## 1 Introduction
[26] exploited the additive homomorphism property of secret sharing and combined it with difference expansion to embed secret data into a pair of pixels, which reduces the time complexity; however, this scheme has only one data hider. [PERSON] et al. [27] proposed a scheme with multiple data hiders, but its ER is limited by the number of hiders. [PERSON] et al. [28] used the Chinese remainder theorem to design an SS-RDHED, which achieves separation and better reversibility. [PERSON] et al. [29] proposed SS-RDHED on the Galois field (GF)(\(p\)) and GF(\(2^{8}\)), which has a low distortion rate and solves the pixel overflow problem. To solve the disadvantage of traditional secret sharing without diffusion, [PERSON] et al. [30] constructed a polynomial encryption method in Cipher Feedback mode that can effectively resist a variety of attacks. Subsequently, [PERSON] et al. [31] proposed a scheme using matrix secret sharing, which can extract data without loss, and its embedding capacity is much larger than that of the existing scheme. To adapt to distributed systems with multiple participants, [PERSON] et al. [32] utilized the comprehensive decoding of Hamming codes and additive homomorphism of polynomials; multiple data hiders can embed the shared data into the shared image. Recently, [PERSON] et al. [33] designed an embedding scheme based on secret sharing without preprocessing, and a reliable key management system was not required to securely manage the secret image. [PERSON] et al. [34] proposed a new SS-RDHED based on the Chinese residual theorem and hybrid coding, which utilizes iterative encryption to preserve the correlation between encrypted blocks and has better embedding performance. [PERSON] et al. [35] used secret sharing to divide secret data and then used the proposed modification-recomputation strategy to generate labeled shared images, obtaining fault tolerance and full separability. [PERSON] et al. [36] used pixel value order technology and secret sharing, allowing multiple data hiders, which can address the limitations of single data hider scenarios, enhancing the scheme's flexibility and security, particularly in applications requiring robust copyright protection. Most of the existing SS-RDHEDs have been developed assuming a static security environment where the threshold remains constant. However, in reality, security landscapes are dynamic, necessitating adjustments to the threshold based on varying situations and risks. For instance, within the financial sector, the information security standards of banks may require modifications to address diverse risks and threats. During normal circumstances, banks might maintain a relatively lower information security level to offer more convenient services and enhance operational efficiency. Yet, in times of financial crises or other significant events, banks may swiftly elevate their information confidentiality protocols and bolster security measures to preempt potential threats and attacks. To adapt to the variable security environment, [PERSON] et al. [37] proposed changing the threshold in a bivariate polynomial-based secret image-sharing scheme (TCSIS), which can provide multiple possible thresholds. When the secret level is low, fewer participants can recover the secret information, and when the secret level is high, more participants are required to recover it. Using this scheme to design RDH-ED has extensive application potential, and there is a large amount of coefficient redundancy available for embedding additional data during the construction of binary polynomial coefficients. However, this scheme uses all the image pixels as the coefficient matrix of the binary polynomial, which is susceptible to exhaustive attacks due to the strong correlation between the pixels in each row and column. Therefore, the resistance to attacks needs to be improved by eliminating the correlations between pixels. The traditional method is to encrypt the carrier image, but the complex encryption process affects its practicality.