Secure performance analysis and detection of pilot attack in massive multiple-input multiple-output system

Abstract

This article studies the physical layer security of massive multiple-input multiple-output system in time-division-duplex mode. Specifically, a single-cell downlink massive multiple-input multiple-output communication system is considered. The resulting achievable secrecy rate is investigated in the presence of passive or active eavesdroppers. The analytical results reveal that the massive multiple-input multiple-output system is naturally immune to passive eavesdroppers, but will be dramatically degraded by active attack. On account of the risk caused by active attack, a simple and effective detection algorithm is proposed. Uplink pilots with random phases are transmitted during the detection operation. By comparing the phase deviation change of the received pilot signals’ cross product, the active eavesdroppers can be detected exactly. The closed-form expressions for probabilities of detection, false alarm, and false rejection are obtained respectively, which can demonstrate the robust performance of the proposed detection scheme. To maximize the secrecy energy efficiency of the system, the optimal power allocation strategy is studied under total power constraints. This optimization problem is efficiently solved by fractional programming. Numerical simulation results are derived to validate the secrecy performance of the massive multiple-input multiple-output system, the active pilot attacker detection performance, and the energy efficiency optimization effect.

Introduction

The fifth-generation (5G) wireless network is expected to support a significantly large amount of mobile data capacity and achieve better quality of service (QoS) in terms of communication rate, delay, reliability, and security.^1,2 Among the emerging technologies, massive multiple-input multiple-output (MIMO) as one of the most potential promising technologies has attracted much attention and has been well assessed through theoretical analysis and laboratory tests.^3,4 Massive MIMO is a special MIMO technology where base station (BS) is equipped with large number of antennas, typically tens or hundreds. Combining with a time-division-duplex (TDD)-based transmission can help resolve many of the issues gaining channel state information.^5,6 In TDD mode, massive MIMO systems could exploit the reciprocity to process the channel estimation on the uplink. In particular, the BS exploits law of large numbers like certainties as it serves each one over a combination of a large number of independent transmission channels. Large arrays can eliminate the noise and interference with simple signal processing, such as basic linear combination operations.^7,8 So the multiplexing gain can be fully used to maximize the channel capacity. The advantages of massive MIMO include higher data rate, better reliability, and less noise and interference. Another advantage of massive MIMO is the potential of physical layer security as the huge degrees of freedom, especially against the passive eavesdroppers. ⁹

The open broadcast nature of wireless medium results in information leakage to some unintended receivers. Physical layer security is promoted to achieve secure transmission by exploiting the inherent uncertainties and independent of wireless channels, noise, fading, and interference.^10,11 And this security is achieved through information theory security rather than traditional encryption and decryption. Accordingly, physical layer security has the potential to push the secrecy performance toward a brand new level with its theoretically uncompromisable security guarantee.

Research on physical layer security in conventional MIMO system has obtained abundant results.^12–14 As a promising candidate technology for meeting the upcoming 5G era, academia and industry has paid extensive attention to massive MIMO communication recently. However, the existing literature on the combination of physical layer security and massive MIMO is not sufficient. In the studies by Zhu et al. ¹⁵ and Wu et al., ¹⁶ reliable secure downlink transmission in multi-cell multi-user massive MIMO systems with maximum ration transmission (MRT) precoding and artificial noise generation at BS was investigated in detail. A single-cell massive MIMO communication in the presence of an adversary capable of jamming and eavesdropping also have attracted attention of some researchers.^9,17 These works show that massive MIMO is naturally resilient to passive eavesdroppers, but harmed by training-phase jamming. In addition, large antenna arrays can achieve excellent directional gain, so excellent beamforming design can be exploited to enhance the security performance.^18,19 Moreover, Relaying is a potential technique to improve legitimate channel capacity and reduce the undesirable information leakage, so it can be introduced into massive MIMO system to improve the secrecy performance.^20,21 Also few works focus some other special systems with massive MIMO, such as Internet of Things (IoT) ²² and Device-to-Device (D2D). ²³ Additionally, the very large freedom of the massive MIMO system can generate extremely long secret encryption keys. Some secret keys generation methods have been proposed for massive MIMO system.^24,25

According to some existing related studies,^9,17 massive MIMO has potentiality of physical layer security against passive eavesdropping attacks. So smart eavesdroppers should switch from a passive mode to an active mode by pretending to be an intended user and transmitting pilot sequences of its own. The impact of this pilot contamination attack is that it will reduce the accuracy of BS’s channel estimation, which will infect the precoding filter design at transmitter to implicitly beamform toward active eavesdropper, leading to information leakage. ²⁶ So it is crucial to detect the active eavesdropper for the secure communication. Some simple detection strategy can be used during the training phase. For instance, BS can detect the active attacker by some slowly changing parameters, such as signal power, large-scale fading, or certain statistics. ⁹ As the large-scale fading and noise power changes slowly at a long period, the received power will converge to some fixed value. If an active attacker exists, the received power will increase, so the active attacker can be detected. ¹⁷ Unfortunately, the eavesdropper can adapt its transmitted power over several coherence intervals and emulate the natural change of the channel propagation environment. Through these operations, the eavesdroppers cannot be detected. Besides, when the eavesdroppers are absent, the covariance matrix of the received pilot signal converges to a rank-one matrix. Otherwise, the matrix will converge to a full rank matrix with a large probability when some eavesdroppers are present. So the eavesdropper can be detected by comparing the largest and second eigenvalues of the covariance matrix. This scheme can provide significant performance, but it needs some more observations and also needs a good estimation of the noise power. As we know, the inner product of the different received signals would converge to a fixed phase value if the pilots adopted phase-shift keying (PSK) signals. ²⁷ If any eavesdropper is present, this inner product value will converge to a different phase at a large probability. This scheme is easy, but the false alarm probability depends on the PSK constellation scale. Only when the size is very large, the detection probability can be close to 1. This scheme is hard for the actual system due to its low energy efficiency and high symbol error rate. In the study by Hou et al., ²⁸ some random orthogonal pilot signals were used to operate the detection, and this scheme was robust even without any prior channel knowledge. However, these pilots should be designed carefully, which improves the complexity of the system.

Energy efficiency is a crucial design objective to reduce operating costs for mobile operators. ⁴ The development of massive antennas array can achieve diversity gain with some simple beamforming techniques, such as maximum ratio transmission (MRT), zero forcing (ZF), and minimum mean square error (MMSE). Meanwhile, massive MIMO technology can provide high energy efficiency which is suitable for the future wireless communication network. Ngo et al. ⁷ analyzed potential of energy efficiency of the massive MIMO in detail. In secure communication system, due to the simultaneous impact of legitimate users’ and eavesdroppers’ signal quality, transmit power has a bidirectional effect on the secrecy performance. Hence, power allocation scheme is significant for enhancing wireless security. Power allocation between desired signal and artificial noise is investigated by maximizing a lower bound on the achievable secrecy rate. ²⁹ In the study by Chen et al., ³⁰ a joint optimization of source and amplify-and-forward (AF) relay power allocation scheme had been proposed to maximize the secrecy energy efficiency (SEE). And the optimal source and decode-and-forward (DF) massive array relay power allocation was analyzed in the study by Chen et al. ³¹ Moreover, the optimal energy efficient power allocation for secure communication in massive MIMO had also been studied. ³² These analysis methods and mathematical tools also can be used to analyze the SEE of the system in this article.

Active attacker detection and power allocation are all crucial factors that affect physical layer security performance in massive MIMO system. To the best of our knowledge, overall there is limited attention to these problems in existing works. ³³ In this article, we consider all these issues and give some valuable formulas, data, and conclusions. The contributions of this article are as follows:

• The secrecy capacity analysis for the basic three-node massive MIMO model is provided. It reveals that the massive MIMO is robust against the passive eavesdroppers and harmed by active pilot attack. Also the nonnegative secrecy capacity condition is obtained. Results suggest that the secrecy capacity increases with the number of antennas. Moreover, the limit of the secure rate has been obtained as antenna number increase toward infinity.

• A new active eavesdropping detection scheme based on pilots with random phases is proposed at training phase. The scheme only needs uplink pilots with random phases. If an active eavesdropper are present, the phase of the received signals’ cross product will converge to a fixed value. So the detection can be operated by comparing the cross product’s phase to some threshold. System overhead will be very small. Then, the probabilities of detection, false alarm, and false rejection are analyzed respectively.

• To maximize the SEE under some constraints, the optimal source power allocation scheme is studied in this article. The fractional programming and Lagrange multiplier method are adopted to solve the fractional programming problem. And the optimal value can be easily gained by limited iterations.

The remainder of this article is organized as follows. The system model and the secrecy performance are described in section “System model.” The proposed active attacker detection algorithm is presented in detail in section “Detection for active eavesdropper.” In section “SEE analysis,” the energy-efficient power allocation scheme is analyzed. In section “Numerical results,” the numerical and simulation results are presented to validate the effectiveness of the schemes in the article. Finally, section “Conclusion” draws the concluding remarks.

Notations: Bold face letters denote matrices and vectors, for example, $\mathbf{Y}$ , $\mathbf{h}$ . And $(\mathbf{h}{)}^{*}$ , $(\mathbf{h}{)}^T$ , $(\mathbf{h}{)}^H$ , and $\Vert \mathbf{h}\Vert$ denote complex conjugate, transpose, Hermitian-transpose, and Euclidean norm of the vector, respectively. $\mathbb{E}[x]$ and $\mathrm{var}(x)$ represent the expectation and variance of variable $x$ . The notation $\mathbf{x}~C\mathcal{N}(\mathbf{m},\mathbf{H})$ stands for a circularly symmetric complex Gaussian random vector $\mathbf{x}$ with mean $\mathbf{m}$ and covariance $\mathbf{H}$ . In addition, i.i.d indicates the abbreviation of independent and identically distributed. $\angle Z$ denotes the phase of the complex variable $Z$ .

System model

We consider a typical three-node transmission system, consisting a legitimate transmitter (Alice), an intended receiver (Bob), an eavesdropper (Eve). The TDD massive MIMO system is considered, in which channel reciprocity holds perfectly in data block (one slot at least). As depicted in Figure 1, Alice is a massive antenna BS equipped with $M$ antennas where the value of $M$ is always over tens or hundreds, $M>>1$ . In addition, Eve usually is a legitimate node or pretends to be a normal destination in the local cell, so we can assume that Eve is equipped with only one antenna as same as Bob, that is, $N_B=N_E=1$ . The wireless channels between the three terminals experience large-scale path loss as well as small-scale fading. In general, the large-scale fading coefficients change very slowly over time, which can be assumed invariant in a large period (one data block at least). The channels from Alice to Bob and from Alice to Eve are given as $\sqrt{{\alpha }_{\mathrm{AB}}}{\mathbf{h}}_{\mathrm{AB}}$ , $\sqrt{{\alpha }_{\mathrm{AE}}}{\mathbf{h}}_{\mathrm{AE}}$ , where ${\alpha }_{\mathrm{AB}}$ , ${\alpha }_{\mathrm{AE}}$ stand for the large-scale path fading including path loss and shadowing. ${\mathbf{h}}_{\mathrm{AB}}$ and ${\mathbf{h}}_{\mathrm{AE}}$ are the small-scale fading $1\times M$ vectors, and each element of the two vectors is independent and identically distributed (i.i.d) complex Gaussian random variables with zero mean and unit variance.

Figure 1.

The system model, one base station with massive antennas (Alice), one single antenna receiver (Bob), and one single antenna eavesdropper (Eve).

Channel estimation and precoding

The goal of the system is to provide reliable communication between Alice and Bob. Due to TDD model, it is effective for transmitter Alice to focus on the user Bob by precoding or beamforming design. That design requires the knowledge of ${\mathbf{h}}_{\mathrm{AB}}$ to be obtained at Alice but mostly unknown to Eve. As the uplink and downlink channels are reciprocal, ${\mathbf{h}}_{\mathrm{AB}}={\mathbf{h}}_{\mathrm{BA}}$ . This process can be realized using uplink training, that Bob send pilot signals to allow the channel estimation at Alice. We assume the channel fading gains are constant during one data transmission block and change to some independent values in the other blocks. So before data transmission, the uplink training is operated in advance.

During this training phase, Bob sends the pilot sequence $\sqrt{\tau }\omega \in {\mathbb{C}}^{\tau \times 1}$ of length $\tau$ to Alice, where ${\omega }^H\omega =1$ . In addition, Eve may also send pilots to Alice for active attacking. In this article, we consider the worst case that the pilots sent by Bob and Eve are all same and perfectly synchronous. Now Alice’s received training signals, ${\mathbf{Y}}_A\in {\mathbb{C}}^{\tau \times M}$ , can be expressed as

$$Y_A=\sqrt{P_B{\alpha }_{BA}}\omega h_{BA}+\sqrt{P_E{\alpha }_{EA}}\omega h_{EA}+N_A$$ (1)

where $P_B$ , $P_E$ denote the training power at Bob and the pilot attack power at Eve, ${\mathbf{N}}_A\in {\mathbb{C}}^{\tau \times M}$ is the additive Gaussian noise matrix with zero mean, variance ${\delta }_A^2$ , ${\mathbf{N}}_A~C\mathcal{N}(0,{\delta }_A^{2}I)$ . If Eve is passive, $P_E=0$ . In addition, assuming Alice knows Bob’s training power and the received noise power ${\delta }_A^2$ , the linear minimum mean-square error (LMMSE) method ¹⁵ is applied to estimate the channel $h_{\mathrm{AB}}$ , which can be given by

$$\begin{array}{c}\multicolumn{1}{c}{{\hat{\mathbf{h}}}_{\mathrm{AB}}=\sqrt{P_B{\alpha }_{\mathrm{BA}}\tau }{\omega }^H({\delta }_A^2{\mathbf{I}}_{\tau }+\omega (P_B{\alpha }_{\mathrm{BA}}\tau {+P_E{\alpha }_{\mathrm{EA}}\tau ){\omega }^H)}^{-1}{\mathbf{Y}}_A}\\ =\sqrt{P_B{\alpha }_{\mathrm{BA}}\tau }{(P_B{\alpha }_{\mathrm{BA}}+P_E{\alpha }_{\mathrm{EA}}+{\delta }_A^2)}^{-1}{\omega }^H{\mathbf{Y}}_A\hfill \end{array}$$ (2)

We note that parameter $P_B{\alpha }_{\mathrm{BA}}+P_E{\alpha }_{\mathrm{EA}}+{\delta }_A^2$ is required for this LMMSE channel estimation. This value can be obtained by the mean power of the received signals, that is, $P_B{\alpha }_{\mathrm{BA}}+P_E{\alpha }_{\mathrm{EA}}+{\delta }_A^2=\mathbb{E}[\Vert y_A{\Vert }^2]$ , where $y_A$ is the arbitrary element of the matrix ${\mathbf{Y}}_A$ . In addition, training power of Eve also can be calculated by this formula if BS can determine the presence of an eavesdropper.

Then Alice treats the estimated channel ${\hat{\mathbf{h}}}_{\mathrm{AB}}$ as the true parameters to design the precoding vector $w$ . Since maximum ratio transmission (MRT) can achieve asymptotically optimal performance in massive MIMO system with low complexity,^17,20 we utilize this simplest linear precoding scheme in this article, that is

$$\mathbf{w}=\frac{{\hat{\mathbf{h}}}_{\mathrm{AB}}^H}{\Vert {\hat{\mathbf{h}}}_{\mathrm{AB}}^H\Vert }$$ (3)

where $\mathbf{w}$ denotes the precoding vector. Thus, the received signals at Bob and Eve can be given by

$$y_B=\sqrt{P_A{\alpha }_{\mathrm{AB}}}{\mathbf{h}}_{\mathrm{AB}}\mathbf{w}{x}_d+n_B$$ (4)

$$y_E=\sqrt{P_A{\alpha }_{\mathrm{AE}}}{\mathbf{h}}_{\mathrm{AE}}\mathbf{w}{x}_d+n_E$$ (5)

where $n_B~C\mathcal{N}(0,1)$ , $n_E~C\mathcal{N}(0,1)$ represent the normalized additive white Gaussian noises at Bob and Eve respectively. $P_A$ is the transmit effective power of Alice.

Secrecy performance analysis

Accordingly, the instantaneous signal-to-noise ratio (SNR) at Bob is given by

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{SN}{R}_B=P_A{\alpha }_{\mathrm{AB}}\mathbb{E}\left[{\Vert {\mathbf{h}}_{\mathrm{AB}}\mathbf{w}\Vert }^2\right]}\\ =\frac{P_A{\alpha }_{\mathrm{AB}}\mathbb{E}\left[{\Vert {\mathbf{h}}_{\mathrm{AB}}{\hat{\mathbf{h}}}_{\mathrm{AB}}^H\Vert }^2\right]}{\mathbb{E}\left[{\Vert {\hat{\mathbf{h}}}_{\mathrm{AB}}^H\Vert }^2\right]}\hfill \end{array}$$ (6)

Similarly, the SNR at Eve can be expressed as

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{SN}{R}_E=P_A{\alpha }_{\mathrm{AE}}\mathbb{E}\left[{\Vert {\mathbf{h}}_{\mathrm{AE}}w\Vert }^2\right]}\\ =\frac{P_A{\alpha }_{\mathrm{AE}}\mathbb{E}\left[{\Vert {\mathbf{h}}_{\mathrm{AE}}{\hat{\mathbf{h}}}_{\mathrm{AB}}^H\Vert }^2\right]}{\mathbb{E}\left[{\Vert {\hat{\mathbf{h}}}_{\mathrm{AB}}^H\Vert }^2\right]}\hfill \end{array}$$ (7)

In the massive MIMO system, the antenna arrays’ distance and the distance between Bob and Eve are typically longer than the half of the wave length $\lambda /2$ , so that the channel responses ${\mathbf{h}}_{\mathrm{AB}}$ and ${\mathbf{h}}_{\mathrm{AE}}$ are less correlated. Larger antenna spacing and a sufficiently complex propagation environment will provide independent and identically (i.i.d) channel vectors, which are satisfied for Rayleigh fading channels. This channel model has been considered in the vast majority of works on massive MIMO.^{3–9,15–17,30–33} Recent channel measurement campaigns have testified that massive MIMO systems have characteristics that approximate the above assumption fairly well.^2,6 We know that the antenna number of Alice is very large, $M>>1$ . Then, according to the law of large numbers,^7,30 we can obtain that

$$\frac{1}{M}{\mathbf{h}}_{\mathrm{AB}}{\mathbf{h}}_{\mathrm{AB}}^H{\to }^{a.s.}1,\frac{1}{M}{\mathbf{h}}_{\mathrm{AE}}{\mathbf{h}}_{\mathrm{AE}}^H{\to }^{a.s.}1,\frac{1}{M}{\mathbf{h}}_{\mathrm{AB}}{\mathbf{h}}_{\mathrm{AE}}^H{\to }^{a.s.}0$$ (8)

as $M\to \infty$ , where ${\to }^{a.s.}$ represents the almost sure convergence.

In addition, according to the Lindeberg-Levy central limit theorem, we get this

$$\frac{1}{\sqrt{M}}{\mathbf{h}}_{\mathrm{AB}}{\mathbf{h}}_{\mathrm{AE}}^H{\to }^{d}C\mathcal{N}(0,1),\mathrm{as}M\to \infty$$ (9)

where the ${\to }^d$ denotes convergence in distribution.

Using the above conclusions, we can simplify the $\mathrm{SN}{R}_B$ as equation (10)

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{SN}{R}_B=P_A{\alpha }_{\mathrm{AB}}\mathbb{E}\left[{\Vert {\mathbf{h}}_{\mathrm{AB}}\mathbf{w}\Vert }^2\right]=\frac{P_A{\alpha }_{\mathrm{AB}}\mathbb{E}\left[{\Vert {\mathbf{h}}_{\mathrm{AB}}{\hat{\mathbf{h}}}_{\mathrm{AB}}^H\Vert }^2\right]}{\mathbb{E}\left[{\Vert {\hat{\mathbf{h}}}_{\mathrm{AB}}^H\Vert }^2\right]}}\\ =\frac{P_A{\alpha }_{\mathrm{AB}}{P}_B^2{\alpha }_{\mathrm{AB}}^2\mathbb{E}\left[{\Vert {\mathbf{h}}_{\mathrm{AB}}\Vert }^4\right]}{(P_B{\alpha }_{\mathrm{AB}}+P_E{\alpha }_{\mathrm{AE}}+{\delta }_A^2)\mathbb{E}\left[{\Vert P_B{\alpha }_{\mathrm{AB}}{\mathbf{h}}_{\mathrm{AB}}+\sqrt{P_B{\alpha }_{\mathrm{AB}}{P}_E{\alpha }_{\mathrm{AE}}}{\mathbf{h}}_{\mathrm{AE}}+\sqrt{P_B{\alpha }_{\mathrm{AB}}}{\mathbf{n}}_A\Vert }^2\right]}\hfill \\ {\to }^{a.s.}\frac{M{P}_A{P}_B{\alpha }_{\mathrm{AB}}^2}{{(P_B{\alpha }_{\mathrm{AB}}+P_E{\alpha }_{\mathrm{AE}}+{\delta }_A^2)}^2}\hfill \end{array}$$ (10)

Similar to Bob, average SNR at Eve is given by

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{SN}{R}_E=P_A{\alpha }_{\mathrm{AE}}\mathbb{E}\left[{\Vert {\mathbf{h}}_{\mathrm{AE}}\mathbf{w}\Vert }^2\right]}\\ =\frac{P_A{\alpha }_{\mathrm{AE}}\mathbb{E}\left[{\Vert {\mathbf{h}}_{\mathrm{AE}}{\hat{\mathbf{h}}}_{\mathrm{AB}}^H\Vert }^2\right]}{\mathbb{E}\left[{\Vert {\hat{\mathbf{h}}}_{\mathrm{AB}}^H\Vert }^2\right]}\hfill \\ {\to }^{a.s.}\frac{M{P}_A{P}_E{\alpha }_{\mathrm{AE}}^2}{{(P_B{\alpha }_{\mathrm{AB}}+P_E{\alpha }_{\mathrm{AE}}+{\delta }_A^2)}^2}\hfill \end{array}$$ (11)

As a result, the achievable secrecy rate is formulated as

$$\begin{array}{c}\multicolumn{1}{c}{R_s=\mathbb{E}\left[{[{\log }_2(1+\mathrm{SN}{R}_B)-{\log }_2(1+\mathrm{SN}{R}_E)]}^{+}\right]}\\ ={\left[{\log }_2\left(\frac{a+M{P}_A{P}_B{\alpha }_{\mathrm{AB}}^2}{a+M{P}_A{P}_E{\alpha }_{\mathrm{AE}}^2}\right)\right]}^{+}\hfill \end{array}$$ (12)

where $[x{]}^{+}=max\{x,0\}$ , as the secrecy rate is a nonnegative quantity. For simplicity, we set $a=(P_B{\alpha }_{\mathrm{AB}}+P_E{\alpha }_{\mathrm{AE}}+{\delta }_A^2{)}^2$ .

We can observe that if Eve is passive, $P_E=0$ , the system secrecy rate is almost equal with the Bob’s received information rate. If Eve is active, when $P_E{\alpha }_{\mathrm{AE}}^2\ge P_B{\alpha }_{\mathrm{AB}}^2$ , the achievable secrecy rate is zero. So only at the condition $P_E{\alpha }_{\mathrm{AE}}^2<P_B{\alpha }_{\mathrm{AB}}^2$ , the Bob can get the secrecy rate under the pilot attack.

The first-order derivative of $R_s$ with respect to $M$ can be calculated as (consider the $M$ as an continuous variable)

$$\frac{\partial \left(R_s\right)}{\partial M}=\frac{a(P_A{P}_B{\alpha }_{\mathrm{AB}}^2-P_A{P}_E{\alpha }_{\mathrm{AE}}^2)}{\ln 2(a+M{P}_A{P}_B{\alpha }_{\mathrm{AB}}^2)(a+M{P}_A{P}_E{\alpha }_{\mathrm{AE}}^2)}$$ (13)

From equation (13), it can be derived easily that $(\partial (R_s)/\partial M)>0$ , when the Bob has nonzero secrecy rate, $P_B{\alpha }_{\mathrm{AB}}^2>P_E{\alpha }_{\mathrm{AE}}^2$ . Then, as $M$ increases, the secrecy capacity $R_S$ also increases. ¹⁷

Meanwhile, we can get the upper limiting secrecy capacity when BS antennas number reach infinity

$$\begin{array}{c}\multicolumn{1}{c}{\underset{M\to \infty }{lim}{R}_s=\underset{M\to \infty }{lim}{\left[{\log }_2\left(\frac{a+M{P}_A{P}_B{\alpha }_{\mathrm{AB}}^2}{a+M{P}_A{P}_E{\alpha }_{\mathrm{AE}}^2}\right)\right]}^{+}}\\ ={\left[{\log }_2\left(\frac{P_B{\alpha }_{\mathrm{AB}}^2}{P_E{\alpha }_{\mathrm{AE}}^2}\right)\right]}^{+}\hfill \end{array}$$ (14)

Detection for active eavesdropper

From the above analysis, we can understand that if Eve attempt to eavesdrop the useful message in massive MIMO system, a more effective strategy is to actively contaminate the pilot training phase. For fixed training scheme and most practical applications, as the cellular communication, the pilot set is public and used repeatedly. The eavesdroppers can easily survey and imitate the pilot signals. And once if Eve emits the same pilots as Bob, then the uplink channel estimation ${\hat{\mathbf{h}}}_{\mathrm{AB}}$ is correlated with ${\mathbf{h}}_{\mathrm{AE}}$ . And it will change the direction of beamforming to Eve, so Eve can intercept signaling exchange between Alice and Bob. The consequence of this active attack is that the promising secrecy benefits of massive MIMO are lost if the attacker cannot be detected. Even worse, if Eve’s training power is higher, it will dominate the uplink channel estimation and the secrecy capacity will become zero.

As described previously, detection of an active attacker is crucial for secure communication in massive MIMO system. Since the channel vectors $\sqrt{{\alpha }_{\mathrm{AB}}}{\mathbf{h}}_{\mathrm{AB}}$ , $\sqrt{{\alpha }_{\mathrm{AE}}}{\mathbf{h}}_{\mathrm{AE}}$ are uncorrelated, the imperfect estimated channel would not result in decoding errors and packet loss. So the conventional pilot contamination detection, as packet reception, decoding error rate, does not apply successfully in massive MIMO system. As we know, if the active pilot attack cannot be detected effectively and timely at the training phase, it will be detrimental for the secure data transmission. Hence, attacking detection should be operated in the uplink training phase.

The large-scale fading changes very slowly at a long period, while the small-scale fading changes rapidly. The antenna number is very large, so the statistics of Alice’s received vector will converge to some fixed numerical value. In this section, we utilize phase of some statistics to detect the existence of active eavesdroppers.

In the detection scheme, Bob sends the normal uplink pilot signals to BS (Alice) in the training phase, but delivers some changed pilots with a known random phase symbol occasionally when BS tries to detect presence of active eavesdroppers. We assume two training slots are used for detection. $s_P$ denotes the normal pilot and $s_R$ represents the Bob’s detection pilot with random known phase. Meanwhile, the statistical characteristics of $s_R$ should be the same as the normal pilots to prevent Eve from being conscious of the detection. It is given that $\mathbb{E}[s_R]=\mathbb{E}[s_P]=0$ , $\mathrm{var}(s_R)=\mathrm{var}(s_P)=1$ . And the received signal at Alice is represented as

$$\{\begin{array}{c}\hfill {\mathbf{y}}_A^1=\sqrt{P_B{\alpha }_{\mathrm{BA}}}{s}_R^1{\mathbf{h}}_{\mathrm{BA}}+\sqrt{P_E{\alpha }_{\mathrm{EA}}}{s}_P^1{\mathbf{h}}_{\mathrm{EA}}+{\mathbf{n}}_A^1\hfill \\ \multicolumn{1}{c}{{\mathbf{y}}_A^2=\sqrt{P_B{\alpha }_{\mathrm{BA}}}{s}_R^2{\mathbf{h}}_{\mathrm{BA}}+\sqrt{P_E{\alpha }_{\mathrm{EA}}}{s}_P^2{\mathbf{h}}_{\mathrm{EA}}+{\mathbf{n}}_A^2}\end{array}$$ (15)

where the superscripts 1 and 2 represent different pilot moments.

Then, we can derive a statistical quantity (scaled cross product of the received vectors ${\mathbf{y}}_A^1$ , ${\mathbf{y}}_A^2$ ), $Z=({\mathbf{y}}_A^1({\mathbf{y}}_A^2{)}^H)/M$ .

The scalar product can also be denoted as equation (16)

$$\begin{array}{c}\multicolumn{1}{c}{Z=\frac{{\mathbf{y}}_A^1{\left({\mathbf{y}}_A^2\right)}^H}{M}=\frac{(\sqrt{P_B{\alpha }_{\mathrm{BA}}}{s}_R^1{\mathbf{h}}_{\mathrm{BA}}+\sqrt{P_E{\alpha }_{\mathrm{EA}}}{s}_P^1{\mathbf{h}}_{\mathrm{EA}}+{\mathbf{n}}_A^1){(\sqrt{P_B{\alpha }_{\mathrm{BA}}}{s}_R^2{\mathbf{h}}_{\mathrm{BA}}+\sqrt{P_E{\alpha }_{\mathrm{EA}}}{s}_P^2{\mathbf{h}}_{\mathrm{EA}}+{\mathbf{n}}_A^2)}^H}{M}}\\ =\underset{\mathbb{E}\left[Z\right]}{\underbrace{\frac{(\sqrt{P_B{\alpha }_{\mathrm{BA}}}{s}_R^1{\mathbf{h}}_{\mathrm{BA}}+\sqrt{P_E{\alpha }_{\mathrm{EA}}}{s}_P^1{\mathbf{h}}_{\mathrm{EA}}){(\sqrt{P_B{\alpha }_{\mathrm{BA}}}{s}_R^2{\mathbf{h}}_{\mathrm{BA}}+\sqrt{P_E{\alpha }_{\mathrm{EA}}}{s}_P^2{\mathbf{h}}_{\mathrm{EA}})}^H}{M}}}\hfill \\ +\underset{n_Z}{\underbrace{\frac{{\mathbf{n}}_A^1{(\sqrt{P_B{\alpha }_{\mathrm{BA}}}{s}_R^2{\mathbf{h}}_{\mathrm{BA}}+\sqrt{P_E{\alpha }_{\mathrm{EA}}}{s}_P^2{\mathbf{h}}_{\mathrm{EA}}+{\mathbf{n}}_A^2)}^H+(\sqrt{P_B{\alpha }_{\mathrm{BA}}}{s}_R^1{\mathbf{h}}_{\mathrm{BA}}+\sqrt{P_E{\alpha }_{\mathrm{EA}}}{s}_P^1{\mathbf{h}}_{\mathrm{EA}}+{\mathbf{n}}_A^1){\left({\mathbf{n}}_A^2\right)}^H}{M}}}\hfill \end{array}$$ (16)

Recalling channel reciprocity and equation (8), we can simplify this statistic as equation (17)

$$\begin{array}{c}\multicolumn{1}{c}{\mathbb{E}\left[Z\right]=\frac{\sqrt{P_B{\alpha }_{\mathrm{BA}}{P}_E{\alpha }_{\mathrm{EA}}}{s}_R^1{\left(s_P^2\right)}^{*}{\mathbf{h}}_{\mathrm{BA}}{\mathbf{h}}_{\mathrm{EA}}^H+\sqrt{P_B{\alpha }_{\mathrm{BA}}{P}_E{\alpha }_{\mathrm{EA}}}{s}_P^1{\left(s_R^2\right)}^{*}{\mathbf{h}}_{\mathrm{EA}}{\mathbf{h}}_{\mathrm{BA}}^H}{M}}\\ +\frac{P_B{\alpha }_{\mathrm{BA}}{s}_R^1{\left(s_R^2\right)}^{*}{\mathbf{h}}_{\mathrm{BA}}{\mathbf{h}}_{\mathrm{BA}}^H+P_E{\alpha }_{\mathrm{EA}}{s}_P^1{\left(s_P^2\right)}^{*}{\mathbf{h}}_{\mathrm{EA}}{\mathbf{h}}_{\mathrm{EA}}^H}{M}{\to }^{a.s.}{P}_B{\alpha }_{\mathrm{BA}}{s}_R^1{\left(s_R^2\right)}^{*}+P_E{\alpha }_{\mathrm{EA}}{s}_P^1{\left(s_P^2\right)}^{*}\hfill \end{array}$$ (17)

Meanwhile, according to the central limit theorem, we can get that

$$\frac{n_Z}{\sqrt{\mathrm{var}\left(n_Z\right)}}{\to }^{d}C\mathcal{N}(0,1),\mathrm{as}M\to \infty$$ (18)

where the variance $\mathrm{var}(n_Z)$ can be given by

$$\begin{array}{c}\multicolumn{1}{c}{\mathrm{var}\left(n_Z\right)=\frac{{\delta }_A^2}{M^2}(M{\delta }_A^2+\Vert \sqrt{P_B{\alpha }_{\mathrm{BA}}}{s}_R^1{\mathbf{h}}_{\mathrm{BA}}+\sqrt{P_E{\alpha }_{\mathrm{EA}}}{s}_P^1{\mathbf{h}}_{\mathrm{EA}}{\Vert }^2}\\ \multicolumn{1}{c}{+\Vert \sqrt{P_B{\alpha }_{\mathrm{BA}}}{s}_R^2{\mathbf{h}}_{\mathrm{BA}}+\sqrt{P_E{\alpha }_{\mathrm{EA}}}{s}_P^2{\mathbf{h}}_{\mathrm{EA}}{\Vert }^2)}\end{array}$$ (19)

Furthermore, we can easily get that

$$\mathrm{var}\left(n_Z\right){\to }^{a.s.}\frac{{\delta }_A^2({\delta }_A^2+2P_B{\alpha }_{\mathrm{BA}}+2P_E{\alpha }_{\mathrm{EA}})}{M}$$ (20)

When $M$ is extremely large, $\mathrm{var}(n_Z)$ will be almost close to zero

$$\underset{M\to \infty }{lim}\mathrm{var}\left(n_Z\right)=0$$ (21)

From equations (16), (18), and (21), we know that the variance of $Z$ is very small, even it can be treated as a constant. This feature makes $Z$ suitable to be used for some judgment

Now we set that

$$\{\begin{array}{c}\varphi =\angle s_R^1{\left(s_R^2\right)}^{*}\hfill \\ \psi =\angle Z\hfill \end{array}$$ (22)

To summarize, the procedure of the detection algorithm is as follows:

1. Bob send the pilots $s_R^1$ , $s_R^2$ with random phase at two different time slots;

2. Compute the cross product $Z=y_A^1(y_A^2{)}^H/M$ ;

3. Compare the phase $\psi$ to phase $\varphi$ , obtain the phase difference $\psi -\varphi$ ;

4. According to the phase difference, we can make a judgment. If $|\psi -\varphi |>\theta$ , we predicate that the active eavesdropper exist. Else if $|\psi -\varphi |\le \theta$ , the result of the judgment is that there is no active attacker at training phase. Where the phase $\theta$ is the judgment threshold that we set before, $0<\theta <(\pi /2)$ .

From the above analysis, we can realize that $Z$ approximately obeys the Gauss distribution, and the mean and variance can be given as

$$\mathbb{E}\left[Z\right]=P_B{\alpha }_{\mathrm{BA}}{s}_R^1{\left(s_R^2\right)}^{*}+P_E{\alpha }_{\mathrm{EA}}{s}_P^1{\left(s_P^2\right)}^{*}$$ (23)

$$\mathrm{var}\left(Z\right)=\frac{{\delta }_A^2({\delta }_A^2+2P_B{\alpha }_{\mathrm{BA}}+2P_E{\alpha }_{\mathrm{EA}})}{M}$$ (24)

Without loss of generality, we can assume ${\delta }_A^2=1$ to simplify the analysis.

Detection algorithm depends on whether the scalar product $Z$ is outside or inside the detection regions, as Figure 2. Following this, we can analyze the probabilities of false alarm and false rejection, respectively.

Figure 2.

Illustration of the detection for active pilot attacker.

Under the hypothesis that there is no active eavesdropper in the cell, the product $Z$ is a random variable obeying the Gauss distribution with mean $\mathbb{E}[Z]$ and variance $\mathrm{var}(Z)$ . Hence, BS can operate a simple hypothesis test from which it infers the active eavesdropper’s presence. Therefore, the threshold $\theta$ determines how large the pilot power of Eve that can be undetected. Statistic $Z$ can be denoted as the sum of the mean and the error components, that is, $Z=\mathbb{E}[Z]+n_Z$ , which also can be represented by vector graph as Figure 2. In the case of no active Eve present, $\overrightarrow{\mathrm{OR}}$ and $\overrightarrow{\mathrm{RT}}$ denote $\mathbb{E}[Z]$ and $n_Z$ . Only the amplitude of the noise component $\left|n_Z\right|=\left|\overrightarrow{\mathrm{RT}}\right|$ exceeds value $\left|\overrightarrow{\mathrm{RS}}\right|=\left|\mathbb{E}\left[Z\right]\right|\sin \theta =P_B{\alpha }_{\mathrm{BA}}\sin \theta$ and phase $\angle n_Z$ lies in $\angle \mathrm{MRN}$ and $\angle \mathrm{PRQ}$ , a false alarm occurs. Angle $\theta$ is the threshold phase. In addition, the phase $\angle n_Z$ obeys uniform distribution. For fixed noise power $\left|n_Z\right|\ge \left|\mathbb{E}\left[Z\right]\right|\sin \theta$ , the false alarm probability is that $\angle \mathrm{MRN}/\pi =2\arccos ((\left|E\left(Z\right)\right|\sin \theta )/(\left|n_Z\right|))/\pi$ . In order to obtain insightful results, we can derive a closed-form upper bound for the false alarm probability as equation (25), where the function $f(·)$ is Gauss probability density function and $Q(·)$ is Q function

$$\begin{array}{c}\mathrm{Pr}(|\psi -\varphi |>\theta |P_E=0)=2\underset{\left|\mathbb{E}\left[Z\right]\right|\sin \theta }{\overset{+\infty }{\int }}f\left(n\right)·\arccos \left(\frac{\left|\mathbb{E}\left[Z\right]\right|\sin \theta }{n}\right)/\mathrm{Pr}(|\psi -\varphi |>\theta |P_E=0)=2\underset{\left|\mathbb{E}\left[Z\right]\right|\sin \theta }{\overset{+\infty }{\int }}f\left(n\right)·\arccos \left(\frac{\left|\mathbb{E}\left[Z\right]\right|\sin \theta }{n}\right)\frac{\pi }{2}\mathrm{dn}\frac{\pi }{2}\mathrm{dn}\hfill \\ \le \mathrm{Pr}(\left|n_Z\right|>\left|\mathbb{E}\left[Z\right]\right|·\sin \theta |P_E=0)=2Q\left(\frac{\sqrt{M}{P}_B{\alpha }_{\mathrm{BA}}\sin \theta }{\sqrt{2P_B{\alpha }_{\mathrm{BA}}+1}}\right)\hfill \end{array}$$ (25)

Moreover, if $M$ is large enough, $\mathrm{var}(n_Z)$ goes to zero as $1/M$ , and the phase difference $|\angle Z-\angle \mathbb{E}\left[Z\right]|$ would also be quite small. Thus, for the massive MIMO system, the false alarm probability will be very small, and even need not consider the noise effects.

The detection probability and false rejection probability can also be analyzed using vector graph, as depicted in Figure 2. If Eve’s effective energy $\left|\overrightarrow{\mathrm{RT}}\right|=P_E{\alpha }_{\mathrm{EA}}$ is not larger than the radius of the small circle, $P_E{\alpha }_{\mathrm{EA}}\le \left|\overrightarrow{\mathrm{RS}}\right|=P_B{\alpha }_{\mathrm{BA}}\sin \theta$ , the active attack cannot be detected at all. Besides, when $\overrightarrow{\mathrm{OT}}$ lies between the two dashed lines $\overrightarrow{\mathrm{OQ}}$ and $\overrightarrow{\mathrm{ON}}$ (two tangent lines of the small circle), Eve also cannot be detected. Otherwise, if $\overrightarrow{\mathrm{OT}}$ lies outside of the region between lines $\overrightarrow{\mathrm{OQ}}$ and $\overrightarrow{\mathrm{ON}}$ , the eavesdropper can be successfully detected.

Based on above description and discussion, we can easily obtain the probability of false rejection, which is given by

$$\begin{array}{c}\mathrm{Pr}(|\psi -\varphi |<\theta |P_E\ne 0)\hfill \\ =\{\begin{array}{c}1,P_E\le \frac{P_B{\alpha }_{\mathrm{BA}}\sin \theta }{{\alpha }_{\mathrm{EA}}}\hfill \\ \frac{2}{\pi }\arcsin \left(\frac{P_B{\alpha }_{\mathrm{BA}}\sin \theta }{P_E{\alpha }_{\mathrm{EA}}}\right),P_E>\frac{P_B{\alpha }_{\mathrm{BA}}\sin \theta }{{\alpha }_{\mathrm{EA}}}\hfill \end{array}\hfill \end{array}$$ (26)

As the above analysis, we can easily derive the detection probability, that is, $\mathrm{Pr}(|\psi -\varphi |\ge \theta |P_E\ne 0)=1-\mathrm{Pr}(|\psi -\varphi |<\theta |P_E\ne 0)$ .

From these expressions, we can get the following conclusions:

1. The threshold phase $\theta$ is a key parameter for the detection performance, the probability of false alarm will decrease when $\theta$ increases. Otherwise, the probability of false rejection will increase. So we should choose an appropriate parameter.

2. Only when the antenna number is extremely large, we can ignore the influence of noise and obtain the detection performance. So this detection is naturally suitable for the massive MIMO system.

3. Eve’s effective power $P_E{\alpha }_{\mathrm{EA}}$ is another determining factor. With $P_E{\alpha }_{\mathrm{EA}}$ increasing, the detection probability will be improved rapidly.

In addition, if the probability of false rejection cannot exceed a certain proportion $\zeta \in (0,1)$ , that is, $(2/\pi )\arcsin (P_B{\alpha }_{\mathrm{BA}}\sin \theta /P_E{\alpha }_{\mathrm{EA}})\le \zeta$ . Hence, we can further deduce that the Bob’s pilot signal effective power needs to meet the following condition

$$P_B{\alpha }_{\mathrm{BA}}\le \frac{P_E{\alpha }_{\mathrm{EA}}\sin \left(\frac{\pi }{2}\zeta \right)}{\sin \theta },\zeta \in (0,1)$$ (27)

Evidently, the above results provide valuable reference for the pilot signal design of massive MIMO.

SEE analysis

The development of massive number of antennas can significantly enhance the spectral and energy efficiency of the future wireless network.^3–7 We provide performance analysis of the secure massive MIMO system in previous section. The results suggest that powers of Alice and Bob are interrelated under a total power limit, and both of them have a major influence on the secrecy performance. So an optimal power allocation scheme is preferred. In this section, the SEE is considered, which is defined as sum secrecy rate divided by transmit power.

As a matter of fact, we derive the spectral efficiency capacity (bits/s/Hz) in section “System model,” as we have not considered the width of the system. Without loss of generality, we can replace the secrecy rate with the spectral efficiency capacity to calculate the SEE (bits/J/Hz).

In the first place, total power consumption of the system should be investigated prior to analysis. According to some existing related literatures,^30,32 we can model the total energy consumption as a sum of some components, which can be given by

$$P_T=\frac{P_A}{{\eta }_1}+\frac{P_B}{{\eta }_2}+P_C$$ (28)

where $P_T$ represents total power consumption, $P_A$ and $P_B$ indicate the effective power of Alice and Bob, respectively; the factors ${\eta }_1$ and ${\eta }_2$ denote the power amplifiers efficiency of Alice and Bob. Apparently, we know that ${\eta }_1,{\eta }_2\in (0,1]$ . Without loss of generality, we assume ${\eta }_1,{\eta }_2=1$ . In addition, $P_C$ is the total circuit power consumption in the transmission system, which can be regarded as a constant.

The SEE can be expressed as

$$\mathrm{SEE}=\frac{R_s}{P_T}$$ (29)

Recalling equations (12), (28), and (29), both powers of Alice and Bob have a great impact on SEE. In this article, we only focus on maximizing SEE in the scenario where the total power is constrained, which can be formulated as

$$\begin{array}{c}{J}_1:\underset{P_A}{max}\frac{R_S}{P_T}\hfill \\ s.t:R_s\ge R_{min}\hfill \\ P_A+P_B+P_C\le P_{max}\hfill \end{array}$$ (30)

where $R_{min}$ is the minimum secrecy outage capacity which satisfy the requirement of the secure communication and $P_{max}$ is the maximum power of the communication system (total power limit). Since the objective function is a fractional form, the problem $J_1$ is non-convex.

According to some previous related works,^30,32 problem $J_1$ can be equivalently transformed into a parameterized polynomial subtractive form. Then, the objective function is equivalent to $R_s-q^{*}{P}_T$ , where $q^{*}$ is the maximum SEE, namely $q^{*}=\underset{P_A}{max}(R_s/P_T)$ . In the actual system setting, Bob’s power can be designed related to the false detection probability, Eve’s effective power, channel estimation quality. Thus, we can transform the jointly convex optimization problem to a sub-problem which is to optimize the Alice’s power $P_A$ for a given Bob’s power $P_B$ . Thus, the optimization problem can be transformed to

$$\begin{array}{c}{J}_2:\underset{P_A}{max}{R}_S-q^{*}{P}_T\hfill \\ s.t:R_s\ge R_{min}\hfill \\ P_A\le P_{max}-P_B-P_C\hfill \end{array}$$ (31)

This optimization problem $J_2$ has one property that the function $R_S-q^{*}{P}_T$ is a concave function with respect to the transmit power at the feasible set.

Proof

Intuitively, since $q^{*}{P}_T$ is affine, function $R_S-q^{*}{P}_T$ has the same concavity as $R_S$ . So we can focus on the concavity of $R_S$ with respect to $P_A$ . The secrecy capacity can be denoted by

$$R_s=\underset{2}{\log }\left(\frac{a+M{P}_A{P}_B{\alpha }_{\mathrm{AB}}^2}{a+M{P}_A{P}_E{\alpha }_{\mathrm{AE}}^2}\right)=\underset{2}{\log }\left(K\left(P_A\right)\right)$$ (32)

We define a new function $K(P_A)=(a+M{P}_A{P}_B{\alpha }_{\mathrm{AB}}^2)/(a+M{P}_A{P}_E{\alpha }_{\mathrm{AE}}^2)$ , which can be transformed as

$$\begin{array}{c}\multicolumn{1}{c}{K\left(P_A\right)=\frac{a+M{P}_A{P}_B{\alpha }_{\mathrm{AB}}^2}{a+M{P}_A{P}_E{\alpha }_{\mathrm{AE}}^2}}\\ =\frac{P_B{\alpha }_{\mathrm{AB}}^2}{P_E{\alpha }_{\mathrm{AE}}^2}+\frac{a(P_E{\alpha }_{\mathrm{AE}}^2-P_B{\alpha }_{\mathrm{AB}}^2)}{P_E{\alpha }_{\mathrm{AE}}^2(a+M{P}_A{P}_E{\alpha }_{\mathrm{AE}}^2)}\hfill \end{array}$$ (33)

Since these parameters satisfy that $a>0$ , $P_B{\alpha }_{\mathrm{AB}}^2>0$ , $P_E{\alpha }_{\mathrm{AE}}^2>0$ , and $P_E{\alpha }_{\mathrm{AE}}^2-P_B{\alpha }_{\mathrm{AB}}^2<0$ (Bob has nonzero secrecy rate), it is clearly seen that function $K(P_A)$ has the same convexity as $-(1/P_A)$ . Therefore, $K(P_A)$ is a concave function and $K\prime (P_A)>0$ , $K″(P_A)<0$ . Based on this, we can derive that

$$R_s\prime =\left[{\log }_2\left(K\left(P_A\right)\right)\right]\prime =\frac{K\prime \left(P_A\right)}{\ln 2·K\left(P_A\right)}>0$$ (34)

$$R_s″=\frac{K\left(P_A\right)K\left(P_A\right)-K\prime \left(P_A\right)K\prime \left(P_A\right)}{\ln 2K^2\left(P_A\right)}<0$$ (35)

Thus, $R_S$ is a concave function, which finish the proof.

As proved before, the objective function is concave with the feasible set of $P_A$ . Hence, $J_2$ is a convex optimization problem. Thus, we can easily derive the optimal power allocation $P_A$ utilizing the standard convex algorithms.

Similar to previous works, ³² we can use the Lagrange multipliers algorithm to resolve this problem. The objective function can be replaced with

$$\begin{array}{c}\multicolumn{1}{c}{L(P_A,\lambda ,\mu )=R_s-q^{*}{P}_T+\lambda (R_s-R_{min})}\\ \multicolumn{1}{c}{+\mu (P_{max}-P_B-P_C-P_A)}\end{array}$$ (36)

where the Lagrange multipliers $\lambda \ge 0$ , $\mu \ge 0$ associated with the constraints.

Meanwhile, the dual optimization problem is represented by

$$\underset{\lambda ,\mu }{min}\underset{P_A}{max}L(P_A,\lambda ,\mu )$$ (37)

The optimal power $P_A^{*}$ for some fixed $\lambda$ and $\mu$ can be obtained by solving the Karush–Kuhn–Tucker (KKT) condition, that is

$$\frac{\partial L(P_A,\lambda ,\mu )}{\partial P_A}=(1+\lambda )\frac{\partial R_S}{\partial P_A}-q^{*}-\mu =0$$ (38)

Then, the optimal solution can be denoted by

$$P_A^{*}={\left[\frac{\sqrt{M^2{a}^2{c}^2-4M^{2}d(a^2-l)}-\mathrm{Mac}}{2M^{2}d}\right]}^{+}$$ (39)

where parameters are given by $c=P_B{\alpha }_{\mathrm{AB}}^2+P_E{\alpha }_{\mathrm{AE}}^2,d=P_B{P}_E{\alpha }_{\mathrm{AB}}^2{\alpha }_{\mathrm{AE}}^2,$ $l=((1+\lambda )\mathrm{aM}(P_B{\alpha }_{\mathrm{AB}}^2-P_E{\alpha }_{\mathrm{AE}}^2))/(\ln 2(q^{*}+\mu ))$ .

Then, SEE can be obtained by $q^{*}=(R_s/P_T),(P_A=P_A^{*})$ . The gradient method can be used to update the Lagrange multipliers, which are given by

$${\lambda }_{n+1}={[{\lambda }_n-{\Delta }_{\lambda }^n(R_s-R_{min})]}^{+}$$ (40)

$${\mu }_{n+1}={[{\mu }_n-{\Delta }_{\mu }^n(P_{max}-P_B-P_C)]}^{+}$$ (41)

where ${\Delta }_x^n={\alpha }_x/n$ , $x=\lambda ,\mu$ are update step sizes, ${\alpha }_x$ are fixed values, and $n$ is the iteration index.

We need repeated computing of the optimal solution $P_A^{*}$ and $q^{*}$ until $q^{*}$ has converged $q_{n+1}^{*}-q_n^{*}\le {\epsilon }_q$ , where ${\epsilon }_q$ is the threshold value (precision). Then, $P_A^{*}$ can be regarded the optimal energy-efficient transmit power.

Numerical results

To examine and evaluate the accuracy and effectiveness of the results in our massive MIMO secure system, Monte Carlo simulations are operated in this section. In all simulations, we set the noise power is ${\delta }_A^2={\delta }_B^2={\delta }_E^2=1$ , and the power of the three terminals are all effective SNR in fact, the large-scale fading factor ${\alpha }_{\mathrm{AB}}={\alpha }_{\mathrm{BA}}={\alpha }_{\mathrm{AE}}={\alpha }_{\mathrm{EA}}=1$ .

First, we show the secure capacity of the massive MIMO system with the increasing of the Alice’s antenna number $M$ , as Figure 3. Here, we set $P_A=10\mathrm{dBW}$ , $P_B=10\mathrm{dBW}$ , where the simulation is operated at $P_E=0\mathrm{W},2\mathrm{dBW},5\mathrm{dBW},8\mathrm{dBW}$ . Also we can get the secure capacity at $P_E=2\mathrm{dBW},5\mathrm{dBW},8\mathrm{dBW}$ with unlimited antenna number $M\to \infty$ , depicted in Figure 3. Given fixed $P_A$ , $P_E$ , we can see that the secrecy capacity improves with the increase in $M$ , approaching the limitation value (the three horizontal lines in Figure 3). In addition, when Bob has much more training power than Eve, system will achieve an excellent performance (obtain more secrecy rate). Simulation results confirm our analysis before.

Figure 3.

Achievable secrecy rate against antenna number $M$ and the limitation of $R_s$ when $M\to \infty$ , with passive Eve $P_E=0$ and active Eve $P_E=2,5,\mathrm{and}8\mathrm{dBW}$ .

To evaluate our new active pilot attack detection algorithm, we simulate the detection probability, the false alarm probability, and false rejection probability. A high probability of false alarm bring pointless communication pause event. Additional false rejection will impair the communication quality severely. As we know, parameter $P_E{\alpha }_{\mathrm{EA}}/P_B{\alpha }_{\mathrm{BA}}$ is the decisive factor of the detection performance. Without loss of generality, we also set ${\alpha }_{\mathrm{BA}}={\alpha }_{\mathrm{EA}}=1$ for the sake of simplicity. Figure 4 shows the detection probability and false rejection probability of our detection scheme versus $P_E/P_B$ with $M=200$ , the judgment thresholds $\theta =0.05\pi ,0.04\pi$ . As expected, the false rejection probability decreases sharply with $P_E/P_B$ , while the detection probability increases. And from the diagram we can also find that if $P_E/P_B$ exceeds –4 dB, detection probability will be improved above 98%, also these features can be demonstrated in Figure 5, which displays the detection probabilities versus parameter $P_E(\mathrm{dB})$ at $P_B=2,5,8,\mathrm{and}10\mathrm{dBW}$ . The results shown in Figure 5 can also prove the correctness of obtained probability expressions. Emulating a passive Eve $(P_E=0\mathrm{W})$ , Figure 6 displays the correct decision probability and false alarm probability versus massive antennas number $M$ . As $M$ increases, the effect of noise diminishesand the false alarm probability will decrease gradually. These also confirm the analytical results and our reasonable inferences at above-mentioned section. As expected from the discussions, Figures 4 and 6 reveal that threshold phase $\theta$ has an important influence on the detection performance. We can observe that false rejection probability is an increasing function of $\theta$ , whereas false alarm probability is decreasing in $\theta$ .

Figure 4.

Detection probability and false rejection probability versus parameter $P_E/P_B(\mathrm{dB})$ , when Eve is active.

Figure 5.

Detection probability versus the parameter $P_E(\mathrm{dBW})$ , when $P_B=2,5,8,\mathrm{and}10\mathrm{dBW}$ .

Figure 6.

Correct decision probability and false alarm probability versus antennas number $M$ , when Eve is passive $P_E=0\mathrm{W}$ and $P_B={\delta }_A^2$ .

Finally, some simulation results are presented to verify effectiveness of the energy-efficient power allocation scheme. Parameters are set to $P_B=10\mathrm{dBW}$ , $P_E=3\mathrm{dBW}$ , $P_C=5\mathrm{W}$ , ${\eta }_1={\eta }_2=1$ , $P_{max}=20\mathrm{dBW}$ , $M=200$ , and $R_{min}=0.6\mathrm{bits}/\mathrm{s}/\mathrm{Hz}$ . Moreover, initial value and threshold are set to $P_A=15\mathrm{dBW}$ and ${\epsilon }_q={10}^{-4}$ . The power optimization algorithm converges after 22 iterations. As seen in Figure 7, SEE is increasing as Alice power $P_A$ at lower power level. But when the power $P_A$ increases higher, the energy efficiency decreases rapidly. The reason is that the secrecy capacity increases logarithmically versus the power level. From the energy efficiency ergodic curve, we can easily observe that function $R_S$ is convex, which is in the agreement with previous analysis. Therefore, the optimal energy-efficient power allocation scheme can be obtained in theory.

Figure 7.

Illustration of the energy efficiency optimization iterative process and the energy efficiency curve versus Alice’s power $P_A$ .

Conclusion

In this article, we investigate the secure downlink transmission in massive MIMO system. For the considered system model, we provide the secrecy capacity analysis in closed form. From the analysis results, we know that massive MIMO is robust against passive eavesdroppers, but will be damaged by active pilot attack. Detection of the active attack is important for secure communication. A new detection scheme is presented. The legitimate receiver sends uplink pilots with random phases, then the detection can be operated comparing phase deviation change of the received pilot signals cross product. To maximize the SEE, an optimization scheme of power allocation is presented in detail. Simulation results validate the secrecy performance, effect of the detection scheme, and SEE of this massive MIMO system.