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

We say a polynomial P over ZZM 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 (mod M) for all x ∈ {0, 1}n . Lower bounds are obtained on the degree and the number of monomials of polynomials over ZZM , which strongly or one-sidedly M -represent the Boolean function deciding if a given nbit 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.

• Type: Conference
• Year: 2001
• Where: FSTTCS
• Authors: Erion Plaku, Igor Shparlinski
• Keywords: Boolean Function, FSTTCS 2001, Mod M, Polynomials, Software Engineering