Abstract

We propose a novel optical information encryption and authentication scheme that uses asymmetric keys generated by the phase-truncation approach and the phase-retrieval algorithm. Multiple images bonded with random phase masks are Fourier transformed, and obtained spectra are amplitude- and phase-truncated. The phase-truncated spectra are encoded into a single random intensity image using the phase-retrieval algorithm. Unlike most of the authentication schemes, in this study, only one encrypted reference image is required for verification of multiple secured images. The conventional double random phase encoding and correlation techniques are employed for authentication verification. Computer simulation results and theoretical explanation prove the effectiveness of the proposed scheme.

© 2014 Optical Society of America

1. INTRODUCTION

Optical security techniques have generated a considerable amount of interest among researchers because of their unique advantages, such as parallel processing and multidimensional capability. A number of optical information security techniques have been proposed in the last few decades, including encryption [122], watermarking [23,24], image hiding [25], and information authentication [2633]. Double random phase encoding (DRPE), a classic optical encryption technique [1], has been found to be vulnerable to some attacks [11,12]. Therefore to overcome the vulnerability issues associated with the DRPE scheme, optical encryption methods based on phase-truncation and phase-retrieval algorithms have been proposed [1322]. These security schemes break the linearity of the DRPE technique and show immunity against existing attacks. The weaknesses of the DRPE scheme have also been overcome by using photon counting imaging techniques [2933], when these techniques are used for information authentication and encoding.

Optical information authentication is also emerging as an important field of research in the domain of optical security. The validation of valuable documents such as credit cards, passports, and other identities is often required. Most of the authentication schemes reported in the literature are based on optical correlation methods [2628]. But recently, authentication and verification based on DRPE and photon counting imaging techniques have attracted the interest of the research community [2933]. These schemes provide an additional layer of security by applying photon counting to input or encrypted images. However, in most of the previously proposed encryption-based information authentication schemes [2933], when multiple images are to be verified, multiple encrypted images are required in the authentication system. Thus, verification of multiple images/data increases the storage (memory) requirement and thus increases the complexity to the system. Hence, new schemes that have smaller memory requirements, are easily optically implementable, and have enhanced security need to be investigated.

In this paper, we propose an information encryption and authentication scheme that uses asymmetric keys that requires storage of only one encrypted image that corresponds to all of the multiple encrypted images. The scheme contains two parts: a key generator and an authentication system. In the key generator, two asymmetric keys are generated using the phase-truncation approach and phase-retrieval algorithm with a properly fixed random intensity image (RII) for security verification. In the authentication system, with the help of a generated asymmetric key, the authenticity of multiple object/information is checked. The phase-retrieval algorithm is designed in such a way that by the use of a single RII, authentication of several images can be carried out. Also, the asymmetric keys are suitable for implementing the authentication system using the conventional DRPE scheme. The proposed method can also be used for other security applications, such as image watermarking and hiding.

2. PROPOSED SCHEME

The block diagram of the key generator for the proposed optical authentication scheme is shown in Fig. 1. This is used for generation of the first and second asymmetric keys. First, an image to be verified is modulated with a random phase mask (RPM), and the resultant complex function is Fourier transformed. Suppose fn(x,y) denotes input images to be verified and exp{i2πr1n(x,y)} denotes RPMs to be used with multiple images, where n=1,2,s, s is an integer. The Fourier spectrum of the resultant complex function is obtained as [13]

Fn(u,v)=I[fn(x,y)×exp{i2πr1n(x,y)}].
Here, I represents the Fourier transform operation. The amplitude-truncated value (ATV) and phase-truncated value (PTV) of Eq. (1) are represented by Pn(u,v) and An(u,v), respectively:
An(u,v)=PT{Fn(u,v)},
Pn(u,v)=AT{Fn(u,v)}.
Here PT and AT denote the phase- and amplitude-truncation operations, respectively. The function, Pn(u,v), helps generate the first asymmetric key. For obtaining the second asymmetric keys for multiple images, the Gerchberg–Saxton (G-S) phase-retrieval algorithm is used [22,34,35]. Unlike most of the cases, the RII is used for multiple encoded images. This modification makes the scheme suitable for multiple users. Now the PTV, An(u,v), and another RPM, exp{i2πr2n(u,v)}, are used as inputs to the phase-retrieval algorithm. The flowchart of the phase-retrieval algorithm for the second asymmetric key generation procedure is shown in the second part of Fig. 1. Here, for deciding the input to the phase-retrieval algorithm, the complex function Anm(u,v)=An(u,v)×exp{i2πr2n(u,v)}, after mth iteration, is written by modulating the phase-truncated function, An(u,v), with the help of the RPM, exp{i2πr2n(u,v)}. The inverse Fourier transformation of this modulated complex function, An(u,v), is performed as
A(n)m+1(x,y)=I1[Anm(u,v)]=|A(n)m+1(x,y)|×exp{iφnm(x,y)}.
The amplitude of Eq. (4), |A(n)m+1(x,y)|, is now replaced with any other RII, say R(x,y), and a new complex function Am+1(x,y)=R(x,y)×exp{iφnm(x,y)} is obtained. This RII, R(x,y), can be fixed for all images whose authenticity is to be verified. The Fourier spectrum of this complex function can be recorded as
A(n)m+1(u,v)=I[A(n)m+1(x,y)]=|A(n)m+1(u,v)|×exp{iφ(n)m(u,v)}.
In this case, the amplitude is replaced with the PTV, An(u,v), and a new complex function, A(n)m+1(u,v)=|An(u,v)|×exp{iφnm(u,v)}, is obtained. The phase function, exp{iφnm(u,v)}, corresponds to the input phase in the phase-retrieval algorithm, exp{ir2(n)m+1(u,v)}, for the next iterations. Here, the convergence of the iteration process is completed by computing the mean square error (MSE) between abs[R(x,y)] and abs[A(n)m+1(x,y)]. The MSE is defined as
MSE=x=0N1y=0N1{|R(x,y)||A(n)m+1(x,y)|}2N×N.
The purpose of iteration is to match amplitude of the RII with the PTV. This step is different from other phase-retrieval algorithms, and, hence, it is used for the novel information authentication scheme. This constraint makes the scheme suitable for securing multiple images [2,5]. It can be seen from the key generation procedure that the process is nonlinear and gives asymmetric keys.

 

Fig. 1. Block diagram of key generator. PT, phase truncation; AT, amplitude truncation; RPM, random phase mask; FT, Fourier transform; FT1, inverse FT; P1n and P2n, phase values for nth image; RII, random intensity image.

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For verification of the original information, the asymmetric keys are calculated with the help of ATV, Pn(u,v), and phases, exp{ir2(n)m+1(u,v)} and exp{iφnm(u,v)}, as follows:

K1n(u,v)=Pn(u,v)×exp{ir2(n)m+1(u,v)},
K2n(x,y)=exp{iφnm(x,y)}.
It is straightforward to relate Eqs. (4) and (5) with the key generation procedure of the phase-truncation and phase-retrieval algorithms. The aim of generating keys in this way is to verify the authenticity of the original image by employing the conventional DRPE scheme. For generation of the first asymmetric key, instead of the amplitude-truncation and phase-truncation approach, the phase-retrieval algorithm can also be used but will enhance the complexity and computational cost to the system. The asymmetric keys are generated analytically. On the other hand, authentication can be implemented optically.

For obtaining the original images whose authenticity is to be verified, RII, which is fixed in the system bonded with the second asymmetric key, is Fourier transformed, and the obtained spectrum is further multiplied with the first asymmetric key and inverse Fourier transformed. Theoretically, the original image whose authenticity is to be verified is obtained by using asymmetric keys as obtained in Eqs. (7) and (8):

dn(x,y)=I1[I{Rn(x,y)×K2n(x,y)}×K1n(u,v)].
It has already been proved that the security schemes based on the phase-truncation approach, phase-retrieval algorithm, and photon counting imaging show immunity against existing attacks [20,22,2933].

For information authentication and verification conventional, nonconventional, and nonlinear correlators have been used [2628,36]. Amongst these correlators, the nonlinear correlator is proved to be more suitable for authentication applications. Therefore, we implement the nonlinear correlation approach, as demonstrated in Ref. [36]. The correlation between original input images and retrieved images can be obtained as

Cn(x,y)=I1[Q1×Q2],
where
Q1=|I{dn(x,y)}×[I{fn(x,y)}]|t,
Q2=exp{iarg[I{dn(x,y)}]arg[I{fn(x,y)}]}.
Here t represents the nonlinearity factor. Equations (9)–(12) can be realized optically with the help of an optoelectronic setup as shown in Fig. 2. An RII, which is fixed in a system bonded with the second asymmetric key, can be displayed on the first spatial light modulator (SLM) and illuminated with the laser light source. The Fourier spectrum multiplied with the second asymmetric key can be displayed on the second SLM. The intensity of the original image whose authenticity is to be checked can be recorded at the Fourier plane through a charge-coupled device (CCD) camera. Finally, auto-correlation and cross-correlation peaks can be obtained with a nonlinear correlation filter. The block diagram of Fig. 1 and first part of the schematic diagram of Fig. 2 can be used for multiple image encryption and decryption processes, respectively.

 

Fig. 2. Schematic diagram of authentication system. SLM, spatial light modulator; L, lens; f, focal length; CF, correlation filters; CCD, charge-coupled device camera.

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3. COMPUTER SIMULATIONS AND DISCUSSION

Computer simulations have been performed to check the validity of the proposed optical encryption and authentication scheme using the MATLAB 7.10 platform. Two gray-scale images of size 256×256 pixels are used. The simulation results for the first gray-scale image are shown in Figs. 3 and 4. Figures 3(a)3(d) show the input image of Capsicum to be encrypted and verified, the RII, the relation showing matching of RII with PTV, and the retrieved original input image obtained in the verification plane, respectively. From Fig. 3(c), it can be inferred that RII completely matches with PTV when the MSE value is zero. Figures 4(a)4(c) show the auto-correlation peak, the cross-correlation when RII is wrong, and the cross-correlation when asymmetric keys are wrong, respectively.

 

Fig. 3. Simulation results for first gray-scale image: (a) an input image to be verified, (b) RII, (c) relation showing matching of random intensity image with PTV, and (d) retrieved input image obtained in verification plane. PTV, phase-truncated value.

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Fig. 4. (a) Auto-correlation peak, (b) cross-correlation peak when RII is wrong, and (c) cross-correlation peak when keys are wrong.

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To check the multiuser applicability of the scheme, another gray-scale image is used for the same RII [Fig. 3(b)]. Figure 5(a) shows the gray-scale image of Baboon to be encrypted and verified. Figure 5(b) shows the relation between MSE and the number of iterations for matching the RII with the encoded image. Figure 5(c) shows the retrieved original image obtained in the verification plane. Figure 5(d) shows the auto-correlation peak. Figure 5(e) shows the cross-correlation when the wrong RII is used. Figure 5(f) shows the cross-correlation when the wrong asymmetric keys are used. For the second gray-scale image, it can also be seen that the MSE value is zero after some iterations, which shows exact matching. The proposed scheme has been checked successfully for several images. Here results of only two gray-scale images have been shown. Figure 6 shows a plot in which the cumulative average of MSEs has been plotted against the number of images. From this plot, it can be observed that the average of MSE is approximately constant for multiple images. From these simulation results, it can be inferred that the authentication scheme is suitable for multiple gray-scale images with high security.

 

Fig. 5. Simulation results for another gray-scale image: (a) an image of cameraman to be verified, (b) relation showing matching of random intensity image with PTV, (c) retrieved input image obtained in verification plane, (d) auto-correlation peak, (e) cross-correlation peak when keys are wrong, and (f) cross-correlation peak when RII is wrong.

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Fig. 6. Plot between cumulative average of MSE for multiple images and number of images. MSE, mean square error; AMSE, average of MSE.

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To check the immunity of the proposed security scheme to special attack, we generated two asymmetric keys with the help of encryption keys and an encrypted image [15]. Special attack results are shown in Figs. 7(a)7(c). Figure 7(a) shows the plot between MSE and the number of iterations during generations of the first asymmetric key. Figure 7(b) shows the plot between MSE and the number of iterations during generations of the second asymmetric key. Figure 7(c) shows the corresponding decrypted image. Here, we performed special attack for the first image and similarly it can be applied to the second image. It is found that the value of the MSE even after 300 iterations has not converged to close to zero. Hence, it can be claimed that the proposed scheme is free from special attack.

 

Fig. 7. Attack results for first gray-scale image: (a) plot between MSE and number of iterations during generations of first asymmetric key, (b) plot between MSE and number of iterations during generations of second asymmetric key, and (c) corresponding decrypted image.

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All the optical systems that use the laser as a light source suffer from the inherent problem of speckle noise. In security applications, because of speckle noise, the qualities of the retrieved images are poor. For overcoming this problem, images can be converted into QR code prior to processing by using approach proposed in Ref. [6]. The proposed scheme can also use benefits such as the invisibility of photon counting imaging [2933] for verification. This is because photon counting techniques can be applied to intensity images in addition to complex images. Also, the proposed scheme can use any optical transform domains, such as the fractional Fourier [2], fractional wavelet [3], gyrator [4], and Fresnel [22] domains. These optical transforms provides additional security to the system.

The proposed security scheme can be used for multiple image watermarking and multiple image hiding applications with multiple levels of security [2325]. For watermarking applications, RII can be watermarked into a host image using any existing watermarking techniques. During retrieval of original images the reverse concept of the watermarking technique and proposed verification concept can be used. This scheme can also be suitably used for image hiding applications, in which multiple images to be hidden are encoded into a gray-scale image by replacing RII with a gray-scale image. In this way multiple images can be hidden into a gray-scale image and all original images can be retrieved with the generated asymmetric keys.

4. CONCLUSION

We have proposed an information encryption and authentication scheme using asymmetric keys generated by the phase-truncation approach and the phase-retrieval algorithm. Only one encrypted reference image is required for verification of multiple images/data designed for the multiple users’ case. The proposed scheme is not only digitally implementable, but it can also be implemented optically by using the conventional DRPE technique and optical correlation approach. The validity of the scheme is proved through the simulation results. It is also shown that the proposed method is free from special attack. The proposed scheme has the potential to enlarge the research area of optical information security.

ACKNOWLEDGMENTS

The authors acknowledge funding from the Defence Research & Development Organisation, Government of India, under Grant No. ERIP/ER/1200428/M/01/1473.

REFERENCES

1. P. Réfrégier and B. Javidi, “Optical image encryption based on input plane encoding and Fourier plane random encoding,” Opt. Lett. 20, 767–769 (1995). [CrossRef]  

2. N. K. Nishchal and T. J. Naughton, “Flexible optical encryption with multiple users and multiple security levels,” Opt. Commun. 284, 735–739 (2011). [CrossRef]  

3. L. Chen and D. Zhao, “Optical image encryption based on fractional wavelet transform,” Opt. Commun. 254, 361–367 (2005). [CrossRef]  

4. J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Applications of gyrator transform for image processing,” Opt. Commun. 278, 279–284 (2007). [CrossRef]  

5. 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). [CrossRef]  

6. J. F. Barrera, A. Mira, and R. Torroba, “Optical encryption and QR codes: secure and noise-free information retrieval,” Opt. Express 21, 5373–5378 (2013). [CrossRef]  

7. W. Chen, X. Chen, A. Anand, and B. Javidi, “Optical encryption using multiple intensity samplings in the axial domain,” J. Opt. Soc. Am. A 30, 806–812 (2013). [CrossRef]  

8. A. Alfalou and C. Brosseau, “Optical image compression and encryption methods,” Opt. Photon. 1, 589–636 (2009). [CrossRef]  

9. A. Alfalou and C. Brosseau, “Dual encryption scheme of images using polarized light,” Opt. Lett. 35, 2185–2187 (2010). [CrossRef]  

10. 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). [CrossRef]  

11. A. Carnicer, M. M. Usategui, S. Arcos, and I. Juvells, “Vulnerability to chosen-cyphertext attacks of the optical encryption schemes based on double random phase keys,” Opt. Lett. 30, 1644–1646 (2005). [CrossRef]  

12. Y. Frauel, A. Castro, T. Naughton, and B. Javidi, “Resistance of the double random phase encryption against various attacks,” Opt. Express 15, 10253–10265 (2007). [CrossRef]  

13. W. Qin and X. Peng, “Asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Lett. 35, 118–120 (2010). [CrossRef]  

14. S. K. Rajput and N. K. Nishchal, “Image encryption based on interference that uses fractional Fourier domains asymmetric keys,” Appl. Opt. 51, 1446–1452 (2012). [CrossRef]  

15. X. Wang and D. Zhao, “A special attack on the asymmetric cryptosystem based on phase-truncated fractional Fourier transforms,” Opt. Commun. 285, 1078–1081 (2012). [CrossRef]  

16. S. K. Rajput and N. K. Nishchal, “Asymmetric color cryptosystem using polarization selective diffractive optical element and structured phase mask,” Appl. Opt. 51, 5377–5386 (2012). [CrossRef]  

17. S. K. Rajput and N. K. Nishchal, “Known-plaintext attack-based optical cryptosystem using phase-truncated Fresnel transform,” Appl. Opt. 52, 871–878 (2013). [CrossRef]  

18. S. K. Rajput and N. K. Nishchal, “Image encryption using polarized light encoding and amplitude- and phase-truncated Fresnel transform,” Appl. Opt. 52, 4343–4352 (2013). [CrossRef]  

19. S. K. Rajput and N. K. Nishchal, “Known-plaintext attack on encryption domain independent optical asymmetric cryptosystem,” Opt. Commun. 309, 231–235 (2013). [CrossRef]  

20. X. Wang and D. Zhao, “Amplitude-phase retrieval attack free cryptosystem based on direct attack to phase-truncated Fourier transform-based encryption using a random amplitude mask,” Opt. Lett. 38, 3684–3686 (2013). [CrossRef]  

21. X. Wan, Y. Chen, C. Dai, and D. Zhao, “Discussion and a new attack of the optical asymmetric cryptosystem based on phase-truncated Fourier transform,” Appl. Opt. 53, 208–213 (2014). [CrossRef]  

22. S. K. Rajput and N. K. Nishchal, “Fresnel domain nonlinear image encryption scheme based on Gerchberg-Saxton phase-retrieval algorithm,” Appl. Opt. 53, 418–425 (2014). [CrossRef]  

23. S. Kishk and B. Javidi, “Watermarking of three-dimensional objects by digital holography,” Opt. Lett. 28, 167–169 (2003). [CrossRef]  

24. N. K. Nishchal, “Hierarchical encrypted image watermarking using fractional Fourier domain random phase encoding,” Opt. Eng. 50, 097003 (2011). [CrossRef]  

25. Y.-Y. Chen, J.-H. Wang, C.-C. Lin, and H.-E. Hwang, “Lensless optical data hiding system based on phase encoding algorithm in the Fresnel domain,” Appl. Opt. 52, 5247–5257 (2013). [CrossRef]  

26. B. Javidi and J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33, 1752–1756 (1994). [CrossRef]  

27. D. Abookasis, O. Arazi, J. Rosen, and B. Javidi, “Security optical systems based on joint transform correlator with significant output images,” Opt. Eng. 40, 1584–1589 (2001). [CrossRef]  

28. S. K. Rajput and N. K. Nishchal, “Image encryption and authentication verification based on nonconventional fractional joint transform correlator,” Opt. Lasers Eng. 50, 1474–1483 (2012). [CrossRef]  

29. E. Perez-Cabre, M. Cho, and B. Javidi, “Information authentication using photon-counting double-random-phase encrypted images,” Opt. Lett. 36, 22–24 (2011). [CrossRef]  

30. M. Cho, A. Mahalanobis, and B. Javidi, “3D passive photon counting automatic target recognition using advanced correlation filters,” Opt. Lett. 36, 861–863 (2011). [CrossRef]  

31. E. Perez-Cabre, H. C. Abril, M. S. Millan, and B. Javidi, “Photon-counting double-random-phase encoding for secure image verification and retrieval,” J. Opt. 14, 094001 (2012). [CrossRef]  

32. M. Cho and B. Javidi, “Three-dimensional photon counting double-random-phase encryption,” Opt. Lett. 38, 3198–3201 (2013). [CrossRef]  

33. A. Markman and B. Javidi, “Full phase photon counting double-random-phase encryption,” J. Opt. Soc. Am. A 31, 394–403 (2014). [CrossRef]  

34. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane,” Optik 35, 237–246 (1972).

35. Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, “Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21, 842–844 (1996). [CrossRef]  

36. B. Javidi, “Nonlinear joint power spectrum based optical correlation,” Appl. Opt. 28, 2358–2367 (1989). [CrossRef]  

References

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  • |

  1. P. Réfrégier and B. Javidi, “Optical image encryption based on input plane encoding and Fourier plane random encoding,” Opt. Lett. 20, 767–769 (1995).
    [Crossref]
  2. N. K. Nishchal and T. J. Naughton, “Flexible optical encryption with multiple users and multiple security levels,” Opt. Commun. 284, 735–739 (2011).
    [Crossref]
  3. L. Chen and D. Zhao, “Optical image encryption based on fractional wavelet transform,” Opt. Commun. 254, 361–367 (2005).
    [Crossref]
  4. J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Applications of gyrator transform for image processing,” Opt. Commun. 278, 279–284 (2007).
    [Crossref]
  5. 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).
    [Crossref]
  6. J. F. Barrera, A. Mira, and R. Torroba, “Optical encryption and QR codes: secure and noise-free information retrieval,” Opt. Express 21, 5373–5378 (2013).
    [Crossref]
  7. W. Chen, X. Chen, A. Anand, and B. Javidi, “Optical encryption using multiple intensity samplings in the axial domain,” J. Opt. Soc. Am. A 30, 806–812 (2013).
    [Crossref]
  8. A. Alfalou and C. Brosseau, “Optical image compression and encryption methods,” Opt. Photon. 1, 589–636 (2009).
    [Crossref]
  9. A. Alfalou and C. Brosseau, “Dual encryption scheme of images using polarized light,” Opt. Lett. 35, 2185–2187 (2010).
    [Crossref]
  10. 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).
    [Crossref]
  11. A. Carnicer, M. M. Usategui, S. Arcos, and I. Juvells, “Vulnerability to chosen-cyphertext attacks of the optical encryption schemes based on double random phase keys,” Opt. Lett. 30, 1644–1646 (2005).
    [Crossref]
  12. Y. Frauel, A. Castro, T. Naughton, and B. Javidi, “Resistance of the double random phase encryption against various attacks,” Opt. Express 15, 10253–10265 (2007).
    [Crossref]
  13. W. Qin and X. Peng, “Asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Lett. 35, 118–120 (2010).
    [Crossref]
  14. S. K. Rajput and N. K. Nishchal, “Image encryption based on interference that uses fractional Fourier domains asymmetric keys,” Appl. Opt. 51, 1446–1452 (2012).
    [Crossref]
  15. X. Wang and D. Zhao, “A special attack on the asymmetric cryptosystem based on phase-truncated fractional Fourier transforms,” Opt. Commun. 285, 1078–1081 (2012).
    [Crossref]
  16. S. K. Rajput and N. K. Nishchal, “Asymmetric color cryptosystem using polarization selective diffractive optical element and structured phase mask,” Appl. Opt. 51, 5377–5386 (2012).
    [Crossref]
  17. S. K. Rajput and N. K. Nishchal, “Known-plaintext attack-based optical cryptosystem using phase-truncated Fresnel transform,” Appl. Opt. 52, 871–878 (2013).
    [Crossref]
  18. S. K. Rajput and N. K. Nishchal, “Image encryption using polarized light encoding and amplitude- and phase-truncated Fresnel transform,” Appl. Opt. 52, 4343–4352 (2013).
    [Crossref]
  19. S. K. Rajput and N. K. Nishchal, “Known-plaintext attack on encryption domain independent optical asymmetric cryptosystem,” Opt. Commun. 309, 231–235 (2013).
    [Crossref]
  20. X. Wang and D. Zhao, “Amplitude-phase retrieval attack free cryptosystem based on direct attack to phase-truncated Fourier transform-based encryption using a random amplitude mask,” Opt. Lett. 38, 3684–3686 (2013).
    [Crossref]
  21. X. Wan, Y. Chen, C. Dai, and D. Zhao, “Discussion and a new attack of the optical asymmetric cryptosystem based on phase-truncated Fourier transform,” Appl. Opt. 53, 208–213 (2014).
    [Crossref]
  22. S. K. Rajput and N. K. Nishchal, “Fresnel domain nonlinear image encryption scheme based on Gerchberg-Saxton phase-retrieval algorithm,” Appl. Opt. 53, 418–425 (2014).
    [Crossref]
  23. S. Kishk and B. Javidi, “Watermarking of three-dimensional objects by digital holography,” Opt. Lett. 28, 167–169 (2003).
    [Crossref]
  24. N. K. Nishchal, “Hierarchical encrypted image watermarking using fractional Fourier domain random phase encoding,” Opt. Eng. 50, 097003 (2011).
    [Crossref]
  25. Y.-Y. Chen, J.-H. Wang, C.-C. Lin, and H.-E. Hwang, “Lensless optical data hiding system based on phase encoding algorithm in the Fresnel domain,” Appl. Opt. 52, 5247–5257 (2013).
    [Crossref]
  26. B. Javidi and J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33, 1752–1756 (1994).
    [Crossref]
  27. D. Abookasis, O. Arazi, J. Rosen, and B. Javidi, “Security optical systems based on joint transform correlator with significant output images,” Opt. Eng. 40, 1584–1589 (2001).
    [Crossref]
  28. S. K. Rajput and N. K. Nishchal, “Image encryption and authentication verification based on nonconventional fractional joint transform correlator,” Opt. Lasers Eng. 50, 1474–1483 (2012).
    [Crossref]
  29. E. Perez-Cabre, M. Cho, and B. Javidi, “Information authentication using photon-counting double-random-phase encrypted images,” Opt. Lett. 36, 22–24 (2011).
    [Crossref]
  30. M. Cho, A. Mahalanobis, and B. Javidi, “3D passive photon counting automatic target recognition using advanced correlation filters,” Opt. Lett. 36, 861–863 (2011).
    [Crossref]
  31. E. Perez-Cabre, H. C. Abril, M. S. Millan, and B. Javidi, “Photon-counting double-random-phase encoding for secure image verification and retrieval,” J. Opt. 14, 094001 (2012).
    [Crossref]
  32. M. Cho and B. Javidi, “Three-dimensional photon counting double-random-phase encryption,” Opt. Lett. 38, 3198–3201 (2013).
    [Crossref]
  33. A. Markman and B. Javidi, “Full phase photon counting double-random-phase encryption,” J. Opt. Soc. Am. A 31, 394–403 (2014).
    [Crossref]
  34. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane,” Optik 35, 237–246 (1972).
  35. Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, “Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21, 842–844 (1996).
    [Crossref]
  36. B. Javidi, “Nonlinear joint power spectrum based optical correlation,” Appl. Opt. 28, 2358–2367 (1989).
    [Crossref]

2014 (3)

2013 (9)

M. Cho and B. Javidi, “Three-dimensional photon counting double-random-phase encryption,” Opt. Lett. 38, 3198–3201 (2013).
[Crossref]

Y.-Y. Chen, J.-H. Wang, C.-C. Lin, and H.-E. Hwang, “Lensless optical data hiding system based on phase encoding algorithm in the Fresnel domain,” Appl. Opt. 52, 5247–5257 (2013).
[Crossref]

J. F. Barrera, A. Mira, and R. Torroba, “Optical encryption and QR codes: secure and noise-free information retrieval,” Opt. Express 21, 5373–5378 (2013).
[Crossref]

W. Chen, X. Chen, A. Anand, and B. Javidi, “Optical encryption using multiple intensity samplings in the axial domain,” J. Opt. Soc. Am. A 30, 806–812 (2013).
[Crossref]

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).
[Crossref]

S. K. Rajput and N. K. Nishchal, “Known-plaintext attack-based optical cryptosystem using phase-truncated Fresnel transform,” Appl. Opt. 52, 871–878 (2013).
[Crossref]

S. K. Rajput and N. K. Nishchal, “Image encryption using polarized light encoding and amplitude- and phase-truncated Fresnel transform,” Appl. Opt. 52, 4343–4352 (2013).
[Crossref]

S. K. Rajput and N. K. Nishchal, “Known-plaintext attack on encryption domain independent optical asymmetric cryptosystem,” Opt. Commun. 309, 231–235 (2013).
[Crossref]

X. Wang and D. Zhao, “Amplitude-phase retrieval attack free cryptosystem based on direct attack to phase-truncated Fourier transform-based encryption using a random amplitude mask,” Opt. Lett. 38, 3684–3686 (2013).
[Crossref]

2012 (5)

S. K. Rajput and N. K. Nishchal, “Image encryption based on interference that uses fractional Fourier domains asymmetric keys,” Appl. Opt. 51, 1446–1452 (2012).
[Crossref]

X. Wang and D. Zhao, “A special attack on the asymmetric cryptosystem based on phase-truncated fractional Fourier transforms,” Opt. Commun. 285, 1078–1081 (2012).
[Crossref]

S. K. Rajput and N. K. Nishchal, “Asymmetric color cryptosystem using polarization selective diffractive optical element and structured phase mask,” Appl. Opt. 51, 5377–5386 (2012).
[Crossref]

E. Perez-Cabre, H. C. Abril, M. S. Millan, and B. Javidi, “Photon-counting double-random-phase encoding for secure image verification and retrieval,” J. Opt. 14, 094001 (2012).
[Crossref]

S. K. Rajput and N. K. Nishchal, “Image encryption and authentication verification based on nonconventional fractional joint transform correlator,” Opt. Lasers Eng. 50, 1474–1483 (2012).
[Crossref]

2011 (4)

E. Perez-Cabre, M. Cho, and B. Javidi, “Information authentication using photon-counting double-random-phase encrypted images,” Opt. Lett. 36, 22–24 (2011).
[Crossref]

M. Cho, A. Mahalanobis, and B. Javidi, “3D passive photon counting automatic target recognition using advanced correlation filters,” Opt. Lett. 36, 861–863 (2011).
[Crossref]

N. K. Nishchal, “Hierarchical encrypted image watermarking using fractional Fourier domain random phase encoding,” Opt. Eng. 50, 097003 (2011).
[Crossref]

N. K. Nishchal and T. J. Naughton, “Flexible optical encryption with multiple users and multiple security levels,” Opt. Commun. 284, 735–739 (2011).
[Crossref]

2010 (3)

2009 (1)

A. Alfalou and C. Brosseau, “Optical image compression and encryption methods,” Opt. Photon. 1, 589–636 (2009).
[Crossref]

2007 (2)

J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Applications of gyrator transform for image processing,” Opt. Commun. 278, 279–284 (2007).
[Crossref]

Y. Frauel, A. Castro, T. Naughton, and B. Javidi, “Resistance of the double random phase encryption against various attacks,” Opt. Express 15, 10253–10265 (2007).
[Crossref]

2005 (2)

2003 (1)

2001 (1)

D. Abookasis, O. Arazi, J. Rosen, and B. Javidi, “Security optical systems based on joint transform correlator with significant output images,” Opt. Eng. 40, 1584–1589 (2001).
[Crossref]

1996 (1)

1995 (1)

1994 (1)

B. Javidi and J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33, 1752–1756 (1994).
[Crossref]

1989 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane,” Optik 35, 237–246 (1972).

Abookasis, D.

D. Abookasis, O. Arazi, J. Rosen, and B. Javidi, “Security optical systems based on joint transform correlator with significant output images,” Opt. Eng. 40, 1584–1589 (2001).
[Crossref]

Abril, H. C.

E. Perez-Cabre, H. C. Abril, M. S. Millan, and B. Javidi, “Photon-counting double-random-phase encoding for secure image verification and retrieval,” J. Opt. 14, 094001 (2012).
[Crossref]

Alfalou, A.

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).
[Crossref]

A. Alfalou and C. Brosseau, “Dual encryption scheme of images using polarized light,” Opt. Lett. 35, 2185–2187 (2010).
[Crossref]

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).
[Crossref]

A. Alfalou and C. Brosseau, “Optical image compression and encryption methods,” Opt. Photon. 1, 589–636 (2009).
[Crossref]

Alieva, T.

J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Applications of gyrator transform for image processing,” Opt. Commun. 278, 279–284 (2007).
[Crossref]

Anand, A.

Arazi, O.

D. Abookasis, O. Arazi, J. Rosen, and B. Javidi, “Security optical systems based on joint transform correlator with significant output images,” Opt. Eng. 40, 1584–1589 (2001).
[Crossref]

Arcos, S.

Barrera, J. F.

Brosseau, C.

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).
[Crossref]

A. Alfalou and C. Brosseau, “Dual encryption scheme of images using polarized light,” Opt. Lett. 35, 2185–2187 (2010).
[Crossref]

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).
[Crossref]

A. Alfalou and C. Brosseau, “Optical image compression and encryption methods,” Opt. Photon. 1, 589–636 (2009).
[Crossref]

Calvo, M. L.

J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Applications of gyrator transform for image processing,” Opt. Commun. 278, 279–284 (2007).
[Crossref]

Carnicer, A.

Castro, A.

Chen, L.

L. Chen and D. Zhao, “Optical image encryption based on fractional wavelet transform,” Opt. Commun. 254, 361–367 (2005).
[Crossref]

Chen, W.

Chen, X.

Chen, Y.

Chen, Y.-Y.

Cho, M.

Dai, C.

Dorsch, R. G.

Frauel, Y.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane,” Optik 35, 237–246 (1972).

Horner, J. L.

B. Javidi and J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33, 1752–1756 (1994).
[Crossref]

Hwang, H.-E.

Javidi, B.

A. Markman and B. Javidi, “Full phase photon counting double-random-phase encryption,” J. Opt. Soc. Am. A 31, 394–403 (2014).
[Crossref]

M. Cho and B. Javidi, “Three-dimensional photon counting double-random-phase encryption,” Opt. Lett. 38, 3198–3201 (2013).
[Crossref]

W. Chen, X. Chen, A. Anand, and B. Javidi, “Optical encryption using multiple intensity samplings in the axial domain,” J. Opt. Soc. Am. A 30, 806–812 (2013).
[Crossref]

E. Perez-Cabre, H. C. Abril, M. S. Millan, and B. Javidi, “Photon-counting double-random-phase encoding for secure image verification and retrieval,” J. Opt. 14, 094001 (2012).
[Crossref]

E. Perez-Cabre, M. Cho, and B. Javidi, “Information authentication using photon-counting double-random-phase encrypted images,” Opt. Lett. 36, 22–24 (2011).
[Crossref]

M. Cho, A. Mahalanobis, and B. Javidi, “3D passive photon counting automatic target recognition using advanced correlation filters,” Opt. Lett. 36, 861–863 (2011).
[Crossref]

Y. Frauel, A. Castro, T. Naughton, and B. Javidi, “Resistance of the double random phase encryption against various attacks,” Opt. Express 15, 10253–10265 (2007).
[Crossref]

S. Kishk and B. Javidi, “Watermarking of three-dimensional objects by digital holography,” Opt. Lett. 28, 167–169 (2003).
[Crossref]

D. Abookasis, O. Arazi, J. Rosen, and B. Javidi, “Security optical systems based on joint transform correlator with significant output images,” Opt. Eng. 40, 1584–1589 (2001).
[Crossref]

P. Réfrégier and B. Javidi, “Optical image encryption based on input plane encoding and Fourier plane random encoding,” Opt. Lett. 20, 767–769 (1995).
[Crossref]

B. Javidi and J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33, 1752–1756 (1994).
[Crossref]

B. Javidi, “Nonlinear joint power spectrum based optical correlation,” Appl. Opt. 28, 2358–2367 (1989).
[Crossref]

Juvells, I.

Kishk, S.

Lin, C.-C.

Mahalanobis, A.

Markman, A.

Mendlovic, D.

Millan, M. S.

E. Perez-Cabre, H. C. Abril, M. S. Millan, and B. Javidi, “Photon-counting double-random-phase encoding for secure image verification and retrieval,” J. Opt. 14, 094001 (2012).
[Crossref]

Mira, A.

Naughton, T.

Naughton, T. J.

N. K. Nishchal and T. J. Naughton, “Flexible optical encryption with multiple users and multiple security levels,” Opt. Commun. 284, 735–739 (2011).
[Crossref]

Nishchal, N. K.

S. K. Rajput and N. K. Nishchal, “Fresnel domain nonlinear image encryption scheme based on Gerchberg-Saxton phase-retrieval algorithm,” Appl. Opt. 53, 418–425 (2014).
[Crossref]

S. K. Rajput and N. K. Nishchal, “Image encryption using polarized light encoding and amplitude- and phase-truncated Fresnel transform,” Appl. Opt. 52, 4343–4352 (2013).
[Crossref]

S. K. Rajput and N. K. Nishchal, “Known-plaintext attack on encryption domain independent optical asymmetric cryptosystem,” Opt. Commun. 309, 231–235 (2013).
[Crossref]

S. K. Rajput and N. K. Nishchal, “Known-plaintext attack-based optical cryptosystem using phase-truncated Fresnel transform,” Appl. Opt. 52, 871–878 (2013).
[Crossref]

S. K. Rajput and N. K. Nishchal, “Asymmetric color cryptosystem using polarization selective diffractive optical element and structured phase mask,” Appl. Opt. 51, 5377–5386 (2012).
[Crossref]

S. K. Rajput and N. K. Nishchal, “Image encryption based on interference that uses fractional Fourier domains asymmetric keys,” Appl. Opt. 51, 1446–1452 (2012).
[Crossref]

S. K. Rajput and N. K. Nishchal, “Image encryption and authentication verification based on nonconventional fractional joint transform correlator,” Opt. Lasers Eng. 50, 1474–1483 (2012).
[Crossref]

N. K. Nishchal, “Hierarchical encrypted image watermarking using fractional Fourier domain random phase encoding,” Opt. Eng. 50, 097003 (2011).
[Crossref]

N. K. Nishchal and T. J. Naughton, “Flexible optical encryption with multiple users and multiple security levels,” Opt. Commun. 284, 735–739 (2011).
[Crossref]

Peng, X.

Perez-Cabre, E.

E. Perez-Cabre, H. C. Abril, M. S. Millan, and B. Javidi, “Photon-counting double-random-phase encoding for secure image verification and retrieval,” J. Opt. 14, 094001 (2012).
[Crossref]

E. Perez-Cabre, M. Cho, and B. Javidi, “Information authentication using photon-counting double-random-phase encrypted images,” Opt. Lett. 36, 22–24 (2011).
[Crossref]

Qin, W.

Rajput, S. K.

Réfrégier, P.

Rodrigo, J. A.

J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Applications of gyrator transform for image processing,” Opt. Commun. 278, 279–284 (2007).
[Crossref]

Rosen, J.

D. Abookasis, O. Arazi, J. Rosen, and B. Javidi, “Security optical systems based on joint transform correlator with significant output images,” Opt. Eng. 40, 1584–1589 (2001).
[Crossref]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane,” Optik 35, 237–246 (1972).

Torroba, R.

Usategui, M. M.

Wan, X.

Wang, J.-H.

Wang, X.

X. Wang and D. Zhao, “Amplitude-phase retrieval attack free cryptosystem based on direct attack to phase-truncated Fourier transform-based encryption using a random amplitude mask,” Opt. Lett. 38, 3684–3686 (2013).
[Crossref]

X. Wang and D. Zhao, “A special attack on the asymmetric cryptosystem based on phase-truncated fractional Fourier transforms,” Opt. Commun. 285, 1078–1081 (2012).
[Crossref]

Zalevsky, Z.

Zhao, D.

Appl. Opt. (8)

J. Opt. (1)

E. Perez-Cabre, H. C. Abril, M. S. Millan, and B. Javidi, “Photon-counting double-random-phase encoding for secure image verification and retrieval,” J. Opt. 14, 094001 (2012).
[Crossref]

J. Opt. Soc. Am. A (2)

Opt. Commun. (6)

N. K. Nishchal and T. J. Naughton, “Flexible optical encryption with multiple users and multiple security levels,” Opt. Commun. 284, 735–739 (2011).
[Crossref]

L. Chen and D. Zhao, “Optical image encryption based on fractional wavelet transform,” Opt. Commun. 254, 361–367 (2005).
[Crossref]

J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Applications of gyrator transform for image processing,” Opt. Commun. 278, 279–284 (2007).
[Crossref]

S. K. Rajput and N. K. Nishchal, “Known-plaintext attack on encryption domain independent optical asymmetric cryptosystem,” Opt. Commun. 309, 231–235 (2013).
[Crossref]

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).
[Crossref]

X. Wang and D. Zhao, “A special attack on the asymmetric cryptosystem based on phase-truncated fractional Fourier transforms,” Opt. Commun. 285, 1078–1081 (2012).
[Crossref]

Opt. Eng. (3)

B. Javidi and J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33, 1752–1756 (1994).
[Crossref]

D. Abookasis, O. Arazi, J. Rosen, and B. Javidi, “Security optical systems based on joint transform correlator with significant output images,” Opt. Eng. 40, 1584–1589 (2001).
[Crossref]

N. K. Nishchal, “Hierarchical encrypted image watermarking using fractional Fourier domain random phase encoding,” Opt. Eng. 50, 097003 (2011).
[Crossref]

Opt. Express (2)

Opt. Lasers Eng. (1)

S. K. Rajput and N. K. Nishchal, “Image encryption and authentication verification based on nonconventional fractional joint transform correlator,” Opt. Lasers Eng. 50, 1474–1483 (2012).
[Crossref]

Opt. Lett. (11)

E. Perez-Cabre, M. Cho, and B. Javidi, “Information authentication using photon-counting double-random-phase encrypted images,” Opt. Lett. 36, 22–24 (2011).
[Crossref]

M. Cho, A. Mahalanobis, and B. Javidi, “3D passive photon counting automatic target recognition using advanced correlation filters,” Opt. Lett. 36, 861–863 (2011).
[Crossref]

S. Kishk and B. Javidi, “Watermarking of three-dimensional objects by digital holography,” Opt. Lett. 28, 167–169 (2003).
[Crossref]

A. Alfalou and C. Brosseau, “Dual encryption scheme of images using polarized light,” Opt. Lett. 35, 2185–2187 (2010).
[Crossref]

M. Cho and B. Javidi, “Three-dimensional photon counting double-random-phase encryption,” Opt. Lett. 38, 3198–3201 (2013).
[Crossref]

Z. Zalevsky, D. Mendlovic, and R. G. Dorsch, “Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21, 842–844 (1996).
[Crossref]

P. Réfrégier and B. Javidi, “Optical image encryption based on input plane encoding and Fourier plane random encoding,” Opt. Lett. 20, 767–769 (1995).
[Crossref]

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).
[Crossref]

W. Qin and X. Peng, “Asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Lett. 35, 118–120 (2010).
[Crossref]

A. Carnicer, M. M. Usategui, S. Arcos, and I. Juvells, “Vulnerability to chosen-cyphertext attacks of the optical encryption schemes based on double random phase keys,” Opt. Lett. 30, 1644–1646 (2005).
[Crossref]

X. Wang and D. Zhao, “Amplitude-phase retrieval attack free cryptosystem based on direct attack to phase-truncated Fourier transform-based encryption using a random amplitude mask,” Opt. Lett. 38, 3684–3686 (2013).
[Crossref]

Opt. Photon. (1)

A. Alfalou and C. Brosseau, “Optical image compression and encryption methods,” Opt. Photon. 1, 589–636 (2009).
[Crossref]

Optik (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane,” Optik 35, 237–246 (1972).

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Figures (7)

Fig. 1.
Fig. 1.

Block diagram of key generator. PT, phase truncation; AT, amplitude truncation; RPM, random phase mask; FT, Fourier transform; FT1, inverse FT; P1n and P2n, phase values for nth image; RII, random intensity image.

Fig. 2.
Fig. 2.

Schematic diagram of authentication system. SLM, spatial light modulator; L, lens; f, focal length; CF, correlation filters; CCD, charge-coupled device camera.

Fig. 3.
Fig. 3.

Simulation results for first gray-scale image: (a) an input image to be verified, (b) RII, (c) relation showing matching of random intensity image with PTV, and (d) retrieved input image obtained in verification plane. PTV, phase-truncated value.

Fig. 4.
Fig. 4.

(a) Auto-correlation peak, (b) cross-correlation peak when RII is wrong, and (c) cross-correlation peak when keys are wrong.

Fig. 5.
Fig. 5.

Simulation results for another gray-scale image: (a) an image of cameraman to be verified, (b) relation showing matching of random intensity image with PTV, (c) retrieved input image obtained in verification plane, (d) auto-correlation peak, (e) cross-correlation peak when keys are wrong, and (f) cross-correlation peak when RII is wrong.

Fig. 6.
Fig. 6.

Plot between cumulative average of MSE for multiple images and number of images. MSE, mean square error; AMSE, average of MSE.

Fig. 7.
Fig. 7.

Attack results for first gray-scale image: (a) plot between MSE and number of iterations during generations of first asymmetric key, (b) plot between MSE and number of iterations during generations of second asymmetric key, and (c) corresponding decrypted image.

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

Fn(u,v)=I[fn(x,y)×exp{i2πr1n(x,y)}].
An(u,v)=PT{Fn(u,v)},
Pn(u,v)=AT{Fn(u,v)}.
A(n)m+1(x,y)=I1[Anm(u,v)]=|A(n)m+1(x,y)|×exp{iφnm(x,y)}.
A(n)m+1(u,v)=I[A(n)m+1(x,y)]=|A(n)m+1(u,v)|×exp{iφ(n)m(u,v)}.
MSE=x=0N1y=0N1{|R(x,y)||A(n)m+1(x,y)|}2N×N.
K1n(u,v)=Pn(u,v)×exp{ir2(n)m+1(u,v)},
K2n(x,y)=exp{iφnm(x,y)}.
dn(x,y)=I1[I{Rn(x,y)×K2n(x,y)}×K1n(u,v)].
Cn(x,y)=I1[Q1×Q2],
Q1=|I{dn(x,y)}×[I{fn(x,y)}]|t,
Q2=exp{iarg[I{dn(x,y)}]arg[I{fn(x,y)}]}.

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