ieee a secure algorithm for image based information hiding with one-dimensional chaotic systems.It used 1 dimensional chaotic system.ieee paper related for image encryption

1. A Secure Algorithm for Image Based Information Hiding with One-dimensional
Chaotic Systems
Yinglei Song
School of Electronics and Information Science
Jiangsu University of Science and Technology
Zhenjiang, China
e-mail: syinglei2013@163.com
Jia Song
Department of Electronic and Information Technology
Suzhou Vocational University
Suzhou, 215104, China
e-mail: yigibird@yahoo.com.cn
Junfeng Qu
Department of Computer Science and Information Technology
Clayton State University
Morrow, GA 30260, USA
e-mail: junfengqu@clayton.edu
Abstract—The goal of information hiding is to securely hide
important information within digital documents such as files,
images and videos. In this paper, we develop a new method
that can efficiently and securely embed a sequence of binary
bits into a gray-scale image in two stages. In the first stage, the
sequence is divided into a few subsequences of equal length and
a set of one-dimensional logistic mappings are used to shuffle
the binary bits in the sequence. In the second stage, the
shuffled sequence is divided into regions of equal length and
the regions are sequentially embedded into the corresponding
pixels in the given gray-scale image. The embedding of a region
in the shuffled sequence is performed by replacing the least
significant bits of the gray value of the corresponding pixel
with the bits in the region. Analysis and testing results show
that this information hiding algorithm can securely embed
information into a gray-scale image without altering the
content of image significantly and thus can potentially be used
in practice for secure image-based information hiding.
Keywords-information hiding; gray-scale image; one-
dimensional logistic systems; key space
I. INTRODUCTION
The past two decades have witnessed rapid development
in information technologies. As a result, a large amount of
digital information and data has been generated. In order to
protect the intellectual property rights associated with crucial
data and information, researchers have developed numerous
methods and techniques for the encryption of important data
and information. Information hiding is one of such
techniques [1]. Specifically, the goal of information hiding is
to embed important data and information into digital
documents such as files, images and videos without changing
the content of the documents significantly. A document
where the embedding is performed is often referred to as a
cover media and the document that results from an
embedding is a stego media. In general, a stego media is
highly similar to the corresponding cover media and the
difference between them often cannot be recognized by
human eyes.
Since images are important for the storage and
transmission of multimedia data, the embedding of
information into gray scale images is an important
technology in the area of information hiding. A large number
of methods for image based information hiding have been
developed, including the information hiding technology
based on pixel value difference and LSB replacement [2], the
side match technology [3], the hiding technology based on
pixel difference expansion and modulus function [4]. Most
of these methods compute the difference in gray values of
certain neighboring pixels. The gray values of both pixels are
then changed to encode a part of the data that needs to be
embedded. In [5], a technique based on similar sample
blocks is developed for image based information hiding. The
technique searches in a standard image for a sample block
and uses the sample block as the reference for the embedding
of information.
Most of these methods can successfully embed the
information into a given image without altering the content
of the image significantly. However, most of them either
need the cover media or a standard image to recover the
embedded information from the stego media, which may not
be convenient in certain applications. In addition, the
information embedded into the image can be obtained with a
straightforward approach when both the stego media and
cover media are available, the security of the embedded
information is thus weak and not guaranteed.
Recently, a large number of algorithms have been
developed to encrypt images with chaotic systems. In [6], an
algorithm that can encrypt an image with three sets of
pseudorandom sequences generated with a high-dimensional
chaotic system is developed. In [7], an image encryption
algorithm that combines permutation and diffusion is
developed. In [8], an image encryption algorithm for color
images by shuffling pixels is developed. The shuffling of
pixels is performed with chaotic systems. Our previous work
[9] develops a new method that can robustly encrypt an
image with two sets of one-dimensional logistic systems. It is
thus interesting to consider whether chaotic systems can be
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2017 3rd IEEE International Conference on Computer and Communications
978-1-5090-6352-9/17/$31.00 ©2017 IEEE

2. utilized to enhance the security of an information hiding
algorithm.
In this paper, we develop a novel image based
information hiding algorithm that can efficiently and
securely embed a sequence of binary bits into a gray scale
image with a set of one-dimensional logistic mappings. In
the first stage of the algorithm, the sequence is divided into
subsequences of equal length and every bit in the sequence
can then be mapped to a pair of two integers, which
correspond to a data point in a two-dimensional space. The
whole sequence thus corresponds to a grid in the two-
dimensional space and the bits in the sequence can be
shuffled based on a set of one-dimensional logistic systems.
In the second stage of the algorithm, the shuffled sequence is
divided into regions of equal length and the regions are
sequentially embedded into the gray scale image to complete
the embedding. The embedding of a region is performed by
replacing the least significant bits of the gray value of the
corresponding pixel with the bits in the region.
The keys for recovering the embedded sequence are the
parameters of the one-dimensional logistic mappings used in
the first stage of the algorithm. The algorithm does not need
the cover image or a standard image to recover the
information. Our analysis shows that the key space of this
algorithm is large and robust against attacks based on
exhaustive search. In addition, our testing results show that
this algorithm can efficiently generate secure embedded
results and the stego images generated by this algorithm is
closer to the cover images than those generated by two other
methods for image based information hiding.
II. THE ALGORITHM
A. The Suffling of the Sequence
Wherever we use B to denote a sequence of binary bits
that need to be embedded into a gray scale image. A
sequence of real numbers ,...,,...,, 110 kk yyyy can be
defined based on a one dimensional logistic mapping R as
follows.
)1(1 kkk yyy (1)
where satisfies 40 and is the parameter of the
mapping, 0k is the integer index. The initial value 0y
of the sequence satisfies 10 0y . It is straightforward to
see that 10 ky holds for each 0k . All numbers in
the sequence are thus positive and less than 1.
A well known fact on logistic mappings is that the
sequence period becomes infinite and the initial value 0y
significantly affects the numbers in the sequence when
457.3 [4], A significant amount of difference in ky
can be created by a tiny perturbation in 0y for large enough
k . A logistic mapping thus has the property of a one-
dimensional chaotic system. A large number of image
encryption algorithms have been developed based on the
chaotic property of one-dimensional logistic mappings [10,
11].
Given a positive integer p and sequence B , we use
BL to denote the length of B and )(iB to denote the i th
bit in B . We assume BLp and 1/ pLq B . It is
not difficult to see that B can be divided into q
subsequences and each subsequence contains p bits. The
last subsequence may contain less than p bits and we can
pad enough bits of 0 to the end of B such that the last
subsequence also contains p bits. We therefore assume that
B contains q subsequences of length p in the rest of the
paper.
We thus can map )(iB to a pair of integers ),( ts such
that pis / and pit mod . s is the first coordinate
of the )(iB and t is its second coordinate. Each bit in B
can thus be uniquely mapped to a point in a two-dimensional
space. All bits that have the same value of the first
coordinate together form a row. Similarly, all bits that have
the same value of the second coordinate together form a
column. Two bits are in the same row if they have the same
first coordinate and in the same column if their second
coordinates are equal. We use integers 2,1 …, q to number
the rows formed by bits in B and p,...,2,1 for columns.
For row s in B , a logistic mapping suR is needed to
shuffle the pixels in the row. Similarly, for column t in the
image, a logistic mapping
tvR is needed to shuffle the pixels
in the column. Parameters pq vvvuuu ,...,,,,...,, 2121 are set
to be numbers between 3.6 and 4.0 and are the keys of the
information hiding algorithm.
The relocation of bits in B starts by shuffling the bits
row by row. For row s ( qs1 ) in B , a sequence of
real numbers pyyy ,...,, 21 can be computed with a given
initial value 0y from suR . From pyyy ,...,, 21 , the
algorithm computes a sequence of integers plll ,...,, 21 as
follows.
pMyl kk mod)( (2)
where pk1 , M is a large integer and p is not a
factor of M . The bits in row s are then shuffled based on
plll ,...,, 21 . Specifically, the bit whose first and second
coordinates are s and k respectively is moved to the
location where the first and second coordinates are s and kl
in B . We denote the resulting sequence by 1B .
The bits in 1B are then shuffled column by column. For
column t ( pt1 ) in 1B , a sequence of real numbers
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3. qzzz ,...,, 21 can be computed based on a given initial value
0z and
tvR . The algorithm computes a sequence of integers
qrrr ,...,, 21 from qzzz ,...,, 21 as follows.
qNyr uu mod)( (3)
where qu1 and N is a large integer and q is not a
factor of N . The bits in column t are then shuffled based
on qrrr ,...,, 21 . The bit whose first and second coordinates
are u and t respectively is relocated to the location where
the first and second coordinates are ur and t . We denote the
resulting sequence by 2B .
B. The Embedding of the Sequence
Let I be the gray scale image where B needs to be
embedded into. We assume I contains m rows and n
columns and ),( hgI is the gray value for the pixel in row
g and column h in I . Since the bits in B have been
relocated and the resulting sequence is 2B , the algorithm
sequentially embeds the bits in 2B into I in the second
stage.
The bits in 2B are embedded into I in groups. Each
group contains equal number of bits. We use w to denote
the number of bits in each group. Since the gray value of a
pixel contains 8 bits, wmust satisfy 81 w . The bits in
2B are thus sequentially divided into 1/ wpqb
subsequences, each subsequence contains w bits. We use
integers b,...,2,1 to sequentially number the subsequences.
Let bmne / , the bits in subsequence l is embedded
into the gray value of pixel ),( ll hg in I , where lg and
lh are computed as follows.
1/ nlegl (4)
nlehl mod (5)
The gray value of pixel ),( ll hg is updated to include
the w bits in subsequence l as follows.
l
w
llll ChgIhgI 2/),(),(2 (6)
where ),(2 ll hgI is the revised gray value for pixel
),( ll hg and lC is the binary value of the w bits in
subsequence l . It is clear from equation (6) that the least
significant bits are replaced by the bits in subsequence after
the embedding is complete.
C. The Recovery of the Embedded Sequence
Let 2I be the resulting stego image after B has been
embedded into I . Given the values of wqp ,, and the keys
used in the shuffling of B , the recovery of B from 2I can
be performed in two stages. In the first stage, the bits in
sequence 2B can be efficiently recovered by accessing
certain pixels in 2I and obtaining the w least significant
bits from the gray values of these pixels. Specifically, the
binary value encoded by the w bits in the subsequence l in
2B can be computed with equation (7).
w
lll hgIC 2mod),(2 (7)
where lC is the binary value encoded by the w bits in the
subsequence l in 2B ; the values of lg and lh are
computed as shown equations (4) and (5). The subsequence
l in 2B can then be obtained from lC .
We assume the initial values used for the shuffling of
rows and columns in are 0y and 0z respectively while the
keys for the shuffling of the mapped rows and columns in
B are quuu ,...,, 21 and pvvv ,...,, 21 . In the second stage
of the recovery process, the algorithm shuffles the bits in 2B
column by column. For column t ( pt1 ) in 2B , a
sequence of real numbers qzzz ,...,, 21 are computed with
the initial value 0z and tvR . The algorithm computes a
sequence of integers qrrr ,...,, 21 from qzzz ,...,, 21 as
follows.
qNzr uu mod)( (8)
where qu1 . The bits in column t are then relocated
with qrrr ,...,, 21 . Specifically, the bit whose first and
second coordinates are ur and t respectively is moved to the
location where the first and second coordinates are u and t .
We denote the resulting sequence by 1B .
Finally, The bits in 1B are shuffled row by row. For row
s ( qs1 ) in 1B , a sequence of real
numbers pyyy ,...,, 21 are computed with the initial value
0y and suR . A sequence of integers plll ,...,, 21 can be
computed from pyyy ,...,, 21 as follows.
pMxl kk mod)( (9)
where pk1 . The bits in row s are then relocated with
plll ,...,, 21 . The bit whose first and second coordinates are
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4. s and kl respectively is moved to the location where the
first and second coordinates are s and k . The resulting
sequence is the original sequence B . It is straightforward to
see that the recovery of the embedded information does not
require the availability of the cover image or a standard
image.
III. EXPERIMENTAL RESULTS AND DISCUSSIONS
We have implemented this information hiding algorithm
into a computer program in MATLAB and tested its
performance on embedding information into four benchmark
gray-scale images. One of the images is the well known
benchmark image Lena, the other three images are
downloaded from the Berkeley Segmentation Data Set and
Benchmarks 500 (BSD500), which can be accessed at site
https://www2.eecs.berkeley.edu/Research/Projects/CS/vision
/grouping/resources.html. To compare the overall
performance of this algorithm with existing methods, we use
our algorithm, two other existing methods to embed
randomly generated binary sequences into the benchmark
images and compare the performance of three methods. The
keys for the shuffling of the embedded sequences are real
numbers randomly generated between 3.7 and 4.0. The
values of 00 , zy are set to be 0.3 and NM , are both set to
be
7
10 . All images are scaled to the same size, which is
512×512. Figure 1 (a) (b) (c) and (d) show the four tested
images.
(a) (b) (c) (d)
Figure 1. (a) image Lena; (b) image 3063 from BSD500; (c) image 5096 from BSD500; (d) image 8068 from BSD500.
We randomly generate 100 sequences of binary bits for
embedding tests. Each sequence contains 100 randomly
generated binary bits. qp, are both set to be 10. The effects
of won stego images are also evaluated by embedding the
same sequence into the same benchmark image with
different values of w. The quality of a stego image can be
evaluated based on Peak Signal of Noise Ratio (PSNR).
Table I, II and III compare the means and standard
deviations of the PSNRs of the stego images obtained with
our algorithm and the method developed in [12] on the 100
testing sequences when the value of w is 1, 3 and 5
respectively.
It is clear from Tables I, II and III that our approach can
achieve higher mean values of PSNRs on all benchmark
images, which suggests that our approach outperforms the
method developed in [12]. In addition, the standard
deviations of PSNRs confirm that the performance of our
approach is reliable and robust. To compare the performance
of our approach with the method developed in [13], we
embed the testing sequences into the four benchmark images
with the method developed in [13] and compute the values of
PSNRs and their standard deviations for the stego images
obtained with the method. The results are shown in Tables
IV, V, and VI.
It can be seen from Tables IV, V, and VI that the mean
values of PSNRs obtained with the method in [13] are lower
than those obtained with our approach on all benchmark
images. This shows that our approach also outperforms the
method developed in [13].
TABLE I. A COMPARISON OF OUR ALGORITHM WITH THE METHOD
DEVELOPED IN [12] WHEN wIS 1.
Stego
Image
Our Approach The Method in [12]
PSNR STD PSNR STD
Lena 85.42 0.40 80.33 0.37
3063 85.27 0.42 81.42 0.39
5096 85.32 0.41 79.61 0.35
8068 85.46 0.43 80.58 0.34
TABLE II. A COMPARISON OF OUR ALGORITHM WITH THE METHOD
DEVELOPED IN [12] WHEN wIS 3.
Stego
Image
Our Approach The Method in [12]
PSNR STD PSNR STD
Lena 84.58 0.41 79.52 0.40
3063 84.44 0.40 80.60 0.38
5096 84.47 0.42 78.79 0.36
8068 84.63 0.41 79.74 0.33
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5. TABLE III. A COMPARISON OF OUR ALGORITHM WITH THE METHOD
DEVELOPED IN [12] WHEN wIS 5.
Stego
Image
Our Approach The Method in [12]
PSNR STD PSNR STD
Lena 83.97 0.42 78.92 0.38
3063 83.81 0.43 80.03 0.39
5096 83.89 0.41 78.13 0.35
8068 84.03 0.40 79.11 0.36
TABLE IV. A COMPARISON OF OUR ALGORITHM WITH THE METHOD
DEVELOPED IN [13] WHEN wIS 1.
Stego
Image
Our Approach The Method in [13]
PSNR STD PSNR STD
Lena 85.42 0.40 81.69 0.43
3063 85.27 0.42 80.31 0.45
5096 85.32 0.41 81.53 0.46
8068 85.46 0.43 80.62 0.39
TABLE V. A COMPARISON OF OUR ALGORITHM WITH THE METHOD
DEVELOPED IN [13] WHEN wIS 3.
Stego
Image
Our Approach The Method in [13]
PSNR STD PSNR STD
Lena 84.58 0.41 80.83 0.41
3063 84.44 0.40 79.48 0.44
5096 84.47 0.42 80.70 0.38
8068 84.63 0.41 79.75 0.42
TABLE VI. A COMPARISON OF OUR ALGORITHM WITH THE METHOD
DEVELOPED IN [13] WHEN wIS 5.
Stego
Image
Our Approach The Method in [13]
PSNR STD PSNR STD
Lena 83.97 0.42 80.10 0.42
3063 83.81 0.43 78.92 0.41
5096 83.89 0.41 80.04 0.45
8068 84.03 0.40 79.03 0.43
It is also clear that the algorithm needs qp keys in
total to embed a sequence that contains up to pq bits. The
key space is thus of size
qp
c , where c is the number of
real numbers that can be selected by a computer from the
interval between 3.6 and 4.0. It is straightforward to see that
10
2c and the size of the key space is thus at least
)10
2 qp
.
For a sequence that contains more than 100 binary bits, this
number is at least
200
2 , which suggests that our approach is
secure against attacks based on exhaustive search when the
embedded sequence is of moderate length.
IV. CONCLUSIONS
In this paper, we develop a new algorithm for image
based information hiding. The algorithm embeds a sequence
of binary bits into a given gray scale image in two stages. In
the first stage, the binary bits in the sequence are shuffled
based on a set of one-dimensional logistic mappings. In the
second stage, the shuffled sequence is divided into
subsequences of equal lengths and the subsequences are
sequentially embedded into the gray values of the
corresponding pixels in the given gray scale image. Our
analysis and testing results show that this approach does not
need the cover image or a standard image to recover the
embedded information and outperforms two other methods
for image based information hiding. In addition, our analysis
shows that the embedded information is secure against
attacks based on exhaustive search. It can be seen clearly
from the description of the algorithm that its time complexity
is linear, which suggests that the algorithm is
computationally efficient and is thus potentially useful for
image based information hiding in internet applications.
ACKNOWLEDGMENT
Y. Song’s work is fully supported by The Fund of
Specially Appointed Professors in Jiangsu Province with the
grant number: 1034901501.
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