Cryptography Reference
In-Depth Information
5.6.2 (2;n)-Threshold Probabilistic Schemes with Any Pixel
Expansion
In this section a (2;n)-threshold probabilistic scheme with pixel expansion,
for n 3 is provided, starting from the (2;n)-threshold deterministic scheme
S
D
described in [7]. Recall that S
D
has pixel expansion m =
n
, and
b
n
2
c
thresholds ` =
n
1
b
n
2
c1
. The scheme has a white base
matrix consisting of ` columns of all 1s and m` columns with all 0s, and a
black base matrix consisting of all binary vectors with exactly bn=2c elements
equal to 1.
For example, for n = 4, scheme S
D
has m = 6, ` = 3, and h = 5 and is
given by:
and h =
n
b
n
2
c
n
2
b
n
2
c
2
3
2
3
1
1
1
0
0
0
1
1
1
0
0
0
4
5
4
5
:
1
1
1
0
0
0
1
0
0
1
1
0
M
W
=
M
B
=
1
1
1
0
0
0
0
1
0
1
0
1
1
1
1
0
0
0
0
0
1
0
1
1
Let S be the (2;n; 0; 0; 1)-VCS obtained applying Lemma 3 to scheme S
D
.
Scheme S is a canonical scheme, and the cardinality of the collections C
B
and
C
W
is r = m. The values of the probabilities are p
bjb
= h=m, p
wjb
= 1h=m,
p
bjw
= `=m, p
wjw
= 1 `=m. Since S is canonical the scheme S
0
obtained
with Construction 1 is canonical. The value of the pixel expansion m
0
can be
arbitrarily fixed between 1 and r.
Consider the case of m
0
= 2. For such a value of m
0
there are only two
possible choices for the thresholds: `
0
= 0;h
0
= 1 or `
0
= 1;h
0
= 2. It is easy
to see from the structure of M
W
and M
B
that the following two properties
hold:
PW: For any set Q of two participants, the matrix M
W
consists of ` columns
of 2 ones and m` columns of 2 zeroes.
PB: For any set Q of two participants, the matrix M
B
consists of s =
n2
b
n
2
c
columns of 2 zeroes and ms columns with at least a 1 (this is because
for a qualified set Q of 2 participants matrix M
B
reconstructs a pixel with
exactly h black subpixels, hence the remaining mh = s are white).
Let us now determine the parameters of the resulting scheme in the two con-
sidered cases.
Case `
0
= 0;h
0
= 1.
To compute p
0
bjb
, consider that in this case a reconstructed pixel is black if
at least one subpixel is black (and white otherwise). If a qualified set Q is
fixed, by property PB, it is easy to see that the matrices of the collection
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