computational complexity

, Volume 9, Issue 1, pp 39–51

The average sensitivity of square-freeness

  • A. Bernasconi
  • C. Damm
  • I. Shparlinski

DOI: 10.1007/PL00001600

Cite this article as:
Bernasconi, A., Damm, C. & Shparlinski, I. Comput. complex. (2000) 9: 39. doi:10.1007/PL00001600


We study combinatorial complexity characteristics of a Boolean function related to a natural number theoretic problem. In particular we obtain an asymptotic formula, having a linear main term, for the average sensitivity of the Boolean function deciding whether a given integer is square-free. This result allows us to derive a quadratic lower bound for the formula size complexity of testing square-free numbers and a linear lower bound on the average decision tree depth. We also obtain lower bounds on the degrees of exact and approximate polynomial representations of this function.

Keywords. Square-freeness, average sensitivity, combinatorial complexity.

Copyright information

© Birkhäuser Verlag, Basel 2000

Authors and Affiliations

  • A. Bernasconi
    • 1
  • C. Damm
    • 2
  • I. Shparlinski
    • 3
  1. 1.Istituto di Matematica Computazionale, Consiglio Nazionale delle Ricerche, 56010 San Cataldo - Pisa, Italy, e-mail: IT
  2. 2.Institut für Numerische und Angewandte Mathematik, Georg-August-Universität Göttingen, D-37083 Göttingen, Germany, e-mail: DE
  3. 3.Department of Computing, Macquarie University, NSW 2109, Australia, e-mail: AU