XOR-Based Visual Cryptography - Visual Cryptography and Secret Image Sharing - page 173

The AND operation for integer scalar operands and matrix operands can be
defined similarly.
In all computations, every pixel is handled individually, separated from
other pixels. Therefore, to make the context clear, we denote pixel A i (s;t)
simply as A i . With the above construction procedure, for a "0" pixel in A and
any i, we have Ci i = B i &0 = 0 and A i = B n+1 C i = B n+1 , thus
A 0 = A i A j = B n+1 B n+1 = 0:
For a "1" pixel in A, C i = B i &1 = B i and A i = B n+1 B i , thus
A 0 = A i A j = B n+1 B n+1 B i B j = B i B j
which could be 0 or 1. In other words, between the original image A and
a reconstructed image A 0 , the "0" bits are kept the same and the "1" bits
may or may not changed. With any single shadow image, no information of
A is revealed because of the random nature of the matrices B 0 is It is easy
to verify that the n matrices A 1 ;A 2 ; ;A n are n distinct random matrices
from construction method above, each A i (i = 1; ;n) does not contain any
information of the original matrix A.
Associated Shamir's secret sharing scheme and a gradual search algorithm
for a single bitmap block truncation coding with the above (2;n) scheme, the
secret sharing scheme for color images was proposed in Reference [3]. Com-
bined the above (2;n) scheme with voting strategy, the probabilistic visual se-
cret sharing scheme for grayscale images was provided in Reference [4]. Based
on the above proposed (2;n) scheme, the matrices B 1 ;B 2 ; ;B n are chosen
according to Yang's into "probabilistic VC scheme and the (2;n) probabilistic
scheme with improved contrast was given in Reference [19].
6.5.2 (n,n) Scheme
The construction steps are given in the context of grayscale images. It is triv-
ially applicable to binary images and can be easily extended to color images.
Wang et al.'s [23] algorithm for (n,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 n1 .
Step 2: Compute the shadow images as below:
A 1 = B 1 ;A 2 = B 1 B 2 ; ;A n1 = B n2 B n1 ;
A n = B n1 A:
Revealing: A 0 = A 1 A 2 A n :
Theorem 11 [23] A 1 A 2 A n = A.