20120140506008 2

International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –
6480(Print), ISSN 0976 – 6499(Online) Volume 5, Issue 6, June (2014), pp. 45-52 © IAEME
45
EFFICIENT WAY OF IMAGE ENCRYPTION USING GENERALIZED
WEIGHTED FRACTIONAL FOURIER TRANSFORM WITH DOUBLE
RANDOM PHASE ENCODING
Priyanka Chauhan1
, Girish Chandra Thakur2
M. Tech. Scholar1
, Associate Professor2
,
Dept. of Electronics & Communication Engg., Institute of Engineering & Technology,
Alwar-301030 (Raj.), India
ABSTRACT
Data transfer between the users over different network technologies is increasing day by day.
With the data transfer, concern about the information theft by any eavesdropper is also increasing.
To prevent this information theft, our information is converted in to unreadable form using some
algorithm and it can only be converted into readable form using key. This process of conversion the
information into secured form is called encryption. At present information transfer is also in the form
of images. In this paper we are going to discuss some of the methods of encrypting the images. In
any encryption technique, the major role is played by its key i.e. larger the number of parameters in
the key stronger is the encryption technique. So in the thesis we will develop the GWFRFT which is
based on the properties of conventional FRFT and discuss random phase encoding technique in
FRFT domain. Then study random phase encoding technique in GWFRFT domain. And at last to
make the scheme more efficient and strong we encrypt two images in to single image.
Keywords: Encryption, FRFT, FT, GWFRFT, Image.
1. INTRODUCTION
In cryptography, encryption is the process of transforming information (referred to as
plaintext) using an algorithm (called a cipher) to make it unreadable to anyone except those
possessing extra ordinary information, more often than not referred to while a key. This product of
the process is encrypted in order in cryptography, referred to cipher text. The invalidate process, i.e.,
to make the encrypted information readable again, is referred to as decryption (i.e. to make it
unencrypted). In recent years there have been numerous reports of confidential data such as
INTERNATIONAL JOURNAL OF ADVANCED RESEARCH
IN ENGINEERING AND TECHNOLOGY (IJARET)
ISSN 0976 - 6480 (Print)
ISSN 0976 - 6499 (Online)
Volume 5, Issue 6, June (2014), pp. 45-52
© IAEME: http://www.iaeme.com/IJARET.asp
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46
customers’ personal records being exposed through loss or theft of laptops or endorsement drives.
Encrypting like files at rest help guard them should physical security events fail. Digital liberties
organization systems which prevent unauthorized use or reproduction of copyrighted material and
protect software against reverse engineering (see also copy protection) are another somewhat
different example of using encryption on data at rest. There have been numerous reports of data in
transit being intercepted in current years. Encrypting information in shipment also helps to secure it
as it is often difficult to physically secure all access to networks. In this paper we are using
Generalized Weighted Fractional Fourier Transform with random phase encoding technique for the
encryption of two images in to single image. GWFRFT will be derived from the FRFT. The
fractional Fourier transform (FRFT) is a superset of the Fourier transform and has been applied in
optics, quantum mechanics, and signal processing areas [15,20,21].In this paper, we discuss a
generalized weighted fractional Fourier transform (GWFRFT) which is weighted sum of the integer
order fractional Fourier transform.
2. DEFINITION OF FRFT AND GWFRFT
FRFT is a generalization of FT [1, 13]. It is not only richer in theory and more flexible in
application, but is also not expensive in implementation. It is a powerful tool for the analysis of time-
varying signals. With the advent of FRFT and related concepts, it is seen that the properties and
applications of the conventional FT are special cases of those of the FRFT. However, in every area
where FT and frequency domain concepts are used, there exists the potential for generalization and
implementation by using FRFT. Mathematically, α th
order FRFT is the α th
power of FT operator.
Hence α th
order FRFT of any signal 2
( ) ( )s t L R∈ is given by
[ ( )] ( ) ( ; , )F s t s t K t dtα
α ω
+∞
−∞
= ∫ (1)
Where
2 2
( ; , ) ( )exp[( ( ) ( ) ( ) )]K t K B C t B tα ω α α ω α ω α= − + (2)
Is the kernel of Fα
, α is the fraction, and
[1 cot( )]
( )
2
j
K
α
α
π
−
= (3)
( ) cot( )B α α= (4)
1
( )
sin( )
C α
α
= (5)
2
aπ
α = (6)

47
For integer values of α it can be deduced that any fractional Fourier transform repeat itself at
an interval of 4 that can be shown as
4
[ ( )] [ ( )]m l l
F s t F s t+
= (7)
Where m is any integer and l can be any real number.
As we discussed the periodic nature of integral order fractional Fourier transform in the
previous chapter i.e. integer order fractional Fourier transform is periodic with period 4. So now it is
given that α th
order FRFT of any signal can be represented in terms of these four integral order
FRFT i.e.
0 1 2 3
0 1 2 3[ ( )] ( ) [ ( )] ( ) [ ( )] ( ) [ ( )] ( ) [ ( )]wF s t A F s t A F s t A F s t A F s tα
α α α α= + + + (8)
Where the coefficients ( )lA α , l = 0,1,2,3 are continuous functions ofα .
Generalized weighted fractional Fourier transform obeys the following axioms.
I. Continuity axiom: 2 2
: ( ) ( )wF L R L Rα
→ is continuous in fractional order α .
II. Boundary axiom: wFα
degenerates into Fα
when α is an integer multiple of
2
π
or α is an
integer. i.e. for integer order the weighted fraction Fourier transform reduces into the normal
Fourier transform of signal s(t ) or the signal itself. For different integer values of order a, the
values of weighted coefficient is shown in the Table 1.
III. Additive axiom: For real numbers α and β , the GWFRFT operator wFα
has the following
additive property
w w w w wF F F F Fα β β α α β+
= = (9)
So using these properties and calculating the value of coefficient ( )lA α the final
equation of generalized weighted fractional Fourier transform comes out to be
3 3
( , )
0 0
[ ( )] exp{( 2 / 4)[(4 1) (4 ) ]} [ ( )]l
w M N k k
l k
F s t j m n k kl F s tα
π α
= =
= − + + −∑∑ (10)
A(or α) A0 A1 A2 A3
0 (or 0) 1 0 0 0
1 (or π/2) 0 1 0 0
2 (or π) 0 0 1 0
3 (or 3π/2) 0 0 0 1

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3. PROPOSED IDEA OF DOUBLE IMAGE ENCRYPTION BASED ON RANDOM PHASE
ENCODING USING GWFRFT
Most of the encryption techniques in the literature are dealing with a single amplitude image
(i.e. binary text, grayscale or color image). However, in somecases, one need to encrypt two images
with certain relations such as two pictures of an object or a picture and its binary text for description,
etc., which are often stored or transmitted together. Encrypting the two images directly using the
single-image encryption algorithms mentioned in the literature results in two separate encrypted
images with random noise distribution, and sometimes, there arises the difficulty in distinguishing
between them before decryption and the decryption has to be conducted twice to recover the two
images. Multiple images can also be encrypted by using wavelength multiplexing [18] or position
multiplexing methods [21], but the qualities of decrypted images are not perfect due to the cross-talk
effects between images. Inspired by the architecture of the fully phase encoding [2, 17] and pixel
scrambling (random shifting) techniques [18, 20]. The schematic of a method which can convert two
images into one encrypted image based on the random phase encoding in the generalized weighted
fractional Fourier domain is given in Fig 1.
Let f (x0) and g(x0) represent the two primary normalized amplitude images to be encrypted
together, J denotes the pixel scrambling operation [20], which interchanges the pixels according to
some random permutation, and the image can be retrieved by repositioning the disturbed image
according to the special order. For encryption, the pixel scrambling is applied to one of the primary
images g(x0), and then, the scrambled result J [g(x0)] is encoded into a phase only function, which
can be mathematically expressed as exp[jπ J [g(x0)]],within the range [0,π ] of the phase variation.
Fig 1: Schematic of double image encryption
The resulting function is multiplied by the other primary image f (x0) to obtain the complex
combination signal
0 0 0( ) ( )exp[ [ ( )]]C x f x j J g xπ= (11)
Let us consider R1=exp[jφ1(x0)] and R2=exp[jφ2(xa)] are the two random phase masks,where
φ1(x0) and φ2(xa) are statistically independent white sequences uniformly distributed in [ 0,2π ] . As
shown in Fig 1, C( x0 ) is first multiplied by the first random phase mask R1 , then transformed by
GWFRFT with order α with two 4-D vectors (M1 , N1) , multiplied by the second random phase mask
R2 , and then transformed by GWFRFT with order β with two 4-D vectors (M2 , N2). Thus, we obtain
the final expression of the output encrypted image ψ(xb) , which is given by
2 2 1 1( , ) ( , ) 0 1 0 2( ) {[ ( ( ) [ ( ( )])] [ ( ( )]}b w M N w M N ax F F C x exp j x exp j xβ α
ψ φ φ= ⋅ ⋅ ⊗ ⊗ (12)
Where ⊗ represent the element-by-element multiplication operation of matrices and the
input image for encryption ``f ” is assumed to be real and nonnegative.

49
The Decryption shown in Fig 2 is the reverse process of the encryption. For decryption, ψ(xb)
is transformed by the -βth GWFRFT and the resultant function is multiplied by the conjugate of the
second random phase mask R2 , then GWFRFT of order -αth is then applied, resulting in the
following decrypted combination function C(x0) after multiplying by the conjugate of the first
random phase mask R1 i.e.
1 1 2 20 ( , ) ( , ) 1 0 2( ) { [( ( )) [ ( ( )]]} [ ( ( )]w M N w M N b aC x F F x exp j x exp j xα β
ψ φ φ− −
= ⋅ ⋅ ⊗ − ⊗ − (13)
Now the amplitude-based image f (x0)can be simply retrieved from the decoded C(x0)by
taking amplitude, whereas the phase-based image g(x0) requires extracting the phase of C(x0), and
dividing it by π, then taking the inverse pixel scrambling transform denoted by J-1
as shown in Fig 2.
Fig 2: Schematic of double image decryption
3. SIMULATION RESULTS
Computer simulations are performed to verify the proposed encryption technique for double
images. Two types of amplitude image, image of college main gate and college logo (Fig 3), are
serving as the two primary images to be encrypted together. Each of them has a size of 256 × 256
pixels. We use college main gate image as the amplitude-based image f(x0), and college logo image
as the phase-based image g(x0). The two random phase masks, order and 4D vector are as follow(a,
b, c, d) is (0.4,3.8,1.7,2.9) and 4D vector are M1=[ 1309 ], M2=[ 102338 ], M3=[12309],
M4=[42338], N1=[8257], N2=[79464], N 3 =[ 925111 ] , N 4 =[ 79164 ] .

50
Then the encrypted and decrypted images are shown in fig 4 and 5 respectively.
Fig 3: Primary Images which are used for encryption
Fig 4: Single encrypted image using double random phase encoding using GWFRFT
Fig 5: Decrypted Images after applying the reverse process

51
4. COMPARISION WITH THE EXISTING TECHNOLOGY
Advanced encryption technique is the symmetric key algorithm similar to double image
encryption where we are using the same key for encryption as well as decryption. AES is introduced
in year 2001 and since then it is the standard technique for encryption. AES comes for different key
sizes like 128 bit, 192 bits or 256 bits.
Number of permutations for AES-128 = 2128
Number of permutations for AES-192 = 2192
Now let us calculate the number of permutation required in double image encryption.
Let M1 = [0-63, 0-63, 0-63, 0-63] = [26
, 26
, 26
, 26
]
Similar to M1 other vectors i.e. M2, M3, M4, N1, N2, N3, N4 also can take any value between 0-63.
Number of possible permutations from vectors = (26
)32
= 2192
Now order “a” can take any value between 0-100 with an interval of 0.1 i.e. 0.1 0.2, 0.3, 0.4 and so
on.
So Number of possible permutation for order “a” = 1000 ≈ 210
Number of possible permutation for order “(a, b, c,d )” ≈ 240
So, Total number of possible permutation = 2192
* 240
= 2232
As from this the key size is 232 bits and for this key size AES also require 2232
permutations
so the performance of this encryption technique is comparable to the AES.
5. CONCLUSION
In this paper, a novel approach to implement the image encryption is discussed. As in any
encryption the key which is used for encryption serves an important role means larger the number of
parameters in the key stronger the encryption techniques. So in GWFRFT, order as well as 4D
vectors served as the key making encryption stronger than the encoding in FRFT. To make this
encryption technique more efficient and stronger an encryption technique is proposed in which two
images is converted into a single complex distribution and finally encrypted into a single image. This
proposed encoding technique is successfully implemented for encryption and decryption. And in the
last the double image phase random encoding encryption technique is compared with the advanced
encryption techniques (AES) using the number of permutation required by the eavesdropper to
decode the key in a brute force method of attacking. By this comparison it is clearly shown that
proposed encryption technique is more efficient and robust against any attack.
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