We prove an exponential lower bound ($2^{\Omega(n/\log n)}$) on the size of any {\em randomized} ordered read-once branching program computing {\em integer multiplication}. Our proof depends on proving a new lower bound on Yao's randomized one-way communication complexity of certain boolean functions. It generalizes to some other common models of randomized branching programs. In contrast, we prove that {\em testing integer multiplication}, contrary even to nondeterministic situation, can be computed by {\em randomized} ordered read-once branching program in polynomial size. It is also known that computing the latter problem with deterministic read-once branching programs is as hard as factoring integers.