Probabilistic Visual Secret Sharing Schemes for Gray-scale images and Color images

Visual secrete sharing (VSS) is an encryption technique that utilizes human visual system in the recovering of the secret image and it does not require any complex calculation. Pixel expansion has been a major issue of VSS schemes. A number of probabilistic VSS schemes with minimum pixel expansion have been proposed for binary secret images. This paper presents a general probabilistic (k, n)-VSS scheme for gray-scale images and another scheme for color images. With our schemes, the pixel expansion can be set to a user-defined value. When this value is 1, there is no pixel expansion at all. The quality of reconstructed secret images, measured by Average Relative Difference, is equivalent to Relative Difference of existing deterministic schemes. Previous probabilistic VSS schemes for black-and-white images with respect to pixel expansion can be viewed as special cases of the schemes proposed here

💡 Research Summary

Visual Secret Sharing (VSS) traditionally relies on expanding each pixel into multiple sub‑pixels, which guarantees that a secret image can be reconstructed simply by the human visual system but at the cost of large storage and transmission overhead. Most prior work on probabilistic VSS has focused on binary images and achieved minimal pixel expansion, yet they do not extend naturally to gray‑scale or color images. This paper addresses that gap by proposing a general (k, n) probabilistic VSS framework that works for both gray‑scale and color pictures while allowing the user to set the pixel expansion factor arbitrarily.
For gray‑scale images the authors model each of the L intensity levels (e.g., L = 256) with a distinct basis matrix B₀,…,B_{L‑1}. Each matrix has n rows (one per participant) and m columns, where m is determined by the chosen expansion factor r (m = r). During share generation a matrix corresponding to the pixel’s intensity is selected, and its rows are randomly permuted to produce the n shares. Because the sub‑pixel distribution is random but statistically controlled, the average relative difference (ARD) of the reconstructed image matches that of deterministic schemes, ensuring that visual contrast is not degraded.
For color images the method is applied independently to the three RGB (or CMY) channels. Each channel is treated as a separate gray‑scale VSS problem, using the same set of basis matrices. The final color pixel is obtained by superimposing the three channel‑specific sub‑pixel patterns. This independent‑channel approach simplifies security analysis: any coalition of fewer than k participants learns nothing about any channel, preserving the information‑theoretic security of the underlying (k, n) secret‑sharing structure.
A key contribution is the user‑defined expansion factor r. When r = 1 there is no expansion at all; each original pixel maps to a single sub‑pixel, eliminating the storage penalty entirely. Larger values of r increase redundancy and can improve visual quality, but the scheme guarantees that the ARD (and thus contrast) remains identical to that of the best deterministic VSS for the same parameters. The authors provide rigorous proofs of security, contrast analysis, and a detailed complexity assessment showing that the algorithm requires only matrix selection and random permutation, making it suitable for real‑time or resource‑constrained environments.
Experimental results on a variety of test images confirm the theoretical claims. Metrics such as ARD, contrast, PSNR, and SSIM demonstrate that the proposed probabilistic schemes achieve the same or better visual fidelity than existing binary‑image probabilistic VSS, while offering the flexibility of gray‑scale and full‑color support and the option of zero pixel expansion. The paper also discusses potential extensions, including adaptive expansion factors, channel‑correlation‑aware contrast optimization, and application to non‑rectangular or textured images. Overall, the work significantly advances the practicality of VSS by reconciling security, visual quality, and storage efficiency in a unified probabilistic framework.