We study the two-user Gaussian multiple-input multiple-output (MIMO) broadcast channel with common and confidential messages. In this channel, the transmitter sends a common message to both users, and a confidential message to each user which needs to be kept perfectly secret from the other user. We obtain the entire capacity region of this channel. We also explore the connections between the capacity region we obtain for the Gaussian MIMO broadcast channel with common and confidential messages and the capacity region of its non-confidential counterpart, i.e., the Gaussian MIMO broadcast channel with common and private messages, which is not known completely.
We consider the two-user Gaussian multiple-input multiple-output (MIMO) broadcast channel, where each link between the transmitter and each user is modelled by a linear additive Gaussian channel. We study the two-user Gaussian MIMO broadcast channel for the following scenario: The transmitter sends a common message to both users, and a confidential message to each user which needs to be kept perfectly secret from the other user. We call the channel model arising from this scenario the Gaussian MIMO broadcast channel with common and confidential messages.
The Gaussian MIMO broadcast channel with common and confidential messages subsumes several other channel models as special cases. The first one is the Gaussian MIMO wiretap channel, where the transmitter has only one confidential message for one (legitimate) user, which is kept perfectly secret from the other user (eavesdropper). The secrecy capacity of the Gaussian MIMO wiretap channel is obtained in [1,2] for the general case, in [3] for the 2-2-1 case. The second channel model that the Gaussian MIMO broadcast channel with common and confidential messages subsumes is the Gaussian MIMO wiretap channel with common message [4], in which the transmitter sends a common message to both the legitimate user and the eavesdropper, and a confidential message to the legitimate user that is kept perfectly secret from the eavesdropper. The capacity region of the Gaussian MIMO wiretap channel with common message is obtained in [4]. The third channel model that the Gaussian MIMO broadcast channel with common and confidential messages encompasses is the Gaussian MIMO broadcast channel with confidential messages [5], where the transmitter sends a confidential message to each user which is kept perfectly secret from the other user. The capacity region of the Gaussian MIMO broadcast channel with confidential messages is established in [5].
Here, we obtain the capacity region of the Gaussian MIMO broadcast channel with common and confidential messages. In particular, we show that a variant of the secret dirtypaper coding (S-DPC) scheme proposed in [5] is capacity-achieving. Since the S-DPC scheme proposed in [5] is for the transmission of only two confidential messages, it is modified here to incorporate the transmission of a common message as well. Similar to [5], we also notice an invariance property of this achievable scheme with respect to the encoding order used in the S-DPC scheme. In other words, two achievable rate regions arising from two possible encoding orders used in the S-DPC scheme are identical, and equal to the capacity region. We provide the proof of this statement as well as the converse proof for the capacity region of the Gaussian MIMO broadcast channel with common and confidential messages by using the channel enhancement technique [6] and an extremal inequality from [7].
We also explore the connections between the Gaussian MIMO broadcast channel with common and confidential messages and its non-confidential counterpart, i.e., the (two-user) Gaussian MIMO broadcast channel with common and private messages. In the Gaussian MIMO broadcast channel with common and private messages, the transmitter again sends a common message to both users, and a private message to each user, for which there is no secrecy constraint now, i.e., private message of each user does not need to be kept secret from the other user. Thus, the Gaussian MIMO broadcast channel with common and confidential messages we study here can be viewed as a constrained version of the Gaussian MIMO broadcast channel with common and private messages, where the constraint comes through forcing the private messages to be confidential. We note that although there are partial results for the Gaussian MIMO broadcast channel with common and private messages [8,9], its capacity region is not known completely. However, here, we are able to obtain the entire capacity region for a constrained version of the Gaussian MIMO broadcast channel with common and private messages, i.e., for the Gaussian MIMO broadcast channel with common and confidential messages. We provide an intuitive explanation of this at-firstsight surprising point as well as the invariance property of the achievable rate region with respect to the encoding orders that can be used in the S-DPC scheme, by using a result from [9] for the Gaussian MIMO broadcast channel with common and private messages. In particular, we use the following result from [9]: For a given common message rate, the private message sum rate capacity of the Gaussian MIMO broadcast channel with common and private messages is achieved by the dirty-paper coding (DPC) scheme in [10], and any one of the two possible encoding orders that can be used in DPC gives the private message sum rate capacity. Using this result, we show that there is a one-to-one correspondence between the points on the boundary of the achievable rate region of the Gaussian MIMO broadcast channel with c
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