Cryptography Reference
In-Depth Information
in [11] it has been shown that any (k;k)-threshold VCS has a pixel expansion
of at least 2
k1
and the relative dierence (h`)=m between reconstructed
white and reconstructed black is at most 1=2
k1
. This means that, in the
reconstructed white pixel, for any 2
k1
subpixels, there is at most a white
subpixel.
Visual cryptography schemes with perfect reconstruction of black pixels
have been analyzed in [12, 4, 3]. In particular, in [3] it was shown how to
construct (
(ΓiQual,Γ
;
Forb
)-VCSs with perfect reconstruction of black pixels having
a pixel expansion m =
P
X2
Qual
2
jXj1
, by using a technique proposed in [1].
More precisely, such a technique to constructs visual cryptography schemes
with a perfect reconstruction of black pixels using small schemes as building
blocks in the construction of larger schemes, as explained in the following.
For i = 1;:::;q, let (
i
(ΓiQual,Γ
;
i
Forb
) be an access structure on a set P of n
participants. If a participant j 2P is not essential for the i-th access structure,
we assume that j 62
i
Forb
and that j does not receive any share. Suppose there
exists a (
i
(ΓiQual,Γ
;
i
Forb
)-VCS with a pixel expansion mi
i
and basis matrices T
i
and T
i
, for each i = 1;:::;q. The basis matrix S
0
(S
1
, resp.) of a VCS for
the access structure (
(ΓiQual,Γ
;
Forb
) where
Qual
=
1
(ΓiQual,Γ
[:::[
q
Qual
and
Forb
=
1
Forb
\:::\
q
Forb
is constructed as the concatenation of some auxiliary matrices
T
i
( T
i
, resp.), for each i = 1;:::;q. Such matrices are obtained as follows:
for each j = 1;:::;n, the j-th row of T
i
( T
i
, resp.) has all ones as entries if
the participant j is not essential for (
i
(ΓiQual,Γ
;
i
Forb
), otherwise it is the row of
T
i
(T
i
, resp.) corresponding to participant j. Hence, S
0
T
1
T
2
::: T
q
=
T
1
T
2
::: T
q
, where denotes the concatenation of matrices.
and S
1
=
The resulting VCS has a pixel expansion m =
P
i=1
m
i
. For a special class of
access structures, such as threshold access structures and graph-based access
structures, it is possible to design VCSs with the perfect reconstruction of
black pixels achieving a smaller value of m, as shown in [1, 12, 3].
n
f1; 2g;f2; 3g;f3; 4g
o
Example 2 Let P = f1; 2; 3; 4g and
0
=
. We can
construct a VCS with a perfect reconstruction of black pixels for the access
structure having basis
0
by using three (2; 2)-threshold VCSs on the sets of
participants f1; 2g;f2; 3g; and f3; 4g. The basis matrices T
0
and T
1
for the
(2; 2)-threshold VCS described in
Section 9.2.1
are
1
1
0
0
T
0
=
T
1
=
:
1
0
0
1
T
i
; T
i
), for i = 1;:::; 3
From T
0
and T
1
we construct the pair of matrices (
having four rows each, as follows:
2
3
2
3
1
0
1
0
4
5
4
5
;
1
0
0
1
T
1
T
1
=
=
1
1
1
1
1
1
1
1
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