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., MUL/sub n - 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 MUL/sub n - 1, n/ with k = O(log n) (implying time O(n log n)), space O(log n) and error probability n/sup -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 /spl Omega/(n/sup 3/2//log n) and /spl Omega/(n/sup 3/2/), respectively, are obtained. Moreover; by bounding the number of subfunctions of MUL/sub n - 1, n/, it is proven that Nechiporuk's technique cannot provide larger lower bounds than O(n/sup 7/4//log n) and O(n/sup 7/4/), respectively.