Extended Visual Cryptography for Photograph Images - Visual Cryptography and Secret Image Sharing - page 96

FIGURE 4.2
An example of visual secret sharing scheme (VSSS). Two shadow images of
random patterns (left and middle) and reconstructed secret image (right).

Let us consider a special case of (2; 2) VSSS. Each share consists of
4 subpixels of a 2 2 array in a physical implementation, where two of
them are white and the rest two are black. The Boolean matrix of this
scheme is 2 4 where each row consisting of two 0's and two 1's repre-
sents an arrangement of subpixels in a share. For instance, six possible pat-
terns of shares having 50% gray as shown in Figure 4.1 are represented as
f[0 0 1 1]; [1 1 0 0]; [0 1 0 1]; [1 0 1 0]; [0 1 1 0]; [1 0 0 1]g. The scheme is
accomplished by the following two collections:

C w = fmatrices obtained by permuting the columns of S w g;
C b = fmatrices obtained by permuting the columns of S b g;

where S w and S b are given as below:

0 0

0 1 1 0 1 1

S w = ; S b = :

0 0 1 1 1 1 0 0

The above matrices S w and S b are called basis matrices. Because the collec-
tions are obtained by permutation of subpixels, each share may have randomly
arranged two white and two black subpixels, which looks 50% gray. A pair of
shares from C w has the same arrangement of subpixels. The combined result
is the same pattern, which looks 50% gray. A pair of shares from C b has the
complementary arrangement of subpixels. The combined result consists of four
black subpixels, which looks completely black. Figure 4.2 shows an example
of resulting shadow images and a reconstructed secret image. The size of all
images are 128 128 pixels, because the original secret image has 64 64
pixels.
The original scheme proposed by Naor and Shamir is uniform, such that
any combined shares from q < k shadow images yield "OR"ed m-D vector
M q with H(M q ) = f(q) with uniform probability distribution, regardless of
if the matrices were taken from C w or C b . Suppose the case of q = 1, the
above-mentioned combined share is a single share of each shadow image. It