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An image hiding method based on cascaded iterative Fourier
transform and public-key encryption algorithm
Bing Zhanga, Jun Sang
*a, Mohammad. S. Alam
b
aSchool of Software Engineering, Chongqing University, Chongqing, China, 401331; bDepartment of Electrical and Computer Engineering, University of South Alabama,
Mobile, AL, USA, 36688-0002
Abstract
An image hiding method based on cascaded iterative Fourier transform and public-key encryption
algorithm was proposed. Firstly, the original secret image was encrypted into two phase-only masks
1M and 2M via cascaded iterative Fourier transform (CIFT) algorithm. Then, the public-key
encryption algorithm RSA was adopted to encrypt 2M into 2'M . Finally, a host image was
enlarged by extending one pixel into 2 2 pixels and each element in 1M and 2'M was
multiplied with a superimposition coefficient and added to or subtracted from two different elements in
the 2 2 pixels of the enlarged host image. To recover the secret image from the stego-image, the
two masks were extracted from the stego-image without the original host image. By applying
public-key encryption algorithm, the key distribution was facilitated, and also compared with the image
hiding method based on optical interference, the proposed method may reach higher robustness by
employing the characteristics of the CIFT algorithm. Computer simulations show that this method has
good robustness against image processing.
Keywords: image hiding, cascaded iterative Fourier transform (CIFT), public key, RSA
1. Introduction
Since the double random phase encoding technique was proposed by Refregier and Javidi in 1995
[1], the optical technique based image encryption [2, 3] and information hiding [4, 5] has been widely
used. Due to the rapid development of the Internet, a large amount of data needs to be transferred and
for data security the information hiding method has a very high value these years [6, 7].
In this paper, we proposed an image hiding method based on cascaded iterative Fourier transform
and public-key encryption algorithm. Cascaded iterative Fourier transform was used to convert one
secret image into two phase-only masks, while the RSA algorithm was used to encrypt one of the
masks. Computer simulations show that the method has a very good robustness and performance.
This paper is organized as follows: in Section 2, the proposed image hiding method is briefly
introduced and analyzed. Section 3 performs the numerical simulation experiment. In Section 4, the
final conclusions are presented.
*corresponding author: [email protected]
Optical Pattern Recognition XXIV, edited by David Casasent, Tien-Hsin Chao, Proc. of SPIE Vol. 8748, 87480H · © 2013 SPIE · CCC code: 0277-786X/13/$18 · doi: 10.1117/12.2018257
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2. The proposed method
2.1 Transform the secret image with the cascaded iterative Fourier transform
The cascaded iterative Fourier transform (CIFT) algorithm [8, 9] is an algorithm based on DRPE
technology. Suppose ( , )f x y is the secret image that to be encrypted. For CIFT, two uniformly
distributed random arrays 0
1P and 0
2P (valued between and ) are selected. Then, the
following iteration is applied.
1
1 2( , ) { {exp[ ( , )]}exp[ ( , )]}.k k kx y F F jP x y jP u v (1)
1
3 1 2( , )exp[ ( , )] { {exp[ ( , )]}exp[ ( , )]}.k k k kf x y jP x y F F jP x y jP u v (2)
( , ), ( , ) 0( , ) .
( , ), ( , ) 0
k
k
f x y if f x yg x y
f x y if f x y
(3)
1 32
1
{ ( , )exp[ ( , )]}( , ) ( ).
{exp[ ( , )]}
k kk
k
F g x y jP x yP u v angle
F jP x y
(4)
1 1 1
1 3 2( , ) [ { { ( , )exp[ ( , )]}exp[ ( , )]}].k k k kP x y angle F F g x y jP x y jP u v (5)
where 0,1,2,...k ; F and 1F are represent the Fourier transform and the inverse Fourier
transform; ( , )x y and ( , )u v are represent the space coordinate and frequency coordinate;
3 ( , )kP x y is the thk iterative phase on the output plane, i.e., the phase of ( , )k x y ; ( , )kf x y is
an approximate amplitude image reconstructed from the thk iteration.
Because of high convergence speed of CIFT algorithm, ( , )kf x y will convergent to ( , )f x y
after about twenty iterations.
2.2 Encrypt one of the obtained masks with RSA
The iteration process yields two phrase-only masks 1( , )P x y and 2 ( , )P u v . One mask
1( , )P x y is chosen and encrypted with the public-key encryption algorithm RSA. After the encryption,
1( , )P x y becomes 1 '( , )P x y .
2.3 Embed the secret image into the host image
In this step, the host image ( , )H x y is enlarged by extending one pixel into 2 2 pixels, and
( , )H x y becomes '( , )H x y . Then the two masks are embedded as follows.
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'
fÔf
.dir111
I I IM ir 16 tr\ ...
w,
1
1
2
2
'(2 1,2 1) '(2 1,2 1) '( , )
'(2 ,2 ) '(2 ,2 ) '( , )
'(2 ,2 1) '(2 ,2 1) ( , )
'(2 1,2 ) '(2 1,2 ) ( , )
H i j H i j t P i j
H i j H i j t P i j
H i j H i j t P i j
H i j H i j t P i j
(6)
Where t is the coefficient to embed the two masks. After embedding, the stego-image S( , )x y
is obtained.
3. Numerical simulations
Numerical simulations were conducted to verify the proposed method. The 256 256 image
Lena is chosen to be the secret image, which is shown in Fig.1.
Fig.1 the secret image
After encrypting the secret image with the CIFT algorithm described in Section 2.1, two
phase-only masks are obtained as Fig.2 shows.
Fig.2 the two phase-only masks
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1'i`;4k
_,.
-c
Next, we choose the key of the RSA algorithm as:
public key: 2773, 847n d
private key: 2773, 63n e
After encrypting the mask 1( , )P x y with the RSA algorithm, 1( , )P x y becomes 1 '( , )P x y
shown in Fig.3.
Fig.3 the phase-only mask after encrypting with the RSA algorithm
Then, the two masks 1 '( , )P x y and 2 ( , )P u v are embedded into the host image (Fig.4) and the
stego-image shown in Fig.5 is obtained.
Fig.4 the host image
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Fig.5 the stego-image
With the correct keys of the RSA algorithm, the secret image can be recovered, which is shown in
Fig.6.
Fig.6 the recovered secret image
By adding salt-pepper noise with density 0.02 to the stego-image, the stego-image is shown in
Fig.7(a). Due to the salt-pepper noise, the secret image recovered from the stego-image will be
somewhat different from the original secret image. However, it can still be recognized visually as
Fig.7(b) shows.
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,;
' - ,-;L ' '
(a) (b)
Fig.7 adding salt-pepper noise with density 0.02 to the stego-image (a) the stego-image (b) the
recovered secret image
By cropping 1/16 of the top left corner of the stego-image, the stego-image is shown in Fig.8(a).
Due to the cropping operation, the secret image recovered from the stego-image will be somewhat
different from the original secret image. However, it can still be recognized visually as Fig.8(b) shows.
(a) (b)
Fig.8 cropping 1/16 of the top left corner of the stego-image (a) the stego-image (b) the recovered
secret image
4. Conclusions
An image hiding method based on cascaded iterative Fourier transform and public-key encryption
algorithm is proposed in this paper. With the CIFT algorithm, two random phase masks are obtained
from the secret image. Then, one of the masks is encrypted by the RSA algorithm. At last, these two
masks are embedded into a host image. Because of the RSA algorithm, the key distribution was
facilitated. The simulation experimental results demonstrate that the proposed method has a very good
robustness.
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Acknowledgments
This work was supported by National Natural Science Foundation of China (No. 60972105). THE
AUTHOR GRATEFULLY ACKNOWLEDGES THE SUPPORT OF K.C.WONG EDUCATION
FOUNDATION, HONG KONG.
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