Reversible data hiding in encrypted images (RDH-EI) schemes based on secret sharing have emerged as a significant area of research in privacy protection. However, existing algorithms have limitations, such as low embedding capacity and insufficient privacy protection. To address these challenges, this paper proposes an RDH-EI scheme based on adaptive median edge detection (AMED) and matrix-based secret sharing (MSS). The algorithm creatively leverages the AMED technique for precise image prediction and then integrates the (r, n)-threshold MSS scheme to partition the image into n encrypted images. Simultaneously, it embeds identifying information during segmentation to detect potential attacks during transmission. The algorithm allows multiple data hiders to embed secret data independently. Experimental results demonstrate that the proposed algorithm significantly enhances the embedding rate while preserving reversibility compared to current algorithms. The average maximum embedding rates achieved are up to 5.8142 bits per pixel (bpp) for the (3, 4)-threshold scheme and up to 7.2713 bpp for the (6, 6)-threshold scheme. With disaster-resilient features, the algorithm ensures $(n-r)$ storage fault tolerance, enabling secure multi-party data storage. Furthermore, the design of the identifying information effectively evaluates the security of the transmission environment, making it suitable for multi-user cloud service scenarios.