International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 5, May 2013
ISSN 2319 - 4847
A robust and efficient scheme for image
encryption-decryption
Prerana Sharma1, Dr. Shiv Kumar2
1
M.Tech. Scholar, ACEIT, Kukas, Jaipur (Raj.), India
2
Professor, ACEIT, Kukas, Jaipur (Raj.), India
ABSTRACT
Fractional Fourier transform and chaos functions play a key role in many of encryption-decryption algorithms. In this work
performance of image encryption-decryption algorithms is quantified and compared using the resemblance of input image to
the restored image, quantified by MSE. This work verifies robustness of the proposed image encryption-decryption algorithms.
Further, simulations are performed using MATLAB7.7 to quantify performance of the proposed algorithms using MSE. The
results obtained in these simulations prove that for the proposed algorithms MSE between restored and original images has a
very high value, if decrypted at an incorrect thereby maintaining the robustness of the algorithms. These algorithms are found
sensitive to a variation of 1x10-1 in the fractional orders used in encryption-decryption process.
Keywords: Fractional Fourier Transform, Discrete Wavelet Transform, Tent map, Robust algorithm, Computation
Complexity.
1. INTRODUCTION
Optical systems process 2-dimensional complex data in parallel and at a high speed. Therefore, these systems are of
growing interest for image encryption. Many optical methods have been proposed in past [1] [2] [3] [4] [5] [6] [7].
Double random phase encoding as presented in [1], [4] has been widely used and remains highly successful. The
fractional Fourier transform also has been potentially used in the field of optical security [2] [3] [4] [5] [7].
2. FRACTIONAL FOURIER TRANSFORM
The ath order fractional Fourier transform is a linear operation defined by the integral

f  u  
 K  u, u' f u 'du '

 
K a u , u '  A exp i cot u 2  2 csc uu ' cot u ' 2
 A

 1  i cot   a 2
When a ≠ 2 j and Ka(u, u’) Ξ  (u – u’) when a = 4j and Ka(u,u’)Ξ  (u + u’) when a = 4j±2, where j is an integer. The
ath order transform is sometimes referred to as the  th order transform [7].
3. CHAOS FUNCTIONS
Chaos functions have been mainly used to develop the mathematical models of non-linear systems. These functions
generate completely random iterative values. But these values are limited between bounds. Convergence of the iterative
values after any value of iterations is kept hidden. Chaos functions are sensitive to the initial conditions. Three chaos
functions are discussed below:
(a) The first chaos function is the logistic map [8] and is defined as:
f ( x)  p.x.(1  x )
This function is bounded for 0 < p < 4 and can be written in the iterative form as:
xn1  p.x n .(1  xn )
with ‘x0’ as the initial value. This is also known as the seed value for the chaos function.
(b) The second chaos function is the tent map [8] and is defined as:
0  x 0  0.5
 a.x
f ( x)  
a.(1  x) 0.5  x 0  1
This function is bounded for 0< a ≤ 2 and can be written in the iterative form as:
Volume 2, Issue 5, May 2013
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 5, May 2013
ISSN 2319 - 4847
0  x 0  0.5
 a.x n
x n1  
a.(1  x n ) 0.5  x 0  1
with ‘x0’ as the initial value.
(c) The third chaos function is the Kaplan–Yorke map [8] and is defined as:
f ( x )  a.x. mod 1
0  x0  0.5

f ( y )  by  cos(4x) 0.5  x0  1
This is bounded for 0 ≤ a ≤ 2 and 0 ≤ b ≤ 1 and can be written in the iterative form as:
xn 1  a.xn mod1
y n 1  byn  cos(4xn )
with ‘x0’ as the initial value. These chaos functions are used to generate random phase masks. Logistic map and tent
map are one-dimensional chaos functions and the Kaplan–Yorke map is a two-dimensional chaos function. For the
two-dimensional chaos function, two seed values are required to generate the CRPM (Chaotic random phase mask).
4. PROPOSED SCHEME
Here we are considering that an image I = f(x, y) of size MXN consists of three primary color channels viz. RED,
GREEN and BLUE. For simplicity, we assume the image size to be N X N. As presented in [1], the image encryption
and decryption algorithms in this work are based on the Discrete Wavelet Transform, fractional Fourier transform and
use chaotic logistic map, Tent map and Kaplan-Yorke map. These algorithms shall be further referred as Algorithm 1,
Algorithm 2 and Algorithm 3 respectively throughout the paper. Each of these algorithms follows a similar sequence of
operations but they differ in their implementation details. A general overview of the encryption and decryption process
is as follows.
For an input image
3
I   f i  x, y ,
where 1  x, y  N
i 1
The encryption process of I produces I1 as below [1]
 

 
 

F
I  g ( x, y )  IDWT  F
DWT f ( x, y) * exp i S1 ( x, y )  * exp i S 2 ( x, y)  

1
2  ,    ,
2 i
 2

 2
 



where α, β, γ, δ denote orders of the fractional Fourier transform.
The decryption
of I1 is carried out as per the following formulae [1]

'
'

 
 
 
  
I  f ( x, y )  IDWT2 F ,  F ,   DWT2 g ( x, y )* conj exp i S 2 ( x, y )   * conj exp i S1 ( x, y)   
 2
 
 2
  


-α, -β, -γ and –δ denote fractional orders used for decryption.
5. SIMULATION RESULTS
In order to investigate the quality of encryption, decryption i.e. robustness of proposed algorithms, digital simulations
were performed in an environment as under:
Processor – Intel® Core™ i3 CPU, M380 2.53GHz.
RAM – 3.00 GB. Operating System – 64-bit Windows 7.
Simulation tool – MATLAB® R2008b.
Fig 1 Input image to algorithm 1, 2 and 3(Lena.tif, 256X256, color).
Volume 2, Issue 5, May 2013
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 5, May 2013
ISSN 2319 - 4847
The input image chosen for analysis is Lena (TIFF image; Size = 193 KB; pixel by pixel resolution = 256X256) as
shown in Fig 1. Lena was encrypted using algorithm 1, 2 and 3 for α = β = γ = δ = 0.5 fractional orders of FRT. The
resulting encrypted images obtained after application of algorithm 1, 2 and 3 are shown in Fig 2(a), 2(b) and 2(c)
respectively. These images can be decrypted on any fractional order of FRT, but the restored image may differ from the
input image depending on the order of FRT. The decrypted images obtained using incorrect fractional orders α = β = γ
= δ = 0.49 and α = β = γ = δ = 0.51 of inverse FRT are shown in Fig 3(a) – 3(c) and Fig 4(a) – Fig 4(c). Fig. 5(a) - 5(c)
show decrypted images obtained when correct fractional orders of inverse FRT are used i.e. α = β = γ = δ = -0.5.
(a)
(b)
(c)
Fig 2. Images encrypted at α = β = γ = δ = 0.5 using (a) Algorithm 1, (b) Algorithm 2 and (c) Algorithm 3.
(a)
(b)
(c)
Fig 3. Images decrypted at α = β = γ = δ = -0.49 using (a) Algorithm 1, (b) Algorithm 2 and (c) Algorithm 3.
(a)
(b)
(c)
Fig 4. Images decrypted at α = β = γ = δ = -0.51 using (a) Algorithm 1, (b) Algorithm 2 and (c) Algorithm 3.
(a)
(b)
(c)
Fig 5. Images decrypted at α = β = γ = δ = 0.5 using (a) Algorithm 1, (b) Algorithm 2 and (c) Algorithm 3.
Volume 2, Issue 5, May 2013
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 5, May 2013
ISSN 2319 - 4847
(a)
(b)
(c)
Fig 6. Graph, showing MSE between input image and decrypted image using (a) Algorithm 1, (b) Algorithm 2
and (c) Algorithm 3. The order of decryption ranging between 0.4-0.6
Fig 6(a) – 6(c) show the variation of MSE between decrypted (restored from the encrypted) image and the input image
at different fractional orders. From the graphs it is observed that the MSE value at α = β = γ = δ = -0.49 orders of
decryption is much higher than MSE value at α = β = γ = δ = -0.5 orders of decryption. This ratio of these MSEs is of
the order of 1029, which can be observed from the graph.
6. CONCLUSION
From the results shown in previous section, the algorithms presented in this work are found to be robust in nature. The
algorithms are sensitive to a variation of fractional order which is of the order of 1×10-1. Besides this, these algorithms
are efficient also [1]. Due to their efficient and robust behavior, the algorithms may find various applications.
References
[1.] P. Sharma, D. Mishra, A. Agarwal, “Efficient Image Encryption and Decryption Using Discrete Wavelet
Transform and Fractional Fourier Transform”, SIN’12, pp 153-157, October 2012.
[2.] M. Joshi, Chandrashakhar, K. Singh, “Color image encryption and decryption using fractional fourier transform”,
Optics communications, Vol. 279 Issue 1, pp 35-42, 1 November 2007.
Volume 2, Issue 5, May 2013
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International Journal of Application or Innovation in Engineering & Management (IJAIEM)
Web Site: www.ijaiem.org Email: editor@ijaiem.org, editorijaiem@gmail.com
Volume 2, Issue 5, May 2013
ISSN 2319 - 4847
[3.] N. Singh, A. Sinha, “Optical image encryption using fractional Fourier transform and chaos”, Optics and Lasers in
Engineering”, Vol. 46 Issue 2, pp 117 – 123, February 2008.
[4.] R. Tao, Y. Xin, Y. Wang, “Double image encryption based on random phase encoding in the fractional fourier
domain”, Optics Express, Vol. 15 Issue 24, 2007, pp 16067-16079.
[5.] B. M. Hennelly, J. T. Sheridan, “Image encryption and the fractional Fourier transform”, Optik - International
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[7.] H. M. Ozaktas, M. AlperKutay, Z. Zalevsky, The fractional Fourier transform: with applications in optics and
signal processing, New York, John Wiley & Sons, 2001.
[8.] R. C. Alligood, T. D. Sauer, J. A. Yorke, “Chaos; an introduction to dynamical systems”, New York, Springer,
2001
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