## 1 Introduction
### Related Work  
As a potential future communication network architecture, the integrated satellite multiple terrestrial relay network (ISMTRN) absorbs and combines the advantages of SatCom and terrestrial communication, which guarantees the communication demands of the users and diversies the communication ways [3, 4, 5]. It is becoming the research hotspot and has important practical significance for future communication development. In [6], the paper studied the performance of non-orthogonal multiple access (NOMA)-based overlay cognitive ISMTRN with secondary network selection. A hybrid satellite-terrestrial relay network (HSTRN) consisting of a satellite and multiple terrestrial nodes was considered in [7]. Moreover, the authors of [8] investigated the secure transmission in an HSTRN. In [9], the authors studied the impacts of joint relay selection and user scheduling schemes on the physical layer security for the HSTRN. The performance of an ISMTRN with the threshold-based decode-and-forward (DF) protocol was investigated in [10]. The authorsof [11] established the multi-relay hybrid satellite-terrestrial systems (HSTSs) with artificial noise (AN) to investigate their reliability and security. However, due to the broad area of satellite beams and the openness of wireless communication, communication security risks are inevitable, such as information eavesdropping, information tampering, malicious interruption interference, source location listening, and others. Traditionally, secure communication is usually achieved by high-level encryption and decryption technology. The authors of [12] used the features of quantum walk to construct a new substitution box method which played a significant role in block cipher techniques for 5G-IoT technologies. In [13], a forward secure public-key searchable encryption scheme was proposed. Besides, physical layer security (PLS) technology offers another promising direction for the security of communication networks by exploiting the inherent randomness of the physical layer wireless fading channel [1; 2; 9], and some other new security techniques are also widely adopted. In [14], the authors presented a method to obtain visible light communication PLS. An adaptive physical-layer key generation scheme was proposed in Smart Homes in [15]. The authors of [16] considered designs that mask the user's identity during communication, resulting in anonymous communications, and in particular, examined the recent interest in physical layer anonymous solutions. Moreover, in [17], the authors provided a tutorial overview of the promising intelligent reflection communication (IRC) technologies and articulated multiple ways of enhancing the security of Internet-of-Everything (IoE) by the IRC. The authors of [18] discussed the challenges and opportunities of information security for next-generation integrated sensing and communication (ISAC) networks. The authors introduced the concept of physical layer anonymity and revealed some related conclusions in [19]. Nevertheless, none of those security techniques mentioned above can completely solve the security problems because eavesdroppers can obtain critical encrypted information by analyzing the eavesdropped traffic data. Thus, to further the safety of the ISMTRN, we utilize covert communication in this paper. The \"prisoner model\" proposed by [PERSON] in 1983 is a classic model in the field of covert communication. It assumed that in prison, two prisoners need to communicate. However, due to a malicious eavesdropper, [PERSON] (passive or active listening), they could not communicate directly and needed to hide the information by some means. Hence, they exploited the randomness of the wireless communication channel and the uncertainty of noise to achieve covert information transmission to prevent the communication signal from being detected by [PERSON] [20]. Namely, if the malicious eavesdropper cannot confirm the existence of the signal, it is difficult for it to carry out further illegal acts.