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On Polynomial Representations of Boolean Functions Related to Some Number Theoretic Problems

  • Erion Plaku
  • Igor E. Shparlinski
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2245)

Abstract

We say a polynomial P over ℤM strongly M-represents a Boolean function F if F(x) ≡P(x) (mod M) for all x∈ {0,1}n. Similarly, P one-sidedly M-represents F if F(x) = 0 ⇔ P(x) ≡ 0 (modM) for all x ∈ {0,1}n. Lower bounds are obtained on the degree and the number of monomials of polynomials over ℤ M, which strongly or one-sidedly M-represent the Boolean function deciding if a given n- bit integer is square-free. Similar lower bounds are also obtained for polynomials over the reals which provide a threshold representation of the above Boolean function.

Keywords

Lower Bound Boolean Function Great Common Divisor Polynomial Representation Real Polynomial 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Erion Plaku
    • 1
  • Igor E. Shparlinski
    • 2
  1. 1.Department of Mathematics and Computer ScienceClarkson UniversityPotsdamUSA
  2. 2.Department of ComputingMacquarie UniversityAustralia

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