New Results on the Complexity of the Middle Bit of Multiplication

Abstract

It is well known that the hardest bit of integer multiplication is the middle bit, i.e., MUL n?1,n . This paper contains several new results on its complexity. First, the size s of randomized read-k branching programs, or, equivalently, their space (log s) is investigated. A randomized algorithm for MUL n?1,n with $$k = {\mathcal{O}}(\hbox{log}\, n)$$ (implying time $${\mathcal{O}}(n\, \hbox{log}\, n))$$ , space $${\mathcal{O}}(\hbox{log}\, n)$$ and error probability n ?c for arbitrarily chosen constants c is presented. Second, the size of general branching programs and formulas is investigated. Applying Nechiporuk's technique, lower bounds of $$\Omega (n^{3/2}/ \hbox{log}\, n)$$ and ? (n 3/2), respectively, are obtained. Moreover, by bounding the number of subfunctions of MUL n?1,n , it is proven that Nechiporuk's technique cannot provide larger lower bounds than $${\mathcal{O}}(n^{5/3}/ \hbox{log}\, n)$$ and $${\mathcal{O}}(n^{5/3})$$ , respectively.