between two symmetric probability distributions over n bits by observing k bits of a sample, subject to the constraint that all (k-1)-wise marginal distributions of the two distributions are identical to each other. Previous works of Bogdanov et al. [Bogdanov et al., 2019] and of Huang and Viola [Huang and Viola, 2019] have established approximately tight results on the maximal possible statistical distance between the k-wise marginals of such distributions when k is at most a small constant fraction of n. Naor and Shamir [Naor and Shamir, 1994] gave a tight bound for all k in the special case k = n and when distinguishing with the OR function; they also derived a non-tight result for general k and n. Krause and Simon [Krause and Simon, 2000] gave improved upper and lower bounds for general k and n when distinguishing with the OR function, but these bounds are exponentially far apart when k = Ω(n). In this work we provide sharp upper and lower bounds on the maximal statistical distance that hold for all k and n. Upper bounds on the statistical distance have typically been obtained by providing uniform low-degree polynomial approximations to certain higher-degree polynomials. This is the first work to construct suitable non-uniform approximations for this purpose; the sharpness and wider applicability of our result stems from this non-uniformity. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Christopher Williamson 2022 Is Part Of LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.STACS.2022.59 urn:nbn:de:0030-drops-158692 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.59 eng https://creativecommons.org/licenses/by/4.0/legalcode