Cryptography Reference
In-Depth Information
Corollary 3 is an immediate result from Theorem 2.
Corollary 3 (T(S
E
[B(0)]);T(S
E
[B(1)])) = (1=2
n1
; 0) whereE =
fR
1
;R
2
;:::;R
n
g is produced by Algorithm 4 with respect to B and S
E
=
R
1
R
2
R
n
.
We give a simple example for n = 3 to explain the relationship among
corresponding pixels b 2 B, r
1
2 R
1
, r
2
2 R
2
, r
3
2 R
3
, a
1
2 A
1
, and a
2
2 A
2
in Algorithm 4.
Example 1. Consider n = 3 and a secret image B. Algorithm 4 first gener-
ates R
1
and R
2
independently; then produces A
1
and A
2
by formula (7.12);
at last finds R
3
depending on B and A
2
according to formula (7.13). Let
E
= fR
1
;R
2
;R
3
g. Table 7.4 summarizes all of the possible combinations of
corresponding pixels b 2 B, r
1
2 R
1
, and r
2
2 R
2
; and their corresponding
results of a
1
(= r
1
) 2 A
1
, a
2
(= f(r
2
;a
1
)) 2 A
2
, and r
3
(= f(b;a
2
)) 2 R
3
; as
well as the superimposed results of r
1
r
2
, r
1
r
3
, r
2
r
3
, and r
1
r
2
r
3
.
Table 7.5
lists the light transmissions of the corresponding results in Table
7.4.
It is seen from Tables 7.4 and 7.5 that all of R
1
, R
2
, R
3
, A
1
, A
2
are
random grids with a light transmission of 1=2. Besides,T(S
D
[B(0)]) =
T(S
D
[B(1)]) = 1=4 andT(S
D
[B(0)]) = 1=4 > 0 =T(S
D
[B(1]) where
D(
E) = fR
1
;R
2
g, fR
1
;R
3
g or fR
2
;R
3
g. It means that each group of two
random grids obtains no information about B(0) or B(1) when superimposed
(thus, no information about the colors of pixels in B can be found), while B(0)
and B(1) can be identified from S
E
due to the difference of their light trans-
missions, that is, we see B out of S
E
because of
(S
E
[B(0)]) >
(S
E
[B(1)]).
T
T
TABLE 7.4
All possible combinations of b, r
1
, and r
2
, the corresponding results
of a
1
(= r
1
), a
2
= f(r
2
;a
1
), r
3
= f(b;a
2
), and r
1
r
2
, r
1
r
3
,
r
2
r
3
as well as r
1
r
2
r
3
by Algorithm 4 for (3, 3)-VCRG.
b r
1
a
1
r
2
a
2
r
3
r
1
r
2
r
1
r
3
r
2
r
3
r
1
r
2
r
3
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
0
0
1
1
1
1
1
0
0
0
0
1
0
1
1
1
1
1
0
1
0
1
1
1
1
0
1
0
1
1
0
1
1
0
1
1
1
1
1
Algorithm 4 can be viewed as the generalization from the idea in Algorithm
1. Such knowledge can be easily adapted to generalize the ideas in Algorithms
2 and 3. Algorithms 5 and 6 are the results accordingly.
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