Cryptography Reference
In-Depth Information
white. From Property 1. of Denition 1, there exists an index j 2f1;:::;mg
for which the encoded pixels in each transparency t
i
1
;j
;:::;
t
i
p
;
j
are all white.
It follo
w
is that for the same j,
j
= 0 and = OR(
1
;:::;
m
) is equal to 1.
Then, is zero, i.e., the reconstructed pixel will be white. Now, assume that
the secret pixel shared by the dealer was black. Since the underlying scheme
is a (
Qual
;
Forb
)-VCS with a perfect reconstr
uct
ion
of b
lack pixels, it
h
olds
that
j
= 1, for j = 1;:::;m. Hence, = OR(
1
;:::;
m
) is zero and = 1,
i.e., the reconstructed pixel will be black. The security of the scheme directly
follows from the security of the underlying VCS.
Compared to the scheme described in
Figure 9.1,
the one of
Figure 9.3
requires each participant to store m transparencies, each having the same
number of pixels as the original image. Furthermore, the scheme requires a
qualified set of p participants to execute exactly m + 1 reversing operations
and mp 1 stacking operations. Finally, the reconstructed image has no loss
of resolution.
Example 5 Let P = f1; 2; 3; 4g and
0
=
n
f1; 2g;f2; 3g;f3; 4g
o
. The basis
matrices S
0
and S
1
in a VCS realizing the access structure
Qual
whose basis
is
0
are:
2
3
2
3
0
1
1
0
1
0
0
1
4
5
4
5
:
0
1
1
1
1
1
1
0
S
0
=
S
1
=
0
1
1
1
1
1
0
1
0
1
0
1
1
0
1
0
The collections C
0
and C
1
are obtained by permuting the columns of the corre-
sponding basis matrix (S
0
for C
0
, and S
1
for C
1
) in all possible ways. Let S
0
be the matrix chosen by the dealer to share a white pixel. The corresponding
pixels generated for the transparencies t
1;1
;:::;t
1;4
for participant 1 are 0, 1,
1, and 0, whereas, the pixels generated for the transparencies t
2;1
;:::;t
2;4
for
participant 2 are 0, 1, 1, and 1. During the reconstruction phase, the two par-
ticipants stack their shares and retrieve
1
= 0;
2
= 1;
3
= 1;
4
= 1. By
applying the r
eve
r
s
i
n
g operation to each
j
and stacki
ng
the results, they ob-
tain = OR(0; 1; 1; 1) = 1. By reversing , they obtain = 0, and reconstruct
a white pixel.
Let S
1
be the matrix chosen by the dealer to share a black pixel. The cor-
responding pixels generated for the transparencies t
1;1
;:::;t
1;4
for participant
1 are 1, 0, 0; and 1, whereas, the corresponding pixels generated for the trans-
parencies t
2;1
;:::;t
2;4
for participant 2 are 1 1 1; and 0. During the reconstruc-
tion phase, the two participants will perform the same ope
ra
t
ion s
equence as
be
fore, retrieving
1
= 1;
2
= 1;
3
= 1;
4
= 1, = OR(1; 1; 1; 1) = 0, and
= 1, reconstructing a black pixel.
Notice that if the qualied set X 2
Qual
is a superset of an X
0
2
0
,
then, in the reconstruction phase, participants in X can recover the secret
image by performing the given sequence of operations on the subset of trans-
parencies held by participants in X
0
only, thus reducing the total number of
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