Online linear extractors for independent sources

Yevgeniy Dodis, Siyao Guo, Noah Stephens-Davidowitz, Zhiye Xie

Research output: Chapter in Book/Report/Conference proceedingConference contribution


In this work, we characterize linear online extractors. In other words, given a matrix A ∈ Fn2×n, we study the convergence of the iterated process S ← AS ⊕ X, where X ∼ D is repeatedly sampled independently from some fixed (but unknown) distribution D with (min)-entropy k. Here, we think of S ∈ {0, 1}n as the state of an online extractor, and X ∈ {0, 1}n as its input. As our main result, we show that the state S converges to the uniform distribution for all input distributions D with entropy k > 0 if and only if the matrix A has no non-trivial invariant subspace (i.e., a non-zero subspace V ⊆ Fn2 such that AV ⊆ V ). In other words, a matrix A yields a linear online extractor if and only if A has no non-trivial invariant subspace. For example, the linear transformation corresponding to multiplication by a generator of the field F2n yields a good linear online extractor. Furthermore, for any such matrix convergence takes at most Oe(n2(k + 1)/k2) steps. We also study the more general notion of condensing - that is, we ask when this process converges to a distribution with entropy at least ℓ, when the input distribution has entropy at least k. (Extractors corresponding to the special case when ℓ = n.) We show that a matrix gives a good condenser if there are relatively few vectors w ∈ Fn2 such that w, AT w,..., (AT )nkw are linearly dependent. As an application, we show that the very simple cyclic rotation transformation A(x1,..., xn) = (xn, x1,..., xn−1) condenses to ℓ = n− 1 bits for any k > 1 if n is a prime satisfying a certain simple number-theoretic condition. Our proofs are Fourier-analytic and rely on a novel lemma, which gives a tight bound on the product of certain Fourier coefficients of any entropic distribution.

Original languageEnglish (US)
Title of host publication2nd Conference on Information-Theoretic Cryptography, ITC 2021
EditorsStefano Tessaro
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959771979
StatePublished - Jul 1 2021
Event2nd Conference on Information-Theoretic Cryptography, ITC 2021 - Virtual, Bertinoro, Italy
Duration: Jul 23 2021Jul 26 2021

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
ISSN (Print)1868-8969


Conference2nd Conference on Information-Theoretic Cryptography, ITC 2021
CityVirtual, Bertinoro


  • Feasibility of randomness extraction
  • Fourier analysis
  • Randomness condensers

ASJC Scopus subject areas

  • Software


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