Degrees of Freedom and Secrecy in Wireless Relay Networks

We translate the problem of designing a secure communications protocol for several users communicating through a relay in a wireless network into understanding certain subvarieties of products of Grassmannians. We calculate the dimension of these subvarieties and provide various results concerning their defining equations. When the relay and all of the users have the same number of antennas, this approach places fundamental limits on the amount of data that can be passed through such a network.