Provable Unlinkability Against Traffic Analysis Already After $\mathcal{O}(\log(n))~$ Steps!

We consider unlinkability of communication problem: given n users, each sending a message to some destination, encode and route the messages so that an adversary analyzing the traffic in the communication network cannot link the senders with the recipients. A solution should have a small communication overhead, that is, the number of additional messages should be kept low.