Cryptography Reference
In-Depth Information
8
<
2
1
0
0
3
2
0
1
0
3
2
0
0
1
3
9
=
4
5
;
4
5
;
4
5
C
1
=
0
1
0
0
0
1
1
0
0
:
:
;
0
0
1
1
0
0
0
1
0
(hl)
h+l
2
4
1
(C
0
;C
1
) is a [(2; 3); 3; 3; 1] scheme. The contrast =
=
=
2
.
6.3.2 (n,n) Scheme
In this section, we show how to construct a (n;n) XOR-based visual cryptog-
raphy scheme.
Proposition 2 [17] Let C
0
and C
1
be the set of all binary vectors of length n
with even, odd number of ones, respectively. Then (C
0
;C
1
) is an [(n;n); 1; 1; 0]
scheme.
An example is given as follows.
Example 2 Let C
0
and C
1
be
8
<
9
=
8
<
9
=
2
3
2
3
2
3
2
3
2
3
2
3
2
3
2
3
0
0
0
1
1
0
1
0
1
0
1
1
1
0
0
0
1
0
0
0
1
1
1
1
4
5
;
4
5
;
4
5
;
4
5
4
5
;
4
5
;
4
5
;
4
5
C
0
=
;C
1
=
:
;
:
;
(hl)
h+l
(C
0
;C
1
) is a [(3; 3); 1; 1; 0] scheme with contrast =
= 1.
6.3.3 (k,n) Scheme
In this section we will introduce three (k;n) XOR-based visual cryptography
schemes. Construction 1 and Construction 2 are given by Tuyls (see more
detail in Reference [17]). Construction 3 comes from Droste's OR-based visual
cryptography [9] and Liu et al. [13] shows that it is also a (k;n) XOR-based
visual cryptography scheme.
Before introducing Construction 1 and Construction 2, some definitions
and theorems will be given, which are used in both constructions.
For describing the constructions, we use the following notation. If A is a
binary matrix, then P(A) is the multi set of matrices obtained by permuting
the columns of A. Moreover, we use the concept of (k;n) pairs, dened as
follows.
Denition 2 [17]A pair (A;B) of binary n m matrices is called a (k;n)
pair if there exist numbers a
1
; ;a
k
and b
1
; ;b
k
such that
1. For each i with 1 i k, the weight of the sum of any i rows from
A equals a
i
and the weight of the sum of any i rows from B equals
b
i
, and
2. a
i
= b
i
for 1 i < k, and a
k
6= b
k
.
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