Journal of Optics

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Nonlinear QR code based optical image encryption using spiral phase

transform, equal modulus decomposition and singular value
decomposition
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Page 1 of 13 AUTHOR SUBMITTED MANUSCRIPT - JOPT-104612.R2

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4 Nonlinear QR code based optical image encryption using spiral
5
6 phase transform, equal modulus decomposition and singular
7
8 value decomposition

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9
10 Ravi Kumar1, Basanta Bhaduri1,* and Naveen K Nishchal2
11
1
12 Optical Imaging and Image Processing Laboratory, Department of Applied Physics
13 Indian Institute of Technology (Indian School of Mines) Dhanbad, Jharkhand 826 004, India

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14 2
Department of Physics, Indian Institute of Technology Patna, Bihta, Patna 801 106, India
15
16 Email: basanta.ism@gmail.com
17
18 Abstract
19
In this study, we propose a quick response (QR) code based non-linear optical image encryption technique using
20
spiral phase transform (SPT), equal modulus decomposition (EMD) and singular value decomposition (SVD).
21

us
22 First, the primary image is converted into a QR code and then multiplied with a spiral phase mask (SPM). Next,
23 the product is spiral phase transformed with particular spiral phase function (SPF), and further, the EMD is
24 performed on the output of SPT, which results into two complex images, Z1 and Z2. Among these, Z1 is further
25 Fresnel propagated with distance d, and Z2 is reserved as a decryption key. Afterwards, SVD is performed on
26 Fresnel propagated output to get three decomposed matrices i.e. one diagonal matrix and two unitary matrices.
The two unitary matrices are modulated with two different SPMs and then, the inverse SVD is performed using
27
28
29
an
the diagonal matrix and modulated unitary matrices to get the final encrypted image. Numerical simulation results
confirm the validity and effectiveness of the proposed technique. The proposed technique is robust against noise
30 attack, specific attack, and brutal force attack. Simulation results are presented in support of the proposed idea.
31
32 Keywords: Spiral phase transform; optical image encryption; QR code; singular value decomposition.
33
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34
35 1. Introduction
36
37 In the modern world, the information security is quite a critical issue. The data transmitted through
38 public networks must be safe and secure from the existing attacks by unauthorized users. Thus, in order
39 to prevent illegal access of data, it is necessary to develop secure encryption methods. Recently, optical
40
image encryption (OIE) techniques attracted the attention of many researchers and have been studied
41
42 widely [1-50]. The optical methods for encryption have their own advantages like parallel processing,
43 fast computing, various degree of freedoms (amplitude, phase, polarization etc.) [1, 2]. Double random
44 phase encryption (DRPE) technique proposed by Réfrégier and Javidi [3] is the pathway to the optical
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45 methods for encryption. Several modifications in the DRPE scheme were proposed to improve the
46 encryption strength such as using fractional Fourier transform [4], Fresnel transform [5, 6], Hartley
47
transform [7-10], gyrator transform [11, 12] and Mellin transform [13, 14]. However, the DRPE and its
48
49 integral techniques suffer from inherent linearity issue and they were found vulnerable to several kind
50 of attacks such as known-plaintext attack (KPA) [15], chosen plaintext attack (CPA) [16] and chosen-
51 ciphertext attack (CCA) [17]. Recently, a new technique using Kronecker product of random matrices
52 and hybrid phase masks have been proposed which is robust against KPA and provides added security
ce

53 with large number of security keys [18]. Further, to improve the security of optical encryption systems
54
55
the schemes based on other aspects of optics were also proposed, such as encryption based on joint
56 transform correlator [19], space-based optical encryption [20], diffractive imaging [21], interferometry
57 [22-24], wavelength multiplexing [25], position multiplexing [26], and computational ghost imaging
58 [27].
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59 Most of these techniques comes under the category of symmetric cryptosystems where the
60 encryption and decryption keys are identical. In order to enhance the security, asymmetric
cryptosystems [28, 29] were proposed. But, shortly it was reported that these cryptosystems also have

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4 the limitations against iterative phase retrieval algorithms (PRA) based attacks [30]. Some asymmetric
5 OIE techniques using equal modulus decomposition (EMD) [31-35] and singular value decomposition
6 (SVD) [36, 37] have also been reported to overcome the issue of linearity. Recently, an encryption
7
system based on spiral phase transform and nonlinear power function has been presented in order to
8
strengthen the security [38]. In starting, most of the encryption techniques were presented with only the

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9
10 computer simulation results, but nowadays, some encryption schemes with the experimental
11 implementations have also been presented along with the numerical simulations. In a recent
12 experimental study, asymmetric OIE based on the double random phase and computer generated
13 holograms has been proposed [39]. Some other techniques with experimental verification such as

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14
encryption system based on a single-lens imaging architecture and PRA [40], verification of encryption
15
16 based on interference [41] and experimental optical encryption of grayscale information [42] were also
17 presented.
18 Usually, in OIE techniques, the input image is transformed into a noise like encrypted image and
19 then successfully decrypted by using the correct encryption/decryption keys. However, there are still
20 various shortcomings and limitations in the optical techniques which need to be resolved [43]. One of
21

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the most brutal issues in OIE is the presence of speckle noise in the decrypted image which degrades
22
23 the quality of the technique. Efforts have been made to reduce the speckle noise in OIE techniques,
24 such as multiple image capture with different RPMs [43], thresholding [44] and fully phase encryption
25 [45]. But, the speckle noise can only be reduced to a certain limit, instead of being fully removed [43].
26 Recently, the encryption schemes using quick response (QR) code were proposed and the results reveal
an
27 that the data stored through a QR code can be correctly retrieved by a reader such as a smart phone
28
under noisy environment [46-48]. The QR code is basically a two-dimensional (2D) code which carries
29
30 the information of an image and can be detected as a 2D digital image by a semiconductor image sensor,
31 and is further digitally analysed by using a programmed processor [46]. Generally, the QR code of an
32 image provides a static or dynamic link to a domain where the input image is efficiently stored. Thus,
33 after scanning the QR code the user will get the link of that domain and the original image can be
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34 accessed from there without any information loss. Barrera et al. used QR code for the first time as input
35
for DRPE architecture and the finally retrieved the noise free original image from their technique [46].
36
37 Afterwards, the QR code based noise-free OIE scheme was combined with several existing techniques
38 [47, 48], to enhance the security level. Encoding with QR codes has their own advantages such as strong
39 fault tolerance and error correction capability [48]. Recently, the experimental demonstration of some
40 QR code based techniques have been also reported, such as, experimental QR code optical encryption
41 for noise free retrieval [49] and QR code based encryption using spatially incoherent illumination [50].
42
43
In this paper, we propose a new QR code based non-linear optical image encryption technique using
44 SPT, EMD and SVD. Here EMD is used for decomposing an input QR code into two complex images,
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45 and SVD is used to further process one of these complex images to get the final encrypted image. These
46 decompositions add additional security to the proposed cryptosystem. Also, the SPMs are used instead
47 of random phase masks (RPMs) as SPMs are useful in overcoming the axis alignment problem and
48 enlarges the key space for enhanced security [51, 52]. In the proposed method, the primary image is
49
50
first converted into a QR code which is modulated with a spiral phase mask (SPM). The product is then
51 spiral phase transformed with a particular spiral phase function (SPF), and further equal modulus
52 decomposed to get two complex functions, Z1 and Z2. The complex function, Z2 from EMD is reserved
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53 as the decryption key, and Z1 is Fresnel propagated with distance d. The output after Fresnel propagation
54 is afterwards decomposed using SVD to get the three matrices i.e. one diagonal matrix and two unitary
55 matrices. The two unitary matrices are then modulated with two different SPMs. The encrypted image
56
57 is finally obtained by performing inverse SVD by combining the diagonal matrix and modulated unitary
58 matrices. The security keys in the proposed technique are the order of MSPF, Fresnel propagation
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59 distance, private key Z2, SPMs, and order of multiplication in SVD. The presented simulation results
60 depicts the efficiency of the proposed technique which is also robust against various attacks such as
noise attack, specific attack, and brute force attack.

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5 2. Theoretical Background
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7
2.1 Spiral phase transform (SPT)
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In SPT, a two-dimensional (2-D) signum function, sgn(u, v), also called spiral phase function (SPF) is

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9
10 used for 2-D Hilbert transform [53, 54]. The 2-D signum function can be defined as a pure SPF in
11 spatial frequency space (u, v) as [53]:
12
u + iv
13 SPF = sgn(u , v) = = exp{iφ (u , v)} (1)
u 2 + v2

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15
16 where, the phase, ɸ is the polar angle in frequency space. The SPF function is not defined at the origin,
17 at which the function can have values either zero or one, which addresses the singularity [54].
18
In this study, we have used a modified SPF (MSPF) function depending on the parameter, q, the
19
20 number of singularities or the order of SPF, which corresponds to the points where MSPF has undefined
21 values (i.e. either zero or one). This (MSPF) can be written as:

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23 MSPF = exp{iqφ (u , v)} (2)
24
Now, the SPT of a function, f(x, y), can be written as [53]:
25
26 −1
SPT { f ( x , y )} = ℑ [ MSPF × ℑ{ f ( x , y )}]. (3)
27
28
29
an
where, the ℑ and ℑ−1 denote the forward and inverse 2-D Fourier transforms, respectively. The inverse
30 SPT can be given by:
31
32 −1
33 ISPT { f ( x , y )} = ℑ [ conj ( MSPF ) × ℑ{ f ( x , y )}]. (4)
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34 where, conj () represents the complex conjugate.
35
36
37 2.2 Equal modulus decomposition (EMD)
38 In EMD, a two-dimensional (2D) complex function, Z(u, v), can be decomposed into two independent
39 complex components i.e. Z1(u, v) and Z2(u, v), having equal moduli [32, 33]. The principle of EMD can
40 be illustrated with the help of an Argand diagram as shown in figure 1 [34, 35].
41
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56 Figure 1. Schematic of equal modulus decomposition.
57
58 Analytically, Z(u, v), can be decomposed into two equal moduli components by defining a random
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59 phase, θ(u, v), distributed uniformly in the interval [0, 2π], which would be the argument of one of the
60 components [34, 35]. Now, the two components can be given as [34]:

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4 A ( u ,v )/2
5 Z1 ( u , v ) = exp( i 2θ ( u , v )). (5)
6 cos[ϕ ( u ,v ) −θ ( u ,v )]
7
A ( u ,v )/2
8
Z 2 ( u ,v ) = exp{i [2ϕ ( u ,v ) −θ ( u ,v )]}. (6)

pt
9 cos[ϕ ( u ,v ) −θ ( u ,v )]
10
11 where A(u, v) and φ(u, v) are the amplitude and phase of Z(u, v), respectively.
12
13
2.3 Singular value decomposition (SVD)

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15 SVD is a well-known technique of linear algebra which decompose the system into a set of linearly
16 independent components with its own energy contribution [55, 56]. Any image I of size M × N can be
17 decomposed as [37]:
18
19 I =USV T . (7)
20
21 where, T denotes the transpose of a matrix. The matrix U and V are orthogonal matrices (i.e. UUT=1

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22 and VVT=1) having sizes M × M and N × N, respectively, whereas matrix S is a diagonal matrix with
23 size M × N [37]. The decomposed three components (i.e. U, S and V) of a given image and their order
24
25
of multiplication play a crucial role in decryption to correctly recover the input image.
26
3. Proposed encryption technique
27
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29 3.1 Image Encryption
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30 The schematic diagram of the proposed technique is shown in figure 2. Coordinates of the input image
31
32
plane are denoted by variables (x, y) and the coordinates after SPT are represented by (u, v). The primary
33 image with size 256 × 256 pixels2 is first converted into a QR code which is used as the input for the
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34 proposed encryption scheme. The encryption process involves the following steps:
35
36 i. First, the input image, I(x, y) is converted into QR code, IQR(x, y) [48]. Then, the QR code is
37 modulated with a spiral phase mask, SPM1, to get the modulated signal, I'(x, y), as:
38
39 I ′ ( x , y ) = I QR ( x , y ). SPM1 . (8)
40
41 ii. SPT is then performed on the modulated signal, I'(x, y), with a MSPF of particular order, q (20) to
42
43 get the complex image, Z(u, v), as:
44 −1
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45 Z ( u , v ) = SPT { I ′ ( x , y )} = ℑ [ MSPF × ℑ{ I ′ ( x , y )}]. (9)

46
47 iii. Next, Z(u, v), is decomposed into two complex images, Z1(u, v) and Z2(u, v), using the EMD principle
48 as discussed in section 2.2. Further, the Z1(u, v) is Fresnel propagated with distance, d (5 cm) to get
49 A1(u', v') and Z2(u, v) is reserved as the decryption key. The value of A1(u', v') can be obtained as [5,
50
51
6]:
52
ce

53 A1 ( u′, v′ ) = FrTd { Z 1 ( u , v )}
54
55 exp{i 2π d / λ } iπ
56
57
=
iλ d ∫∫ Z1 ( u ,v ) exp  λ d (( u −u′)2 +( v−v′)2 )dudv (10)

58
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59
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where, FrTd is the Fresnel propagation with distance d = 5 cm and wavelength, λ = 632 nm.

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4 iv. Then, SVD operation is performed on A1(u', v'), to get the two orthogonal matrices U and V, and one
5 diagonal matrix S, as:
6
7 T
8 SVD{ A1 ( u′, v′ )} = USV . (11)

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10 These orthogonal matrices are further modulated with two different SPMs; SPM2 and SPM3, to get
11 modulated matrices U' and V' as follow,
12
13 U ′=U . SPM 2 , (12)

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15 V ′=V . SPM 3 . (13)
16
17 v. Finally, The encrypted image, E(u', v') is finally obtained by performing inverse SVD operation
18 with two modulated matrices U' and V', and the diagonal matrix S, as,
19 T
20 E ( u′ , v′ ) = U ′SV ′ . (14)
21

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23 3.2 Decryption
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25
The decryption part follows the following steps:
26
i. First, the encrypted image, E(u', v'), is decomposed using SVD to get the U1, S1 and V1 matrices:
27
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29
an
SVD{ E ( u′, v′ )} = U 1 S1V1 .
T
(15)
30
31
The complex conjugates of SPM2 and SPM3 are multiplied with U1 and V1, respectively, to get back
32
33 the U1' and V1' matrices,
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35 U1′ =U1 . SPM 2* , (16)
36
37
38
V1′ =V1 . SPM 3* . (17)
39
Further, inverse SVD is performed using the U1', V1' and S1 matrices to get, D(u, v), as:
40
41
′ ′T
42 D ( u′, v′ ) = U 1 S1V1 . (18)
43
44
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ii. Next, D(u', v') is Fresnel propagated in the reverse direction with distance d, to get, D1(u, v):
45
46
47 D1 ( u , v ) = FrT− d { D ( u′, v′ )}. (19)
48
49 The complex image, D1(u, v) is further combined with the private decryption key Z2(u, v), to get,
50 D2(u, v) as:
51
52 D2 ( u ,v ) = D1 ( u ,v ) + Z 2 ( u ,v ). (20)
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53
54 iii. The inverse SPT of order q (20) is performed on D2(u, v) as:
55
56 DQR ( x , y ) = ISPT { D2 ( u ,v )}, (21)
57
58 The amplitude part of DQR(x, y) provides the decrypted QR code, I'QR(x, y) as:
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59
60 ′ ( x , y ) = DQR ( x , y ) .
I QR (22)

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4 iv. Finally, the original image can be retrieved by scanning the decrypted QR code I'QR(x, y) using a QR
5 code scanner.
6
7
8

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13

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26 an
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29 Figure 2. Schematic diagram of the proposed technique: (a) encryption process and (b) decryption process.
30 SPM, Spiral phase mask; SPT, Spiral phase transform; EMD, equal modulus decomposition; FrTd, Fresnel
31 propagation with distance d; SVD, Singular value decomposition and ‘*’ represents the complex conjugate.
32
33 It is worth mentioning that the encryption is carried out digitally, while the decryption process can
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34 be performed using the proposed optoelectronic setup, shown in Fig. 3. The two complex images D1
35 and Z2 are interfered to get a composite image, which is then subjected to inverse SPT. The inverse SPT
36 can be realized optically by using the two Fourier lenses L1 and L2 (having equal focal lengths, for
37
example, f1 = f2 =200 mm [38], respectively), and a spatial light modulator (SLM1) displaying the
38
39 complex conjugated of MSPF. A collimated coherent light can be used for illumination. Further, the
40 output image after inverse SPT is multiplied with the complex conjugate of the SPM1 displayed on
41 SLM2, and the resultant intensity is recorded by using a charged coupled device (CCD) camera and
42 stored in the computer system. The SLMs and CCD are connected to a personal computer (PC). In the
43 present study, we have only shown the results from numerical simulations. Although the proposed
44
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optoelectronic setup for decryption is simple and can be implemented easily, only numerical simulation
45
46 results are presented due to lack of experimental facilities. It should also be noted that though the Fresnel
47 transform has been implemented in the present study, one can use other optical transforms such as
48 fractional Fourier transform, gyrator transform, etc. We have used the combination of SPT, EMD and
49 SVD in the proposed scheme. All the three techniques have their own advantages. The pre-processing
50 of the QR code using SPT provides a secure path as the order of MSPF used is arbitrary and only known
51
to authorized users which is an additional security key in the proposed method. Afterwards, the EMD
52
ce

53 is used to introduce the nonlinearity in the proposed scheme as one part after EMD is reserved as the
54 private key and other part is transmitted for further processing. The SVD applied to one part of EMD
55 to enhance the security as the two matrices after SVD are also modulated by using the SPMs which
56 increases the number of security keys and play important role in making the proposed method robust to
57 specific attack. The proposed scheme is easy to implement numerically even with the combination of
58
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59
SPT, EMD and SVD which is also an advantage. Further, the SPT and EMD can be easily realized
60 optically.

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Figure 3. Proposed optoelectronic setup for decryption. BS, beam splitter; L, lens; SLM, spatial light
22 modulator; MSPF, spiral phase function, f1,2, focal length of the lenses and ‘*’ represents the complex conjugate.
23
24 4. Simulation Results and Discussion
25
26
4.1 Encryption results
27
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29
an
To verify the validity and feasibility of the proposed technique, the numerical study is performed
using MATLABTM (version R2016a). In this study, we have used an arbitrary image as the input image
30 (256×256 pixels) as shown in figure 4(a) and its corresponding QR code is shown in figure 4(b). Figures
31 4(c) – 4(e) shows three spiral phase masks used during encryption process and figures 4(f) and 4(g)
32 show the complex images, Z1 and Z2. The final encrypted image is shown in figure 4(h). The Fresnel
33
propagation parameters used in the encryption process are d = 5cm and λ = 632nm, whereas the order
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34
35 of MSPF (i.e. q) used was 20.
36
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59 Figure 4. (a) Original image, (b) corresponding QR code, (c) SPM1, (d) SPM2, (e) SPM3, (f) and (g) are outputs
60 after EMD (Z1 and Z2), and (h) final encrypted image.

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4 4.2 Decryption results and key sensitivity analysis
5 In this section, the decryption results using at least one wrong key as well as with all the correct keys
6 are shown, and the sensitivity of security keys are analysed. Figure 5(a) shows the decrypted QR code
7
while decryption is carried out on Z2(u, v) and figures 5(b) and 5(c) show the decrypted image with all
8
correct keys but wrong SPT order, q (increased by 5) and wrong Fresnel propagation distance, d

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9
10 (increased by 2 × 10-4 mm), respectively. Figures 5(d) and 5(e) show the decrypted images with wrong
11 SPM2 and SPM3, respectively, but with all other correct keys. Figure 5(f) shows the decrypted QR code
12 with all correct keys and figure 5(g) show the retrieved image from the correct decrypted QR code in
13 figure 5(f).

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21

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22
23
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26 an
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Figure 5. Decrypted QR code with: (a) only using Z2, (b) wrong order of SPT (changed by 3), (c) wrong Fresnel
38
propagation distance (changed by 2 × 10-4 mm), (d) wrong SPM2, (e) wrong SPM3, (f) with all correct keys and
39
(g) retrieved image from f).
40
41
42 For quantitative analysis of retrieved image, various parameters such as correlation coefficient (CC),
43 peak signal to noise ratio (PSNR) and normalised mean squared error (NMSE) are calculated with
44 variation of one of the security keys, the Fresnel propagation distance, d. Mathematically, the CC, PSNR
pte

45 and NMSE can be expressed as [46, 48]:

46
47 E {[ I 0 − E [ I 0 ]][ I d − E [ I d ]]} , (23)
48 CC =
3 E { I − E [ I ]]2 } E { I − E [ I ]]2 }
0 0 d d
49
50
2552 M × N (24)
51 PSNR ( I 0 , I d ) = 10 log 10
∑ [ I d ( x , y ) − I 0 ( x , y )]
2
52
ce

53 ∀x , y
54
I 0 ( x, y ) − I d ( x, y )
2
M N
NMSE ( I 0 , I d ) = ∑∑
55
(25)
56
m =1 n =1 M ×N
57
58 where, I0 and Id are the input and decrypted images, E[.] is the expected value operation and M, N are
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59 the pixel coordinates of the image.

60
Figure 6(a) shows the effect of change in CC value with change in Fresnel propagation distance, d,
whereas figure. 6(b) show the effect of change in PSNR and NMSE values with change in d. It can be

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4 easily seen that a small change in the propagation distance leads to a large change in the CC, PSNR and
5 NMSE values, which signifies that the proposed scheme is much sensitive to the variation of Fresnel
6 propagation distance. Hence, it plays a very important part in the security of the system.
7
8

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25 Figure 6. Variation in (a) CC Values (b) PSNR values and (c) NMSE values with change in Fresnel propagation
26 distance, d.
an
27
28 It is important to note that the computation time for encryption and decryption processes were 93.57
29
ms and 75.88 ms, respectively, on a laptop with 5thgeneration i5 processor (2.23 GHz) and 8 GB RAM.
30
31
32 4.3 Attack analysis
33 The robustness of the proposed technique is also analysed under various attacks such as noise attack,
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34 specific attack, and brute force attack.
35
36
4.3.1 Noise attack analysis. There is always a possibility of contamination of unwanted noise in the
37
38 encrypted data during transmission, thus, we have checked the robustness of the proposed technique
39 against noise attack. If the encrypted image is contaminated with the Gaussian noise in the following
40 way:
41
42 E ' = E (1 + ρ G ) (26)
43
44 where E and E' are the encrypted and noise polluted encrypted image, respectively, G is a Gaussian
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45 random noise with zero mean and 0.05 variance, and ρ is the coefficient denoting the strength of the
46 Gaussian noise.
47 Figure 7(a) and 7(d) show the noisy encrypted images with ρ = 0.08, and 0.12, respectively. The
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corresponding decoded QR code are shown in figures 7(b) and 7(e), respectively. The retrieved image
50 from figure 7(b) is shown in figure 7(c), whereas the image cannot be retrieved from the decoded QR
51 code shown in figure 7(e) because it is very noisy. From the results, it was found that the noise-free
52 original image can be retrieved from decrypted QR code with noise strength even up to 0.11. To verify
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53 this, the variation of CC and NMSE values with noise strength are also analysed and it is found that the
54 NMSE value increases and CC value decreases sharply for noise strength variation from 0.11 to 0.12
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23 Figure 7. Noise attack results: (a) encrypted image with ρ = 0.08, (b) decrypted QR code from a), (c) retrieved
24 image from b), (d) encrypted image with ρ = 0.12, (e) decrypted QR code from d).
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43 Figure 8. Variation with noise strength: (a) NMSE, and (b) CC values.
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4.3.2. Specific attack. The performance of the proposed technique against specific attack is analysed to
47 validate its efficiency. In this attack, the attacker uses all the constraints of the proposed cryptosystem
48 except for the private key, Z2. According to the principle of the specific attack, the iterative process is
49 started with a random function that is used as the guessed private key Zk(u, v) [34]. This is combined
50 with the known ciphertext Z1(u, v) and subjected to inverse SPT. This gives the following:
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52 I k ( x , y ) = ISPT { Z1 ( u ,v ) + Z k ( u ,v )}. (27)
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Now, since there are no other constraints available at this point, the function is subjected to the SPT and
55 the output is subtracted from the known value of Z1(u, v), as:
56 Z k′′+1 = SPT { I k ( x , y )}− Z1 ( u ,v ). (28)
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4 The iteration steps in Equations (27) - (29) are repeated to check the convergence of the value to
5 give the private key. At the end of each iteration step, the obtained key, i.e. Zk(u, v), is used to retrieve
6 the input to the EMD stage of the proposed technique which is Z(u, v). For each iteration step, CC was
7 calculated to validate the retrieval of the correct private key. The relationship of the CC value to the
8 iteration number is shown in figure 9 and from that it can be seen that the CC value saturates after

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31 Figure 9. Relationship between iteration number and correlation coefficients during the specific attack on the
32 proposed scheme. The decrypted image after 1000 iteration is shown in the inset.
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4.3.3. Brute force attack. In case of brute force attack, the phase key search can be used to retrieve the
36 security keys. The number of combinations can be quantified by using the image size and the number
37 of possible values for the keys. As each pixel of the two SPMs (each of size 256×256 pixels2) can have
38 L (=2n, where n is the image bit size) possible values, the number of attempts required to retrieve both
39 SPMs is of the order of L2(256×256). For L = 256 grey levels (8-bit image), the number of possible SPMs
40 to try would be 256131072. Thus, for any practical values of L, the number of trials are very large and it
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increases exponentially with the size of the image which makes brute force attack very difficult to
43 execute [57-59]. We carried out numerical simulations to assess the immunity of the encryption scheme
44 against brute force attack. The CC values were calculated for every trial of spiral phase key and the
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45 results are shown in figure 10. It can be seen that the proposed technique is also robust to this type of
46 attack as the calculated CC values are always around 0.021 for all 1000 trials.
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22 Figure 10. CC values between original image and the decrypted image by testing 1000 possibilities for the keys.
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24 It is noteworthy to mention that there is always a possibility for occlusion of some information in
25 the proposed method during transmission of encrypted image. This could be due to the singularities
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present in MSPF and the combination of EMD and SPT.
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29 5. Conclusions
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31 In conclusion, a non-linear QR code based optical image encryption technique through spiral phase
32 transform using equal modulus decomposition and singular value decomposition is proposed. The
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security keys in the technique includes Fresnel propagation distance, three SPMs, order of MSPF used
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35 in SPT, the private key obtained using EMD, and the order of multiplication in SVD. The CC, NMSE
36 and PSNR are calculated between primary QR code and decrypted QR code with the variation in Fresnel
37 propagation distance, d and it is found that a small variation in value of d leads to a very large change
38 in the values of these parameters. An optoelectronic setup is also proposed for the decryption. The
39 proposed technique is resistant to various attacks, such as noise, specific and brute force attacks.
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42 Acknowledgements
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44 The work was supported by the Indian Institute of Technology (Indian School of Mines) Dhanbad, India
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45 (project no. FRS(100)/2015-2016/APH) and the Department of Science and Technology, Science &
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Engineering Research Board, Govt. of India (file no. DST(SERB) ECR/2016/000224).
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