Probabilistic Visual Cryptography Schemes - Visual Cryptography and Secret Image Sharing - page 148

will be always mapped to a black pixel in each of the computed shares A i
because of the and operation performed in the computation of the C j (for a
'0' pixel, C j = B j & 0 = 0) and the XOR operation in the reconstruction
phase (the contribution for C j = 0 means that A j = B n+1 C j = B n+1
and then A 0 = A j A k = B n+1 B n+1 = 0. Then p bjb = 1 and p wjb = 0.
A white pixel will be reconstructed according to the pixel returned after the
XOR between the two matrices Bi i and B k , which are randomly selected. Then
p wjw = p bjw = 1=2. Hence, the construction returns a = 1=2 probabilistic
scheme with contrast 1=2.
The same construction has been extended by Wang et al. in [21] to deal
with grayscale images. Basically each pixel is coded with a binary string g1
bits long, where g denotes the gray level number, and having a number of "1"s
equivalent to the 1s present in the binary string obtained by the grey level g
(without respecting the order). A pixel having grey level k in a range from 0
to g1, is represented by a binary string having gk '0's and k1 '1's. The
construction returns a probabilistic scheme with m = g 1.
5.8 Conclusions and Open Problems
The probabilistic model for VCS schemes has been first described in Yang [22],
where the reconstruction of the secret pixel has been given in probabilistic
terms, no more guaranteeing the correctness property of the traditional VCS
schemes. A generalization of the model has been given then in [11], where
probabilistic schemes with pixel expansion have been described, showing that
there is a one-to-one correspondence between probabilistic schemes with no
pixel expansion and deterministic schemes and that such a one-to-one map-
ping trades the probabilistic nature of the scheme with the contrast of the
deterministic scheme. Probabilistic schemes with pixel expansion can be ob-
tained from deterministic schemes and their probabilistic factors can be stud-
ied. While for (n;n)-threshold schemes it has been proved that there is a
linear relation between the pixel expansion and the probabilistic factor, for
(2;n)-threshold schemes no closed expression for the probabilistic factor has
been found. Finally, a probabilistic model has been extended and alternative
operations (instead of OR) have been considered for the distribution of the
shares and the reconstruction of the secret images.