Cryptography Reference
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are merely random grids and none of them individually leaks any information
about B. The security condition of Definition 2 is met.
Consider B(0) (the ares of transparent pixels in B). Since R1[i,
2
[i;j] = R
1
[i;j]
for each B[i;j] = 0 (or B[i;j] 2 B(0)), thus S[B(0)] = R
1
[B(0
)]
R
2
[
B(0)] =
R
1
[B(0)]. Thus
(S[B(0)]) = 1=2. Regarding B(1), R
2
[i;j] = R
1
[i;j] for each
B[i;j] = 1 (or B[i;j] 2 B(1)). We have S[B(1)] = R
1
[B(1)] R
2
[B(1)] = 1
(an area with all opaque pixels), i.e.,T(S[B(1)]) = 0. ThusT(S[B(0)]) >
T(S[B(1)]). The light contrast condition of Definition 2 is satisfied. There-
fore, fR
1
;R
2
g produced by Algorithm 1 is a set of (2, 2)-VCRG of B and
(
T
T
(S[B(0)]),
(S[B(1)])) = (1=2; 0).
For Algorithm 2, we have
T
T
(R
2
[B(0)])
=
T
(R
1
[B(0)])
=
1=2 and
R
2
[B(1)] is purely a random grid with
T
(R
2
[B(1)]) = 1=2. Thus, R
2
is a
random grid with
T
(R
2
) = 1=2. Since R
2
[B(0)] = R
1
[B(0)],
T
(S[B(0)]) =
T
((R
1
[B(0)]) = 1=2; while R
1
[B(1)] and R
2
[B(1)]
are two independent random grids so that
(R
1
[B(0)]R
2
[B(0)]) =
T
T
(S[B(1)]) =
T
(R
1
[B(1)]
R
2
[B(1)]) =
T
(R
1
[B(1)])
T
(R
2
[B(1)]) = 1=4 (by Lemma 1(1)). We ob-
tain
T
(S[B(0)]) >
T
(S[B(1)]).
For Algorithm 3,
T
(R
2
[B(0)])
=
1=2 because R
2
[B(0)] is a purely ran-
dom grid and
T
(R
2
[B(1)]) =
T
(R
1
[B(1)]) = 1
T
((R
1
[B(1)]) = 1=2. We
have
(R
2
) = 1=2. Further, due to the fact that R
1
[B(0)] and R
2
[B(0)]
are independent,
T
T
(S[B(0)]) =
T
(R
1
[B(0)]
T
R
2
[B(0)
]) =
T
(R
1
[B(0)])
T
(R
2
[B(0)]) = 1=4; on the other hand, since R
2
[B(1)] = R
1
[B(1)], S[B(1)] =
R
1
[B(1)] (R
2
[B(1)] = 1, i.e.
T
(S[B(1)] = 0. We have
T
(S[B(0)]) >
T
(S[B(1)]).
Based upon the above statements, we realize that both security and con-
trast conditions in Definition 2 hold for all of the three algorithms. We con-
clude that no information of B can be obtained from random grids R
1
or
R
2
individually, while S reveals B in our visual system for all of the three
algorithms. Theorem 1 is proved.
The following corollary is an immediate consequence from the statements
in the proof of Theorem 1.
Corollary 1 (
(S[B(1)])) = (1=2; 0); (1=2; 1=4) or (1=4; 0) for
Algorithms 1{3, respectively where S = R
1
R
2
and fR
1
;R
2
g is a set of
(2; 2)-VCRG produced by Algorithms 1{3 with respect to secret image B.
T
(S[B(0)]);
T
7.3.3 Experiments for (2, 2)-VCRG
Figure 7.1
illustrates the results of the implementation of the above three
algorithms. Figure 7.1(a) is secret binary image B, Figures 7.1(b) and (c)
present the two random grids produced by Algorithm 1 and Figure 7.1(d) is
the superimposed result of these two shares ((b) and (c)). Figures 7.1(e){g)
illustrate the corresponding results by Algorithm 2, while Figures 7.1(h){(j)
are the corresponding results by Algorithm 3. It can be easily seen from Figure
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