Cryptography Reference
In-Depth Information
2.If k is odd, then x
(W)
i
= x
(B)
ni
for i = 0;:::;n.
3 x
(W)
i
x
(B)
i
= 0 for all i 2f0;:::;ng.
We will say the solution of the linear program is in a canonical form.
Proof
We first show that replacing each transparency of a k-out-of-n visual cryptog-
raphy scheme by its complement again gives a solution of a k-out-of-n visual
cryptography scheme. In the language of linear programming:
Let x
(W)
i
, x
(B)
i
be a solution of the linear program (8.14){(8.16). We claim
that
(
x
(W)
ni
if k is even
x
(B)
x
(W)
i
=
ni
if k is odd
and
(
x
(B)
ni
if k is even
x
(W)
ni
x
(B)
i
=
if k is odd
is also a solution of the linear program.
Since
i
=
ni
the variables x
(W)
m x
(B)
i
satisfy (8.16).
i
We claim that equation 8.16 implies
nj
ih
nj
ih
nj
nj
X
X
x
(W)
i
x
(B)
i
=
i=0
i=0
for all h j. For j = 0 this is trivial and for j 1, h 1 it follows from
nj
ih
=
nj+1
ih+1
nj
ih+1
by induction
nj + 1
ih + 1
nj + 1
ih + 1
n
X
n
X
x
(W)
i
x
(B)
i
=
i=0
i=0
and
nj
ih + 1
nj
ih + 1
n
X
n
X
x
(W)
i
x
(B)
i
=
:
i=0
i=0
To simplify the notation let us assume in the following that k is even.
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