New results on the complexity of the middle bit of multiplication

It is well known that the hardest bit of integer multiplication is the middle bit, i.e. MULn−1,n. This paper contains several new results on its complexity. First, the size s of randomized read-k branching programs, or, equivalently, its space (log s) is investigated. A randomized algorithm for MULn−1,n with k = O(log n) (implying time O(n log n)), space O(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 Ω ` n 3/2 / log n ´ and Ω ` n 3/2 ´ , respectively, are obtained. Moreover, by bounding the number of subfunctions of MULn−1,n, it is proven that Nechiporuk’s technique cannot provide larger lower bounds than O(n 7/4 / log n) and O(n 7/4), respectively