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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