We prove an exponential lower bound ($2^{\Omega(n/\log n)}$) on the
size of any randomized ordered read-once branching program
computing 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 testing integer multiplication, contrary even to
nondeterministic situation, can be computed by 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.