## 3 Main Scheme
### Algorithm III. Meaningful Shares Scheme
At the same time, there are two parameters \(\gamma\) and \(p\) in the proposed VCSs that will affect the light transmittance of the shares and restored images. These two values can also be determined by the user, which makes the proposed VCSs more flexible. Therefore, the user can decide the appropriate method and parameters according to the situation. The time complexity of the encryption process of the proposed VCSs is O(\(wln\)), and the time complexity of the decryption process is O(\(wlnN\)). Theoretical analysis and experimental results show that the proposed VCSs are safe and effective. All our proposed VCSs have the following characteristics:

1. No pixel expansion. 2. Multiple secrets can be encrypted at the same time. 3. Each share can be disguised as a different meaningful image. 4. Both shares and reconstructed images have good visual quality. 5. Parameters \(\gamma\) and \(p\) can be adjusted as required. Table 6 shows the comparison of the proposed scheme with some previous research. Among these four proposed methods, according to the analysis in Section 4, if the user needs to obtain one of the clearer restored images, then Method 3 is the best recommendation; otherwise, Method 1 performs better than other methods in all aspects. However, Method 4 can obtain perfect black restoration, which is helpful for visual recognition of the restored image, and is also a recommended method when \(N\) and \(n\) are large. The VCS presented in this paper has many topics for further study. For example, how to enhance the proposed VCSs to apply to grayscale or color images, modify the proposed VCSs to extend the encrypt ability of \((n,n)\)-threshold to \((k,n)\)-threshold for any \(k<n\), and consider whether the ability of the proposed VCSs can be increased, so that the OR operation can also be used to recover the secret image. Besides, since we initially encrypt the \(N\) secret images into two shares in Algorithm I, although these two shares are subsequently encrypted into \(n\) shares in Algorithm II, this process adds distortion of the schemes. Therefore, it will be fascinating to work on in the future to directly encrypt \(N\) secret images into \(n\) shares in order to reduce distortion. Conceptualization, J.S.-T.J.; methodology, J.S.-T.J. and S.-Y.H. ; software, S.-Y.H. ; validation, S.-Y.H., A.-h.L. and J.S.-T.J.; formal analysis, S.-Y.H. and J.S.-T.J.; investigation, J.S.-T.J.; resources, S.-Y.H. ; data curation, S.-Y.H. ; writing--original draft preparation, S.-Y.H. and A.-h.L. ; writing--review and editing, [PERSON]-T.J.; visualization, S.-Y.H. and A.-h.L. ; supervision, [PERSON]-[PERSON]; project administration, [PERSON]-T.J.; funding acquisition, [PERSON]-T.J. All authors have read and agreed to the published version of the manuscript. This research was funded by the Ministry of Science and Technology of Taiwan, ROC, Grants number MOST 110-2221-E-260-003, and 111-2115-M-260-001. Not applicable. The authors would like to thank the reviewer for the constructive feedback. The authors declare no conflict of interest. ## References

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