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A novel approach to secure image based
steganography by using Eigenvalue and Eigenvector
principles
S. Abbas Hosseini-pourShahid Bahonar University
Kerman, Iranmember of young researcher
society,s.abbas.hosseinipour@gmail.com
Mohadeseh SoleimanpourShahid Bahonar University
Kerman, Iranmember of young researcher
society,mohadese.soleimanpour@yahoo.c
om
Hossein Nezamabadi-pourShahid Bahonar University
Kerman, Irannezam@mail.uk.ac.ir
Abstract In this paper, a novel steganography technique by
using the concepts of eigenvalues and eigenvectors is presented.
In our proposed method, first a symmetric transformation
matrix is calculated based on the information of the cover image
and a combination of the cover image and the secret image.
Afterward, the eigenvalues and eigenvectors of thetransformation matrix are extracted and sent as a secret key. In
this approach, in order to improve the performance of proposed
method a special symmetric matrix is proposed. In receiver, the
secret image is extracted from cover image by using secret key.
The main privilege of the proposed method is high capacity and
security. In the proposed method as another advantage the cover
image is not changed. The experimental results demonstrate the
superiority of proposed method in terms of data capacity and
image quality.
I. INTRODUCTIONThe word of Steganography is made from the Greek words
steganos meaning "covered or protected" and graphei
meaning "writing" [1]. Steganography describes the art and
science of communicating in a way that the presence of a
secret message apart from the identity of sender and intended
recipient could not be detected. Hiding information have been
in use for hundreds of years, however, nowadays by
increasing the use of file transfers in the electronic format,
application of the steganography and watermarking methods
are growing to hide important (secret) information
undetectably and/or irremovably in audios, videos and images
[2, 3].
In this paper, the image that is used for inserting and
hiding secure data is called cover image, and the final image
after hiding secret bits into the cover image is known as
stego image. Steganography methods depend on in which
domain the insertion is performed, could be divided into two
main groups, namely transform domain and spatial domain.
In the transform domain, Discrete Cosine Transform
(DCT), Discrete Fourier Transform (DFT) and Discrete
Wavelet Transform (DWT) are the most common transforms
that are used for data hiding [4]. The transform domain
steganography methods hide messages in more significantareas of the cover images. In order to reach this aim, the
cover image is split into high, middle and low frequency
component. Since most of the signals energy is concentrated
in the lower frequencies (they are very important in visibility)
therefore secret data is embedded in the higher frequencies
(often middle frequency components) in order to avoid image
distortion. Independency of the image format, and as a result
high resistance against compression, is the most important
privilege of transform domain methods [5]. However, high
computational complexity is the main disadvantage of these
methods.
The second group belongs to spatial domain methods. In
these methods, messages are directly embedded in theintensity value of the pixels. The first attempt in spatial
domain is LSB substitution. LSB substitution steganography
is a simple technique that hides message bits in LSB of image
pixels. Against simplicity of this method, the major drawback
of this method is that the secret message could be detected
very easily [6].
Popular steganography tools based on LSB embedding
vary in the existing approaches for hiding information. Some
algorithms change LSB of pixels visited in a random walk
[7], others modify pixels in certain areas of images [8], or
instead of just changing the last bit they increment or
decrement the pixel value [9].
In the proposed method, the principle of eigenvalues and
eigenvectors of a matrix is used for hiding data in cover
image. In order to reach this aim, a special transformation
matrix is proposed which causes real and limited parameters
for eigenvalues and eigenvectors. Afterward, two secret keys
978-1-4673-5634-3/13/$31.00 2013 IEEE
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are evaluated and they are sent separately. The main
privileges of the proposed method are high security and high
capacity.
The rest of this paper is organized as follows. Section 2
details the proposed method. Section 3 presents the
experimental results and a comparison with other methods.
Finally, a conclusion is given in Section 4.
.
II. THE PROPOSED METHODIn the proposed method, the secret data is sent by using the
eigenvalues and eigenvectors of a transformation matrix. To
reach this aim, at first, the concepts of eigenvalues and
eigenvectors are described. Afterward, the method of creating
converter and what will be send is presented. Fig.1 shows the
block diagram of the proposed method.
A. The concepts of Eigenvalue and eigenvectorIn general, for any matrix
n nA there are non-equal vectors
ix that satisfies Equation (1). In this equation the constantcoefficients
i and vectors ix are called eigenvalues and
eigenvectors of matrixA, respectively.
(1)Ax x=
Equation (2) is derived from Equation (1), where I
represent the identity matrix.
(2)( ) 0i iI A x =
Therefore, the eigenvaluesi are the roots of Monique
Polynomial of degree nthat yields from Equation (3).
(3)
11 11 1
21 11 11
11 11 11
1
1 1
....
....
....
n
n n
n n
a a a
a a aI A
a a a
c c c
=
= + + + +
Now, two features of eigenvalues and eigenvectors are
described that have been used in the proposed method.
a.If matrix has symmetry hermitian (hermitian is equalto its own conjugate transpose), then eigenvalues are
real numbers and the corresponding eigenvectors
will be orthogonal. This principle is demonstrated by
Equation (4).
(4)
*
i i
H
i ji j
A Ax x
==
b.If A is real and symmetric and simultaneously theresults in the previous state eigenvectors are also
real as illustrated in Equation (5).
(5)
**
*
i i
T
i i
A A
A A x x
==
= =
It should be noted that matrix A is diagonal and the
eigenvalues are sorted in descending order on main diameter
from left up to right down.
(6)
min
max
0 0 0
0 0 0
0 0 0
0 0 0
B. The transformation matrix and secret keysAs mentioned before, Steganography is the art and science
of hiding information such that no one, apart from the sender
and receiver, suspects the existence of the message. In this
paper, for convenient the cover image, secret image and thecombined image are noted by letters CI, SI and MI,
respectively.
In the proposed method the cover image and the secret
image should be grayscale and in the same size. At first, a
proper cover image is selected; afterward combined image is
made by averaging the cover image and the secret image.
(7)( )1
2I CI SI= +
Fig1. The block diagram of proposed method
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It should be noted that is considered as a normalization
factor.
In the next step, transformation matrix A which convert
cover image into combined image is calculated as follow:
(8)1 1m mm mI A CI =
Where MI
and CI
are made by scanning theMIand CI
matrix row by row.
Afterward, the eigenvalues and eigenvectors of matrix A
must be calculated respectively. It should be mentioned that
to avoid of generating the complex numbers in eigenvalues
and eigenvectors, matrixAshould be evaluated by respecting
to the two considerations which is mentioned in previous
section.
In other words, it should be selected real and symmetric.
Thus, the matrixAis considered as follows.
(9)
1 1 2 1
2 3 4 2
3 3
1
1
4 3
2 1
0 0 0 0 00 0 0 0 0
0 0 0 0 0 0
. 0 0 0 0 0 0 .
. 0 0 .
. 0 0 0 0 0 0 .
0 0 0 0
0 0 0 0 0
m m
m m
m m
I a a CIMI a a CI
I CI
a a
a a
a a
I a a CI
=
In order to decrease the complexity of calculating elements
of transformation matrix A, in each rows just the two
variables is considered and the rest elements of the row is setzero. Symmetrical manner of this row to the center of matrix
is repeated in other row.
For simplify the explanation the structure of matrix A is
shown in Equation (9).
Afterward, by solving each group of equations that yields
from each both symmetrical rows, unknowns in
transformation matrix obtained and desired converter to be
achieved. To clarify the above explanation about the
procedure of how to calculate the variables, an example is
illustrated in Equation (10).
(10)1 1 2 1
2 1 1
m
m m
aCI a CI MI
a CI aCI MI
+ =
+ =
If the arrays of transformation matrix to be integer then
numbers that appears in eigenvectors belong to one of the
following categories.
(11){ }
{ }
0.7071 0.7071 1 0 2 1
0.7071 0.7071 0 2
m k
m k
= +
=
Thus, constant 100 multiplied in transformation matrix
then floating term of arrays is removed. The error that derived
it is very small and negligible.
(12)' (100 )A round A=
Now, the converter is ready to extract the eigenvalues and
eigenvectors by Equations (3) and (1). After calculating the
eigenvalues and eigenvectors we send cover image,
eigenvectors and eigenvalues separately that way of send will
express in the next section. After sending data, at destination
by received cover image and reproduction matrix A with
eigenvalues and eigenvectors, the secret data will be extracted
from cover image by following Equations.
(13)' '2 ( )
2
CISI MI =
(14)' 'I A CI=
(15)' 10.01 ( )A x x =
C. Sending informationIf one of three factors ,x, CIwere not received, secret
data from cover image cannot be extracted. Afterward, each
factor is sent independently, which causes high security in
data sending. In the proposed method, vectors in Equation
(16), (17) which are made from eigenvalues and eigenvectors
will be sent as keys separately. The first vector contains the
eigenvalues that are the elements on the main diagonal of
(16)[ ]min 2 1 maxm
The second is the vector containing the eigenvectors of
transformation matrix A.To create the second vector, matrix
x which contains eigenvectors is swept from upper left to
lower right. If the dimension of A is even. We encounter
along with numbers 0, 0.7071 and -0.071. In order to
decrease the number of required bits for sending data, we
send 1, -1 instead of 0.701 and -0.701 respectively. For
sending zeros, zeros between two non-zero along math are
counted that matrix swept and number of zeros be send.
If there is just one zero, afterward zero will be sent. For
example, the bellow vector is created in Equation (17) for one
matrixxwhile its dimension is considered 4*4.
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(17)[
]
1 2 1 0 1 1
2 1 1 0 1 2 1
As mentioned above, in receiver with the received cover
image and transformation matrix Aby using Equations (13),(14), and (15) secret image is extracted.
In the proposed method, the cover image is not changed,therefore the third part is not suspected on existence of secret
data.
III. EXPERIMENTAL RESULTSThe proposed method is executed on four 256*256 images.
Fig.2 shows examples of the data hiding and the dataextracting process. Then three Images Bridge, boat andpentagon embedded in three cover image Lena, Pepper andBaboon. In table 1, the proposed method is compared with twohiding data methods. The PSNR and capacity are term in db
and bits, respectively. It can be seen that the proposed methodhas higher capacity than the other two methods, and the PSNRof the proposed method due to the cover image did not changeis very high.
Fig 1: the experimental result on four images
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TABLE I: A comparison of proposed method and 2-LSB and 3-LSB on capacity and PSNR
CoverImage
Secretimage
2-bit LSB
Capacity PSNR stegoImage
3-bit LSB
Capacity PSNR stegoImage
Proposed method
Capacity PSNR stego PSNR ExtractedImage Image
Baboon
Lena
Peppers
BridgeBoat
PentagonBridgeBoat
PentagonBridge
BoatPentagon
524,288 44.37524,288 44.52
524,288 44.51524,288 44.37524,288 44.51524,288 44.52524,288 44.39
524,288 44.53524,288 44.54
786,432 37.16786,432 37.17
786,432 37.10786,432 37.03786,432 37.05786,432 36.97786,432 37.05
786,432 37.09786,432 37.02
2,097,152 Inf 48.382,097,152 Inf 48.61
2,097,152 Inf 48.482,097,152 Inf 47.922,097,152 Inf 47.862,097,152 Inf 48.012,097,152 Inf 48.53
2,097,152 Inf 48.632,097,152 Inf 48.51
IV. CONCLUSIONIn this paper, a new method based on the fundamental
concepts of matrix eigenvalues and eigenvectors is presented.This method introduced a symmetric transformation matrix
by the properties of the matrix. The key is provided by usingits and send to the receiver separately. In destination, secret
data is extracted by having the cover image and the secret
key. It should be mentioned that if one of them is not
received, the secret data cannot be extracted.
This method has high security because the cover image
was not changed through the hiding process and the secret
key and cover image send separately. Also, this method has
higher data capacity than previous works because information
is not embedded in cover image. In other words, information
is inserted on the secret key.
The main drawback of proposed method is change of
extracted image in comparison with secret image. It should be
mentioned that it is negligible against capacity and robustnessof proposed method.
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