Cryptography Reference
In-Depth Information
if fewer than k transparencies are stacked together or analyzed by any other
method.
Naor and Shamir [11] proposed a construction for (k;k)-threshold VCSs
using two basis matrices S
0
and S
1
defined as follows: S
0
is the matrix whose
columns are all the Boolean k-vectors having an even number of 1's, and
S
1
is the matrix whose columns are all the Boolean k-vectors having an odd
number of 1's. The pixel expansion m of such a scheme is equal to 2
k1
and the
relative dierence (h`)=m between reconstructed white and reconstructed
black is equal to 1=2
k1
. The above construction is optimal with respect to
the pixel expansion and the relative difference between reconstructed white
and reconstructed black.
Example 1 The basis matrices S
0
and S
1
in a (4; 4)-threshold VCS are
2
3
2
3
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
4
5
4
5
:
0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
S
0
=
S
1
=
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1
1
0
1
0
0
1
1
0
0
1
0
1
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. In this
scheme each pixel of the secret image is encoded into m = 8 subpixels.
It is easy to see that the integers h = 1 and ` = 0 satisfy Property 1 of
Definition 1. Let S
0
be the matrix chosen by the dealer to share a white pixel;
by stacking the transparencies held by all four participants we get the vector
(0; 1; 1; 1; 1; 1; 1; 1; 1). On the other hand, let S
1
be the matrix chosen by the
dealer to share a black pixel; by stacking the transparencies held by all four
participants we get the vector (1; 1; 1; 1; 1; 1; 1; 1; 1). Property 2 of Denition
1 can also be easily verified. Indeed, consider what happens when less than
four participants stack their together transparencies. For example, consider
the first three participants and notice that, by stacking their shares, we get the
vector (0; 1; 1; 1; 1; 1; 1; 1; 1) in both cases when the shared pixel is either white
or black. Thus, the participants are not able to distinguish the color of the
shared pixel by inspecting their shares.
9.2.2 Perfect Black Visual Cryptography Schemes
Visual cryptography schemes such that, in the reconstructed image, all the
subpixels associated with a black pixel are black, i.e., ` = 0, are referred
either to as visual cryptography schemes with perfect reconstruction of black
pixels or to as maximal contrast schemes. For example, the (k;k)-threshold
VCS described in
Section 9.2.1
has perfect reconstruction of black pixels. A
perfect reconstruction of both black and white pixels, which would correspond
to visual cryptography schemes having ideal contrast, is impossible. Indeed,
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