Cryptography Reference
In-Depth Information
means all the shadow images consist of uniformly random pattern of black and
white subpixels. Naor and Shamir pointed out an extension of this scheme for
concealing the very existence of the secret image.
4.2.2 Extended Visual Cryptography Scheme
Ateniese et al. extended the VSSS in the sense of a General Access Structure
(GAS) [1] and extended capability. [2] A General Access Structure controls
the qualified set of transparencies with which one can recover the secret image,
while any k or more transparencies can reconstruct the secret image in (k;n)
VSSS. An extended capability is able to introduce a meaningful image as a
shadow image which Naor and Shamir pointed out in their very first paper
[31]. An innocent-looking image of a house, dog, or something else would be
much less suspicious than a random-dotted image as a shadow image.
In the Extended Visual Cryptography Scheme (EVCS), for an access struc-
ture (
Qual
;
Forb
) on a set of n participants, the shared (secret) image can be
recovered by any qualied set X 2
Qual
with no trace of the shadow images,
but any forbidden set X 2
Forb
has no information on the secret image. More-
over, the shadow images are meaningful so that each participant can recognize
the image on one's transparency.
Similar to the (k;n) VSSS, an EVCS can be constructed in a pixel-wise
manner. Since n participants share one secret image and have their own images
in the n shadow images, we have to consider n+ 1 colors, c;c
1
; ;c
n
2fw;bg
where w and b stands for white and black, respectively. The value c denotes
the color of the secret image pixel and ci
i
denotes the color of the original
image pixel for i-th participant's shadow image. In order to realize an EVCS
that obtains a c pixel when transparencies associated to a set X 2
Qual
,
we need 2
n
pairs of collections of nm Boolean matrices, (C
c
1
c
n
w
;C
c
1
c
b
),
one for each possible combination of white and black pixels in the n original
images for the shadow images.
An EVCS for an access structure (
Qual
;
Forb
) for n participants is valid
if it fulfills the following conditions.
1.
For any X 2
Qual
and for any c
1
;;c
n
2fb;wg, the threshold t
X
and the relative difference
R
exist, which satisfy H(M
X
) t
X
R
m for any M 2C
c
1
c
n
w
and H(M
X
) t
X
for any M 2C
c
1
c
b
.
Here M
X
denotes the m-D vector obtained by taking Boolean "OR"
of the row vectors of M corresponding to the participants in X and
H(M
X
) denotes the Hamming weight of the vector M
X
.
2.
For any X = fi
1
;;i
q
g 2
Forb
and for any c
1
;;c
n
2 fb;wg,
the two collections of qm matrices, D
c
1
c
n
w
and D
c
1
c
b
, obtained
by extracting rows i
1
;;i
q
from each nm matrix in C
c
1
c
n
w
and
C
c
1
c
b
, respectively, are indistinguishable so that the collections
contain the same matrices with the same frequencies.
3.
For any i 2f1; 2; ;ng and any c
1
; ;c
i1
;c
i+1
; ;c
n
2fb;wg,
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