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.