On Derandomized Composition of Boolean Functions

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The (block-)composition of two Boolean functions f: { 0 , 1 } m→ { 0 , 1 } , g: { 0 , 1 } n→ { 0 , 1 } is the function f⋄ g that takes asinputs m strings x1, … , xm∈ { 0 , 1 } n and computes (f⋄ g) (x1, … , xm) = f(g(x1) , … , g(xm)). This operation has been used several times in the past for amplifyingdifferent hardness measures of f and g. This comes at a cost: thefunction f⋄ g has input length m· n rather than m or n, which is a bottleneck for some applications. In this paper, we propose to decrease this cost by “derandomizing” the composition: instead of feeding into f⋄ g independent inputs x1, … , xm, we generate x1, … , xm using a shorter seed. We show that this idea can be realized in the particular setting of the composition of functions and universal relations (Gavinsky et al. in SIAM J Comput 46(1):114–131, 2017; Karchmer et al. in Computat Complex 5(3/4):191–204, 1995b). To this end, we provide two different techniques for achieving such a derandomization: a technique based on averaging samplers and a technique based on Reed–Solomon codes.

Original languageEnglish
Pages (from-to)661-708
Number of pages48
JournalComputational Complexity
Issue number4
StatePublished - 1 Dec 2019

Bibliographical note

Publisher Copyright:
© 2019, Springer Nature Switzerland AG.


  • 68Q15
  • Circuit complexity
  • KRW conjecture
  • Karchmer–Wigderson relations
  • circuit lower bounds
  • communication complexity
  • derandomization
  • formula complexity
  • formula lower bounds

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Mathematics
  • Computational Theory and Mathematics
  • Computational Mathematics


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