Many image hiding schemes based on least significant bit (LSB) transformation have been proposed. One of the LSBbased image hiding schemes that employs diamond encoding was proposed in 2008. In this scheme, the binary secret data is converted into base
n
representation, and the converted secret data is concealed in the cover image. Here, we show that this scheme has two vulnerabilities: noticeable spots in the stegoimage, i.e., a nonsmooth embedding result, and inefficiency caused by rough readjustment of fallingoffboundary value and impractical base translation. Moreover, we propose a new scheme that is efficient and produces a smooth and high quality embedding result by restricting
n
to power of 2 and using a sophisticated readjustment procedure. Our experimental results show that our scheme yields high quality stegoimages and is secure against RS detection attack.
1. INTRODUCTION
Currently, the internet is the mostused channel among the various data communication channels. However, communication through the internet is vulnerable to many forms of attack, including eavesdropping and impersonation. These vulnerabilities cause security and privacy problems. To ensure security and privacy in communication via the internet, a cryptosystem is usually employed
[1

3]
. A cryptosystem modifies the secret message into a series of meaningless bits so that an observer (or eavesdropper) cannot understand the message. However, cryptosystems have a problem that results from the fact that encrypted messages are random bit strings: an observer can notice that the two entities communicate in secret
[1
,
4]
.
However, data hiding conceals the secret communication itself. Data hiding is a part of steganography with watermarking. Steganography is a set of methods that embed information into cover media. Usually, the cover medium is an image; in this case, we call the embedded result a stegoimage. One branch of image steganography(in short, steganography), watermarking, embeds a symbol to protect copyright or ownership. The hidden symbol in the embedded result does not have to be invisible, but it should tolerate many image manipulations and still contains the symbol. In another branch of steganography, image hiding, a secret message is embedded, and an observer should not detect the existence of a secret message in the stegoimage, even if a large amount of secret information is hidden
[1
,
4
,
5

8
,
14]
. In fact, the size of the secret message and the imperceptibility of the secret data in the stegoimage are two criteria of the embedded result; however, improving one requires sacrificing the other. Enhancing both criteria simultaneously is an open problem.
One method to easily obtain a stegoimage is a simple LSB substitution scheme. In this scheme, the secret message is substituted for LSBs in each pixel. The number of substituted LSBs of each pixel depends on the length of the secret message and the size of the cover image. This scheme is based on the fact that changing the least significant bits (LSBs) has only a small effect on the whole image. However, an observer can easily detect the existence of secret data using steganalysis, such as RS detection attack
[4
,
5
,
9]
. In spite of flaws of simple LSB substitution scheme, it has inspired many researchers to study an image hiding scheme in the spatial domain.
In 2006, the EMD scheme
[10]
was proposed that greatly reduces the perceptibility of secret data in the embedded result by adding or subtracting 1 to the pixel value of the cover image. This image hiding scheme starts by translating secret data into a base 2
n
+1 representation, where
n
is an embedding parameter. One digit of the secret data in the base 2
n
+1 numeral system is embedded in
n
pixels. This scheme increases the quality of the embedded result by adding or subtracting 1 to the pixel value, but the total size of the secret message is limited.
To increase the allowed size of the secret message, Chao et al. proposed diamond encoding as an image hiding scheme
[11]
. However, we demonstrate that their scheme has poor efficiency and that the stegoimage from their scheme has many gray stains, which allows an observer to detect the existence of secret data if the cover image has many pixels whose values are close to either 0 or 255, as shown in Section Ⅱ. We propose a new image hiding scheme in Section Ⅲ that provides a high quality embedded result efficiently using a square function. In Section Ⅳ, we present the experimental results and security analysis against RS detection attack
[9]
. Finally, we conclude in Section Ⅴ.
2. REVIEW OF THE DIAMOND ENCODING IMAGE HIDING METHOD
In this section, we review the steganographic method with diamond encoding (in short, diamond scheme)
[11]
. In addition, we analyze the diamond scheme.
 2.1 Description of the diamond scheme
In the diamond scheme, a grayscale cover image is divided into non overlapping blocks of two successive pixels. A block carries one digit in the base(2
k
^{2}
+2
k
+1) numeral system, where
k
is an embedding parameter. For simplicity, we let 2
k
^{2}
+2
k
+1 be represented by
l
. For each
k
, there exists a diamond function
f_{k}
:
Z
_{256}
×
Z
_{256}
→
Z_{l}
where
f_{k}
(
p
,
q
)=(2
k
+1)
p
+
q
mod
l
and a diamond matching function
D_{k}
:
Z_{l}
→
Z
_{2k+1}
×
Z
_{2k+1}
. The diamond matching function
D_{k}
is specified by a diamond shaped table, such as those in
Fig. 1
and
Fig. 2
. Assuming that the coordinate of 0∈
Z_{l}
is (0,0)∈
Z
_{2k+1}
×
Z
_{2k+1}
,
D_{k}
(
a
) is the coordinate of
a
in the diamond shaped table
D_{k}
for all
a
∈
Z_{l}
. For instance,
D
_{2}
(4) is (1,1).
The diamond matching function D_{k} and its coordinates, where k is 1 and is 2.
The diamond matching function D_{k}, where k is 3 and is 6.
The secret embedding procedure starts by selecting the embedding parameter
k
. The embedding parameter
k
is determined according to the length of the secret message s and the size of the cover image
R
×
C
:
k
is the smallest integer that satisfies
Then, the binary secret message
s
is translated into base
l
representation (
s
)
_{l}
,
s
↦(
s
)
_{l}
(=
s
_{1}
║
s
_{2}
║⋯║
s_{t}
║⋯║
s_{m}
), where
s_{t}
is the
t
th digit of (
s
)
_{l}
,
m
is an integer that is less than half the image size
R
×
C
and ║ represents concatenation. The value of the function
f_{k}
,
f_{k}
(
b_{t}
)=
f_{k}
(
p_{t}
,
q_{t}
)=(2
k
+1)
p_{t}
+
q_{t}
mod
l
, is computed for every block
b_{t}
, where
b_{t}
=(
p_{t}
,
q_{t}
) is the
t
th block in the cover image and both
p_{t}
,
q_{t}
are pixels. Then, the difference
d_{t}
=
s_{t}

f_{k}
(
b_{t}
) is calculated for all
t
. Finally, the pair of stegopixel values (
p_{t}′
,
q_{t}′
)=(
p_{t}
,
q_{t}
)+(
a
,
b
) are computed, where (
a
,
b
)=
D_{k}
(
d_{t}
). Here, some stegopixels may not fall between 0 and 255. We call these pixels the fallingoffboundary pixels. These pixels require readjustment so that they are included in [0,255]: if the stegopixel
x′
>255(resp.
x′
<0), then
x′
=
x′

l
(resp.
x′
=
x′
+
l
). The secret embedding procedure ends if all the fallingoffboundary pixels are readjusted.
In the secret data extracting procedure, the stegoimage is divided into nonoverlapping blocks, as in the cover image. Then, the series of outputs of the diamond function for each block is the secret data (
s
)
_{l}
in the base
l
numeral system. In other words,
f_{k}
(
b
_{1}
′
)║
f_{k}
(
b
_{2}
′
)║⋯║
f_{k}
(
b_{t}
′
)║⋯║
f_{k}
(
b′_{m}
) is the secret message that is converted into base
l
representation, where
b_{t}
′
is the
t
th block of stegoimage and
Thus, the binary secret message can be obtained by translating the series of
f_{k}
(
b_{t}
′
) into base2 representation.
Example. We give an example of producing a stegoblock according to the diamond scheme. Assuming that the embedding parameter
k
is 3, then we determine that
l
is 25. We let the
t
th digit
s_{t}
of the secret message in the base25 numeral system be 6∈
Z
_{25}
, and the
t
th block
b_{t}
=(
p_{t}
,
q_{t}
) of the cover image be (200,190). The diamond function value of
b_{t}
is
f
_{3}
(
b_{t}
)=
f
_{3}
(200,190)=7×200+190 mod. 25= 15.Then,
d_{t}
=
s_{t}

f
_{3}
(
b_{t}
) mod
l
=615 mod 25 =16. The output of the diamond matching function
D
_{3}
(
d_{t}
) is (1,2), as shown in
Fig. 2.
(a). Thus, the
t
th block of the stegoimage is
b_{t}
+
D
_{3}
(
b_{t}
)=(199,188). The secret digit
s_{t}
concealed in
t
th stegoblock is extracted by the diamond function as follows:
s_{t}
=
f_{k}
(
b_{t}
′
)=
f
(199,188)=7×199+188mod25=6.
 2.2 Analysis of the diamond scheme
This section shows the two drawbacks of the diamond scheme : a nonsmooth embedded result and inefficiency.
 2.2.1 Nonsmooth stegoimage
The diamond scheme solves the fallingoffboundary problem by simply adding or subtracting
l
to the fallingoffboundary pixel value. This readjustment procedure is simple to apply, but the difference between the original cover pixel and stegopixel is remarkable when
l
is large. This large difference makes the embedded result look nonsmooth.
For example, let
k
=3(
l
=25),
s_{t}
=9,
b_{t}
=(253,264). Then, stegoblock
b_{t}′
is (254,256). This block requires readjustment because the second pixel of this block overflows the boundary value 255. The readjusted block is (254,256)(0,25)=(254,231). As a result, the difference between the original block and the stegoblock is (1,23), which is a significant difference that causes the embedded result to have a perceptible spot.
Fig. 3
. shows a nonsmooth stegoimage when the secret data
s
is a series of random bits with 
s
=777350, which is less than 3×
R
×
C
, where
R
×
C
is the image size. Compared with the cover image, the embedded result has many perceptible spots in Tiffany’s hair and teeth so that the image is not smooth. Even if the peak of the signaltonoise ratio (PSNR) that is employed to estimate the quality of the stegoimage is pretty high (35.21 dB), we can still detect the existence of a secret message
[1
,
11
,
12]
. In short, the embedded result looks nonsmooth when the cover image has many pixels whose values are in the neighborhood of a boundary value(0 or 255) and
l
is large.
Nonsmoothness of a stegoimage generated by the diamond scheme.
In the diamond scheme, binary secret data is converted to its representation in base
l
. Because
l
is not a power of 2, the base conversion is not easy when the secret data is large. In fact, the time required to convert 
s
bit secret data into base
l
representation is
O
(
s

^{2}
)
[13]
. Moreover, the naïve conversion procedure has difficulty translating the base, even if the secret data has about 200,000 bits.
 2.2.2 Inefficiency
In the diamond scheme, binary secret data is converted to its representation in base
l
. Because
l
is not a power of 2, the base conversion is not easy when the secret data is large. In fact, the time required to convert 
s
bit secret data into base
l
representation is
O
(
s

^{2}
)
[13]
. Moreover, the naïve conversion procedure has difficulty translating the base, even if the secret data has about 200,000 bits.
3. PROPOSED SCHEME WITH SQUARE FUNCTION
In this section, we propose an image hiding scheme that employs a square function to address two problems of the diamond scheme. In order to solve the nonsmoothness of the stegoimage, we employ a refined readjustment step for fallingoffboundary pixels rather than the rough readjustment procedure, such as adding or subtracting a relatively large number
l
. Moreover, to solve the inefficiency, we set the base
l
to a power of 2.
Notations. The followings are notations used in this paper.

k: the embedding parameter

l: 22k

s: the secret data

s : the length of secret data

(s)l: the secret data translated into baselrepresentation

st: thetth digit of (s)l

bt: thetth block of an image.bt=(pt,qt),ptandqtare two consecutive pixels. In particular, we leaveb′tfor thetth block of the stegoimage.

dt: the difference betweenstandBk(bt)

Bk: for eachk, there is ablock functionBk:Z256×Z256→Zl, whereBk(p,q)=2kp+qmodl

Sk: for eachk, there is asquare functionSk:Zl→Z2k×Z2k, whereSkis the inverse function ofBkZ2k×Z2k, with the restriction ofBk:Z256×Z256→ZltoZ2k×Z2k. In other words,Sk1(a,b)=2ka＋bmodl, for all (a,b)∈Z2k×Z2k. This function is explicitly represented by a table like those inFig. 4.(a), (b), and (c). We let the coordinate of 0∈Zlbe (0,0)∈Z2k×Z2k. Then,Sk(n) is the coordinate ofnin the table. For example,S2(12) is (1,0), as shown inFig. 4.(a) and (b).
The square function S_{2} and its coordinates, the square function S_{3}, and construction of blocks and the direction of the embedding progress
 3.1 Data embedding procedure
In proposed scheme, a grayscale cover image is divided into nonoverlapping blocks, and one block consists of two consecutive pixels, as in the diamond scheme. Furthermore, one digit of the secret data that is translated into base2
^{2k}
representation can be hidden in each block. In this section, we describe proposed data embedding procedure in detail.
 3.1.1 Setup phase
 Cover image division : the cover image with
R
×
C
pixels is divided into nonoverlapping blocks of two pixels, as shown in
Fig. 4
(d). Simply we represent the cover image as series of blocks
b
_{1}
║
b
_{2}
║⋯║
b_{t}
║⋯
.
 Determining the embedding parameter
k
: the embedding parameter
k
depends on the image size
R
×
C
and the length of the secret message 
s
 . In detail,
k
is the smallest integer that satisfies
 Base translation : the secret message
s
should be converted into (
s
)
_{l}
. In other words, the base of
s
must be translated into
l
.
 3.1.2 Encoding phase
The embedding phase is described below. It is similar to the diamond scheme, except for the readjustment step.

a. Find the block function valueBk(bt) for eachbt.

b. Compute the differencedt=stBk(bt) modlfor everyt.

c. FindSk(dt)=(a,b) for everyt.

d. Compute the prestegoblock

bt′=bt+Sk(dt)=(pt,qt)+(a,b) for every t.

e. The prestegoblock may contain the fallingoffboundary pixels; thus, the prestegoblock should be readjusted as follows.

i. Ifp′t≤0 andqt′<0, thenpt′=pt′+2k1,qt′=qt′+2k.

ii. Ifp′t<0 and 0≤qt′≤255, thenpt′=pt′+2k,qt′=qt′.1

iii. Ifp′t<0 andqt′>255, thenpt′=pt′+2k+1,qt′=qt′2k.

iv. If 0

v. If 0≤pt′<255 andqt′>255, thenpt′=pt′+1,qt′=qt′2k.

vi. Ifp′t>255 and 0≤qt′≤255, thenpt′=pt′2k,qt′=qt′.

vii. Ifp′t>256 andqt′<0, thenpt′=pt′2k1,qt′=qt′+2k.

viii. Ifp′t≥255 andqt′>255, thenpt′=pt′2k+1,qt′=qt′2k.
Because the block function value of the stegoblock includes information about the secret message, block function value should not be changed after the block is readjusted. One solution is to set , where is a fallingoffboundary pixel, as in the diamond scheme. However, there is large gap between the cover image and stegoimage with this rough readjustment step; thus, it allows the secret message to be perceived. The proposed sophisticated readjustment step employs a value less than , rather than employing ; as a result, we reduce the difference between the cover image and the stegoimage. Moreover, there is no difference in the block function value before and after the readjustment step.

f. The stegoimage is a composition of stegoblocks. Namely, the stegoimage is

b1′║⋯║bt′║⋯║.
 3.1.3 Data extraction procedure
Extraction of the secret message from the stegoimage is simpler than embedding.

a. The stegoimage is divided into nonoverlapping blocks of two pixels, as in the embedding procedure. As a result, the stegoimage is represented asb1′║⋯║bt′║⋯║.

b. Find the block function valueBk(bt′) for eachbt′.

c. Concatenate all block function values, and then obtain the presecret message (s)l. Namely, the presecret message (s)lisBk(b1′)║⋯║Bk(bt′)║⋯║Bk.

d. The secret messagesis obtained by converting the presecret message (s)lto binary data.
Note. The following illustrates why
B_{k}
(
b_{t}
′
)=
s_{t}
.
B_{k}
(
b_{t}
′
)=2
^{k}
p_{t}′
+
q_{t}′
mod
l
=2
^{k}
(
p_{t}
+
a
)+(
q_{t}
+
b
) mod
l
=2
^{k}
p_{t}
+
q_{t}
+2
^{k}
a
+
b
mod
l
=
B_{k}
(
b_{t}
)+
S_{k}
^{1}
(
a
,
b
)
=
B_{k}
(
b_{t}
)+
d_{t}
=
s_{t}
.
Example. We give an example of producing a stegoblock with proposed scheme. Assuming that the embedding parameter
k
and
l
are 2 and 16, respectively. We let the
t
th digitnof secret data
s_{t}
be 12∈
Z
_{16}
, and the
t
th block
b_{t}
=(253,254). Then the block function value
B
_{2}
(253,254) is 2
^{2}
×253+254 mod 16=2. And
d_{t}
=
s_{t}

B
_{2}
(
b_{t}
) mod
l
=10,
S
_{2}
(
d_{t}
)=
S
_{2}
(10)=(2,2) as shown in
Fig. 4.
(a) and (b). Thus the
t
th prestegoblock
b_{t}
′
is
b_{t}
+
S
_{2}
(
d_{t}
)=9255,256). This block requires to be readjusted because the second pixel of the block overflows the boundary value 255. By e.viii of embedding phase, the final stegoblock
b_{t}
′
is (252,252). The extraction procedure is
s_{t}
=
B
_{2}
(
b_{t}
′
)=2
^{2}
×252+252 mod 16=12.
4. EXPERIMENTAL RESULTS
In this section, we present our experimental results, including a quality comparison with other schemes and security analysis against RS detection attack. We experimented with the wellknown 8bit grayscale images “Couple", “Zelda", “Lena", “Tiffany", “F16", “Baboon", and “Pepper" in
Fig. 5
,
6
, and
7
; the size of first two images is 256×256, and the others are 512×512. Moreover, we performed the experiment with MATLAB, and the secret message was generated by the pseudorandom function in MATLAB.
The cover images and stegoimages with 256×256 images generated by the diamond scheme and the proposed scheme where s=196608 bits.
The cover images and stegoimages with 512×512 images generated by the diamond scheme and the proposed scheme where s=786432 bits.
The cover images and stegoimages with 512×512 images generated by the diamond scheme and the proposed scheme where s=786432bits.
 4.1 Quality and capacity
We employ peaks of the signaltonoise ratio (PSNR) to estimate the quality of the stegoimages. The capacities and relative PSNRs for various test images are shown in
Table 1
and
Table. 2
.
Table 1
and
Table 2
, respectively, represent the PSNRs when the capacity is 2×
R
×
C
bits and 3×
R
×
C
bits with an image size of
R
×
C
.
Comparison of PSNR with simple LSB substitution scheme, the diamond scheme, and ours when s=2×R×C, whereR×Cis the image size
Comparison of PSNR with simple LSB substitution scheme, the diamond scheme, and ours when s=2×R×C, where R×C is the image size
Comparison of PSNR with simple LSB substitution scheme, the diamond scheme, and ours whens=3×R×Cwith an image size ofR×C
Comparison of PSNR with simple LSB substitution scheme, the diamond scheme, and ours whens=3×R×C with an image size of R×C
The tables show that our scheme provides consistent PSNRs over various images. On the other hand, we find that the PSNRs of stegoimages generated by the diamond scheme are inconsistent and depend on the cover images. In detail, the diamond scheme and the proposed method yield similar outputs, except “Couple”, “Zelda”, and “Tiffany”: the suit of a man in
Fig. 5.
(b) and the hair and the cloth in Zelda in
Fig. 5.
(e) have many gray stains. Moreover, the teeth and hair of Tiffany in
Fig. 6.
(e) have many gray spots. These three cover images have multiple pixels whose values are close to either 0 or 255. As cited in Section Ⅱ.2.A, the diamond scheme provides a poor embedding result when the size of the embedded message is large and the cover image have many neighborhoods of pure white or pure black (pixel values of 255 and 0, respectively). In the meantime, our scheme yields reasonable embedding results irrespective of the images.
 4.2 Security : RS detection attack
This section shows that our scheme is secure against RS detection attack
[9]
. The RS detection attack was proposed by Fridrich et al. and detects stegoimages. In their paper, a discrimination function
f
is defined that estimates the smoothness of a pixel group, an invertible operation flipping
F
, and a row vector of length
n
mask
M
. To determine whether or not an image is a stegoimage, the image is divided into disjoint groups of
n
adjacent pixels. Furthermore,
f
(
G
) is compared with
f
(
F_{M}
(
G
)), where
G
is a pixel group and
F_{M}
(
G
) is flipped
G
with mask
M
. If
f
(
G
) is larger(or smaller) than
f
(
F_{M}
(
G
)), the pixel group
G
is defined as a regular group
R
(or singular group
S
). Otherwise, it is defined as an unusable group
U
. The value of
R_{M}
(or
S_{M}
and
U_{M}
) is the number of regular(or singular and unusable) groups with mask
M
. Finally, the image is judged to be a stegoimage if it does not satisfy the following properties:
R_{M}
≅
R_{M}
and
S_{M}
≅
S_{M}
.
We subjected the stegoimages generated by our scheme to an RS detection attack with
f
(
G
)=
f
(
x
_{1}
,
x
_{2}
,⋯,
x_{n}
)=

x
_{i+1}

x_{i}
 as a discrimination function and [0 1 1 0] as the mask
M
.
Fig. 8
. is one attack result when the cover image is “Lena”, and we obtained similar results for the other images.
Security of proposal scheme against RS attack.
5. CONCLUSION
To secure communication, research on an image hiding scheme that hides the communication itself in the images has been proposed in addition to cryptography. In image hiding, the capacity and imperceptibility of secret data in the stegoimage are two important barometers of the quality of the scheme. Many researchers have proposed LSBbased image hiding schemes to conceal a large quantity of secret data while maintaining the imperceptibility of the secret data in the stegoimage. Among the image hiding schemes, we reviewed the diamond scheme proposed by Chao et al. and revealed two drawbacks of the diamond scheme: poor embedding results for images that have many pixels whose values are close to 0 or 255 when there is a large quantity of secret data and inefficiency caused by base conversion. We proposed a new image hiding scheme that employs a square function. In addition, the proposed scheme employs a sophisticated readjustment procedure for fallingoffboundary pixels and efficient base conversion. In detail, one digit in the base 2
^{2k}
numeral system is embedded in two pixels. Moreover, we showed that the proposed scheme provides higher quality results than the diamond scheme according to the experimental results.
BIO
Hyejin Kwon
Feb. 2007: Kyungpook National University, Dept. of Mathematics, B.A.
Feb. 2009: Kyungpook National University, Dept. of Information Security, M.S.
Feb. 2009~: Kyungpook National University, Dept. of Electronics Engineering, PhD Candidate
Research Interest: Information security, Cryptography, Network security, Steganography
Haemun Kim
Feb. 2002: Kyungpook National University, Dept. of Electronics Engineering, B.S
Feb. 2004: Kyungpook National University, Dept. of Electronics Engineering, M.S..
Feb. 2005: Kyungpook National University, Dept. of Electronics Engineering, PhD Candidate
Research Interest: Information security, Cryptography, Network security, Steganography
Soonja Kim
Feb. 1975: Kyungpook National University, Dept. of Mathematics Education, B.S.
Feb. 1977: Kyungpook National University, Dept. of Mathematics Education, M.S.
Feb. 1988: Keimyung University, Dept. of Mathematics, PhD.
Apr. 1993~: Kyungpook National University, Dept. of Electronics Engineering, Professor
Research Interest: nformation security, Cryptography, Network security
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