Tight Bounds for Testing k-Linearity

  • Eric Blais
  • Daniel Kane
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7408)

Abstract

The function \(f : \mathbb{F}_2^n \to \mathbb{F}_2\) is k-linear if it returns the sum (over \(\mathbb{F}_2\)) of exactly k coordinates of its input. We introduce strong lower bounds on the query complexity for testing whether a function is k-linear. We show that for any \(k \le \frac n2\), at least k − o(k) queries are required to test k-linearity, and we show that when \(k \approx \frac n2\), this lower bound is nearly tight since \(\frac43 k + o(k)\) queries are sufficient to test k-linearity. We also show that non-adaptive testers require 2k − O(1) queries to test k-linearity.

We obtain our results by reducing the k-linearity testing problem to a purely geometric problem on the boolean hypercube. That geometric problem is then solved with Fourier analysis and the manipulation of Krawtchouk polynomials.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Eric Blais
    • 1
  • Daniel Kane
    • 2
  1. 1.School of Computer ScienceCarnegie Mellon UniversityUSA
  2. 2.Department of MathematicsStanford UniversityUSA

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