*4.2. AN as a Countermeasure*

We find that the AN beamforming can compensate for the detriment of multiple eavesdroppers. As shown in Figure 7, the increase in *λ<sup>p</sup>* causes an STP (*P*(*CS* ≥ 0)) reduction from 0.85 to 0.75 and 0.5 to 0.1, respectively. However, with the introduction of AN in the non-colluding case, the STP (*P*(*CS* ≥ <sup>0</sup>)) rises to 0.95, leading to an improvement

of nearly 27%. In the colluding case, the STP rises to 0.7, corresponding to an improvement of 600%. It is noteworthy that the detriment of multiple eavesdroppers in the colluding case is more than that in the non-colluding case. In the non-colluding case, for *R* > 2, the STP with AN beamforming (*λ<sup>p</sup>* = 0.02) is higher than that with traditional beamforming (*λ<sup>p</sup>* = 0.01). However, in the colluding case, the situation is reversed for *R* > 2 which means that colluding eavesdroppers cause greater damage to transmission security.

**Figure 7.** The benefit of AN on STP for (**a**) a non-colluding case; and a (**b**) colluding case. Other parameters are given by: *G* = 25 dBi, *N* = 3, *f* = 300 GHz, *P* = −10 dBm, *RS* = 15 m, *η* = 0.3.

In Figure 8, we find that the optimal *η* depends on both density *λ<sup>p</sup>* and the number of antennas *N*. For *λ<sup>p</sup>* = 0.01 (blue and yellow line), the optimal *η* in the non-colluding and colluding cases is 0.28 and 0.22, respectively, which are larger than 0.21 and 0.15 for *λ<sup>p</sup>* = 0.02. More Eves around PEC signify stronger information attacks. Therefore, Alice must allocate more transmission power to AN to resist the adverse effect of the added Eves. Additionally, the optimal value of *η* increases with *N*. As shown in the inset, the optimal *η* are 0.34 and 0.27 for *N* = 6 while 0.28 and 0.22 for *N* = 2. We stress that only Bob benefits from the increase in antennas since the transmitter maximizes the signal strength to Bob and the signal power at Eves' side remains unchanged.

**Figure 8.** The optimal *η* under different *λp* and *N*. The solid line describes non-colluding cases while the dashed line describes colluding cases. The main figure for *N* = 2 while the inset for *N* = 6. Other parameters are given by: *G* = 27 dBi; *f* = 300 GHz; *P* = −10 dBm; and *RS* = 15 m.

In Figure 9, we find that the ESC decreases with the *f* while increasing with the *P* when *η* = 1. For a system without AN, the ESC will not be influenced by *P* since SNRB and SNRE benefit from them to the same extent, as shown by Equations (7) and (11). However, with the introduction of AN, *P* can no longer influence the supremum of SNRE but still impacts SNRB. Additionally, we also find *P* and *f* cannot significantly change the optimal *η*. In Figure 9a,b, the optimal *η* varies in the ranges of 0.27∼0.31 and 0.26∼0.3 with standard deviations (STD) of 1.13 × <sup>10</sup>−<sup>2</sup> and 1.14 × <sup>10</sup><sup>−</sup>2, respectively, lending to a tiny change. We note that despite Figure 9 only showing a non-colluding case, the same rule can also be applied to the colluding scenario.

**Figure 9.** Secrecy performance in a non-colluding case. (**a**) The ECS as a function of *η* and *f* with *P* = −10 dBm; (**b**) The ECS as a function of *η* and *P* with *f* = 300 GHz. Other parameters are given by: *G* = 25 dBi, *N* = 3, *RS* = 15 m, *λ<sup>p</sup>* = 0.02.
