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Optical encryption and authentication scheme based on phase-shifting interferometry in a joint transform correlator
Optics and Laser Technology 126 (2020) 106108
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Optical encryption and authentication scheme based on phase-shifting interferometry in a joint transform correlator Y. Xiong, J. Du, C. Quan
T
⁎
Department of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, Singapore 117576, Singapore
H I GH L IG H T S
drawbacks of the cryptosystem based on phase-shifting in a JTC. • Demonstrated an advanced JTC-based cryptosystem free from these drawbacks. • Proposed an intensity modulator based on histogram equalization and photon-counting. • Employed • Employed a post-processing operator to extract phase part of ciphertexts and authenticate.
A R T I C LE I N FO
A B S T R A C T
Keywords: Optical image encryption and authentication Photon-limited binary intensity pattern Phase-shifting Joint transform correlator
In this paper, the effectiveness of the cryptosystem based on phase-shifting technique (PST) in a joint transform correlator (JTC) is evaluated. It is shown that the cryptosystem has some inherent drawbacks, for example, since the plaintexts are required to obtain the phase-shifting intensity patterns in the first phase extraction step, which means that the information of the plaintexts should be known by the authorized users before decryption processing. It seems impossible and paradoxical in the practical case. Moreover, the cryptosystem can only achieve binary plaintexts encryption while the inputs of this scheme cannot be the gray-scale or color images. To address these issues, we propose an encryption and authentication scheme based on PST in a JTC. By introducing a preprocessing operation including an intensity modulator based on histogram equalization and photon-counting (PC) operator, a photon-limited binary distribution of the plaintexts is obtained and used as the input of the cryptosystem. Due to this design, our proposed scheme can achieve encryption for different kinds of images such as binary, gray-scale and color images using same optical setup without any changes, which can simplify the optical implementation. Furthermore, a post-processing operator to extract phase information of encoded images and authenticate the decoded images is introduced. Since no useful information will be released from the photon-limited binary decoded images obtained using correct decryption keys and the interface of the database storing the plaintexts is only given to the authorized users, an additional security layer is established. This is the first time to report the use of binary intensity pattern in a scheme based on PSI in a JTC for secured verification of different images.
1. Introduction With the widespread development and use of computer and internet, different encryption techniques for transmission of secure information via digital communication channels have attracted growing interest [1–4]. Compared to the digital encryption techniques, optical image encryption techniques have noticeable advantages, such as highspeed parallel processing and multidimensional capabilities [5]. A pioneering work in the field of optical image encryption, named double random phase encoding (DRPE), has been proposed by Refregier and
⁎
Javidi in 1995 [6]. In the classical DRPE system [6], an input image is converted into the stationary white noise by using statistically independent random phase-only masks (RPMs) respectively located at the input and Fourier planes. Sequentially, DRPE algorithm has been extended from Fourier domain into other domains, such as fractional Fourier [7–9], Fresnel [10,11] and gyrator [12–14] domains, to further improve security of the traditional DRPE system [1] by using structure parameters as additional private keys to enlarge the key space. In addition to the aforementioned works, cryptoanalysis is important and necessary for any security system including DRPE. The DRPE system
Corresponding author. E-mail address: [email protected] (C. Quan).
https://doi.org/10.1016/j.optlastec.2020.106108 Received 13 September 2019; Received in revised form 30 November 2019; Accepted 28 January 2020 0030-3992/ © 2020 Elsevier Ltd. All rights reserved.
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has been demonstrated that it is vulnerable to various attacks [15–20] due to the inherent linearity introduced by the Fourier transform. In addition, since the encrypted image obtained using DRPE system is complex matrix including both amplitude and phase information, the encrypted data needs to be registered holographically. It means that DRPE system requires accurate optical alignment, which in practice is difficult to attain. To alleviate this constraint, JTC has been introduced in the DRPE structure [21]. In the JTC-based cryptosystem [21], the plaintext bonded with a RPM is placed side by side with the encryption key in the input plane and the intensity distribution of the joint power spectrum (JPS) as the encrypted data can be recorded using a common power-law sensor, such as a charge-coupled device (CCD). Consequently, various encryption schemes based on JTC [22–28] have been proposed. On the other hand, various authentication schemes based on optical techniques have also been proposed [29–34]. Compared to the traditional encryption schemes in which the information of plaintexts is directly visible from decoded images obtained using correct private keys, decoded images cannot visually render information about plaintexts in the authentication schemes. Then optical authentication methods with a separated or remote database where stores plaintexts are further applied to verify decoded images. Since the interface of the database is only given to the authorized users, an additional security layer is established to resist the potential attacks. Various attacks have proposed to crack the JTC-based encryption schemes [35–38] while few attacks have been investigated to break the JTC-based authentication schemes. Cecilia and Iemmi [39] proposed a DRPE-based optical encryption technique in which the PST in a JTC system is used. Compared to the traditional DRPE in a JTC structure [21] in which complex distribution representation for the input plane is required, only a pure phase modulator is used to display the input in the cryptosystem proposed in [39]. It further simplifies the optical alignment for the JTC-based cryptosystems. However, there are some existing drawbacks in the cryptosystem [39] which can be summarized as: (1) only binary amplitude information can be introduced due to the mathematic principle of the specific technique, which means the proposed cryptosystem in [39] can be only used to encrypt binary plaintexts, (2) since two phase extraction processes based on three-step PST are used to retrieve the information of binary plaintexts in the decryption process, the plaintext is required to obtain the intensity patterns in the first phase extraction process; however, it is impossible in the practical decryption process due to unknown knowledge of plaintexts. On the other hand, if the first threestep PST-based extraction process is considered as the part of the encryption process, three corresponding intensity patterns for a binary plaintext have been recorded as ciphertexts, which may burden the transmission of ciphertexts. On the one hand, the experimental work in [39] is attractive and promising. On the other hand, the cryptosystem in [39] has some drawbacks. Hence, the motivation of our work is to propose a novel encryption and authentication scheme based on the optical implementation in [39] with some modifications. In this work, we propose an JTC-based optical image encryption and authentication algorithm using an elaborately designed operator and PC. It is noteworthy that our proposed cryptosystem has advantages. Firstly, with the help of the elaborately designed pre-processing operator including an intensity modulator based on histogram equalization and a PC operation, our proposed cryptosystem can achieve encryption and authorization not only for binary images but also for gray-scale and color plaintexts. It is noteworthy that the cryptosystem can achieve different kinds of images (binary, gray-scale, color images) encryption and authorization for the first time. Secondly, since the decoded images do not visually render the input information due to the designed optical encoding strategy using only photon-limited binary intensity pattern, an additional security layer is established. The rest of this paper is organized as follows. In Section 2, a brief
Fig. 1. Schematic diagram of the system for (a) encryption and (b) decryption processes in [39]. SF and L are spatial filter and lens, respectively. P1 and P2 are polarizers while WP1 and WP2 are wave plates. The programmable liquidcrystal television display (LCTV) is used to display input signals and achieve phase shifting. The computer is used to drive a LCTV and a charge-coupled device (CCD).
introduction and analysis for the optical cryptosystem in [39] is given. In Section 3, an advanced JTC-based encryption and authentication scheme using an elaborately designed operator and PC is proposed with support by a number of simulations, and the paper is concluded in Section 4. 2. The cryptosystem under study The schematic diagram of the JTC-based optical cryptosystem for binary data encryption proposed in [39] is shown in Fig. 1. To display the binary images (which is amplitude information) and phase-only mask in a pure modulator, the technique proposed in [39] is described as follows. The scene distribution is given by
g (x , y ) = exp{i2π [f (x , y ) n (x , y )]} × exp{l (x − a, y − b)[1 − f (x , y )]}, (1) where f (x , y ) is the binary plaintext, n (x , y ) is a random function distributed uniformly in the interval [0, 1]. l (x , y ) is a quadratic phase distribution corresponding to a convergent lens of focal distance d for wavelength λ . x and y are indices in input plane. Since f (x , y ) is a binary image, g (x , y ) given in Eq. (1) can be expressed as
when f (x , y ) = 1 ⎧ g (x , y ) = exp{i2πn (x , y )}, ⎨ g ( x , y ) = exp{ l ( x − a , y − b )}, when f (x , y ) = 0 ⎩
(2)
The small region obtained when g (x , y ) is illuminated by a uniform plane wave and then the light passing through the area with f (x , y ) = 0 followed by the free propagation is centered at coordinates p = a , q = b ( p and q are indices of the Fresnel domain), which can be considered that no lights arrive at the output plane with properly set for a and b in the practical encryption process. A complex distribution given by Eq. 2
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Fig. 2. (a) The binary image to be encoded, (b) a phase-only mask n (x , y ) , (c) a reference wave r (x , y ) , (d–f) three intensity patterns (I0 (u, v ) , I2 3 (u, v ) and I4 3 (u, v ) ), (g) the phase difference between G (p , q) and R (p , q) , (h) the decoded binary image.
process, a constant distribution C (p , q) = exp[iφC ] is used to replace the unknown distribution E (p , q) at the output plane, while the Fresnel spectrum R (p , q) remains as a reference. The intensity patterns are given by
(3) with the amplitude f (x , y ) and the phase n (x , y ) can be emulated using the phase-only distribution g (x , y ) given in Eq. (1).
g (x , y ) = f (x , y ) exp[i2πn (x , y )]
(3)
The reference wave r (x , y ) is a random-phase distribution and can be described as
r (x , y ) = exp[i2πφr (x , y )],
I ′j (p , q) = 1 + |R (p , q)|2 + 2 |R (p , q)| × cos[φR (p , q) − φC + jπ − 2cp]
(4)
where φC is a known constant. Similarly, the phase difference between C (p , q) and R (p , q) is given by
where φr (x , y ) is a random function distributed uniformly in the interval [0, 1] and it is statistically independent from n (x , y ) . In the cryptosystem [39], g (x , y ) and r (x , y ) respectively centered at (c, 0) and (−c, 0) are displayed side by side at the input plane. The resultant input is uniformly illuminated and then freely propagated a distance, the encrypted image I0 (p , q) in the output plane is given by
I0 (p , q) = |G (p , q) + R (p , q)|2 , G (p , q) = FrT {g (x , y ); λ, d}, R (p , q) = FrT {r (x , y ); λ, d},
[I4′ 3 (p , q) − I2′ 3 (p , q)] 3 ⎫ φR (p , q) − φC − 2cp = arctan ⎧ ⎨ ′ ′ ′ ⎬ 2 I ( p , q ) − I ( p , q ) − I ( p , q ) 0 4 3 2 3 ⎩ ⎭ (10) Subtracting Eq. (8) from Eq. (10), φG (p , q) added to a known constant term is obtained. The decrypted image fd (x , y ) is given by
(5)
fd (x , y ) = IFrT {exp[−i2πφG (p , q)]; λ, −d}, (6)
(11)
where IFrT {·} denotes the inverse Fresnel transform. From the procedures mentioned above, it can be seen that some inherent drawbacks exist in the cryptosystem [39]. In the encryption process, since the phase-only distribution by Eq. (1) is used to emulate the complex distribution by Eq. (3) under the approximation, the cryptosystem in [39] can only achieve encryption for binary plaintexts. Additionally, the authors [39] claimed that the binary plaintext is required to obtain the intensity patterns by PST in the first phase extraction step of the decryption process. However, in the practical case, only the ciphertexts and decryption keys are given to the authorized receivers. And then, using the correct decryption keys, information of the original images can be retrieved from the encoded image. Hence, it seems impossible to obtain the information of plaintexts before the decryption process is achieved with correct private keys; consequently, it seems impossible to obtain the intensity patterns using the first phase extraction process without any knowledge of the plaintexts. On the other hand, if the three intensity patterns were obtained in the encryption process, a binary plaintext corresponds to three intensity patterns which should be recorded as ciphertexts and transmitted to the authorized receivers. This process will burden the problem of transmission. To further demonstrate the drawbacks of the cryptosystem in [39], a
where denotes the Fresnel transform, FrT {·} R (p , q) = |R (p , q)| G (p , q) = |G (p , q)| exp[i2πφG (p , q)] and exp[i2πφR (p , q)] are the Fresnel spectrums of g (x , y ) and r (x , y ) , respectively. The decryption process [39] includes two phase extraction processes using the three-step PST. In the first phase extraction process, the phase difference between G (p , q) and R (p , q) is retrieved by performing phase shifts on the reference distribution r (x , y ) . The responding intensity patterns at the Fresnel plane are given by
I j (p , q) = |G (p , q)|2 + |R (p , q)|2 + 2 |G (p , q)||R (p , q)| × cos[φR (p , q) − φG (p , q) + jπ − 2cp]
(9)
(7)
where j takes the values j = 0 , 2 , and 4 . The phase difference between 3 3 G (p , q) and R (p , q) is given by
[I4 3 (p , q) − I2 3 (p , q)] 3 ⎫ φR (p , q) − φI (p , q) − 2cp = arctan ⎧ ⎨ 2 ( I ⎩ 0 p , q) − I4 3 (p , q) − I2 3 (p , q) ⎬ ⎭ (8) To retrieve the phase information of G (p , q) , φR (p , q) is needed to be retrieved firstly. Consequently, in the second phase extraction 3
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Fig. 3. (a) The gray-scale image to be encoded, (b-d) three intensity patterns (I0 (u, v ) , I2 3 (u, v ) and I4 3 (u, v ) ), (e) the phase difference between G (p , q) and R (p , q) , (f) the decoded gray-scale image.
to achieve optical image encryption and authorization. Our proposed algorithm has advantages which can be summarized as: (1) binary, gray-scale and color images can be encrypted and authorized, (2) an additional security layer is established for the developed optical system, since optical verification is conducted based on optical encoding systems without direct observation of input information from the decoded images.
numerical simulation is carried out. A binary image with 256 × 256 pixels used as the plaintext is shown in Fig. 2(a). The random phase mask bonded with the binary image is shown in Fig. 2(b) and the reference wave with random-phase distribution is shown in Fig. 2(c). Fig. 2(d–f) show three corresponding intensity patterns with phase 2 4 shifts of 0 , 3 π and 3 π , respectively. Using the intensity patterns obtained by three-step PST, the phase difference between E (p , q) and R (p , q) is shown in Fig. 2(g). The final decrypted image is shown in Fig. 2(h). To evaluate the reliability of the cryptosystem in [39], a correlation coefficient (CC) between the retrieved plaintext (Ir ) and the original image (Io ) is calculated as follows:
CC =
cov(Io, Ir ) (σIo, σIr )
3. The proposed encryption and authentication scheme Fig. 4 shows a schematic diagram of the proposed optical system. In the proposed optical system, the encryption process consists of following steps:
(12) (1) Since Eq. (1) is used to emulate a complex distribution given by Eq. (3) under the approximation, f (x , y ) with only two quantization levels (0 and 1) are acceptable. Consequently, to convert the grayscale or color images to binary intensity distributions, an elaborately designed operator based on histogram equalization [40] is used.
where cov(Io, Ir ) denotes the cross-covariance, and σ denotes the standard deviation. The CC value between Fig. 2(a) and (h) is 0.8013, which can be seen that the quality of the decrypted image is low due to noise and errors introduced. It is claimed that the cryptosystem in [39] is utilized to encrypt a binary image, a simulation is carried out to demonstrate effectiveness of the cryptosystem in [39] used to encrypt a gray-scale image. A gray-scale image with 256 × 256 pixels shown in Fig. 3(a) is utilized as a plaintext while three intensity patterns are shown in Fig. 3(b–d), respectively. The phase difference between φE (p , q) and φR (p , q) is shown in Fig. 3(e) while the decrypted image is shown in Fig. 3(f). The CC value between Fig. 3(a) and (f) is 0.2862, which means most detailed information of the decrypted image has lost. It is shown that the cryptosystem in [39] cannot be used to encrypt gray-scale images. From simulation results and analysis shown above, it can be seen that the cryptosystem in [39] has some drawbacks which can be summarized as: (1) the cryptosystem can only be used to encrypt binary images, (2) the binary plaintext is required to obtain the intensity patterns in the decryption process, (3) too many ciphertexts are required to be transmitted. To address these issues in the cryptosystem in [39], we propose an advanced algorithm based on PC and PST in a JTC
1 f (x , y )⩾threshold f ′ (x , y ) = histeq {f (x , y )} = ⎧ ⎨ ⎩ 0 f (x , y )
(13)
where histeq {·} denotes an operation to transform the image f (x , y ) , returning in image f ′ (x , y ) with two discrete gray levels. The threshold is flexibly adjusted using histogram equalization. In addition, the indexed format which consists of a data matrix and a colormap matrix is utilized to represent a color image and only the data matrix is regarded as the input of the proposed cryptosystem. (1) A PC operation is performed on f ′ (x , y ) to generate a photon-limited binary intensity distribution, a sparse intensity matrix f ′ ′ (x , y ) is given by
4
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Fig. 4. Schematic diagram of the proposed advanced method for (a) encryption and (b) authentication processes.
f ′ ′ (x , y ) = PC {f ′ (x , y )} = f ′ (x , y ) Pd (l j ; λj ) =f ′ (x , y )
[I4 3 (p , q) − I2 3 (p , q)] 3 ⎫ E (p , q) = arctan ⎧ ⎨ ⎩ 2I0 (p , q) − I4 3 (p , q) − I2 3 (p , q) ⎬ ⎭
[λj ]l j e−λj lj !
(14)
It is noteworthy that steps 1 and 2 are used as the pre-processing operators while the step 4 is used as the post-processing operator. For the authentication process, a three-step phase shifting technique is applied in the first step.
where Pd (l j ; λj ) is the probability of counting l j photons at pixel j , l j (l j = 0, 1, 2, ⋯) is the number of photons detected at pixel j and the Poisson parameter λj is given by λj = Np x j . x j is the normalized irradiance at pixel j and Np is the number of counts in the entire scene.
(1) Employing Eq. (10), the phase difference E′ (p , q) between φ R0 (p , q) and φC is given by
(2) The intensity distribution patterns obtained in the CCD plane is given by
[I ′4 3 (p , q) − I ′2 3 (p , q)] 3 ⎫ E′ (p , q) = arctan ⎧ ⎨ ′ 2 I ⎩ 0 (p , q) − I ′4 3 (p , q) − I ′2 3 (p , q) ⎬ ⎭
I0 (p , q) = |E′ ′ (p , q) + R 0 (p , q)|2 , I2 3 (p , q) = |E′ ′ (p , q) + R2 3 (p , q)|2 , I4 3 (p , q) = |E′ ′ (p , q) + R 4 3 (p , q)|2 .
(16)
(17)
where I0′ (p , q) , I2′ 3 (p , q) and I4′ 3 (p , q) are given by Eq. (9).
(15)
(2) The retrieved image fd (x , y ) is given by
where E′ ′ (p , q) is the Fresnel spectrum of e (x , y ) which is given by e (x , y ) = f ′ ′ (x , y ) exp[i2πn (x , y )], R 0 (p , q) , R2 3 (p , q) and R 4 3 (p , q) are the Fresnel spectrums of r (x , y ) which is given by 2 4 r (x , y ) = exp[i2πφr (x , y )] with phase shifts of 0 , 3 π and 3 π , respectively. Note that the optical setup for this step is the same as Fig. 1.
fd (x , y ) = |IFrT {exp[i2π (E′ (p , q) − E (p , q) + φC )]}|
(18)
(3) Since only one sparse intensity pattern is available for the decoding, information of the original image is not visible. Here, optical verification based on nonlinear optical correlation (NOC) algorithm
(4) The encoded image E (p , q) recorded as the ciphertext is given by 5
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Fig. 5. (a) The original gray-scale image to be encoded, (b) the binary image obtained by performing intensity modulation on (a), (c) the photon-limited binary image, (d) the intensity histogram of (a), (e) the intensity histogram of (c), (f-h) three intensity patterns (I0 (p , q) , I2 3 (p , q) and I4 3 (p , q) ), (i) the encoded image E (p , q) , (j) the decoded image fd (x , y ) , (k) the generated nonlinear optical correlation distribution corresponding to (j).
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Fig. 6. (a) The original color image to be encoded, (b) the data matrix of the indexed format, (c) the image obtained by intensity modulation and photon-counting, (d) the intensity histogram of (a), (e) the intensity histogram of (c), (f–h) three intensity patterns (I1 (u, v ) , I2 (u, v ) and I3 (u, v ) ), (i) the encoded image C (u, v ) , (j) the decoded image, (k) the generated nonlinear correlation distribution corresponding to (j).
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Fig. 7. Robustness against the contaminations: (a) the encoded image polluted by random noise, (b) the decoded image obtained from (a), (c) the encoded image is partially occluded, (d) the decoded image obtained from (c), (e, f) the generated nonlinear correlation distributions, respectively, corresponding to (b) and (d).
image with Np = 103 is shown in Fig. 5(c). The intensity histogram of the image in Fig. 5(a) is shown as Fig. 5(d) while the intensity histogram of the photon-limited binary image in Fig. 5(c) is shown as Fig. 5(e). From the histograms in Fig. 5(d) and 5(e), it can be seen that the intensity matrix of the photon-limited binary image is sparse. Three intensity patterns (I0 (p , q) , I2 3 (p , q) and I4 3 (p , q) ) obtained using the three-step phase-shifting technique are shown in Fig. 5(f–h) and the ciphertext is shown in Fig. 5(i). Using the correct reference wave without phase shifting (r0 (x , y ) ), wavelength (λ) and diffraction distance (d), a decoded image is shown in Fig. 5(j). It can be seen that no useful information of the plaintext is visible, which means the attackers cannot obtain the original information from the decrypted image even with correct private keys. The decoded image is further verified by using nonlinear optical correlation algorithm, and the generated nonlinear optical correlation distribution corresponding to Fig. 5(j) is shown in Fig. 5(k). It can be seen that only one remarkable peak is obtained in the generated nonlinear optical correlation map, which means that the decoded image being authenticated. A simulation is carried out further to validate the feasibility and effectiveness of the proposed algorithm used to encode the color image. A color image with 256 × 256 pixels used as the plaintext is shown in Fig. 6(a) and the corresponding data matrix is shown in Fig. 6(b). Fig. 6(c) shows the image obtained by performing the intensity modulation and PC operation (Np = 103 ) on the image shown in Fig. 6(b). Fig. 6(d) and (e) show the intensity histograms of the data matrix and the photon-limited image, respectively. Fig. 6(f–h) show the intensity patterns (I0 (p , q) , I2 3 (p , q) and I4 3 (p , q) ) obtained using the three-step PST. The decoded image is shown in Fig. 6(j) and the generated nonlinear optical correlation map corresponding to Fig. 6(j) is shown in Fig. 6(k). In our proposed encryption and authentication scheme, the decrypting operation in which the decoded images are obtained should be first implemented. Compared to the conventional cryptosystems [6–14] in which the original information of plaintexts can be directly visible
[41] is applied to authenticate the decoded image, and the verification function is described by
NOC (x , y ) = |IFT (|{FT [f (x , y )]}{FT [fd (x , y )]}∗|α − 1 {FT [f (x , y )]}{FT [fd (x , y )]}∗)|2 (19) where NOC {·} denotes the generated nonlinear optical correlation map, f (x , y ) and fd (x , y ) are the plaintext and the decoded image, respectively, α denotes the strength of applied nonlinearity [41]. Here, the parameter α is set as 0.4. Compared to the traditional image encryption systems [6–9] in which the information of the original image can be clearly observed from the decoded image, the information of original image is not directly released in the decrypted image obtained using the proposed decryption algorithm. In practice, a separated or remote database used as an additional security layer is introduced to store the original input images [42], and only one interface is given for the authorized receivers to conduct the verification without direct disclosure of the original input images [42]. A flow chart is shown in Fig. 4 to clearly illustrate the optical encoding, decoding, and verification processes described previously. 4. Numerical simulation and discussion of results 4.1. Feasibility of the proposed method A simulation is numerically conducted to show feasibility and effectiveness of the proposed method. The light wavelength is 532 nm and the diffraction distance d is 108 cm. In the simulation, only grayscale and color images are used as examples while the simulation results of binary images are similar and not shown in this paper. A grayscale image with 256 × 256 pixels used as the plaintext is shown in Fig. 5(a) and the binary image obtained using the operation based on histogram equalization is shown in Fig. 5(b). The photon-limited binary 8
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Fig. 8. (a) The decoded image obtained when number of optical photons is 101.5 , (b) the decoded image obtained when number of optical photons is 10 2.5 , (c) the decoded image obtained when number of optical photons is 103.5 , (d–f) the generated nonlinear correlation distributions, respectively, corresponding to (a–c).
from the decoded images, the decoded images obtained in our algorithm should be compared and correlated with the original input images stored in the remote or separated database. Since the interface of the database is elaborately designed that only allow the verification operation for the receiver, an additional security layer has been established [39].
σ (p , q) generates random numbers uniformly distributed in the interval [0, 1]. Fig. 7(a) shows the noisy encoded image with β = 1 and Fig. 7(b) shows the decoded image retrieved from Fig. 7(a). Fig. 7(c) shows the encoded image in which 25% pixels are occluded and Fig. 7(d) shows the decoded image retrieved from Fig. 7(c). It can be seen in Fig. 7(c) and (d) that the decoded images can still be correctly verified, and high robustness against contaminations is achieved in the proposed optical system.
4.2. Robustness of the proposed method 4.2.1. Robustness of noise and occlusion In practice, due to some external and unexpected influences, such as noise and occlusion, the obtained encoded image may be imperfect. Here, system robustness against the contaminations is investigated. The encoded image contaminated by random noise is given by
Enoise (p , q) = E (p , q)[1 + β ·σ (p , q)]
4.3. Security analysis 4.3.1. Selection of Np Performance of the expected number of photons Np is tested. Fig. 8(a–c) show the decoded images, when Np is 101.5 , 102.5 and 103.5, respectively. It is shown that the larger Np is, the more visible information of the original image. As an example, when Np = 103.5, the silhouette information of the original image can be seen from the
(20)
where parameter β represents a noise magnification factor. Function 9
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Fig. 9. (a) The decoded image obtained using the wrong wavelength, (b) the decoded image obtained using the wrong diffraction distance, (c) the decoded image obtained using the wrong phase-only mask r2 (x , y ) , (d–f) the generated nonlinear correlation distributions, respectively, corresponding to (a–c).
Fig. 10. Discrimination capability: (a) an arbitrary image, (b) the generated nonlinear optical correlation between Fig. 5(j) and Fig. 10(a). 10
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Fig. 8(c). The generated nonlinear optical correlation maps corresponding to Fig. 8(a–c) are shown in Fig. 8(d–f). It is shown that only a noisy background is obtained without any remarkable correlation peak in Fig. 8(d), when Np = 101.5 . A sharp peak points out over a noisy background in Fig. 8(e) and 8(f). It can be seen that the larger Np is, the higher peak value and less noise. Consequently, to ensure that no useful information of the original image would be released and effectiveness of authorization, the value of Np is set as 103 .
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4.3.2. Key sensitivity test Since information verification is developed based on optical encoding, system parameters, such as r0 (x , y ) , d and λ, play an important role as those in the conventional optical encoding systems. Performance of the parameters is further analyzed here. Fig. 9(a–c) show the decoded images, only when λ or d or r0 (x , y ) is wrongly used during the decoding, respectively. The generated nonlinear optical correlation distributions corresponding to Fig. 9(a–c) are shown in Fig. 9(d–e), respectively. From simulation results shown in Fig. 9(d–f), it can be seen no remarkable correlation peak is observed in the noisy background, which means that the receiver does not process correct keys and is not the authorized receiver. 4.3.3. Discrimination test Discrimination capability of the proposed method is further tested. Another arbitrary input image in the database is shown in Fig. 10(a). The generated nonlinear optical correlation distribution between Figs. 5(j) and 10(a) is shown in Fig. 10(b). As seen in Fig. 10(b), only noise background is generated in the nonlinear correlation distribution map, which means the proposed method processes high discrimination capability. 5. Conclusion In this paper, the inherent drawbacks of the optical encryption system [39] using PST in a JTC are demonstrated and an advanced encryption and authentication scheme is proposed based on the analysis. In the proposed algorithm, optical verification based on nonlinear correlation is used as a post-processing operator to authenticate the decoded images without direct disclosure of original information. Since the interface of the database in which plaintexts are stored is only given to the authorized users, an additional security layer has been established compared to the conventional encryption and decryption schemes. In addition, to achieve the encryption and authorization for different kinds of images, such as binary, gray-scale and color images, an intensity modulator based on histogram equalization and PC is used as the pre-processing operator. This is the first time to report that a cryptosystem can achieve encryption for gray-scale and color plaintexts using same optical setup without any changes. Numerical simulations are carried out to demonstrate the feasibility and effectiveness of proposed algorithm. 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. Acknowledgements The authors acknowledge the financial support provided by the National University of Singapore under research project R-265-000589-114. References [1] D.E. Denning, A lattice model of secure information flow, Commun. ACM 19 (5)
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joint transform correlator, Opt. Lett. 31 (17) (2006) 2562–2564. [40] R.C. Gonzalez, R.E. Woods, Digital Image Processing, 3rd ed., 2008. [41] W. Chen, X. Chen, A. Stern, B. Javidi, Phase-modulated optical system with sparse representation for information encoding and authentication, IEEE Photon. J. 5 (2) (2013) 6900113. [42] W. Chen, X. Chen, Grayscale object authentication based on ghost imaging using binary signals, Eur. Phys. Lett 110 (4) (2015) 44002.
attack on a joint transform correlator encrypting system, Opt. Lett. 35 (21) (2010) 3553–3555. [37] M. Liao, W. He, X. Peng, X. Liu, X. Meng, Cryptanalysis of optical encryption with a reference wave in a joint transform correlator architecture, Opt. Laser Technol. 45 (2013) 763–767. [38] Y. Xiong, A. He, C. Quan, Security analysis of a double-image encryption technique based on an asymmetric algorithm, J. Soc. Am. A 35 (2018) 320–326. [39] C. La Mela, C. Iemmi, Optical encryption using phase-shifting interferometry in a
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Contents lists available at ScienceDirect
Optics and Laser Technology journal homepage: www.elsevier.com/locate/optlastec
Optical encryption and authentication scheme based on phase-shifting interferometry in a joint transform correlator Y. Xiong, J. Du, C. Quan
T
⁎
Department of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, Singapore 117576, Singapore
H I GH L IG H T S
drawbacks of the cryptosystem based on phase-shifting in a JTC. • Demonstrated an advanced JTC-based cryptosystem free from these drawbacks. • Proposed an intensity modulator based on histogram equalization and photon-counting. • Employed • Employed a post-processing operator to extract phase part of ciphertexts and authenticate.
A R T I C LE I N FO
A B S T R A C T
Keywords: Optical image encryption and authentication Photon-limited binary intensity pattern Phase-shifting Joint transform correlator
In this paper, the effectiveness of the cryptosystem based on phase-shifting technique (PST) in a joint transform correlator (JTC) is evaluated. It is shown that the cryptosystem has some inherent drawbacks, for example, since the plaintexts are required to obtain the phase-shifting intensity patterns in the first phase extraction step, which means that the information of the plaintexts should be known by the authorized users before decryption processing. It seems impossible and paradoxical in the practical case. Moreover, the cryptosystem can only achieve binary plaintexts encryption while the inputs of this scheme cannot be the gray-scale or color images. To address these issues, we propose an encryption and authentication scheme based on PST in a JTC. By introducing a preprocessing operation including an intensity modulator based on histogram equalization and photon-counting (PC) operator, a photon-limited binary distribution of the plaintexts is obtained and used as the input of the cryptosystem. Due to this design, our proposed scheme can achieve encryption for different kinds of images such as binary, gray-scale and color images using same optical setup without any changes, which can simplify the optical implementation. Furthermore, a post-processing operator to extract phase information of encoded images and authenticate the decoded images is introduced. Since no useful information will be released from the photon-limited binary decoded images obtained using correct decryption keys and the interface of the database storing the plaintexts is only given to the authorized users, an additional security layer is established. This is the first time to report the use of binary intensity pattern in a scheme based on PSI in a JTC for secured verification of different images.
1. Introduction With the widespread development and use of computer and internet, different encryption techniques for transmission of secure information via digital communication channels have attracted growing interest [1–4]. Compared to the digital encryption techniques, optical image encryption techniques have noticeable advantages, such as highspeed parallel processing and multidimensional capabilities [5]. A pioneering work in the field of optical image encryption, named double random phase encoding (DRPE), has been proposed by Refregier and
⁎
Javidi in 1995 [6]. In the classical DRPE system [6], an input image is converted into the stationary white noise by using statistically independent random phase-only masks (RPMs) respectively located at the input and Fourier planes. Sequentially, DRPE algorithm has been extended from Fourier domain into other domains, such as fractional Fourier [7–9], Fresnel [10,11] and gyrator [12–14] domains, to further improve security of the traditional DRPE system [1] by using structure parameters as additional private keys to enlarge the key space. In addition to the aforementioned works, cryptoanalysis is important and necessary for any security system including DRPE. The DRPE system
Corresponding author. E-mail address: [email protected] (C. Quan).
https://doi.org/10.1016/j.optlastec.2020.106108 Received 13 September 2019; Received in revised form 30 November 2019; Accepted 28 January 2020 0030-3992/ © 2020 Elsevier Ltd. All rights reserved.
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has been demonstrated that it is vulnerable to various attacks [15–20] due to the inherent linearity introduced by the Fourier transform. In addition, since the encrypted image obtained using DRPE system is complex matrix including both amplitude and phase information, the encrypted data needs to be registered holographically. It means that DRPE system requires accurate optical alignment, which in practice is difficult to attain. To alleviate this constraint, JTC has been introduced in the DRPE structure [21]. In the JTC-based cryptosystem [21], the plaintext bonded with a RPM is placed side by side with the encryption key in the input plane and the intensity distribution of the joint power spectrum (JPS) as the encrypted data can be recorded using a common power-law sensor, such as a charge-coupled device (CCD). Consequently, various encryption schemes based on JTC [22–28] have been proposed. On the other hand, various authentication schemes based on optical techniques have also been proposed [29–34]. Compared to the traditional encryption schemes in which the information of plaintexts is directly visible from decoded images obtained using correct private keys, decoded images cannot visually render information about plaintexts in the authentication schemes. Then optical authentication methods with a separated or remote database where stores plaintexts are further applied to verify decoded images. Since the interface of the database is only given to the authorized users, an additional security layer is established to resist the potential attacks. Various attacks have proposed to crack the JTC-based encryption schemes [35–38] while few attacks have been investigated to break the JTC-based authentication schemes. Cecilia and Iemmi [39] proposed a DRPE-based optical encryption technique in which the PST in a JTC system is used. Compared to the traditional DRPE in a JTC structure [21] in which complex distribution representation for the input plane is required, only a pure phase modulator is used to display the input in the cryptosystem proposed in [39]. It further simplifies the optical alignment for the JTC-based cryptosystems. However, there are some existing drawbacks in the cryptosystem [39] which can be summarized as: (1) only binary amplitude information can be introduced due to the mathematic principle of the specific technique, which means the proposed cryptosystem in [39] can be only used to encrypt binary plaintexts, (2) since two phase extraction processes based on three-step PST are used to retrieve the information of binary plaintexts in the decryption process, the plaintext is required to obtain the intensity patterns in the first phase extraction process; however, it is impossible in the practical decryption process due to unknown knowledge of plaintexts. On the other hand, if the first threestep PST-based extraction process is considered as the part of the encryption process, three corresponding intensity patterns for a binary plaintext have been recorded as ciphertexts, which may burden the transmission of ciphertexts. On the one hand, the experimental work in [39] is attractive and promising. On the other hand, the cryptosystem in [39] has some drawbacks. Hence, the motivation of our work is to propose a novel encryption and authentication scheme based on the optical implementation in [39] with some modifications. In this work, we propose an JTC-based optical image encryption and authentication algorithm using an elaborately designed operator and PC. It is noteworthy that our proposed cryptosystem has advantages. Firstly, with the help of the elaborately designed pre-processing operator including an intensity modulator based on histogram equalization and a PC operation, our proposed cryptosystem can achieve encryption and authorization not only for binary images but also for gray-scale and color plaintexts. It is noteworthy that the cryptosystem can achieve different kinds of images (binary, gray-scale, color images) encryption and authorization for the first time. Secondly, since the decoded images do not visually render the input information due to the designed optical encoding strategy using only photon-limited binary intensity pattern, an additional security layer is established. The rest of this paper is organized as follows. In Section 2, a brief
Fig. 1. Schematic diagram of the system for (a) encryption and (b) decryption processes in [39]. SF and L are spatial filter and lens, respectively. P1 and P2 are polarizers while WP1 and WP2 are wave plates. The programmable liquidcrystal television display (LCTV) is used to display input signals and achieve phase shifting. The computer is used to drive a LCTV and a charge-coupled device (CCD).
introduction and analysis for the optical cryptosystem in [39] is given. In Section 3, an advanced JTC-based encryption and authentication scheme using an elaborately designed operator and PC is proposed with support by a number of simulations, and the paper is concluded in Section 4. 2. The cryptosystem under study The schematic diagram of the JTC-based optical cryptosystem for binary data encryption proposed in [39] is shown in Fig. 1. To display the binary images (which is amplitude information) and phase-only mask in a pure modulator, the technique proposed in [39] is described as follows. The scene distribution is given by
g (x , y ) = exp{i2π [f (x , y ) n (x , y )]} × exp{l (x − a, y − b)[1 − f (x , y )]}, (1) where f (x , y ) is the binary plaintext, n (x , y ) is a random function distributed uniformly in the interval [0, 1]. l (x , y ) is a quadratic phase distribution corresponding to a convergent lens of focal distance d for wavelength λ . x and y are indices in input plane. Since f (x , y ) is a binary image, g (x , y ) given in Eq. (1) can be expressed as
when f (x , y ) = 1 ⎧ g (x , y ) = exp{i2πn (x , y )}, ⎨ g ( x , y ) = exp{ l ( x − a , y − b )}, when f (x , y ) = 0 ⎩
(2)
The small region obtained when g (x , y ) is illuminated by a uniform plane wave and then the light passing through the area with f (x , y ) = 0 followed by the free propagation is centered at coordinates p = a , q = b ( p and q are indices of the Fresnel domain), which can be considered that no lights arrive at the output plane with properly set for a and b in the practical encryption process. A complex distribution given by Eq. 2
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Fig. 2. (a) The binary image to be encoded, (b) a phase-only mask n (x , y ) , (c) a reference wave r (x , y ) , (d–f) three intensity patterns (I0 (u, v ) , I2 3 (u, v ) and I4 3 (u, v ) ), (g) the phase difference between G (p , q) and R (p , q) , (h) the decoded binary image.
process, a constant distribution C (p , q) = exp[iφC ] is used to replace the unknown distribution E (p , q) at the output plane, while the Fresnel spectrum R (p , q) remains as a reference. The intensity patterns are given by
(3) with the amplitude f (x , y ) and the phase n (x , y ) can be emulated using the phase-only distribution g (x , y ) given in Eq. (1).
g (x , y ) = f (x , y ) exp[i2πn (x , y )]
(3)
The reference wave r (x , y ) is a random-phase distribution and can be described as
r (x , y ) = exp[i2πφr (x , y )],
I ′j (p , q) = 1 + |R (p , q)|2 + 2 |R (p , q)| × cos[φR (p , q) − φC + jπ − 2cp]
(4)
where φC is a known constant. Similarly, the phase difference between C (p , q) and R (p , q) is given by
where φr (x , y ) is a random function distributed uniformly in the interval [0, 1] and it is statistically independent from n (x , y ) . In the cryptosystem [39], g (x , y ) and r (x , y ) respectively centered at (c, 0) and (−c, 0) are displayed side by side at the input plane. The resultant input is uniformly illuminated and then freely propagated a distance, the encrypted image I0 (p , q) in the output plane is given by
I0 (p , q) = |G (p , q) + R (p , q)|2 , G (p , q) = FrT {g (x , y ); λ, d}, R (p , q) = FrT {r (x , y ); λ, d},
[I4′ 3 (p , q) − I2′ 3 (p , q)] 3 ⎫ φR (p , q) − φC − 2cp = arctan ⎧ ⎨ ′ ′ ′ ⎬ 2 I ( p , q ) − I ( p , q ) − I ( p , q ) 0 4 3 2 3 ⎩ ⎭ (10) Subtracting Eq. (8) from Eq. (10), φG (p , q) added to a known constant term is obtained. The decrypted image fd (x , y ) is given by
(5)
fd (x , y ) = IFrT {exp[−i2πφG (p , q)]; λ, −d}, (6)
(11)
where IFrT {·} denotes the inverse Fresnel transform. From the procedures mentioned above, it can be seen that some inherent drawbacks exist in the cryptosystem [39]. In the encryption process, since the phase-only distribution by Eq. (1) is used to emulate the complex distribution by Eq. (3) under the approximation, the cryptosystem in [39] can only achieve encryption for binary plaintexts. Additionally, the authors [39] claimed that the binary plaintext is required to obtain the intensity patterns by PST in the first phase extraction step of the decryption process. However, in the practical case, only the ciphertexts and decryption keys are given to the authorized receivers. And then, using the correct decryption keys, information of the original images can be retrieved from the encoded image. Hence, it seems impossible to obtain the information of plaintexts before the decryption process is achieved with correct private keys; consequently, it seems impossible to obtain the intensity patterns using the first phase extraction process without any knowledge of the plaintexts. On the other hand, if the three intensity patterns were obtained in the encryption process, a binary plaintext corresponds to three intensity patterns which should be recorded as ciphertexts and transmitted to the authorized receivers. This process will burden the problem of transmission. To further demonstrate the drawbacks of the cryptosystem in [39], a
where denotes the Fresnel transform, FrT {·} R (p , q) = |R (p , q)| G (p , q) = |G (p , q)| exp[i2πφG (p , q)] and exp[i2πφR (p , q)] are the Fresnel spectrums of g (x , y ) and r (x , y ) , respectively. The decryption process [39] includes two phase extraction processes using the three-step PST. In the first phase extraction process, the phase difference between G (p , q) and R (p , q) is retrieved by performing phase shifts on the reference distribution r (x , y ) . The responding intensity patterns at the Fresnel plane are given by
I j (p , q) = |G (p , q)|2 + |R (p , q)|2 + 2 |G (p , q)||R (p , q)| × cos[φR (p , q) − φG (p , q) + jπ − 2cp]
(9)
(7)
where j takes the values j = 0 , 2 , and 4 . The phase difference between 3 3 G (p , q) and R (p , q) is given by
[I4 3 (p , q) − I2 3 (p , q)] 3 ⎫ φR (p , q) − φI (p , q) − 2cp = arctan ⎧ ⎨ 2 ( I ⎩ 0 p , q) − I4 3 (p , q) − I2 3 (p , q) ⎬ ⎭ (8) To retrieve the phase information of G (p , q) , φR (p , q) is needed to be retrieved firstly. Consequently, in the second phase extraction 3
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Fig. 3. (a) The gray-scale image to be encoded, (b-d) three intensity patterns (I0 (u, v ) , I2 3 (u, v ) and I4 3 (u, v ) ), (e) the phase difference between G (p , q) and R (p , q) , (f) the decoded gray-scale image.
to achieve optical image encryption and authorization. Our proposed algorithm has advantages which can be summarized as: (1) binary, gray-scale and color images can be encrypted and authorized, (2) an additional security layer is established for the developed optical system, since optical verification is conducted based on optical encoding systems without direct observation of input information from the decoded images.
numerical simulation is carried out. A binary image with 256 × 256 pixels used as the plaintext is shown in Fig. 2(a). The random phase mask bonded with the binary image is shown in Fig. 2(b) and the reference wave with random-phase distribution is shown in Fig. 2(c). Fig. 2(d–f) show three corresponding intensity patterns with phase 2 4 shifts of 0 , 3 π and 3 π , respectively. Using the intensity patterns obtained by three-step PST, the phase difference between E (p , q) and R (p , q) is shown in Fig. 2(g). The final decrypted image is shown in Fig. 2(h). To evaluate the reliability of the cryptosystem in [39], a correlation coefficient (CC) between the retrieved plaintext (Ir ) and the original image (Io ) is calculated as follows:
CC =
cov(Io, Ir ) (σIo, σIr )
3. The proposed encryption and authentication scheme Fig. 4 shows a schematic diagram of the proposed optical system. In the proposed optical system, the encryption process consists of following steps:
(12) (1) Since Eq. (1) is used to emulate a complex distribution given by Eq. (3) under the approximation, f (x , y ) with only two quantization levels (0 and 1) are acceptable. Consequently, to convert the grayscale or color images to binary intensity distributions, an elaborately designed operator based on histogram equalization [40] is used.
where cov(Io, Ir ) denotes the cross-covariance, and σ denotes the standard deviation. The CC value between Fig. 2(a) and (h) is 0.8013, which can be seen that the quality of the decrypted image is low due to noise and errors introduced. It is claimed that the cryptosystem in [39] is utilized to encrypt a binary image, a simulation is carried out to demonstrate effectiveness of the cryptosystem in [39] used to encrypt a gray-scale image. A gray-scale image with 256 × 256 pixels shown in Fig. 3(a) is utilized as a plaintext while three intensity patterns are shown in Fig. 3(b–d), respectively. The phase difference between φE (p , q) and φR (p , q) is shown in Fig. 3(e) while the decrypted image is shown in Fig. 3(f). The CC value between Fig. 3(a) and (f) is 0.2862, which means most detailed information of the decrypted image has lost. It is shown that the cryptosystem in [39] cannot be used to encrypt gray-scale images. From simulation results and analysis shown above, it can be seen that the cryptosystem in [39] has some drawbacks which can be summarized as: (1) the cryptosystem can only be used to encrypt binary images, (2) the binary plaintext is required to obtain the intensity patterns in the decryption process, (3) too many ciphertexts are required to be transmitted. To address these issues in the cryptosystem in [39], we propose an advanced algorithm based on PC and PST in a JTC
1 f (x , y )⩾threshold f ′ (x , y ) = histeq {f (x , y )} = ⎧ ⎨ ⎩ 0 f (x , y )
(13)
where histeq {·} denotes an operation to transform the image f (x , y ) , returning in image f ′ (x , y ) with two discrete gray levels. The threshold is flexibly adjusted using histogram equalization. In addition, the indexed format which consists of a data matrix and a colormap matrix is utilized to represent a color image and only the data matrix is regarded as the input of the proposed cryptosystem. (1) A PC operation is performed on f ′ (x , y ) to generate a photon-limited binary intensity distribution, a sparse intensity matrix f ′ ′ (x , y ) is given by
4
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Fig. 4. Schematic diagram of the proposed advanced method for (a) encryption and (b) authentication processes.
f ′ ′ (x , y ) = PC {f ′ (x , y )} = f ′ (x , y ) Pd (l j ; λj ) =f ′ (x , y )
[I4 3 (p , q) − I2 3 (p , q)] 3 ⎫ E (p , q) = arctan ⎧ ⎨ ⎩ 2I0 (p , q) − I4 3 (p , q) − I2 3 (p , q) ⎬ ⎭
[λj ]l j e−λj lj !
(14)
It is noteworthy that steps 1 and 2 are used as the pre-processing operators while the step 4 is used as the post-processing operator. For the authentication process, a three-step phase shifting technique is applied in the first step.
where Pd (l j ; λj ) is the probability of counting l j photons at pixel j , l j (l j = 0, 1, 2, ⋯) is the number of photons detected at pixel j and the Poisson parameter λj is given by λj = Np x j . x j is the normalized irradiance at pixel j and Np is the number of counts in the entire scene.
(1) Employing Eq. (10), the phase difference E′ (p , q) between φ R0 (p , q) and φC is given by
(2) The intensity distribution patterns obtained in the CCD plane is given by
[I ′4 3 (p , q) − I ′2 3 (p , q)] 3 ⎫ E′ (p , q) = arctan ⎧ ⎨ ′ 2 I ⎩ 0 (p , q) − I ′4 3 (p , q) − I ′2 3 (p , q) ⎬ ⎭
I0 (p , q) = |E′ ′ (p , q) + R 0 (p , q)|2 , I2 3 (p , q) = |E′ ′ (p , q) + R2 3 (p , q)|2 , I4 3 (p , q) = |E′ ′ (p , q) + R 4 3 (p , q)|2 .
(16)
(17)
where I0′ (p , q) , I2′ 3 (p , q) and I4′ 3 (p , q) are given by Eq. (9).
(15)
(2) The retrieved image fd (x , y ) is given by
where E′ ′ (p , q) is the Fresnel spectrum of e (x , y ) which is given by e (x , y ) = f ′ ′ (x , y ) exp[i2πn (x , y )], R 0 (p , q) , R2 3 (p , q) and R 4 3 (p , q) are the Fresnel spectrums of r (x , y ) which is given by 2 4 r (x , y ) = exp[i2πφr (x , y )] with phase shifts of 0 , 3 π and 3 π , respectively. Note that the optical setup for this step is the same as Fig. 1.
fd (x , y ) = |IFrT {exp[i2π (E′ (p , q) − E (p , q) + φC )]}|
(18)
(3) Since only one sparse intensity pattern is available for the decoding, information of the original image is not visible. Here, optical verification based on nonlinear optical correlation (NOC) algorithm
(4) The encoded image E (p , q) recorded as the ciphertext is given by 5
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Fig. 5. (a) The original gray-scale image to be encoded, (b) the binary image obtained by performing intensity modulation on (a), (c) the photon-limited binary image, (d) the intensity histogram of (a), (e) the intensity histogram of (c), (f-h) three intensity patterns (I0 (p , q) , I2 3 (p , q) and I4 3 (p , q) ), (i) the encoded image E (p , q) , (j) the decoded image fd (x , y ) , (k) the generated nonlinear optical correlation distribution corresponding to (j).
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Fig. 6. (a) The original color image to be encoded, (b) the data matrix of the indexed format, (c) the image obtained by intensity modulation and photon-counting, (d) the intensity histogram of (a), (e) the intensity histogram of (c), (f–h) three intensity patterns (I1 (u, v ) , I2 (u, v ) and I3 (u, v ) ), (i) the encoded image C (u, v ) , (j) the decoded image, (k) the generated nonlinear correlation distribution corresponding to (j).
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Fig. 7. Robustness against the contaminations: (a) the encoded image polluted by random noise, (b) the decoded image obtained from (a), (c) the encoded image is partially occluded, (d) the decoded image obtained from (c), (e, f) the generated nonlinear correlation distributions, respectively, corresponding to (b) and (d).
image with Np = 103 is shown in Fig. 5(c). The intensity histogram of the image in Fig. 5(a) is shown as Fig. 5(d) while the intensity histogram of the photon-limited binary image in Fig. 5(c) is shown as Fig. 5(e). From the histograms in Fig. 5(d) and 5(e), it can be seen that the intensity matrix of the photon-limited binary image is sparse. Three intensity patterns (I0 (p , q) , I2 3 (p , q) and I4 3 (p , q) ) obtained using the three-step phase-shifting technique are shown in Fig. 5(f–h) and the ciphertext is shown in Fig. 5(i). Using the correct reference wave without phase shifting (r0 (x , y ) ), wavelength (λ) and diffraction distance (d), a decoded image is shown in Fig. 5(j). It can be seen that no useful information of the plaintext is visible, which means the attackers cannot obtain the original information from the decrypted image even with correct private keys. The decoded image is further verified by using nonlinear optical correlation algorithm, and the generated nonlinear optical correlation distribution corresponding to Fig. 5(j) is shown in Fig. 5(k). It can be seen that only one remarkable peak is obtained in the generated nonlinear optical correlation map, which means that the decoded image being authenticated. A simulation is carried out further to validate the feasibility and effectiveness of the proposed algorithm used to encode the color image. A color image with 256 × 256 pixels used as the plaintext is shown in Fig. 6(a) and the corresponding data matrix is shown in Fig. 6(b). Fig. 6(c) shows the image obtained by performing the intensity modulation and PC operation (Np = 103 ) on the image shown in Fig. 6(b). Fig. 6(d) and (e) show the intensity histograms of the data matrix and the photon-limited image, respectively. Fig. 6(f–h) show the intensity patterns (I0 (p , q) , I2 3 (p , q) and I4 3 (p , q) ) obtained using the three-step PST. The decoded image is shown in Fig. 6(j) and the generated nonlinear optical correlation map corresponding to Fig. 6(j) is shown in Fig. 6(k). In our proposed encryption and authentication scheme, the decrypting operation in which the decoded images are obtained should be first implemented. Compared to the conventional cryptosystems [6–14] in which the original information of plaintexts can be directly visible
[41] is applied to authenticate the decoded image, and the verification function is described by
NOC (x , y ) = |IFT (|{FT [f (x , y )]}{FT [fd (x , y )]}∗|α − 1 {FT [f (x , y )]}{FT [fd (x , y )]}∗)|2 (19) where NOC {·} denotes the generated nonlinear optical correlation map, f (x , y ) and fd (x , y ) are the plaintext and the decoded image, respectively, α denotes the strength of applied nonlinearity [41]. Here, the parameter α is set as 0.4. Compared to the traditional image encryption systems [6–9] in which the information of the original image can be clearly observed from the decoded image, the information of original image is not directly released in the decrypted image obtained using the proposed decryption algorithm. In practice, a separated or remote database used as an additional security layer is introduced to store the original input images [42], and only one interface is given for the authorized receivers to conduct the verification without direct disclosure of the original input images [42]. A flow chart is shown in Fig. 4 to clearly illustrate the optical encoding, decoding, and verification processes described previously. 4. Numerical simulation and discussion of results 4.1. Feasibility of the proposed method A simulation is numerically conducted to show feasibility and effectiveness of the proposed method. The light wavelength is 532 nm and the diffraction distance d is 108 cm. In the simulation, only grayscale and color images are used as examples while the simulation results of binary images are similar and not shown in this paper. A grayscale image with 256 × 256 pixels used as the plaintext is shown in Fig. 5(a) and the binary image obtained using the operation based on histogram equalization is shown in Fig. 5(b). The photon-limited binary 8
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Fig. 8. (a) The decoded image obtained when number of optical photons is 101.5 , (b) the decoded image obtained when number of optical photons is 10 2.5 , (c) the decoded image obtained when number of optical photons is 103.5 , (d–f) the generated nonlinear correlation distributions, respectively, corresponding to (a–c).
from the decoded images, the decoded images obtained in our algorithm should be compared and correlated with the original input images stored in the remote or separated database. Since the interface of the database is elaborately designed that only allow the verification operation for the receiver, an additional security layer has been established [39].
σ (p , q) generates random numbers uniformly distributed in the interval [0, 1]. Fig. 7(a) shows the noisy encoded image with β = 1 and Fig. 7(b) shows the decoded image retrieved from Fig. 7(a). Fig. 7(c) shows the encoded image in which 25% pixels are occluded and Fig. 7(d) shows the decoded image retrieved from Fig. 7(c). It can be seen in Fig. 7(c) and (d) that the decoded images can still be correctly verified, and high robustness against contaminations is achieved in the proposed optical system.
4.2. Robustness of the proposed method 4.2.1. Robustness of noise and occlusion In practice, due to some external and unexpected influences, such as noise and occlusion, the obtained encoded image may be imperfect. Here, system robustness against the contaminations is investigated. The encoded image contaminated by random noise is given by
Enoise (p , q) = E (p , q)[1 + β ·σ (p , q)]
4.3. Security analysis 4.3.1. Selection of Np Performance of the expected number of photons Np is tested. Fig. 8(a–c) show the decoded images, when Np is 101.5 , 102.5 and 103.5, respectively. It is shown that the larger Np is, the more visible information of the original image. As an example, when Np = 103.5, the silhouette information of the original image can be seen from the
(20)
where parameter β represents a noise magnification factor. Function 9
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Fig. 9. (a) The decoded image obtained using the wrong wavelength, (b) the decoded image obtained using the wrong diffraction distance, (c) the decoded image obtained using the wrong phase-only mask r2 (x , y ) , (d–f) the generated nonlinear correlation distributions, respectively, corresponding to (a–c).
Fig. 10. Discrimination capability: (a) an arbitrary image, (b) the generated nonlinear optical correlation between Fig. 5(j) and Fig. 10(a). 10
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Fig. 8(c). The generated nonlinear optical correlation maps corresponding to Fig. 8(a–c) are shown in Fig. 8(d–f). It is shown that only a noisy background is obtained without any remarkable correlation peak in Fig. 8(d), when Np = 101.5 . A sharp peak points out over a noisy background in Fig. 8(e) and 8(f). It can be seen that the larger Np is, the higher peak value and less noise. Consequently, to ensure that no useful information of the original image would be released and effectiveness of authorization, the value of Np is set as 103 .
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4.3.2. Key sensitivity test Since information verification is developed based on optical encoding, system parameters, such as r0 (x , y ) , d and λ, play an important role as those in the conventional optical encoding systems. Performance of the parameters is further analyzed here. Fig. 9(a–c) show the decoded images, only when λ or d or r0 (x , y ) is wrongly used during the decoding, respectively. The generated nonlinear optical correlation distributions corresponding to Fig. 9(a–c) are shown in Fig. 9(d–e), respectively. From simulation results shown in Fig. 9(d–f), it can be seen no remarkable correlation peak is observed in the noisy background, which means that the receiver does not process correct keys and is not the authorized receiver. 4.3.3. Discrimination test Discrimination capability of the proposed method is further tested. Another arbitrary input image in the database is shown in Fig. 10(a). The generated nonlinear optical correlation distribution between Figs. 5(j) and 10(a) is shown in Fig. 10(b). As seen in Fig. 10(b), only noise background is generated in the nonlinear correlation distribution map, which means the proposed method processes high discrimination capability. 5. Conclusion In this paper, the inherent drawbacks of the optical encryption system [39] using PST in a JTC are demonstrated and an advanced encryption and authentication scheme is proposed based on the analysis. In the proposed algorithm, optical verification based on nonlinear correlation is used as a post-processing operator to authenticate the decoded images without direct disclosure of original information. Since the interface of the database in which plaintexts are stored is only given to the authorized users, an additional security layer has been established compared to the conventional encryption and decryption schemes. In addition, to achieve the encryption and authorization for different kinds of images, such as binary, gray-scale and color images, an intensity modulator based on histogram equalization and PC is used as the pre-processing operator. This is the first time to report that a cryptosystem can achieve encryption for gray-scale and color plaintexts using same optical setup without any changes. Numerical simulations are carried out to demonstrate the feasibility and effectiveness of proposed algorithm. 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. Acknowledgements The authors acknowledge the financial support provided by the National University of Singapore under research project R-265-000589-114. References [1] D.E. Denning, A lattice model of secure information flow, Commun. ACM 19 (5)
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