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Procedia Computer Science 132 (2018) 1636–1645

International Conference on Computational Intelligence and Data Science (ICCIDS 2018)

Data Computation and Secure Encryption Based on Gyrator Transform using Singular Value Decomposition and Randomization a

Mehak Khuranaa, Hukum Singhb a

Department of Computer Science,The NorthCap University, Gurugram, India

b

Department of Applied sciences, The NorthCap University, Gurugram, India

Abstract A digital scheme based on singular value decomposition (SVD) encrypted using Gyrator transform is proposed to enhance the security of the system. In this technique, plain input image is multiplied with another image and is then masked with random phase mask (RPM) along with Double random Phase Encoding (DRPE) in gyrator domain. Then obtained image is decomposed into segments [U, S, V] using SVD and is then each decomposed segment is randomized to get three encrypted image. It will increases the key space and also increases the number of parameters which makes it difficult for an attacker to find exact key to recover original image. The robustness of the proposed cryptosystem has been analysed on the basis of different parameters by simulating on MATLAB 8.1.0 (R2012b). The experimental results are provided to highlight the effectiveness, robustness and suitability of the proposed cryptosystem and to prove the feasibility and validity of the proposal. © © 2018 2018 The The Authors. Authors. Published Published by by Elsevier Elsevier Ltd. B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/3.0/) Peer-review under responsibility of the scientific committee of the International Conference on Computational Intelligence and Peer-review under responsibility of the scientific committee of the International Conference on Computational Intelligence and Data Science Science (ICCIDS (ICCIDS 2018). 2018). Data Keywords:Singular Value Decomposition, Randomization, Gyrator Transform, Noise Attack, Occlusion Attack.

Corresponding author: [email protected] 1877-0509© 2018 The Authors. Published by Elsevier B.V.

Peer-review under responsibility of the scientific committee of the International Conference on Computational Intelligence and Data Science (ICCIDS 2018).

1877-0509 © 2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/3.0/) Peer-review under responsibility of the scientific committee of the International Conference on Computational Intelligence and Data Science (ICCIDS 2018). 10.1016/j.procs.2018.05.129

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1. Introduction The digital communication has become a challenge to transmit the data securely. Digital images are used in many applications such as biometric, military applications, police identification etc. These images can easily be accessed and used by unauthorized person or hacker and can be transmitted over a communication network. Therefore optical image encryption is one of the secure method which can be used for communication. In 1995, Refregier and Javidi proposed a 4f scheme based double random phase encoding (DRPE) in Fourier domain to encrypt a plain image and to convert it into a white noise [1-4]. This suffered from many attacks and some security issues so therefore DRPE with different transformations such as Fractional Fourier transform (FrFT) [5-8], Fresnel transform (FrT) [9-11], Gyrator transforms (GT) [12], Cosine transform (CT) [13], Hartley transform (HT) [14], Arnoldtransform (AT) [15] and Mellin transform (MT) [16-18] were proposed to enhance security and to enhance the resistance against possible attacks. Many other optical image encryption schemes for reliable and secure transmission in this direction have been reported [19-23]. GT is one of the canonical transforms proposed by Rodrigo in the field of image processing and applications, properties and its implementation were presented subsequently [24-25]. Then Pei and Ding expanded the discussion and presented digital implementation of GT. It is known, that in conventional GT methods, the plain image is only encrypted by using random or chaotic mask, and there remains correlation between plain and ciphered image. Due to that attacker can easily attack and important information can be revealed [26-28]. To overcome the issues it was further extended by using another structured phase mask. In this paper a new scheme has been proposed where the plain image is combined with another image to form one single image, and then that image is encoded by multiplying pixel by pixel with the random key mask in GT domain [29-30]. GT rotation angles and are additional keys to ensure the security [31-35]. And two images are combined here to introduce the pixel scrambling of image which reduces the image correlation and give good effect which further enhances the security. This technique adds scrambling of image by randomizing which makes it resistant to many conventional attacks and gives good performance with an avalanche effect. It also adds non linearity in the path of encryption and decryption which makes it difficult for an attacker to find the actual key. The non-linear operation in this technique amplifies the error, as while decrypting the attacker needs to find another secret image to decrypt the plain image with correct parameters. This new proposed scheme includes diffusion property by introducing scrambling effect in encryption path which adds non linearity and makes it more secure from traditional attacks and increases the key space and enhances the security. It also uses SVD which decomposes into three segments which are required in the same order for multiplication at the time of decryption. If the order is not same then the correct image will not be recovered. SVD also increases the key space in the algorithm. The robustness of our proposed cryptosystem has been analysed and verified on the basis of various parameters by simulating on MATLAB 8.1.0 (R2012b) and experimental results are presented below to highlight the effectiveness of the algorithm. 2. Gyrator Transform 40]

The gyrator transform has only two-dimensional format. The 2-D function

for GT is given as [36-

where is the transform angle and GT is represented as the output of Gyrator Transform. When the value for , the gyrator transform reduces to a Fourier transform with the rotation of the coordinates . The is represented as . inverse of gyrator transform 3. Singular Value Decomposition Singular value decomposition (SVD) is one of the reliable matrix decomposition method [41-46]. It is analogous to the diagonalization of Hermitian or symmetric matrix based of eigen vectors. SVD is an effective method to decompose the system into a set of independent linear components. Let be a real matrix, where size of matrix is . This matrix can be decomposed into sets of three linear components .

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Mathematically it can be represented as is a left singular orthogonal matrix of

. And is mathematically represented by

is a m diagonal matrix with non-diagonal entries all zero and And is mathematically represented by

is a right singular orthogonal matrix of

.

T is a transpose. SVD is mostly used in image processing because of its number of advantages. It is used in decomposing image into three components for encrypting image and their order of multiplication of these three components is important for decryption process to recover original image and hence enhances the security by introducing additional keys. 4. Image Encryption algorithm using SVD and randomization in Gyrator Domain In this paper, a symmetric image encryption technique based on Gyrator Transform using singular value decomposition is proposed. In the Encryption scheme, the original plain image is first multiplied withanother secret image which changes the image pixel value and then obtained image is multiplied by with Random Phase Mask (RPM) in GT and inverse GT domain. Then obtained result is decomposed into three segments to enhance the security. Then three segments are individually randomized to scramble the image and to introduce diffusion property in it. These segments can be communicated through different channels to secure the data but for decryption, these three segments are required to decrypt the image. The three cipher images are first de randomized and then three segments are multiplied to get one single image and decrypted using with random phase Mask in GT and inverse GT domain. To obtain the actual image the secret image is conjugated from obtained result. 4.1 Encryption Technique image

In encryption process, the original plain image .

Then multiplied with random phase mask

that is to be encrypted is multiplied with another

in Gyrator domain.

The obtained result is then multiplied by another random phase mask

in inverse gyrator domain.

denotes the order by which image is transformed. Then obtained result is decomposed into three segments to enhance the security. Then three segments are individually randomized to obtain encrypted image property.

and to introduce diffusion

The three segments of encrypted image are obtained is secure and can be transmitted through three different insecure channel.This is symmetric approach so the secret key used for encryption and decryption is same.

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I(x, y)

x

G(u, v)

GT(α )

H(x, y)

E1(x, y)

U(x, y)

R2(x,y)

R1(x,y)

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S(x, y)

G(u, v)

GT(β)

R

E1(x, y)

V(x, y)

E3(x, y)

I1(x, y)

Fig. 1. (a) Encryption Flowchart

4.2 Decryption Technique In the decryption scheme, three encrypted images received at the receiver end is firstly decrypted by applying de-randomization function. Then three segments are multiplied in given order to form one single image. Then obtained image is multiplied with conjugate of Random Phase Mask (RPM) in GT and inverse GT domain. Finally conjugate of secret image used to obtain original image. If three segments are not received the original image cannot be recovered. Each segments of encrypted image is de_randomized to obtain U, S and V

U, S and V are multiplied in same order to obtain intermediate image The

.

is multiplied by

The intermediate result image . The obtained result

is now multiplied with another in Inverse GT domain to obtain final decrypted is then conjugate by secret image

to obtain the final image

Conjugate is represented by . The flow graph of proposed encryption and decryption using random key mask based on gyrator transform is shown in Fig 1(a) and 1(b)

U(x, y)

E1(x, y) E1(x, y) E3(x, y)

R

S(x, y)

R1*(x,y)

R2*(x,y) G(u, v)

GT(β)

G(u, v)

GT(α )

H(x, y)

x

I(x, y)

V(x, y)

I1*(x, y)

Fig. 1. (b) Decryption Flowchart


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5. Simulations Results This paper demonstrates and verifies that proposed algorithm using SVD based on gyrator transform achieves better performance in terms of quality of the decrypted image. Consider an original input grayscale Lena image shown in Fig 2(a) and peeper Image and secondary image in Fig 2(b) of size pixels. RPM is shown in Fig 2(c), after applying SVD, it is decomposed into three segments are shown in fig 2(d-f). The encrypted image or stationary white noise is obtained after applying randomization in Fig 2(g). Fig 2(h-i) are the decrypted image for Lena and pepper respectively.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Fig. 2. (a-b)Input Image (c)RPM (d-f) U,S,V (g) CipherImage (h-i) Decrypted Images

5.1 Quality Metrics The quality performance has been analysed against Structural Similarity Index (SSIM) for measuring image quality, Mean Square Error (MSE), Peak Signal to Noise Ratio (PSNR) and Relative Error (RE). SSIM: Structural Similarity Index measures the image quality of the recovered image. It compares the Image to be encrypted with recovered image to check its quality. It calculates the global SSIM value for the image and local SSIM values for each pixel. The SSIM value obtained for our algorithm is 0.9898 MSE: The Mean Square Error (MSE) parameter measures the effectiveness and verifies the average squared difference between original image and recovered image pixel value. Mathematically MSE can be represented in equation (19)

is the size of the image in the pixels. Our paper uses Lena image of size 256 256. Smaller value of MSE states higher similarity between the recovered and original image and it shows that recovered image’s quality is high. For our paper MSE computed for Lena is which is very low which shows difference between the original and recovered image is very small. This also demonstrates the quality image has been recovered and difference between images is not visible from human eye. RMSE: The root mean square error also verifies the quality of the recovered image. It is the root of MSE. The RMSE between original image and its corresponding decrypted image is computed and analysed by varying topological charge, wavelength, focal length and are plotted and by calculating RMSE values. The corresponding RMSE from deviations of their original

Mehak Khurana et al. / Procedia Computer Science 00 (2018) 000–000 Mehak Khurana et al. / Procedia Computer Science 132 (2018) 1636–1645 180

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Lena Pepper

80

RMSE

RMSE

6

100

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0

(a) 0

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(c) 400

500

600

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Wavelength (nm)

800

900

Fig. 3. (a) Topological Charge Vs RMSE (b) Focal Length Vs RMSE (c) Wavelength Vs RMSE

PSNR: It weighs the difference between original plain image and the decrypted recovered image obtained using our proposed algorithm. Equation (20) shows mathematical expression of the PSNR

Larger the value of PSNR means better is the quality of recovered image. For our proposed algorithm value of PSNR for Lena is 387.953 respectively. These values indicate high quality of the decrypted image. Relative Error: It can be computed between original image and the decrypted image obtained using our proposed algorithm. Equation (21) shows mathematical expression of the RE [47] If value of the RE is near about zero it indicates the image is perfectly recovered. The value of RE from our algorithm for images Lena is respectively which is nearly zero. These values indicate that through our algorithm images can be perfectly recovered. The obtained values of MSE, SSIM, PSNR and RE for our proposed algorithm shows that it recover a high quality image. 5.2 Noise attack In noise attack, the proposed algorithm, the algorithm has been verified and analysed on the basis of impact on the quality of the recovered image. The proposed algorithm has also been evaluated on the basis of resistance from noise. Let be the cipher image without noise and be the noise effected cipher image, so the noise interference can be represented as [47] represents noise strength and represents Gaussian noise with mean zero and unity standard deviation. The quality of the recovered images gets affected as noise strength increases as shown in Fig 4(a-c) with ranging from with increment of . The recovered images are low quality images but the still visible from normal human eye which means the proposed algorithm can tolerates the data loss and is resistant to noise.

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(a)

(b)

(c)

Fig. 4. Effect of Gaussian noise on image by varying

MSE is also calculated by varying the noise intensity which indicates algorithm is secure and robust.

(a-c)

and is plotted for Lena image are shown in Fig 5,

7

2

x 10

1.8 1.6 1.4

MSE

1.2 1 0.8 Lena Pepper

0.6 0.4 0.2 0 0

0.5

1

1.5

2

2.5

Noise

3

3.5

4

4.5

5

Fig. 5. Noise Vs MSE graph

5.3 Occlusion Attack In occlusion attack [48], the proposed algorithm has been verified and analysed against on the basis of blocked part in images. The images recovered after decryption has loss of information due to which quality images are not recovered but are still visible from human eye. Different cases with block size of around 5%, 25% and 50% is cropped and are recovered image can be shown in Fig 6[a-f] for Lena. The MSE has also been evaluated for 5%, 25% and 50% for Lena image , ,4 . Larger the value of MSE means larger the loss of information which means larger the occluded the area of encrypted image larger will be the loss of information and good quality image is not recovered.

(a)

(b)

(c)

(d)

(e)

(f) Fig. 6. (a-f) Occlusion and decryption for 5%, 25%, 50%

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5.4 Key Sensitivity The algorithm is secure only if the key is strong and attacker is not able to recover the original image by applying wrong keys and wrong values for other parameters. The proposed algorithm has been verified against key sensitivity. In simulation, value of the one parameter has been changed and values of the other parameters have been kept fixed and results are shown in Fig 7 and MSE also changes which shows the quality of the recovered image. Fig 7(a) shows the original figures which is taken as input to the algorithm. In first case fig 7(b) the order of alpha ( is changed from to . In second case 7(c) the order of topological charge is changed to . In third case 7(d) the wavelength is changed to . In forth case 6(e) the focal length ) is changed to

(b)

(c)

(a)

(d)

(e)

Fig. 7. (a) Original Image (b-e) Output obtained by applying wrong keys

6. Conclusion and Future Work The proposed cryptosystem with Gyrator transform enhances the security of the system as encryption involves randomization which introduces the scrambling of image and adds diffusion which makes difficult for an attacker to find relation between cipherimage and key. This makes it resistant from various attacks. It also uses SVD which decomposes into three segments. With this decomposition three segments can be kept at three different places so even if attacker finds one segment cannot find the original image. And for decryption the order of three segments for multiplication has to be same to recover correct image. The system is simulated for greyscale and binary image and simulation results provided in this report demonstrates that proposed algorithm recovers quality images with computation in terms of MSE, SSIM and PSNR. The scheme is also verified against various attacks such as occlusion and noise attack which proves that loss of information during transmission over a channel does not affect decryption of the image to some extent. Simulation result demonstrates the feasibility and robustness of symmetric cryptosystem. In future, the encryption scheme can be modified by introducing Equal Modulus Decomposition so as to enhance the security which will add more number of parameters. Attacker will need more number of parameters to find the actual key and original image. References [1] Matoba O, Nomura T, Perez-Cabre E, Millan M S, Javidi B, (2009) Optical techniques for information security, Proc IEEE, 97, 1128-1148 [2] Alfalou A, Brosseau C, (2009) Optical image compression and encryption methods, Adv Opt Photon, 1:589-536 [3] Millan M S, Perez-Cabre E, (2011) Optical data encryption, Optical and Digital Image Processing: Fundamentals and Applications, G. [4] [5]

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