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Color image encryption using MSVD, DWT and Arnold transform in fractional Fourier domain
Accepted Manuscript Title: Color image encryption using MSVD, DWT and Arnold transform in fractional Fourier domain Author: Ankita Vaish Manoj Kumar PII: DOI: Reference:
S0030-4026(17)30858-6 http://dx.doi.org/doi:10.1016/j.ijleo.2017.07.041 IJLEO 59437
To appear in: Received date: Revised date: Accepted date:
11-10-2016 11-5-2017 16-7-2017
Please cite this article as: Ankita Vaish, Manoj Kumar, Color image encryption using MSVD, DWT and Arnold transform in fractional Fourier domain, (2017), http://dx.doi.org/10.1016/j.ijleo.2017.07.041 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Color image encryption using MSVD, DWT and Arnold transform in fractional Fourier domain Manoj Kumar1 , Ankita Vaish2∗ Babasaheb Bhimrao Ambedkar University, Lucknow, (U.P.), India.
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1, 2
Abstract
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This paper presents, a new image encryption technique using Multiresolution Singular Value Decomposition (MSVD), Discrete Wavelet Transform (DWT) and Arnold transform in frac-
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tional domain. Images are encrypted using keys of FrFT, subbands of MSVD and DWT along with the use of parameters generated by MSVD in various DWT subbands and Arnold
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transform. We have used keys of FrFT, the arrangement of MSVD and wavelet subbands, values and arrangement of parameters generated by MSVD in various DWT subbands, the use of Arnold transform and the way of arrangement of several subimages as encryption and
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decryption keys. For the correct decryption of encrypted images it is required to have correct
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knowledge of all the keys in correct order along with their exact values. The effectiveness of proposed work is analyzed by comparing it with some related works, the comparison verifies
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that the proposed work has obtained an increased level of security and robustness when compared to exiting works.
Keywords: Arnold transform, Image encryption, Discrete wavelet transform, Fractional fourier transform, Multiresolution singular value decomposition. 1. Introduction
With the ever growing field of multimedia applications, a lot of information is being transferred all around the world over the public networks. However, direct transmission of the secret information over the public networks is not preferable due to security reasons. ∗
Corresponding author. URL: [email protected] (Ankita Vaish2 ) 1 Assistant Professor, Babasaheb Bhimrao Ambedkar University, Lucknow, (U.P.), India. 2 Research Scholar, Babasaheb Bhimrao Ambedkar University, Lucknow, (U.P.), India.
Preprint submitted to Optik
May 11, 2017 Page 1 of 23
Considerable work has been done by the researchers to keep the information secure from unauthorized users. Optical encryption techniques are gaining a lot attention in information security. Since Refregier et al. [1] first proposed an optical encryption technique based on double random phase, many techniques for secure transmission of sensitive images have been
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proposed based on digital holography (DH) [4], Hartley transform (HT) [5, 6, 7], Gyrator
Fourier transform (FrFT) [18, 19, 20, 21, 22, 23, 24, 25].
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transform (GT) [8, 9, 11, 10, 12, 13], Arnold Transform (AT) [15, 16, 17] and Fractional
Several techniques are reported in literature based on Hartley transform viz. Singh et
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al. [5] proposed an image encryption technique using logistic map. An image encryption technique in Hartley transform domain is proposed by Chen et al. [6] which is based on
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interferometer. A color image encryption technique using chaotic map in Hartley transform domain is given in [7] where baker mapping is used to scramble each of the color plane of
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the image.
Many color image encryption techniques are given in literature based on Gyrator transform such as: Singh et al. [8] proposed a chaos based image encryption technique in Gyrator
d
transform. Plenty of work have been proposed by Abuturab [9, 11, 10, 12, 13] in Gyrator
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transform domain such as Abuturab [9] proposed an image encoding technique using double random phase. Again, a new encryption technique using Arnold transform is given by
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Abuturab [10]. Another technique based on Hartley transform in Gyrator transform domain is introduced by Abuturab [12]. More recently, Abuturab [14] proposed a color image encryption technique using Singular value decomposition in Gyrator transform domain. A lot of work has been reported in literature for image encryption using Arnold transform, it scrambles the intensity values of an image which results to an image in unrecognizable form. A color image encryption technique in Intensity (I), Hue (H) and Saturation (S) color space using Arnold and Discrete Fractional Random Transforms is given by Guo et al. [15], in this work I component is encrypted using discrete fractional random transform and H and S components are encrypted by using Arnold transform. Liu et al. [16] proposed an image encryption technique using Arnold transform and discrete cosine transform. Again a color image encryption technique using Arnold and discrete angular transforms is given by Liu et al. [17]. 2 Page 2 of 23
Various techniques are proposed in literature that have used Fractional fourier transform (FrFT) for image encoding, FrFT is the generalization of conventional Fourier transform. Further, a random FrFT is given by Liu et al. [18], which is obtained by randomizing the conventional FrFT. Liu et al [19] proposed an image encryption technique using multi
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channel and multi stage fractional Fourier domain filtering. An iterative fractional Fourier transform based image encryption technique is proposed by Zhang et al. [20]. Chen et al. [21]
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introduced a fractional wavelet packet based color image encoding technique. A new color image encoding technique using random phases in dual fractional Fourier-wavelet domain
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is proposed by Chen et al. [22]. Prasad et al. [24] proposed a color image encoding using DWT in FrFT domain, this work provides good security but doubles the size of encrypted
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image that is not a suitable choice from transmission point of view. Recently, a color image encryption technique is introduced by Chen et al. [25] using SVD and Arnold transform
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in FrFT domain, this work has obtained enough security but transmits three encrypted images of dimension equal to the original image and at the time of decryption all the three encrypted images must be required for correct decryption. More recently, Kumar et al.
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[26, 27] introduced a new color image encoding technique using DWT and SVD in Discrete
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Cosine Stockwell transform domain. Though, the works given in [24, 25] provide good security, but the arrangement used in [24] doubles the size of encoded image while the work
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introduced in [25] has increased the transmission cost by transmitting three encrypted images of dimension equal to the original image, which is not appropriate from transmission point of view as extra space and time costs is required. However, in our work an attempt has been made to provide a more secure encryption technique by overcoming the disadvantages of [24, 25].
In this paper, a new color image encryption technique using MSVD, DWT and Arnold transform in FrFT domain is proposed. The purpose of this work is to propose a more secure encryption technique for secure transmission over unsecure networks. An original color image is first divided into its primary color components i.e. Red (R), Green (G) and Blue (B). Further each of the component is encrypted independently using DWT, MSVD and Arnold transform in fractional domain. The keys of FrFT, arrangement of MSVD and DWT subbands, values and arrangement of values of 4 × 4 matrix, generated by MSVD in DWT 3 Page 3 of 23
subbands, the use of Arnold transform and the way of arrangement of several subimages are termed as encryption and decryption keys. At the time of decryption, correct knowledge of all the keys along with the way of arrangement are necessary for correct decryption of encrypted images.
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This paper is organized as follows: some key terms related to the proposed work are given in section 2, the proposed encryption and decryption technique is introduced in section
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3, section 4 addresses the experimental results and analysis of proposed work with some standard test images, comparison of proposed work with some recently published works is
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discussed in section 5, finally, conclusions are drawn in section 6.
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2. Related theories 2.1. Fractional Fourier Transform (FrFT)
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FrFT is a generalized version of Fourier transform with parameter α, which is used most frequently in optical and digital information processing. It is used in many areas of image
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processing such as optical image encryption, watermarking [35, 36, 37]. In mathematics,
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FrFT is known for more than last 70 years, but the definition of FrFT in optics is given by Mendlovic and Ozaktas in 1993. It is based on the propagation of light within a medium
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with continuously varying refractive index [36, 37]. The 2D-FrFT with parameters α1 and α2 of an image f (x, y) is represented by Fα1 ,α2 (f (x, y))(u, v) is given as: ∫ ∫ i(x2 +u2 ) cot α1 1 √ −ixu csc α1 2 Fα1 ,α2 (f (x, y))(u, v) = (1 − i cot α1 )(1 − i cot α2 ) e 2Π R R ×e
i(y 2 +v 2 ) cot α2 −ixu 2
csc α2
f (x, y)dxdy
(1)
As the FrFT has property of linearity and continuity, hence the inverse of FrFT can be obtained with corresponding negative fractional orders. 2.2. Multiresolution Singular Value Decomposition (MSVD) For a given matrix A of size M × N , Singular value decomposition (SVD) [32] can be written by using equation: A = U SV T , where U and V are orthogonal matrices of size M ×M and N × N respectively and S represents a diagonal matrix containing sorted singular values in descending order. 4 Page 4 of 23
MSVD [33] of an image (I) of size M × N can be obtained by first reshaping I to a matrix I1 of size 4 × M N/4. SVD of I1 is calculated as: [U, S] = SV D(I1 )
(2)
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where, U is of size 4 × 4 and S is 4 × M N/4 now, matrix T is calculated as:
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T = U T I1
(3)
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where, T is a matrix of size [4 × M N/4]. The first, second, third and fourth rows of matrix T are reshaped to obtain matrices of size M/2 × M/2, the reshaped matrices are called LL,
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LH, HL and HH subbands. The low frequency details are in LL subband, while the LH, HL and HH subbands contain the high frequency details. These subbands (LL, LH, HL
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and HH) and matrix U of size 4 × 4 is used in our work for as encryption keys. 2.3. Discrete Wavelet Transform (DWT)
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Discrete Wavelet Transform (DWT) has been used more frequently in many applications of image processing like image compression, image encryption, watermarking etc. DWT
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is localized both in time and frequency domains, which reveal the spatial and frequency
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views at the same time. It decomposes the image at multiresolution level, which helps in analyzing an image at different resolutions. Due to the multiresolution property of DWT, the information that seems to be unnoticed at one level may become noticeable at another level.
The 2-D DWT can be obtained by first applying 1-D DWT in horizontal direction and then in vertical direction. 2-D DWT decomposes an image into four parts: approximation (A), horizontal (H), vertical (V ) and diagonal (D). Approximation part contains low frequency subband, while the horizontal, vertical and diagonal parts contain the high frequency subbands. For an image of size M × N , 2-D DWT can be defined as [42]: M −1 N −1 1 ∑∑ WΦ (j0 , m, n) = f (x, y)Φj0 ,m,n (x, y) M N x=0 y=0
(4)
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WΨi (j, m, n)
M −1 N −1 1 ∑∑ = f (x, y)Ψij,m,n (x, y) M N x=0 y=0
(5)
where, WΦ (j0 , m, n) represents approximation part and WΨi (j, m, n) represents horizontal,
The inverse of 2-D DWT can be defined as:
∞ ∑∑ ∑ ∑ i=H,V,D j=j0 m
WΨi (j, m, n)Ψij,m,n (x, y)
(6)
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1 MN
1 ∑∑ WΦ (j0 , m, n)Φj0 ,m,n (x, y) + MN m n
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f (x, y) =
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vertical and diagonal parts of image f (x, y).
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In 1993, Mendlovic and Konforti gives the optical realization of 2D- DWT [43]. Optically, it can be obtained using conventional coherent correlator along with multi reference matched
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filter.
Table 1: Arnold transform period (T) Vs. varying values of (N)
3
4
T
3
4
3
N
32
64
T
24
48
T
6
7
8
9
10
10
12
8
6
12
30
128
256
512
1024
2048
4096
8192
96
192
384
768
1536
3072
6144
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N
5
d
2
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N
33
65
100
257
513
1025
2049
4097
8193
20
70
150
258
1432
100
684
360
780
2.4. Arnold Transform
Arnold transform changes the positions of pixels rather than masking their values. It is used most frequently in image encryption and watermarking [38, 39, 40, 41]. It can be
defined as:
where,
x
′
′
x y
′
′
=
1 1
1 2
x y
mod N
(7)
represents the position vector of an image after shifting while
x
rep-
y y resents the position vector before shifting and mod shows the modulus after division with 6 Page 6 of 23
N . The parameter N that is used over here is the size of input image, which determines the period of Arnold transform. Table-1 shows the period of transform vs. the values of N . Arnold transform is applied on an original image shown in fig. 1(a), which consists 512 × 512 pixels, from the table-1 it can be seen that its period is 384. Fig. 1(b) shows the
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result after 50 times Arnold transform and fig. 1(c) shows the result after 100 times Arnold transform while the fig. 1(d) shows the image after 384 times Arnold transform. Hence, the
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results shown in fig. 1 validates the period property of Arnold transform shown in table-1. The original image information can be recovered using period property but as the value
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of N becomes larger, the period will also become larger and it will take more time to recover
2 −1
a
-1 1
c
(8)
1
1
1
2
.
d
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b
is used instead of A =
d
where, A−1 =
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the original information. Therefore, an inverse of Arnold transform is defined as: ′ x 2 -1 x = mod N y′ −1 1 y
Figure 1: (a) Original image; (b) 50 times Arnold transform; (c) 100 times Arnold transform; (d) after 384 times Arnold transform.
An original Lena image is shown in fig. 2(a), fig. 2(b) shows the result after one time Arnold transform, while fig. 2(c) results from inverse of Arnold transform. It can be analyzed from figures 1 and 2 that the original image information can be recovered from both the period and inverse Arnold transform. The histogram corresponding to figures 2(a) and 2(b) are shown in figures 3(a) and 3(b) respectively. Hence, it can be analyzed from figures 3(a) and 3(b) that the Arnold transform does not masks the values of original image.
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a
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b
b
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a
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Figure 2: (a) Original image; (b) after one time Arnold transform; (c) inverse Arnold transform.
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Figure 3: Histogram of (a) Original image; (b) after Arnold transform.
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3. Proposed Encryption and Decryption Scheme
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The detailed description of proposed encryption and decryption scheme is shown in figs 4(a) and (b) respectively. In the encryption procedure, at first an original colored image of size M × N is decomposed into three primary color components i.e Red (R), Green (G) and Blue (B). FrFT is applied on each of the R, G and B planes independently, and then MSVD is applied on all of the three planes, which results LL, LH, HL and HH subbands for each plane. Again, 2-D DWT is applied on each of the MSVD subbands i.e. LL, LH, HL and HH subbands, which further results four subbands: A, H, V and D corresponding to each subbands of the MSVD (i.e. LL, LH, HL and HH) for all planes. Now the values of each of the subband obtained from DWT are masked by the the values of U matrix (generated by MSVD (shown in subsection 2.2)) as:
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a
U4* 4
U(1,1) *ALL ALL VLL
A'LL
U(1,2)*H LL
H'LL
HLL DLL
U(1,3)*V LL
V'LL
V''LL
U(1,4)*DLL
D'LL
D''LL
H'LH
HLH DLH
U(2,3)*V LH
V'LH
V''LH
U(2,4)*D LH
D'LH
D''LH
U(3,1)*A HL
A'HL
A''HL
AHL V HL
U(3,2)*H HL
H'HL
HHL DHL
U(3,3)*VHL
V'HL
V''HL
U(3,4)*DHL
D'HL
D''HL
U LH
HH
H''LH IDWT
H''HL IDWT
U(4,1)*A HH
A'HH
U(4,2)*H HH
H'HH
H''HH
U(4,3)*V HH
V'HH
V''HH
U(4,4)*D HH
D'HH
D''HH
AT
E3
IDWT
E4
Encrypted Image
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HHH DHH
E2
A''HH
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AHH VHH
AT
A''LH
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FRFT
DWT
AT
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A'LH
U(2,2)*H LH
Arrangemen of subimages in order E4, E2, E1 and E3
U(2,1)*A LH
E1
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Original image
HL
H''LL IDWT
ALH VLH MSVD LL
A''LL AT
U4* 4
Encrypted Image
E2
E3
E4
d A'LL
A'LL/U (1,1)
ALL
H'LL
H'LL/U(1,2)
HLL
V'LL
V'LL/U(1,3)
VLL
D'LL
D'LL/U(1,4)
DLL
A'LH
A'LH/U(2,1)
ALH
H'LH
H'LH/U(2,2)
HLH
V'LH
V'LH/U(2,3)
VLH
D'LH
D'LH/U(2,4)
DLH
A'HL
A'HL/U(3,1)
AHL
H'HL
H'HL/U(3,2)
HHL
V'HL
V'HL/U(3,3)
VHL
D'HL
D'HL/U(3,4)
DHL
te
E1
DWT A''LL V''LL IAT H''LL D''LL
DWT A'' LH V''LH IAT
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Decompostion into various subimages according to their indexing
b
H''LH D''LH
DWT A'' V'' IAT HL HL H''HL D''HL
DWT A''HH V''HH IAT H''HH D''HH
A'HH
A'HH/U(4,1)
AHH
H'HH
H'HH/U(4,2)
HHH
V'HH
V'HH/U(4,3)
VHH
D'HH
D'HH/U (4,4)
DHH
IDWT
LL
IDWT
LH IMSVD
IFRFT
Decrypted Image
IDWT
HL
IDWT
HH
Figure 4: Proposed (a) Encryption scheme (b) Decryption scheme.
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(9)
Hl′ (i, j) = U (v, 2) × Hl (i, j)
(10)
Vl′ (i, j) = U (v, 3) × Vl (i, j)
(11)
where, v = 1, 2, 3, 4 and 1 ≤ i ≤
M 4
and 1 ≤ j ≤
N , 4
(12)
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Dl′ (i, j) = U (v, 4) × Dl (i, j)
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A′l (i, j) = U (v, 1) × Al (i, j)
suffix l used here corresponding to
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LL, LH, HL and HH subbands of MSVD. The use of values of U matrix in various subimages adds an extra layer of security. All the modified subbands are Arnold transformed
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and results to respective subimages i. e. A′′l , Hl′′ , Vl′′ and Dl′′ for all the DWT subbands for all planes. Inverse of DWT is applied on Arnold transformed subimages, which results to subimages E1 , E2 , E3 and E4 . Further, all the subimages obtained from inverse of DWT
E(2i, 2j) = E1 (i, j)
(13)
E(2i − 1, 2j) = E2 (i, j)
(14)
E(2i, 2j − 1) = E3 (i, j)
(15)
E(2i − 1, 2j − 1) = E4 (i, j)
(16)
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d
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are arranged according to their indexing as:
where, E represents the final encrypted image of size M ×N . We have used keys of FrFT, arrangement of MSVD and DWT sub-bands, values of 4×4 matrix and its use in various DWT subbands, arrangement of various subimages obtained from inverse of DWT as encryption and decryption keys.
The procedure for decryption is just reverse of encryption. Fig. 4(b) shows the decryption process of proposed work. All the decrypted R, G and B planes are further combined to get the reconstructed color image. 4. Experimental Results and Security Analysis The efficiency of proposed work is evaluated by testing it on various standard test images each of size 512 × 512. The various standard test images used in the proposed work are 10 Page 10 of 23
c
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a
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Figure 5: (a) Barbara; (b) Lena; (c) Baboon; (d) Peppers; (e) Girl; (f) Splash.
shown in fig. 5. An original image of Barbara and its encrypted images are shown in figures
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6(a) and (b) respectively with fractional orders 0.4 and 0.33, arrangement of values of 4 × 4 matrix obtained from MSVD in various DWT subbands, use of an Arnold transform and
d
arrangement of wavelet subbands in order D, V, A and H (when applying inverse DWT in
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encryption phase). The correctly decrypted image with all the correct keys along with the correct arrangement is shown in fig. 6(c).
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a
c
Figure 6: (a) an original image; (b) encrypted image; (c) correctly decrypted image.
The robustness of proposed work with the change of each and every keys is demonstrated in fig. 7. We have used Barbara image to demonstrate the robustness of proposed work. Fig. 7(a) shows an incorrectly decrypted Barbara image using incorrect fractional orders 11 Page 11 of 23
b
c
e
f
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a
Figure 7: decrypted images (a) incorrect fractional order; (b) incorrect arrangement of MSVD subbands;
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(c) incorrect arrangement of DWT subbands obtained from MSVD subbands; (d) incorrect values of 4 × 4 matrix; (e) without division of values of U matrix; (f) incorrect decomposition of final encrypted image.
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(incorrect values are -0.98, -1.48) while the rest of the parameters are correct. An incorrectly
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decrypted image using incorrect arrangement of MSVD subbands is shown in fig. 7(b). Figs. 7(c) and (d) show the incorrectly decrypted image using incorrect arrangement of
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DWT subbands and incorrect values of 4 × 4 matrix respectively. An incorrectly decrypted image is shown in fig. 7(e) when the values of 4 × 4 matrix are not used for division at the time of decryption. Fig. 7(f) demonstrate an incorrectly decrypted image when the subimages E1 , E2 , E3 and E4 are incorrectly arranged (incorrect way used is E4 E2 , E1 , and E3 ). It can be analyzed from the various incorrectly decrypted images shown in fig. 7 that it is nearly impossible to even guess the original image. Hence, the proposes scheme has obtained a desired level of security as if any one of the parameter get wrong (either values or arrangement) at the time of decryption even though the reconstructed image is not recognizable. The security analysis of proposed work on combination of two incorrect keys is given in fig. 8. An incorrectly decrypted image is shown in fig. 8(a) when incorrect fractional order along with incorrect arrangement of MSVD subbands are used. Fig. 8(b) shows a wrongly decrypted image when way of arrangements of MSVD and DWT subbands are incorrect. A 12 Page 12 of 23
b
c
e
f
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d
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a
Figure 8: decrypted images (a) incorrect fractional order with wrong arrangements MSVD subbands; (b) incorrect arrangement of MSVD and DWT subbands; (c) incorrect arrangement of DWT along with incorrect
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values of 4 × 4 matrix; (d) incorrect values of 4 × 4 matrix with incorrect decomposition of final encrypted image;(e) incorrect decomposition and without division of values of U matrix; (f) incorrect arrangement of
d
MSVD subbands with incorrect values of 4 × 4 matrix.
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wrongly decrypted image is shown on fig. 8(c) when arrangement of DWT subbands and the values of 4 × 4 matrix are wrong. Fig. 8(d) shows an incorrectly decrypted image when
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values of 4 × 4 matrix and subimages E1 , E2 , E3 and E4 are incorrectly arranged. An incorrectly decoded image is shown in fig. 8(e) when subimages E1 , E2 , E3 and E4 are incorrectly arranged and values of 4 × 4 matrix are not used in division. A wrongly decoded image is shown in fig. 8(f) when way of arrangements of MSVD subbands and values of 4 × 4 matrix are incorrect. It can be observed from the several images shown in fig. 8 that the proposed scheme has maintained the image security even if the two parameters are wrong. Further, we have performed the incorrect key test when three parameters are wrong and the respective results are shown in fig. 9. A wrongly decrypted image with incorrect fractional order along with incorrect arrangements of MSVD and DWT subbands is shown in fig. 9(a). Fig. 9(b) shows an incorrectly decrypted image with incorrect way of arrangements of MSVD and DWT subbands along with wrong arrangement of subimages E1 , E2 , E3 and E4 . An incorrectly decrypted image with wrong way of arrangements of MSVD and DWT 13 Page 13 of 23
b
c
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a
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Figure 9: decrypted images (a) incorrect fractional order with incorrect arrangement of MSVD and DWT subbands; (b) incorrect arrangement of MSVD, DWT subbands with incorrect decomposition of final en-
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crypted image; (c) incorrect arrangement of MSVD, DWT subbands with incorrect values of 4 × 4 matrix.
subbands along with wrong values of 4 × 4 matrix is shown in fig. 9(c). Although, we
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have tested the robustness of proposed work on the change of one, two and three incorrect parameters while the rest are exactly correct and it is obvious that if the proposed work
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can provide secure information on one, two and three incorrect parameters then it would be more difficult to recognize the incorrectly reconstructed image when more than three
over the networks.
d
parameters are wrong. Hence, the proposed work can transmit the images more securely
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Further, we have also analyzed the effectiveness of proposed work on some other type of data such as noisy data, which are not in 8 bit. To accomplish this, first of all, we have added
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Gaussian noise (standard deviation 35) to the original source image (shown in fig. 5(a)), the noisy image thus obtained is shown in fig. 10(a) and then the proposed encryption algorithm is applied on noisy data. The robustness of proposed work on noisy data is analyzed and the corresponding results are shown in figs. 10(b)-10(e). An incorrectly decrypted image is shown in fig. 10(b) when wrong fractional order of FrFT are used. Fig. 10(c) shows a wrongly decrypted image when subbands of MSVD are wrongly arranged. An incorrectly decrypted image using wrong arrangements of DWT subbands and incorrect values of 4 × 4 matrix are shown in figs. 10(d) and (e) respectively. However, a correctly decrypted image when all the encryption keys are correct is shown in fig. 10(f). It is found from our experiments that the proposed scheme is applicable on noisy data too. For an instance, when the proposed encryption algorithm is applied on noisy Barbara image then mean square error between input and correctly decrypted images for each of the R, G and B planes are obtained as: 14 Page 14 of 23
5.3131 × 10−30 , 4.4630 × 10−30 and 5.1443 × 10−30 respectively, which are quite similar to the MSE’s obtained in case of 8-bit images. Hence the proposed scheme is also applicable on other type of data. a
c
e
f
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d
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b
d
Figure 10: decrypted images (a) noisy image; (b) incorrect fractional order; (c) incorrect arrangement of MSVD subbands; (d) incorrect arrangement of DWT subbands obtained from MSVD subbands; (e)incorrect
b
c
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values of 4 × 4 matrix; (f) correctly decoded image.
Figure 11: Histogram analysis of figure 6 (a) Histogram of Barbara image; (b) Histogram of Encrypted Barbara (shown in fig. 6(b)) (c) Histogram of correctly decrypted Barbara.
4.1. Security analysis 4.1.1. Histogram analysis The histogram analysis of proposed work is discussed in this subsection. The distribution of intensity values in the three color plane namely; R, G and B is demonstrated by using 15 Page 15 of 23
Table 2: Mean Square Errors of existing and proposed schemes on various test images
4
5
6
Girl
Splash
7.5757 × 10−25
1.2020 × 10−29
9.2770 × 10−30
Green
5.7824 × 10−25
1.2046 × 10−29
5.6062 × 10−30
Blue
4.9245 × 10−25
8.4432 × 10−30
6.5952 × 10−30
Red
1.6581 × 10−24
4.5817 × 10−29
3.0951 × 10−29
Green
3.6301 × 10−25
7.5088 × 10−30
7.5336 × 10−30
Blue
3.6723 × 10−25
9.7892 × 10−30
8.9521 × 10−30
Red
5.4772 × 10−25
2.4168 × 10−29
2.2753 × 10−29
Green
2.5581 × 10−25
1.8396 × 10−29
8.1118 × 10−30
Blue
6.5808 × 10−25
7.3902 × 10−30
6.3833 × 10−30
Red
1.2662 × 10−24
2.2560 × 10−29
1.2661 × 10−29
Green
2.1288 × 10−24
9.9477 × 10−30
9.3511 × 10−30
Blue
8.0634 × 10−25
3.8857 × 10−30
3.1224 × 10−30
Red
2.3666 × 10−24
5.3887 × 10−30
4.2351 × 10−30
Green
1.4014 × 10−25
4.2230 × 10−30
1.8933 × 10−30
Blue
1.4152 × 10−25
2.5194 × 10−30
2.7763 × 10−30
1.587 × 10−24
5.3492 × 10−29
3.3456 × 10−29
1.3361 × 10−25
4.6212 × 10−29
4.0512 × 10−30
1.4581 × 10−25
4.4682 × 10−29
3.8679 × 10−30
Red Green
Ac ce p
Blue
ip t
Red
cr
Barbara
Peppers
Proposed Scheme
us
planes
Baboon
Scheme in [25]
an
3
Images
Lena
Scheme in [24]
M
2
RGB color
d
1
Input
te
S. no.
histogram. Fig. 11(a) shows the histogram of original Barbara, the histogram of encrypted Barbara using proposed technique is shown in fig. 11(b), while the histogram of correctly decrypted image is demonstrated in fig. 11(c). The histogram shown in fig. 11(b) is completely different from the histogram shown in fig. 11(a), which is an indication that the original information can not be revealed from the histogram of encrypted image. However, the histogram of correctly decrypted image (shown in fig.11(c)) is almost identical to the histogram of original image (shown in fig. 11(a)).
16 Page 16 of 23
Input Images
RGB color planes
SSIM
1
Barbara
Red
0.0288
Green
0.0430
Blue
0.0490
Red
0.0194
Green
0.0448
Blue
0.0475
Red
0.0237
3
Lena
Baboon
us
2
cr
S. no.
Green
0.0243
Blue Peppers
0.0286
Red
0.0280
an
4
Green
0.0400
Red
0.1280
Blue
0.1517
Red
0.0275
Green
0.0952
d
Splash
0.0884
Green
te
6
Girl
0.0811
M
Blue 5
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Table 3: SSIM between original and encrypted images of proposed scheme on various test images
0.0941
Ac ce p
Blue
4.2. Numerical Analysis
4.2.1. Mean Square Error (MSE)
′
Mean Square Error (MSE) between the original (I) and correctly decrypted images (I ) of size M × N can be defined as:
M −1 N −1 ∑ ∑ 1 M SE = [|I(m∆x, n∆y) − I ′ (m∆x, n∆y)|2 ] M × N i=0 j=0
(17)
where, ∆x and ∆y represent the pixel size. The quality of correctly decrypted image is verified using MSE. The proposed scheme is applied on several test images shown in fig. 5 and the MSE values for R, G and B planes of correctly decrypted images are given in table-2. It is evident from table-2 that the MSE values of proposed work are very small (almost zero), which indicates that our scheme can reconstruct the images more accurately with less information loss. 17 Page 17 of 23
4.2.2. Structural Similarity Index Metric (SSIM) To evaluate the quality of reconstructed images several measures are available such as: PSNR and MSE etc. Wang and Bovik [34] proposed an other method to evaluate the degradation in the quality of reconstructed images. Its values lie within the range of -1 to
ip t
1. If SSIM is 1, it indicates that the reconstructed image is identical to the original one. Structural Similarity of two images is calculated as: (2µI µI ′ + C1 )(2σII ′ + C2 ) + µ2I ′ + C1 )(σI2 + σI2′ + C2 )
cr
SSIM =
(µ2I
(18)
′
′
us
where, µI and µI ′ are the averages corresponding to images I and I respectively while, σI2 and σI2′ represent the variances of images I and I respectively. σII ′ is the covariance between ′
an
I and I images and C1 and C2 are the predefined constants. The SSIM values of various test images are shown in table-3. It is evident from table-3 that the proposed scheme gives
M
sufficient security as the SSIM values between original and encrypted images, for all the test images are much away from 1 (close to zero), However, the SSIM values between the original
te
is equal to 1.
d
and correctly decrypted images is calculated and for all the test images the calculated value
Ac ce p
5. Comparison of proposed work with some existing works This section is devoted to comparison of our work with some related works. We have compared our work with the work of Prasad et al.[24] and Chen et al.[25]. The use of FrFT, MSVD and DWT along with the use of U matrix in various subbands of DWT and Arnold transform have added extra layers of security when compared to [24] and [25]. As discussed in section 1 that the rearrangement step used in [24] doubles the size of encoded image which is not a good approach as it will take extra time and memory costs for transmission. On the other hand in [25], three encrypted images of dimensions equal to the original image are transmitted that will add an extra overhead of transmission of three encrypted images independently. Though, the schemes given in [24, 25] provides good security but scheme in [24] doubles the size of encoded image, while the scheme in [25] transmits three encrypted images of dimension equal to the original image, which is not appropriate for transmission purpose as it will increase the transmission time and cost. The proposed scheme overcomes 18 Page 18 of 23
the drawbacks of these existing schemes along with the increased security, by transmitting a single encrypted image of dimension equal to the original image. The MSE values of proposed and existing schemes on various test images are shown in table-2. It can be observed from table-2 that the proposed scheme can reconstruct the encrypted images with less information
ip t
loss as the MSE values between original and correctly decrypted images of proposed work are smaller than the MSE values of work given in [24] and [25]. The effectiveness of proposed
cr
work is also analyzed on incorrect key test and it is found that proposed work has obtained
us
an increased security on each step when compared to the existing. 6. Conclusions
an
In this paper, a color image encryption technique using MSVD and DWT along with Arnold transform in FrFT domain is proposed. The use of MSVD and DWT in FrFT domain
M
along with the values of 4 × 4 matrix in various subbands of DWT with the use of Arnold transform gives more secure information. For the correct decryption of encrypted images it
d
is necessary to have correct information of all the keys along with the way of arrangement. If any one of the parameter is wrong while the rest are exactly correct even thought it is
te
not possible to guess original image information, which shows that it is indeed necessary to
Ac ce p
have exact knowledge of all the keys in correct order. The comparison of proposed work with some existing works show that the proposed work can reconstruct the encrypted image more efficiently as the MSE values of correctly decrypted images for all the color planes using our approach are smaller than the MSE values of existing schemes. However, the structural similarity between the original and encrypted images shows that proposed scheme can transmit images more securely without revealing original information. References [1] P. Refregier, B. Javidi, Optical image encryption based on input plane and Fourier plane random encoding. Optics Letters 20(1995) 767–769. [2] G. Unnikrishnan, J. Joseph, K.Singh, Optical encryption by double-random phase encoding in the fractional Fourier domain. Optics Letters 25(2000) 887–889. 19 Page 19 of 23
[3] G. Situ, J. Zhang, Double random-phase encoding in the Fresnel domain. Optics Letters 29(2004) 1584. [4] B. Javidi, N. Takanori, Securing information by use of digital holography. Optics Letters
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25(2000) 28-30. [5] N. Singh, A. Sinha, Optical image encryption using Hartley transform and logistic map.
cr
Optics Communications 282(6) (2009) 1104–1109.
us
[6] L. Chen, D.Zhao, Optical image encryption with Hartley transforms, Optics Letters 31(2006) 3438-3440.
an
[7] Z. Liu, Y.Zhang, W.Liu, F.Meng, S.Liu, Optical color image hiding scheme based on chaotic mapping and Hartley transform, Opt. Lasers Eng. 51(2013) 967-972.
Lasers Eng. 47(2009) 539-546.
M
[8] N. Singh, A. Sinha, Gyrator transform-based optical image encryption using chaos, Opt.
d
[9] M. R. Abuturab, Color image security system using double random-structured phase
te
encoding in gyrator transform domain, Appl. Opt. 51(2012) 3006-3016.
Ac ce p
[10] M. R. Abuturab, Securing color information using Arnold transform in gyrator transform domain, Opt. Lasers Eng. 50(2012) 772-779. [11] M. R. Abuturab, Securing color image using discrete cosine transform in gyrator transform domain structured-phase encoding, Opt. Lasers Eng. 50(2013) 1383-1390. [12] M.R. Abuturab, Color image security system based on discrete Hartley transform in gyrator transform domain, Opt. Lasers Eng. 51(2013) 317-324. [13] M. R. Abuturab, Noise-free recovery of color information using a joint-extended gyrator transform correlator, Opt. Lasers Eng. 51(2013) 230-239. [14] M. R. Abuturab, Color information verification system based on singular value decomposition in gyrator transform domains, Optics and Lasers in Engineering 57 (2014) 13-19 20 Page 20 of 23
[15] Q. Guo, Z. Liu, S. Liu, Color image encryption by using Arnold and discrete fractional random transform in IHS space, Opt. Lasers Eng. 48(2010) 1174-1181. [16] Z. Liu, L. Xu, T. Liu, H. Chen, P. Li, S.Liu, Color image encryption by using Arnold
ip t
transform and color-blend operation in Discrete Cosine Transform domains, Opt. Commun. 284(2011) 123-128.
cr
[17] Z. Liu, M. Gong, Y. Dou, F. Liu, S. Lin, M. A. Ahmad, et al., Double image encryption by using Arnold transform and discrete fractional angular transform, Opt. Lasers
us
Eng.50(2012) 248-255.
an
[18] Z. Liu, S. Liu, Random fractional Fourier transform, Opt. Lett. 32(2007) 2088-2090. [19] S. Liu, Q. Mi, B. Zhu,Optical image encryption with multistage and multichannel frac-
M
tional Fourier-domain filtering, Opt. Lett. 26(2001) 1242-1244. [20] Y. Zhang, C. H. Zheng, N. Tanno, Optical encryption based on iterative fractional
d
Fourier transform, Opt. Commun. 202(2002) 277-285.
Ac ce p
119(2008) 286-291.
te
[21] L. Chen, D. Zhao, Image encryption with fractional wavelet packet method, Optik
[22] L. Chen, D. Zhao, Color image encoding in dual fractional Fourier-wavelet domain with random phases, Opt. Commun. 282(2009) 3433-3438. [23] M. Singh, A. Kumar, K. Singh, Encryption by using matrix-added, or matrix-multiplied input images placed in the input plane of a double random phase encoding geometry, Opt. Laser Eng. 47(2009) 1293-1300. [24] A. Prasad, M. Kumar, D. R. Choudhury, Color image encoding using fractional Fourier transformation associated with wavelet transformation, Opt. Commun. 285 (2012) 10051009. [25] L. Chen, D. Zhao, F. Ge, Image encryption based on singular value decomposition and Arnold transform in fractional domain, Opt. Commun. 291(2013) 98-103.
21 Page 21 of 23
[26] M. Kumar, S. Agrawal, Color image encoding in DOST domain using DWT and SVD, Optics and Laser Technology. 75(2015) 138-145. [27] M. Kumar, A. Vaish, Encryption of color images using MSVD in DCST domain. Optics
ip t
and laser in engineering, 88 (2017) 51- 59. [28] John Ladan, An Analysis of Stockwell Transforms, with Applications to Image Process-
cr
ing,(Ph.D. thesis), University of Waterloo, Ontario, Canada (2014).
us
[29] Y.Wang, Efficient Stockwell Transform with applications to image processing (Ph.D.thesis), University of Waterloo, Ontario, Canada, (2011).
an
[30] R.G. Stockwell, A basis for efficient representation of the S-transform, Digit. Signal Process. 17(2007) 371393.
M
[31] J. Ladan and Edward R. Vrscay, The Discrete Orthonormal Stockwell Transform and Variations, with Applications to Image Compression, ICIAR 2013, LNCS 7950(2013)
d
235-244,.
te
[32] J. L. Kamm, SVD-based methods for signal and image restoration (Ph.D.thesis), 1998.
Ac ce p
[33] S. Malini, R. S. Moni, Image Denoising Using Multiresolution Singular value Decomposition Transform, ICICT 46(2015) 1708- 1715. [34] Z. Wang, A. C. Bovik, H. R. Sheikh, E. P. Simoncelli, Image quality assessment: from error visibility to structural similarity, IEEE Trans. Image Process. 13(4)(2004) 600–612. [35] A. W. Lohmann, Image rotation, Wigner rotation, and the fractional Fourier transform, Journal of the Optical Society of America 10(10) (1993) 2181-2186. [36] D. Mendlovic, H.M.Ozaktas, Fractional Fourier transformations and their optical implementation: Part I, Journal of the Optical Society of America 10(10) (1993) 1875-1881 [37] A.W.Lohmann, D. Mendlovic, Z. Zalevsky, Progress in Optics 38 (1998) 263342.
22 Page 22 of 23
[38] Z. J. Liu, L. Xu, T. Liu, H.Chen, P. F. Li, C. Lin, S. T. Liu, Color image encryption by using Arnold transform and color-blend operation in discrete cosine transform domains, Optics Communications 284(1) (2011) 123128.
ip t
[39] W. Chen, C. Quan, C. J. Tay, Optical color image encryption based on Arnold transform and interference method, Optics Communications 282 (2009) 3680-3685.
cr
[40] X. P. Deng, D. M. Zhao, Color component 3D Arnold transform for polychromatic
us
pattern recognition, Optics Communications 284 (2011) 56235629.
[41] Z. J. Liu, M. Gong, Y. K. Dou, F. Liu, S. Lin, M. A. Ahmad, J. M. Dai, S. T. Liu,
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Double image encryption by using Arnold transform and discrete fractional angular transform, Optics and Lasers in Engineering 50 (2012) 248-255.
M
[42] R.C. Gonzalez, R. E. Woods, Digital Image Processing, third edition, Pearson Education, New Delhi, India, 2013.
d
[43] D. Mendlovic, N.Konforti, Optical realization of the wavelet transform for two-
Ac ce p
te
dimensional objects, Appl. Opt. 32 (32) (1993) 65426546.
23 Page 23 of 23
S0030-4026(17)30858-6 http://dx.doi.org/doi:10.1016/j.ijleo.2017.07.041 IJLEO 59437
To appear in: Received date: Revised date: Accepted date:
11-10-2016 11-5-2017 16-7-2017
Please cite this article as: Ankita Vaish, Manoj Kumar, Color image encryption using MSVD, DWT and Arnold transform in fractional Fourier domain, (2017), http://dx.doi.org/10.1016/j.ijleo.2017.07.041 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Color image encryption using MSVD, DWT and Arnold transform in fractional Fourier domain Manoj Kumar1 , Ankita Vaish2∗ Babasaheb Bhimrao Ambedkar University, Lucknow, (U.P.), India.
cr
ip t
1, 2
Abstract
us
This paper presents, a new image encryption technique using Multiresolution Singular Value Decomposition (MSVD), Discrete Wavelet Transform (DWT) and Arnold transform in frac-
an
tional domain. Images are encrypted using keys of FrFT, subbands of MSVD and DWT along with the use of parameters generated by MSVD in various DWT subbands and Arnold
M
transform. We have used keys of FrFT, the arrangement of MSVD and wavelet subbands, values and arrangement of parameters generated by MSVD in various DWT subbands, the use of Arnold transform and the way of arrangement of several subimages as encryption and
d
decryption keys. For the correct decryption of encrypted images it is required to have correct
te
knowledge of all the keys in correct order along with their exact values. The effectiveness of proposed work is analyzed by comparing it with some related works, the comparison verifies
Ac ce p
that the proposed work has obtained an increased level of security and robustness when compared to exiting works.
Keywords: Arnold transform, Image encryption, Discrete wavelet transform, Fractional fourier transform, Multiresolution singular value decomposition. 1. Introduction
With the ever growing field of multimedia applications, a lot of information is being transferred all around the world over the public networks. However, direct transmission of the secret information over the public networks is not preferable due to security reasons. ∗
Corresponding author. URL: [email protected] (Ankita Vaish2 ) 1 Assistant Professor, Babasaheb Bhimrao Ambedkar University, Lucknow, (U.P.), India. 2 Research Scholar, Babasaheb Bhimrao Ambedkar University, Lucknow, (U.P.), India.
Preprint submitted to Optik
May 11, 2017 Page 1 of 23
Considerable work has been done by the researchers to keep the information secure from unauthorized users. Optical encryption techniques are gaining a lot attention in information security. Since Refregier et al. [1] first proposed an optical encryption technique based on double random phase, many techniques for secure transmission of sensitive images have been
ip t
proposed based on digital holography (DH) [4], Hartley transform (HT) [5, 6, 7], Gyrator
Fourier transform (FrFT) [18, 19, 20, 21, 22, 23, 24, 25].
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transform (GT) [8, 9, 11, 10, 12, 13], Arnold Transform (AT) [15, 16, 17] and Fractional
Several techniques are reported in literature based on Hartley transform viz. Singh et
us
al. [5] proposed an image encryption technique using logistic map. An image encryption technique in Hartley transform domain is proposed by Chen et al. [6] which is based on
an
interferometer. A color image encryption technique using chaotic map in Hartley transform domain is given in [7] where baker mapping is used to scramble each of the color plane of
M
the image.
Many color image encryption techniques are given in literature based on Gyrator transform such as: Singh et al. [8] proposed a chaos based image encryption technique in Gyrator
d
transform. Plenty of work have been proposed by Abuturab [9, 11, 10, 12, 13] in Gyrator
te
transform domain such as Abuturab [9] proposed an image encoding technique using double random phase. Again, a new encryption technique using Arnold transform is given by
Ac ce p
Abuturab [10]. Another technique based on Hartley transform in Gyrator transform domain is introduced by Abuturab [12]. More recently, Abuturab [14] proposed a color image encryption technique using Singular value decomposition in Gyrator transform domain. A lot of work has been reported in literature for image encryption using Arnold transform, it scrambles the intensity values of an image which results to an image in unrecognizable form. A color image encryption technique in Intensity (I), Hue (H) and Saturation (S) color space using Arnold and Discrete Fractional Random Transforms is given by Guo et al. [15], in this work I component is encrypted using discrete fractional random transform and H and S components are encrypted by using Arnold transform. Liu et al. [16] proposed an image encryption technique using Arnold transform and discrete cosine transform. Again a color image encryption technique using Arnold and discrete angular transforms is given by Liu et al. [17]. 2 Page 2 of 23
Various techniques are proposed in literature that have used Fractional fourier transform (FrFT) for image encoding, FrFT is the generalization of conventional Fourier transform. Further, a random FrFT is given by Liu et al. [18], which is obtained by randomizing the conventional FrFT. Liu et al [19] proposed an image encryption technique using multi
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channel and multi stage fractional Fourier domain filtering. An iterative fractional Fourier transform based image encryption technique is proposed by Zhang et al. [20]. Chen et al. [21]
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introduced a fractional wavelet packet based color image encoding technique. A new color image encoding technique using random phases in dual fractional Fourier-wavelet domain
us
is proposed by Chen et al. [22]. Prasad et al. [24] proposed a color image encoding using DWT in FrFT domain, this work provides good security but doubles the size of encrypted
an
image that is not a suitable choice from transmission point of view. Recently, a color image encryption technique is introduced by Chen et al. [25] using SVD and Arnold transform
M
in FrFT domain, this work has obtained enough security but transmits three encrypted images of dimension equal to the original image and at the time of decryption all the three encrypted images must be required for correct decryption. More recently, Kumar et al.
d
[26, 27] introduced a new color image encoding technique using DWT and SVD in Discrete
te
Cosine Stockwell transform domain. Though, the works given in [24, 25] provide good security, but the arrangement used in [24] doubles the size of encoded image while the work
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introduced in [25] has increased the transmission cost by transmitting three encrypted images of dimension equal to the original image, which is not appropriate from transmission point of view as extra space and time costs is required. However, in our work an attempt has been made to provide a more secure encryption technique by overcoming the disadvantages of [24, 25].
In this paper, a new color image encryption technique using MSVD, DWT and Arnold transform in FrFT domain is proposed. The purpose of this work is to propose a more secure encryption technique for secure transmission over unsecure networks. An original color image is first divided into its primary color components i.e. Red (R), Green (G) and Blue (B). Further each of the component is encrypted independently using DWT, MSVD and Arnold transform in fractional domain. The keys of FrFT, arrangement of MSVD and DWT subbands, values and arrangement of values of 4 × 4 matrix, generated by MSVD in DWT 3 Page 3 of 23
subbands, the use of Arnold transform and the way of arrangement of several subimages are termed as encryption and decryption keys. At the time of decryption, correct knowledge of all the keys along with the way of arrangement are necessary for correct decryption of encrypted images.
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This paper is organized as follows: some key terms related to the proposed work are given in section 2, the proposed encryption and decryption technique is introduced in section
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3, section 4 addresses the experimental results and analysis of proposed work with some standard test images, comparison of proposed work with some recently published works is
us
discussed in section 5, finally, conclusions are drawn in section 6.
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2. Related theories 2.1. Fractional Fourier Transform (FrFT)
M
FrFT is a generalized version of Fourier transform with parameter α, which is used most frequently in optical and digital information processing. It is used in many areas of image
d
processing such as optical image encryption, watermarking [35, 36, 37]. In mathematics,
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FrFT is known for more than last 70 years, but the definition of FrFT in optics is given by Mendlovic and Ozaktas in 1993. It is based on the propagation of light within a medium
Ac ce p
with continuously varying refractive index [36, 37]. The 2D-FrFT with parameters α1 and α2 of an image f (x, y) is represented by Fα1 ,α2 (f (x, y))(u, v) is given as: ∫ ∫ i(x2 +u2 ) cot α1 1 √ −ixu csc α1 2 Fα1 ,α2 (f (x, y))(u, v) = (1 − i cot α1 )(1 − i cot α2 ) e 2Π R R ×e
i(y 2 +v 2 ) cot α2 −ixu 2
csc α2
f (x, y)dxdy
(1)
As the FrFT has property of linearity and continuity, hence the inverse of FrFT can be obtained with corresponding negative fractional orders. 2.2. Multiresolution Singular Value Decomposition (MSVD) For a given matrix A of size M × N , Singular value decomposition (SVD) [32] can be written by using equation: A = U SV T , where U and V are orthogonal matrices of size M ×M and N × N respectively and S represents a diagonal matrix containing sorted singular values in descending order. 4 Page 4 of 23
MSVD [33] of an image (I) of size M × N can be obtained by first reshaping I to a matrix I1 of size 4 × M N/4. SVD of I1 is calculated as: [U, S] = SV D(I1 )
(2)
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where, U is of size 4 × 4 and S is 4 × M N/4 now, matrix T is calculated as:
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T = U T I1
(3)
us
where, T is a matrix of size [4 × M N/4]. The first, second, third and fourth rows of matrix T are reshaped to obtain matrices of size M/2 × M/2, the reshaped matrices are called LL,
an
LH, HL and HH subbands. The low frequency details are in LL subband, while the LH, HL and HH subbands contain the high frequency details. These subbands (LL, LH, HL
M
and HH) and matrix U of size 4 × 4 is used in our work for as encryption keys. 2.3. Discrete Wavelet Transform (DWT)
d
Discrete Wavelet Transform (DWT) has been used more frequently in many applications of image processing like image compression, image encryption, watermarking etc. DWT
te
is localized both in time and frequency domains, which reveal the spatial and frequency
Ac ce p
views at the same time. It decomposes the image at multiresolution level, which helps in analyzing an image at different resolutions. Due to the multiresolution property of DWT, the information that seems to be unnoticed at one level may become noticeable at another level.
The 2-D DWT can be obtained by first applying 1-D DWT in horizontal direction and then in vertical direction. 2-D DWT decomposes an image into four parts: approximation (A), horizontal (H), vertical (V ) and diagonal (D). Approximation part contains low frequency subband, while the horizontal, vertical and diagonal parts contain the high frequency subbands. For an image of size M × N , 2-D DWT can be defined as [42]: M −1 N −1 1 ∑∑ WΦ (j0 , m, n) = f (x, y)Φj0 ,m,n (x, y) M N x=0 y=0
(4)
5 Page 5 of 23
WΨi (j, m, n)
M −1 N −1 1 ∑∑ = f (x, y)Ψij,m,n (x, y) M N x=0 y=0
(5)
where, WΦ (j0 , m, n) represents approximation part and WΨi (j, m, n) represents horizontal,
The inverse of 2-D DWT can be defined as:
∞ ∑∑ ∑ ∑ i=H,V,D j=j0 m
WΨi (j, m, n)Ψij,m,n (x, y)
(6)
us
1 MN
1 ∑∑ WΦ (j0 , m, n)Φj0 ,m,n (x, y) + MN m n
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f (x, y) =
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vertical and diagonal parts of image f (x, y).
n
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In 1993, Mendlovic and Konforti gives the optical realization of 2D- DWT [43]. Optically, it can be obtained using conventional coherent correlator along with multi reference matched
M
filter.
Table 1: Arnold transform period (T) Vs. varying values of (N)
3
4
T
3
4
3
N
32
64
T
24
48
T
6
7
8
9
10
10
12
8
6
12
30
128
256
512
1024
2048
4096
8192
96
192
384
768
1536
3072
6144
Ac ce p
N
5
d
2
te
N
33
65
100
257
513
1025
2049
4097
8193
20
70
150
258
1432
100
684
360
780
2.4. Arnold Transform
Arnold transform changes the positions of pixels rather than masking their values. It is used most frequently in image encryption and watermarking [38, 39, 40, 41]. It can be
defined as:
where,
x
′
′
x y
′
′
=
1 1
1 2
x y
mod N
(7)
represents the position vector of an image after shifting while
x
rep-
y y resents the position vector before shifting and mod shows the modulus after division with 6 Page 6 of 23
N . The parameter N that is used over here is the size of input image, which determines the period of Arnold transform. Table-1 shows the period of transform vs. the values of N . Arnold transform is applied on an original image shown in fig. 1(a), which consists 512 × 512 pixels, from the table-1 it can be seen that its period is 384. Fig. 1(b) shows the
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result after 50 times Arnold transform and fig. 1(c) shows the result after 100 times Arnold transform while the fig. 1(d) shows the image after 384 times Arnold transform. Hence, the
cr
results shown in fig. 1 validates the period property of Arnold transform shown in table-1. The original image information can be recovered using period property but as the value
us
of N becomes larger, the period will also become larger and it will take more time to recover
2 −1
a
-1 1
c
(8)
1
1
1
2
.
d
Ac ce p
te
b
is used instead of A =
d
where, A−1 =
M
an
the original information. Therefore, an inverse of Arnold transform is defined as: ′ x 2 -1 x = mod N y′ −1 1 y
Figure 1: (a) Original image; (b) 50 times Arnold transform; (c) 100 times Arnold transform; (d) after 384 times Arnold transform.
An original Lena image is shown in fig. 2(a), fig. 2(b) shows the result after one time Arnold transform, while fig. 2(c) results from inverse of Arnold transform. It can be analyzed from figures 1 and 2 that the original image information can be recovered from both the period and inverse Arnold transform. The histogram corresponding to figures 2(a) and 2(b) are shown in figures 3(a) and 3(b) respectively. Hence, it can be analyzed from figures 3(a) and 3(b) that the Arnold transform does not masks the values of original image.
7 Page 7 of 23
a
c
cr
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b
b
M
an
a
us
Figure 2: (a) Original image; (b) after one time Arnold transform; (c) inverse Arnold transform.
d
Figure 3: Histogram of (a) Original image; (b) after Arnold transform.
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3. Proposed Encryption and Decryption Scheme
Ac ce p
The detailed description of proposed encryption and decryption scheme is shown in figs 4(a) and (b) respectively. In the encryption procedure, at first an original colored image of size M × N is decomposed into three primary color components i.e Red (R), Green (G) and Blue (B). FrFT is applied on each of the R, G and B planes independently, and then MSVD is applied on all of the three planes, which results LL, LH, HL and HH subbands for each plane. Again, 2-D DWT is applied on each of the MSVD subbands i.e. LL, LH, HL and HH subbands, which further results four subbands: A, H, V and D corresponding to each subbands of the MSVD (i.e. LL, LH, HL and HH) for all planes. Now the values of each of the subband obtained from DWT are masked by the the values of U matrix (generated by MSVD (shown in subsection 2.2)) as:
8 Page 8 of 23
a
U4* 4
U(1,1) *ALL ALL VLL
A'LL
U(1,2)*H LL
H'LL
HLL DLL
U(1,3)*V LL
V'LL
V''LL
U(1,4)*DLL
D'LL
D''LL
H'LH
HLH DLH
U(2,3)*V LH
V'LH
V''LH
U(2,4)*D LH
D'LH
D''LH
U(3,1)*A HL
A'HL
A''HL
AHL V HL
U(3,2)*H HL
H'HL
HHL DHL
U(3,3)*VHL
V'HL
V''HL
U(3,4)*DHL
D'HL
D''HL
U LH
HH
H''LH IDWT
H''HL IDWT
U(4,1)*A HH
A'HH
U(4,2)*H HH
H'HH
H''HH
U(4,3)*V HH
V'HH
V''HH
U(4,4)*D HH
D'HH
D''HH
AT
E3
IDWT
E4
Encrypted Image
M
HHH DHH
E2
A''HH
an
AHH VHH
AT
A''LH
cr
FRFT
DWT
AT
ip t
A'LH
U(2,2)*H LH
Arrangemen of subimages in order E4, E2, E1 and E3
U(2,1)*A LH
E1
us
Original image
HL
H''LL IDWT
ALH VLH MSVD LL
A''LL AT
U4* 4
Encrypted Image
E2
E3
E4
d A'LL
A'LL/U (1,1)
ALL
H'LL
H'LL/U(1,2)
HLL
V'LL
V'LL/U(1,3)
VLL
D'LL
D'LL/U(1,4)
DLL
A'LH
A'LH/U(2,1)
ALH
H'LH
H'LH/U(2,2)
HLH
V'LH
V'LH/U(2,3)
VLH
D'LH
D'LH/U(2,4)
DLH
A'HL
A'HL/U(3,1)
AHL
H'HL
H'HL/U(3,2)
HHL
V'HL
V'HL/U(3,3)
VHL
D'HL
D'HL/U(3,4)
DHL
te
E1
DWT A''LL V''LL IAT H''LL D''LL
DWT A'' LH V''LH IAT
Ac ce p
Decompostion into various subimages according to their indexing
b
H''LH D''LH
DWT A'' V'' IAT HL HL H''HL D''HL
DWT A''HH V''HH IAT H''HH D''HH
A'HH
A'HH/U(4,1)
AHH
H'HH
H'HH/U(4,2)
HHH
V'HH
V'HH/U(4,3)
VHH
D'HH
D'HH/U (4,4)
DHH
IDWT
LL
IDWT
LH IMSVD
IFRFT
Decrypted Image
IDWT
HL
IDWT
HH
Figure 4: Proposed (a) Encryption scheme (b) Decryption scheme.
9 Page 9 of 23
(9)
Hl′ (i, j) = U (v, 2) × Hl (i, j)
(10)
Vl′ (i, j) = U (v, 3) × Vl (i, j)
(11)
where, v = 1, 2, 3, 4 and 1 ≤ i ≤
M 4
and 1 ≤ j ≤
N , 4
(12)
cr
Dl′ (i, j) = U (v, 4) × Dl (i, j)
ip t
A′l (i, j) = U (v, 1) × Al (i, j)
suffix l used here corresponding to
us
LL, LH, HL and HH subbands of MSVD. The use of values of U matrix in various subimages adds an extra layer of security. All the modified subbands are Arnold transformed
an
and results to respective subimages i. e. A′′l , Hl′′ , Vl′′ and Dl′′ for all the DWT subbands for all planes. Inverse of DWT is applied on Arnold transformed subimages, which results to subimages E1 , E2 , E3 and E4 . Further, all the subimages obtained from inverse of DWT
E(2i, 2j) = E1 (i, j)
(13)
E(2i − 1, 2j) = E2 (i, j)
(14)
E(2i, 2j − 1) = E3 (i, j)
(15)
E(2i − 1, 2j − 1) = E4 (i, j)
(16)
Ac ce p
te
d
M
are arranged according to their indexing as:
where, E represents the final encrypted image of size M ×N . We have used keys of FrFT, arrangement of MSVD and DWT sub-bands, values of 4×4 matrix and its use in various DWT subbands, arrangement of various subimages obtained from inverse of DWT as encryption and decryption keys.
The procedure for decryption is just reverse of encryption. Fig. 4(b) shows the decryption process of proposed work. All the decrypted R, G and B planes are further combined to get the reconstructed color image. 4. Experimental Results and Security Analysis The efficiency of proposed work is evaluated by testing it on various standard test images each of size 512 × 512. The various standard test images used in the proposed work are 10 Page 10 of 23
c
d
e
f
ip t
b
us
cr
a
an
Figure 5: (a) Barbara; (b) Lena; (c) Baboon; (d) Peppers; (e) Girl; (f) Splash.
shown in fig. 5. An original image of Barbara and its encrypted images are shown in figures
M
6(a) and (b) respectively with fractional orders 0.4 and 0.33, arrangement of values of 4 × 4 matrix obtained from MSVD in various DWT subbands, use of an Arnold transform and
d
arrangement of wavelet subbands in order D, V, A and H (when applying inverse DWT in
te
encryption phase). The correctly decrypted image with all the correct keys along with the correct arrangement is shown in fig. 6(c).
b
Ac ce p
a
c
Figure 6: (a) an original image; (b) encrypted image; (c) correctly decrypted image.
The robustness of proposed work with the change of each and every keys is demonstrated in fig. 7. We have used Barbara image to demonstrate the robustness of proposed work. Fig. 7(a) shows an incorrectly decrypted Barbara image using incorrect fractional orders 11 Page 11 of 23
b
c
e
f
an
us
cr
d
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a
Figure 7: decrypted images (a) incorrect fractional order; (b) incorrect arrangement of MSVD subbands;
M
(c) incorrect arrangement of DWT subbands obtained from MSVD subbands; (d) incorrect values of 4 × 4 matrix; (e) without division of values of U matrix; (f) incorrect decomposition of final encrypted image.
d
(incorrect values are -0.98, -1.48) while the rest of the parameters are correct. An incorrectly
te
decrypted image using incorrect arrangement of MSVD subbands is shown in fig. 7(b). Figs. 7(c) and (d) show the incorrectly decrypted image using incorrect arrangement of
Ac ce p
DWT subbands and incorrect values of 4 × 4 matrix respectively. An incorrectly decrypted image is shown in fig. 7(e) when the values of 4 × 4 matrix are not used for division at the time of decryption. Fig. 7(f) demonstrate an incorrectly decrypted image when the subimages E1 , E2 , E3 and E4 are incorrectly arranged (incorrect way used is E4 E2 , E1 , and E3 ). It can be analyzed from the various incorrectly decrypted images shown in fig. 7 that it is nearly impossible to even guess the original image. Hence, the proposes scheme has obtained a desired level of security as if any one of the parameter get wrong (either values or arrangement) at the time of decryption even though the reconstructed image is not recognizable. The security analysis of proposed work on combination of two incorrect keys is given in fig. 8. An incorrectly decrypted image is shown in fig. 8(a) when incorrect fractional order along with incorrect arrangement of MSVD subbands are used. Fig. 8(b) shows a wrongly decrypted image when way of arrangements of MSVD and DWT subbands are incorrect. A 12 Page 12 of 23
b
c
e
f
an
us
cr
d
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a
Figure 8: decrypted images (a) incorrect fractional order with wrong arrangements MSVD subbands; (b) incorrect arrangement of MSVD and DWT subbands; (c) incorrect arrangement of DWT along with incorrect
M
values of 4 × 4 matrix; (d) incorrect values of 4 × 4 matrix with incorrect decomposition of final encrypted image;(e) incorrect decomposition and without division of values of U matrix; (f) incorrect arrangement of
d
MSVD subbands with incorrect values of 4 × 4 matrix.
te
wrongly decrypted image is shown on fig. 8(c) when arrangement of DWT subbands and the values of 4 × 4 matrix are wrong. Fig. 8(d) shows an incorrectly decrypted image when
Ac ce p
values of 4 × 4 matrix and subimages E1 , E2 , E3 and E4 are incorrectly arranged. An incorrectly decoded image is shown in fig. 8(e) when subimages E1 , E2 , E3 and E4 are incorrectly arranged and values of 4 × 4 matrix are not used in division. A wrongly decoded image is shown in fig. 8(f) when way of arrangements of MSVD subbands and values of 4 × 4 matrix are incorrect. It can be observed from the several images shown in fig. 8 that the proposed scheme has maintained the image security even if the two parameters are wrong. Further, we have performed the incorrect key test when three parameters are wrong and the respective results are shown in fig. 9. A wrongly decrypted image with incorrect fractional order along with incorrect arrangements of MSVD and DWT subbands is shown in fig. 9(a). Fig. 9(b) shows an incorrectly decrypted image with incorrect way of arrangements of MSVD and DWT subbands along with wrong arrangement of subimages E1 , E2 , E3 and E4 . An incorrectly decrypted image with wrong way of arrangements of MSVD and DWT 13 Page 13 of 23
b
c
ip t
a
cr
Figure 9: decrypted images (a) incorrect fractional order with incorrect arrangement of MSVD and DWT subbands; (b) incorrect arrangement of MSVD, DWT subbands with incorrect decomposition of final en-
us
crypted image; (c) incorrect arrangement of MSVD, DWT subbands with incorrect values of 4 × 4 matrix.
subbands along with wrong values of 4 × 4 matrix is shown in fig. 9(c). Although, we
an
have tested the robustness of proposed work on the change of one, two and three incorrect parameters while the rest are exactly correct and it is obvious that if the proposed work
M
can provide secure information on one, two and three incorrect parameters then it would be more difficult to recognize the incorrectly reconstructed image when more than three
over the networks.
d
parameters are wrong. Hence, the proposed work can transmit the images more securely
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Further, we have also analyzed the effectiveness of proposed work on some other type of data such as noisy data, which are not in 8 bit. To accomplish this, first of all, we have added
Ac ce p
Gaussian noise (standard deviation 35) to the original source image (shown in fig. 5(a)), the noisy image thus obtained is shown in fig. 10(a) and then the proposed encryption algorithm is applied on noisy data. The robustness of proposed work on noisy data is analyzed and the corresponding results are shown in figs. 10(b)-10(e). An incorrectly decrypted image is shown in fig. 10(b) when wrong fractional order of FrFT are used. Fig. 10(c) shows a wrongly decrypted image when subbands of MSVD are wrongly arranged. An incorrectly decrypted image using wrong arrangements of DWT subbands and incorrect values of 4 × 4 matrix are shown in figs. 10(d) and (e) respectively. However, a correctly decrypted image when all the encryption keys are correct is shown in fig. 10(f). It is found from our experiments that the proposed scheme is applicable on noisy data too. For an instance, when the proposed encryption algorithm is applied on noisy Barbara image then mean square error between input and correctly decrypted images for each of the R, G and B planes are obtained as: 14 Page 14 of 23
5.3131 × 10−30 , 4.4630 × 10−30 and 5.1443 × 10−30 respectively, which are quite similar to the MSE’s obtained in case of 8-bit images. Hence the proposed scheme is also applicable on other type of data. a
c
e
f
M
an
d
us
cr
ip t
b
d
Figure 10: decrypted images (a) noisy image; (b) incorrect fractional order; (c) incorrect arrangement of MSVD subbands; (d) incorrect arrangement of DWT subbands obtained from MSVD subbands; (e)incorrect
b
c
Ac ce p
a
te
values of 4 × 4 matrix; (f) correctly decoded image.
Figure 11: Histogram analysis of figure 6 (a) Histogram of Barbara image; (b) Histogram of Encrypted Barbara (shown in fig. 6(b)) (c) Histogram of correctly decrypted Barbara.
4.1. Security analysis 4.1.1. Histogram analysis The histogram analysis of proposed work is discussed in this subsection. The distribution of intensity values in the three color plane namely; R, G and B is demonstrated by using 15 Page 15 of 23
Table 2: Mean Square Errors of existing and proposed schemes on various test images
4
5
6
Girl
Splash
7.5757 × 10−25
1.2020 × 10−29
9.2770 × 10−30
Green
5.7824 × 10−25
1.2046 × 10−29
5.6062 × 10−30
Blue
4.9245 × 10−25
8.4432 × 10−30
6.5952 × 10−30
Red
1.6581 × 10−24
4.5817 × 10−29
3.0951 × 10−29
Green
3.6301 × 10−25
7.5088 × 10−30
7.5336 × 10−30
Blue
3.6723 × 10−25
9.7892 × 10−30
8.9521 × 10−30
Red
5.4772 × 10−25
2.4168 × 10−29
2.2753 × 10−29
Green
2.5581 × 10−25
1.8396 × 10−29
8.1118 × 10−30
Blue
6.5808 × 10−25
7.3902 × 10−30
6.3833 × 10−30
Red
1.2662 × 10−24
2.2560 × 10−29
1.2661 × 10−29
Green
2.1288 × 10−24
9.9477 × 10−30
9.3511 × 10−30
Blue
8.0634 × 10−25
3.8857 × 10−30
3.1224 × 10−30
Red
2.3666 × 10−24
5.3887 × 10−30
4.2351 × 10−30
Green
1.4014 × 10−25
4.2230 × 10−30
1.8933 × 10−30
Blue
1.4152 × 10−25
2.5194 × 10−30
2.7763 × 10−30
1.587 × 10−24
5.3492 × 10−29
3.3456 × 10−29
1.3361 × 10−25
4.6212 × 10−29
4.0512 × 10−30
1.4581 × 10−25
4.4682 × 10−29
3.8679 × 10−30
Red Green
Ac ce p
Blue
ip t
Red
cr
Barbara
Peppers
Proposed Scheme
us
planes
Baboon
Scheme in [25]
an
3
Images
Lena
Scheme in [24]
M
2
RGB color
d
1
Input
te
S. no.
histogram. Fig. 11(a) shows the histogram of original Barbara, the histogram of encrypted Barbara using proposed technique is shown in fig. 11(b), while the histogram of correctly decrypted image is demonstrated in fig. 11(c). The histogram shown in fig. 11(b) is completely different from the histogram shown in fig. 11(a), which is an indication that the original information can not be revealed from the histogram of encrypted image. However, the histogram of correctly decrypted image (shown in fig.11(c)) is almost identical to the histogram of original image (shown in fig. 11(a)).
16 Page 16 of 23
Input Images
RGB color planes
SSIM
1
Barbara
Red
0.0288
Green
0.0430
Blue
0.0490
Red
0.0194
Green
0.0448
Blue
0.0475
Red
0.0237
3
Lena
Baboon
us
2
cr
S. no.
Green
0.0243
Blue Peppers
0.0286
Red
0.0280
an
4
Green
0.0400
Red
0.1280
Blue
0.1517
Red
0.0275
Green
0.0952
d
Splash
0.0884
Green
te
6
Girl
0.0811
M
Blue 5
ip t
Table 3: SSIM between original and encrypted images of proposed scheme on various test images
0.0941
Ac ce p
Blue
4.2. Numerical Analysis
4.2.1. Mean Square Error (MSE)
′
Mean Square Error (MSE) between the original (I) and correctly decrypted images (I ) of size M × N can be defined as:
M −1 N −1 ∑ ∑ 1 M SE = [|I(m∆x, n∆y) − I ′ (m∆x, n∆y)|2 ] M × N i=0 j=0
(17)
where, ∆x and ∆y represent the pixel size. The quality of correctly decrypted image is verified using MSE. The proposed scheme is applied on several test images shown in fig. 5 and the MSE values for R, G and B planes of correctly decrypted images are given in table-2. It is evident from table-2 that the MSE values of proposed work are very small (almost zero), which indicates that our scheme can reconstruct the images more accurately with less information loss. 17 Page 17 of 23
4.2.2. Structural Similarity Index Metric (SSIM) To evaluate the quality of reconstructed images several measures are available such as: PSNR and MSE etc. Wang and Bovik [34] proposed an other method to evaluate the degradation in the quality of reconstructed images. Its values lie within the range of -1 to
ip t
1. If SSIM is 1, it indicates that the reconstructed image is identical to the original one. Structural Similarity of two images is calculated as: (2µI µI ′ + C1 )(2σII ′ + C2 ) + µ2I ′ + C1 )(σI2 + σI2′ + C2 )
cr
SSIM =
(µ2I
(18)
′
′
us
where, µI and µI ′ are the averages corresponding to images I and I respectively while, σI2 and σI2′ represent the variances of images I and I respectively. σII ′ is the covariance between ′
an
I and I images and C1 and C2 are the predefined constants. The SSIM values of various test images are shown in table-3. It is evident from table-3 that the proposed scheme gives
M
sufficient security as the SSIM values between original and encrypted images, for all the test images are much away from 1 (close to zero), However, the SSIM values between the original
te
is equal to 1.
d
and correctly decrypted images is calculated and for all the test images the calculated value
Ac ce p
5. Comparison of proposed work with some existing works This section is devoted to comparison of our work with some related works. We have compared our work with the work of Prasad et al.[24] and Chen et al.[25]. The use of FrFT, MSVD and DWT along with the use of U matrix in various subbands of DWT and Arnold transform have added extra layers of security when compared to [24] and [25]. As discussed in section 1 that the rearrangement step used in [24] doubles the size of encoded image which is not a good approach as it will take extra time and memory costs for transmission. On the other hand in [25], three encrypted images of dimensions equal to the original image are transmitted that will add an extra overhead of transmission of three encrypted images independently. Though, the schemes given in [24, 25] provides good security but scheme in [24] doubles the size of encoded image, while the scheme in [25] transmits three encrypted images of dimension equal to the original image, which is not appropriate for transmission purpose as it will increase the transmission time and cost. The proposed scheme overcomes 18 Page 18 of 23
the drawbacks of these existing schemes along with the increased security, by transmitting a single encrypted image of dimension equal to the original image. The MSE values of proposed and existing schemes on various test images are shown in table-2. It can be observed from table-2 that the proposed scheme can reconstruct the encrypted images with less information
ip t
loss as the MSE values between original and correctly decrypted images of proposed work are smaller than the MSE values of work given in [24] and [25]. The effectiveness of proposed
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work is also analyzed on incorrect key test and it is found that proposed work has obtained
us
an increased security on each step when compared to the existing. 6. Conclusions
an
In this paper, a color image encryption technique using MSVD and DWT along with Arnold transform in FrFT domain is proposed. The use of MSVD and DWT in FrFT domain
M
along with the values of 4 × 4 matrix in various subbands of DWT with the use of Arnold transform gives more secure information. For the correct decryption of encrypted images it
d
is necessary to have correct information of all the keys along with the way of arrangement. If any one of the parameter is wrong while the rest are exactly correct even thought it is
te
not possible to guess original image information, which shows that it is indeed necessary to
Ac ce p
have exact knowledge of all the keys in correct order. The comparison of proposed work with some existing works show that the proposed work can reconstruct the encrypted image more efficiently as the MSE values of correctly decrypted images for all the color planes using our approach are smaller than the MSE values of existing schemes. However, the structural similarity between the original and encrypted images shows that proposed scheme can transmit images more securely without revealing original information. References [1] P. Refregier, B. Javidi, Optical image encryption based on input plane and Fourier plane random encoding. Optics Letters 20(1995) 767–769. [2] G. Unnikrishnan, J. Joseph, K.Singh, Optical encryption by double-random phase encoding in the fractional Fourier domain. Optics Letters 25(2000) 887–889. 19 Page 19 of 23
[3] G. Situ, J. Zhang, Double random-phase encoding in the Fresnel domain. Optics Letters 29(2004) 1584. [4] B. Javidi, N. Takanori, Securing information by use of digital holography. Optics Letters
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25(2000) 28-30. [5] N. Singh, A. Sinha, Optical image encryption using Hartley transform and logistic map.
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Optics Communications 282(6) (2009) 1104–1109.
us
[6] L. Chen, D.Zhao, Optical image encryption with Hartley transforms, Optics Letters 31(2006) 3438-3440.
an
[7] Z. Liu, Y.Zhang, W.Liu, F.Meng, S.Liu, Optical color image hiding scheme based on chaotic mapping and Hartley transform, Opt. Lasers Eng. 51(2013) 967-972.
Lasers Eng. 47(2009) 539-546.
M
[8] N. Singh, A. Sinha, Gyrator transform-based optical image encryption using chaos, Opt.
d
[9] M. R. Abuturab, Color image security system using double random-structured phase
te
encoding in gyrator transform domain, Appl. Opt. 51(2012) 3006-3016.
Ac ce p
[10] M. R. Abuturab, Securing color information using Arnold transform in gyrator transform domain, Opt. Lasers Eng. 50(2012) 772-779. [11] M. R. Abuturab, Securing color image using discrete cosine transform in gyrator transform domain structured-phase encoding, Opt. Lasers Eng. 50(2013) 1383-1390. [12] M.R. Abuturab, Color image security system based on discrete Hartley transform in gyrator transform domain, Opt. Lasers Eng. 51(2013) 317-324. [13] M. R. Abuturab, Noise-free recovery of color information using a joint-extended gyrator transform correlator, Opt. Lasers Eng. 51(2013) 230-239. [14] M. R. Abuturab, Color information verification system based on singular value decomposition in gyrator transform domains, Optics and Lasers in Engineering 57 (2014) 13-19 20 Page 20 of 23
[15] Q. Guo, Z. Liu, S. Liu, Color image encryption by using Arnold and discrete fractional random transform in IHS space, Opt. Lasers Eng. 48(2010) 1174-1181. [16] Z. Liu, L. Xu, T. Liu, H. Chen, P. Li, S.Liu, Color image encryption by using Arnold
ip t
transform and color-blend operation in Discrete Cosine Transform domains, Opt. Commun. 284(2011) 123-128.
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[17] Z. Liu, M. Gong, Y. Dou, F. Liu, S. Lin, M. A. Ahmad, et al., Double image encryption by using Arnold transform and discrete fractional angular transform, Opt. Lasers
us
Eng.50(2012) 248-255.
an
[18] Z. Liu, S. Liu, Random fractional Fourier transform, Opt. Lett. 32(2007) 2088-2090. [19] S. Liu, Q. Mi, B. Zhu,Optical image encryption with multistage and multichannel frac-
M
tional Fourier-domain filtering, Opt. Lett. 26(2001) 1242-1244. [20] Y. Zhang, C. H. Zheng, N. Tanno, Optical encryption based on iterative fractional
d
Fourier transform, Opt. Commun. 202(2002) 277-285.
Ac ce p
119(2008) 286-291.
te
[21] L. Chen, D. Zhao, Image encryption with fractional wavelet packet method, Optik
[22] L. Chen, D. Zhao, Color image encoding in dual fractional Fourier-wavelet domain with random phases, Opt. Commun. 282(2009) 3433-3438. [23] M. Singh, A. Kumar, K. Singh, Encryption by using matrix-added, or matrix-multiplied input images placed in the input plane of a double random phase encoding geometry, Opt. Laser Eng. 47(2009) 1293-1300. [24] A. Prasad, M. Kumar, D. R. Choudhury, Color image encoding using fractional Fourier transformation associated with wavelet transformation, Opt. Commun. 285 (2012) 10051009. [25] L. Chen, D. Zhao, F. Ge, Image encryption based on singular value decomposition and Arnold transform in fractional domain, Opt. Commun. 291(2013) 98-103.
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[26] M. Kumar, S. Agrawal, Color image encoding in DOST domain using DWT and SVD, Optics and Laser Technology. 75(2015) 138-145. [27] M. Kumar, A. Vaish, Encryption of color images using MSVD in DCST domain. Optics
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and laser in engineering, 88 (2017) 51- 59. [28] John Ladan, An Analysis of Stockwell Transforms, with Applications to Image Process-
cr
ing,(Ph.D. thesis), University of Waterloo, Ontario, Canada (2014).
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[29] Y.Wang, Efficient Stockwell Transform with applications to image processing (Ph.D.thesis), University of Waterloo, Ontario, Canada, (2011).
an
[30] R.G. Stockwell, A basis for efficient representation of the S-transform, Digit. Signal Process. 17(2007) 371393.
M
[31] J. Ladan and Edward R. Vrscay, The Discrete Orthonormal Stockwell Transform and Variations, with Applications to Image Compression, ICIAR 2013, LNCS 7950(2013)
d
235-244,.
te
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