*2.3. Scatter Channel*

The scatter channel of Eve *hE* is given by:

$$h\_{\mathbf{E}\_l} = l\_{\mathbf{E}\_l} \mathbf{s}\_{\mathbf{E}\_{l'}} \tag{8}$$

where *l <sup>i</sup>* and *sEi* are, respectively, the large-scale factor and small-scale random vector of *i*-th Eve. The *lE* and *sE* are totally different from *lB* and *sB* owing to the PEC between Alice and Bob. The PEC between Alice and Bob is a kind of material with infinite conductivity and zero electric field inside. When the incident field *Ei* strikes the surface of PEC, it provokes a surface current *JZ* that generates a scattered field *Es* and total reflection occurs. By adopting the method of moments (MoM) [42], the scatter field *Es* around the PEC at *i*-th Eve is given by (see Appendix A):

$$E\_s = \frac{-k\eta\_0}{4\pi} \sqrt{\frac{\eta\_0 P G\_t}{k d\_{2\_i}}} \exp\{-j(k d\_{2\_i} - \frac{\pi}{4})\} \mathbf{C}^T \mathbf{A}^{-1} \mathbf{D},\tag{9}$$

where *k* is the wave number, *η*<sup>0</sup> 377 Ω is the intrinsic impedance of free space, *d*2*<sup>i</sup>* is the distance between the PEC and *i*-th Eve and the matrices **C**, **A**, **D** are determined by the shape, size, and location of the PEC. Here, we assume that the PEC is a cylinder with sufficient height. As such, we can denote the scattering coefficient *K*(*a*, *d*3) = **C***T***A**−1**D**, where *a* is the radius of PEC and *d*<sup>3</sup> is the distance between Alice and PEC. Therefore, the *lEi* can be derived as:

$$I\_{E\_i} = \sqrt{\frac{|E\_s|^2}{2\eta\_0} \frac{G\_r \lambda^2}{4\pi P}} = \frac{\eta\_0 \lambda K(a, d\_3)}{8\pi} \sqrt{\frac{kG\_l G\_r}{2\pi d\_{2\_i}}},\tag{10}$$

where we assume that Bob and all Eves have the same antenna gain *Gr*.

The scattering coefficient *K* is influenced by *a* and *d*3. As shown in Figure 2, the THz wave nearly scatters uniformly around the PEC center (*d*<sup>2</sup> *λ*, [42]) and the scattered field gradually fades along as it becomes farther away from the center. The scattering coefficient *K* increases with radius *a* and decreases with *d*3, as we can see since the color in Figure 2b is deeper than that in Figure 2a.

**Figure 2.** The scattered fields of PEC for (**a**) *a* = 20 mm, *d*<sup>3</sup> =2m(**b**) *a* = 40 mm, *d*<sup>3</sup> =2m (**c**) *a* = 40 mm, *d*<sup>3</sup> = 1.5 m. The maximum values were cut off at 8 since only a few values exceed it.

Unlike the main channel wherein a direct line-of-sight (LOS) link exists between Alice and Bob, Eve indirectly receives the signal information from non-line-of-sight (NLOS)

transmission. Many rays will scatter from PEC and finally converge on Eve's side as each point on the surface of PEC can generate an electromagnetic field. As such, a tiny move of PEC or Eve may tremendously change the received signal strength. Therefore, we assume *sEi* ∼ *Nakagami*(1, 1), which is also a *Rayleigh* distribution. Based on Equation (4), the SNR of *i*-th Eve is given by:

$$\text{SNR}\_{E\_i} = \frac{S\_{E\_i} L\_{E\_i} P \eta}{\frac{AL\_{E\_i} P (1 - \eta)}{N\_A - 1} + \sigma\_n^2} \stackrel{(a)}{\leq} \phi S\_{equal\_i} \tag{11}$$

where *A* ∼ *Gamma*(*N* − 1), *SEi* ∼ *Exp*(1), *LEi* = *l* 2 *Ei* , the PDF of random variable *Sequali* is given by *fSequal*(*x*) = *<sup>N</sup>*−<sup>1</sup> (1+*x*)*<sup>N</sup>* and *<sup>φ</sup>* <sup>=</sup> *<sup>η</sup>*(*N*−1) <sup>1</sup>−*<sup>η</sup>* , (*a*) holds for considering the worst-case situation where the normalized noise *σ<sup>n</sup>* are arbitrarily small. Note that this approach was also taken in [16,35,37].

### **3. Secrecy Performance**

In this section, we introduce STP and ECS which are both secrecy performance metrics. Then, we analyze the secrecy performance with and without AN in both non-colluding and colluding cases.
