Cryptography Reference
In-Depth Information
8
<
:
2
0
0
0
1
3
2
0
0
1
0
3
2
0
1
0
0
3
2
1
0
0
0
3
2
1
1
1
0
3
2
1
1
0
1
3
2
1
0
1
1
3
2
0
1
1
1
3
9
=
;
4
5
;
4
5
;
4
5
;
4
5
;
4
5
;
4
5
;
4
5
;
4
5
C
W
=
:
Fix an m
0
, 2 m
0
r, and let S
0
be the probabilistic scheme with pixel
expansion m
0
obtained using Construction 1 from scheme S. The threshold `
0
for S
0
also needs to be selected, remembering also that the threshold h
0
will
be obtained if the assumption h
0
= `
0
+ 1 is satisfied. Observe that `
0
must
be m
0
1 because there is only one column with all white pixels and thus it
would be never possible to get more than 1 white subpixel in a reconstructed
pixel (equivalently always at least m
0
1 black subpixels in a reconstructed
pixel will be obtained). This implies that if by selecting `
0
< m
0
1 all recon-
structed pixels will always be considered black and thus no valid scheme can
be obtained. To construct schemes, values `
0
= m
0
1 and h
0
= m
0
must be
selected.
To compute the probabilistic factor of scheme S
0
it is necessary to compute
p
0
bjb
and p
0
bjw
. All the columns of the black base matrix M
B
of the deterministic
scheme S
D
have at least a 1; hence, all columns of C
B
(S
0
) have at least a 1.
Thus, m
0
black subpixels when reconstructing a black secret pixel are always
returned. This means that p
0
bjb
= 1 and p
0
wjb
= 0. To compute p
0
bjw
consider the
white base matrix M
W
of scheme S
D
, which determines C
W
(S
0
). Since M
W
has one column with all 0s there will be some matrices of C
W
(S
0
) that include
such a column. In order for a reconstructed pixel to be considered black it
must have h
0
= m
0
black subpixels. Among the
m
0
possible distribution
matrices of C
W
(S
0
), exactly
r1
m
0
do not include the unique column with all
0s. Hence,
r 1
m
0
r
m
0
p
0
bjw
=
= 1 m
0
=r
=
and
p
0
wjw
= m
0
=r
The formula expressing the probability factor of S
0
is then the following:
m
0
2
n1
:
=
The formula shows that there is a linear relation between the pixel expan-
sion of the scheme and the probability factor. A probabilistic scheme with no
pixel expansion implies a small probability factor ( = 1=r) and then a recon-
structed image with many errors; the probability factor of the scheme can be
increased by increasing the pixel expansion to obtain a better reconstructed
image. Clearly for m
0
= 2
n1
one would get = 1, that is, a deterministic
scheme.
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