Optical noise-free image encryption based on quick response code and high dimension chaotic system in gyrator transform domain

Optics and Lasers in Engineering 91 (2017) 106–114

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Optics and Lasers in Engineering journal homepage: www.elsevier.com/locate/optlaseng

Optical noise-free image encryption based on quick response code and high dimension chaotic system in gyrator transform domain

MARK

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Liansheng Suia, , Minjie Xua, Ailing Tianb a b

School of Computer Science and Engineering, Xi'an University of Technology, Xi'an 710048, China Shannxi Province Key Lab of Thin Film Technology and Optical Test, Xi'an Technological University, Xi'an 710048, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Double random phase encoding Image encryption Phase retrieval algorithm Gyrator transform

A novel optical image encryption scheme is proposed based on quick response code and high dimension chaotic system, where only the intensity distribution of encoded information is recorded as ciphertext. Initially, the quick response code is engendered from the plain image and placed in the input plane of the double random phase encoding architecture. Then, the code is encrypted to the ciphertext with noise-like distribution by using two cascaded gyrator transforms. In the process of encryption, the parameters such as rotation angles and random phase masks are generated as interim variables and functions based on Chen system. A new phase retrieval algorithm is designed to reconstruct the initial quick response code in the process of decryption, in which a priori information such as three position detection patterns is used as the support constraint. The original image can be obtained without any energy loss by scanning the decrypted code with mobile devices. The ciphertext image is the real-valued function which is more convenient for storing and transmitting. Meanwhile, the security of the proposed scheme is enhanced greatly due to high sensitivity of initial values of Chen system. Extensive cryptanalysis and simulation have performed to demonstrate the feasibility and effectiveness of the proposed scheme.

1. Introduction As we know, image security that utilizes optical techniques has become an important research topic during the past decades. Since Refregier and Javidi proposed the famous image encryption architecture based on double random phase encoding (DRPE) in Fourier transform domain [1], a large number of schemes which make use of different optical techniques such as diffractive imaging [2,3], integral imaging [4,5], ghost imaging [6–8], photon-counting [9,10], polarized light encoding [11], interferometer [12,13], compressive sensing [14– 16] and ptychography [17] have been suggested. It is worth noting that the plain image can be encrypted and compressed simultaneously by using special optical schemes which are introduced by Alfalou and Brosseau [18]. Numerous kinds of optical image encryption techniques are analyzed [19,20], which provide potential solutions to purely optical cryptosystem. Recently, Javidi et al. [21] present an overview of the potential, recent advances and challenges of optical security and encryption using free space optics in different aspects such as novel encryption approaches, compression for compressed data, phase retrieval algorithms, implementation at nano- or micro-scale, ghost imaging and quantum imaging. Due to intrinsic linearity, image encryption schemes based on DRPE

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architecture have serious security risks, namely they are vulnerable to several forms of attack such as known plaintext attack [22–24]. In order to enhance security, the DRPE-based architecture has been extended into various transform domains such as fractional Fourier transform domain [25,26], Fresnel transform domain [27–29], gyrator transform (GT) domain [30–32], fractional angular transform domain [33,34], fractional random transform domain [35], fractional Mellin transform domain [36], gyrator wavelet transform domain [37] and liftingwavelet transform frequency domain [38] in which additional parameters can be employed as the private keys. The output of these schemes usually is the complex amplitude distribution, which is not convenient to store and transmit because optical elements such as spatial light modulator cannot record the phase and amplitude data simultaneously. To avoid considering the complex data as ciphertext, a plenty of schemes based on interference is suggested since Zhang and Wang [39] originally proposed to encrypt a plain image into two phaseonly masks (POMs) without using iterative calculations. However, an inherent problem among these optical cryptosystems cannot be resolved efficiently, where the silhouette information of the plain image can be detected presumably when any one of resultant masks is deployed in the process of decryption [40–45]. Additionally, a plenty of image encryption schemes based on different chaotic maps are

Corresponding author.

http://dx.doi.org/10.1016/j.optlaseng.2016.11.017 Received 5 August 2016; Received in revised form 17 November 2016; Accepted 17 November 2016 0143-8166/ © 2016 Elsevier Ltd. All rights reserved.

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Let the function f (x, y ) with size of M × N pixels denotes the consequential QR code, which is illuminated by a coherent parallel light beam and encrypted by using two chaotic RPMs represented by r1 (x, y ) and r2 (x′, y′), respectively. Initially, f (x, y ) is multiplied with the first RPM r1 (x, y ), and then the product is optically transformed by using the first GT with the rotation angle α1. The transformed result is multiplied with the second RPM r2 (x′, y′), and then the product is optically transformed by using the second GT with the rotation angle α2 . Finally, by superimposing the transformed result on the plane reference beam, the intensity distribution in the output plane is considered as the ciphertext which can be captured in the output plane by recording the holographic interference fringe as an off-axis hologram with CCD camera. The process can be expressed mathematically as

presented [46–49], in which the plain image can be encoded as the realvalued function. Recently, approaches have been reported to solve the silhouette problem by making use of additional techniques such as time-consuming transformation, encoding with larger number of POMs and phase retrieval algorithms. In the scheme suggested by Kumar et al. [50], this shortcoming is surmounted by employing the jigsaw transformation in a single step. Wang and Zhao [51] generalized earlier interferencebased encryption scheme with two POMs and suggested to hide the information of plain image into three POMs, in which it is still possible that the remnant information can be intercepted by an unauthorized user when two of POMs are known simultaneously. Wang et al. [52] encrypted the plain image into two complex ciphertexts, in which it makes the decryption process complicated that the random POM and related complex field distribution should be modulated by two predesigned phase-only factors. Wang et al. [53] presented an image hiding method to encrypt the plain image into two POMs by using the phase retrieval algorithm under the framework of nonlinear DRPE, which is time-consumed. Due to the property of fast read capacity, Quick response (QR) code has been widely deployed in the field of image encryption. Wang et al. [54] proposed an optical encryption technology by using the known positions of QR code as support constraint, in which the original image can be decrypted with the phase retrieval process. Barrera et al. [55] reported an experimental implementation of a noisefree data recovering based on the joint transform correlator architecture, in which the QR code is used as a container of plain image. Wang et al. [56] designed a secured information retrieval scheme of triple images based on two authenticated phase-only masks, which are calculated with three QR codes of plain images as amplitude constraints. Different from aforementioned schemes, a novel optical encryption system based on QR code and high dimension chaotic system is proposed under the architecture of DRPE in GT domain in this paper. In the process of encryption, the real-valued ciphertext image can be obtained by only recording the intensity distribution of two cascaded GTs, which are performed on the QR code of the plain image. In the process of decryption, a new phase retrieval algorithm is designed to reconstruct the corresponding QR code from the ciphertext image, in which the known three position detection patterns of QR code are used as the support constraint to achieve high convergence speed. Importantly, although the reconstructed QR code may be destroyed seriously under some kinds of attack, the plain image can be visualized perfectly without any energy loss of information due to high error correction capability of QR code. Finally, numerical simulation results are carried out to demonstrate the feasibility and effectiveness of the proposed scheme. The rest of this paper is organized as follows. In Section 2, the encryption and decryption processes are introduced in detail. In Section 3, numerical simulation results and security analysis are given. Finally, the conclusion is given in Section 4.

C (x″, y″) = G α2 {G α1 {f (x, y ) × r1 (x, y )} × r2 (x′, y′)} 2 ,

(1)

G α {⋅}

where denotes the GT with the rotation angle α and ⋅ represents the modulus operation. Due to its excellent properties such as the fixed distance between generalized lenses and input-output planes, GT has been widely employed to image encryption [57,58]. Additionally, GT can be implemented optically by utilizing the FT, inverse FT and phaseonly filtering with the help of convolution operation [59]. It should be pointed out that the proposed scheme only records the intensity information of the transformed result as the ciphertext with stationary white noise distribution and is different from other encryption schemes under the framework of DRPE where the encrypted results usually are the complex amplitude functions. As a high-dimension chaos function, Chen system has more complicated dynamical property, which makes it more suitable for practical applications in the fields of optical image encryption [60]. In order to further improve the security of the proposed scheme, two chaotic RPMs r1 (x, y ) and r2 (x′, y′) used in Eq. (1) are generated randomly based on Chen system which is expressed as follows

⎧ x ̇ = a (y − x ) ⎪ ⎨ y ̇ = (c − a ) x − xz + cy , ⎪ z ̇ = xy − bz ⎩

(2)

where a , b and c denote control parameters. Given the initial values x 0 , y0 and z 0 , when the control parameters are set to a = 35, b = 3 and c ∈ [20, 28.4], three different random value sequences with non-periodic and non-convergent properties can be engendered by iterating this chaotic system with large iterations. In the process of iteration, the fourth order Runge-Kutta algorithm with the small step value such as 0.001 is repeatedly performed. Let three random value sequences are denoted as xi , yi and zi , respectively. According to first two sequence xi and yi , only the last M × N iterative values are preserved to form two new sequences s1 and s2 as follows

s1 = 2π × ((abs (xi ) − floor (abs (xi ))) × 1014 mod(256))/255,

(3)

s2 = 2π × ((abs (yi ) − floor (abs (yi ))) × 1014 mod(256))/255.

(4)

where abs (⋅) is used to compute the absolute value and floor (⋅) is used to obtain the nearest integer of the argument. Through rearranging the s1, sequence the two-dimensional matrix denoted as {s1′(i , j )|i = 1, 2, …, M ; j = 1, 2, …, N} is obtained, with which the first RPM r1 (x, y ) is produced as exp(is1′(i , j )). Similarly, the second RPM r2 (x′, y′) is generated as exp(is2′(i , j )) with the matrix {s2′(i , j )|i = 1, 2, …, M ; j = 1, 2, …, N} after rearranging the sequence s2 . Apparently, s′1 and s′2 are two independent random phase functions distributed in the range [0, 2π ]. Obviously, two RPMs r1 (x, y ) and r2 (x′, y′) are used as the interim functions and not considered as the private phase keys directly. Because no random phase masks are used as the private keys, the proposed scheme has high convenience to the management of secret keys in the process of storage and transmission. Different from other encryption schemes in gyrator transform domain where the rotation angles are predefined as the fixed values, α1 and α2 used in Eq. (1) are respectively calculated from the matrices s′1

2. Encryption and decryption processes As an important two-dimensional barcode with many excellent properties such as fast readability with mobile devices such as smart phones, large storage capacity and high error tolerance capability, QR codes have been demonstrated great potential for image encryption. In the proposed scheme, the plain image to be encrypted is first transformed to the corresponding QR code by making use of some generation tools. The consequential QR code as an information container is considered as the input image of the DRPE architecture in the gyrator transform domain. The optical setup is schematically shown in Fig. 1, in which tow chaotic random phase masks (RPMs) are placed in the spatial plane and frequency plane, respectively, and the intensitysensitive device such as charge-coupled device (CCD) camera in the output plane. 107

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Fig. 1. Optical setup of encryption system.

Fig. 2. (a) Plain image “Baboon”, (b) generated QR code, (c) support constraint, (d) ciphertext, (e) decrypted QR code and (f) retrieved “Baboon”.

variables in the proposed scheme, which are not employed as the private keys. Only the initial values x 0 , y0 and z 0 of Chen system are used as the private keys. So, the proposed scheme has camouflage property to some extent. Most image encryption schemes under the framework of DRPE have an obvious property that the encrypted results are comprised of the amplitude and phase part, where the phase part contains more information than the amplitude part in the different transform domain. So, the plain image can be easily restored from the ciphertext with the simple decryption process that are basically the inverse of the encryption process, in which the decryption keys are usually identical to the encryption keys. However, because no any phase information is included in the ciphertext in the proposed scheme, the plain image should be decrypted by using other measurement process such as the phase retrieval algorithm. As we all know, the phase retrieval algorithm has the fundamental drawback that the stagnation problem may be encountered when two

and s′2 by making use of the third random sequence zi of Chen system, where some elements of the matrices are randomly chosen from s′1 and s′2 . Let the number of the selected elements is K , only the last 2K values of the sequence zi are preserved to form the sequence s3 as follows

s3 = floor ((abs (zi ) − floor (abs (zi ))) × 1014).

(5)

Supposing the coordinates of the selected elements are denoted as (mi , ni ) , i = 1, 2, … K , these coordinates can be determined as mi = s3 (2i )mod M and ni = s3 (2i + 1)mod N . Then, α1 and α2 are computed mathematically as K

α1 =

∑ s1′(mi, ni )/(2πK ), i =1

(6)

K

α2 =

∑ s2′(mi, ni )/(2πK ). i =1

(7)

Obviously, the rotation angles are engendered as the interim 108

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Fig. 3. (a) Chaotic RPM r1 (x, y ), (b) chaotic RPM r 2 (x′, y′) and (c) CC curve between the initial QR code and the decrypted one.

or less intensity data are used in the iterative process. Though only one intensity data is used in the proposed scheme, the stagnation problem can be eliminated efficiently by using additional information as the

support constraint, namely three position detection patterns located in the top-left, top-right and bottom-left in the QR code f (x, y ). Let the function P (x, y ) denotes the support constraint image which includes

Fig. 4. Decrypted QR code with (a) incorrect x 0 , (b) incorrect y0 and (c) incorrect z0 .

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Fig. 5. CC curve with (a) incorrect x 0 , (b) incorrect y0 and (c) incorrect z0 .

fˆk (x, y ) = G−α1 {G−α2 {gˆk (x″, y″)} × conj (r2 (x′, y′))} × conj (r1 (x, y )),

three position detection patterns, a new phase retrieval algorithm is designed to retrieve the initial QR from the ciphertext with high fidelity, which can be described as below:

(10) where G−α {⋅} denotes the inverse GT with the rotation angle α , and conj (⋅) returns the conjugate value of the argument. (5) In next iteration, with the help of the support constraint image P (x, y ), the estimation of the original QR code is updated to fk+1 (x, y ), which is expressed mathematically as

(1) Guess an initial estimation f0 (x, y ) of the original QR code, and then start the following iterative process. In the proposed scheme, f0 (x, y ) is set as a two-dimension array of all ones, which has M × N elements. (2) For the kth iteration, the estimation fk (x, y ) of the original QR code is illuminated by the coherent parallel light beam, and then modulated with two chaotic RPMs r1 (x, y ) and r2 (x′, y′) in the spatial and frequency plane, respectively. In the output plane, the transformed result by using two cascaded GTs with the rotation angles α1 and α2 is acquired as

gk (x″, y″) = G α2 {G α1 {fk (x, y ) × r1 (x, y )} × r2 (x′, y′)}.

fk +1 (x, y ) = (β × P (x, y ) × ( fˆk (x, y ) + fk (x, y ) × P (x, y )))/(P (x, y ) + γ ), (11)

where β and γ are the adjustment parameters. The parameter β is used to adjust the convergent speed of the iteration process and γ is exploited to ensure that the denominator of Eq. (11) is not equal to zero. (6) In order to determine when the iteration stops, the correlation coefficient (CC) between the amplitude of light distribution function fˆk (x, y ) and the original QR code f (x, y ) is used as convergent criterion, which can be computed mathematically as

(8)

(3) Substitute the amplitude part of interim function gk (x″, y″) with the recorded intensity distribution, namely the ciphertext C (x″, y″) and hold the phase part unchanged, the light distribution function is modified as follows

gˆk (x″, y″) = C (x″, y″) 1/2 × arg(gk (x″, y″)),

CC =

E {[f − E [f ]][|fk | − E [|fk |]] } E {[f − E [f ]]2 } E {[|fk | − E [|fk |]]2 }

, (12)

where E [⋅] denotes the expected value operator. If CC value is larger than a predefined threshold which is close to 1, the best iteration result is achieved. (7) Repeat the above steps (2) through (5) until the CC value reach the predefined threshold. Once the iteration process is finished, the last updated result fk+1 (x, y ) will be considered as the decrypted QR code.

(9)

where arg(⋅) is used to extract the phase part of the argument. (4) Take the inverse process of the encryption as indicated in Eq. (8) according to the modified light distribution gˆk (x″, y″), in which gˆk (x″, y″) is transformed by using two cascaded GTs with the rotation angles −α1 and −α2 , respectively. In the input plane, the light distribution function is obtained as 110

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Fig. 6. Decrypted QR code with the coefficient (a) k = 0.2 , (b) k = 0.4 , (c) k = 0.6 , (d) k = 0.8, (e) k = 1.0 and (f) retrieved plain image “Baboon”.

QR code by using the free software tool [65]. The QR code is then attached in the input plane of the optical encryption system shown in Fig. 1, in which it is illuminated by the coherent parallel light beam and transformed with two cascaded GTs. In the output plane, only the intensity distribution is preserved as the ciphertext. For Chen system used to generate two RPMs and two rotation angles of GTs, the initial values x 0 , y0 and z 0 are set to 0.25, 1.005 and 0.35, respectively. In the phase retrieval process, the adjustment parameters β , γ are set to 0.09, 0.0001, respectively. Fig. 2(b) displays the profile of the corresponding QR code, with which three position detection patterns are extracted to constitute the picture of the support constraint as shown in Fig. 2(c). Fig. 2(d) displays the encrypted result of the QR code, namely the ciphertext that is noiselike. Fig. 2(e) is the decrypted QR code by using the proposed phase retrieval process, which has high quality as the same as the initial one. As demonstrated in Fig. 2(f), the original image can be recovered without any loss of information visually by scanning Fig. 2(e) with a mobile device. Figs. 3(a) and (b) show the chaotic RPMs with noise-like distribution, respectively, which are independent random functions. Fig. 3(c) shows the calculated CC curve between the initial QR code and the decrypted one, from which it is known that the CC value is larger than 0.9978 after 200 iterations. So, the proposed scheme has better performance on the convergence speed. Though only three initial values of Chen system are employed as the private keys, which is convenient to key management, the proposed scheme has enhanced security due to high sensitivity of these keys. So, the sensitivity of the private keys to the tiny deviation is investigated. Figs. 4(a), (b) and (c) show the decrypted QR code when one of the initial values has a little deviation, from which it is obvious that any

Once the iteration process stops, the plain image can be revealed without any noise destruction by scanning the decrypted QR code with mobile devices. It is worth noting that the security of the proposed scheme is enhanced greatly due to high sensitivity of the initial values of Chen system, which are used to generate the system parameters such as two RPMs and two rotation angles of GTs. These parameters are interim functions and variables produced in the process of encryption and an unauthorized user cannot access them directly, which makes that the proposed scheme has high resistance against to various possible attacks. Different from the optical compression approaches to simultaneously encrypt the original image [61–63], a new encryption scheme to encode the QR code containing the plain image is described in this discussion, where the volume of the original data is not compressed in the encryption process. Although only the intensity of cascaded transformed results is recorded as the ciphertext, which indicates the scheme can be easily implemented with optical setup, the volume of the ciphertext is not decreased. However, the size of QR code can be reduced to some extent by using certain generation tools, which means the encryption results can be reduced. It is obvious that the robustness of the proposed scheme will be deteriorated when the size of the QR code decreasing. There is a trade-off between these two aspects, which will be the research work in the future. 3. Numerical simulation and security analysis A series of numerical simulations have been carried out to verify the feasibility and security of the proposed scheme. As shown in Fig. 2(a), the gray-scale image “Baboon” selected from USC-SIPI image database [64] is chosen as the plain image to be encoded into the corresponding 111

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Fig. 7. (a) Ciphertext with 20% occlusion, (b) decrypted QR code, (c) improved QR code and (d) retrieved plain image “Baboon”.

proposed scheme with correct keys, the resultant QR code images are displayed in Figs. 6(a)–(e), respectively, in which k equals 0.2, 0.4, 0.6, 0.8 and 1.0. It is obvious that the quality of the decrypted QR code becomes worse gradually with the noise strength increasing. According to the decrypted results with recognizable pattern depicted in Figs. 6(a)–(d), the original plain image shown in Fig. 6(f) can be retrieved by scanning them directly with mobile devices. For the worst one depicted in Fig. 6(e), some post-processing operations are employed to improve its quality. When analyzing the effects of occlusion attack, certain percentage of the pixels in the ciphertext is discarded while the remainder is decrypted to obtain the initial QR code with correct keys. Fig. 7(a) shows the occluded ciphertext with 20% occlusion size from the top side. Fig. 7(b) shows the decrypted QR code with poor quality, which is almost the same as the image shown in Fig. 6(e) due to the effect of energy loss. As depicted in Fig. 7(c), the QR code has better appearance after performing median filtering with the 3-by-3 neighborhood and binarization sequentially, from which the plain image can be retrieved efficiently without visible loss of information as shown in Fig. 7(d). In a nutshell, the results have demonstrated that the proposed scheme has exhibited good performance to resist the noise and occlusion attacks. It is well known that the histogram as a statistical evaluation tool is usually utilized to estimate the information of an image, which shows the intensity distribution of the pixels. When the histograms of the different ciphertext images have the same profile approximately, the

information about the initial QR code cannot be discerned visually and the original plain image cannot be cracked. According to Figs. 4(a), (b) and (c), the deviation of initial values are set as x 0 = 0.25 + 1.0e − 14 , y0 = 1.005 + 1.0e − 15 and z 0 = 0.35 + 1.0e − 14 , respectively. The corresponding CC curves are displayed in Figs. 5(a), (b) and (c), respectively, in which the max CC values after 300 iterations only are 0.1137, 0.1171 and 0.1154, respectively. Let the deviation of the initial values are denoted as Δx0 , Δy0 and Δz0 , respectively. Generally, any valid useful information cannot be retrieved from the decrypted QR code when one of initial values satisfies Δx0 ≥ 1.0e − 14 , Δy0 ≥ 1.0e − 15 or Δz0 ≥ 1.0e − 14 . Additionally, the entire key space has reached 1.0e + 43, which is enormous enough to resist against the brute-force attack. Usually, the ciphertext is vulnerable to pixel errors caused by noise and occlusion attacks in the process of storage and transmission. So, it is necessary to investigate the robustness against these undesirable situations. When considering the effects of noise attack, the ciphertext is supposed to be polluted with an additive Gaussian random noise, denoted as G (x″, y″) with zero-mean and standard deviation 0.5, which is expressed as

C′(x″, y″) = C (x″, y″) × (1 + kG (x″, y″)),

(13)

where C′(x″, y″) is the noise-polluted ciphertext and k is the coefficient to describe noise strength. After the modified ciphertext images polluted with different noise strength are decrypted by using the 112

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Fig. 8. Histogram of (a) the ciphertext shown in Fig. 2(d), (b) the ciphertext encrypted with QR code of the image “Lena” and (c) Retrieved image by using the known plaintext attack in the DRPE architecture.

attackers cannot intercept any valid information by this way. Fig. 8(a) shows the histogram of the ciphertext image depicted in Fig. 2(d), which is encrypted result of the QR code of the plain image “Baboon”. Fig. 8(b) shows the histogram of the ciphertext encrypted with another QR code of the plain image “Lena”, which also is selected from the image database [64]. Compared with two results, it can be concluded that the histograms of the ciphertext encrypted with different QR code of plain images have similar distribution, which means that the proposed scheme can defeat this kind of statistical attack efficiently. Because only the intensity distribution in the output plane of optical system is considered as the ciphertext, the linear relationship between the plain image and its ciphertext can be broken thoroughly. So, the proposed scheme has high security to resist several forms of attacks such as known plaintext attack. Moreover, though two RPMs in Eq. (1) can be retrieved approximately by using the known plaintext attack in the DRPE architecture, the accurate rotation angles α1 and α2 cannot be computed mathematically by using Eqs. (6) and (7), even the coordinates of pixels selected from two phase functions are known. Fig. 8(c) shows the decrypted result by using the known plaintext attack, in which the retrieved image is random noise-like. So, the proposed scheme can well resist unauthorized user's attack.

generated by using Chen system. Because only the intensity distribution in the output plane is recorded as the ciphertext, a new phase retrieval process is designed to reconstruct the initial QR code. By scanning the decoded QR code, the original plain image can be recovered without any energy loss. The proposed scheme has obvious properties: the ciphertext is real-valued function, which makes the storage and transmission of ciphertext more convenient; only three initial values of Chen system are used as the private keys, which means the key management also is convenient; the security is enhanced greatly because of high sensitivity of the initial values of Chen system; because the parameters such as rotation angles of gyrator transform are produced as interims and cannot be accessed directly, the proposed scheme has high resistance against to various possible attacks such as known plaintext attack and so on.

4. Conclusions

References

Acknowledgments This work was supported by Xi'an Science and Technology Bureau under Grant CXY1509(3), and Key Laboratory Science Research Plan of Education Department of Shaanxi Province under Grant Number 16JS079.

[1] Refregier P, Javidi B. Optical image encryption based on input plane and Fourier plane random encoding. Opt Lett 1995;20(7):767–9. [2] Chen W, Chen X, Sheppard CJR. Optical color-image encryption and synthesis using coherent diffractive imaging in the Fresnel domain. Opt Express 2012;20(4):3853–65. [3] Mehra I, Nishchal NK. Optical asymmetric watermarking using modified wavelet fusion and diffractive imaging. Opt Lasers Eng 2015;68:74–82. [4] Li X, Cho S, Kim S. High security and robust optical image encryption approach based on computer generated integral imaging pickup and iterative back-projection

In summary, a novel optical image encryption scheme is proposed based on QR code and high dimension chaotic system in the gyrator transform domain, in which only the intensity information of encoded result is employed as the ciphertext. In the process of encryption, the QR code containing the plain image is placed in the input plane of DRPE architecture and then encoded with cascaded gyrator transforms, in which two chaotic RPMs and two rotation angles of the transforms are 113

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