# Verifiable (2, \(n\)) Image Secret Sharing Scheme Using Sudoku Matrix
## 1 Introduction  
As Internet technology becomes more sophisticated, users have grown accustomed to backing up digital content, such as text files, photos, and videos, to cloud platforms. These technologies allow users to easily access their photos from anywhere; however, if the platform is not trustworthy, personal privacy can be jeopardized. Image encryption [1; 2] is a reliable tool for traditional information security. Files containing sensitive data are usually encrypted prior to uploading. However, traditional encryption methods face shortcomings; for instance, an encrypted image is meaningless and thus easily attracts the attention of hackers, who try to decrypt them through data analysis and data mining. The meaningless content is difficult to manage. In order to facilitate management, image information can be carried under the encrypted image [3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28], so as to identify the content according to the extracted information later. However, since traditional data are hidden in a single carrier, once the carrier is discovered, the probability of the code being cracked greatly increases. In order to reduce the risk that comes from using a single carrier, [PERSON] and [PERSON] [29] proposed a \((t,n)\) threshold secret sharing approach that first decomposes the secret into \(n\) parts, which then jointly provide their share to reconstruct the secret, where \(t\) is less than or equal to \(n\). It then distributes them to different participants of \(n\). Afterward, participants are greater than \(t-1\). However, the shares are still meaningless to users, which avenues the suspicion of malicious attackers. Therefore, secure Internet usage and data privacy have become major issues and given rise to various data-hiding technologies that embed shared content into images. [PERSON] et al. [30] proposed a reversible secret image sharing system (SIS) using an exploiting modification direction (EMD) matrix. The shared images are embedded into thecover images as stego images. After embedding, the stego image has high visual quality. The shared image can be extracted, and then the secret image can be reconstructed through calculation. To improve the visual quality, [PERSON] et al. [31] proposed an SIS scheme based on the Sudoku matrix and Lagrange polynomial. Later, [PERSON] et al. proposed a reversible image secret sharing scheme [32]. To guarantee the integrity of the shares, the authentication mechanism authenticates which pixels are modified or lost during transmission. In 2018, [PERSON] and [PERSON] [33] proposed applying the authentication mechanism to secret sharing to authenticate whether the shadow images have been tampered with (modified) or not. In 2021, [PERSON] et al. [34] proposed a high-capacity secret sharing scheme with an authentication mechanism. To achieve reversibility, a reversible and verifiable SIS scheme has also been proposed. [PERSON] et al. [35] designed a stick insect matrix to verify the shared image, and embedded the verification code and the secret image into the shared image. However, the shared images generated by these reversible SIS schemes rely on a single cover image, which may increase the possibility that confidential images may be identified through statistical analysis of subtle differences between share images. In order to prevent the discovery of secret images by statistical analysis, [PERSON] et al. [36] and [PERSON] et al. [37] generated shares from different cover images. Nevertheless, these methods cannot construct secret images losslessly. In this paper, we encode a secret image into \(n\) shares using the concept of Sudoku. The schemes [33; 34; 35; 36; 38] are not allowed more than three participants. The scheme allows \(n\) joint participants, so it is more flexible than the above schemes.