The sum secure degrees of freedom (s.d.o.f.) of two fundamental multi-user
network structures, the K-user Gaussian multiple access (MAC) wiretap channel
and the K-user interference channel (IC) with secrecy constraints, have been
determined recently as K(K-1)/(K(K-1)+1) [1,2] and K(K-1)/(2K-1) [3,4],
respectively. In this paper, we determine the entire s.d.o.f. regions of these
two channel models. The converse for the MAC follows from a middle step in the
converse of [1,2]. The converse for the IC includes constraints both due to
secrecy as well as due to interference. 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 4-user case, i.e., the 2 or 3-user cases do not
illustrate the 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. Specifically, 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>=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.Comment: Submitted to IEEE Transactions on Information Theory, April 201
