Collective Secrecy over the K -Transmitter Multiple Access Channel

This paper studies the problem of secure communication over a K -transmitter multiple access channel (MAC) in the presence of an external eavesdropper, subject to a collective secrecy constraint (i.e., information leakage rate to an eavesdropper on a collection of messages that are from a pre-specified subset of the K transmitters, say \mathcal {S}\subseteq \mathcal {K}=\{1,2,\ldots, K\} , is made vanishing). Since secrecy is of concern only to transmitters \{i|i\in \mathcal {S}\} but not to transmitters \{i|i\in \mathcal {S}^{c}\} , where \mathcal {S}^{c}=\mathcal {K}\backslash \mathcal {S} , different transmission strategies could be employed at transmitters \{i|i\in \mathcal {S}^{c}\}. Consider the following two scenarios: 1) transmitters \{i|i\in \mathcal {S}^{c}\} use deterministic encoders (which are conventionally used for MAC without secrecy), competing for the channel resource (i.e., being competitive) and 2) transmitters \{i|i\in \mathcal {S}^{c}\} use stochastic encoders, helping to hide other transmitters' messages from the eavesdropper (i.e., being cooperative). As a result, we establish the respective \mathcal {S} -collective secrecy achievable rate regions and demonstrate the advantage of being cooperative theoretically and numerically. To this end, in addition to the standard techniques, our results build upon two techniques. The first is a generalization of Chia-El Gamal's lemma on entropy bound for a set of codewords given partial information. The second is to utilize a compact representation of a list of sets that, together with submodular properties of mutual information functions involved, leads to an efficient Fourier-Motzkin elimination. These two approaches allow us to derive achievable regions in this work, and could also be of independent interest in other context.