Secure routerless routing

Suppose that there are n Senders and r Receivers. Our goal is to design a communication network such that long messages can be sent from Sender i to Receiver π(i) such that no other receiver can retrieve the message intended for Receiver π(i). The task can easily be completed using some classical interconnection network and routers in the network. Alternatively, if every Receiver is directly connected to all $n$ Senders, then the Senders can choose which channel to use for communication, without using any router. Fast optical networks are slowed down considerably if routers are inserted in their nodes. Moreover, handling queues or buffers at the routers is extremely hard in all-optical setting. An obvious routerless solution, connecting each possible Sender-Receiver pairs with direct channels seems to be infeasible in most cases. The main result of the present work is the mathematical model of two networks and corresponding network-protocols in which the Senders and the Receivers are connected with only r^o(1) channels (in practice no more than 32 channels in both networks); there are no switching or routing-elements in the network, just linear combinations of the signals are computed. Such designs would be usable in fast all-optical networks. In the proof of the security of the networks we do not use any unproven cryptographical or complexity theoretical assumptions: the security is information-theoretically proved, and does not depend on cryptographical primitives.