Complexity of some arithmetic problems for binary polynomials

Abstract

We study various combinatorial complexity measures of Boolean functions related to some natural arithmetic problems about binary polynomials, that is, polynomials over $\mathbb{F}_2$. In particular, we consider the Boolean function deciding whether a given polynomial over $\mathbb{F}_2$ is squarefree. We obtain an exponential lower bound on the size of a decision tree for this function, and derive an asymptotic formula, having a linear main term, for its average sensitivity. This allows us to estimate other complexity characteristics such as the formula size, the average decision tree depth and the degrees of exact and approximative polynomial representations of this function. Finally, using a different method, we show that testing squarefreeness and irreducibility of polynomials over $\mathbb{F}_2$ cannot be done in $\textrm{AC}^0[p]$ for any odd prime p. Similar results are obtained for deciding coprimality of two polynomials over $\mathbb{F}_2$ as well.