XOR-Based Visual Cryptography - Visual Cryptography and Secret Image Sharing - page 172

G(green), and B(blue). If each R, G, or B takes a value between 0 and 255,
we have c = 256 3 . This integer matrix is also called A and will be treated
as equivalent to the secret image A itself. The symbol "&" denotes an AND
operation in the following construction.
6.5.1 (2,n) Scheme
To solve the pixel expansion problem with VSS schemes, Ito et al. [11], Yang
[24] and Cimato et al. [6] proposed probabilistic VSS models. The frequency
of white pixels in a white (or black) area is used to display the contrast of the
recovered image. In reconstructing the secret image, the "OR"-ed operation of
pixels of the shadows is the same as the stacking operation of subpixels in the
nonprobabilistic VSS schemes. They defined p 0 (resp. p 1 ) as the appearance
probability of a white pixel in a white (resp. black) area of the recovered
image. For a xed threshold probability 0 p TH 1 and relative contrast
0, if p 0 p TH and p 1 p TH , the frequency of white pixels in a white
area of the recovered image should be higher than that in a black area.
Wang et al. [23] proposed a probabilistic (2;n) secret sharing scheme for
binary images. Boolean XOR and AND operations are employed, and n + 1
distinct random matrices are generated as intermediate results. The scheme
is described in a pseudo-code style below in terms of its input, output, the
construction procedure, and the revealing procedure.
Wang et al.'s [23] algorithm for (2,n) scheme
Input: an integer n with n 2, and the secret image A.
Output: n distinct matrices A 1 ;A n , called shadow images.
Construction:
Step 1: Generate n + 1 random matrices B 1 ; ;B n+1 .
Step 2: Compute n intermediate matrices C 1 ; ;C n with C i = B i &A
for i = 1; ;n.
Step 3: Compute n shadow images A 1 ; ;A n with A i = B n+1 C i for
i = 1; ;n.
Revealing: A 0 = A i A j where i;j 2f1; 2; ;ng and i 6= j.
For integer scalar inputs between 0 and c1, each operand is represented in
binary and the operation is carried out bit-by-bit. For example, when a = 125
and b = 18, the XOR between these two integers is
ab = (125) 10 (18) 10 = (01111101) 2 (00010010) 2 = (01101111) 2 = (111) 10
For matrix inputs, the XOR operation of two N R N C matrices is defined
pixel-wise. That is,
AB = [a ij b ij ]; where i = 1; 2; ;N R ;j = 1; 2; ;N c :