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
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