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 Ω (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.