An elementary proof of a 3n - o(n) lower bound on the circuit complexity of affine dispersers

Evgeny Demenkov, Alexander S. Kulikov

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

23 Scopus citations

Abstract

A Boolean function f: double-struck F2n → double-struck F 2 is called an affine disperser of dimension d, if f is not constant on any affine subspace of double-struck F2 n of dimension at least d. Recently Ben-Sasson and Kopparty gave an explicit construction of an affine disperser for sublinear d. The main motivation for studying such functions comes from extracting randomness from structured sources of imperfect randomness. In this paper, we show another application: we give a very simple proof of a 3n-o(n) lower bound on the circuit complexity (over the full binary basis) of affine dispersers for sublinear dimension. The same lower bound 3n-o(n) (but for a completely different function) was given by Blum in 1984 and is still the best known. The main technique is to substitute variables by linear functions. This way the function is restricted to an affine subspace of F2n. An affine disperser for sublinear dimension then guarantees that one can make n-o(n) such substitutions before the function degenerates. It remains to show that each such substitution eliminates at least 3 gates from a circuit.

Original languageEnglish
Title of host publicationMathematical Foundations of Computer Science 2011 - 36th International Symposium, MFCS 2011, Proceedings
Pages256-265
Number of pages10
DOIs
StatePublished - 1 Sep 2011
Event36th International Symposium on Mathematical Foundations of Computer Science, MFCS 2011 - Warsaw, Poland
Duration: 22 Aug 201126 Aug 2011

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume6907 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference36th International Symposium on Mathematical Foundations of Computer Science, MFCS 2011
CountryPoland
CityWarsaw
Period22/08/1126/08/11

Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

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