Cryptography Reference
In-Depth Information
Let A be the binary matrix representing the secret image.
Distributionphase.
The dealer:
generates n+1 random matrices B
1
;:::;B
n+1
; compute intermediate
matrices C
1
;:::;C
n
where C
i
= B
i
&A, for i = 1;:::;n;
distributes the n shares A
1
;:::;A
n
where A
I
= B
n+1
C
i
for i =
1;:::;n .
Reconstructionphase. A qualified set fi
j
;i
k
g of two participants
reconstructs the secret image as follows:
superimpose their shares executing binary XOR operation to obtain
A
0
= A
j
A
k
.
FIGURE 5.1
A construction for (2,n) Boolean probabilistic VCS.
In Wang's scheme, a reverse coding of the pixel is adopted, assigning `0'
to black pixel and `1' to white pixels. In this scheme it is easy to see that
a black pixel in the secret image 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). The
probabilities of reconstructing a black pixel are then p
bjb
= 1 and p
wjb
= 0.
A white pixel will be reconstructed on the basis of the result of the XOR
operation performed on 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 above presented scheme has been modified by Ulutas et al. in [18] and
Chang et al. in [8]. In the first work the construction is modified by adding a
mechanism to augment the probability of selecting a white (black) pixel given
that a white (black) pixel was present in the secret image. Basically, matrices
B
i
are no more randomly generated as in Wang's scheme, but are selected in
two different sets, one for white pixels and the other for black pixels, in order
to maximize the probability of reconstructing a pixel of the right color. Then
a random matrix R of the same size of the secret image is generated and used
to compute the share. The reconstruction phase is not modified. The modified
scheme is given in
Figure 5.2.
In Chang et al's construction, a voting strategy is adopted to apply the
Wang method to grayscale images [8]. In such a method, for each original
pixel, the binary representation of its grayscale value is considered, and for
each bit composing the representation, the Wang scheme is executed and
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