Perfect secrecy in wireless networks

In a classic paper, among a multitude of other security issues, Claude E Shannon defined perfect secrecy for a pair of secure computers communicating over an insecure link. The present paper extends Shannon's notion of perfect secrecy to ad hoc wireless networks of computers. All of the wireless transmissions are insecure. This paper assumes a secure base-station communicating wirelessly using symmetric encryption with physically insecure autonomous computers. These wireless communications use a common symmetric encryption algorithm. Each of the computers has a unique fixed key. The base station has a copy of each key. This scenario gives way to a generalization of Shannon's perfect secrecy. This generalizes to t-perfect secrecy which is directly based on t-wise independence. This gives another natural application of t-wise independence to cryptography. Finally, transmission noise, message collisions, and other issues are briefly discussed with respect to this generalization of Shannon's perfect secrecy.