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Allerton Conference on Communication, Control, and Computing

Real interference alignment for the K-user Gaussian interference compound wiretap channel

Jianwei Xie, Şennur Ulukuş

Published Sep 1, 2010• 14 citations
Publisher

Abstract

We study the K-user Gaussian interference wiretap channel with N external eavesdroppers. All the transmitters, receivers and eavesdroppers have a single antenna each. We propose an achievable scheme to lower bound the secure degrees of freedom (d.o.f.) for each transmitter-receiver pair. Our approach is based on the (real) interference alignment technique. Our achievable scheme not only aligns the interference at each receiver to prevent the d.o.f. from vanishing, but also aligns the signals observed by the eavesdroppers to reduce the secrecy penalty. The achievable secure d.o.f. of each transmitter-receiver pair is shown to be equations for almost all channel gains.

DOI
10.1109/allerton.2010.5707057

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We study the K-user Gaussian interference wiretap channel with N external eavesdroppers. All the transmitters, receivers and eavesdroppers have a single antenna each. We propose an achievable scheme to lower bound the secure degrees of freedom (d.o.f.) for each transmitter-receiver pair. Our approach is based on the (real) interference alignment technique. Our achievable scheme not only aligns the interference at each receiver to prevent the d.o.f. from vanishing, but also aligns the signals observed by the eavesdroppers to reduce the secrecy penalty. The achievable secure d.o.f. of each transmitter-receiver pair is shown to be equations for almost all channel gains.
In this work, a Gaussian two-user MIMO interference channel is considered in the presence of an external eavesdropper whose channel is completely unknown to the legitimate communication parties and can be varying in an arbitrary fashion from one channel use to the next. We improve our recent result by deriving a larger achievable secrecy degrees of freedom region. In the achievable scheme, the transmitter injects artificial noise to confuse the eavesdropper, and at the same time aligns the injected noise with the interference from the other user at the intended receiver. The achieved secure degrees of freedom are shown to be closely connected to the rank of the effective channel matrix through which the eavesdropper observes the artificial noise. The precoding matrix is then designed to ensure that this rank cannot be reduced under any possible Eve's channel state.
We consider a two-user multiple-input multiple-output multiple access wiretap channel with N antennas at each transmitter, N antennas at the legitimate receiver, and K antennas at the eavesdropper. We determine the optimal sum secure degrees of freedom (s.d.o.f.) for this model for all values of N and K. We subdivide our problem into several regimes based on the values of N and K, and provide achievable schemes based on vector space alignment and real alignment techniques for fixed and fading channel gains. To prove the optimality of the achievable schemes, we provide matching converses for each regime. Our results show how the number of eavesdropper antennas affects the optimal sum s.d.o.f. of the multiple access wiretap channel.
We revisit the recent secure degrees of freedom (s.d.o.f.) results for one-hop multiuser wireless networks by considering three fundamental wireless network structures: Gaussian wiretap channel with helpers, Gaussian multiple access wiretap channel, and Gaussian interference channel with secrecy constraints. We present main enabling tools and resulting communication schemes in an expository manner, along with key insights and design principles emerging from them. The main achievable schemes are based on real interference alignment, channel prefixing via cooperative jamming, and structured signalling. Real interference alignment enables aligning the cooperative jamming signals together with the message carrying signals at the eavesdroppers to protect them akin to one-time-pad protecting messages in wired systems. Real interference alignment also enables decodability at the legitimate receivers by rendering message carrying and cooperative jamming signals separable, and simultaneously aligning the cooperative jamming signals in the smallest possible subspace. The main converse techniques are based on two key lemmas which quantify 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; and the role of a helper by developing a direct relationship between the cooperative jamming signal of a helper and the message rate. These two lemmas when applied according to the unique structure of individual networks provide tight converses. Finally, we present a blind cooperative jamming scheme for the helper network with no eavesdropper channel state information at the transmitters that achieves the same optimal s.d.o.f. as in the case of full eavesdropper channel state information.
We study a wiretap channel model where the sender has MMM transmit antennas and there are two groups consisting of J1J_1J1​ and J2J_2J2​ receivers respectively. Each receiver has a single antenna. We consider two scenarios. First we consider the compound wiretap model -- group 1 constitutes the set of legitimate receivers, all interested in a common message, whereas group 2 is the set of eavesdroppers. We establish new lower and upper bounds on the secure degrees of freedom. Our lower bound is based on the recently proposed \emph{real interference alignment} scheme. The upper bound provides the first known example which illustrates that the \emph{pairwise upper bound} used in earlier works is not tight. The second scenario we study is the compound private broadcast channel. Each group is interested in a message that must be protected from the other group. Upper and lower bounds on the degrees of freedom are developed by extending the results on the compound wiretap channel.
We study the secure degrees of freedom (d.o.f.) of one-hop wireless networks by considering four fundamental wireless network structures: 1) Gaussian wiretap channel; 2) Gaussian broadcast channel with confidential messages; 3) Gaussian interference channel with confidential messages; and 4) Gaussian multiple access wiretap channel. 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.We then generalize this approach to more general network structures with multiple messages. We show that the sum secure d.o.f. of the Gaussian broadcast channel with confidential messages and M helpers is 1, the sum secure d.o.f. of the twouser interference channel with confidential messages is 2/3, the sum secure d.o.f. of the two-user interference channel with confidential messages and M helpers is 1, and the sum secure d.o.f. of the K-user multiple access wiretap channel is K(K-1)/K(K-1)+1.