Cryptography Reference
In-Depth Information
Because, under the OR operation, the stacking of two black subpixels
results in a black subpixel, i.e., 1 OR 1 = 1, so this means that there is a black
subpixel that is ineffective (overlapped). Now we define a black pixel (value 1)
to be ineffective if it does not contribute to the total number of black pixels
in the recovered secret image. There are three cases when a black subpixel
1 is ineffective: (1) when that is in the top right corner of the matrix M
0
(or M
1
) and another 1 is shifted in, this results in a overlap; (2) when that
is in the bottom left corner of the matrix which is then shifted out (appear
below an asterisk `*'); and (3) when an overlap happens after a shift, which is
possible on the rst m 1 positions (and mr positions in general). Hence,
the total Hamming weight of the stacking of the shifted share matrices can
be calculated by the total number of the 1's subtracted by the number of
the 1's that are ineective (when two 1's overlapped, we only count one as
ineffective).
In the rst case, denote by s
0;c
and s
1;c
the number of 1's that are in-
eective for the collections C
0
and C
1
, respectively when the subpixel c is
shifted in. Since there are m! share matrices in the collection C
0
and C
1
, so
the total number of 1's that are ineective in the top right corner of all the
share matrices in C
0
and C
1
is is
0;1
= s
1;1
=
a
0
+c+
m
m! (when a 1 is shifted in)
and s
0;0
= s
1;0
= 0 (when a 0 is shifted in), where
a
0
+c+e
is the probability
m
of the 1's in the top right corner of the rst row.
In the second case, denote by s
0;c
and s
1;c
the number of 1's that are
ineffective for the collections C
0
and C
1
, respectively when the subpixel c is
shifted in. So the total number of 1's that are ineective in the bottom left
corner of all the share matrices in C
0
and C
1
is is
1;1
= s
0;1
= s
1;0
= s
0;0
=
a
0
+d+e
m
m!, where
a
0
+d+e
m
is the probability of the 1's in the bottom left corner
of the second row.
In the third case, denote by s
0;c
and s
1;c
the number of 1's that are ineec-
tive for the collections C
0
and C
1
respectively when the subpixel c is shifted
in. Note that, the pattern
1
1
in the shifted share matrices is the shifted
11
01
10
11
10
01
11
11
result of the following four patterns,
in
the collections C
0
and C
1
. We calculate the probability of the first pattern
,
,
, and
11
01
for example, and the other three patterns can be calculated similarly.
1
0
The probability that the pattern
appears at the column i in the matri-
ces of the collection C
1
is
c+
m
, and, fixing this pattern, the probability that
the pattern
1
1
a
0
appears at the column i + 1 of the collection C
1
is
m1
,
11
01
where 1 i m1. So the probability that the pattern
appears both
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