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to prevent message extraction. A game model between data-hider and data-attacker and
the game equilibria are given in [13].
As opponents of attackers, data-hiders always try to design steganographic strategies
for resisting statistical analyses [14]. For example, by carefully choosing replacement
of the host pixel values, an LSB-based method can effectively withstand the RQP
analysis [15]. In the present paper, a novel steganographic approach is proposed, which
avoids the asymmetric characteristic inherent in conventional LSB embedding tech-
niques so that distortion to the image histogram is kept minimum. The proposed tech-
nique is capable of resisting several powerful steganalytic methods including the RS
analysis, the
test, and a new technique termed GPC analysis as introduced in Section
2 of this paper.
χ
2
2
Analyses of Steganographic Techniques
2.1
Chi-Square Test [4, 5]
Secret message for encoding or encryption can be considered a pseudo-random bit
stream consisting of 0s and 1s. After replacing the LSBs of a cover image with these
hidden bits, occurrences with gray values 2
i
and 2
i
+1 tend to become equal. Supposing
that
n
j
is the number of pixels with a gray value
j
, the χ
2
test calculates
[
(
)
]
2
k
n
−
n
+
n
/
2
2
=
χ
=
2
i
2
i
2
i
+
1
.
(1)
(
)
n
+
n
/
2
i
1
2
i
2
i
+
1
and
k
−
1
1
t
2
−
−
1
χ
p
=
1
−
]
∫
exp
t
dt
,
2
(2)
[
(
k
−
1
/
2
2
2
Γ
(
k
−
1
/
2
0
where
p
represents the probability that the distributions of
n
2i
and
n
2i
+1
are equal. This
can be used to decide the presence of secret information.
2.2
RS Analysis [2, 6]
This method defines two mappings:
F
1
for 0↔1, 2↔3, …, 254↔255 and
F
−
1
for
−1↔0, 1↔2, …, 255↔256. In other words,
F
1
is used when the LSB of cover image is
different from the hidden bit. Also, the following function is defined to measure the
smoothness of a pixel group (
x
1
,
x
2
,
…
,
x
n
):
n
−
1
(
)
∑
f
x
,
x
,
,
x
=
x
+
−
x
.
(3)
1
2
n
i
1
i
i
=
1
Divide the received image into small blocks of the same size. Define
R
M
as the ratio
of blocks in which
f
increases when
F
1
is applied to a part of each block, and
S
M
as the
