**1. Introduction**

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Wireless multi-user MIMO communications are used more and more often to exchange sensitive data. Because of the broadcast nature of the physical medium, unauthorized receivers located within the transmission range can observe the signals sent by the transmitter to a legitimate receiver and eavesdrop them. Therefore, security has become an extremely important issue to deal with. Multiuser MIMO communications are particularly sensitive to the problem of security, because each confidential message must be kept secret not only from external nodes, but also from all the users other than the intended one.

Traditionally, wireless security is ensured by network-layer cryptography techniques. However, these techniques may not be suitable in the case of large dynamic wireless networks, since they raise issues like key distribution and management (for symmetric cryptosystems), and high computational complexity (for asymmetric cryptosystems). Moreover, these schemes are potentially vulnerable, since they rely on the limited resources of the eavesdropper and on the unproven assumption that certain encryption algorithms are hard to invert. Methods exploiting the randomness inherent in noisy channels, known as physical layer security, have been proposed to enhance the protection of transmitted data and achieve perfect secrecy [1, 2]. Physical-layer security allows secret communications over a wireless channel without requiring an encryption key, and it works by limiting the amount of information that can be extracted at the physical level by an unintended receiver. This is performed by designing appropriate coding and precoding schemes, and by exploiting the channel state information available at the network nodes [3].

Physical layer security for communications was proposed in the 1970's by Wyner [4], who studied a three-terminal network consisting of a transmitter, an intended user and an eavesdropper, known as the wiretap channel. For this network, the secrecy capacity was defined as the maximum rate at which a message can be transmitted reliably to the intended

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user while the rate of information leakage to the eavesdropper vanishes asymptotically with the code length. For the case when the eavesdropper's channel is a degraded version of the intended user's channel, Wyner showed that it is possible to have secret communication without using an encryption key. This can be achieved by a randomized coding scheme where the information is hidden in the additional noise seen by the eavesdropper. Each message is mapped to many codewords, thus inducing maximal equivocation at the eavesdropper. Csizar and Korner generalized Wyner's work by considering a nondegraded version of the wiretap channel [5].

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Physical Layer Security for Multiuser MIMO Communications

the first receiver but needs to be kept secret from the second receiver, and viceversa. This scenario was studied in [15] for the multiple-input single-output (MISO) Gaussian case and in [16] for general MIMO Gaussian case. In this case it was shown that both confidential messages can be simultaneously transmitted at their respective maximum secrecy rates, and

Let us cosider now a larger multiuser network with more than two malicious users. For this network, it is required that the base station (BS) securely transmits each confidential message, ensuring that none of the other unintended users receive any information. Since in general the behavior of the users cannot be determined by the transmitter, a conservative worst-case scenario can be assumed for each user, where all the remaining users can cooperate to jointly eavesdrop. In this case, for each user, the alliance of the cooperating eavesdropper

The MISO BCC with a generic number of malicious receivers was studied in [17, 18], and it consists of a BS with *M* antennas that simultaneously transmit independent confidential messages to *K* spatially dispersed single-antenna users, which can cooperate and eavesdrop on each other. Although determining the secrecy capacity region for the generic MISO BCC is still an open problem, suboptimal transmission schemes have been proposed to achieve high secrecy sum-rates by controlling the amount of crosstalk between the users [19]. These schemes are based on linear precoding, and unlike dirty-paper coding, their low complexity makes them suitable for practical implementation. In the following sections, we present some

new results on the secrecy sum-rates achieved by multiuser MIMO linear precoding.

**3. Physical layer security with multi-user MIMO linear precoding**

Let the transmitted signal be denoted by **x**, then the received signal is given by

Although suboptimal, linear precoding schemes are of particular interest because of their low-complexity implementations and because they can control the amount of crosstalk between the users to maintain a high sum-rate in the broadcast channel [20–27]. In the MISO BCC, linear precoding can be employed to control the amount of interference and information leakage to the unintended receivers introduced by the transmission of each

where **<sup>H</sup>** <sup>=</sup> [**h**1,..., **<sup>h</sup>***K*] is the *<sup>K</sup>* <sup>×</sup> *<sup>M</sup>* channel matrix, **<sup>h</sup>***<sup>k</sup>* is the *<sup>k</sup>*-th column of **<sup>H</sup>** and it represents the channel between the BS and the *k*-th user, and **n** is complex Gaussian noise. In linear precoding, the transmitted vector **x** is derived from the vector containing

(precoding) [22–25]. Let **<sup>W</sup>** <sup>=</sup> [**w**1,..., **<sup>w</sup>***K*] be the *<sup>M</sup>* <sup>×</sup> *<sup>K</sup>* precoding matrix, where **<sup>w</sup>***<sup>k</sup>* is the

*K* ∑ *k*=1

**x** = **Wu** =

**y** = **Hx** + **n** (1)

*<sup>T</sup>* through a deterministic linear transformation

**w***kuk*. (2)

the achievability was obtained using the dirty-paper coding.

is equivalent to a single multi-antenna eavesdropper.

confidential message [17–19].

the confidential messages **u** = [*u*1,..., *uK*]

*k*-th column of **W**. Then the transmitted signal is

Physical layer security was then applied to Gaussian channels [6], and it was observed that a secret transmission can be achieved only if the channel at the eavesdropper is noisier than the channel at the intended user. The presence of slow fading was shown to significantly change the situation, since it allows the transmitter to employ a variable-rate transmission, thus achieving secrecy even when the eavesdropper's channel is better than the intended receiver's channel on average [7]. Also the use of multiple antennas can enhance the secrecy capability, because it enables the transmitter to beamform in a direction as orthogonal to the eavesdropper and as close to the intended user as possible [8–10]. Even when the channel at the eavesdropper is unknown by the transmitter, artificial noise can be transmitted to degrade the eavesdropper's channel and thus reduce its signal-to-noise ratio, while being harmless to the intended receiver [11–13].

More recently, physical layer security has also been extended to multiuser MIMO channels. In this chapter, we will survey the research in the field of physical layer security for multiuser MIMO communications, especially focusing on the case when multiple malicious users are present in the network, and they can eavesdrop on each other. For these complex scenarios, we will present some suboptimal low-complexity transmission schemes, discuss their performance and quantify the sum-rate penalties imposed by the secrecy requirements and by the presence of multiple users. We will discuss the challenges that arise in networks with a large number of malicious receivers, we will identify potential ways to deal with these challenges, and present an outlook on future directions for research.
