# Cooperative Jamming-Based Physical-Layer Group Secret and Private Key Generation
## 5 Numerical Results  
To verify the security performance of the proposed CJ-based key generation scheme, we numerically analyze the analytical results presented in previous sections. In this section, we consider a scenario with a group size of \(N(\geq 3)\). Especially, when \(N=3\), \(U_{1}\), \(U_{2}\), and \(U_{3}\) represent [PERSON], [PERSON], and [PERSON], respectively. Moreover, the channel \(G_{i}(i=1,2,\ldots,N)\) is modeled as \(G_{i}=g_{i}d_{G_{i}}^{-l/2}\), where \(g_{i}\sim\mathcal{CN}(0,1)\), \(d_{G_{i}}\) is the distance between the relay and \(U_{i}\), and \(l\) is the pass-loss exponent. Therefore, \(G_{i}\sim\mathcal{CN}(0,d_{G_{i}}^{l})\). Additionally, to more accurately portray the influence of the parameter \(T\), the metric unit utilized for measuring the key rate in this section is \"bits per symbol time\" (BPST). For brevity, the coherence time is divided into \(N\) time slots, each with a duration of \(T_{0}=5\); all nodes have the same transmit power \(P\); the distances between the group members \(U_{i}(i=1,2,\ldots,N)\) and the relay are assumed to be equal, denoted by \(d\) in meters, with a path loss coefficient of \(l=3\). The distance value is within the range of 10 to 100 m, and the noise power is set as \(\sigma^{2}=-50\) dBm [27]. Figure 4 shows the achievable SK/PK rate of the proposed group key generation scheme versus the distance \(d\), with a transmit power of \(P=10\) dBm and \(N=3\) or 5. As depicted in the figure, both the SK and PK rates decrease as \(d\) increases, as longer distances result in lower received SNR at each receiver during the channel measurement process. For the scenario where \(N=3\) and \(d=10\) m, the achievable SK and PK rates are approximately 0.7 and 0.6 BPST, respectively. Furthermore, with \(N=5\) and \(d=10\) m, the SK and PK rates decrease to around 0.33 and 0.43 BPST, respectively. This indicates a difference of about 0.1 BPST in the achievable SK and PK rates for small values of \(d\). Figure 5 showcases the SK and PK rates versus the transmit power \(P\), with \(d=10\) m and \(N=3\) or 5. As shown in Figure 5, when \(N=3\) and \(P=-10\) dBm, the SK and PK rates are 0.275 BPST and 0.174 BPST, respectively. In addition, when \(N=3\) and \(P=0\) dBm, the SK and PK rates become 0.493 BPST and 0.387 BPST, respectively. One can observe that the achievable rates of the SK and PK increase as the transmit power \(P\) increases. This is because high transmit power improves the accuracy of channel measurement. Furthermore, when \(N=3\) or 5, a noticeable difference of about 0.1 BPST exists between the SK and PK rates, illustrating the amount of information revealed to the relay, as described in (21). Figure 6 shows the impact of the group size on the achievable SK/PK rate, where \(d=10\) m and \(P=0\) or 10 dBm. As shown in this figure, when \(P=0\) dBm, the SK and PK rates with \(N=3\) are 0.493 BPST and 0.387 BPST, respectively. Furthermore, the achievable SK and PK rates with \(N=6\) are 0.246 BPST and 0.161 BPST, respectively. One can observe that as the group size increases, both the SK and PK rates decrease. This decline can be attributed to the necessity for more time slots for channel measurement as the group size grows. Specifically, when \(N=3\), the coherence time required is \(T=15\). However, when \(N=6\), the coherence time extends to \(T=30\). Figure 4: The achievable key rates of SK and PK versus the distance \(d\). Figure 5: The achievable key rates of SK and PK versus the transmit power \(P\). Figure 7 shows the SK/PK upper and lower bounds as they vary with the transmit power \(P\), considering \(N=3\) and \(d=10\) m. The representation clearly shows that both these capacity bounds increase as the transmit power \(P\) increases. Furthermore, in scenarios characterized by high transmit power (i.e., high transmit SNR), a consistent gap of around \(\frac{1}{15}\log_{2}\left(\frac{3}{2}\right)\) BPST is maintained between the SK/PK upper and lower bounds. This observation aligns with Theorem 1 through the transformation of the metric unit.