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Phase-retrieval attack free cryptosystem based on cylindrical asymmetric diffraction and double-random phase encoding
Optics Communications 410 (2018) 468–474
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Phase-retrieval attack free cryptosystem based on cylindrical asymmetric diffraction and double-random phase encoding Jun Wang a , Xiaowei Li a , Yuhen Hu b , Qiong-Hua Wang a, * a b
School of Electronics and Information Engineering, Sichuan University, Chengdu 610065, China Department of Electrical & Computer Engineering, University of Wisconsin-Madison, Madison, WI 53706, USA
a r t i c l e
i n f o
Keywords: Optical encryption Phase-retrieval attack free cryptosystem Cylindrical diffraction method Computer holography
a b s t r a c t A phase-retrieval attack free cryptosystem based on the cylindrical asymmetric diffraction and double-random phase encoding (DRPE) is proposed. The plaintext is abstract as a cylinder, while the observed diffraction and holographic surfaces are concentric cylinders. Therefore, the plaintext can be encrypted through a two-step asymmetric diffraction process with double pseudo random phase masks located on the object surface and the first diffraction surface. After inverse diffraction from a holographic surface to an object surface, the plaintext can be reconstructed using a decryption process. Since the diffraction propagated from the inner cylinder to the outer cylinder is different from that of the reversed direction, the proposed cryptosystem is asymmetric and hence is free of phase-retrieval attack. Numerical simulation results demonstrate the flexibility and effectiveness of the proposed cryptosystem. © 2017 Elsevier B.V. All rights reserved.
1. Introduction The network globalization and active social networking make the information security technologies increasingly important. Among them, optical information processing is one of the most promising approaches for data securing, encryption, and authentication [1–3]. The field of optical encryption was pioneered by Refregier and Javidi, who used double-random phase encoding (DRPE) in the standard 4f optical system [4,5]. DRPE has developed to one of the most widely used and studied optical encryption techniques. It has been extended to the Fresnel transform [6–8], the fractional Fourier transformation [9,10], gyrator transform [11], and phase truncation operations [12–14] by many researchers. To improve security, DRPE has been combined with other imaging techniques such as iterative computational algorithms [15–17], compressive sensing [18,19], photon-counting imaging [20–22], and so on. These methods have either reinforced the decoding resistance or simplified the implementation of the operation. Unfortunately, the conventional DRPE encryption approach has vulnerability against specific types of attacks, such as chosen-plaintext attacks and known-plaintext attacks [23–26]. Several cryptosystems have been proposed to improve the weakness of conventional DRPE based approaches. An asymmetric cryptosystem was proposed, which is based on phase-truncated Fourier transforms [27]. Later, an improved
asymmetric cryptosystem was developed by using random binary phase modulation and a mixture phase retrieval algorithm [28]. Also, coherent superposition and equal modulus decomposition were proposed for the asymmetric optical cryptosystem [29]. More recently, the distorted wavefront beam illumination was adopted to develop a novel cryptosystem based on double-random phase encoding [30]. And a novel optical image encryption method was proposed by employing divergent illumination [31]. Although they are easy to implement in digital simulation, they are difficult to implement in optical experiments. Moreover, they are generally vulnerable to the collision algorithm [32,33]. This is because these cryptosystems are derived from mathematical formulas and most of them lack a reasonable explanation from an optical viewpoint. Therefore, a cryptosystem with asymmetric diffraction based on optical principles that is practical to implement and exhibits a high resistance against specific attacks is desirable. Inspired by the aforementioned research, we proposed a phaseretrieval attack free cryptosystem based on cylindrical asymmetric diffraction and DRPE in this paper. It is noticed that the majority of existing schemes of optical image encryption based on DRPE at present are propagations between two or more planers. To the best of our knowledge, it is the first time to apply the cylindrical diffraction to holographic encryption. In this proposal, the plaintext image is abstract as a cylinder, while the observed diffraction and holographic surfaces are concentric
* Corresponding author.
E-mail address: [email protected] (Q. Wang). https://doi.org/10.1016/j.optcom.2017.10.061 Received 28 September 2017; Received in revised form 23 October 2017; Accepted 24 October 2017 0030-4018/© 2017 Elsevier B.V. All rights reserved.
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Optics Communications 410 (2018) 468–474
Fig. 1. Illustration of geometrical relation in cylindrical diffraction with top-view. (a) IOP Model, (b) OIP Model.
Rayleigh–Sommerfeld integral equation in its form can be written by
cylinders. Therefore, the plaintext can be encrypted through a two-step diffraction process with double pseudo random phase masks located on the object surface and the first diffraction surface. The distribution of complex amplitude on the second diffraction surface can be recorded and recovered by the two-step phase-shifting holography. The plaintext image can be reconstructed by inverse diffraction from a holographic surface to an object surface in a decryption process. The initial value and control parameter of double pseudo random phase masks, wavelength of the cylindrical wave, and parameters (inner and outer radii and height) of the cylinders are highly sensitive keys for authorized users. Compared with the conventional DRPE based cryptosystems, our proposed scheme has two additional keys, which enhance the security of the proposed cryptosystem. Since the diffraction calculation of propagation from inner cylinder to outer one is different from that of the reversed direction, the cylindrical diffraction is asymmetric, and hence the proposed cryptosystem is free of phase-retrieval attack. Numerical simulation results demonstrate the effectiveness and flexibility of the proposed cryptosystem.
𝑢𝑅 (𝜃𝑅 , 𝑧𝑅 ) = 𝐶
∬𝑠
𝑢𝑟 (𝜃𝑟 , 𝑧𝑟 )
exp(𝑖𝑘𝑑𝑃𝑟 𝑄𝑅 ) 𝑑𝑃𝑟 𝑄𝑅
𝑑𝜃𝑟 𝑑𝑧𝑟
⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ IOP, (1) exp(𝑖𝑘𝑑𝑃𝑅 𝑄𝑟 )[𝑟 − 𝑅 cos(𝜃𝑟 − 𝜃𝑅 )] 𝑢𝑟 (𝜃𝑟 , 𝑧𝑟 ) = 𝐶 𝑢 (𝜃 , 𝑧 ) 𝑑𝜃𝑅 𝑑𝑧𝑅 ∬𝑠 𝑅 𝑅 𝑅 𝑑𝑃2 𝑄 𝑅
𝑟
(2)
⋯ ⋯ OIP, 2
2
𝑑 = 𝑑𝑃𝑟 𝑄𝑅 = 𝑑𝑃𝑅 𝑄𝑟 = [𝑅 + 𝑟 − 2𝑅𝑟 cos(𝜃𝑅 − 𝜃𝑟 ) + (𝑧𝑅 − 𝑧𝑟 )2 ]1∕2 ,
(3)
where 𝑘 and 𝐶 denote the wavenumber of the incident light and a constant, respectively. The 𝑑 represents the distance between two points of 𝑃 and 𝑄 on the object and observation surfaces, respectively. The 𝑠 denotes the object surface. Note that these two equations can be accelerated by FFT algorithm [34,35]. 2.2. DRPE with cylindrical asymmetric diffraction The optical setup for the realization of the DRPE with cylindrical asymmetric diffraction is depicted as Fig. 2. The laser, special filter (SF), and lens (L1) make up a collimating system to generate the uniform plane wave. M1 and M2 are mirrors. The beam splitter (BS1) between two mirrors and the beam splitter (BS2) just before CCD divide the wave light into two beams. The random phase masks (RPM1 and RPM2) are cylindrical surfaces, which have radii of 𝑅1 , 𝑅2 and 𝑟, respectively. The input object, which is bonded to RPM1, is also cylindrical surface with radius of 𝑅1 . The object wave, 𝑢𝑅1 (𝜃𝑅1 , 𝑧𝑅1 ), modulates by the RPM1 and propagates to the intermediate surface, on which the distributions is 𝑢𝑟 (𝜃𝑟 , 𝑧𝑟 ). After reflection and modulation by the RPM2, the wave field continuously propagates to the destination surface, on which the distributions is 𝑢𝑅2 (𝜃𝑅2 , 𝑧𝑅2 ). Finally, it interferes with the reference light, and the holograms are captured by the CCD. The two-step phaseshifting holography is employed here [36]. The holograms (PSH1 and PSH2) are sent to computer for processing and transmittance. Note that, the optical path lengths of the two wavefront sensing optical paths of the object and reference lights should be equal.
2. Principle In this section, the principle of our proposal is descripted. Firstly, the basic diffraction theory between a pair of concentric cylindrical surfaces is introduced. Then, the optical setup of our proposed DRPE with cylindrical asymmetric diffraction is given. Lastly, how to encrypt and decrypt an object image applying the cylindrical diffraction theory is presented.
2.1. Diffraction between two concentric cylinders In the cylindrical diffraction theory [34,35], the object and the observation surfaces are concentric cylindrical surfaces as shown in Fig. 1, where 𝑅 and 𝑟 denote the radii of the inner and outer surfaces, respectively. Obviously, there are two, inside-out and outside-in, propagation models in the cases that object are placed on the inside and outside surfaces as shown in Figs. 1(a) and 1(b), respectively. In the case of inside-out propagation (IOP) model, the object and observation points are represented by 𝑃𝑟 (𝜃𝑟 , 𝑧𝑟 ) and 𝑄𝑅 (𝜃𝑅 , 𝑧𝑅 ) in cylindrical coordinate, respectively. While in the case of outside-in propagation (OIP) model, the object and observation points are represented by 𝑃𝑅 (𝜃𝑅 , 𝑧𝑅 ) and 𝑄𝑟 (𝜃𝑟 , 𝑧𝑟 ) in cylindrical coordinate, respectively. Here, 𝑧𝑟 and 𝑧𝑅 are in range of −𝐻/2 to 𝐻/2, where 𝐻 is the height of the cylindrical surface. If the distributions on the inner and outer surfaces are represented by 𝑢𝑟 (𝜃𝑟 , 𝑧𝑟 ) and 𝑢𝑅 (𝜃𝑅 , 𝑧𝑅 ), respectively, the
2.3. Encryption and decryption The flow chart of encryption and decryption can be depicted in Fig. 3. In encryption, the object image is located at the cylindrical surface with radius of 𝑅1 as input and is modulated by the first random phase mark (RPM1). After propagation a distance of 𝑑1 in OIP model, the wavefront reaches the first intermediate surface and is modulated by the second random phase mark (RPM2). After propagation a distance of 𝑑2 in IOP model, the wavefront reaches the destination surface and is captured by 469
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algorithm [37]. The 𝑢′𝑅2 is reconstructed complex distribution by using two-step phase-shifting holography [36].
3. Simulation results 3.1. Results of the proposal For the system discussed in Section 2, numerical experiments are carried out to verify the proposed cryptosystem based on cylindrical asymmetric diffraction and DRPE, which is refer to CAD-DRPE. Let the values of 𝑟, 𝑅1 , 𝑅2 , 𝐻 be 1, 60, 65, 5 mm, respectively. The wavelength use in the simulation is 842 nm. Then, the pixel numbers in the azimuthal and vertical directions are at least 15178 and 503, respectively, according to the Shannon sampling theory [38]. Since the cylindrical surface in azimuthal direction is much longer than that in vertical one, it is better to select an equal pixel size in both directions which should be greater than 503. Therefore, we design the sampling number of plaintext image to be 512 × 512 pixels, and 15178/512 is nearly 30. That is the reason we chose a part of the cylindrical surface in azimuthal direction with angle of 2𝜋/30, ex. [−2𝜋/60, 2𝜋/60], to present the plaintext image. In fact, we have many other options for the sampling number of plaintext image. If we chose angle of 2𝜋/15, the pixels should be 512×1024, and if we chose angle of 2𝜋∕60, the pixels should be 512×256, and so on. The 𝑅2 , 𝐻, 𝑟, and 𝜆 are independent and can be used as the cylindrical diffraction keys, and the random phase mask generation keys of 𝑥2 and 𝜇2 are DRPE encryption keys. We employed Rulers (512 × 512 pixels) as input plaintext image and performed the simulations using matlab R2010b platform to demonstrate the correctness and the security of the cryptosystem, as shown in Figs. 4–6. The correlation coefficient (𝐶𝐶) values between the original image, 𝑓 (𝑚, 𝑛), and the different images, 𝑓 ′ (𝑚, 𝑛), shown in Figs. 4(a2)–(a5), Figs. 4(b2)–(b5), Figs. 5(b)–(j) and Figs. 6(b)–(j), are calculated using
Fig. 2. Optical setup for realization of proposed DRPE with cylindrical asymmetric diffraction.
Fig. 3. Encryption and decryption of the proposed cryptosystem.
𝐶𝐶 = [cov(𝑓 , 𝑓 ′ )]∕(𝜎𝑓 × 𝜎𝑓 ′ ) CCD camera. The output is holograms (PSH1 = 𝐼0 and PSH2 = 𝐼𝜋 ∕∕2) through the imaging system of two-step phase-shifting holography. Decryption is the inverse of encryption. However, the first inverse propagation of IOP is OIP model in optical setup. And the second inverse propagation of OIP is IOP model in optical setup. Therefore, this cryptosystem is asymmetric, and is expected to have higher security than that of conventional DRPE based on propagation between several parallel planes. The computer simulations can employ the inverse propagation to replace the optical process. The encryption and decryption can be abstract in mathematic as
where cov denotes the cross-covariance and 𝜎 denotes standard deviation. For the sake of brevity, the coordinate (𝑚, 𝑛) is omitted in this equation. The diffraction and encryption results of the proposed method without and with RPMs are shown in Figs. 4(a2)–(a5) and (b2)–(b5), respectively. The CC values of corresponding to Figs. 4(a2)–(a5) are 0.1530, 0.2285, 0.2453, and 0.2624, respectively. The CC values of corresponding to Figs. 4(b2)–(b5) are 0.0230, 0.0025, 0.0186, and 0.0103, respectively. The CC values of diffraction and encryption results without RPMs show that the results images are related to the original image. And it also verified by subjective observation that the some information of the original object can be recognized. When the cylindrical diffraction is combined with the DPRE method by adding two RPMs, the encrypted images show good noise-like property. And it is also verified by their CC values. The CC values of corresponding to Figs. 5(b)–(j) are 0.2285, 0.2613, 0.1964, 0.2849, 0.1587, 0.1338, 0.0752, 0.2946, 0.0146, respectively. These results show that wavelength, radii and height of the cylinder can be employed as secret keys. However, only cylindrical asymmetric diffraction is not enough to achieve good encryption performance. The CC values of corresponding to Figs. 6(b)–(j) are 0.0186, 0.0103, 0.0007, 0.0009, 0.0338, 0.0362, 0.0462, 0.0474, 0.9999, respectively. These results show that known any part of keys, whatever they are double-random phase decryption keys or cylindrical asymmetric diffraction decryption keys, cannot alone successfully decrypt the original image. Therefore, it is demonstrated that the proposed cryptosystem has very high validity and robustness against different attacks.
𝑢𝑅2 (𝜃𝑅2 , 𝑧𝑅2 ) = 𝐶𝐼𝑂𝑃𝑟,𝑅2,𝐻,𝜆 {𝐶𝑂𝐼𝑃𝑅1,𝑟,𝐻,𝜆 [𝑢𝑅1 (𝜃𝑅1,𝑧𝑅1 ) ∗ 𝑀𝑥1,𝜇1 (𝜃𝑅1,𝑧𝑅1 )] ∗ 𝑀𝑥2,𝜇2 (𝜃𝑟,𝑧𝑟 )},
(4)
𝑢′𝑅1 (𝜃𝑅1 , 𝑧𝑅1 ) = 𝐼𝐶𝑂𝐼𝑃𝑅1,𝑟,𝐻,𝜆 {𝐼𝐶𝐼𝑂𝑃𝑟,𝑅2,𝐻,𝜆 [𝑢′𝑅2 (𝜃𝑅2 , 𝑧𝑅2 )] ∗ ∗ ∗ 𝑀𝑥2,𝜇2 (𝜃𝑟 , 𝑧𝑟 )} ∗ 𝑀𝑥1,𝜇1 (𝜃𝑅1 , 𝑧𝑅1 ),
(5)
𝑀𝑥1 ,𝜇1 (𝜃𝑅 , 𝑧𝑅 ) = exp[𝑖2𝜋 × 𝑅𝑥1 ,𝜇1 (𝜃𝑅 , 𝑧𝑅 )], 𝑀𝑥2 ,𝜇2 (𝜃𝑟 , 𝑧𝑟 ) = exp[𝑖2𝜋 × 𝑅𝑥2 ,𝜇2 (𝜃𝑟 , 𝑧𝑟 )],
(7)
(6)
where (𝜃𝑅1 , 𝑧𝑅1 ), (𝜃𝑟 , 𝑧𝑟 ), and (𝜃𝑅2 , 𝑧𝑅2 ) are the coordination of the input, intermediate, and output surfaces, respectively. The 𝐶𝑂𝐼𝑃 𝑅1,r,𝐻,𝜆 and 𝐶𝐼𝑂𝑃 𝑟,𝑅2,𝐻,𝜆 are the diffractions of cylindrical outside-in and inside-out propagation models under conditions of (𝑅1 , 𝑟, 𝐻, 𝜆) and (𝑟, 𝑅2 , 𝐻, 𝜆), which can be calculated by Eqs. (2) and (1), respectively. The ICOIP and ICIOP are their reverse calculations, respectively. The 𝑀𝑥1,𝜇1 and 𝑀𝑥2 , , 𝜇2 represent RPM1 and RPM2, respectively. The operator of ∗ represents conjugation. The 𝑅𝑥1,𝜇1 and 𝑅𝑥2,𝜇2 follow pseudo random distribution in a range of 0 to 1, which can be generated by Chaos 470
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Fig. 4. Diffraction and encryption results under different circumstances using Rulers (512 × 512 pixels). Without RPM1 and RPM2, (a1) input plaintext, (a2) 1st round diffraction, (a3) 2nd round diffraction, (a4) PSH1 (a5) PSH2; with RPM1 and RPM2, (b1) input plaintext, (b2) 1st round diffraction, (b3) 2nd round diffraction, (b4) PSH1 (b5) PSH2.
Fig. 5. (a) Input plaintext (Rulers, 512 × 512 pixels); only cylindrical asymmetric diffraction keys are used; 𝜆 = 842 nm; (b) [𝑟/𝐻/𝑅1 ∕𝑅2 ]=[1/5/60/65] mm, (c) [𝑟/𝐻/𝑅1 ∕𝑅2 ]=[1/5/60/75] mm, (d) [𝑟/𝐻/𝑅1 ∕𝑅2 ]=[1/6/90/95] mm; 𝜆 = 632 nm; (e) [𝑟/𝐻/𝑅1 ∕𝑅2 ]=[0.7/5/80/85]mm; (f) [𝑟/𝐻/𝑅1 ∕𝑅2 ]=[0.7/5/80/100]mm; (g) [𝑟/𝐻/𝑅1 ∕𝑅2 ]=[0.7/6/120/130]mm; 𝜆 = 538 nm; (h) [𝑟/𝐻/𝑅1 ∕𝑅2 ]=[0.6/5/100/110]mm; (i) [𝑟/𝐻/𝑅1 ∕𝑅2 ]=[0.6/5/100/120]mm; (j) [𝑟/𝐻/𝑅1 ∕𝑅2 ]=[0.6/6/140/150]mm.
3.2. Key space and sensitivity
3.3. Security analysis
The high sensitive to initial conditions is inherent in any chaos system. To provide an encryption algorithm with high security, the key space should be large enough to make any brute force attack ineffective. The total key space is generally decided by the keys in the algorithm include the initial conditions of chaotic system (𝜇2 , 𝑥2 ) and reconstruction system parameters (𝜆, 𝑟, 𝐻, 𝑅2 ). Compared with convention DRPE, the proposed scheme of CAD-DRPE has two additional keys. The dependences of 𝐶𝐶 on the change of (𝜆, 𝑟, 𝐻, 𝑅2 ) are shown in Figs. 7(a)–7(c), respectively. Only when the values of (𝜆, 𝑟, 𝐻, 𝑅2 ) are the correct does the 𝐶𝐶 reach or very close to 1. In the case of a small derivations, the 𝐶𝐶 decreases quickly and there is a failure to recognize the original image visually. The sensitivity of the chaos sequence on the factors (𝜇2 , 𝑥2 ) is shown as Fig. 7(d). When the precision decimal value of keys varies from 10−13 to 10−14 , the 𝐶𝐶 value of the two arrays is very small. Therefore, the total key space is at least 10(14+14+6+6+5+10) . The high sensitivity and huge key space result in great difficulty in duplicating the decryption system.
Finally, the input–output algorithm (HIO) proposed by Fienup [39] is employed to test the effectiveness of phase-retrieval attack free. The sum square error (SSE) is used as the evaluation index: } {𝑀 𝑁 ∑ ∑[ ]2 , (8) SSE = 10 log10 𝑓 (𝑚, 𝑛) − 𝑓 𝑁 (𝑚, 𝑛) 𝑚=1 𝑛=1
where 𝑓 (𝑚, 𝑛) denotes the known amplitude distribution on the object surface, and 𝑓 𝑁 (𝑚, 𝑛) denotes the computed amplitude distribution after 𝑁 iterations. Fig. 8 shows the effectiveness of the system when the phase-retrieval algorithm is implemented. Figs. 8(a)–(c) show the input plaintext and the result images retrieved using the proposed CAD-DRPE and the conventional DRPE schemes, respectively. Fig. 8(d) shows the SSEs of phase-key retrieval with respect to the number of iterations for the CAD-DRPE and DRPE. The results of the proposed scheme show higher robustness against phase-retrieval attack than the DRPE scheme, demonstrating the validity of our phase-retrieval attack free method. 471
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Fig. 6. (a) Input plaintext (Rulers, 512 × 512 pixels); Encryption with two encryption keys, (b) PSH1; (c) PSH2; (d) Two decryption keys are both wrong; (e) False double-random phase decryption keys and true cylindrical asymmetric diffraction decryption keys; True double-random phase decryption keys and false cylindrical asymmetric diffraction decryption keys, only false in: (f) 𝜆; (g) 𝑟; (h) 𝑅2 ; (i) 𝐻; (j) Two decryption keys are both true.
Fig. 7. The correlation coefficient (𝐶𝐶) varies with (a) wavelength shift 𝛥𝜆 at the wavelength of 842 nm when 𝑅2 = 65 mm, 75 mm, 95 mm, 𝑟/𝐻/𝑅1 =1/5/60 mm; (b) 𝑅2 shift 𝛥𝑅2 at 𝑅2 = 120 mm when 𝜆 = 842 nm, 632 mm, 538 mm, 𝑟/𝐻/𝑅1 =1/5/60 mm; (c) shifts of 𝛥𝑟/𝛥𝐻/𝛥𝑅2 at 𝑟/𝐻/𝑅2 =1/5/65 mm when 𝜆 = 842 nm, 𝑅1 = 60 mm; (d) shifts of 𝛥𝜇2 ∕𝛥𝑥2 at 𝜇2 ∕𝑥2 = 3.674∕0.7 when 𝜆 = 842 nm, 𝑟∕𝐻∕𝑅1 ∕𝑅2 = 1∕5∕60∕65 mm.
4. Conclusions
conventional DRPE based methods. Furthermore, it is free of phaseretrieval attack because of the cylindrical asymmetric diffraction between two concentric cylinders. Numerical simulation results demonstrate the effectiveness and flexibility of the proposed cryptosystem. This cylindrical diffraction based DRPE scheme could be extended to 3D object encryption and multi or color image encryption.
We proposed a cryptosystem based on the cylindrical asymmetric diffraction and DRPE, which is free of phase-retrieval attack. In this proposal, the plaintext image can be encrypted through a two-step diffraction process with double pseudo random phase masks located on the object surface and the first diffraction surface. The two-step phaseshifting holography is employed to record and recover the distribution of complex amplitude. The security of the proposed cryptosystem is improved in key space due to two additional keys compared with the
Acknowledgments This work is supported by the National Science Foundation of China (NSFC) (Grant Nos. 61405130, and 61320106015). 472
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Fig. 8. Results of phase-retrieval attack. (a) Plaintext image (Mona, 256 × 256 pixels); (b) retrieval result of the proposed CAD-DRPE; (c) retrieval result of DRPE; (d) retrieval results of SSE for CAD-DRPE (red) and DRPE (blue). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Phase-retrieval attack free cryptosystem based on cylindrical asymmetric diffraction and double-random phase encoding Jun Wang a , Xiaowei Li a , Yuhen Hu b , Qiong-Hua Wang a, * a b
School of Electronics and Information Engineering, Sichuan University, Chengdu 610065, China Department of Electrical & Computer Engineering, University of Wisconsin-Madison, Madison, WI 53706, USA
a r t i c l e
i n f o
Keywords: Optical encryption Phase-retrieval attack free cryptosystem Cylindrical diffraction method Computer holography
a b s t r a c t A phase-retrieval attack free cryptosystem based on the cylindrical asymmetric diffraction and double-random phase encoding (DRPE) is proposed. The plaintext is abstract as a cylinder, while the observed diffraction and holographic surfaces are concentric cylinders. Therefore, the plaintext can be encrypted through a two-step asymmetric diffraction process with double pseudo random phase masks located on the object surface and the first diffraction surface. After inverse diffraction from a holographic surface to an object surface, the plaintext can be reconstructed using a decryption process. Since the diffraction propagated from the inner cylinder to the outer cylinder is different from that of the reversed direction, the proposed cryptosystem is asymmetric and hence is free of phase-retrieval attack. Numerical simulation results demonstrate the flexibility and effectiveness of the proposed cryptosystem. © 2017 Elsevier B.V. All rights reserved.
1. Introduction The network globalization and active social networking make the information security technologies increasingly important. Among them, optical information processing is one of the most promising approaches for data securing, encryption, and authentication [1–3]. The field of optical encryption was pioneered by Refregier and Javidi, who used double-random phase encoding (DRPE) in the standard 4f optical system [4,5]. DRPE has developed to one of the most widely used and studied optical encryption techniques. It has been extended to the Fresnel transform [6–8], the fractional Fourier transformation [9,10], gyrator transform [11], and phase truncation operations [12–14] by many researchers. To improve security, DRPE has been combined with other imaging techniques such as iterative computational algorithms [15–17], compressive sensing [18,19], photon-counting imaging [20–22], and so on. These methods have either reinforced the decoding resistance or simplified the implementation of the operation. Unfortunately, the conventional DRPE encryption approach has vulnerability against specific types of attacks, such as chosen-plaintext attacks and known-plaintext attacks [23–26]. Several cryptosystems have been proposed to improve the weakness of conventional DRPE based approaches. An asymmetric cryptosystem was proposed, which is based on phase-truncated Fourier transforms [27]. Later, an improved
asymmetric cryptosystem was developed by using random binary phase modulation and a mixture phase retrieval algorithm [28]. Also, coherent superposition and equal modulus decomposition were proposed for the asymmetric optical cryptosystem [29]. More recently, the distorted wavefront beam illumination was adopted to develop a novel cryptosystem based on double-random phase encoding [30]. And a novel optical image encryption method was proposed by employing divergent illumination [31]. Although they are easy to implement in digital simulation, they are difficult to implement in optical experiments. Moreover, they are generally vulnerable to the collision algorithm [32,33]. This is because these cryptosystems are derived from mathematical formulas and most of them lack a reasonable explanation from an optical viewpoint. Therefore, a cryptosystem with asymmetric diffraction based on optical principles that is practical to implement and exhibits a high resistance against specific attacks is desirable. Inspired by the aforementioned research, we proposed a phaseretrieval attack free cryptosystem based on cylindrical asymmetric diffraction and DRPE in this paper. It is noticed that the majority of existing schemes of optical image encryption based on DRPE at present are propagations between two or more planers. To the best of our knowledge, it is the first time to apply the cylindrical diffraction to holographic encryption. In this proposal, the plaintext image is abstract as a cylinder, while the observed diffraction and holographic surfaces are concentric
* Corresponding author.
E-mail address: [email protected] (Q. Wang). https://doi.org/10.1016/j.optcom.2017.10.061 Received 28 September 2017; Received in revised form 23 October 2017; Accepted 24 October 2017 0030-4018/© 2017 Elsevier B.V. All rights reserved.
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Fig. 1. Illustration of geometrical relation in cylindrical diffraction with top-view. (a) IOP Model, (b) OIP Model.
Rayleigh–Sommerfeld integral equation in its form can be written by
cylinders. Therefore, the plaintext can be encrypted through a two-step diffraction process with double pseudo random phase masks located on the object surface and the first diffraction surface. The distribution of complex amplitude on the second diffraction surface can be recorded and recovered by the two-step phase-shifting holography. The plaintext image can be reconstructed by inverse diffraction from a holographic surface to an object surface in a decryption process. The initial value and control parameter of double pseudo random phase masks, wavelength of the cylindrical wave, and parameters (inner and outer radii and height) of the cylinders are highly sensitive keys for authorized users. Compared with the conventional DRPE based cryptosystems, our proposed scheme has two additional keys, which enhance the security of the proposed cryptosystem. Since the diffraction calculation of propagation from inner cylinder to outer one is different from that of the reversed direction, the cylindrical diffraction is asymmetric, and hence the proposed cryptosystem is free of phase-retrieval attack. Numerical simulation results demonstrate the effectiveness and flexibility of the proposed cryptosystem.
𝑢𝑅 (𝜃𝑅 , 𝑧𝑅 ) = 𝐶
∬𝑠
𝑢𝑟 (𝜃𝑟 , 𝑧𝑟 )
exp(𝑖𝑘𝑑𝑃𝑟 𝑄𝑅 ) 𝑑𝑃𝑟 𝑄𝑅
𝑑𝜃𝑟 𝑑𝑧𝑟
⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ ⋯ IOP, (1) exp(𝑖𝑘𝑑𝑃𝑅 𝑄𝑟 )[𝑟 − 𝑅 cos(𝜃𝑟 − 𝜃𝑅 )] 𝑢𝑟 (𝜃𝑟 , 𝑧𝑟 ) = 𝐶 𝑢 (𝜃 , 𝑧 ) 𝑑𝜃𝑅 𝑑𝑧𝑅 ∬𝑠 𝑅 𝑅 𝑅 𝑑𝑃2 𝑄 𝑅
𝑟
(2)
⋯ ⋯ OIP, 2
2
𝑑 = 𝑑𝑃𝑟 𝑄𝑅 = 𝑑𝑃𝑅 𝑄𝑟 = [𝑅 + 𝑟 − 2𝑅𝑟 cos(𝜃𝑅 − 𝜃𝑟 ) + (𝑧𝑅 − 𝑧𝑟 )2 ]1∕2 ,
(3)
where 𝑘 and 𝐶 denote the wavenumber of the incident light and a constant, respectively. The 𝑑 represents the distance between two points of 𝑃 and 𝑄 on the object and observation surfaces, respectively. The 𝑠 denotes the object surface. Note that these two equations can be accelerated by FFT algorithm [34,35]. 2.2. DRPE with cylindrical asymmetric diffraction The optical setup for the realization of the DRPE with cylindrical asymmetric diffraction is depicted as Fig. 2. The laser, special filter (SF), and lens (L1) make up a collimating system to generate the uniform plane wave. M1 and M2 are mirrors. The beam splitter (BS1) between two mirrors and the beam splitter (BS2) just before CCD divide the wave light into two beams. The random phase masks (RPM1 and RPM2) are cylindrical surfaces, which have radii of 𝑅1 , 𝑅2 and 𝑟, respectively. The input object, which is bonded to RPM1, is also cylindrical surface with radius of 𝑅1 . The object wave, 𝑢𝑅1 (𝜃𝑅1 , 𝑧𝑅1 ), modulates by the RPM1 and propagates to the intermediate surface, on which the distributions is 𝑢𝑟 (𝜃𝑟 , 𝑧𝑟 ). After reflection and modulation by the RPM2, the wave field continuously propagates to the destination surface, on which the distributions is 𝑢𝑅2 (𝜃𝑅2 , 𝑧𝑅2 ). Finally, it interferes with the reference light, and the holograms are captured by the CCD. The two-step phaseshifting holography is employed here [36]. The holograms (PSH1 and PSH2) are sent to computer for processing and transmittance. Note that, the optical path lengths of the two wavefront sensing optical paths of the object and reference lights should be equal.
2. Principle In this section, the principle of our proposal is descripted. Firstly, the basic diffraction theory between a pair of concentric cylindrical surfaces is introduced. Then, the optical setup of our proposed DRPE with cylindrical asymmetric diffraction is given. Lastly, how to encrypt and decrypt an object image applying the cylindrical diffraction theory is presented.
2.1. Diffraction between two concentric cylinders In the cylindrical diffraction theory [34,35], the object and the observation surfaces are concentric cylindrical surfaces as shown in Fig. 1, where 𝑅 and 𝑟 denote the radii of the inner and outer surfaces, respectively. Obviously, there are two, inside-out and outside-in, propagation models in the cases that object are placed on the inside and outside surfaces as shown in Figs. 1(a) and 1(b), respectively. In the case of inside-out propagation (IOP) model, the object and observation points are represented by 𝑃𝑟 (𝜃𝑟 , 𝑧𝑟 ) and 𝑄𝑅 (𝜃𝑅 , 𝑧𝑅 ) in cylindrical coordinate, respectively. While in the case of outside-in propagation (OIP) model, the object and observation points are represented by 𝑃𝑅 (𝜃𝑅 , 𝑧𝑅 ) and 𝑄𝑟 (𝜃𝑟 , 𝑧𝑟 ) in cylindrical coordinate, respectively. Here, 𝑧𝑟 and 𝑧𝑅 are in range of −𝐻/2 to 𝐻/2, where 𝐻 is the height of the cylindrical surface. If the distributions on the inner and outer surfaces are represented by 𝑢𝑟 (𝜃𝑟 , 𝑧𝑟 ) and 𝑢𝑅 (𝜃𝑅 , 𝑧𝑅 ), respectively, the
2.3. Encryption and decryption The flow chart of encryption and decryption can be depicted in Fig. 3. In encryption, the object image is located at the cylindrical surface with radius of 𝑅1 as input and is modulated by the first random phase mark (RPM1). After propagation a distance of 𝑑1 in OIP model, the wavefront reaches the first intermediate surface and is modulated by the second random phase mark (RPM2). After propagation a distance of 𝑑2 in IOP model, the wavefront reaches the destination surface and is captured by 469
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algorithm [37]. The 𝑢′𝑅2 is reconstructed complex distribution by using two-step phase-shifting holography [36].
3. Simulation results 3.1. Results of the proposal For the system discussed in Section 2, numerical experiments are carried out to verify the proposed cryptosystem based on cylindrical asymmetric diffraction and DRPE, which is refer to CAD-DRPE. Let the values of 𝑟, 𝑅1 , 𝑅2 , 𝐻 be 1, 60, 65, 5 mm, respectively. The wavelength use in the simulation is 842 nm. Then, the pixel numbers in the azimuthal and vertical directions are at least 15178 and 503, respectively, according to the Shannon sampling theory [38]. Since the cylindrical surface in azimuthal direction is much longer than that in vertical one, it is better to select an equal pixel size in both directions which should be greater than 503. Therefore, we design the sampling number of plaintext image to be 512 × 512 pixels, and 15178/512 is nearly 30. That is the reason we chose a part of the cylindrical surface in azimuthal direction with angle of 2𝜋/30, ex. [−2𝜋/60, 2𝜋/60], to present the plaintext image. In fact, we have many other options for the sampling number of plaintext image. If we chose angle of 2𝜋/15, the pixels should be 512×1024, and if we chose angle of 2𝜋∕60, the pixels should be 512×256, and so on. The 𝑅2 , 𝐻, 𝑟, and 𝜆 are independent and can be used as the cylindrical diffraction keys, and the random phase mask generation keys of 𝑥2 and 𝜇2 are DRPE encryption keys. We employed Rulers (512 × 512 pixels) as input plaintext image and performed the simulations using matlab R2010b platform to demonstrate the correctness and the security of the cryptosystem, as shown in Figs. 4–6. The correlation coefficient (𝐶𝐶) values between the original image, 𝑓 (𝑚, 𝑛), and the different images, 𝑓 ′ (𝑚, 𝑛), shown in Figs. 4(a2)–(a5), Figs. 4(b2)–(b5), Figs. 5(b)–(j) and Figs. 6(b)–(j), are calculated using
Fig. 2. Optical setup for realization of proposed DRPE with cylindrical asymmetric diffraction.
Fig. 3. Encryption and decryption of the proposed cryptosystem.
𝐶𝐶 = [cov(𝑓 , 𝑓 ′ )]∕(𝜎𝑓 × 𝜎𝑓 ′ ) CCD camera. The output is holograms (PSH1 = 𝐼0 and PSH2 = 𝐼𝜋 ∕∕2) through the imaging system of two-step phase-shifting holography. Decryption is the inverse of encryption. However, the first inverse propagation of IOP is OIP model in optical setup. And the second inverse propagation of OIP is IOP model in optical setup. Therefore, this cryptosystem is asymmetric, and is expected to have higher security than that of conventional DRPE based on propagation between several parallel planes. The computer simulations can employ the inverse propagation to replace the optical process. The encryption and decryption can be abstract in mathematic as
where cov denotes the cross-covariance and 𝜎 denotes standard deviation. For the sake of brevity, the coordinate (𝑚, 𝑛) is omitted in this equation. The diffraction and encryption results of the proposed method without and with RPMs are shown in Figs. 4(a2)–(a5) and (b2)–(b5), respectively. The CC values of corresponding to Figs. 4(a2)–(a5) are 0.1530, 0.2285, 0.2453, and 0.2624, respectively. The CC values of corresponding to Figs. 4(b2)–(b5) are 0.0230, 0.0025, 0.0186, and 0.0103, respectively. The CC values of diffraction and encryption results without RPMs show that the results images are related to the original image. And it also verified by subjective observation that the some information of the original object can be recognized. When the cylindrical diffraction is combined with the DPRE method by adding two RPMs, the encrypted images show good noise-like property. And it is also verified by their CC values. The CC values of corresponding to Figs. 5(b)–(j) are 0.2285, 0.2613, 0.1964, 0.2849, 0.1587, 0.1338, 0.0752, 0.2946, 0.0146, respectively. These results show that wavelength, radii and height of the cylinder can be employed as secret keys. However, only cylindrical asymmetric diffraction is not enough to achieve good encryption performance. The CC values of corresponding to Figs. 6(b)–(j) are 0.0186, 0.0103, 0.0007, 0.0009, 0.0338, 0.0362, 0.0462, 0.0474, 0.9999, respectively. These results show that known any part of keys, whatever they are double-random phase decryption keys or cylindrical asymmetric diffraction decryption keys, cannot alone successfully decrypt the original image. Therefore, it is demonstrated that the proposed cryptosystem has very high validity and robustness against different attacks.
𝑢𝑅2 (𝜃𝑅2 , 𝑧𝑅2 ) = 𝐶𝐼𝑂𝑃𝑟,𝑅2,𝐻,𝜆 {𝐶𝑂𝐼𝑃𝑅1,𝑟,𝐻,𝜆 [𝑢𝑅1 (𝜃𝑅1,𝑧𝑅1 ) ∗ 𝑀𝑥1,𝜇1 (𝜃𝑅1,𝑧𝑅1 )] ∗ 𝑀𝑥2,𝜇2 (𝜃𝑟,𝑧𝑟 )},
(4)
𝑢′𝑅1 (𝜃𝑅1 , 𝑧𝑅1 ) = 𝐼𝐶𝑂𝐼𝑃𝑅1,𝑟,𝐻,𝜆 {𝐼𝐶𝐼𝑂𝑃𝑟,𝑅2,𝐻,𝜆 [𝑢′𝑅2 (𝜃𝑅2 , 𝑧𝑅2 )] ∗ ∗ ∗ 𝑀𝑥2,𝜇2 (𝜃𝑟 , 𝑧𝑟 )} ∗ 𝑀𝑥1,𝜇1 (𝜃𝑅1 , 𝑧𝑅1 ),
(5)
𝑀𝑥1 ,𝜇1 (𝜃𝑅 , 𝑧𝑅 ) = exp[𝑖2𝜋 × 𝑅𝑥1 ,𝜇1 (𝜃𝑅 , 𝑧𝑅 )], 𝑀𝑥2 ,𝜇2 (𝜃𝑟 , 𝑧𝑟 ) = exp[𝑖2𝜋 × 𝑅𝑥2 ,𝜇2 (𝜃𝑟 , 𝑧𝑟 )],
(7)
(6)
where (𝜃𝑅1 , 𝑧𝑅1 ), (𝜃𝑟 , 𝑧𝑟 ), and (𝜃𝑅2 , 𝑧𝑅2 ) are the coordination of the input, intermediate, and output surfaces, respectively. The 𝐶𝑂𝐼𝑃 𝑅1,r,𝐻,𝜆 and 𝐶𝐼𝑂𝑃 𝑟,𝑅2,𝐻,𝜆 are the diffractions of cylindrical outside-in and inside-out propagation models under conditions of (𝑅1 , 𝑟, 𝐻, 𝜆) and (𝑟, 𝑅2 , 𝐻, 𝜆), which can be calculated by Eqs. (2) and (1), respectively. The ICOIP and ICIOP are their reverse calculations, respectively. The 𝑀𝑥1,𝜇1 and 𝑀𝑥2 , , 𝜇2 represent RPM1 and RPM2, respectively. The operator of ∗ represents conjugation. The 𝑅𝑥1,𝜇1 and 𝑅𝑥2,𝜇2 follow pseudo random distribution in a range of 0 to 1, which can be generated by Chaos 470
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Fig. 4. Diffraction and encryption results under different circumstances using Rulers (512 × 512 pixels). Without RPM1 and RPM2, (a1) input plaintext, (a2) 1st round diffraction, (a3) 2nd round diffraction, (a4) PSH1 (a5) PSH2; with RPM1 and RPM2, (b1) input plaintext, (b2) 1st round diffraction, (b3) 2nd round diffraction, (b4) PSH1 (b5) PSH2.
Fig. 5. (a) Input plaintext (Rulers, 512 × 512 pixels); only cylindrical asymmetric diffraction keys are used; 𝜆 = 842 nm; (b) [𝑟/𝐻/𝑅1 ∕𝑅2 ]=[1/5/60/65] mm, (c) [𝑟/𝐻/𝑅1 ∕𝑅2 ]=[1/5/60/75] mm, (d) [𝑟/𝐻/𝑅1 ∕𝑅2 ]=[1/6/90/95] mm; 𝜆 = 632 nm; (e) [𝑟/𝐻/𝑅1 ∕𝑅2 ]=[0.7/5/80/85]mm; (f) [𝑟/𝐻/𝑅1 ∕𝑅2 ]=[0.7/5/80/100]mm; (g) [𝑟/𝐻/𝑅1 ∕𝑅2 ]=[0.7/6/120/130]mm; 𝜆 = 538 nm; (h) [𝑟/𝐻/𝑅1 ∕𝑅2 ]=[0.6/5/100/110]mm; (i) [𝑟/𝐻/𝑅1 ∕𝑅2 ]=[0.6/5/100/120]mm; (j) [𝑟/𝐻/𝑅1 ∕𝑅2 ]=[0.6/6/140/150]mm.
3.2. Key space and sensitivity
3.3. Security analysis
The high sensitive to initial conditions is inherent in any chaos system. To provide an encryption algorithm with high security, the key space should be large enough to make any brute force attack ineffective. The total key space is generally decided by the keys in the algorithm include the initial conditions of chaotic system (𝜇2 , 𝑥2 ) and reconstruction system parameters (𝜆, 𝑟, 𝐻, 𝑅2 ). Compared with convention DRPE, the proposed scheme of CAD-DRPE has two additional keys. The dependences of 𝐶𝐶 on the change of (𝜆, 𝑟, 𝐻, 𝑅2 ) are shown in Figs. 7(a)–7(c), respectively. Only when the values of (𝜆, 𝑟, 𝐻, 𝑅2 ) are the correct does the 𝐶𝐶 reach or very close to 1. In the case of a small derivations, the 𝐶𝐶 decreases quickly and there is a failure to recognize the original image visually. The sensitivity of the chaos sequence on the factors (𝜇2 , 𝑥2 ) is shown as Fig. 7(d). When the precision decimal value of keys varies from 10−13 to 10−14 , the 𝐶𝐶 value of the two arrays is very small. Therefore, the total key space is at least 10(14+14+6+6+5+10) . The high sensitivity and huge key space result in great difficulty in duplicating the decryption system.
Finally, the input–output algorithm (HIO) proposed by Fienup [39] is employed to test the effectiveness of phase-retrieval attack free. The sum square error (SSE) is used as the evaluation index: } {𝑀 𝑁 ∑ ∑[ ]2 , (8) SSE = 10 log10 𝑓 (𝑚, 𝑛) − 𝑓 𝑁 (𝑚, 𝑛) 𝑚=1 𝑛=1
where 𝑓 (𝑚, 𝑛) denotes the known amplitude distribution on the object surface, and 𝑓 𝑁 (𝑚, 𝑛) denotes the computed amplitude distribution after 𝑁 iterations. Fig. 8 shows the effectiveness of the system when the phase-retrieval algorithm is implemented. Figs. 8(a)–(c) show the input plaintext and the result images retrieved using the proposed CAD-DRPE and the conventional DRPE schemes, respectively. Fig. 8(d) shows the SSEs of phase-key retrieval with respect to the number of iterations for the CAD-DRPE and DRPE. The results of the proposed scheme show higher robustness against phase-retrieval attack than the DRPE scheme, demonstrating the validity of our phase-retrieval attack free method. 471
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Fig. 6. (a) Input plaintext (Rulers, 512 × 512 pixels); Encryption with two encryption keys, (b) PSH1; (c) PSH2; (d) Two decryption keys are both wrong; (e) False double-random phase decryption keys and true cylindrical asymmetric diffraction decryption keys; True double-random phase decryption keys and false cylindrical asymmetric diffraction decryption keys, only false in: (f) 𝜆; (g) 𝑟; (h) 𝑅2 ; (i) 𝐻; (j) Two decryption keys are both true.
Fig. 7. The correlation coefficient (𝐶𝐶) varies with (a) wavelength shift 𝛥𝜆 at the wavelength of 842 nm when 𝑅2 = 65 mm, 75 mm, 95 mm, 𝑟/𝐻/𝑅1 =1/5/60 mm; (b) 𝑅2 shift 𝛥𝑅2 at 𝑅2 = 120 mm when 𝜆 = 842 nm, 632 mm, 538 mm, 𝑟/𝐻/𝑅1 =1/5/60 mm; (c) shifts of 𝛥𝑟/𝛥𝐻/𝛥𝑅2 at 𝑟/𝐻/𝑅2 =1/5/65 mm when 𝜆 = 842 nm, 𝑅1 = 60 mm; (d) shifts of 𝛥𝜇2 ∕𝛥𝑥2 at 𝜇2 ∕𝑥2 = 3.674∕0.7 when 𝜆 = 842 nm, 𝑟∕𝐻∕𝑅1 ∕𝑅2 = 1∕5∕60∕65 mm.
4. Conclusions
conventional DRPE based methods. Furthermore, it is free of phaseretrieval attack because of the cylindrical asymmetric diffraction between two concentric cylinders. Numerical simulation results demonstrate the effectiveness and flexibility of the proposed cryptosystem. This cylindrical diffraction based DRPE scheme could be extended to 3D object encryption and multi or color image encryption.
We proposed a cryptosystem based on the cylindrical asymmetric diffraction and DRPE, which is free of phase-retrieval attack. In this proposal, the plaintext image can be encrypted through a two-step diffraction process with double pseudo random phase masks located on the object surface and the first diffraction surface. The two-step phaseshifting holography is employed to record and recover the distribution of complex amplitude. The security of the proposed cryptosystem is improved in key space due to two additional keys compared with the
Acknowledgments This work is supported by the National Science Foundation of China (NSFC) (Grant Nos. 61405130, and 61320106015). 472
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Fig. 8. Results of phase-retrieval attack. (a) Plaintext image (Mona, 256 × 256 pixels); (b) retrieval result of the proposed CAD-DRPE; (c) retrieval result of DRPE; (d) retrieval results of SSE for CAD-DRPE (red) and DRPE (blue). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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