Cryptography Reference
In-Depth Information
it results that
min
M2M
b
H(M
i
) max
M2M
w
H(M
i
)
S
m;
where
M
b
= C
c
1
c
i1
bc
i+1
c
n
[C
c
1
c
i1
bc
i+1
c
n
w
;
b
M
w
= C
c
1
c
i1
wc
i+1
c
n
[C
c
1
c
i1
wc
i+1
c
n
w
;
b
and H(M
i
) denotes the Hamming weight of the i-th row vector Mi
i
of a matrix M.
The values
R
> 0 and
S
> 0 are referred to as the relative difference of
the reconstructed image and relative difference of shadow images, respectively.
The number
R
m 1 and
S
m 1 are contrasts of the reconstructed image
and the shadow images. People would like both
R
and
S
to be as large as
possible.
The first condition is the contrast condition that indicates any qualified
set X 2
Qual
can recover the secret image. The secret image can be re-
covered by stacking the transparencies of a qualified set, belonging to
Qual
.
The second condition is the security condition that states any forbidden set
X = fi
1
;;i
q
g2
Forb
has no information on the secret image. People can-
not get any information on the secret image by inspecting the shadow images
of a forbidden set. The third condition is the extended condition that im-
plies that the shadows images are still meaningful after the original images
are encoded. Any participant can recognize the shadow image on one's trans-
parency. Although the collection M
b
is obtained by combining two collections
C
c
1
c
i1
bc
i+1
c
n
b
and C
c
1
c
i1
bc
i+1
c
w
, we have the same set of fM
i
g only with
one of the collections, because fM
i
: M 2C
c1c
n
w
gfM
i
: M 2C
c1c
b
g for
any c
1
;;c
n
2fb;wg and any i 2f1;;ng due to the second condition.
Here we show how to accomplish a 2 out of 2 EVCS. Each share consists
of 4 subpixels like (2; 2) VSSS. However, it contains either two 1's or three
1's depending on the colors of pixels of the corresponding original image,
white or black, respectively. The scheme is given by the 4 pairs of collections
(C
c
1
c
w
;C
c
1
c
b
), namely 8 collections C
c
1
c
c
, where c;c
1
;c
2
2 fb;wg. The collec-
tions are obtained by permuting the columns of the following 8 basic matrices,
S
c
1
c
2
c
:
0
0
0
1
1
0
1
1
S
ww
w
; S
ww
b
=
=
;
0
1
0
1
1
1
0
0
0
0
0
1
1
0
1
1
S
wb
w
=
; S
wb
b
=
;
0
1
1
1
1
1
1
0
0
0
1
1
1
1
1
1
S
b
w
=
; S
bw
b
=
;
0
1
0
1
1
1
0
0
0
0
1
1
1
1
1
1
S
b
w
=
; S
bb
b
=
:
0
1
1
1
1
1
1
0
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