# Cooperative Jamming-Based Physical-Layer Group Secret and Private Key Generation
## 1 Introduction  
With the widespread application of wireless devices, the demand for information sharing and communication among multiple devices is increasing, making ensuring information security especially crucial. Physical layer key generation (PLKG), based on information theory, has gained extensive attention as an effective approach for safeguarding wireless communication [1; 2; 3; 4; 5; 6; 7; 8]. Most of the existing studies focus on PLKG for two legitimate users. However, the generation of a group key for secure group communications is essential in various scenarios, such as corporate meetings, military operations, healthcare teams, family or personal groups, and more. For example, in a business environment, a team of employees working on a confidential project may require secure communication. The generation of a group key ensures that only authorized team members can access sensitive information and discussions. Military units often rely on secure communication channels to share strategic information, where a group key guarantees that only authorized personnel can access and exchange mission-critical data. Medical professionals engaged in patient care or research may need a secure communication channel to exchange patient data and medical information, ensuring privacy and compliance with regulations. Similarly, a group of travelers may wish to share travel plans and photos exclusively within their group, ensuring that this information remains private and inaccessible to individuals outside the group [9]. Compared to the key generation involving just two parties, the key generation among multiple terminals faces more challenges. These challenges primarily stem from two mainreasons. First, due to the large number of users in the group and varying distances between them, the efficiency of protocols for secure group communication is of greater concern. Second, the presence of numerous random channels among group users adds complexity to ensure that only authorized users within the group can obtain the final group key. Ref. [10] first studied the problem of group key generation for multiple users, considering a scenario where a group of users generate keys shared by all users within the group with the assistance of other users. Since then, refs. [11; 12; 13; 14] investigated pairwise key generation schemes to generate keys shared within the group. These schemes first generate pairwise keys between two users and then generate the final group key by exploiting these pairwise keys. Additionally, ref. [15] studied the collaborative secret key generation scheme based on received signal strength, applicable to both star and chain network typologies. Ref. [16] investigated the cooperative group key generation based on secure network coding (SNC) via star topology. Moreover, ref. [17] introduced an efficient scheme for group key generation utilizing multiple-input multiple-output (MIMO) technology, which can improve the overall efficiency of key generation. Most of the studies mentioned above have focused on the generation of keys with a single security level [9; 11; 12; 13; 14; 15; 16; 17]. However, in practice, the generation of keys with different security levels is also important. For example, in a group of terminals, each terminal may have a different security clearance level, and access to confidential information is based on individual clearance levels. Terminals with the same clearance level should share a common key while remaining unaware of keys at higher levels. Several studies have addressed the generation of multiple keys with different security levels [18; 19; 20; 10; 21; 22; 23; 24]. Refs. [18; 19; 20] investigated schemes for generating a secret key (SK) and a private key (PK) among three terminals. In these schemes, the three terminals aim to share an SK that remains secret from an eavesdropper (Eve), while two legitimate terminals aim to generate a PK that is also concealed from the third terminal. Since then, refs.