Cryptography Reference
In-Depth Information
ticipants. Two collections of n Boolean matrices C
0
and C
1
constitute a
VSS scheme if there exists a value () and value t
X
for every X in
Qual
satisfying [3]:
1. Contrast condition: any (qualified) subset X = fi
1
;i
2
; ;i
u
g2
Qual
of u participants can recover the secret image by stacking the
corresponding transparencies. Formally, for a matrix M 2C
j
, (j =
0; 1) the row vectors v
j
(X; M) = OR(r
i
1
; r
i
2
; ; r
i
u
). It holds that:
w(v
0
(X; M)) t
X
() for all M 2C
0
and w(v
1
(X; M)) t
X
for all M 2 C
1
. () is called the relative dierence referred to as
the contrast of the decoded image and t
X
is the threshold to visually
interpret the reconstructed pixel as black or white.
2. Security condition: Any (forbidden) subset X = fi
1
;i
2
; ;i
v
g2
Forb
has no information of the secret image. Formally, the two col-
lections D
j
(j = 0; 1), obtained by extracting rows i
1
;i
2
; ;i
v
from
each matrix in C
j
, are indistinguishable.
1.2.2 Construction of VSS Scheme
If the given secret pixel p is black (white), the matrix M is randomly selected
from matrices collections C
1
(C
0
). The matrix collections can be obtained by
permuting the columns of the corresponding basis matrix S
0
or S
1
in all
possible ways [3]. The basis matrices are defined below.
Definition 2 Two matrices S
0
and S
1
are called basis matrices, if S
0
and S
1
satisfy the following tow conditions [3]:
1. Contrast condition: If X = fi
1
;i
2
; ;i
u
g 2
Qual
, the row
vectors v
0
and v
1
, obtained by performing OR operation on rows
i
1
;i
2
; ;i
u
of S
0
and S
1
respectively, satisfy w(v
0
) t
X
()
and w(v
1
) t
X
.
2. Security condition: If X = fi
1
;i
2
; ;i
v
g 2
Forb
, one of
the two v matrices, formed respectively by extracting rows
i
1
;i
2
; ;i
v
from S
0
and S
1
, equals to a column permutation of
the other.
The algorithm to construct the basis matrices for a given VSS scheme can
be found in [2, 5]. See [5] for the construction algorithm of basis matrices
that leads to the best contrast. As an example, the S
0
and S
1
in a 2-out-of-2
scheme are shown below:
0
0
1
1
S
0
=
; S
1
=
:
(1.3)
0
1
1
0
S
0
corresponds to the encoding of a white secret pixel and S
1
corresponds to
the encoding of a black secret pixel.
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