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arXiv:0809.4804 · Allerton Conference on Communication, Control, and Computing

On the secure degrees of freedom of wireless X networks

Tiangao Gou, Syed A. Jafar

Published Sep 1, 2008• 37 citations
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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 × 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.

DOI
10.1109/allerton.2008.4797643

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Asilomar Conference on Signals, Systems and Computers (ACSSC)•Nov 1, 2015•10 citations
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 × 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.
This paper studies the K-user Gaussian interference channel with secrecy constraints. Two distinct network models, namely the interference channel with confidential messages and the one with an external eavesdropper, are analyzed. Using interference alignment along with secrecy pre-coding at each transmitter, it is shown that each user in the network can achieve non-zero secure Degrees of Freedoms (DoFs) in both scenarios. In particular, the proposed coding scheme achieves K−2/2K−2 secure DoFs for each user in the interference channel with confidential messages model, and K−2/2K secure DoFs in the case of an external eavesdropper. The fundamental difference between the two scenarios stems from the lack of channel state information (CSI) about the external eavesdropper. Remarkably, the results establish the positive impact of interference on the secrecy capacity of wireless networks.
The problem of transmitting confidential messages in M x K wireless X networks is considered in which each transmitter intends to send one confidential message to every receiver. In particular, the secure degrees of freedom (SDOF) of the considered network are studied based on an artificial noise alignment (ANA) approach, which integrates interference alignment and artificial noise transmission. At first, an SDOF upper bound is derived for the M x K X network with confidential messages (XNCM) to be K(M-1)/K+M-2. By proposing an ANA approach, it is shown that the SDOF upper bound is tight when K = 2 for the considered XNCM with time-/frequency-varying channels. For K ≥ 3, it is shown that SDOF of K(M-1)/K+M-1 can be achieved, even when an external eavesdropper is present. The key idea of the proposed scheme is to inject artificial noise into the network, which can be aligned in the interference space at receivers for confidentiality. Moreover, for the network with no channel state information at transmitters, a blind ANA scheme is proposed to achieve SDOF of K(M-1)/K+M-1 for K, M ≥ 2, with reconfigurable antennas at receivers. The proposed method provides a linear approach to secrecy coding and interference alignment.
This dissertation studies the security of wireless interference networks from an information-theoretic point of view. In this setting, several transmitter-receiver pairs wish to have secure communication against the eavesdropper(s). The central goal of this dissertation is to develop a framework based on information-theoretic principles to determine the complete solutions for the signaling schemes in different wireless interference networks with large transmit powers, and derive the corresponding fundamental limits in terms of the secure degrees of freedom (s.d.o.f.). Next, we study one-hop wireless networks by considering four fundamental wireless network structures: Gaussian wiretap channel with helpers, Gaussian broadcast channel (BC) with confidential messages, Gaussian interference channel (IC) with confidential messages, and Gaussian multiple access (MAC) wiretap channel. The secrecy capacity of the canonical Gaussian wiretap channel does not scale with the transmit power, and hence, the s.d.o.f. of the Gaussian wiretap channel with no helpers is zero. We show that the exact s.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 s.d.o.f. in this case is M/(M+1). We then generalize this approach to more general network structures with multiple messages. We show that the sum s.d.o.f. of the Gaussian BC with confidential messages and M helpers is 1, the sum s.d.o.f. of the two-user IC with confidential messages is 2/3, the sum s.d.o.f. of the two-user IC with confidential messages and M helpers is 1, and the sum s.d.o.f. of the K-user MAC wiretap channel is K(K-1)/(K(K-1)+1). Then, we study the sum s.d.o.f. of multi-receiver networks. In this dissertation, we determine the exact sum s.d.o.f. of the K-user Gaussian IC. We consider three different secrecy constraints: 1) K-user IC with one external eavesdropper (IC-EE), 2) K-user IC with confidential messages (IC-CM), and 3) K-user IC with confidential messages and one external eavesdropper (IC-CM-EE). We show that for all of these three cases, the exact sum secure d.o.f. is K(K-1)/(2K-1). We show converses for IC-EE and IC-CM, which imply a converse for IC-CM-EE. We show achievability for IC-CM-EE, which implies achievability for IC-EE and IC-CM. We develop the converses by relating the channel inputs of interfering users to the reliable rates of the interfered users, and by quantifying the secrecy penalty in terms of the eavesdroppers' observations. Our achievability uses structured signaling, structured cooperative jamming, channel prefixing, and asymptotic real interference alignment. While the traditional interference alignment provides some amount of secrecy by mixing unintended signals in a smaller sub-space at every receiver, in order to attain the optimum sum s.d.o.f., we incorporate structured cooperative jamming into the achievable scheme, and intricately design the structure of all of the transmitted signals jointly. Next, we study the entire s.d.o.f. regions of multi-user network structures. In this dissertation, we determine the entire s.d.o.f. regions of the K-user MAC wiretap channel and the K-user IC with secrecy constraints. The converse for the MAC follows from a middle step in the converse of the sum s.d.o.f. 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. As a byproduct of determining the entire s.d.o.f. regions of MAC and IC channels, we show 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. Then, we determine the sum s.d.o.f. of two-unicast layered wireless networks. Without any secrecy constraints, the sum d.o.f. of this class of networks was shown to take only one of three possible values: 1, 3/2 and 2, for all network configurations. We consider the setting where, in addition to being reliably transmitted, each message is required to be kept information-theoretically secure from the unintended receiver. We show that the sum s.d.o.f. can only take one of five possible values: 0, 2/3, 1, 3/2,2, for all network configurations. To determine the sum s.d.o.f., we divide the class of two-unicast layered networks into several sub-classes, and propose an achievable scheme based on the specific structure of the networks in each sub-class. Our achievable schemes are based on real interference alignment, cooperative jamming, interference neutralization and cooperative jamming neutralization techniques. Next, we consider the Gaussian wiretap channel with M helpers, where no eavesdropper channel state information (CSI) is available at the legitimate entities. One of the key ingredients of our optimal achievable scheme with perfect CSI is to align cooperative jamming signals with the information symbols at the eavesdropper to limit the information leakage rate. This requires perfect eavesdropper CSI at the transmitters. We propose a new achievable scheme in which cooperative jamming signals span the entire space of the eavesdropper, but are not exactly aligned with the information symbols. We show that this scheme achieves the same s.d.o.f. of M/(M+1) but does not require any eavesdropper CSI; the transmitters blindly cooperative jam the eavesdropper. Then, we study the separability of the parallel MAC wiretap channel. Separability, when exists, is useful as it enables us to code separately over parallel channels, and still achieve the optimum overall performance. It is well-known that the parallel single-user channel, parallel MAC and parallel BC are all separable, however, the parallel IC is not separable in general. In this dissertation, we show that, while MAC is separable MAC wiretap channel is not separable in general. We prove this via a specific linear deterministic MAC wiretap channel. We then show that even the Gaussian MAC wiretap channel is inseparable in general. Finally, we show that, when the channel gains are drawn from continuous distributions, and when the s.d.o.f. region is considered, then the Gaussian MAC wiretap channel is almost surely separable. Finally, we study the two-user one-sided IC with confidential messages. In this IC, in addition to the usual selfishness of the users, the relationship between the users is adversarial in the sense of both receivers' desires to eavesdrop on the communication of the other pair. We develop a game-theoretic model for this setting. We start with a model where each pair's payoff is their own secrecy rate. We then propose a refinement for the payoff function by explicitly accounting for the desire of the receiver to eavesdrop on the other party's communication. This payoff function captures the adversarial relationship between the users better. We determine the Nash equilibria for the binary deterministic channel for both payoff functions.
The problem of transmitting confidential messages in the M × K wireless X network is considered, in which each transmitter intends to send one confidential message to every receiver. In particular, the secrecy degrees of freedom (SDOF) of the considered network are studied by an artificial noise alignment (ANA) approach, which integrates interference alignment and artificial noise transmission. At first, an SDOF upper bound is derived for the M × K X network with confidential messages (XNCM) K(M-1)/K+M-2 to be equation. By proposing an ANA approach, it is shown that the SDOF upper bound is tight when either K = 2 or M = 2 for the considered XNCM with time/frequency varying channels. For K, M ≥ 3, it is shown that an SDOF of K(M-1)/K+M1 equation can be achieved, even when an external eavesdropper appears. The key idea of the proposed scheme is to inject artificial noise into the network, which can be aligned in the interference space at receivers for confidentiality. The proposed method provides a linear approach for secure interference alignment.
The achievable rate region and degrees of freedom are studied for interference channels with confidential messages under strong secrecy constraints. The problem is studied based on the framework of Han and Verdacutetextrmu\\acute{\\textrm{u}}acutetextrmu's channel resolvability theory. It is shown that if the random binning rate for securing a confidential message is slightly above the resolution of its corresponding wiretapped channel, strong secrecy can be guaranteed. At first, the information-spectrum method introduced by Han and Verdacutetextrmu\\acute{\\textrm{u}}acutetextrmu is generalized to an arbitrary interference channel to obtain a direct channel resolvability result. After specifying the channel transition probability, the derived resolvability results are shown to be the achievable strong secrecy rates for the stationary and memoryless interference channel. As an application, the secure degrees of freedom (SDOF) for the KKK-user Gaussian interference channel with confidential messages are investigated under strong secrecy constraints. It is shown that the optimal SDOF of the considered network is fracK−12K−1\\frac{K-1}{2K-1}fracK−12K−1 for each user, which equals to its counterpart with weak secrecy constraints. The optimal SDOF scheme is achieved by combining the resolvability-based secrecy coding and artificial noise alignment on the real line. A suboptimal ergodic secure alignment is also proposed for the time-varying interference channels to achieve a sum SDOF of fracK−32K−2\\frac{K-3}{2K-2}fracK−32K−2 for each user. Following the footsteps of Hayashi and of Bloch and Laneman, our results provide further evidence that channel resolvability can be a powerful and general framework for strong secrecy analysis in multiuser networks.
We consider the two-user multiple-input multiple- output (MIMO) interference channel with confidential messages (ICCM). We determine the exact sum secure degrees of freedom (s.d.o.f.) for the symmetric case of M antennas at both transmitters and N antennas at both receivers. We develop the converse by combining the broadcast channel with confidential messages (BCCM) cooperative upper bound, decodability upper bound for the IC with no secrecy constraints, and extensions of the secrecy penalty and role of a helper lemmas. We propose a novel achievable scheme for the 2 × 2 ICCM, which combines asymptotic real interference alignment with spatial interference alignment. Using this scheme, we provide achievable schemes for any M and N by proper vector space operations.