Image Sharing with Steganography and Authentication - Visual Cryptography and Secret Image Sharing - page 419

(3) When any k or more shadows are brought together co-operatively,

the secret image can be restored without any computations based
on the properties of the human visual system.

Following this pioneering research, many VSS schemes have been proposed
over the past decade. Shamir's polynomial-based secret sharing approach was
adopted into secret image sharing by Thien and Lin in 2002 [23]. Then, Wang
and Su [25] made some improvements to decrease the size of the shared images
based on Thien and Lin's scheme [23]. Studies in secret image sharing have
focused on different active topics: decreasing share size [4,5,25], dealing with
attacks by malicious users, and detecting fake shadows [14,18,29]. The research
domain has been also extended from gray images to color images [1,3,4,6,7,21].
And, the generated shares become meaningful instead of noise-like from the
start [24]. Aside from these issues, Yang [27], Cimato et al. [8], and Wang et
al. [26] separately proposed a new VSS scheme, called a ProbVSS scheme, to
deal with problems such as pixel expansion and computational complexity that
typically arise in image sharing. The ProbVSS scheme is based on probabilistic
concepts and is designed solely for binary images. By taking advantage of
Boolean operations, this scheme overcomes the shortcomings inherent in high
computational complexity and large pixel expansion at the same time.
In 2004, Lin and Tsai proposed another kind of secret sharing scheme in-
corporating both steganography and authentication [15]. In Lin-Tsai's scheme,
the secret image is divided into shadows and data hiding is used to embed the
shares and corresponding authentication codes into the cover image. During
the reconstruction and verifying phase, the hidden authentication codes are
then extracted to verify the integrity of the stego-image. However, the size
of the cover image becomes fourfold that of the secret image. The Lin-Tsai
scheme combines steganography and authentication features to avoid trans-
mitting an erroneous stego-image or intentionally providing a false image by
using a parity check bit. However, by the principle of the parity check itself,
the authentication ability is weak.
Yang et al. proposed a new scheme to overcome some of the weaknesses in
the Lin-Tsai scheme in 2007 [28]. Their objective was to prevent cheating by
participants. They also improved the authentication ability and the quality
of the reconstructed grayscale secret image. Although the Yang et al. scheme
can restore a distortion-free secret image, it allows for a significant probabil-
ity that a fake stego-image can be authenticated successfully, and the cover
image size is still four times that of the original secret image. Building on
their predecessors' experience, in 2008 Chang et al. proposed a secret shar-
ing scheme that incorporates authentication based on the Chinese Remainder
Theorem (CRT) [9]. They successfully improved authentication capability by
combining CRT. However, in Chang et al.'s scheme, pixel expansion remains
a problem and the size of the cover image should be at least twice that of the