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MIMO Broadcast Channel with Arbitrarily Varying Eavesdropper Channel: Secrecy Degrees of Freedom

Xiang He, Ashish J. Khisti, Aylin Yener

Published Dec 1, 2011• 19 citations
Publisher

Abstract

A two-receiver MIMO broadcast-wiretap channel is considered where the channel state of the eavesdropper is arbitrarily varying. It is assumed that the eavesdropper knows this channel state perfectly whereas the legitimate nodes have no knowledge of it. It is further assumed that the eavesdropper experiences no additive noise. The channel between the transmitter and the two legitimate receivers is a constant MIMO Gaussian broadcast channel. This paper establishes the secrecy degrees of freedom region for transmitting a common-confidential message as well as a private- confidential message to each receiver. It is observed that a straightforward extension of single user random binning does not achieve the optimal secrecy degrees of freedom (s.d.o.f.) region. The proposed coding scheme that achieves the s.d.o.f. region involves simultaneous diagonalization of the channel matrices of the two legitimate receivers using the generalized singular value decomposition (GSVD) as well as a particular \emph{structured binning} across codebooks that minimizes the rate of the fictitious message. While the focus is on achieving weak secrecy for ease of exposition, an outline is provided on how the results can be extended for achieving strong secrecy.

DOI
10.1109/glocom.2011.6134247

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IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC)•Sep 1, 2010•1 citation
A two-receiver MIMO broadcast-wiretap channel is considered where the channel state of the eavesdropper is arbitrarily varying. It is assumed that the eavesdropper knows this channel state perfectly whereas the legitimate nodes have no knowledge of it. It is further assumed that the eavesdropper experiences no additive noise. The channel between the transmitter and the two legitimate receivers is a constant MIMO Gaussian broadcast channel. This paper establishes the secrecy degrees of freedom region for transmitting a common-confidential message as well as a private- confidential message to each receiver. It is observed that a straightforward extension of single user random binning does not achieve the optimal secrecy degrees of freedom (s.d.o.f.) region. The proposed coding scheme that achieves the s.d.o.f. region involves simultaneous diagonalization of the channel matrices of the two legitimate receivers using the generalized singular value decomposition (GSVD) as well as a particular \emph{structured binning} across codebooks that minimizes the rate of the fictitious message. While the focus is on achieving weak secrecy for ease of exposition, an outline is provided on how the results can be extended for achieving strong secrecy.
We study a multi-antenna broadcast channel with two legitimate receivers and an external eavesdropper. We assume that the channel matrix of the eavesdropper is unknown to the legitimate terminals but satisfies a maximum rank constraint. As our main result we characterize the associated secrecy degrees of freedom for the broadcast channel with common and private messages. We show that a direct extension of the single-user wiretap codebook does not achieve the secrecy degrees of freedom. Our proposed optimal scheme involves decomposing the signal space into a common subspace, which can be observed by both receivers, and private subspaces which can be observed by only one of the receivers, and carefully transmitting a subset of messages in each subspace. We also consider the case when each user's private message must additionally remain confidential from the other legitimate receiver and characterize the s.d.o.f. region in this case.
A two-transmitter Gaussian multiple access wiretap channel with multiple antennas at each of the nodes is investigated. The channel matrices of the legitimate users are fixed and revealed to all the terminals, whereas the channel matrices of the eavesdropper are arbitrarily varying and only known to the eavesdropper. The secrecy degrees of freedom (s.d.o.f.) region under a strong secrecy constraint is characterized. A transmission scheme that orthogonalizes the transmit signals of the two users at the intended receiver, and uses a single-user wiretap code for each user, is shown to achieve the s.d.o.f. region. The converse involves establishing an upper bound on a weighted-sum-rate expression. This is accomplished by using induction, where at each step one combines the secrecy and multiple-access constraints associated with an adversary eavesdropping a carefully selected group of sub-channels.
In this work, a class of information theoretic secrecy problems is addressed where the eavesdropper channel states are completely unknown to the legitimate parties. In particular, MIMO wiretap channel models are considered where the channel of the eavesdropper is arbitrarily varying over time. Assuming that the number of antennas of the eavesdropper is limited, the secrecy rate of the MIMO wiretap channel in the sense of strong secrecy is derived, and shown to match with the converse in secure degrees of freedom. It is proved that there exists a universal coding scheme that secures the confidential message against any sequence of channel states experienced by the eavesdropper. This yields the conclusion that secure communication is possible regardless of the location or channel states of (potentially infinite number of) eavesdroppers. Additionally, it is observed that, the present setting renders the secrecy capacity problems for multi-terminal wiretap-type channels more tractable as compared the case with full or partial knowledge of eavesdropper channel states. To demonstrate this observation, secure degrees of freedom regions are derived for the Gaussian MIMO multiple access wiretap channel (MIMO MAC-WT) and the Gaussian MIMO broadcast wiretap channel (MIMO BC-WT) where the transmitter(s) and the intended receiver(s) have the same number of antennas.
In this paper, a class of information theoretic secrecy problems is addressed where the eavesdropper channel state is completely unknown to the legitimate parties. In particular, a Gaussian MIMO wiretap channel is considered, where the eavesdropper channel state can vary from one channel use to the next, and the overall channel state sequence is known only to the eavesdropper. When the eavesdropper has fewer antennas than the transmitter and its intended receiver, a positive secrecy rate in the sense of strong secrecy is proved to be achievable and shown to match with the converse in secure degrees of freedom. This yields the conclusion that secure communication is possible regardless of the location or channel states of the eavesdropper. Additionally, it is observed that, the present setting renders the secrecy capacity problems for some multiterminal wiretap-type channels more tractable as compared to the case with full or partial knowledge of eavesdropper channel states. To demonstrate this observation, secure degrees of freedom regions are derived for the Gaussian MIMO multiple access (MAC) wiretap channel and the two-user Gaussian MIMO broadcast (BC) wiretap channel, where the transmitter(s) and intended receiver(s) have the same number of antennas.
We investigate a two-transmitter Gaussian multiple access wiretap channel with multiple antennas at each of the nodes. The channel transfer matrices at the legitimate terminals are fixed and revealed to all the terminals, whereas the channel transfer matrix of the eavesdropper is arbitrarily varying and only revealed to the eavesdropper. We characterize the secrecy degrees of freedom (s.d.o.f.) region under a strong secrecy constraint. A transmission scheme that orthogonalizes the transmit signals of the two users at the intended receiver and uses a single-user wiretap code is sufficient to achieve the s.d.o.f. region. The converse involves establishing an upper bound on a weighted-sum-rate expression. This is accomplished by using an induction procedure, where at each step we combine the secrecy and multiple-access constraints associated with an adversary eavesdropping a carefully selected group of sub-channels.
We study the Gaussian multiple-input multiple-output (MIMO) wiretap channel, which consists of a transmitter, a legitimate user, and an eavesdropper. In this channel, the transmitter sends a common message to both the legitimate user and the eavesdropper. Moreover, the legitimate user receives a private message, which is desired to be kept hidden as much as possible from the eavesdropper. We obtain the entire capacity-equivocation region of the Gaussian MIMO wiretap channel. This region contains all achievable common message, private message, and private message's equivocation rates. This capacity-equivocation region is equal to the capacity region of a Gaussian MIMO broadcast channel where the transmitter sends a common message to both the legitimate user and the eavesdropper, a public message to the legitimate user on which there is no secrecy constraint, and a confidential message to the legitimate user which needs to kept perfectly secret from the eavesdropper.