Complexity of some arithmetic problems for binary polynomials

Eric Allender*, Anna Bernasconi, Carsten Damm, Joachim Von Zur Gathen, Michael Saks, Igor Shparlinski

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)

Abstract

We study various combinatorial complexity measures of Boolean functions related to some natural arithmetic problems about binary polynomials, that is, polynomials over Struct F sign 2. In particular, we consider the Boolean function deciding whether a given polynomial over Struct F sign 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 Struct F sign2 cannot be done in AC0[p] for any odd prime p. Similar results are obtained for deciding coprimality of two polynomials over Struct F sign 2 as well.

Original languageEnglish
Pages (from-to)23-47
Number of pages25
JournalComputational Complexity
Volume12
Issue number1-2
Publication statusPublished - 2003

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