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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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