# Cooperative Jamming-Based Physical-Layer Group Secret and Private Key Generation
## 2 System Model  
As shown in Figure 1, for the sake of brevity, a group communication network architecture is considered, which consists of three legitimate nodes (As shown later in this paper, the proposed group key generation scheme can be applied to the models with arbitrary number of legitimated nodes in a straightforward manner.) [PERSON], [PERSON], and [PERSON], along with a relay node and an [PERSON]. Each of these nodes has a single antenna and functions in a time-division duplex (TDD) mode of operation. The relay node can act as either a trusted or untrusted central node. We assume that [PERSON] can intercept information and estimate the channels but cannot interfere with the communication channels or alter any transmitted messages. Moreover, to maintain anonymity, [PERSON] positions himself more than half a wavelength away from group members, leading to the eavesdropping channels remaining independent from the legitimate channels. According to the channel reciprocity, the channel gains between the relay and [PERSON], [PERSON], and [PERSON] are \(G_{1}\), \(G_{2}\), and \(G_{3}\), respectively. Note that channel gains are also used to represent the respective channels, provided that this does not cause any confusion. Moreover, it is assumed that the channels among legitimate nodes cannot be utilized for key generation due to lacking randomness. Key generation relies on the relay channels \(G_{1}\), \(G_{2}\), and \(G_{3}\). Moreover, [PERSON] serves as the reference node. Without loss of generality, the channel between [PERSON] and the relay node, i.e., \(G_{1}\), is considered as the reference channel (To generate common randomness, the reference channel, \(G_{1}\), needs to be estimated by all legitimate nodes. It is worth noting that any relay channel can be chosen as the reference channel.). A typical application scenario is Unmanned Aerial Vehicle (UAV) communication, where [PERSON], [PERSON], and [PERSON] are UAVs, and the relay is a ground base station or control console. Since UAVs operate in the air, only line-of-sight (LoS) channels exist among them. However, the channels between the UAVs and the ground base station or control console include non-line-of-sight (NLoS) components; hence, there exists randomness that is suitable for key generation. For simplicity, the wireless channels are assumed to be Rayleigh (The proposed group key generation scheme can also be simply applied to the models with Rician channel assumption, by exploiting the NLoS components.) block fading channels, where the channel gains remain constant during the coherence time. Furthermore, \(G_{1}\sim\mathcal{CN}(0,p_{A})\), \(G_{2}\sim\mathcal{CN}(0,p_{B})\), and \(G_{3}\sim\mathcal{CN}(0,p_{C})\), where \(p_{A}\), \(p_{B}\), and \(p_{C}\) represent the variances of the channel gains \(G_{1}\), \(G_{2}\), and \(G_{3}\), respectively. In addition to wireless channels, a noiseless and infinite-capacity public channel where the group members communicate with each other is also considered. The public channel is commonly utilized in the majority of related existing works (e.g., [9; 10; 11; 12; 13; 14; 15; 16]). It is important to  
Figure 1: The model of group key generation and the process of training. note that all nodes, including Eve, have access to all the information transmitted over the public channel. Based on the above star network and model assumptions, the DMS model for group SK/PK generation is shown in Figure 2. The observations of [PERSON], [PERSON], and [PERSON] on random source are \(\tilde{Y}_{A}\), \(\tilde{Y}_{B}\), and \(\tilde{Y}_{C}\), respectively. Moreover, the observations of the relay and [PERSON] on random sources are \(\left(\tilde{Y}_{R}^{(1)},\tilde{Y}_{R}^{(2)}\right)\) and \(\tilde{Y}_{E}\), respectively. From their sequence of observations on random sources and information transmitted over the public channel, the legitimate members can generate an SK and a PK shared within the group. The SK is required to be confidential from [PERSON], whereas both the relay and [PERSON] have no knowledge of the PK.