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arXiv:0903.3096 · IEEE Transactions on Information Theory

The Secrecy Capacity Region of the Gaussian MIMO Multi-Receiver Wiretap Channel

Ersen Ekrem, Şennur Ulukuş

Published Mar 15, 2011• 243 citations
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Abstract

In this paper, we consider the Gaussian multiple-input multiple-output (MIMO) multi-receiver wiretap channel in which a transmitter wants to have confidential communication with an arbitrary number of users in the presence of an external eavesdropper. We derive the secrecy capacity region of this channel for the most general case. We first show that even for the single-input single-output (SISO) case, existing converse techniques for the Gaussian scalar broadcast channel cannot be extended to this secrecy context, to emphasize the need for a new proof technique. Our new proof technique makes use of the relationships between the minimum-mean-square-error and the mutual information, and equivalently, the relationships between the Fisher information and the differential entropy. Using the intuition gained from the converse proof of the SISO channel, we first prove the secrecy capacity region of the degraded MIMO channel, in which all receivers have the same number of antennas, and the noise covariance matrices can be arranged according to a positive semi-definite order. We then generalize this result to the aligned case, in which all receivers have the same number of antennas; however, there is no order among the noise covariance matrices. We accomplish this task by using the channel enhancement technique. Finally, we find the secrecy capacity region of the general MIMO channel by using some limiting arguments on the secrecy capacity region of the aligned MIMO channel. We show that the capacity achieving coding scheme is a variant of dirty-paper coding with Gaussian signals.

DOI
10.1109/tit.2011.2111750

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In this paper, we consider the Gaussian multiple-input multiple-output (MIMO) multi-receiver wiretap channel in which a transmitter wants to have confidential communication with an arbitrary number of users in the presence of an external eavesdropper. We derive the secrecy capacity region of this channel for the most general case. We first show that even for the single-input single-output (SISO) case, existing converse techniques for the Gaussian scalar broadcast channel cannot be extended to this secrecy context, to emphasize the need for a new proof technique. Our new proof technique makes use of the relationships between the minimum-mean-square-error and the mutual information, and equivalently, the relationships between the Fisher information and the differential entropy. Using the intuition gained from the converse proof of the SISO channel, we first prove the secrecy capacity region of the degraded MIMO channel, in which all receivers have the same number of antennas, and the noise covariance matrices can be arranged according to a positive semi-definite order. We then generalize this result to the aligned case, in which all receivers have the same number of antennas; however, there is no order among the noise covariance matrices. We accomplish this task by using the channel enhancement technique. Finally, we find the secrecy capacity region of the general MIMO channel by using some limiting arguments on the secrecy capacity region of the aligned MIMO channel. We show that the capacity achieving coding scheme is a variant of dirty-paper coding with Gaussian signals.
We consider the Gaussian multiple-input multiple-output (MIMO) multi-receiver wiretap channel, and derive the secrecy capacity region of this channel for the most general case. We first prove the secrecy capacity region of the degraded MIMO channel, in which all receivers have the same number of antennas, and the noise covariance matrices exhibit a positive semi-definite order. We then generalize this result to the aligned case, in which all receivers have the same number of antennas, however there is no order among the noise covariance matrices. We accomplish this task by using the channel enhancement technique. Finally, we find the secrecy capacity region of the general MIMO channel by using some limiting arguments on the secrecy capacity region of the aligned MIMO channel. We show that a variant of dirty-paper coding with Gaussian signals is optimal.
We study secure communication in which a transmitter wants to send confidential messages to an arbitrary number of receivers in the presence of an external eavesdropper. We consider two classes of Gaussian channels: the Gaussian single-input single-output (SISO) multi-receiver wiretap channel and the degraded Gaussian multiple-input multiple-output (MIMO) multi-receiver wiretap channel. These two channels belong to the class of degraded multi-receiver wiretap channels. In previous literature the secrecy capacity regions of these two channels were established under the conservative joint secrecy constraint. Thus, we focus our work on the less conservative individual secrecy constraint, which is characterized by a higher throughput. We establish the individual secrecy capacity region for these two cases. The achievability follows from the individual secrecy capacity established for the discrete memoryless degraded wiretap channels where coding is performed using Gaussian signals. On the other hand, the converse is established by adapting the techniques used for formulating the joint secrecy capacity region to the individual secrecy constraint.
In this paper, we consider a scenario where a source node wishes to broadcast two confidential messages for two respective receivers via a Gaussian MIMO broadcast channel. A wire-tapper also receives the transmitted signal via another MIMO channel. First we assumed that the channels are degraded and the wire-tapper has the worst channel. We establish the capacity region of this scenario. Our achievability scheme is a combination of the superposition of Gaussian codes and randomization within the layers which we will refer to as Secret Superposition Coding. For the outerbound, we use the notion of enhanced channel to show that the secret superposition of Gaussian codes is optimal. We show that we only need to enhance the channels of the legitimate receivers, and the channel of the eavesdropper remains unchanged. Then we extend the result of the degraded case to non-degraded case. We show that the secret superposition of Gaussian codes along with successive decoding cannot work when the channels are not degraded. we develop a Secret Dirty Paper Coding (SDPC) scheme and show that SDPC is optimal for this channel. Finally, we investigate practical characterizations for the specific scenario in which the transmitter and the eavesdropper have multiple antennas, while both intended receivers have a single antenna. We characterize the secrecy capacity region in terms of generalized eigenvalues of the receivers channel and the eavesdropper channel. We refer to this configuration as the MISOME case. In high SNR we show that the capacity region is a convex closure of two rectangular regions.
In this paper, we consider a scenario where a source node wishes to broadcast two confidential messages for two respective receivers via a Gaussian multiple-input multiple-output (MIMO) broadcast channel. An eavesdropper also receives the transmitted signal via another MIMO channel. We first consider the discrete memoryless channel and obtain the capacity region of the degraded channel. The secret dirty paper coding (SDPC) region as an achievable rate region for the general discrete channel is introduced. Relying on the results for the discrete channel, we fully characterize the secrecy capacity region of MIMO broadcast channel. It is shown that the SDPC scheme is optimal. The converse part of the proof relies on the generalized Costa's entropy power inequality and a new channel enhancement strategy in which we only need to enhance the channels of the legitimate receivers, and the channel of the eavesdropper remains unchanged.
We study the multi-receiver wiretap channel with public and confidential messages. In this channel, there is a transmitter that wishes to communicate with two legitimate users in the presence of an external eavesdropper. The transmitter sends a pair of public and confidential messages to each legitimate user. While there are no secrecy constraints on the public messages, confidential messages need to be transmitted in perfect secrecy. We study the discrete memoryless multi-receiver wiretap channel as well as its Gaussian multi-input multi-output (MIMO) instance. First, we consider the degraded discrete memoryless channel, and obtain an inner bound for the capacity region by using an achievable scheme that uses superposition coding and binning. Next, we obtain an outer bound, and show that this outer bound partially matches the inner bound, providing a partial characterization for the capacity region of the degraded channel model. Second, we obtain an inner bound for the general, not necessarily degraded, discrete memoryless channel by using Marton's inner bound, superposition coding, rate-splitting and binning. Third, we consider the degraded Gaussian MIMO channel, and show that, to evaluate both the inner and outer bounds, considering only jointly Gaussian auxiliary random variables and channel input is sufficient. Since the inner and outer bounds partially match, these sufficiency results provide a partial characterization of the capacity region of the degraded Gaussian MIMO channel. Finally, we provide an inner bound for the capacity region of the general, not necessarily degraded, Gaussian MIMO multi-receiver wiretap channel.
This paper considers the problem of proof for the secrecy capacity region of the Gaussian wiretap channel. In this channel, the transmitter sends confidential messages to the legitimate receivers, which need to be kept asymptotically perfectly secret from the eavesdropper. We characterize the secrecy capacity region of this channel and give a converse proof by using the fundamental relationship between information theory and the estimation theory in the Gaussian channel, relating the derivative of the mutual information to the minimum mean-square-error (MMSE). Our proof is different from the superposition coding and the random binning. We obtain that using the relationship between MMSE and mutual information one can provide insightful and simple proofs for multiuser single antenna problems and the secrecy capacity problem, such as broadcast channel.