Robust Image Encryption Using Discrete Fractional Fourier Transform with Eigen Vector Decomposition Algorithm

Advances in Microelectronic Engineering (AIME) Volume 1 Issue 4, October 2013

www.seipub.org/aime

Robust Image Encryption Using Discrete
Fractional Fourier Transform with Eigen
Vector Decomposition Algorithm

Deepak Sharma* 1 , Rajiv Saxena 2 , Ashutosh Rajput 3

^Department of Electronics and Communication, Jay pee University of Engineering and Technology, Guna (M.P.)
Postcode - 473226, India

department of Electronics and Communication, Modern Institute of Technology and Research Centre, Alwar,
Postcode - 301028, India

Meepakforu23@rediffmail.com; 2 rajiv.saxena@juet.ac.in

Abstract

Numerous methods have been recently proposed in the
literature for the encryption of 2-D information using optical
systems based on the Fractional Fourier Transform.
Encryption is one of the well known techniques to provide
security in transmission of multimedia contents over the
internet and wireless networks. There is a vast use of image
in all areas so its security is of great concern nowadays.
Discrete Fractional Fourier Transform (DFRFT) generalization
of the Discrete Fourier transform (DFT) with an additional
parameter is incorporated in image encryption to achieve a
more robust encryption system. To focus on security aspect,
in this paper a novel method of image encryption has been
proposed based on discrete Fractional Fourier Transform
(DFRFT), using exponential random phase mask. Encryption
with this technique makes it almost impossible to retrieve an
image without using both the correct keys. The technique
has been implemented experimentally and parameters like
security, sensitivity and mean square error (MSE) are
discussed.

Keywords

Fourier Transform (FT); Discrete Fourier Transform (DFT);
Fractionl Fourier Transform (FRFT); Discrete Fractional Fourier
Transform (DFRFT).

Introduction

Fourier transform was first introduced by the French
scientist Jean Baptiste Joseph Fourier, in 1807 [17].
After that FT was applied in all the fields of science
and engineering. Fractional Fourier Transform was
introduced as the generalization of Fourier Transform,
thus creating a scope of improvement where Fourier
transform was applied along with time varying signal
analysis. It has been applied to various areas like
signal processing, optics and quantum mechanics. The
discrete fractional Fourier transform (DFRFT) is a
generalization of the DFT with additional free

parameters defined by Pei and Ozaktas. Pei and Yeh
defined the DFRFT based on the eigen decomposition
of the DFT matrix, and a DFRFT with one fractional
parameter was defined by taking fractional eigen
value powers of an eigen decomposition of the DFT
matrix. The DFT eigenvectors used are Hermit
Gaussian type. These eigenvectors are computed from
a DFT commuting matrix proposed by B. W.
Dickinson and K. Steiglitz. Pei et al. first proposed the
eigen decomposition- based definition of the DFRFT,
and then Candan et al. consolidated this definition .

In recent years, there has been great concern over
information security. Various optical encryption
methods have been proposed by researchers in the
past two decades. More robust encryption schemes are
always required to protect data. Which can be fulfilled
by proposing a more robust transform and applying
this transform in a model to achieve more
unauthorized user protected scheme for encryption.

In proposed encryption scheme, Discrete FRFT was
utilized with the double random phase encoding to
enhance its data security for input grayscale image. An
image was encrypted using DFRFT with double
random phase matrix and an image decrypted by
using same key utilized for encryption. In proposed
scheme, two keys are utilized to encrypt an image
which enhances the robustness and security of the
system. The security key is highly sensitivity to
deviation of correct keys and it has also been
investigated in simulated results.

The outline of this paper is as follows: In section II we
discuss the FRFT and DFRFT briefly. In section III we
discuss the proposed DFRFT based image encryption
model with double random phase matrix. In section IV
the security, sensitivity and MSE of for an image have

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Advances in Microelectronic Engineering (AIME) Volume 1 Issue 4, October 2013

been investigated. In section V the results are
summarized.

Preliminaries

Fourier transform is the rotation of a signal by an
angle of n/2 in time frequency plane. The fractional
Fourier transform removes the constraint of Fourier
transform and allows rotation of angle 'a' where it is a
multiple of n/2. The FRFT of signal x(t) of order 'p' can
be represented as,

X p (u) = jx(t)K p (u,t)dt.

(1)

There is a relation between the angle of rotation 'a'
and order given as a = prc/2. The kernel K P is defined as,

K p (u,t) =

1- /'cot a . J 2 + u 2

exp(y cot ot — jut esc ot) a^nn

In
S(t-u)

S(t + u)

a = Inn
a = (2n± n)

(2)

As the development goes on in the field of digital
signal processing, there is a need for discrete fractional
Fourier transform for all applications using fractional
Fourier transform because all the signals processed in
discrete form. DFRFT is the generalization of the
discrete Fourier transform (DFT) with an additional
parameter. Some basic properties of DFRFT are

1) Unitary

2) Additive

3) Reduction to DFT when order is equal to unity.
The MxM DFT matrix can be expressed as,

1 —y^-km

X=^=e M for K>0,m<M -1
VM ' (3)

Matrix X has only four distinct Eigen values 1,-1, j and
-j. Now a MxM matrix Z is defined whose entries are

Z m ,m = 2 COS

2/T

-n

Z = Z

m,m+l m+l,m

V M J0<n<M-l
= l 0<n<M-2

(4)

7 - 7 - 1

^M-1,0 ^0,M-1 1

Now as ZX=XZ, i.e. they commute with each other,
and have same value of eigen vector but different
eigen values. Now MxM DFRFT matrix can be defined
as,

X a =VD a V T

M-l

-j-ka

2> J ^v k v!

k=0

for M = odd

v k v[ + e ' 2 v M v T M for M = even

Here D is the diagonal matrix and T is the transpose.
The matrix V is defined as,

V = [v ,v p v M _ 2 ,v M _J for M = odd

V = [v ,v v v M _ 2 ,v M ] for M=even (g)

This is known as eigen decomposition form of DFRFT.

Proposed Model for Encryption and
Decryption

Encryption

Encryption, a very ancient technique, is a process in
which information is secured with the help of a key to
protect it from an unauthorized person. The
information which has to be encrypted can be in the
form of a text or an image etc. Encrypted information
should not give any idea about the original
information. In the proposed method, image
encryption of gray scale image is done using double
random discrete fractional Fourier transform. In Gray
scale image, the value of each pixel carries only
intensity information ranging from to 255. They also
known as black-and-white, are composed exclusively
of shades of gray, varying from black at the weakest
intensity to white at the strongest.

In double random matrix DFRFT method, the input
image is multiplied with the first exponential random
matrix S after that the DFRFT of order 'a! is applied
then, the resulting matrix is multiplied by the second
exponential random matrix C and again DFRFT of
order '|3' is applied to get the encrypted image. The
random matrices at encryption and decryption should
be orthogonal to each other.

Input
Image 'P ;

-A-

Encrypted
Image 'E ;

Random

Random

matrix- S

matrix- C

FIG. 1 IMAGE ENCRYPTION MODEL

Exponential random matrix S and C should be
independent of each other. The value of of S and C are.

S = e m , C

= e iPm

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Now let us understand the mathematics of the
proposed encryption process.

W' = P®e j<pn

(7)

where P is input image matrix multiplied by S, the
random matrix

(8)

W = X a (P®e Jgm )
DFRFT of order a is applied.

W m = (X a (Pxe im ))xe m (9)
Then it is multiplied by random matrix C.

E = X /] ((X a (Pxe jgm ))xe iPm ) (1Q)

Then DFRFT of order [3 is applied to get the encrypted
Image E.

Decryption Model

It is the process of retrieving the original information
from the encrypted form as well as reverse process of
the encryption part. Input is an encrypted image in
decrption model. Input image 'E' is taken then DFRFT
of order '-|3' is applied then to it, and random matrix
C* is multiplied which cancel out the effect of random
matrix C as they both are orthogonal to each other. To
the resulting matrix DFRFT of order '-a! is applied
then multiplied by the random matrix S* to get the
decrypted image.

Encrypted
Image E

DFRFT

Order 'f

-A

DFRFT
Order V

Random

Matrix C*

ft i Decrypted
li Image D

Random
Matrix S*

FIG. 2 IMAGE DECRYPTION MODEL
The Decryption model mathematically given as,

E' = X~ P (E)

= X- p (X p (X a (P®e j(pn )®e jPm ))

(11)

Then it is multiplied by the random matrix C* as it will
cancel out the effect of C as they are orthogonal.

E" = (X - p (X p ((X a (P ® )) <S> e iPm ))) <S> e~ j/im
Now, DFRFT of order -a is applied

E m = X~ a ((X - p (X 13 ((X a (P <g> e iv " )) ® e iPm ))) ® e~ iPm

(12)

(13)

Then random matrix S*, is multiplied

D = (X~ a ((X - p (X p ((X a (P®e j(pn )) <8> e jPm ))) ® e~ IPm )) ® e~ j<pn ^
Where D is the decrypted image and after solving it

gives original image T' as output.

Salient Features

Security

Security is the main aim. The key here is formed, by
the combination of the order of discrete fractional
Fourier transform and the random phase matrix. In
encryption model various possible combination
provides formidable key sets, thus providing higher
amount of security. This is also effective against brute
force attack i.e. if all the set of key are known. In this
case as the key set is large and the time taken is very
large in brute force attack thus making it impractical to
crack the correct key.

Sensitivity

The model of encryption and decryption should be
highly sensitive with respect to the variation of correct
key i..e. if the any key other than the correct key is
used, it will not decrypt the correct input image, result
with our simulation. It has been checked that our
image is much sensitive to the deviation in the original
key.

Complexity

If complexity of system increases it usually also
improves the security of system. There is a tradeoff
between the complexity and the security of a system
required. It is also analyzed based on the property of
the utilized discrete fractional Fourier transform
(DFRFT) that the encryption scheme can be realized by
the fast Fourier transform (FFT)-based algorithm. In
proposed algorithm encryption and the decryption
procedures are both realized by the matrix
multiplications. For an image with a size of A*B, the
complexity of the encryption and the decryption is
about equal to A 2 B + B 2 A complex multiplications.

Results

In this paper, an image has successfully encrypted and
decrypted using DFRFT with double random phase
matrix. Simulation of the proposed method has been
done on a grayscale image of Lena of dimension 300 x
300. The image was encrypted for the selected orders a
= 0.8, [3 = 1.2 by using two exponential random
matrices S and C which are independent to each other.
As it can be seen that fig. 3 (a) is the original image
and after encryption an encrypted image is obtained
as shown in the fig 3(b). In decryption process we have
to apply the order a = -0.8 and [3 = -1.2 to get the

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correct decrypted image observed in the figure 3(c)
and if the proper value of the order is not used then it
will not give the correct result as observed in fig 3(d).
Here the orders are a = 0.9, [3 = 1.1. The image can be
decrypted only by using the correct order.

FIG. 3(A) ORIGINAL IMAGE FIG. 3(B) ENCRYPTED IMAGE

FIG. 3(C) DECRYPTED IMAGE (CORRECT KEY)

FIG. 3(D) DECRYPTED IMAGE (WRONG KEY)

FIG.3 IMAGE ENCRYPTION USING DOUBLE RANDOM MATRIX
DFRFT METHOD (A) ORIGINAL IMAGE (B) ENCRYPTED IMAGE
FOR A = 0.8, B = 1.2. (C) DECRYPTED IMAGE WITH CORRECT
PARAMETER A =- 0.8, B = -1.2 (D) DECRYPTED IMAGE WITH
WRONG PARAMETER A=0.9,B=1.1.

As the result shows that encrypted image doesn't give
any idea about the original image and image can be
decrypted only by the use of correct key as shown in
Fig. 3(c) and if correct, it is not used, original image
cannot be retrieved as shown in fig. 3(b). Thus, the
proposed method encrypts and decrypts image
properly. Histogram of the image plots frequency of
each pixel value, and summarizes the intensity of an
image.

FIG. 4(A) HISTOGRAM OF ORIGINAL IMAGE

7000 -
6000 -

5000
^ 4000

ZD

o
o

3000

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

FIG. 4(B) HISTOGRAM OF AN ENCRYPTED IMAGE

FIG. 4 HISTOGRAM OF AN ORIGINAL AND ENCRYPTED
IMAGE

In fig 4. (a) and 4.(b), the histogram of an original
image and the encrypted image of Lena are shown. As
it can be seen that the histogram of the encrypted
image is completely different from that of original
image so it does not give any idea to unauthorized
person to carry on the attack.

MSE (mean square error) is calculated between the
original image and decrypted image given by,

MSE =

Where M and N indicate the size of the image while
P(m,n) and W indicate the original and decrypted
image of pixel (m,n) respectively.

TABLE-l MSE OF DEFFERENT IMAGES

IMAGE

MEAN SQUARE ERROR (MSE)

Min

Max

Aug

Lena

2.6754 x 10-13

4.3435 x 10-13

3.2457 x 10-13

Camera man

3.4593 x 10-13

5.7725 x 10-13

4.3491 x 10-13

Spine MRI

3.5514 x 10-13

4.7220 x 10-13

4.0645 x 10-13

MSE of different images are calculated to check the
consistency of the proposed technique. The size of
images can be different and tested for various size of
images with same scheme. The results are obtaind
over 201 iterations for each image to get the result as
shown in the table-1. The proposed scheme is quite
consistent with different images for encryption
/decryption verified with the results shown in table-1.

In fig. 5 the sensitivity of key is shown, and image will
be correctly decrypted only by the use of original key,
if there is small deviation from the original key then it
will show a sharp increase in the normalized MSE.
Comparison between the sensitivity of FRFT and
DFRFT is done. DFRFT is more sensitive to the

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deviation in the correct key. So proposed scheme is
highly sensitive with respect to their original key.
Small deviation from original key provides large error
and wrong detection of the image.

-0.06 -0.04 -0.02 0.02 0.04 0.06

Deviation

FIG.5 NORMALIZED MSE BETWEEN THE DECRYPTED IMAGE
AND ORIGINAL IMAGE AS A FUNCTION OF DEVIATION IN
THE FRACTIONAL ORDER USED FOR DECRYPTION FOR
DFRFT AND FRFT

Conclusions

In this paper, a novel method has been proposed for
image encryption using discrete fractional Fourier
transform with double random matrix. The DFRFT
used here is eigen vector decomposition type. The
minimum MSE achieved here is 2.6754 x 10 13 for Lena
image. The decryption is highly sensitive towards the
original key as shown in fig. 5 compared to the FRFT
based scheme. The system is more sensitive to the
correct key if the key deviate by more than .01 value
with their original key, MSE increases and original
image cannot be decrypted. The complexity of the
system increases with DFRFT comparatively using
FRFT in the same system. The histogram also verifies
our results that the encrypted image has almost
uniform distribution of data so original image cannot
be decoded successfully. The DFRFT based image
encryption schemes is better and more sensitive but
produces more complexity than FRFT based
encryption scheme.

ACKNOWLEDGMENT

The authors thankfully acknowledge all the authorities
of Jaypee University of Engineering & Technology,
Guna (M.P.) - 473226, INDIA.

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Rajiv Saxena Dr. Saxena, born at Gwalior
in Madhya Pradesh in 1961, obtained B.E.
(Electronics & Telecommunication
Engineering) in the year 1982 from
Jabalpur University, Jabalpur.
Subsequently, Dr. Saxena joined the

Reliance Industries, Ahmedabad, as

Graduate Trainee. In 1984, Dr. Saxena joined Madhav
Institute of Technology & Science, Gwalior as Lecturer in
Electronics Engineering. He obtained his M.E. (Digital
Techniques & Data Processing) from Jiwaji University,
Gwalior in 1990. The Ph. D. degree was conferred on him in
1996-97 in Electronics & Computer Engineering from IIT,
Roorkee (erstwhile UOR, Roorkee). Currently Dr. Saxena is
head and professor in ECE department at JUET, Guna.

Deepak Sharma completed his M. Tech.
(Microwave Engineering) from Madhav
Institute of Technology and Science in
2006. Before joining JUET, he worked as a
Lecturer in Electronics Department,
MITS, Gwalior (M.P). His Research areas
include Antenna Theory, Radar System,
Signal processing and Integral Transforms .

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