Cryptography Reference
In-Depth Information
Proof
Assume that the scheme is in canonical form (see Lemma 13) and let M
W
and
M
B
be the nm Boolean matrices that describe the encoding.
Since the scheme is in canonical form, m is even and exactly m
0
=
2
subpixels on each slide are black. In terms of the linear program this means
n 1
j
n 1
j
n
X
n
X
1
2
:
x
(W)
j
x
(B)
j
=
=
(8.18)
j=0
j=0
By equation 8.17 (with h = 1) the contrast is
n 3
i 1
n 3
i 1
n
X
n
X
x
(B)
i
x
(W)
i
=
i=0
i=0
Since the scheme is in canonical form we have x
(W)
i
= x
(B)
ni
and hence,
n 3
i 1
n 3
i 1
n
X
x
(B)
i
x
(B)
ni
=
i=0
n 3
i 1
n 3
ni 1
X
x
(B)
i
=
i=0
n 3
i 1
n 3
i 2
X
x
(B)
i
=
i=0
Now use equation (8.18) to multiply with 1 and we get
n3
i1
P
i=0
x
(B)
n3
i2
i
=
2
P
i=0
x
(B)
n1
j
i
P
x
f(x)
P
x
g(x)
max
x
f(x)
The inequality
holds for any functions f and g. Hence,
g(x)
n3
i1
n3
i2
2
n1
j
max
0in
The maximum is reached for i = b
n+
4
c, which proves the theorem.
2
In a recent article M. Bose and R. Mukerjee [5] show how to use group
divisible designs and balanced incomplete block designs to construct 3-out-of-n
visual cryptography schemes with optimal contrast and small pixel expansion.
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