## 5 Conclusions  
In conclusion, here we proposed a CGI-based multi-party interactive CKD protocol over a public network, where a QR code image of Version 2 is used as the container of CKs and its functional region is treated as the watermark-embedding region. Since the QR code is used for CK extraction, the content of CKs will no longer be limited to binary streams, and can contain more information, such as video, pictures, hyperlinks and so on. In this protocol, the QR code image is split into multiple FPs, in each of which an independent \"watermark\" image label that corresponds to each user will be embedded in its functional region and then be treated as an original object image to be sampled. The modulated patterns are shared with the legitimate users in advance. The watermark-embedded FPs will be separately encrypted into random bucket value sequences via a CGI setup and be sent to users through public channels. This encryption process makes the \"watermark\" labels undetectable, and ensures the confidentiality and imperceptibility of the CKs. On the receiving end, each user can recover the ghost image and compute its binarized image. The content of the recovered \"watermarks\" will be used to perform users' identity self-authentication. Thus, the watermarking technology used here helps to strengthen the protocol's security. Furthermore, the spatial resolution of region patterns that locate in the watermark-embedding region of the FP can be different from that of non-watermark-embedding region, which can concentrate the wrong points more in or around the watermark-embedding region when an attack occurs. This makes the attack easier to detect. In addition, the use of cascade ablation in the intermediary realizes interactive joint authentication, adding second protection against illegal attacks. By scanning the returned cascade ablation result (recovered QR code), legitimate users can quickly acquire their distributed CKs. Both numerical simulations and optical experiments have demonstrated the feasibility of this protocol and its susceptibility to attacks. We have also performed some noise addition tests to show that this protocol has a certain tolerance for noise in the public channels. Therefore, this protocol may provide a new means of utilizing watermarking-based self-authentication and cascade-ablation-based interactive authentication in high-security CKD applications.  
Conceptualization, W.-K.Y.; methodology, W.-K.Y. and Y.-X.L.; software, W.-K.Y., Y.-X.L. and S.-F.W.; validation, W.-K.Y., Y.Y., Y.-X.L. and S.-F.W.; formal analysis, W.-K.Y., Y.-X.L. and N.W.; investigation, W.-K.Y. and Y.Y.; resources, W.-K.Y.; data curation, W.-K.Y., Y.-X.L. and N.W.; writing--original draft preparation, W.-K.Y., Y.Y. and Y.-X.L.; writing--review and editing, W.-K.Y.; visualization, W.-K.Y.; supervision, W.-K.Y.; project administration, W.-K.Y.; funding acquisition, W.-K.Y. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Beijing Natural Science Foundation (4222016), the Civil Space Project of China (D040301) and the Youth Talent Promotion Project of the Beijing Association for Science and Technology (none).  
Not applicable.  
The authors declare no conflict of interest.