**4. The cost of physical layer security in multi-user MIMO**

Guaranteeing secrecy and serving multiple (and potentially malicious) users at the same time both come at a cost in terms of the per-user transmission rate. In this section, we discuss the cost of achieving physical layer security in multiuser MIMO communications.

#### **4.1. Secrecy loss**

The cost due to the secrecy requirements, which we denote by *secrecy loss*, can be obtained by comparing the secrecy sum-rate *<sup>R</sup>*⋆◦ *<sup>s</sup>* achieved by the RCI precoder to the sum-rate *<sup>R</sup>*⋆◦ achieved by an optimized RCI precoder without secrecy requirements. The gap between *<sup>R</sup>*⋆◦ *s* and *<sup>R</sup>*⋆◦ represents how much guaranteeing secrecy costs in terms of the achievable sum-rate.

The optimal sum-rate *<sup>R</sup>*⋆◦ without secrecy requirements is obtained by using the precoder in (6), and it is given by [29]

$$R^{\star \diamond} = K \log\_2 \left[ 1 + \lg \left( \beta\_\prime \mathfrak{f}\_{\rm ns}^{\star \diamond} \right) \right],\tag{16}$$

8 Recent Trends in Multiuser MIMO Communications

for a single-user system.

**4.1. Secrecy loss**

(6), and it is given by [29]

by comparing the secrecy sum-rate *<sup>R</sup>*⋆◦

(*ρ* + 1) *β*<sup>3</sup>

served with non-zero secrecy sum-rate. Fig. 3 shows that *β*max is a decreasing function of the SNR. The value of *β*max can be found by solving the following cubic equation [19]

The value of *β*max falls between 1 and 2. This means that if *K* ≥ 2*M*, i.e. if *β* ≥ 2, then the secrecy sum-rate is zero for all SNRs. In the limit of large SNR, equation (14) reduces to

yielding to *β*max = 1. These results can be explained as follows. In the worst-case scenario, the alliance of cooperating eavesdroppers can cancel the interference, and its received SINR is the ratio between the signal leakage and the thermal noise. In the limit of large SNR, the thermal noise vanishes, and the only means for the transmitter to limit the eavesdropper's SINR is by reducing the signal leakage to zero by inverting the channel matrix. This can only be accomplished when the number of transmit antennas is larger than or equal to the number of users, hence only if *β* ≤ 1. When *β* > 1 this is not possible, and no positive secrecy sum-rate can be achieved. When *β* ≥ 2, the eavesdroppers are able to drive the secrecy sum-rate to zero irrespective of *ρ*. This is consistent with the results presented in [10]

Guaranteeing secrecy and serving multiple (and potentially malicious) users at the same time both come at a cost in terms of the per-user transmission rate. In this section, we discuss the

The cost due to the secrecy requirements, which we denote by *secrecy loss*, can be obtained

achieved by an optimized RCI precoder without secrecy requirements. The gap between *<sup>R</sup>*⋆◦

and *<sup>R</sup>*⋆◦ represents how much guaranteeing secrecy costs in terms of the achievable sum-rate. The optimal sum-rate *<sup>R</sup>*⋆◦ without secrecy requirements is obtained by using the precoder in

*<sup>R</sup>*⋆◦ <sup>=</sup> *<sup>K</sup>* log2 [<sup>1</sup> <sup>+</sup> *<sup>g</sup>* (*β*, *<sup>ξ</sup>*⋆◦

(*β*max − 1)

**4. The cost of physical layer security in multi-user MIMO**

cost of achieving physical layer security in multiuser MIMO communications.

max <sup>+</sup> <sup>3</sup>*ρβ*max <sup>−</sup> *<sup>ρ</sup>* <sup>=</sup> 0. (14)

<sup>3</sup> = 0, (15)

*<sup>s</sup>* achieved by the RCI precoder to the sum-rate *<sup>R</sup>*⋆◦

ns )] , (16)

*s*

max <sup>−</sup> (3*<sup>ρ</sup>* <sup>+</sup> <sup>2</sup>) *<sup>β</sup>*<sup>2</sup>

**Figure 3.** Asymptotic secrecy sum-rate per transmit antenna with RCI as a function of *β*. Circles denote *β*opt, squares denote *β*max.

with *<sup>ξ</sup>*⋆◦ ns <sup>=</sup> *<sup>β</sup>*/*ρ*. It is easy to show that *<sup>R</sup>*⋆◦ <sup>≥</sup> 0 for all values of *<sup>β</sup>* and *<sup>ρ</sup>*, with equality only for *<sup>ρ</sup>* = 0, and that *<sup>R</sup>*⋆◦ tends to zero as *<sup>β</sup>* → <sup>∞</sup>. Hence, there is no limit to the number of users per transmit antenna *β* that the system can accommodate with a non-zero sum-rate. However if we impose the secrecy requirements, the secrecy sum-rate *<sup>R</sup>*⋆◦ *<sup>s</sup>* is zero for *β* ≥ *β*max, with *β*max given by (14). Therefore, introducing the secrecy requirements will limit to *β*max the number of users per transmit antenna that can be served with a non-zero sum-rate.

We now compare the secrecy sum-rate *<sup>R</sup>*⋆◦ *<sup>s</sup>* to the sum-rate *<sup>R</sup>*⋆◦ in the limit of large SNR. Again by using the regularization parameter *<sup>ξ</sup>*⋆◦ ns = *<sup>β</sup>*/*<sup>ρ</sup>* we obtain the following large-SNR approximation for the secrecy sum-rate without secrecy requirements [19]

$$R^{\*\diamond} \approx \begin{cases} K \log\_2 \frac{1-\beta}{\beta} + K \log\_2 \rho & \text{for } \beta < 1\\ \frac{K}{2} \log\_2 \rho & \text{for } \beta = 1\\ K \log\_2 \frac{\beta}{\beta - 1} & \text{for } \beta > 1 \end{cases} \tag{17}$$

By comparing (17) to (13), we can draw the following conclusions regarding the large-SNR regime. If the number of transmit antennas *M* is larger than the number of users *K*, then *<sup>β</sup>* < 1, *<sup>R</sup>*⋆◦ *<sup>s</sup>* <sup>=</sup> *<sup>R</sup>*⋆◦, and the secrecy requirements do not decrease the sum-rate of the network. Therefore, by using *<sup>ξ</sup>*⋆◦ from (10) one can achieve secrecy while maintaining the same sum-rate, i.e. there is no secrecy loss. If *M* = *K*, then *β* = 1, the secrecy requirements only reduce the sum-rate by a constant value, and the scaling factor *K*/2 remains unchanged. Alternatively, one can achieve secrecy while maintaining the same sum-rate, by increasing the transmitted power by a factor 64/27 ≈ 3.75dB. If *M* < *K*, i.e. *β* > 1, then the secrecy requirements result in a value of *<sup>R</sup>*⋆◦ *<sup>s</sup>* that decreases with the SNR, as opposed to a constant sum-rate *<sup>R</sup>*⋆◦ without secrecy. Therefore if the SNR is too large, then the secrecy sum-rate becomes zero.

#### **4.2. Multiuser Loss**

The cost due the interference caused by the presence of multiple users in the system, which we denote by *multiuser loss*, is given by the gap between the per-user secrecy rate *<sup>R</sup>*⋆◦ *<sup>s</sup>* /*K* and the secrecy capacity *Cs*,SU of the single-user MISOME wiretap channel, where one user is served at a time and the remaining users can eavesdrop.

The value of *Cs*,SU was obtained in [10], and for large SNR it can be approximated by

$$\mathsf{C}\_{\mathsf{s},\mathbf{SU}} \approx \begin{cases} \log\_2 \rho & \text{for } \beta < 1\\ \frac{1}{2} \log\_2 \rho & \text{for } \beta = 1\\ \max\left\{ \log\_2 \frac{1}{(\beta - 1)}, 0 \right\} & \text{for } \beta > 1 \end{cases}, \quad \text{as } \rho \to \infty. \tag{18}$$

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**4.4. Numerical results**

parameters *<sup>ξ</sup>*⋆◦ and *<sup>ξ</sup>*⋆◦

sum-rate to zero.

0

of *β* are considered: 0.8, 1, and 1.2, corresponding to *M* = 15, 12 and 10 antennas.

2

Secrecy

secrecy rate as large as *Cs*,SU.

 sum-rate

4

6

Rs/K R/K Cs,SU

8

analysis.

Fig. 4 compares the simulated ergodic sum-rates *Rs* and *R* of the RCI precoder with and without secrecy requirements, respectively. These were obtained by using the regularization

negligible at large SNR, and secrecy can be achieved without additional costs. For *β* = 1, the two curves tend to have same slope at large SNR, but there is a residual gap between them. Therefore, secrecy can be achieved at a lower sum-rate. We note that in order to achieve secrecy without decreasing the sum-rate, the required additional power is less than 4dB at all SNRs. For *β* > 1, the sum-rate tends to saturate for large SNR, whereas the secrecy sum-rate starts decreasing. If the SNR is too large, then the secrecy requirements force the

Fig. 4 also shows the simulated secrecy capacity *Cs*,SU of the MISOME wiretap channel. For *β* ≤ 1, the RCI precoder achieves a per-user secrecy rate which has the same linear scaling factor as *Cs*,SU. When 1 < *<sup>β</sup>* < 2, *Cs*,SU saturates at high SNR, while the secrecy sum-rate decreases. All these numerical results confirm the ones obtained from the large-system

0 5 10 15 20 25

β = 1.2

β = 1

β = 0.8

ρ [dB]

**Figure 4.** Comparison between the simulated ergodic per-user secrecy rate with RCI (solid) and the two upper bounds: (i) per-user rate without secrecy requirements (dashed) and (ii) MISOME secrecy capacity (dotted), for *K* = 12 users. Three values

Fig. 5 shows the simulated per-user secrecy rate of the RCI-PA precoder from [18], with optimal power allocation. This is compared to the RCI-EP precoder. Fig. 5 also shows that the power allocation scheme reduces the sum-rate loss due to the secrecy requirements. For *ρ* ≥ 15dB, RCI with power allocation achieves a per-user secrecy rate which is even higher than the per-user rate achieved by the optimal RCI-EP without secrecy requirements. Furthermore, Fig. 5 shows the simulated secrecy capacity *Cs*,SU of a MISOME channel with the same per-message transmit power. Although *Cs*,SU is obtained in a single-user and interference-free system [10], at high SNR, RCI with power allocation achieves a per-user

ns , respectively. For *<sup>β</sup>* <sup>&</sup>lt; 1, the difference between *<sup>R</sup>* and *Rs* becomes

Physical Layer Security for Multiuser MIMO Communications

We compare *<sup>R</sup>*⋆◦ *<sup>s</sup>* /*K* to *Cs*,SU in the large-SNR regime. We note that in *Cs*,SU from [10] a single-user system is considered. Therefore, only one message is transmitted to one legitimate user, and the user does not experience any interference. By comparing (18) to *<sup>R</sup>*⋆◦ *<sup>s</sup>* /*K*, we can conclude that for *<sup>β</sup>* <sup>≤</sup> 1, the RCI precoder achieves a per-user secrecy rate which has the same linear scaling factor as the secrecy capacity of a single-user system with no interference. When 1 < *<sup>β</sup>* < 2, the presence of interference results in a value of *<sup>R</sup>*⋆◦ *<sup>s</sup>* that decreases with the SNR, as opposed to a constant value for *Cs*,SU. When *<sup>β</sup>* ≥ 2, the secrecy rate is zero irrespective of the presence of interference.

#### **4.3. Power allocation**

In some cases, the rate loss generated by the secrecy requirements and by the interference due to the presence of multiple users can be compensated by a power allocation scheme. In [18], an iterative power allocation algorithm was proposed to obtain the maximum secrecy sum-rate for a fixed regularization parameter *ξ*. The algorithm was also extended to maximize the secrecy sum-rate by jointly optimizing the regularization parameter *ξ* and the power allocation vector. However, in many cases there is a negligible performance difference between the joint and the separate optimization. As a result, a low-complexity, near-optimal RCI precoder may be implemented by using *<sup>ξ</sup>*⋆◦ from (10) and optimizing the power vector separately [18].

The RCI precoder with optimal power allocation (RCI-PA) outperforms the RCI precoder with equal power (RCI-EP), and the gain does not vanish at high SNR. The RCI-PA precoder thus reduces the rate loss due to secrecy requirements and interference, and in some cases it achieves a per-user rate which is as high as the rate achieved by the optimal RCI-EP precoder without secrecy requirements, and as high as the secrecy capacity of a single-user system [18].

#### **4.4. Numerical results**

10 Recent Trends in Multiuser MIMO Communications

requirements result in a value of *<sup>R</sup>*⋆◦

served at a time and the remaining users can eavesdrop.

 

1

max � log2

rate is zero irrespective of the presence of interference.

*Cs*,SU ≈

becomes zero.

**4.2. Multiuser Loss**

We compare *<sup>R</sup>*⋆◦

**4.3. Power allocation**

separately [18].

[18].

*<sup>R</sup>*⋆◦

*<sup>s</sup>* that decreases with the SNR, as opposed to a constant

*<sup>s</sup>* /*K* and

*<sup>s</sup>* that

, as *ρ* → ∞. (18)

sum-rate *<sup>R</sup>*⋆◦ without secrecy. Therefore if the SNR is too large, then the secrecy sum-rate

The cost due the interference caused by the presence of multiple users in the system, which we denote by *multiuser loss*, is given by the gap between the per-user secrecy rate *<sup>R</sup>*⋆◦

the secrecy capacity *Cs*,SU of the single-user MISOME wiretap channel, where one user is

The value of *Cs*,SU was obtained in [10], and for large SNR it can be approximated by

log2 *<sup>ρ</sup>* for *<sup>β</sup>* <sup>&</sup>lt; <sup>1</sup>

<sup>2</sup> log2 *<sup>ρ</sup>* for *<sup>β</sup>* <sup>=</sup> <sup>1</sup>

a single-user system is considered. Therefore, only one message is transmitted to one legitimate user, and the user does not experience any interference. By comparing (18) to

*<sup>s</sup>* /*K*, we can conclude that for *<sup>β</sup>* <sup>≤</sup> 1, the RCI precoder achieves a per-user secrecy rate which has the same linear scaling factor as the secrecy capacity of a single-user system with no interference. When 1 < *<sup>β</sup>* < 2, the presence of interference results in a value of *<sup>R</sup>*⋆◦

decreases with the SNR, as opposed to a constant value for *Cs*,SU. When *<sup>β</sup>* ≥ 2, the secrecy

In some cases, the rate loss generated by the secrecy requirements and by the interference due to the presence of multiple users can be compensated by a power allocation scheme. In [18], an iterative power allocation algorithm was proposed to obtain the maximum secrecy sum-rate for a fixed regularization parameter *ξ*. The algorithm was also extended to maximize the secrecy sum-rate by jointly optimizing the regularization parameter *ξ* and the power allocation vector. However, in many cases there is a negligible performance difference between the joint and the separate optimization. As a result, a low-complexity, near-optimal RCI precoder may be implemented by using *<sup>ξ</sup>*⋆◦ from (10) and optimizing the power vector

The RCI precoder with optimal power allocation (RCI-PA) outperforms the RCI precoder with equal power (RCI-EP), and the gain does not vanish at high SNR. The RCI-PA precoder thus reduces the rate loss due to secrecy requirements and interference, and in some cases it achieves a per-user rate which is as high as the rate achieved by the optimal RCI-EP precoder without secrecy requirements, and as high as the secrecy capacity of a single-user system

for *β* > 1

*<sup>s</sup>* /*K* to *Cs*,SU in the large-SNR regime. We note that in *Cs*,SU from [10]

1 (*β*−1), 0 � Fig. 4 compares the simulated ergodic sum-rates *Rs* and *R* of the RCI precoder with and without secrecy requirements, respectively. These were obtained by using the regularization parameters *<sup>ξ</sup>*⋆◦ and *<sup>ξ</sup>*⋆◦ ns , respectively. For *<sup>β</sup>* <sup>&</sup>lt; 1, the difference between *<sup>R</sup>* and *Rs* becomes negligible at large SNR, and secrecy can be achieved without additional costs. For *β* = 1, the two curves tend to have same slope at large SNR, but there is a residual gap between them. Therefore, secrecy can be achieved at a lower sum-rate. We note that in order to achieve secrecy without decreasing the sum-rate, the required additional power is less than 4dB at all SNRs. For *β* > 1, the sum-rate tends to saturate for large SNR, whereas the secrecy sum-rate starts decreasing. If the SNR is too large, then the secrecy requirements force the sum-rate to zero.

Fig. 4 also shows the simulated secrecy capacity *Cs*,SU of the MISOME wiretap channel. For *β* ≤ 1, the RCI precoder achieves a per-user secrecy rate which has the same linear scaling factor as *Cs*,SU. When 1 < *<sup>β</sup>* < 2, *Cs*,SU saturates at high SNR, while the secrecy sum-rate decreases. All these numerical results confirm the ones obtained from the large-system analysis.

**Figure 4.** Comparison between the simulated ergodic per-user secrecy rate with RCI (solid) and the two upper bounds: (i) per-user rate without secrecy requirements (dashed) and (ii) MISOME secrecy capacity (dotted), for *K* = 12 users. Three values of *β* are considered: 0.8, 1, and 1.2, corresponding to *M* = 15, 12 and 10 antennas.

Fig. 5 shows the simulated per-user secrecy rate of the RCI-PA precoder from [18], with optimal power allocation. This is compared to the RCI-EP precoder. Fig. 5 also shows that the power allocation scheme reduces the sum-rate loss due to the secrecy requirements. For *ρ* ≥ 15dB, RCI with power allocation achieves a per-user secrecy rate which is even higher than the per-user rate achieved by the optimal RCI-EP without secrecy requirements. Furthermore, Fig. 5 shows the simulated secrecy capacity *Cs*,SU of a MISOME channel with the same per-message transmit power. Although *Cs*,SU is obtained in a single-user and interference-free system [10], at high SNR, RCI with power allocation achieves a per-user secrecy rate as large as *Cs*,SU.

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**<sup>H</sup>** = **Hˆ** + **<sup>E</sup>** (19)

Physical Layer Security for Multiuser MIMO Communications

where the matrix **<sup>E</sup>** represents the channel estimation error, and it is independent from **Hˆ** . The knowledge of **Hˆ** is used by the transmitter to obtain the RCI precoding matrix. The entries of **Hˆ** and **<sup>E</sup>** are i.i.d. complex Gaussian random variables with zero mean and variances 1 − *τ*<sup>2</sup> and *τ*2, respectively. The value of *τ* ∈ [0, 1] depends on the quality and technique used for channel estimation. When *τ* = 0 the CSI is perfectly known, whereas

Future research could analyze the performance of linear precoding in the presence of imperfect CSI, deriving the achievable secrecy sum-rate as a function of the channel estimation error variance *τ*2. This would allow to study how the CSI estimation error must scale with the SNR, in order to maintain a given high-SNR rate gap to the case with perfect CSI, so that the multiplexing gain is not affected. More specifically, the case of frequency division duplex (FDD) systems could be studied. Assuming that users quantize their channel directions by using *B* bits and employing random vector quantization (RVQ), and that they feed the quantization index back to the transmitter [30, 31], it would be interesting to determine how many feedback bits are required by each user in order to maintain a constant

Throughout this chapter, we presented an up-to-date summary of the research in the field of physical layer security for multiuser MIMO communications. Unlike classical cryptography, physical layer security does not require key distribution and management, it does not rely on the limited computational power of the eavesdroppers, and it does not employ complex encryption algorithms. For these reasons, it is suitable for large dynamic wireless networks, and it has been proposed to enhance the protection of confidential messages transmitted over wireless channels. In this chapter, we especially focused on the problem of secret communication in a multiuser MIMO system. We considered the general case where a multiantenna base station transmits independent confidential messages to a generic number of users. We assumed that the users can potentially act maliciously and eavesdrop on each other. For this system set-up, we presented some transmission schemes based on linear precoding. We discussed the performance of these schemes as well as the cost of

It has been recently shown that, in the large SNR regime, a linear precoding scheme based on regularized channel inversion can achieve secrecy without reducing the sum-rate at no additional cost when the number of transmit antennas *M* is larger than the number of users *K*. If *K* = *M*, secrecy can be achieved with a small rate loss or, alternatively, without reducing the sum-rate at a cost of less than 4dB in terms of the power transmitted. However, the secrecy requirements limit the maximum number of users that can be served with a non-zero rate. When *K* > *M*, there is an optimal value of the SNR beyond which the achievable rate starts decreasing, and at large SNR the secrecy sum-rate achievable by RCI precoding is poor. The base station could prevent the secrecy sum-rate from decreasing by reducing the transmit power, and therefore the SNR, to the value that maximizes the secrecy sum-rate. This would

*τ* = 1 corresponds to the case when no CSI is available at all.

simultaneously guaranteeing secrecy to multiple users.

gap to the case with perfect CSI.

**6. Conclusions**

**Figure 5.** Per-user secrecy rate vs. *ρ* for *β* = 1 and *K* = 4 users: with equal power allocation (solid) and with optimal power allocation (dashed). The rate of the optimal RCI precoder without secrecy requirements (squares) and the secrecy capacity of the MISOME channel (circles) are also plotted.
