Conference Paper

On the secure degrees of freedom of wireless X networks

Electr. Eng. & Comput. Sci., Univ. of California, Irvine, CA
DOI: 10.1109/ALLERTON.2008.4797643 Conference: Communication, Control, and Computing, 2008 46th Annual Allerton Conference on
Source: IEEE Xplore

ABSTRACT Previous work showed that the X network with M transmitters, N receivers has MN/M+N-1 degrees of freedom. In this work we study the degrees of freedom of the X network with secrecy constraints, i.e. the X network where some/all messages are confidential. We consider the M times N network where all messages are secured and show that N(M-1)/M+N-1 degrees of freedom can be achieved. Secondly, we show that if messages from only M - 1 transmitters are confidential, then MN/M+N-1 degrees of freedom can be achieved meaning that there is no loss of degrees of freedom because of secrecy constraints. We also consider the achievable secure degrees of freedom under a more conservative secrecy constraint. We require that messages from any subset of transmitters are secure even if other transmitters are compromised, i.e., messages from the compromised transmitter are revealed to the unintended receivers. We also study the achievable secure degrees of freedom of the K user Gaussian interference channel under two different secrecy constraints where 1/2 secure degrees of freedom per message can be achieved. The achievable scheme in all cases is based on random binning combined with interference alignment.

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    ABSTRACT: 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.
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    ABSTRACT: The secrecy capacity of the canonical Gaussian wiretap channel does not scale with the transmit power, and hence, the secure d.o.f. of the Gaussian wiretap channel with no helpers is zero. It has been known that a strictly positive secure d.o.f. can be obtained in the Gaussian wiretap channel by using a helper which sends structured cooperative signals. We show that the exact secure d.o.f. of the Gaussian wiretap channel with a helper is 1/2. Our achievable scheme is based on real interference alignment and cooperative jamming, which renders the message signal and the cooperative jamming signal separable at the legitimate receiver, but aligns them perfectly at the eavesdropper preventing any reliable decoding of the message signal. Our converse is based on two key lemmas. The first lemma quantifies the secrecy penalty by showing that the net effect of an eavesdropper on the system is that it eliminates one of the independent channel inputs. The second lemma quantifies the role of a helper by developing a direct relationship between the cooperative jamming signal of a helper and the message rate. We extend this result to the case of M helpers, and show that the exact secure d.o.f. in this case is M/M+1.
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    ABSTRACT: In this paper, we propose two opportunistic jammer selection schemes for secure communications aided by jammers using the concept of interference alignment. Because jamming signals are interference to both a legitimate receiver and a eavesdropper, the legitimate receiver selects two jammers whose jamming signals are the most aligned in a small dimensional subspace. The alignment is measured by either interference-to-noise ratio (INR) or chordal distance for the opportunistic jammer selection. We find the achievable secure degrees of freedom (DoF) by the proposed jammer selection schemes when the number of jammers goes to infinity and find the computational complexity of each scheme.
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