An asymmetric color-image cryptosystem based on spiral phase transformation and equal modulus decomposition

Optics and Laser Technology 126 (2020) 106106

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An asymmetric color-image cryptosystem based on spiral phase transformation and equal modulus decomposition

T

⁎

Zheng Zhua, Xu-Dong Chena, Chao Wua, Jun Wanga, , Weixing Wangb a b

School of Electronics and Information Engineering, Sichuan University, Chengdu 610065, China School of Information Engineering, Chang'an University, Xi'an 710064, China

H I GH L IG H T S

color-image cryptosystem combining two spiral phase transformations and an equal modulus decomposition. • Asymmetric security cryptosystem with simple structure, reduced data amount and fast speed. • High • Improved sensitivity of order of spiral phase transformation with 100 times compared with traditional method.

A R T I C LE I N FO

A B S T R A C T

Keywords: Spiral phase transformation Equal modulus decomposition Color image cryptosystem

In this paper, we propose an asymmetric color-image cryptosystem based on spiral phase transform (SPT) and equal modulus decomposition (EMD). The original image is divided into three-channel components of R, G and B firstly. Then we encrypt these components by two random phase mask (RPMs) modulations, two spiral phase transformations, and an equal modulus decomposition. Finally, we obtain four encryption keys which are RPM1, RPM2, an EMD’s angle and the order q of the SPT, and produce two private keys (PKs) which are PK1 and PK2. The proposed method has enhanced the key sensitivity of the order q with 100 times, while it maintains the advantages of simple structure, high speed and less data amount. The proposed method is highly robust to iteration attack and other attacks. Numerical simulation results show the effectiveness and feasibility of the proposed cryptosystem.

1. Introduction Nowadays, with the advent of the information age, the demand for information security is gradually increasing, and optical and digital technologies have been widely used in various security fields. Compared with digital technologies, optical systems have the advantages of parallel processing, poor coping ability, and the high degree of freedom such as amplitude, phase, and polarization, etc., which has led to extensive research on optical systems for securing data transmission. The earliest technology was the Double Random Phase Encoding (DRPE) technique proposed by Refregier and Javidi in 1995 [1]. However, due to the inherent linear nature of DRPE, it is proved to be vulnerable to special attacks such as known-plaintext attack (KPA) [2] and chosen-plaintext attack (CPA) [3] etc. [4]. Therefore, to improve the security, modifications [5–8], fractional Fresnel transform [9], gyrator transform [10–13], cylindrical diffraction transform [14,15], and Merlin transform [16] are derived based on DRPE.

⁎

The previously described methods are all symmetric encryption systems, which is not reliable enough because of the similarity between the encryption keys and the decryption keys. Therefore, Peng first proposed an asymmetric cryptosystem based on the phase truncated Fourier transform (PTFT) [17] to improve the security of the cryptosystem, in which the decryption keys are different from the ones in an encryption process. However, PTFT is still susceptible to special attacks, such as Fourier iterations [18,19]. Then, Cai et al. have proposed an equal-mode decomposition (EMD) based asymmetric cryptosystem [20], which is highly secure but proved to be still threatened by iterative retrieval algorithms (IRA) [21]. In order to solve this problem, modifications are derived by introducing Fresnel domain [22], fractional Fourier domain [23,24], cascaded EMD [25], and etc. [26–30]. The encryption methods of spiral phase transform (SPT) [31–34] proposed in recent years, which are widely used in image processing [35–37], can better protect the encryption system from iterative attack [3]. This is due to the emphasizing singularity of the modified spiral

Corresponding author. E-mail address: [email protected] (J. Wang).

https://doi.org/10.1016/j.optlastec.2020.106106 Received 9 October 2019; Received in revised form 23 November 2019; Accepted 28 January 2020 0030-3992/ © 2020 Elsevier Ltd. All rights reserved.

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phase function (MSPF) as the number of iterations increases, resulting in some pixel saturation in the image, which leads to error or random output. However, the SPT based encryption has some issues of complicated cryptosystem and insufficient key sensitivity [31,33,34], which leads to inefficiency and insecurity for a cryptosystem. Moreover, in color image encryption [35–43], the amount of data is larger, which is a challenge to transmission and storage, compared with grey image encryption, and it is effortless to generate cross-talk among channels when decrypting. Therefore, it is critical for color-image encryption to improve the efficacy and security without cross-talk. In this paper, we propose an asymmetric color-image cryptosystem based on SPT and EMD. The original color image is decomposed into three-channel components of R, G and B, which are as the real and imaginary parts modulated by RPM. Then the modulated results are performed by EMD once and SPT twice. In the process of encryption, it has four main encryption keys and produces two private keys (PKs). The proposed method mainly contributes as follows. Firstly, the sensitivity of the order q is 100 times higher than that of previous works. Secondly, the proposed cryptosystem is simple, fast and efficient. Thirdly, the superposition of the real part and imaginary part reduces the amount of data and saves the transmission bandwidth. Fourthly, the singularity of SPT makes the EMD-based method resistant to iteration attack. Finally, the complementarity of three technologies builds a highly secure cryptosystem. The numerical simulation results verify the feasibility and effectiveness of the proposed method. The rest of this paper is organized as follows. The second part outlines the principles of SPT and EMD. Then, the encryption process and optical implementation of the method are described in detail in the third part. The fourth part is numerical simulation and discussion of the results. The final part is the summary.

Fig. 1. Scheme of EMD.

statistically independent random phase mask R(x). Then the I'(x) can be constructed as below:

I ′ (u) = I { I (x ) ∙exp[i2πR (x )]}

(5)

where x and u represent the coordinates of the spatial domain and the Fourier domain, respectively. The I'(u) resolution is then decomposed into two masks P1 and P2 by EMD with the aid of θ. Here θ is a random value which in the range [0, 2π]. The principle of EMD is depicted in Fig. 1, and the two masks can be obtained from the following formulas:

P1 (u) =

A (u)/2 ∙exp[iθ (u)] cos(φ (u) − θ (u))

(6)

P2 (u) =

A (u)/2 ∙exp{i [2φ (u) − θ (u)]} cos(φ (u) − θ (u))

(7)

where A(u) and φ(u) represent the amplitude and phase part of I'(u), respectively.

2. Principle 2.1. Spiral phase transform (SPT)

3. Proposed cryptosystem In SPT, 2-D signum function sgn(u, v) is used to perform a twodimensional Hilbert transform, which is also referred as a SPF. The expression of two-dimensional signum function SPF(u, v) in spatial frequency space [44] can be expressed as:

SPF (u, v ) = sgn(u, v ) =

u + iv u2 + v 2

= exp{iφ (u, v )}

The order of R, G and B channels is not fixed. Here, we introduce the case where the G channel and the B channel are superimposed first, and the other cases are similar. 3.1. Process of encryption

(1)

where the φ represents the polar angle in the frequency domain. The SPF is unassigned at the origin, with a value of 0 or 1, which solves the concept of singularity. Based on SPF, we introduce a parameter q, the number of singular points or the order of SPF, which corresponds to the undefined point of SPF value (i.e. either 0 or 1), and the modulated SPF (MSPF) is defined as [45]:

MSPF = exp(iqφ (u, v ))

The encryption process steps are as follows: I. First, the original image is decomposed into three-channel components of R, G, and B. The G component and the B component are superimposed as the real part and the imaginary part, respectively. And the superimposed result is multiplied by a random phase mask (RPM1) as the input of the SPT. The outputSPT1 is described as follows:

(2)

Then the spiral phase transformation [46] for the function f(x, y) can be written as:

SPT {f (x , y )} =

I−1 {MSPF∙I {f

(x , y )}}

(3)

(8)

SPT 1 = SPT {U ∙RPM 1, q}

(9)

where SPT{·,q} represents the spiral phase transformation of order q.

I−1 {·} and

where I {·} denote the forward and inverse Fourier transforms, respectively. The expression of inverse SPT is given by:

ISPT {f (x , y )} = I−1 {conj (MSPF ) ∙I {f (x , y )}}

U = G + B∙j

II. The results are divided into phase and modulus, where the phase is the first private key (PK1). The modulus and R components of the image are superimposed as the real part and imaginary part, respectively. Then, EMD is performed on the product to obtain a private key (PK2) and The public key θ after the Fourier transformation.

(4)

where conj{·} denotes the complex conjugate. 2.2. Equal modulus decomposition (EMD)

[PK 2, θ , amp] = EMD {I {abs (SPT 1) + R∙j}} EMD is the process of dividing a two-dimensional image into two separate masks [47]. Fig. 1 shows an image decomposition method. Suppose I(x) is the original image. Firstly, I(x) is multiplied by a

(10)

where EMD{·} denotes the equal modulus decomposition, abs{·}means taking the amplitude part. 2

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4. Simulation results and discussion

III. Finally, multiply the output modulus of EMD and another random phase mask (RPM2) to obtain a complex function as the input of the second SPT, and the output result is the cipher expressed by:

cipher = SPT {amp∙RPM 2, q}

4.1. Encryption and decryption results

(11)

The decryption process is the inverse of the encryption process. I: First, the ciphertext is performed by inverse SPT, and the result is multiplied by the conjugate of RPM2.

In order to verify the validity and feasibility of the proposed method, we use MATLAB(R2016a) for numerical simulation. In this paper, we use Fig. 4(a) and (e) as the input primary images for the encryption system. Fig. 4(b) and (f) are the two PK1s, Fig. 4(c) and (g) are the two PK2s generated during the encryption process, Fig. 4(d) and (h) are the encrypted ciphertext images. The value order q of the SPT in this study is set to 30000.

Damp = ISPT {cipher , q} ∙conj (RPM 2)

4.2. Performance

3.2. Process of decryption

(12)

where ISPT{·,q} represents the inverse transformation of SPT of order q. II: The result of the first step and PK2 together act as the input of the inverse EMD, then the product performs inverse Fourier transformation for further decryption.

DIEMD = I−1\{ IEMD {Damp , PK 2}}

In this section, we show the results of decryption Besides, we will analyze the sensitivity of the keys and the time compared with exiting works. Determine the decryption effect and decryption quality by calculating the correlation coefficient (CC), peak signal-to-noise ratio (PSNR) and mean squared error (MSE) between the input and decrypted images. The CC and PSNR can be given by:

(13)

where IEMD{·} denotes the inverse transformation of EMD. III: Take the real and imaginary parts of the product, and then multiply the real part with PK1 as the input to the inverse SPT, and multiply the output with the conjugate of RPM1, respectively.

DU = ISPT {real (DIEMD ) ∙PK 1, q} ∙conj (RPM 1)

CC =

E {[Io − E [Io]][Id − E [Id]]} E {[Io − E [Io]]2 } E {[Id − E [Id]]2 }

(14)

PSNR (Io, Id ) = 10log10

DR = imag (DIEMD )

(15)

where real{·} and imag{·} denote taking the real part and the imaginary part, respectively. Ⅳ: Take the real part and the imaginary part of the DU, respectively to obtain the decrypted B component and G component and then combine them with the imaginary part obtained in the third step to obtain the decrypted image.

Dimg = cat {real (DU ), imag (DU ), DR}

MSE (Io, Id ) =

1 M∙N

M

2552M∙N ∑∀ x , y [Id (x , y ) − Io (x , y )]2

(17)

(18)

N

∑ ∑ |Io − Id |2 i=1 j=1

(19)

where Io and Id are the input and decrypted images, E{·}denotes the expected value of the function and M, N are the pixel coordinates of the image. Fig. 5 shows the results of decryption with the correct keys for each image, where Fig. 5(a)–(d) are the original images and Fig. 5(e)–(h) are the decrypted images. Compared with some well-known methods [48,49], the proposed method has a higher recovery quality. The CC and PSNR values for the original images and corresponding decrypted images are 1 and infinity, respectively. It can be seen that the entire cryptosystem is lossless and has a very good decryption effect with a high quality of decrypted images. On the other hand, it can be seen that the proposed method has no cross-talk among channels when decrypting.

(16)

where cat{·} respresents the combination of pictures. A block diagram of the proposed encryption and decryption process is shown in Fig. 2. The possible optoelectronic setup of the proposed method is shown in Fig. 3. The system is illuminated by a collimating laser beam. The ciphertext passes through two lenses and two spatial light modulators, then the product is combined with the PK2 in the beam splitter, the real and imaginary parts and the inverse SPT are implemented in a computer, which obtains the decrypted image.

Fig. 2. The flow chart of encryption and decryption. 3

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Fig. 3. Proposed optoelectronic setup for decryption. L, SLM, BS are lens, spatial light modulator, beam splitter, respectively, MSPF is modified spiral phase function.

Fig. 4. (a), (e) Primary images; (b), (f) the PK1s, (c), (g) the PK2s; (d), (h) final encrypted images.

Fig. 5. Original images (a)–(d) for Airplane, Woman, Tree, Female, (e)–(h) decrypted images of (a)–(d). 4

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Fig. 6. (a) CC values with change in SPF order q, (b)–(d) decrypted image with q change 0.01, 0.02 and 0.03, respectively. (e), (f) the order sensitivy of FrFT and FrMT in [49]. Table 1 The comparison of encryption and decryption times.

Kumar’s 256 × 256 The proposal 256 × 256 × 3

Table 2 Comparison on the data amount of the proposed and other methods.

Encryption time

Decryption time

81.83 ms 41.35 ms

48.53 ms 21.63 ms

Encryption methods

Kumar’s [29] Wang’s [38] Chen’s [44] Proposed method

4.2.1. Key sensitivity of order q of SPF Fig. 6(a) shows the CC values varying with the SPF order. Fig. 6(d)–(e) depicts the decrypted images with q changing 0.01, 0.02 and 0.03, respectively. In the previous SPT papers [31,33,34], when q value changes ± 2, CC value will change dramatically. As we can see, when q changes by 0.01, the decrypted image has a little blurred outline, but when q changes more than 0.01, the decrypted images do not contain any useful information of the original image. In the method

Data amount (one color image size = 1) Plaintext

Ciphertext

Private keys

Ciphertext + Private keys

1 1 1 1

1 1 1 1/3

1 2 1 2/3

2 3 2 1

proposed in this paper, when q value changes ± 0.02, it will change dramatically, and its sensitivity is 100 times higher than that of previous encryption methods. Because of the high security of EMD, by putting it ahead of SPT, the sensitivity of q value will be improved. 5

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Fig. 7. Polluted ciphertext images with Gaussian noise (a) k = 0.05, (b) k = 0.1, (c) k = 0.17, (d) k = 0.19, (e), (f) the occlusion performance.

Fig. 8. CC values change during phase retrieval.

Fig. 6(e) and (f) show the order sensitivity test of FrFT and FrMT in [49]. It can be seen that the sensitivity of these two methods is not as good as the sensitivity of the proposed improved SPT. 4.2.2. Time comparison with existing works In Kumar's article [32], the structure is complex, and the encryption and decryption speed is slow. The method proposed in this paper has the characteristics of simple structure, fast encryption and decryption speed, and good security performance. The time required for encryption and decryption is shown in Table 1. The encryption speed and the decryption speed of a single image by using the proposed method are about six times and about six times faster than that of Kumar. This work was done on a laptop with Processor Intel (R) Core (TM) i5-4590 @ 3.30 GHz, Memory 4096 MB RAM, and 64-bit OS Win10.

Fig. 9. CC values with the change of different disclosure proportion: (a) PK2's disclosure proportion, (b) PK1's disclosure propotion.

method, the ciphertext is only one-third of the plaintext in data amount, which greatly reduces the data amount. Table 2 shows a clear comparison on the data amount of before and after encryption.

4.2.3. Reduction in data amount In traditional color-image cryptosystems [31,42,47], the data amount of ciphertext is the same as that of the plaintext, and the total data amount of the ciphertext and private keys will increase a lot, which is a challenge for transmission and storage. In our proposed 6

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Fig. 10. (a) The original image, (b) the arbitray image, (c) decrypted image with arbitrary private key.

we set the encrypted image to be contaminated by Gaussian noise as follows:

Table 3 CC value when decrypt with arbitrary private key.

CC value

R

G

B

0.0046

0.0186

0.0120

E′ = E (1 + kG )

(20)

where E and E' represent an encrypted image and a noise-contaminated encrypted image, respectively. G is a Gaussian random noise having a mean of 0 with a variance of 0.05 and k denotes a Gaussian noise intensity coefficient. Fig. 7 shows the decrypted image of the ciphertext in the case of noise pollution, Fig. 7(a)–(d) corresponds to the decrypted images when the Gaussian noise of the intensity k = 0.05, 0.1, 0.17, 0.19, respectively. It can be seen that the strength of noises can reach 0.19, which depicts the system has good anti-noise performance. The test results verify the robustness of the proposed encryption scheme. Due to the separation of phase as well as the modulus in EMD and the combination of SPT, the encryption system has occlusion performance. The Fig. 7(e) and (f) show the occlusion performance of the

4.3. Attack analysis This section aims to analyze the robustness of the proposed cryptosystem under various attacks such as noise attacks, iteration attacks. Besides, we explore the impact of information disclosure on the cryptosystem.

4.3.1. Noise attack analysis The ciphertext is susceptible to noise pollution during transmission. In this section, we tested the effects of noise on the cryptosystem and

Fig. 11. The histgram of original image of (a) Lena, (c) Tree, the histogram of encrypted image of (b) Lena, (d) Tree. 7

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that different images have similar ciphertext histograms. Illegal users can’t attack the cryptosystem according to histograms.

proposed method, which represents the CC values and MSE values of the decrypted images when the encrypted image is obscured by N*N pixels.

5. Conclusions 4.3.2. Phase retrieval attack analysis We test the robustness of the encryption method by using the phase retrieval attacks, also called phase iteration attack. Under the constraint condition, the phase of PK2 is estimated, and the phase retrieval is performed [31,33,34,50]. Firstly, we give PK2 an estimated value PK2′0, according to EMD, P2′0 is expressed as follow:

P 2′0 = Damp ∙exp(i∙PK 2′0),

(21)

P1 = Damp ∙exp(i∙θ),

(22)

In this paper, we propose an asymmetric color-image cryptosystem based on SPT and EMD, which has a large key space due to four main encryption keys and two private keys. By skillfully combining the EMD and SPT, the proposed cryptosystem can improve the sensitivity of q value with 100 times compared with that of the traditional SPT, which finally enhances the security. Moreover, the simple structure and reduction of data amount make the method faster and higher-efficiency, which has remarkable performances such as anti-noise and resisting various attacks. Numerical simulation results verify the effectiveness and feasibility of the proposed cryptosystem.

where the Damp is the decrypted amp. Then, PK2′0 and Damp are taken as the input of IEMD, the output DIEMD is taken as the basis for the next iteration, which is given by:

P 2′k + 1 = DIEMD − P1,

(23)

P 2′k + 1 = Damp ∙exp(i∙arg(P 2′k + 1)),

(24)

CRediT authorship contribution statement Zheng Zhu: Conceptualization, Methodology, Software, Validation, Investigation, Data curation, Writing - original draft. Xu-Dong Chen: Resources, Writing - review & editing. Chao Wu: Conceptualization, Methodology, Resources, Writing - review & editing. Jun Wang: Conceptualization, Writing - review & editing, Writing - review & editing, Project administration. Weixing Wang: Resources.

The CC values changing with the increase of iteration numbers are shown in Fig. 8. As it can be seen from Fig. 8, the CC value is finally converged around 0, which indicates that the phase retrieval does not obtain the plaintext information, and the method is resistant to phase retrieval attack. This test shows that the cryptosystem has effective and feasible robustness.

Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

4.3.3. The disclosure of private key In the process of transmission, it is necessary to consider the information disclosure, illegal users may receive partial or complete private keys. We discuss the disclosure of PK1 and PK2 from partial to full in this section. Fig. 9(a) shows the disclosure of PK2, it can be seen that when PK2 is disclosed partly, the decrypted image has no features of the original image. Only in the case of full disclosure of PK2, the decrypted image barely contains any blurred outline, while the CC value of G channel is about 0.035 that attackers can hardly obtain any useful information. Fig. 9(b) depicts the gradual disclosure of PK1, we can see that no matter how much PK1 discloses, no original image information can be obtained. By comparing these two figures, it can be concluded that PK2 is a highly secure and independent key, and the disclosure of PK1 is not risky. The combination of these two keys enhances the security of the encryption system.

Acknowledgements This work is supported by the National Natural Science Foundation of China (NSFC) under Grant U1933132 and the Sichuan Science and Technology Program under Grant 2018GZ0533. Appendix A. Supplementary material Supplementary data to this article can be found online at https:// doi.org/10.1016/j.optlastec.2020.106106. References

4.3.4. Robustness against traditional attack Generally, there are four types of traditional attack, namely, cipher only attack, known plaintext attack, chosen cipher attack and chosen plaintext attack, among which chosen plaintext attack has the strongest performance. Hence, if a cryptosystem could resist chosen plaintext attack, it can resist other attacks. Based on the proposed method, the illegal user may use an arbitrary image to encrypt and obtain a fake private key, and could try to decrypt the original image with the obtained fake private key. The decrypted image is shown in Fig. 10(c), in which Fig. 10(a) and (b) are the original and arbitrary images, respectively. Table 3 is the CC value of each channel when decrypting with the fake private key. It shows that the proposed method has high security and strong robustness to resist chosen plaintext attack.

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4.3.5. Histogram analysis Histogram analysis is performed on the proposed method, which is an important indicator to describe image features. Fig. 11(a) and (c) are histograms of Lena and Tree, respectively. It shows that the histograms of different images are completely different. Fig. 11(b) and (d) are ciphertext histograms of Lena and Tree, respectively. It can be observed that they are quite similar. After many experiments, it can be concluded 8

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