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Top1. Introduction
Secret image sharing (SIS) for (k, n)-threshold splits a secret image into n noise-like shadows, a.k.a, shares or shadow images, and then assigns the shadows among n participants. The secret can be disclosed by obtaining k or more authorized shadows. However, less than k shadows overall dis- close no information on the secret. Polynomial-based SIS (Shamir, 1979) and visual secret sharing (VSS) (Naor & Shamir, 1995; Wang, Liu & Yan, 2016) are the chief research branches.
In Shamir’s original polynomial-based secret sharing (Shamir, 1979) for (k, n)-threshold, the secret is split into the constant coefficient of a random (k 1)-degree polynomial to obtain n shadows, which are then also assigned to n corresponding participants. The secret can be disclosed by Lagrange interpolation when obtaining any k or more shadows. Following Shamir’s original scheme and using all the coefficients of the polynomial to embed secret pixels, Thien and Lin (Thien & Lin, 2002) reduced the size of each shadow 1/k times to that of the original secret image. Based on Thien and Lin’s work, some other polynomial-based schemes (Yang & Ciou, 2010; Zhou, Lu, Yan, Wang & Liu, 2018; Li, Yang & Kong, 2018) were further proposed to obtain more features. The strength of polynomial-based SIS lies in the secret image can be disclosed with high quality. Although polynomial-based SIS only uses any k shadows to disclose a distortion-less secret image, in the disclosing phase it needs more complicated computations, a.k.a, Lagrange interpolation, and known order of shares.
In a VSS (Cimato, De Prisco & De Santis, 2006; Wang, Zhang, Ma & Li, 2007; Wang, Arce & Di Crescenzo, 2009; Weir & Yan, 2010) for (k, n)-threshold, first the split n shadows are printed onto transparencies and then assigned to n corresponding participants. The merit of VSS is that, the secret image can be disclosed by just superposing any k or more shadows (transparencies) and human eyes with no cryptographic computation. Less than k shadows generally give no clue about the secret even if high computation power is available. Following Naor and Shamir’s original method, the physical properties and corresponding VSS limitations are widely researched, such as threshold (Yan, Wang & Niu, 2014), contrast (Wu & Sun, 2013; Yan, Liu & Yang, 2018), pixel expansion (Cimato, De Prisco & De Santis, 2006; Guo, Liu & Wu, 2013; Fu & Yu, 2014), multiple secrets (Li, Ma, Su & Yang, 2012), meaningful shadows (Yan, Wang, Niu & Yang, 2015a; Wang, Arce & Di Crescenzo, 2009; Liu & Wu, 2011; Yan, Wang, Niu & Yang, 2015b), and so on (Yan & Lu, 2018; Liu, Wang, Yan & Zhang, 2017).
Although SIS can be applied to not only information hiding and watermarking, but also authen- tication, transmitting passwords, access control, distributed storage and computing, etc (Yan, Lu, Liu, Wan, Ding & Liu, 2017a; Belazi & El-Latif, 2017; Yan, Lu, Liu, Wan, Ding & Liu, 2017b), its practical application is an important issue.