Broadcasting in Unlabeled Tori

We consider broadcasting a message from one node to all other nodes of an asynchronous totally unlabeled torus: neither nodes nor links have a priori assigned labels but they know the topology and the size of the torus. Nodes can send messages of arbitrary size and we are interested in minimizing the total number of messages. A naive broadcasting algorithm in a n \Theta n totally unlabeled torus uses 3n + 1 messages, while the obvious lower bound is n \Gamma 1