Cryptography Reference
In-Depth Information
below for the q pixels:
R
i
=
[S
i
T]
"
1
#
1
1
1 M M
=
M M 1
1
1
1
;
(1.7)
1
1 M M 1
1
|
{z
}
|{z}
Z
1
|{z}
Z
2
|{z}
Z
3
Z
0
where each row corresponds to q pixels in one share. The indicates the SIPs
that are determined by Si.
i
. Columns of Ri
i
are partitioned into several sets Zi,
i
,
where the columns with the same configuration are assigned to the same set.
As shown in (1.7), columns are partitioned into 4 sets Zi,
i
, i = 0; 1;:::; 3. The
set Z
0
denotes the distribution of SIPs and contains 1=2 of all the pixels. The
non-SIPs of each share are partitioned between set Z
1
, Z
2
, and Z
3
, where each
set contains 1=6 of all the pixels. Thus, to generate Z
i
on the share, we can set
the parameters for the multitone error diffusion as follows: w = 3, z
0
= 0:5,
and z
1
= z
2
= z
3
= 1=6. The corresponding tone is arbitrarily chosen as:
g
0
= 0, g
1
= 0:3, g
2
= 0:6, and g
3
= 0:9. By using the algorithm proposed in
[4], we obtain homogenous distributions Zi,
i
, i = 0; 1;:::; 3. The combination
of Z
1
and Z
2
is the distribution of ABPs of share 1; the combination of Z
2
and Z
3
is the distribution of ABPs of share 2; and the combination of Z
1
and
Z
3
is the distribution of ABPs of share 3, as shown in (1.8)
8
<
SIPs of all shares ! Z
0
ABPs of share 1 ! Z
1
[Z
2
ABPs of share 2 ! Z
2
[Z
3
ABPs of share 3 ! Z
1
[Z
3
:
(1.8)
:
As an example, assume the share has size 6 8 and is partitioned into
4 halftone cells of size 3 4, each cell corresponding to one secret image
pixel. Within each cell, there are 6 SIPs and 4 ABPs. Suppose the generated
Then, as also shown in Figure 1.6, the distributions of SIPs and ABPs of each
share are determined based on Zi.
i
. Note that without knowing what values the
Ms carry, any two shares can be stacked together to decode the secret image
pixels.
Notice that we may not be able to generate an exact number of pixels as
desired for sets Zi,
i
, i = 0; 1; 2; 3. However, for ABPs, as long as the membership
of ABPs to the sets Zi
i
indicated in (1.8) is maintained, a slight deviation
from its desired number of pixels for Zi,
i
, i = 1; 2; 3 is allowable. The contrast
condition of image decoding is still maintained. Such a point can be clearly
illustrated in
Figure 1.7
where the composition of each share is shown. It is
clear that the relative size of Zi,
i
, i = 1; 2; 3 is not important.
However, the distribution of SIPs, denoted by Z
0
, needs to be rened to
guarantee that there are exactly SIPs in each halftone cell. Each halftone
cell is checked to find out the number of pixels belonging to Z
0
. Assume the
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