## 1 Introduction  
In the information age, people's lives are inseparable from the Internet, and information security has become one of the most critical issues. Especially after the outbreak of COVID-19, online work, meetings and payment have become frequent. The public network brings convenience to people but also has various security risks. As we know, the guarantee of information security relies on the reliable cryptosystems. With the rapid development of optical information technology, many optical encryption schemes have been proposed. For example, [PERSON] et al. [1] proposed a double random phase encoding (DRPE) scheme in 1995 to encode information by using the phase characteristic of light, and this scheme has evolved into many variants [2; 3; 4; 5]. In the same year, the idea of ghost imaging (GI) was proposed by [PERSON] et al. [6]. It was first experimentally demonstrated with quantum entangled photon pairs [6] and later extended to true thermal light [7], pseudothermal light [8], X-ray [9] and particles [10; 11; 12]. Later research found that it can also be simplified from double-arm to single-arm by using a spatial light modulator (SLM) to perform optical encoding, which is called computational ghost imaging (CGI) [13]. Combined with compressed sensing (CS), the qualities of ghost images can be greatly improved [14]. Since GI generally uses completely random patterns to encode the object image, it has been successfully used for optical encryption [15; 16; 17]. It is not enough to just encrypt information, as encryption allows a certain error rate, since even if some part of the encrypted information is missing in the transmission process,it will not affect the overall content too much. If the cryptographic keys (CKs) themselves are distributed directly, any error will have a huge impact on the information to be encrypted, which requires higher security of its distribution. Thus, the cryptographic key distribution (CKD) is a hard nut to crack. In 1984, the famous BB84 protocol [18] was proposed to realize quantum key distribution (QKD). The QKD shows perfect security. Any eavesdropping can be detected because it has quantum mechanics as its theoretical support [19]. Presently though, quantum channels are still too expensive to be used in practice and are difficult to make compatible with traditional optical fiber networks. Furthermore, it is also hard for QKD protocols [18; 19; 20] to realize multi-party CKD. The generation of entangled light and single-photon measurements undoubtedly increase the complexity of the protocols, and impose high requirements on hardware devices, resulting in low key generation rates, high bit error rates, poor stability and low reliability for distributed CKs. Additionally, for long-distance transmission, expensive trusted quantum repeaters are required. Therefore, it is urgent to study a CKD protocol that can work with regular public channels and has the features of low cost, high efficiency and high security comparable with QKD. In our previous work, we designed CGI-based multi-party CKD protocols [21; 22] over a public network, where the modulated patterns are treated as pre-shared initial secret keys for later privacy amplification, the encrypted bucket values sampled by a CGI setup are sent to legitimate users through public channels. Each user can complete independent identity authentication via CS [21] and extract some digits after the decimal points of gray values in the images reconstructed by GI to form a random bit sequence (i.e., the distributed CK) [21; 22]. Since the public network is used, remote transmission is not difficult. Beyond that, the CKD protocol can also be improved in many other ways. For instance, [PERSON] et al. [23] proposed a camouflaged encryption method based on compressive GI, where the secret image is hidden in the camouflaged image to further improve the concealment of information. Later, they also proposed another hybrid encryption scheme based on temporal ghost imaging [24], which uses asymmetric public key cryptography to enhance security.