**1. Introduction**

Wireless traffic volume has exponentially grown in recent years and wireless data rates exceeding 100 Gbit/s will be required in the coming decades [1]. As a result, new frequency spectra are demanded to fulfill the broad bandwidth requirements for future communication. Among others, the THz band (0.06–10 THz) is regarded as a promising candidate to enable ultra-fast and ultra-broadband data transmission [2–5]. Recently, THz wireless communication systems are under rapid development and many wireless transmissions exceeding 100 Gbit/s have already been demonstrated in laboratories and in field environments [6–11], which bring THz communication closer to reality. However, ultrahigh-speed THz communications also pose major challenges to information security [12,13]. Once a malicious eavesdropper tries to intercept the signals, a vast amount of information will be leaked in the blink of an eye which is absolutely unacceptable, particularly in some sensitive fields such as the military and financial industry.

Security mechanisms exist at every layer of a network. Compared to the conventional upper-layer methods [14,15], physical-layer security (PLS) approaches [16–20] do not rely on the assumption that eavesdroppers have limited computational abilities and avoid distributing and managing secret keys [21–24]. In contrast to the broadcast nature of the microwave communication, highly directive THz waves are more prone to the blockage problem caused by the malicious eavesdropper [25,26]. Recently, researchers have comprehensively investigated the blocking effects of an illegal recipient and proposed a hybrid beamforming and reflecting scheme to eliminate the adverse effects [27,28]. In this environment, any eavesdropper intending to hide itself should control its size, otherwise, it may cast a detectable shadow and raise an alarm. Therefore, the performance of eavesdropping is restricted by the size of the illegal receiver. Alternatively, recent works have pointed

**Citation:** He, Y.; Zhang, L.; Liu, S.; Zhang, H.; Yu, X. Secure Transmission of Terahertz Signals with Multiple Eavesdroppers. *Micromachines* **2022**, *13*, 1300. https://doi.org/10.3390/ mi13081300

Academic Editors: Dmitri V. Lioubtchenko and Jeonghyun Kim

Received: 4 July 2022 Accepted: 9 August 2022 Published: 12 August 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

out that an eavesdropper may put a tiny passive object instead of itself, like a metal cup or a mobile phone, inside the narrow beam to scatter THz electromagnetic waves [29,30]. By this mean, the bulky illegal receiver placed outside the THz beam can capture the information signal without raising an alarm, as a consequence. We note that the feasibility of this scheme has already been demonstrated in experiments in which the eavesdropper can even intercept a signal strength as good as that of the intended receiver. Nevertheless, all the aforementioned work using scatter (tiny passive object) only consider a singleeavesdropper scenario while a case with multiple eavesdroppers has not been investigated. The reflector in the narrow beam scattering THz waves to multiple eavesdroppers may bring a greater security threat to the THz communication system.

Compared to the single eavesdropper, multiple eavesdroppers can increase the occurrence of stronger attackers that are closer to the legitimate transmitter due to the random spatial distribution [31,32]. Additionally, multiple eavesdroppers may also combine their own observations and jointly process their received message, which will considerably degrade the secrecy performance [33–35]. From a practical point of view, multi-eavesdropper scenes will be widespread phenomenon in our future, since potential eavesdroppers in the ubiquitous Internet of Things (IoT) may be some curious legitimate devices belonging to different subsystems [36]. However, secrecy performance and secure transmission schemes in highly directive THz communication systems have not been yet analyzed in the presence of multiple eavesdroppers. Moreover, how to safeguard this point-to-point THz system against randomly located eavesdroppers is still unknown.

In this paper, we comprehensively investigated the secrecy performance of a highly directive THz communication link with multiple eavesdroppers. We established the received signal models with two different multi-antenna techniques, namely traditional beamforming and AN beamforming, as transmission schemes for comparison. We note that the received signal mode is affected by the fading channel, where both the large-scale and small-scale effects matter. We emulate the effect of perfect electric conductor (PEC) parameters on the received signal-to-noise ratio (SNR) of Eve in a multiple-eavesdropper environment. We derive the mathematical framework of the STP and ESC in both noncolluding and colluding cases, so that the secrecy performance of the THz wireless link can be characterized. The results show that Eves can successfully intercept a huge amount of information by changing some parameters, such as the density, size, and distance. As a countermeasure, Alice could consider the deployment of the AN beamforming technique to counterbalance the adverse effect of multiple eavesdroppers.

The rest of the paper is organized as follows. In Section 2, we introduce the system model in the presence of multiple eavesdroppers. In Section 3, we analyze the STP and ESC in non-colluding and colluding cases, respectively. In Section 4, we conduct simulation experiments and demonstrate the factors affecting the secrecy performance. In Section 5, we discuss how one may find the attackers. Finally, we give a brief conclusion in Section 6. Additionally, the important notations in this paper are listed in Table 1 to make this paper clearer.


**Table 1.** Parameter settings.


**Table 1.** *Cont.*

#### **2. System Model**

In this section, we first propose a security model for the THz system, in which two transmission schemes, namely traditional beamforming and AN beamforming, are adopted to prevent being overheard by multiple eavesdroppers. Then, the details of the highly directive channel of Bob *hB* and the scatter channel of Eve *hE* are investigated, respectively.

#### *2.1. Signal Model*

As shown in Figure 1a, a transmitter (Alice) sends a highly directive THz wave to the receiver (Bob) in the presence of multiple eavesdroppers (Eves). A PEC on the origin *O* is put inside this narrow beam between Alice and Bob. When there is an incident beam, PEC will scatter the THz signal to Eves in all directions (see Appendix A). We note that the PEC is located at the very edge of the THz beam with only a sliver of THz wave so it will not cast a detectable shadow in the receiver Bob. Additionally, the PEC is a cylinder which has the advantage of being able to scatter light in all directions, giving an attacker more flexibility. We model the locations of multiple eavesdroppers by the homogeneous Poisson point process (PPP) Φ in a circle region of radius *RS* with a density *λp*, as shown in Figure 1b. The total number of Eves *NE* in PPP is a random variable but the average number can be determined by *NE* = *πR*<sup>2</sup> *<sup>S</sup>λp*. Due to the short transmission distance (*RS* < 15 m) in an indoor environment, all receivers are supposed to be in a high SNR regime. Alice has *N* antennas while Bob and all the Eves use only one antenna each for reception.

When traditional beamforming is adopted, the received symbols at Bob and *i*-th Eve are, respectively, given by:

$$y\_B = h\_B \mathfrak{x} + n\_{B'} \tag{1}$$

$$y\_{\text{E}\_l} = h\_{\text{E}\_l} \mathfrak{x} + n\_{\text{E}\_l} \,\, i = 1, 2, \dots, N\_{\text{E}\_l} \tag{2}$$

where *hB* and *hEi* are both 1 × *N* vectors denoting the channel between Alice and Bob and between Alice and the *i*-th Eve, respectively; *NE* is the total number of eavesdroppers; *x* = *pux* is the transmitted signal containing the beamforming vector *p* and signal *ux* with useful information; *nB* and *nE* are i.i.d. additive white Gaussian noise with *<sup>n</sup>* ∼ CN (0, *<sup>σ</sup>*<sup>2</sup> *n*). We assume that both Alice and Bob only know the CSI of *hB*, while Eve knows both *hB* and *hEi* perfectly, which is a more rigorous scenario for the security issue [37–39].

**Figure 1.** System model. (**a**) Alice transmits a highly directive THz signal *x* to Bob with or without AN *w*. A PEC (orange cylinder) located at the edge of beam can scatter the incident THz wave to Eves in all directions. (**b**) The spatial distribution of Eves is modeled as PPP in a circle region. The objects in this indoor scene can scatter THz signals.

With the introduction of AN beamforming, the transmitted THz signal *x* can be carefully designed as: *x* = *s* + *w*. The information signal *s* = *pus*, where the *N* × 1 beamforming vector *p* = *h*† *<sup>B</sup>*/||*hB*|| and signal *us* with a variance of *<sup>σ</sup>*<sup>2</sup> *us* . The AN *w* = *Zv*, where the *N* × (*N* − 1) matrix *Z* is the null space of vector *hB* so that *hBZ* = 0 while *hEZ* = 0 and noise vector *<sup>v</sup>* contains (*<sup>N</sup>* − <sup>1</sup>) random noise elements with a variance of *<sup>σ</sup>*<sup>2</sup> *v* . Consequently, the received signals of Bob and *i*-th Eve are, respectively, given by:

$$y\_B = h\_B(\mathbf{s} + \mathbf{w}) + n\_B = h\_B \mathfrak{p} \mathfrak{u}\_{\mathbf{s}} + n\_{B\prime} \tag{3}$$

$$y\_{E\_l} = h\_{E\_l}(s + \varpi) + n\_E = h\_{E\_l} \mu \iota\_s + h\_{E\_l} \varpi \upsilon + n\_E. \tag{4}$$

The AN *w* passes through the channel *hEi* and finally develops into the additional noise *hEiw*. We stress that, despite the AN, the *w* on Alice's side is sent to both the *i*-th Eve and Bob, whereas on the receiving side, the AN only deteriorates the *i*-th Eve without impacting Bob. As we can see, there is additional noise *hEiZv* on Equation (4) while there is no extra term on Equation (3).

The total transmitter power *P* = *E*[*x*†*x*] = *σ*<sup>2</sup> *us* + (*<sup>N</sup>* − <sup>1</sup>)*σ*<sup>2</sup> *<sup>v</sup>* , where (·)† denotes the conjugate transpose. We define *η* as the fraction of *σ*<sup>2</sup> *us* to the total transmit power *P*. When *η* = 1, the AN beamforming is equivalent to traditional beamforming as the information signal is transmitted with the full power *P*. We note that *η* is an important design parameter that can optimize the secrecy performance.
