High-density and Secure Data Transmission via Linear Combinations

Suppose that there are $n$ Senders and $n$ Receivers. Our goal is to send long messages from Sender $i$ to Receiver $i$ such that no other receiver can retrieve the message intended for Receiver $i$. The task can easily be completed using $n$ private channels between the pairs. Solutions, using one channel needs either encryption or switching elements for routing the messages to their addressee. The main result of the present work is a description of a network in which The Senders and the Receivers are connected with only $n^{o(1)}$ channels; the encoding and de-coding is nothing else just very fast linear combinations of the message-bits; and there are no switching or routing-elements in the network, just linear combinations are computed, with fixed connections (channels or wires). In the proofs we do not use {\em any} unproven cryptographical or complexity theoretical assumptions.