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IEEE Transactions on Information Theory

Interference Alignment for the Multiantenna Compound Wiretap Channel

Ashish J. Khisti

Published Apr 21, 2011• 78 citations
Publisher

Abstract

We study a wiretap channel model where the sender has$M $transmit antennas and there are two groups consisting of$J_{1}$and$J_{2}$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 (d.o.f.). Our lower bound is based on the recently proposed real interference alignment scheme. The upper bound provides the first known example which illustrates that the 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 d.o.f. are developed by extending the results on the compound wiretap channel.

DOI
10.1109/tit.2011.2120130

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We study a wiretap channel model where the sender hasMM Mtransmit antennas and there are two groups consisting ofJ1J_{1}J1​andJ2J_{2}J2​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 (d.o.f.). Our lower bound is based on the recently proposed real interference alignment scheme. The upper bound provides the first known example which illustrates that the 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 d.o.f. are developed by extending the results on the compound wiretap channel.
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 a wiretap channel model where the sender has transmit antennas and there are two groups consisting of and receivers respectively. Each receiver has a single an- tenna. 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 (d.o.f.). Our lower bound is based on the recently proposed real interference alignment scheme. The upper bound provides the first known example which illustrates that the 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 pro- tected from the other group. Upper and lower bounds on the d.o.f. are developed by extending the results on the compound wiretap channel. Index Terms—Common message, compound channels, deter- ministic approximation, information theoretic security, interfer- ence alignment, multi-level coding, private broadcast channels, wiretap channel.
We study the secure degrees of freedom (d.o.f.) of the MISO compound wiretap channel. The transmitter has M antennas, whereas the legitimate receiver and the eavesdropper each have one antenna and the channel vectors take one of finitely many values. If the number of states of either the legitimate receiver or the eavesdropper channel is less than M, then then we achieve full 1 d.o.f. If however the number of states of both the legitimate receiver and the eavesdropper channel are at-least equal to M, then we establish that the d.o.f. is strictly less than 1. Our upper bound is, to our knowledge, the first bound which is strictly tighter than the ¿pairwise upper bound¿. Lower bounds that combine ideas based on time-sharing, noise transmission, signal alignment and multi-level coding schemes are also provided.
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.
We consider three different secure broadcasting scenarios: i) Broadcast channels with common and confidential messages (BCC), ii) multi-receiver wiretap channels with public and confidential messages, and iii) compound wiretap channels. The BCC is a broadcast channel with two users, where in addition to the common message sent to both users, a private message, which needs to be kept hidden as much as possible from the other user, is sent to each user. In this model, each user treats the other user as an eavesdropper. The multi-receiver wiretap channel is a broadcast channel with two legitimate users and an external eavesdropper, where the transmitter sends a pair of public and confidential messages to each legitimate user. Although there is no secrecy concern about the public messages, the confidential messages need to be kept perfectly secret from the eavesdropper. The compound wiretap channel is a compound broadcast channel with a group of legitimate users and a group of eavesdroppers. In this model, the transmitter sends a common confidential message to the legitimate users, and this confidential message needs to be kept perfectly secret from all eavesdroppers. In this paper, we provide a survey of the existing information-theoretic results for these three forms of secure broadcasting problems, with a closer look at the Gaussian multiple-input multiple-output (MIMO) channel models. We also present the existing results for the more general discrete memory-less channel models, as they are often the first step in obtaining the capacity results for the corresponding Gaussian MIMO channel models.
In this paper, we study the information-theoretical security of the K-user MIMO Y wiretap Gaaussian channels and deduce the lower bounds of the secure degrees of freedom (SDOF). Due to the different receiving links of the eavesdropper, we propose two MIMO Y wiretap models which are called Broadcast Wiretap Channel (BWC) and Multi-access Broadcast Wiretap Channel (MBWC), respectively. Each legitimate user equipped with M antennas intends to transmit independent signals via an intermediate amplify-and-forward (AF) relay with N antennas. Besides, there exists an eavesdropper with Ne antennas close to the K users (K ≥ 3). Based on the asymptotic real interference alignment and the structured cooperative jamming, the security of the MIMO Y wiretap channels can be guaranteed. From the asymptotic analysis, we can find that a lower bound of the sum SDOF in BWC is K/4. While in the MBWC model, since the relay working in the full duplex mode broadcasts cooperative jamming signals to all the other nodes especially to the eavesdropper, the lower bound of sum SDOF is K2-3K/4(K-1), which is altered along with the number of users.