Secure Degrees of Freedom Regions of Multiple Access and Interference Channels: The Polytope Structure.

In this paper, we determine the entire secure degrees of freedom (s.d.o.f.) regions of the $K$ -user Gaussian multiple access (MAC) wiretap channel and the $K$ -user interference channel (IC) with secrecy constraints. For the IC, we consider three secrecy constraints: K-user IC with an external eavesdropper (IC-EE), K-user IC with confidential messages (IC-CM), and their combination Kuser IC with confidential messages and external eavesdropper (IC-CM-EE). The converse for the IC includes constraints both due to secrecy as well as due to interference. For the IC, although the portion of the region close to the optimum sum s.d.o.f. point is governed by the upper bounds due to secrecy constraints, the other portions of the region are governed by the upper bounds due to interference constraints. Different from the existing literature, in order to fully understand the characterization of the s.d.o.f. region of the IC, one has to study the four-user case, i.e., the two- or three-user cases do not illustrate the full generality of the problem. In order to prove the achievability, we use the polytope structure of the converse region. In both MAC and IC cases, we develop explicit schemes that achieve the extreme points of the polytope region given by the converse. In particular, the extreme points of the MAC region are achieved by an $m$ -user MAC wiretap channel with $K-m$ helpers, i.e., by setting $K-m$ users’ secure rates to zero and utilizing them as pure (structured) cooperative jammers. The extreme points of the IC region are achieved by a $(K-m)$ -user IC with confidential messages, $m$ helpers, and $N$ external eavesdroppers, for $m\ge 1$ and a finite $N$ . A byproduct of our results in this paper is that the sum s.d.o.f. is achieved only at one extreme point of the s.d.o.f. region, which is the symmetric-rate extreme point, for both MAC and IC channel models. [ABSTRACT FROM AUTHOR]