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Interference Alignment for the Multiantenna

Compound Wiretap Channel, Ashish J. Khisti

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

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.

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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 a wiretap channel model where the sender has$M transmitantennasandtherearetwogroupsconsistingoftransmit antennas and there are two groups consisting oftransmitantennasandtherearetwogroupsconsistingofJ_{1}andandandJ_{2}$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.
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 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.
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.
This paper considers the compound wiretap channel, which generalizes Wyner's wiretap model to allow the channels to the (legitimate) receiver and to the eavesdropper to take a number of possible states. No matter which states occur, the transmitter guarantees that the receiver decodes its message and that the eavesdropper is kept in full ignorance about the message. The compound wiretap channel can also be viewed as a multicast channel with multiple eavesdroppers, in which the transmitter sends information to all receivers and keeps the information secret from all eavesdroppers. For the discrete memoryless channel, lower and upper bounds on the secrecy capacity are derived. The secrecy capacity is established for the degraded channel and the semideterministic channel with one receiver. The parallel Gaussian channel is further studied. The secrecy capacity and the secrecy degree of freedom ( ) are derived for the degraded case with one receiver. Schemes to achieve the for the case with two receivers and two eavesdroppers are constructed to demonstrate the necessity of a prefix channel in encoder design. Finally, the multi-antenna (i.e., MIMO) compound wiretap channel is studied. The secrecy capacity is established for the degraded case and an achievable is given for the general case.