Multiple-image authentication with a cascaded multilevel architecture based on amplitude field random sampling and phase information multiplexing Desheng Fan,1 Xiangfeng Meng,1,* Yurong Wang,1 Xiulun Yang,1 Xuemei Pan,1 Xiang Peng,2 Wenqi He,2 Guoyan Dong,3 and Hongyi Chen4 1
Department of Optics, School of Information Science and Engineering and Shandong Provincial Key Laboratory of Laser Technology and Application, Shandong University, Jinan 250100, China
2
College of Optoelectronics Engineering, Shenzhen University, Shenzhen 518060, China
3
College of Materials Science and Opto-Electronic Techology, University of Chinese Academy of Sciences, Beijing 100049, China 4
College of Electronic Science and Technology, Shenzhen University, Shenzhen 518060, China *Corresponding author: [email protected] Received 28 January 2015; revised 8 March 2015; accepted 8 March 2015; posted 9 March 2015 (Doc. ID 233390); published 8 April 2015
A multiple-image authentication method with a cascaded multilevel architecture in the Fresnel domain is proposed, in which a synthetic encoded complex amplitude is first fabricated, and its real amplitude component is generated by iterative amplitude encoding, random sampling, and space multiplexing for the low-level certification images, while the phase component of the synthetic encoded complex amplitude is constructed by iterative phase information encoding and multiplexing for the high-level certification images. Then the synthetic encoded complex amplitude is iteratively encoded into two phase-type ciphertexts located in two different transform planes. During high-level authentication, when the two phase-type ciphertexts and the high-level decryption key are presented to the system and then the Fresnel transform is carried out, a meaningful image with good quality and a high correlation coefficient with the original certification image can be recovered in the output plane. Similar to the procedure of high-level authentication, in the case of low-level authentication with the aid of a low-level decryption key, no significant or meaningful information is retrieved, but it can result in a remarkable peak output in the nonlinear correlation coefficient of the output image and the corresponding original certification image. Therefore, the method realizes different levels of accessibility to the original certification image for different authority levels with the same cascaded multilevel architecture. © 2015 Optical Society of America OCIS codes: (100.5070) Phase retrieval; (100.3010) Image reconstruction techniques. http://dx.doi.org/10.1364/AO.54.003204
1. Introduction 1559-128X/15/113204-12$15.00/0 © 2015 Optical Society of America 3204
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Optical information security has received much attention in recent years, since Réfrégier and Javidi proposed the double random-phase encoding (DRPE)
technique [1], and it has been successfully combined with other optical information processing techniques or transforms, such as fractional Fourier transform [2–4], digital holography [5,6], phase retrieval [7–10], phase-shifting interferometry [11,12], gyrator transform [13], fractional Mellin transform [14], and twobeam interference [15,16]. To increase the encoding efficiency and facilitate the transmission of the ciphertext, multiple-image encoding, encryption, and authentication systems have received wide attention, since we proposed a multiple-image encryption scheme by random phase matching [17], in which more than one image can be successfully encoded and decoded with the same set of transmitted ciphertexts based on the idea of DRPE and wave field superposition. Subsequently, various approaches, such as wavelength multiplexing [18,19], position multiplexing [20,21], phase-only mask (POM) multiplexing [22], lateral shifting [23,24], and optical data compression [25,26], have been proposed. In 2013, Gong et al. proposed a method of multiple-image encryption and authentication based on space multiplexing and sparse representation strategy [27], in
which the sparse data of multiple encrypted images can be extracted with aid of the random binary amplitude masks and then integrated into a synthesized ciphertext with space multiplexing. Recently, based on a modified Gerchberg–Saxton algorithm and random sampling strategy, Chen and Chen proposed an optical multiple-image authentication scheme in which the system capacity can be arbitrarily controlled by use of random sampling [28]. To decrease the cross talk effect between the multiplexing images, Wang et al. proposed a multiple-image encryption system based on a triple-POM cascaded architecture using a modified Gerchberg–Saxton algorithm in the gyrator domain, by which the primitive images can be converted into common POMs with noise-like phase distribution [29]. More recently, to present a security-enhanced nonlinear cryptosystem for multiple-image encryption based on amplitude and phase modulation, Wang et al. proposed a nonlinear multiple-image encryption method based on a mixture retrieval algorithm and phase mask multiplexing in the Fresnel domain [30]. In summary, according to the result of whether or not the recovered image can be visually recognized,
Fig. 1. Schematic diagram of the proposed cascaded multiple-image authentication system. 10 April 2015 / Vol. 54, No. 11 / APPLIED OPTICS
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the multiple-image authentication systems may be divided into two classes: high-level authentication and low-level authentication. In the former, the reconstructed output image is a meaningful image with good quality and a high correlation coefficient (CC), while in the latter, no significant or meaningful information is retrieved, but it can result in a remarkable peak output in the nonlinear correlation coefficient (NCC) of the output image and the corresponding standard certification image. In this sense, to increase the authority levels, we present here a multiple-image authentication method with a cascaded multilevel architecture based on amplitude field random sampling and phase information multiplexing, which can realize not only low-level authentication but also high-level authentication. We first give the theoretical analysis, description, and procedure of the method, then provide its simulation verification, and finally draw the conclusion. 2. Theoretical Analysis and Description of the Authentication System
The cascaded authentication system in the Fresnel domain is mainly composed of two processes: highlevel authentication and low-level authentication, the schematic diagram of which is shown in Fig. 1; the theoretical analysis, description, and working procedure is described in the following five subsections. x2
x1
A. Real Amplitude and Phase Information Iteratively Generated in the Design of Low-Level Authentication
Given that there are M certification images in the case of low-level authentication, here, the mth individual certification image is denoted by Gm x; y, where subscript m m 1; 2…M is the serial number of the certification images. Here, we attempt to iteratively encode the gray-scale image “airplane” acting as the example individual certification image Gm x; y into real amplitude and phase information, and the iteration process is shown in Fig. 1(a). One statistical independent random phase mask is placed at the input plane x0 ; y0 , whose amplitude transmittance is expj2πψ 0m x0 ; y0 , and ψ 0m is a random distribution matrix in the range of [0, 1]. The mth individual certification image Gm x; y in low-level authentication is located at the output plane x; y. The distance between the input and the output plane is z. When the input plane is illuminated by an onaxis plane wave of wavelength λ, the complex ampli~ m at the output plane under the Fresnel tude G approximation is [10,31,32] ~ m x; y FrTz fexpj2πψ 0m x0 ; y0 g; G where FrT represents the Fresnel transform. x
x0
y1
y
y0
y2
z Input plane
POM2 POM1
z1
Output plane
z2
z
recovered images
decryption keys
N
∏ exp(− jφi )
i =1,i ≠ n
Fn' high-level authentication
RSMm ψm −φ
low-level authentication
Gm'
Fig. 2. Schematic diagram of the authentication process. 3206
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(1)
The flow chart of the iteration process is similar to Fig. 2 in our recent work [31]. The complex amplitude field distribution (including both real amplitude and phase information) in the input plane is updated after each loop, which helps keep the circulation running, and the mth individual certification image Gm x; y serves as the amplitude constraint at the output plane. After the kth iteration (k 1; 2; 3…), assuming that the real amplitude and phase information in the input plane and the complex amplitude ~ km , respectively in the output plane are Rkm , ψ km , and G (for simplicity, the coordinate notations are neglected), then their distributions in the k 1th step can be described as [10,31,32] k1 ~ km g; Rm absIFrTz fGm expj angleG
(2)
k1 ~ km g; angleIFrTz fGm expj angleG ψm
(3)
where IFrT denotes the inverse Fresnel transform and abs• and angle• denote the extraction operator of the real amplitude and phase, respectively. The CC [10] is generally adopted as the convergent criteria to evaluate the similarity between the output ~ km (assuming after the real amplitude image absG kth iteration) and the standard certification image Gm , which is defined as [10] CC
~ km − EabsG ~ km g EGm − EGm − fabsG ; (4) ~ km σGm σabsG
where E• denotes the expected value operator and σ• is the standard deviation of the corresponding image. The iteration cycle process does not stop until the CC is larger than a predefined value. When the iterative cycle stops, the final real amplitude Rm and phase information ψ m located in the input plane are generated and stored, respectively. Repeating above iteration process for all the M individual low-level certification images, one series of real amplitude fields R1 ; R2 ; … Rm …; RM can be obtained, which are used for sampling, superposition, and synthesis; the other series of phase information distributions ψ 1 ; ψ 2 ; … ψ m …; ψ M , can also be determined, which are applied for designing the decryption keys of the individual certification images. The detailed process will be described in the following sections. B. Sampling, Superposition, and Synthesis of the Real Amplitude Fields
As shown in Fig. 1(b), to acquire the sparse data of the individual real amplitude fields R1 ; R2 ; … Rm …; RM ; M binary random sampling masks (RSMs) RSM1 ; RSM2 ; …RSMm …; RSMM are generated; generally, only a small portion of the pixels of each RSM are set as 1, while the other pixels are 0. The random sampling rate, that is, the percent of the value 1, can be arbitrarily controlled. Therefore, the redundancy in space makes it possible to superpose multiple real amplitude information into just one synthesized
amplitude field R with the aid of random sampling and space multiplexing, which can be described by Rx0 ; y0 R1 x0 ; y0 × RSM1 x0 ; y0 R2 x0 ; y0 × RSM2 x0 ; y0 Rm x0 ; y0 × RSMm x0 ; y0 RM x0 ; y0 × RSMM x0 ; y0
M X m1
Rm x0 ; y0 × RSMm x0 ; y0 ;
(5)
P where denotes the summation operator. In addition, to avoid cross talk between various sampled amplitude information, any two of the RSMs should be generated with no overlap with the areas in them that contain 1’s. Finally, the remnant positions that contain zeros in the synthesized amplitude field R are added by stationary white noise; that is, the zeros in these positions are replaced by random decimals distributed in the range of [0, 1], as shown in Fig. 1(c). Obviously, the final real amplitude Rx0 ; y0 integrates the sampled information of all the M low-level certification images. C. Iterative Phase Information Encoding and Multiplexing in the Design of High-Level Authentication
As shown in Fig. 1(d), assuming that there are N certification images in the case of high-level authentication, here, the nth individual certification image is denoted by F n x; y, where subscript n n 1; 2 … N is the serial number of the certification images. Here, we attempt to iteratively encode the grayscale image “Lena” acting as the example individual certification image F n x; y in the output plane into the phase information in the input plane x0 ; y0 based on the phase retrieval algorithm in the Fresnel domain, the iteration process of which is similar to that in Section 2.A. In each iteration, the individual certification image F n x; y serves as the amplitude constraint in the output plane, while the real amplitude in the input plane is constrained to the synthesized amplitude field R in Section 2.B. The iteration cycle process does not stop until the required CC between the approximate output image absF~ kn and the certification image F n x; y is achieved. When the iterative cycle stops, the final iterative phase information ϕn located in the input plane is generated and stored. Repeating the above iteration process for all the N individual high-level certification images, a series of iterative phase information values ϕ1 ; ϕ2 ; …ϕn …; ϕN can be obtained, which are used for multiplexing. Then a synthetic encoded complex amplitude Ex0 ; y0 in the input plane is fabricated by superposing and multiplying all of these phase components together with Rx0 ; y0 as its real amplitude component. The superposed phase information ϕx0 ; y0 and synthetic encoded complex amplitude Ex0 ; y0 then can be expressed as 10 April 2015 / Vol. 54, No. 11 / APPLIED OPTICS
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ϕx0 ; y0 ϕ1 x0 ; y0 ϕ2 x0 ; y0 ϕn x0 ; y0 ϕN x0 ; y0
N X n1
ϕn x0 ; y0 ;
carried out, which is illustrated in Fig. 2 and described as follows. (6)
Ex0 ; y0 Rx0 ; y0 expjϕx0 ; y0 Rx0 ; y0
N Y n1
expjϕn x0 ; y0 ;
(7)
Q where denotes the multiplication operation. Evidently, the superposed phase information ϕx0 ; y0 integrates all the phase information of N high-level certification images, and this process is illustrated in Fig. 1(e). D. Generation of Phase-Type Ciphertexts and Decryption Keys
Based on phase retrieval in the Fresnel domain, the synthetic encoded complex amplitude Ex0 ; y0 used as the target constraint is then iteratively encoded into two phase-type ciphertexts, POM1 x1 ; y1 and POM2 x2 ; y2 , located in two transform planes x1 ; y1 and x2 ; y2 , respectively, as illustrated in Fig. 1(f). The distance between the two transform planes is z1 , and that between the transform planes x2 ; y2 and the input plane x0 ; y0 (the plane that the synthetic encoded complex amplitude E is located in) is z2 . The wavelength of the on-axis plane wave is λ. The iteration process is very similar to that in Section 2.A, and therefore, a detailed description is omitted here. In the case of high-level authentication, the decryption keys for different individual high-level certification images are generated by the following expression: DKhn x0 ; y0
N Y i1;i≠n
F 0n absFrTz FrTz2 fFrTz1 expjPOM1 × expjPOM2 gDKhn Y N abs FrTz R × expjϕ exp−jϕi i1;i≠n
absfFrTz R × expjϕn g;
(10)
and then the authentication center calculates the CC between the recovered image F 0n and the nth standard certification image F n . If the CC is higher than the predetermined threshold (e.g., 0.90), that means the quality of the reconstructed image is so good that it is a success of high-level authentication, while a lower value means it is a failure. 2. Low-Level Authentication Similar to the process of high-level authentication, when the mth low-level decryption key DKlm is located in the input plane x0 ; y0 , a recovered image G0m is generated in the output plane x; y, which is mathematically expressed as G0m absFrTz FrTz2 fFrTz1 expjPOM1 × expjPOM2 gDKlm absFrTz fR × expjϕ × RSMm × expjψ m − ϕg
exp−jϕi x0 ; y0 ;
n 1; 2; 3…N:
1. High-Level Authentication One authentication participant places his/her highlevel decryption key DKhn in the input plane x0 ; y0 . When the system is illuminated by a plane wave with the correct wavelength λ, a recovered image F 0n is obtained in the output plane x; y, which is mathematically expressed as
absfFrTz RSMm × R × expjψ m g: (8)
(11)
Here, RSMm and ψ m denote the corresponding RSM and the iterated phase information for the mth individual low-level certification image, respectively.
As only a small proportion of the real amplitude information is sampled by the mth random sampling mask RSMm , the recovered image G0m is outwardly a noise-like image whose CC is too low to identify any useful information by direct visual inspection; that is, in this circumstance, it cannot successfully pass through the high-level authentication. To provide an additional authentication layer for the high-level authentication and thus achieve a higher discrimination capability, NCC distribution [33] is applied and calculated to compare the recovered noise-like image G0m with the standard certification image Gm , which is defined as [33–35]
E.
NCCξ; η jIFTjfFTGm ξ; ηgfFTG0m ξ; ηgjω−1
During low-level authentication, the decryption keys for different individual low-level certification images are generated as DKlm x0 ; y0 RSMm x0 ; y0 expfjψ m x0 ; y0 − ϕx0 ; y0 g; m 1; 2; 3…M:
(9)
Authentication Process
In the authentication center, after both two phasetype ciphertexts POM1 x1 ; y1 and POM2 x2 ; y2 are placed in two transform planes x1 ; y1 and x2 ; y2 , respectively, the authentication process can be 3208
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×fFTGm ξ; ηgfFTG0m ξ; ηgj2 ;
(12)
where FT and IFT represent the Fourier and inverse Fourier transforms, respectively, and (ξ, η) is the
Fig. 3. Two phase-type ciphertexts: (a) POM1 and (b) POM2 .
coordinate of the spectrum transverse plane. The parameter ω defines the strength of the applied nonlinearity, whose range is in [0.2, 0.6], in which ω best suits the verification application.
As the sampled information is just related to the synthesized amplitude of the mth low-level certification image, after calculating and displaying the 3D NCC distributions between the recovered image G0m and its corresponding certification image Gm , one remarkable peak is generated in the NCC distribution outputs, which can help authenticate the information without the direct visualization of hidden information. In this circumstance, the authentication process can be called low-level authentication, due to the recovered noise-like image with a low CC, if there exists a remarkable peak in its NCC distributions, which means the low-level authentication is successful, while no remarkable peak indicates a failure. 3. Computer Simulations
A set of computer simulations have been carried out to verify the feasibility of our proposed method and investigate its performance. All the images shown in
Fig. 4. (a)–(d) Four preselected low-level certification images, G1 − G4 , respectively.
Fig. 5. In the design of low-level authentication, (a)–(d) the final retrieved real amplitudes R1 − R4 , (e)–(h) the corresponding retrieved phase information ψ 1 − ψ 4 . 10 April 2015 / Vol. 54, No. 11 / APPLIED OPTICS
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Fig. 6. (a)–(d) Four binary random sampling masks RSM1 − RSM4 .
Fig. 7. Final synthesized real amplitude field R after random sampling, space multiplexing, and random noise addition.
the following simulations are 256 × 256 pixels in size. All the resolutions of the digital images in the input, intermediate transform, and output planes are 15 μm, λ 0.532 μm, z1 z2 z 108.3 mm, ω 0.5. Two phase-type ciphertexts, POM1 and POM2 , are shown in Figs. 3(a) and 3(b), respectively.
Four preselected low-level certification images G1 − G4 are shown in Figs. 4(a)–4(d), based on the iterative phase retrieval process mentioned in Section 2.A. The final retrieved real amplitudes R1 − R4 are given in Figs. 5(a)–5(d), respectively, and the corresponding retrieved phase information ψ 1 − ψ 4 is given in Figs. 5(e)–5(h), respectively. Figures 6(a)– 6(d) depict the corresponding four binary random sampling masks RSM1 − RSM4 (the sampling rate for these four RSMs is 12.5%), respectively, and the final synthesized amplitude field R after random noise addition is shown in Fig. 7. Figure 8 shows the decryption keys DKls for different individual lowlevel certification images; its top row depicts the real amplitude parts, random sampling masks RSM1 − RSM4 , while its bottom row depicts the phase information parts, ψ 1 − ϕ − ψ 4 − ϕ, respectively. Based on the low-level authentication method mentioned in Section 2.E.2, when using all the correct low-level decryption keys DKls, the four
Fig. 8. Decryption keys DKls for different individual low-level certification images. (a)–(d) The real amplitude parts, random sampling masks RSM1 − RSM4 ; (e)–(h) the phase information parts, ψ 1 − ϕ − ψ 4 − ϕ. 3210
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Fig. 9. In the case of low-level authentication, (a)–(d) four retrieved noise-like images.
noise-like images in Figs. 9(a)–9(d) with a low CC are retrieved in the output plane, from which we can see that no meaningful information can be identified by direct visual inspection. That means it cannot successfully pass through the high-level authentication, as the CC is far lower than the threshold value set beforehand. However, the NCC distributions of the final retrieved image and the standard certification image can be focused and calculated, and these are
shown in Figs. 10(a)–10(d), respectively; obviously, a remarkable peak can be observed in all the corresponding NCC distributions. That means that the low-level authentication is successful, which provides an additional authentication layer for the high-level authentication. If the low-level decryption key is false or incorrect, it will result in a failure of low-level authentication. Figure 11 shows the 3D NCC distributions in the
Fig. 10. (a)–(d) 3D NCC distributions of Figs. 9(a)–9(d), respectively. 10 April 2015 / Vol. 54, No. 11 / APPLIED OPTICS
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Fig. 11. 3D NCC distributions when using the false low-level decryption key.
output plane when a random phase is adopted as the phase part of the decryption key; clearly, only the noisy NCC distributions without a remarkable peak are generated, which means the low-level authentication has failed. The system robustness analysis for noise attack is tested; here, the authentication process for the lowlevel certification image in Fig. 4(a) is taken as an example. Figures 12(a) and 12(b) show the NCC distributions when the two ciphertexts POM1 and POM2 , respectively, are contaminated by additive white noise (zero mean noise with 0.01 variance), from which it can be seen that a remarkable peak is obtained in both. Next, we further analyze and discuss how the sampling ratio of the low-level decryption keys (RSM1 − RSM4 ) affects the verification results. Here we also take the authentication process for the lowlevel certification image in Fig. 4(a) as an example. Figures 13(a)–13(f) show the NCC distributions
when the sampling ratio of RSM1 is 9.5%, 8.4%, 6.3%, 5.5%, 5.1%, and 4.1%, respectively, after calculation and analysis. If the sampling ratio is less than 5%, the low-level authentication will fail, and only noisy NCC distributions without a remarkable peak will be obtained. In the circumstance of high-level authentication, four preselected high-level certification images F 1 − F 4 are shown in Figs. 14(a)–14(d). Based on the iterative phase encoding process mentioned in Section 2.C, the corresponding retrieved phase information ϕ1 − ϕ4 is given in Figs. 15(a)–15(d), respectively. After phase information multiplexing, four decryption keys DKh1 − DKh4 for different individual high-level certification images are generated, as shown in Figs. 16(a)–16(d). Based on the high-level authentication method mentioned in Section 2.E.1, Figs. 17(a)–17(d) show the final recovered images obtained from each correct high-level decryption key, and the CCs are 0.9701, 0.9497, 0.9711, and 0.9702, respectively. It is clear that the recovered images with good quality are very similar to the corresponding individual certification images; if the CC’s predetermined threshold is set as 0.90 or smaller, that means the high-level authentication is successful. Finally, the system robustness analysis for noise attack in the circumstance of high-level authentication is investigated; here we take the authentication process for the high-level certification image in Fig. 14(d) as an example. Figures 18(a) and 18(b) give the recovered images when the two ciphertexts POM1 and POM2 , respectively, are contaminated by additive white noise (zero mean noise with 0.01 variance), the CCs of which are 0.7586 and 0.7634, respectively. Obviously, both retrieved images can be recognized without doubt, and the proposed method possesses high robustness against noise attack.
Fig. 12. 3D NCC distributions when the two ciphertexts (a) POM1 and (b) POM2 are contaminated by additive white noise. 3212
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Fig. 13. 3D NCC distributions when the sampling ratio of decryption key RSM1 is (a) 9.5%, (b) 8.4%, (c) 6.3%, (d) 5.5%, (e) 5.1%, and (f) 4.1%.
4. Conclusions
In conclusion, we have proposed a multiple-image authentication method with a cascaded multilevel architecture, in which the real amplitude component of the synthetic encoded complex amplitude field is generated by iterative amplitude encoding, random
sampling, and space multiplexing for the low-level certification images, while the phase component of the synthetic encoded complex amplitude is constructed by iterative phase information encoding and multiplexing for the high-level certification images. As a result, the proposed method can 10 April 2015 / Vol. 54, No. 11 / APPLIED OPTICS
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Fig. 14. (a)–(d) Four preselected high-level certification images, F 1 − F 4 , respectively.
Fig. 15. In the design of high-level authentication, (a)–(d) the retrieved phase information ϕ1 − ϕ4 , respectively.
Fig. 16. (a)–(d) Four decryption keys for different individual high-level certification images, DKh1 − DKh4 , respectively.
Fig. 17. In the case of high-level authentication, (a)–(d) the final recovered images obtained from each correct high-level decryption key. 3214
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Fig. 18. Recovered high-level certification images when the two ciphertexts (a) POM1 and (b) POM2 are contaminated by additive white noise.
accomplish not only low-level authentication but also high-level authentication. Furthermore, it can realize different levels of accessibility of the original certification image for different authority levels with the same cascaded multilevel architecture. Theoretical analysis and numerical simulations both validate the feasibility of the proposed method. This work is supported by the National Natural Science Foundation of China (grants 61275014, 61307003, 61171073, 51102148, and 11104188). We also thank the reviewers for some useful suggestions. References 1. P. Refrégier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20, 767–769 (1995). 2. G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption by double-random phase encoding in the fractional Fourier domain,” Opt. Lett. 25, 887–889 (2000). 3. S. T. Liu, Q. L. Mi, and B. H. Zhu, “Optical image encryption with multistage and multichannel fractional Fourier-domain filtering,” Opt. Lett. 26, 1242–1244 (2001). 4. R. Tao, Y. Xin, and Y. Wang, “Double image encryption based on random phase encoding in the fractional Fourier domain,” Opt. Express 15, 16067–16079 (2007). 5. E. Tajahuerce and B. Javidi, “Encrypting three dimensional information with digital holography,” Appl. Opt. 39, 6595– 6601 (2000). 6. X. G. Wang, D. M. Zhao, F. Jing, and X. F. Wei, “Information synthesis (complex amplitude addition and subtraction) and encryption with digital holography and virtual optics,” Opt. Express 14, 1476–1486 (2006). 7. R. K. Wang, I. A. Watson, and C. Chatwin, “Random phase encoding for optical security,” Opt. Eng. 35, 2464–2469 (1996). 8. Y. Li, K. Kreske, and J. Rosen, “Security and encryption optical systems based on a correlator with significant output images,” Appl. Opt. 39, 5295–5301 (2000). 9. G. Situ and J. Zhang, “Image hiding with computer-generated phase codes for optical authentication,” Opt. Commun. 245, 55–65 (2005). 10. X. F. Meng, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Xu, G. Y. Dong, X. X. Shen, H. Zhang, and X. C. Cheng, “Information security system by iterative multiple-phase retrieval and pixel random permutation,” Appl. Opt. 45, 3289–3297 (2006). 11. L. Z. Cai, M. Z. He, Q. Liu, and X. L. Yang, “Digital image encryption and watermarking by phase-shifting interferometry,” Appl. Opt. 43, 3078–3084 (2004). 12. X. F. Meng, L. Z. Cai, X. F. Xu, X. L. Yang, X. X. Shen, G. Y. Dong, and Y. R. Wang, “Two-step phase-shifting interferometry and its application in image encryption,” Opt. Lett. 31, 1414–1416 (2006).
13. Z. J. Liu, Q. Guo, L. Xu, M. A. Ahmad, and S. T. Liu, “Double image encryption by using iterative random binary encoding in gyrator domains,” Opt. Express 18, 12033–12043 (2010). 14. N. R. Zhou, Y. X. Wang, and L. H. Gong, “Novel optical image encryption scheme based on fractional Mellin transform,” Opt. Commun. 284, 3234–3242 (2011). 15. Y. Zhang and B. Wang, “Optical image encryption based on interference,” Opt. Lett. 33, 2443–2445 (2008). 16. N. Zhu, Y. T. Wang, J. Liu, J. H. Xie, and H. Zhang, “Optical image encryption based on interference of polarized light,” Opt. Express 17, 13418–13424 (2009). 17. H. M. He, L. Z. Cai, Q. Liu, X. C. Wang, and X. F. Meng, “Multiple image encryption and watermarking by random phase matching,” Opt. Commun. 247, 29–37 (2005). 18. G. Situ and J. Zhang, “Multiple-image encryption by wavelength multiplexing,” Opt. Lett. 30, 1306–1309 (2005). 19. H. T. Chang, H. E. Huang, C. L. Lee, and M. T. Lee, “Wavelength multiplexing multiple-image encryption using cascaded phase-only masks in the Fresnel transform domains,” Appl. Opt. 50, 710–716 (2011). 20. G. Situ and J. Zhang, “Position multiplexing for multiple-image encryption,” J. Opt. A 8, 391–397 (2006). 21. H. T. Chang, H. E. Huang, and C. L. Lee, “Wavelength multiplexing multiple-image encryption using cascaded phase-only masks in Fresnel transform domains,” Appl. Opt. 284, 4146– 4151 (2011). 22. Q. Wang, Q. Guo, L. Lei, and J. Y. Zhou, “Multiple-image encryption based on interference principle and phase only mask multiplexing in Fresnel transform domain,” Appl. Opt. 52, 6849–6857 (2013). 23. J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiplexing encryption-decryption via lateral shifting of a random phase mask,” Opt. Commun. 259, 532–536 (2006). 24. Y. Qin and Q. Gong, “Multiple-image encryption in an interference-based scheme by lateral shift multiplexing,” Opt. Commun. 315, 220–225 (2014). 25. A. Alfalou and C. Brosseau, “Exploiting root-mean-square time-frequency structure for multiple-image optical compression and encryption,” Opt. Lett. 35, 1914–1916 (2010). 26. A. Alfalou and C. Brosseau, “Implementing compression and encryption of phase-shifting digital holograms for three-dimensional object reconstruction,” Opt. Commun. 307, 67–72 (2013). 27. Q. Gong, X. Y. Liu, G. Q. Li, and Y. Qin, “Multiple-image encryption and authentication with sparse representation by space multiplexing,” Appl. Opt. 52, 7486–7493 (2013). 28. W. Chen and X. D. Chen, “Optical multiple-image authentication based on modified Gerchberg–Saxton algorithm with random sampling,” Opt. Commun. 318, 128–132 (2014). 29. Q. Wang, Q. Guo, and L. Lei, “Multiple-image encryption system using cascaded phase mask encoding and a modified Gerchberg–Saxton algorithm in gyrator domain,” Opt. Commun. 320, 12–21 (2014). 30. Y. Wang, C. Quan, and C. J. Tay, “Nonlinear multiple-image encryption based on mixture retrieval algorithm in Fresnel domain,” Opt. Commun. 330, 91–98 (2014). 31. D. S. Fan, X. F. Meng, Y. R. Wang, X. L. Yang, X. Peng, W. Q. He, G. Y. Dong, and H. Y. Chen, “Optical identity authentication scheme based on elliptic curve digital signature algorithm and phase retrieval algorithm,” Appl. Opt. 52, 5645– 5652 (2013). 32. X. M. Pan, X. F. Meng, Y. R. Wang, X. L. Yang, X. Peng, W. Q. He, G. Y. Dong, and H. Y. Chen, “Multilevel image authentication using shared secret threshold and phase retrieval,” J. Mod. Opt. 61, 1470–1478 (2014). 33. W. Chen, X. D. Chen, A. Stern, and B. Javidi, “Phase-modulated optical system with sparse representation for information encoding and authentication,” IEEE Photon. J. 5, 6900113 (2013). 34. W. Chen and X. D. Chen, “Optical color-image verification using multiple-pinhole phase retrieval,” J. Opt. A 16, 075403 (2014). 35. W. Chen and X. D. Chen, “Double random phase encoding using phase reservation and compression,” J. Opt. A 16, 025402 (2014). 10 April 2015 / Vol. 54, No. 11 / APPLIED OPTICS
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Department of Optics, School of Information Science and Engineering and Shandong Provincial Key Laboratory of Laser Technology and Application, Shandong University, Jinan 250100, China
2
College of Optoelectronics Engineering, Shenzhen University, Shenzhen 518060, China
3
College of Materials Science and Opto-Electronic Techology, University of Chinese Academy of Sciences, Beijing 100049, China 4
College of Electronic Science and Technology, Shenzhen University, Shenzhen 518060, China *Corresponding author: [email protected] Received 28 January 2015; revised 8 March 2015; accepted 8 March 2015; posted 9 March 2015 (Doc. ID 233390); published 8 April 2015
A multiple-image authentication method with a cascaded multilevel architecture in the Fresnel domain is proposed, in which a synthetic encoded complex amplitude is first fabricated, and its real amplitude component is generated by iterative amplitude encoding, random sampling, and space multiplexing for the low-level certification images, while the phase component of the synthetic encoded complex amplitude is constructed by iterative phase information encoding and multiplexing for the high-level certification images. Then the synthetic encoded complex amplitude is iteratively encoded into two phase-type ciphertexts located in two different transform planes. During high-level authentication, when the two phase-type ciphertexts and the high-level decryption key are presented to the system and then the Fresnel transform is carried out, a meaningful image with good quality and a high correlation coefficient with the original certification image can be recovered in the output plane. Similar to the procedure of high-level authentication, in the case of low-level authentication with the aid of a low-level decryption key, no significant or meaningful information is retrieved, but it can result in a remarkable peak output in the nonlinear correlation coefficient of the output image and the corresponding original certification image. Therefore, the method realizes different levels of accessibility to the original certification image for different authority levels with the same cascaded multilevel architecture. © 2015 Optical Society of America OCIS codes: (100.5070) Phase retrieval; (100.3010) Image reconstruction techniques. http://dx.doi.org/10.1364/AO.54.003204
1. Introduction 1559-128X/15/113204-12$15.00/0 © 2015 Optical Society of America 3204
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Optical information security has received much attention in recent years, since Réfrégier and Javidi proposed the double random-phase encoding (DRPE)
technique [1], and it has been successfully combined with other optical information processing techniques or transforms, such as fractional Fourier transform [2–4], digital holography [5,6], phase retrieval [7–10], phase-shifting interferometry [11,12], gyrator transform [13], fractional Mellin transform [14], and twobeam interference [15,16]. To increase the encoding efficiency and facilitate the transmission of the ciphertext, multiple-image encoding, encryption, and authentication systems have received wide attention, since we proposed a multiple-image encryption scheme by random phase matching [17], in which more than one image can be successfully encoded and decoded with the same set of transmitted ciphertexts based on the idea of DRPE and wave field superposition. Subsequently, various approaches, such as wavelength multiplexing [18,19], position multiplexing [20,21], phase-only mask (POM) multiplexing [22], lateral shifting [23,24], and optical data compression [25,26], have been proposed. In 2013, Gong et al. proposed a method of multiple-image encryption and authentication based on space multiplexing and sparse representation strategy [27], in
which the sparse data of multiple encrypted images can be extracted with aid of the random binary amplitude masks and then integrated into a synthesized ciphertext with space multiplexing. Recently, based on a modified Gerchberg–Saxton algorithm and random sampling strategy, Chen and Chen proposed an optical multiple-image authentication scheme in which the system capacity can be arbitrarily controlled by use of random sampling [28]. To decrease the cross talk effect between the multiplexing images, Wang et al. proposed a multiple-image encryption system based on a triple-POM cascaded architecture using a modified Gerchberg–Saxton algorithm in the gyrator domain, by which the primitive images can be converted into common POMs with noise-like phase distribution [29]. More recently, to present a security-enhanced nonlinear cryptosystem for multiple-image encryption based on amplitude and phase modulation, Wang et al. proposed a nonlinear multiple-image encryption method based on a mixture retrieval algorithm and phase mask multiplexing in the Fresnel domain [30]. In summary, according to the result of whether or not the recovered image can be visually recognized,
Fig. 1. Schematic diagram of the proposed cascaded multiple-image authentication system. 10 April 2015 / Vol. 54, No. 11 / APPLIED OPTICS
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the multiple-image authentication systems may be divided into two classes: high-level authentication and low-level authentication. In the former, the reconstructed output image is a meaningful image with good quality and a high correlation coefficient (CC), while in the latter, no significant or meaningful information is retrieved, but it can result in a remarkable peak output in the nonlinear correlation coefficient (NCC) of the output image and the corresponding standard certification image. In this sense, to increase the authority levels, we present here a multiple-image authentication method with a cascaded multilevel architecture based on amplitude field random sampling and phase information multiplexing, which can realize not only low-level authentication but also high-level authentication. We first give the theoretical analysis, description, and procedure of the method, then provide its simulation verification, and finally draw the conclusion. 2. Theoretical Analysis and Description of the Authentication System
The cascaded authentication system in the Fresnel domain is mainly composed of two processes: highlevel authentication and low-level authentication, the schematic diagram of which is shown in Fig. 1; the theoretical analysis, description, and working procedure is described in the following five subsections. x2
x1
A. Real Amplitude and Phase Information Iteratively Generated in the Design of Low-Level Authentication
Given that there are M certification images in the case of low-level authentication, here, the mth individual certification image is denoted by Gm x; y, where subscript m m 1; 2…M is the serial number of the certification images. Here, we attempt to iteratively encode the gray-scale image “airplane” acting as the example individual certification image Gm x; y into real amplitude and phase information, and the iteration process is shown in Fig. 1(a). One statistical independent random phase mask is placed at the input plane x0 ; y0 , whose amplitude transmittance is expj2πψ 0m x0 ; y0 , and ψ 0m is a random distribution matrix in the range of [0, 1]. The mth individual certification image Gm x; y in low-level authentication is located at the output plane x; y. The distance between the input and the output plane is z. When the input plane is illuminated by an onaxis plane wave of wavelength λ, the complex ampli~ m at the output plane under the Fresnel tude G approximation is [10,31,32] ~ m x; y FrTz fexpj2πψ 0m x0 ; y0 g; G where FrT represents the Fresnel transform. x
x0
y1
y
y0
y2
z Input plane
POM2 POM1
z1
Output plane
z2
z
recovered images
decryption keys
N
∏ exp(− jφi )
i =1,i ≠ n
Fn' high-level authentication
RSMm ψm −φ
low-level authentication
Gm'
Fig. 2. Schematic diagram of the authentication process. 3206
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(1)
The flow chart of the iteration process is similar to Fig. 2 in our recent work [31]. The complex amplitude field distribution (including both real amplitude and phase information) in the input plane is updated after each loop, which helps keep the circulation running, and the mth individual certification image Gm x; y serves as the amplitude constraint at the output plane. After the kth iteration (k 1; 2; 3…), assuming that the real amplitude and phase information in the input plane and the complex amplitude ~ km , respectively in the output plane are Rkm , ψ km , and G (for simplicity, the coordinate notations are neglected), then their distributions in the k 1th step can be described as [10,31,32] k1 ~ km g; Rm absIFrTz fGm expj angleG
(2)
k1 ~ km g; angleIFrTz fGm expj angleG ψm
(3)
where IFrT denotes the inverse Fresnel transform and abs• and angle• denote the extraction operator of the real amplitude and phase, respectively. The CC [10] is generally adopted as the convergent criteria to evaluate the similarity between the output ~ km (assuming after the real amplitude image absG kth iteration) and the standard certification image Gm , which is defined as [10] CC
~ km − EabsG ~ km g EGm − EGm − fabsG ; (4) ~ km σGm σabsG
where E• denotes the expected value operator and σ• is the standard deviation of the corresponding image. The iteration cycle process does not stop until the CC is larger than a predefined value. When the iterative cycle stops, the final real amplitude Rm and phase information ψ m located in the input plane are generated and stored, respectively. Repeating above iteration process for all the M individual low-level certification images, one series of real amplitude fields R1 ; R2 ; … Rm …; RM can be obtained, which are used for sampling, superposition, and synthesis; the other series of phase information distributions ψ 1 ; ψ 2 ; … ψ m …; ψ M , can also be determined, which are applied for designing the decryption keys of the individual certification images. The detailed process will be described in the following sections. B. Sampling, Superposition, and Synthesis of the Real Amplitude Fields
As shown in Fig. 1(b), to acquire the sparse data of the individual real amplitude fields R1 ; R2 ; … Rm …; RM ; M binary random sampling masks (RSMs) RSM1 ; RSM2 ; …RSMm …; RSMM are generated; generally, only a small portion of the pixels of each RSM are set as 1, while the other pixels are 0. The random sampling rate, that is, the percent of the value 1, can be arbitrarily controlled. Therefore, the redundancy in space makes it possible to superpose multiple real amplitude information into just one synthesized
amplitude field R with the aid of random sampling and space multiplexing, which can be described by Rx0 ; y0 R1 x0 ; y0 × RSM1 x0 ; y0 R2 x0 ; y0 × RSM2 x0 ; y0 Rm x0 ; y0 × RSMm x0 ; y0 RM x0 ; y0 × RSMM x0 ; y0
M X m1
Rm x0 ; y0 × RSMm x0 ; y0 ;
(5)
P where denotes the summation operator. In addition, to avoid cross talk between various sampled amplitude information, any two of the RSMs should be generated with no overlap with the areas in them that contain 1’s. Finally, the remnant positions that contain zeros in the synthesized amplitude field R are added by stationary white noise; that is, the zeros in these positions are replaced by random decimals distributed in the range of [0, 1], as shown in Fig. 1(c). Obviously, the final real amplitude Rx0 ; y0 integrates the sampled information of all the M low-level certification images. C. Iterative Phase Information Encoding and Multiplexing in the Design of High-Level Authentication
As shown in Fig. 1(d), assuming that there are N certification images in the case of high-level authentication, here, the nth individual certification image is denoted by F n x; y, where subscript n n 1; 2 … N is the serial number of the certification images. Here, we attempt to iteratively encode the grayscale image “Lena” acting as the example individual certification image F n x; y in the output plane into the phase information in the input plane x0 ; y0 based on the phase retrieval algorithm in the Fresnel domain, the iteration process of which is similar to that in Section 2.A. In each iteration, the individual certification image F n x; y serves as the amplitude constraint in the output plane, while the real amplitude in the input plane is constrained to the synthesized amplitude field R in Section 2.B. The iteration cycle process does not stop until the required CC between the approximate output image absF~ kn and the certification image F n x; y is achieved. When the iterative cycle stops, the final iterative phase information ϕn located in the input plane is generated and stored. Repeating the above iteration process for all the N individual high-level certification images, a series of iterative phase information values ϕ1 ; ϕ2 ; …ϕn …; ϕN can be obtained, which are used for multiplexing. Then a synthetic encoded complex amplitude Ex0 ; y0 in the input plane is fabricated by superposing and multiplying all of these phase components together with Rx0 ; y0 as its real amplitude component. The superposed phase information ϕx0 ; y0 and synthetic encoded complex amplitude Ex0 ; y0 then can be expressed as 10 April 2015 / Vol. 54, No. 11 / APPLIED OPTICS
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ϕx0 ; y0 ϕ1 x0 ; y0 ϕ2 x0 ; y0 ϕn x0 ; y0 ϕN x0 ; y0
N X n1
ϕn x0 ; y0 ;
carried out, which is illustrated in Fig. 2 and described as follows. (6)
Ex0 ; y0 Rx0 ; y0 expjϕx0 ; y0 Rx0 ; y0
N Y n1
expjϕn x0 ; y0 ;
(7)
Q where denotes the multiplication operation. Evidently, the superposed phase information ϕx0 ; y0 integrates all the phase information of N high-level certification images, and this process is illustrated in Fig. 1(e). D. Generation of Phase-Type Ciphertexts and Decryption Keys
Based on phase retrieval in the Fresnel domain, the synthetic encoded complex amplitude Ex0 ; y0 used as the target constraint is then iteratively encoded into two phase-type ciphertexts, POM1 x1 ; y1 and POM2 x2 ; y2 , located in two transform planes x1 ; y1 and x2 ; y2 , respectively, as illustrated in Fig. 1(f). The distance between the two transform planes is z1 , and that between the transform planes x2 ; y2 and the input plane x0 ; y0 (the plane that the synthetic encoded complex amplitude E is located in) is z2 . The wavelength of the on-axis plane wave is λ. The iteration process is very similar to that in Section 2.A, and therefore, a detailed description is omitted here. In the case of high-level authentication, the decryption keys for different individual high-level certification images are generated by the following expression: DKhn x0 ; y0
N Y i1;i≠n
F 0n absFrTz FrTz2 fFrTz1 expjPOM1 × expjPOM2 gDKhn Y N abs FrTz R × expjϕ exp−jϕi i1;i≠n
absfFrTz R × expjϕn g;
(10)
and then the authentication center calculates the CC between the recovered image F 0n and the nth standard certification image F n . If the CC is higher than the predetermined threshold (e.g., 0.90), that means the quality of the reconstructed image is so good that it is a success of high-level authentication, while a lower value means it is a failure. 2. Low-Level Authentication Similar to the process of high-level authentication, when the mth low-level decryption key DKlm is located in the input plane x0 ; y0 , a recovered image G0m is generated in the output plane x; y, which is mathematically expressed as G0m absFrTz FrTz2 fFrTz1 expjPOM1 × expjPOM2 gDKlm absFrTz fR × expjϕ × RSMm × expjψ m − ϕg
exp−jϕi x0 ; y0 ;
n 1; 2; 3…N:
1. High-Level Authentication One authentication participant places his/her highlevel decryption key DKhn in the input plane x0 ; y0 . When the system is illuminated by a plane wave with the correct wavelength λ, a recovered image F 0n is obtained in the output plane x; y, which is mathematically expressed as
absfFrTz RSMm × R × expjψ m g: (8)
(11)
Here, RSMm and ψ m denote the corresponding RSM and the iterated phase information for the mth individual low-level certification image, respectively.
As only a small proportion of the real amplitude information is sampled by the mth random sampling mask RSMm , the recovered image G0m is outwardly a noise-like image whose CC is too low to identify any useful information by direct visual inspection; that is, in this circumstance, it cannot successfully pass through the high-level authentication. To provide an additional authentication layer for the high-level authentication and thus achieve a higher discrimination capability, NCC distribution [33] is applied and calculated to compare the recovered noise-like image G0m with the standard certification image Gm , which is defined as [33–35]
E.
NCCξ; η jIFTjfFTGm ξ; ηgfFTG0m ξ; ηgjω−1
During low-level authentication, the decryption keys for different individual low-level certification images are generated as DKlm x0 ; y0 RSMm x0 ; y0 expfjψ m x0 ; y0 − ϕx0 ; y0 g; m 1; 2; 3…M:
(9)
Authentication Process
In the authentication center, after both two phasetype ciphertexts POM1 x1 ; y1 and POM2 x2 ; y2 are placed in two transform planes x1 ; y1 and x2 ; y2 , respectively, the authentication process can be 3208
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×fFTGm ξ; ηgfFTG0m ξ; ηgj2 ;
(12)
where FT and IFT represent the Fourier and inverse Fourier transforms, respectively, and (ξ, η) is the
Fig. 3. Two phase-type ciphertexts: (a) POM1 and (b) POM2 .
coordinate of the spectrum transverse plane. The parameter ω defines the strength of the applied nonlinearity, whose range is in [0.2, 0.6], in which ω best suits the verification application.
As the sampled information is just related to the synthesized amplitude of the mth low-level certification image, after calculating and displaying the 3D NCC distributions between the recovered image G0m and its corresponding certification image Gm , one remarkable peak is generated in the NCC distribution outputs, which can help authenticate the information without the direct visualization of hidden information. In this circumstance, the authentication process can be called low-level authentication, due to the recovered noise-like image with a low CC, if there exists a remarkable peak in its NCC distributions, which means the low-level authentication is successful, while no remarkable peak indicates a failure. 3. Computer Simulations
A set of computer simulations have been carried out to verify the feasibility of our proposed method and investigate its performance. All the images shown in
Fig. 4. (a)–(d) Four preselected low-level certification images, G1 − G4 , respectively.
Fig. 5. In the design of low-level authentication, (a)–(d) the final retrieved real amplitudes R1 − R4 , (e)–(h) the corresponding retrieved phase information ψ 1 − ψ 4 . 10 April 2015 / Vol. 54, No. 11 / APPLIED OPTICS
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Fig. 6. (a)–(d) Four binary random sampling masks RSM1 − RSM4 .
Fig. 7. Final synthesized real amplitude field R after random sampling, space multiplexing, and random noise addition.
the following simulations are 256 × 256 pixels in size. All the resolutions of the digital images in the input, intermediate transform, and output planes are 15 μm, λ 0.532 μm, z1 z2 z 108.3 mm, ω 0.5. Two phase-type ciphertexts, POM1 and POM2 , are shown in Figs. 3(a) and 3(b), respectively.
Four preselected low-level certification images G1 − G4 are shown in Figs. 4(a)–4(d), based on the iterative phase retrieval process mentioned in Section 2.A. The final retrieved real amplitudes R1 − R4 are given in Figs. 5(a)–5(d), respectively, and the corresponding retrieved phase information ψ 1 − ψ 4 is given in Figs. 5(e)–5(h), respectively. Figures 6(a)– 6(d) depict the corresponding four binary random sampling masks RSM1 − RSM4 (the sampling rate for these four RSMs is 12.5%), respectively, and the final synthesized amplitude field R after random noise addition is shown in Fig. 7. Figure 8 shows the decryption keys DKls for different individual lowlevel certification images; its top row depicts the real amplitude parts, random sampling masks RSM1 − RSM4 , while its bottom row depicts the phase information parts, ψ 1 − ϕ − ψ 4 − ϕ, respectively. Based on the low-level authentication method mentioned in Section 2.E.2, when using all the correct low-level decryption keys DKls, the four
Fig. 8. Decryption keys DKls for different individual low-level certification images. (a)–(d) The real amplitude parts, random sampling masks RSM1 − RSM4 ; (e)–(h) the phase information parts, ψ 1 − ϕ − ψ 4 − ϕ. 3210
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Fig. 9. In the case of low-level authentication, (a)–(d) four retrieved noise-like images.
noise-like images in Figs. 9(a)–9(d) with a low CC are retrieved in the output plane, from which we can see that no meaningful information can be identified by direct visual inspection. That means it cannot successfully pass through the high-level authentication, as the CC is far lower than the threshold value set beforehand. However, the NCC distributions of the final retrieved image and the standard certification image can be focused and calculated, and these are
shown in Figs. 10(a)–10(d), respectively; obviously, a remarkable peak can be observed in all the corresponding NCC distributions. That means that the low-level authentication is successful, which provides an additional authentication layer for the high-level authentication. If the low-level decryption key is false or incorrect, it will result in a failure of low-level authentication. Figure 11 shows the 3D NCC distributions in the
Fig. 10. (a)–(d) 3D NCC distributions of Figs. 9(a)–9(d), respectively. 10 April 2015 / Vol. 54, No. 11 / APPLIED OPTICS
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Fig. 11. 3D NCC distributions when using the false low-level decryption key.
output plane when a random phase is adopted as the phase part of the decryption key; clearly, only the noisy NCC distributions without a remarkable peak are generated, which means the low-level authentication has failed. The system robustness analysis for noise attack is tested; here, the authentication process for the lowlevel certification image in Fig. 4(a) is taken as an example. Figures 12(a) and 12(b) show the NCC distributions when the two ciphertexts POM1 and POM2 , respectively, are contaminated by additive white noise (zero mean noise with 0.01 variance), from which it can be seen that a remarkable peak is obtained in both. Next, we further analyze and discuss how the sampling ratio of the low-level decryption keys (RSM1 − RSM4 ) affects the verification results. Here we also take the authentication process for the lowlevel certification image in Fig. 4(a) as an example. Figures 13(a)–13(f) show the NCC distributions
when the sampling ratio of RSM1 is 9.5%, 8.4%, 6.3%, 5.5%, 5.1%, and 4.1%, respectively, after calculation and analysis. If the sampling ratio is less than 5%, the low-level authentication will fail, and only noisy NCC distributions without a remarkable peak will be obtained. In the circumstance of high-level authentication, four preselected high-level certification images F 1 − F 4 are shown in Figs. 14(a)–14(d). Based on the iterative phase encoding process mentioned in Section 2.C, the corresponding retrieved phase information ϕ1 − ϕ4 is given in Figs. 15(a)–15(d), respectively. After phase information multiplexing, four decryption keys DKh1 − DKh4 for different individual high-level certification images are generated, as shown in Figs. 16(a)–16(d). Based on the high-level authentication method mentioned in Section 2.E.1, Figs. 17(a)–17(d) show the final recovered images obtained from each correct high-level decryption key, and the CCs are 0.9701, 0.9497, 0.9711, and 0.9702, respectively. It is clear that the recovered images with good quality are very similar to the corresponding individual certification images; if the CC’s predetermined threshold is set as 0.90 or smaller, that means the high-level authentication is successful. Finally, the system robustness analysis for noise attack in the circumstance of high-level authentication is investigated; here we take the authentication process for the high-level certification image in Fig. 14(d) as an example. Figures 18(a) and 18(b) give the recovered images when the two ciphertexts POM1 and POM2 , respectively, are contaminated by additive white noise (zero mean noise with 0.01 variance), the CCs of which are 0.7586 and 0.7634, respectively. Obviously, both retrieved images can be recognized without doubt, and the proposed method possesses high robustness against noise attack.
Fig. 12. 3D NCC distributions when the two ciphertexts (a) POM1 and (b) POM2 are contaminated by additive white noise. 3212
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Fig. 13. 3D NCC distributions when the sampling ratio of decryption key RSM1 is (a) 9.5%, (b) 8.4%, (c) 6.3%, (d) 5.5%, (e) 5.1%, and (f) 4.1%.
4. Conclusions
In conclusion, we have proposed a multiple-image authentication method with a cascaded multilevel architecture, in which the real amplitude component of the synthetic encoded complex amplitude field is generated by iterative amplitude encoding, random
sampling, and space multiplexing for the low-level certification images, while the phase component of the synthetic encoded complex amplitude is constructed by iterative phase information encoding and multiplexing for the high-level certification images. As a result, the proposed method can 10 April 2015 / Vol. 54, No. 11 / APPLIED OPTICS
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Fig. 14. (a)–(d) Four preselected high-level certification images, F 1 − F 4 , respectively.
Fig. 15. In the design of high-level authentication, (a)–(d) the retrieved phase information ϕ1 − ϕ4 , respectively.
Fig. 16. (a)–(d) Four decryption keys for different individual high-level certification images, DKh1 − DKh4 , respectively.
Fig. 17. In the case of high-level authentication, (a)–(d) the final recovered images obtained from each correct high-level decryption key. 3214
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Fig. 18. Recovered high-level certification images when the two ciphertexts (a) POM1 and (b) POM2 are contaminated by additive white noise.
accomplish not only low-level authentication but also high-level authentication. Furthermore, it can realize different levels of accessibility of the original certification image for different authority levels with the same cascaded multilevel architecture. Theoretical analysis and numerical simulations both validate the feasibility of the proposed method. This work is supported by the National Natural Science Foundation of China (grants 61275014, 61307003, 61171073, 51102148, and 11104188). We also thank the reviewers for some useful suggestions. References 1. P. Refrégier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20, 767–769 (1995). 2. G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption by double-random phase encoding in the fractional Fourier domain,” Opt. Lett. 25, 887–889 (2000). 3. S. T. Liu, Q. L. Mi, and B. H. Zhu, “Optical image encryption with multistage and multichannel fractional Fourier-domain filtering,” Opt. Lett. 26, 1242–1244 (2001). 4. R. Tao, Y. Xin, and Y. Wang, “Double image encryption based on random phase encoding in the fractional Fourier domain,” Opt. Express 15, 16067–16079 (2007). 5. E. Tajahuerce and B. Javidi, “Encrypting three dimensional information with digital holography,” Appl. Opt. 39, 6595– 6601 (2000). 6. X. G. Wang, D. M. Zhao, F. Jing, and X. F. Wei, “Information synthesis (complex amplitude addition and subtraction) and encryption with digital holography and virtual optics,” Opt. Express 14, 1476–1486 (2006). 7. R. K. Wang, I. A. Watson, and C. Chatwin, “Random phase encoding for optical security,” Opt. Eng. 35, 2464–2469 (1996). 8. Y. Li, K. Kreske, and J. Rosen, “Security and encryption optical systems based on a correlator with significant output images,” Appl. Opt. 39, 5295–5301 (2000). 9. G. Situ and J. Zhang, “Image hiding with computer-generated phase codes for optical authentication,” Opt. Commun. 245, 55–65 (2005). 10. X. F. Meng, L. Z. Cai, Y. R. Wang, X. L. Yang, X. F. Xu, G. Y. Dong, X. X. Shen, H. Zhang, and X. C. Cheng, “Information security system by iterative multiple-phase retrieval and pixel random permutation,” Appl. Opt. 45, 3289–3297 (2006). 11. L. Z. Cai, M. Z. He, Q. Liu, and X. L. Yang, “Digital image encryption and watermarking by phase-shifting interferometry,” Appl. Opt. 43, 3078–3084 (2004). 12. X. F. Meng, L. Z. Cai, X. F. Xu, X. L. Yang, X. X. Shen, G. Y. Dong, and Y. R. Wang, “Two-step phase-shifting interferometry and its application in image encryption,” Opt. Lett. 31, 1414–1416 (2006).
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