Polynomial-Based Secret Image Sharing Scheme with Fully Lossless Recovery

Introduction

Security is a big concern when considering the storage and transmission of image information. Traditional cryptography (Assad & Farajallah, 2016; Li, El-Latif, Shi, & Niu, 2012; Zhang & Xiao, 2014) method cannot prevent the data from being lost or corrupted during the transmission. Instead of cryptography, secret sharing is a way to ensure security since it has a property of loss-tolerant. Secret sharing, which comes from key management, was introduced by Shamir (1979) and Blakley (1979) independently. A (k, n) threshold secret sharing encrypts the secret into shares, and distributes the shares among n participants(kn). When any k or more participants collect together, the secret can be revealed. With this property of loss-tolerant, the secret can be decoded under the case even some shares are lost. Furthermore, secret sharing can be applied in many scenarios (Belazi & El-Latif, 2016; Yan et al., 2017). For example, in distributed storage and transmission, an image can be shared into n shares by a (k, n) threshold secret sharing scheme and be stored in n severs, the secret can be decoded under the case even n-k or less servers are broken.

Shamir's polynomial-based scheme (Li, Ma, Su, & Yang, 2012; Li, Yang, Wu, Kong, & Ma, 2013; Lin, S. J., & Lin, J. C., 2007; Shamir, 1979; Thien & Lin, 2002; Yang & Ciou, 2010) encrypts the secret into n shares using a polynomial and distributes the n shares to n participants separately. When any k or more participants with their shares get together, the secret can be reconstructed by Lagrange interpolation (Bleichenbacher & Nguyen, 2000; Chen, Liu, & Wang, 2008; Werner, 1984). Noar and Shamir (1994) extended the concept of secret sharing from number to image. In 2002, Thien and Lin (2002) used Shamir's scheme to share secret images in domain [0,250]. The scheme embedded the secret pixels in all the coefficients of the sharing polynomial, thus reducing the size of shadow images to 1/k times that of the original secret image. However, the scheme in Thien and Lin’s work truncated all gray values larger than 250 to 250 so that the method cannot actually get a lossless secret image (2002).

In terms of actual applications, the ability to recover image losslessly can be useful in a number of scenarios (Devaki & Rao, 2012; Hu & Jeon, 2006; Li, El-Latif, Yan, & Wang, 2012; Tso, Lou, Wang, & Liu, 2008; Yan & Lu, 2017). In real-world applications, like in area of medicine and military (Cheddad, Condell, Curran, & McKevitt, 2010), images details are significant and lossless images are important for the transmission and storage of image data.