Cryptography Reference
In-Depth Information
and C W of the probabilistic scheme with no pixel expansion taken as a starting
point. Extending such a technique, it is possible to obtain schemes with arbi-
trary pixel expansion m 0 , with 1 < m 0 r. The new collections of matrices
C 0 B and C 0 W for the resulting scheme is built by constructing in all possible
ways matrices with m 0 columns from the vectors of the collections C B and
C W of the starting probabilistic scheme (in the particular case of m 0 = r, the
resulting scheme is deterministic). We remark that Construction 1 does not
allow repetition of the same column. It is possible to build schemes by also
allowing repetitions of the columns; however, the resulting schemes have a
worst probabilistic factor. Notice that it is useless to construct probabilistic
schemes with m 0 > r, since if m 0 = r then a deterministic scheme can be
realized.
Construction 1 Let S be a canonical -probabilistic (k;n; 0; 1; 1)-VCS. Fix
1 < m 0 r, where r = jC B (S)j = jC W (S)j. Construct a scheme S 0 whose
collection C B (S 0 ) (resp. C W (S 0 )) consists of all the matrices of dimension n
m 0 that we can build by choosing m 0 vectors of C B (S) (resp. C W (S)).
Notice that we also need to fix the contrast thresholds ` 0 and h 0 of the new
scheme S 0 . There can be several valid choices.
To illustrate the construction, consider the following 1=3-probabilistic
(2; 3; 0; 1; 1)-VCS S.
8
<
2
3
2
3
2
3
9
=
8
<
2
3
2
3
2
3
9
=
1
0
0
0
1
0
0
0
1
1
1
1
0
0
0
0
0
0
4
5 ;
4
5 ;
4
5
4
5 ;
4
5 ;
4
5
C B =
C W =
:
:
;
:
;
For such a scheme, the parameters are p bjb = 2=3 and p bjw = 1=3.
If m 0 = 2 is fixed, a
1
3 -probabilistic (2; 3;;; 2)-VCS is obtained by applying
Construction 1 to scheme S:
8
<
:
2
3
2
3
2
3
2
3
2
3
2
3
9
=
;
1
0
1
0
0
0
0
1
0
1
0
0
4
5 ;
4
5 ;
4
5 ;
4
5 ;
4
5 ;
4
5
C B =
0
1
0
0
1
0
1
0
0
0
0
1
0
0
0
1
0
1
0
0
1
0
1
0
8
<
9
=
2
3
2
3
2
3
2
3
2
3
2
3
1
0
1
0
0
0
0
1
0
1
0
0
4
5 ;
4
5 ;
4
5 ;
4
5 ;
4
5 ;
4
5
C W =
1
0
1
0
0
0
0
1
0
1
0
0
:
:
;
1
0
1
0
0
0
0
1
0
1
0
0
The thresholds of S 0 can be selected in different ways returning in every
case a
1
3 -probabilistic scheme. If values ` 0 = 0 and h 0 = 1 are selected, the
resulting probabilities for the scheme S 0 are p 0 bjb = 1 and p 0 bjw = 2=3. For
 
Search WWH ::




Custom Search