New bounds for randomized busing

We consider anonymous secure communication, where parties not only wish to conceal their communications from outside observers, but also wish to conceal the very fact that they are communicating. We consider the bus framework introduced by Beimel and Dolev (J. Cryptology 16 (2003) 25), where messages are delivered by a bus traveling on a random walk. We generalize this idea to consider more than one bus. We show that if w buses are allowed, then the expected delivery time for a message can be decreased from Θ(n) to Θ(n/w) in the case of a complete graph. Additionally, we introduce a class of graphs called r-partite directed collars and obtain analogous bounds on the expected delivery time for these graphs. We also propose several new features that resolve possible shortcomings in the systems proposed by Beimel and Dolev.