## 1 Introduction
[21] introduced a central folding method, which folded the encoded secret messages into two equal digits before concealing them into two stego images. [PERSON] et al. [22] proposed a similar scheme making use of LSB matching. [PERSON] et al. [23] improved these works [21], offering a larger embedding capacity. Recently, [PERSON] et al. [26] extended the dual-image RDH scheme to include the turtle shell reference matrix, and [PERSON] and [PERSON] [27] improved their work by reducing the pixel distortion. [PERSON] et al. [28] utilized the rhombus prediction and error shifting prediction to produce dual stego images with high fidelity. They embedded a half number of the secret bits based on prediction to render a tentative stego image. They then predicted the blank pixels in the tentative stego image to convey the other half number of bits. Although the current state-of-the-art works [30; 31] can affectively carry secret messages, providing both satisfactory embedding capacity and stego image quality, they are limited to carrying a single level of secret message. We consider a new application scenario called \"multi-hider reversible data hiding\", where several authorized recipients can extract different secret messages using elaborate secret key combinations from dual stego images. In this application, a sender first embeds \(y\) different secret data, where each secret message is injected into the cover image by the corresponding secret keys. The produced dual stego images are then uploaded to a cloud stego server or sent to the authorized receivers. The legitimate recipients who hold authorized key(s) can access the cloud stego server and extract one or any parts of the secret messages. The key combination produces \(L\) number of message extraction types, where \(L=C_{1}^{y}+C_{2}^{y}+\ldots+C_{y}^{y}\). Certainly, the recipient who holds all secret keys denotes the supervisor with the highest rank, able to extract all secret messages. The multi-hider mechanism, together with secret key combinations, provides an advantage to differentially extract assorted secret messages and thus applications therein. In this paper, we propose a multi-hider dual-image reversible data-hiding scheme (MHDI-RDH) satisfying the above application scenario. Our algorithm can convey three secret messages (\(y=3\)) and provide seven levels of secret message extraction (\(L=7\)) from dual stego images. The message embedding in our scheme contains two phases. In the first phase, differing from conventional image data-hiding schemes, which utilize the existing image as the cover one to embed messages, our algorithm conceals the first secret message when constructing a new image through an optimal weighted color-transfer process. Specifically, we synthesize a color-transferred image and concurrently convey the first 3-tuple \(n\)-ary secret message vector (\(\mathbf{d_{sb}}\)) using the first secret key (\(key_{1}\)), thus producing a tentative stego image. We then embed the second \(m\)-ary secret message (\(d_{wm}\)) into this tentative stego image and produce a new tentative stego image, \(l^{\prime\prime}_{qncv}\), using the second secret key (\(key_{2}\)). This message-embedding technique is designed elaborately so that the first hidden message survived intact. Furthermore, in the second phase, we conceal the third 3-tuple \(k\)-ary secret message vector (\(\mathbf{d_{du}}\)) with the reversibility benefit using the third secret key (\(key_{3}\)) and produce the final dual stego images, \(l^{\prime\prime}_{11}\) and \(l^{\prime\prime}_{22}\). Such a multi-hider approach enables us to independently embed a total of three secret messages, \(\mathbf{d_{sb}}\), \(d_{wm}\), and \(\mathbf{d_{du}}\), which, in return, allows the recipients to adopt one of seven types of key combinations to flexibly extract secret messages. This is in contrast to most current state-of-the-art works, which provide the receiver to extract single-level secret data from the dual stego images.