# Secret Image Sharing Revisited: Forbidden Type, Support Type, and Their Two Approaches
## 1 Introduction  
Secret sharing was introduced by [PERSON] [1] and [PERSON] [2] in 1979. In secret sharing, a given secret is encoded and divided into \(n\) shares, and two requirements must be met before a secret can be disclosed: a) any \(t\) of the \(n\) shares can cooperate to unveil the given secret, and b) less than \(t\) shares cannot unveil. Secret image sharing has at least two main streams. The first is visual cryptography (VC) [3], which is used for black-and-white (2-levels) images, and the second is polynomial-based sharing [4], which is used for gray-value or color images. Introduced by [PERSON] and [PERSON] [3] in 1994, visual cryptography uses several transparencies as media to share the secret image, where each transparency is \"larger\" than the secret image in size. Building on [PERSON]'s (t, n) scheme, in 2002, [PERSON] and [PERSON] [4] proposed a (t, n) threshold scheme for sharing 256-level secret images. In [4], each share was t times \"smaller\" than the given secret image. For a (\(t\), \(n\)) threshold scheme, \(t\leq n\) always holds. The security level is controlled by the ratio \(t/n\), where a larger \(t/n\) ratio means that more participants are required for the secret to be deciphered, and hence, prevent the betraying from a small group of participants. If every participant trusts no one else, then \(t=n\) can be set. Smaller \(t/n\) ratios are used in unstable environments (e.g., during a war or if the storage medium or Internet connection is unreliable) in which many participants may lose contact. Thus, a (\(t\), \(n\)) system not only addresses security concerns from betrayal but also allows for tolerance toward missing shares in unstable environments. Because sharing is useful, all four aforementioned papers [1; 2; 3; 4] have been frequently cited, particularly [PERSON] [1], which has had more than 10,000 citations from 1979 to 2019. Many recent studies have also focused on sharing [5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19]. Finally, as for image sharing, some readers may wonder why one cannot just use a key to encrypt the secret image and then share the key withauthorized personnel. This is because the resultant protection of the image will be extremely weak, although the \(n\) shares of the key will be smaller in size than the \(n\) shares of the secret image. In cases of a disk failure or hacker attack on the computer storing the encrypted version of secret image, the entire secret image will be deleted \"forever\".  
In the present study, we attempt to extend our previous foundational study [4] on sharing to introduce two other types of sharing: the forbidden type and cross-department support type. We implement these two types first by using a polynomial approach and then by using a linear-equations approach. Notably, the polynomial approach has been widely used by researchers, but not the linear-equations approach. We list both approaches here to increase security against hackers. For example, the secret image can be divided into several parts, where odd-numbered parts use the polynomial approach, and the even-numbered parts use the linear-equations approach.  
The paper is organized as follows: Section 2 outlines the basics of polynomial-based sharing, Section 3 introduces the two proposed sharing types, including our two approaches to implementing each of the two types. Section 4 discusses some details of the design, such as how to obtain independent equations for the linear-equations approach and how to mix the two approaches to increase security. Section 5 describes practical application and implementation results. Section 6 concludes this study.