# Cooperative Jamming-Based Physical-Layer Group Secret and Private Key Generation
## 3 Group Secret and Private Key Generation Scheme
### Channel Measurement  
Channel measurements are repeatedly collected by the group members, based on the sampling of the legitimate communication channels \(G_{1}\), \(G_{2}\), and \(G_{3}\). Each fading block is divided into three time slots with \(T_{1}\), \(T_{2}\), and \(T_{3}\) symbol times, where \(T_{1}+T_{2}+T_{3}=T\). Additionally, assume that all transmitters employ a uniform transmit power of \(P\) in the training phase, and each receiver experiences a noise power of \(\sigma^{2}\). The details of the training phase for channel measurements are listed as follows.  
* In the first time slot, a predetermined sequence \(\mathbf{S}_{1}=\underbrace{\left[\sqrt{P},\ldots,\sqrt{P}\right]^{\prime}}_{T_{ 1}}\) is transmitted by the relay; thus, [PERSON], [PERSON], and [PERSON] receive \(T_{1}\)-sample sequence: \[\mathbf{Y}_{A}^{(1)} =G_{1}\mathbf{S}_{1}+\mathbf{N}_{A}^{(1)},\] (1) \[\mathbf{Y}_{B}^{(1)} =G_{2}\mathbf{S}_{1}+\mathbf{N}_{B}^{(1)},\] (2) \[\mathbf{Y}_{C}^{(1)} =G_{3}\mathbf{S}_{1}+\mathbf{N}_{C}^{(1)},\] (3) where \(\mathbf{N}_{A}^{(1)}\), \(\mathbf{N}_{B}^{(1)}\), and \(\mathbf{N}_{C}^{(1)}\) are Gaussian noise vectors of size \(T_{1}\times 1\). Then, [PERSON], [PERSON], and [PERSON] obtain the estimates of channels \(G_{1}\), \(G_{2}\), and \(G_{3}\), which are given by \[\tilde{Y}_{A} =G_{1}+N_{A},\] (4) \[\tilde{Y}_{B} =G_{2}+N_{B},\] (5) \[\tilde{Y}_{C} =G_{3}+N_{C},\] (6) respectively, with \(N_{A},N_{B},N_{C}\sim\mathcal{CN}\big{(}0,\sigma_{1}^{2}\big{)}\) and \(\sigma_{1}^{2}\triangleq\frac{\sigma^{2}}{T_{1}^{2}}\).  
Figure 2: Group SK or PK generation in the DMS model.  
* In the second time slot, based on the concept of [PERSON], [PERSON] and [PERSON] send a known sequence \(\mathbf{S}_{2}=[\underbrace{\sqrt{P},\ldots,\sqrt{P}}_{T_{2}}]^{\prime}\) at the same time; thus, the relay receives \[\mathbf{Y}_{R}^{(2)}=G_{1}\mathbf{S}_{2}+G_{2}\mathbf{S}_{2}+\mathbf{N}_{R}^{(2)}.\] (7) Then, the relay obtains the superimposed channel estimates regarding \(G_{1}\) and \(G_{2}\): \[\tilde{Y}_{R}^{(1)}=G_{1}+G_{2}+N_{R}^{(1)},\] (8) where \(N_{R}^{(1)}\sim\mathcal{CN}\big{(}0,\sigma_{2}^{2}\big{)}\) and \(\sigma_{2}^{2}\triangleq\frac{\sigma^{2}}{\tau_{2}^{2}P}\).
* In the third time slot, based on the concept of CJ, [PERSON] and [PERSON] simultaneously transmit a known sequence \(\mathbf{S}_{3}=[\underbrace{\sqrt{P},\ldots,\sqrt{P}}_{T_{3}}]^{\prime}\); thus, the relay receives \[\mathbf{Y}_{R}^{(3)}=G_{1}\mathbf{S}_{3}+G_{3}\mathbf{S}_{3}+\mathbf{N}_{R}^{ (3)}.\] (9) Then, the relay obtains the superimposed channel estimates regarding \(G_{1}\) and \(G_{3}\): \[\tilde{Y}_{R}^{(2)}=G_{1}+G_{3}+N_{R}^{(2)},\] (10) where \(N_{R}^{(2)}\sim\mathcal{CN}\big{(}0,\sigma_{3}^{2}\big{)}\) and \(\sigma_{3}^{2}\triangleq\frac{\sigma^{2}}{\tau_{3}^{2}P}\).  
Note that during each fading block, each member acquires one random channel measurement. Consequently, for every \(n\) fading blocks, [PERSON], [PERSON], and [PERSON] obtain independent and identically distributed (i.i.d.) \(n\)-dimensional vectors \(\mathbf{\tilde{Y}}_{A}\), \(\mathbf{\tilde{Y}}_{B}\), and \(\mathbf{\tilde{Y}}_{C}\), respectively. Since the relay can obtain channel measurements in both \(T_{2}\) and \(T_{3}\) time slots, the relay obtains \(n\)-dimensional sequences \(\mathbf{\tilde{Y}}_{R}^{(1)}\) and \(\mathbf{\tilde{Y}}_{R}^{(2)}\).