Abstract

An encryption and verification method with multiple encrypted keys based on interference principle is proposed. The encryption process is realized on computer digitally and the verification process can be completed optically or digitally. Two different images are encoded into three diffractive phase elements (DPEs) by using two different incident wavelengths. Three DPEs have different distances from output plane. The two wavelength parameters and three distance parameters can be used as encryption keys, which will boost security degree of this system. Numerical simulation proves that the proposed encryption method is valid and has high secrecy level.

©2010 Optical Society of America

1. Introduction

Optical technology has been widely used in information encryption and decryption application, owing to its multiple parameters and parallel processing ability. In the past decade, various algorithms and systems about optical image encryption and security have been proposed [14]. Refregier and Javidi [1] proposed a method that uses double random phases to encrypt an image to stationary white noise. Liu et al. [5] and Zhang et al. [6] extended this method into the fractional Fourier domain. The security of double random phase encoding has also been thoroughly analyzed and a few weaknesses have started to appear [79]. In order to make it easy to decode the encrypted image with intensity-sensitive detectors,Wang et al. have also proposed a method to encrypt the original image into a phase-only mask in the Fourier plane of a 4 f optically system [10]. Li et al. improved this method to encrypt the image into the mask in the input plane [11]. Recently, it has also been expanded to encrypt multiple images [12]. However, all of the above-mentioned works used the iterative algorithm to encode the original image into the pure phase diffraction elements (DPEs). Furthermore, the basic ideas of these methods are based on the diffraction principle. The encryption methods based on the interference principle have also been investigated. Javidi and Nomura used a digital hologram to store an encrypted image and the decryption key [13], and Nishchal et al. proposed a method of securing information using fractional Fourier transform in digital holography [14]. Meng et al. proposed a hybrid cryptosystem, in which one image is encrypted to two interferograms with the aid of double random-phase encoding (DRPE) and two-step phase-shifting interferometry (2-PSI) [15]. Only recently, Zhang et al. [16] proposed a new encryption method that an image is encoded into two pure phase masks based on the optical interference principle. The encryption method is quite simple and does not need the iterative algorithm and the used encryption keys have high sensitivity. They encode information of an image into two pure phase masks by using only one wavelength and the two pure phase masks must have identical distances from the output plane, which make invaders easy to obtain the distance key and the wavelength key by testing repeatedly.

In this article, we proposed an encryption and verification method, which is also based on optical interference. In the new method, information of two different images are encoded into three pure phase diffraction elements (DPEs) by using two different illuminating wavelengths, respectively. The three DPEs have different distances from the output plane. The two illuminating wavelengths and the three distances can be used as encryption keys, which significantly increase security degree. For simple, we only encode image information into three DPEs in this article, more than three DPEs can also be employed to enhance performance of the encryption system. The encryption process is implemented in computer digitally and the verification process can be realized through optical system or digitally.

2. The verification optical system

The verification optical system is schematically shown in Fig. 1 . There are two methods for verification process. In Fig. 1(a), three coherent parallel light beams are modulated by three diffractive phase elements (DPEs) and then are combined by two half-mirrors (HM1 and HM2). Thus three beams interfere with each other on the output plane and generate the image encrypted in the three DPEs. Notice that three DPEs comprise two kinds of encrypted information for two different wavelengths, that is, using two groups of incident light with different wavelengths will generate two different images on the output plane. The two wavelengths can be used as encryption keys. In verification process (or, decryption process), an image detector such as Charge Coupled Device (CCD) can be used to accept image on the output plane. In order that three light beams through three DPEs have appropriate field intensities on the output plane, we use an optical attenuator (OA) on the optical path of DPE3. The distance between DPE1 and output plane l1 is set to be the same as the distance between DPE2 and output plane l2, but the distance between DPE3 and output plane l3 is different from l1 and l2. So these different distance parameters can also be used as encryption keys and enhance secrecy degree. A simpler verification system is shown in Fig. 1(b). Three coherent light beams transmit in different directions through three DPEs, but the three light beams will intersect with each other on the output plane and then generate the encrypted image. In this case, if angles between three light beams α and β are small, the complex field distributions of light beams through DPE1 and DPE3 on the output plane will only need to multiply exp(i2πλαx) and exp(i2πλβx), here, x represents vertical coordinate on the output plane and λ is wavelength of incident light beam.

 figure: Fig. 1

Fig. 1 Schematic of the verification optical system

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The encryption process should be implemented digitally, while the decryption process can be implemented optically or digitally. For highly secure verification, three DPEs can be assigned to three different important users, and the system parameters (three distance parameters and two wavelength parameters) can also set as encryption keys, which not only can be fixed in the verification system, but also set as passwords for different users. As a result, the encrypted image can be obtained at the output plane only when three correct users put three DPEs in the verification system and input correct system parameters.

3. Digital encryption algorithm

The encryption problem is equal to separate information of an image into three phase diffractive elements (DPEs). In the present method, we use two wavelengths to encrypt the image information (here, represented by λ1 andλ2). Firstly, we use the first wavelength λ1.

If the encrypted image is an intensity distribution oλ1(m,n), a new complex distribution can be constructed by adding it with a random phase distribution,

oλ1'(m,n)=oλ1(m,n)exp[i2πrand(m,n)],
where rand(m,n) generates a random distribution between 0 and 1. The new constructed complex distribution can be expressed as the interference of fields generated by DPE1, DPE2 and DPE3,
oλ1'(m,n)=exp(iΦ1λ1)h(x,y,λ1,l1)+exp(iΦ2λ1)h(x,y,λ1,l2)+exp(iΦ3λ1)h(x,y,λ1,l3),
where
h(x,y,λ,l)=exp(i2πl/λ)ilλexp[iπlλ(x2+y2)]
is the point pulse function of the Fresnel transform and ∗ represents the convolution operation. If one wants to use more than three DPEs in the verification system, e.g., four DPEs, exp(iΦ4λ1)h(x,y,λ1,l4) must be added to right of Eq. (2). Here, Φ4λ1 represents phase shift distribution introduced by the fourth DPE and l4 is the distance between the fourth DPE and the output plane.

If the relationship between distance from DPE1 to the output plane and distance from DPE2 to the output plane is l1=l2=l and set the phase distribution Φ3λ1 of DPE3 as a random phase distribution between 0 and 2π, an equation can be obtained after a simple deduction,

exp(iΦ1λ1)+exp(iΦ2λ1)=F1{F{oλ1'(m,n)}F{exp(iΦ3λ1)}F{h(x,y,λ1,l3)}F{h(x,y,λ1,l)}},
where F{}expresses the Fourier transform and F1{}expresses the inverse Fourier transform. For four DPEs, the right portion of the Eq. (4) will becomeF1{F{oλ1'(m,n)}F{exp(iΦ3λ1)}F{h(x,y,λ1,l3)}F{exp(iΦ4λ1)}F{h(x,y,λ1,l4)}F{h(x,y,λ1,l)}}.We set the right portion of the Eq. (4) as
D=F1{F{oλ1'(m,n)}F{exp(iΦ3λ1)}F{h(x,y,λ1,l3)}F{h(x,y,λ1,l)}},
thus we have

exp(iΦ2λ1)=Dexp(iΦ1λ1).

Since module of the left portion of the Eq. (6) is equal to 1, we have

|Dexp(iΦ1λ1)|2=[Dexp(iΦ1λ1)][D*exp(iΦ1λ1)]=1.

At last, we can achieve the phase distributions of DPE1 and DPE2 as

Φ1λ1=arg(D)arccos(abs(D)/2),
Φ2λ1=arg(Dexp(iΦ1λ1)).

According to the same principle, the phase distributions of three DPEs for incident light with wavelength λ2 can be obtained. Notice that the three phase distributions Φ1λ2,Φ2λ2and Φ3λ2 correspond to an interference intensity distribution oλ2(m,n) on the output plane that is different with the intensity distribution oλ1(m,n) for the first wavelengthλ1.

Now, we have three phase distributions for λ1 and three phase distributions forλ2, but only three DPEs can be used to generate these phase distributions. We express the height distribution of surface-relief structure of three DPEs ash1, h2 andh3, respectively. For DPE1, the phase distribution corresponding to incident wavelength λ1 can be expressed as

Φ1λ1=2πλ1(n(λ1)1)h1,
here, n(λ1) is the refractive index of DPE1 for incident wavelength λ1. Similarly, we have
Φ1λ2=2πλ2(n(λ2)1)h1
for incident wavelengthλ2. If values of Φ1λ1 and Φ1λ2 is limited between 0 and 2π, it is difficult to satisfy Eqs. (9) and (10) simultaneously. So, we enlarge the value scope of Φ1λ1 and Φ1λ2 by adding 2Pπ and 2Qπ (P and Q are two arbitrary integers), respectively, which will not influence the actual field distribution. Through carefully choosing P and Q, the equation
h1(x,y)=Φ1λ1(x,y)+2P(x,y)π2π(n(λ1)1)λ1Φ1λ2(x,y)+2Q(x,y)π2π(n(λ2)1)λ2
can be satisfied. For example, we can limit values of P and Q to be integers between 0 and 6, then by varying P and Q respectively, a couple of proper values of P and Q could be found for the situation that two items in right of Eq. (11) are very close to each other. Based on the same method, we can achieve the height distribution of relief structure of DPE2 and DPE3.

4. Numerical simulations

Computer simulations are carried out to show validity of our proposed new method. The images to be encrypted are two 256×256 pixel gray images which correspond to incident wavelengths λ1 and λ2, respectively. As shown in Figs. 2(a) and 2(b), two images comprise different letters. The sizes of two images are both 5cm×5cm, and two wavelengths of the illuminating light are set as λ1=633nm and λ2=514nm, respectively. The three distances are l1=l2=20cm and l3=40cm. According to the encryption method mentioned above, we can obtain two groups of phase distributions for two incident wavelength, then the height distributions of three DPEs can be achieved based on Eq. (11). Here, the largest height of relief structures of three DPEs is set as 6μm. The computed height distributions of three DPEs are displayed in Figs. 3 . It is obvious that no information about the original images can be found in height distributions of three DPEs.

 figure: Fig. 2

Fig. 2 Encryption results. (a) The first original image for encryption, (b) the second original image for encryption, (c) and (d) two reconstructed images with correct encryption keys, (e) and (f) two reconstructed images with incorrect encryption keys.

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 figure: Fig. 3

Fig. 3 Height distributions of three encrypted DPEs

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The reconstructed images with correct height distributions of three DPEs and right system parameters are displayed in Figs. 2(c) and 2(d), respectively. It can be seen that the two primary images have been retrieved well, but only some fluctuations exist. These fluctuations can be eliminated by increasing values of P and Q in Eq. (11), i.e., enlarging the largest heights of three DPEs. However, too large height will make fabrication of DPEs difficult. If we set the height distribution of DPE3 as incorrect values, the reconstructed images for two correct wavelengths are shown as Figs. 2(e) and 2(f). It is clear that we cannot observe any useful information from the reconstructed images.

To evaluate the reliability of the encryption method quantitatively, the relative error (RE) between the two reconstructed images and two original images is introduced as

RE=m=1Nn=1N||rλ1(m,n)||oλ1'(m,n)||2+m=1Nn=1N||rλ2(m,n)||oλ2'(m,n)||2m=1Nn=1N|oλ1'(m,n)|2+m=1Nn=1N|oλ2'(m,n)|2,
where N×N is the size of the two images, oλ1'(m,n) and rλ1(m,n) denote the amplitude value of the original and reconstructed images corresponding to the first wavelength λ1, respectively, at the pixel (m,n). Similarly, oλ2'(m,n) and rλ2(m,n) correspond to the second wavelength λ2.

In the proposed method, the two wavelengths of the illuminating light λ1 and λ2 can be used as encryption keys. We have calculated the dependence relationship of RE on the two wavelengths, which is shown in left portion of Fig. 4(a) . It can be seen in left portion of Fig. 4(a) that only when both of the two illuminating wavelengths have correct values, RE has the smallest value RE=0.0095; value of RE will sharply rise with increasing of wavelength error.

 figure: Fig. 4

Fig. 4 (a) Dependence of RE on wavelength difference Δλ1 and Δλ2, (b) decryption result with Δλ1=1×104nm, (c) decryption result with Δλ2=1×104nm.

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More precise simulation is done to evaluate the evolution of the smallest RE value with Δλ1, as shown in right portion of Fig. 4 (a). It can be seen from right figure of Fig. 4(a) that RE quickly increases with Δλ1even in magnitude order of 104nm. If the first illuminating wavelength has a tiny difference from the correct value, e.g., Δλ1=1×104nm, which corresponds toRE=0.25, the first reconstructed image will not display any useful information about original image, as shown in Fig. 4(b). The same phenomenon can appear for the second illuminating wavelength λ2, whose corresponding reconstructed image is shown in Fig. 4(c) for Δλ2=1×104nm. It can be proved that the decrypted image cannot be distinguished with the naked eye for RE>0.2. The wavelength sensitivity of the proposed encryption method is about 1×104nm.

The distances between each DPE and the output plane can also be used as encryption keys. In our proposed method, the three distances have the relationships l1=l2=l and l3l. In order to validate effect of the two distance encryption keys, we have also calculated the influence of change of l and l3 on RE, as shown in left portion of Fig. 5(a) . When l=20cm and l3=40cm, RE has smallest value 0.0095. If l and l3 differs from correct value little, value of RE will significantly increase. More precise simulation is done to evaluate the evolution of the smallest RE value with Δl, as shown in right portion of Fig. 5 (a). It is obvious that value of RE quickly rise increases with Δleven in so small varying scope. The same evolution of the smallest RE value is obtained for Δl3. Figures 5(b) and 5(c) show the reconstructed images for the case of Δl=2nm and Δl3=2nm, respectively. It is obvious that the two reconstructed images are entirely different from two original images, that is, tiny difference of distance will fail to decrypt correct information. Further simulation can obtain a conclusion that, whenRE>0.2, there is a failure to distinguish the decrypted image. The distance sensitivity of the proposed encryption method is about 2nm.

 figure: Fig. 5

Fig. 5 (a) Dependence of RE on distance difference Δl and Δl3, (b) decryption result withΔl=2nm, (c) decryption result withΔl3=2nm.

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The high wavelength and distance sensitivity will cause great difficulty in copying the encryption system, which make the proposed encryption system have high security degree. Meanwhile, a more precise mechanical supporting system for determining distance parameters l and l3 is needed to implement the verification process. It can also be concluded from further detailed simulation that transversal displacement of three DPEs will significantly influence quality of reconstructed images. For example, a transversal displacement with one-tenth pixel of DPE1 (which corresponds to about 20μm) can cause an increase of relative error for ΔRE=0.1. Therefore, precise control of the transversal positions of three DPEs is also very important for practical optical verification system. Finally, we also calculate influence of surface-relief deviation of three DPEs on reconstructed images. Tolerance scope of surface-relief height of three DPEs is about ±0.4%of surface-relief height. The scope lies in the region of state of the art of optical fabrication and means that the optical encryption process can be implemented conveniently.

5. Conclusion

In conclusion, we proposed a encryption and verification method based on interference principle. The encryption process is digitally implemented on computer and the verification process can realize digitally and optically. Two different original images are encrypted into three DPEs by using two illuminating wavelengths, respectively. The three DPEs comprise encrypted information and are assigned to three different users as encryption keys. The three DPEs have different distances from the output plane. The two wavelength parameters and the three distance parameters can also be used as encryption keys. In verification process, three correct users put their own DPE in the verification system and input two wavelength parameter and three distance parameter. The verification will set correct positions of three DPEs and illuminating lights, and the correct encrypted images can be generated on the output plane only when all of encryption keys are correct. Because of multiple encryption keys in our proposed method and high sensitive of these encryption keys, the interloper will difficulty in copying the decryption system or passing through the verification system. The simulation results have shown the validity of the proposed method.

Acknowledgement

This work is supported by Project of Beijing Education Committee (Grant No. KM200910772005).

References and links

1. P. Refregier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20(7), 767–769 (1995). [CrossRef]   [PubMed]  

2. O. Matoba and B. Javidi, “Encrypted optical memory system using three-dimensional keys in the Fresnel domain,” Opt. Lett. 24(11), 762–764 (1999). [CrossRef]  

3. J. W. Han, C. S. Park, D. H. Ryu, and E. S. Kin, “Optical image encryption based on XOR operations,” Opt. Eng. (Bellingham) 38(1), 47–54 (1999). [CrossRef]  

4. J. W. Han, S. H. Lee, and E. S. Kin, “Optical key bit stream generator,” Opt. Eng. (Bellingham) 38(1), 33–38 (1999). [CrossRef]  

5. S. T. Liu, Q. L. Mi, and B. H. Zhu, “Optical image encryption with multistage and multichannel fractional Fourier-domain filtering,” Opt. Lett. 26(16), 1242–1244 (2001). [CrossRef]  

6. Y. Zhang, C. H. Zheng, and N. Tanno, “Optical encryption based on iterative fractional Fourier transform,” Opt. Commun. 202(4-6), 277–285 (2002). [CrossRef]  

7. A. Carnicer, M. Montes-Usategui, S. Arcos, and I. Juvells, “Vulnerability to chosen-cyphertext attacks of optical encryption schemes based on double random phase keys,” Opt. Lett. 30(13), 1644–1646 (2005). [CrossRef]   [PubMed]  

8. U. Gopinathan, D. S. Monaghan, T. J. Naughton, and J. T. Sheridan, “A known-plaintext heuristic attack on the Fourier plane encryption algorithm,” Opt. Express 14(8), 3181–3186 (2006). [CrossRef]   [PubMed]  

9. Y. Frauel, A. Castro, T. J. Naughton, and B. Javidi, “Resistance of the double random phase encryption against various attacks,” Opt. Express 15(16), 10253–10265 (2007). [CrossRef]   [PubMed]  

10. P. K. Wang, L. A. Watson, and C. Chatwin, “Random phase encoding for optical security,” Opt. Eng. (Bellingham) 35(9), 2464–2469 (1996). [CrossRef]  

11. Y. Li, K. Kreske, and J. Rosen, “Security and encryption optical systems based on a correlator with significant output images,” Appl. Opt. 39(29), 5295–5301 (2000). [CrossRef]  

12. Z. J. Liu and S. T. Liu, “Double image encryption based on iterative fractional Fourier transform,” Opt. Commun. 275(2), 324–329 (2007). [CrossRef]  

13. B. Javidi and T. Nomura, “Securing information by use of digital holography,” Opt. Lett. 25(1), 28–30 (2000). [CrossRef]  

14. N. K. Nishchal, J. Joseph, and K. Singh, “Securing information using fractional Fourier transform in digital holography,” Opt. Commun. 235(4-6), 253–259 (2004). [CrossRef]  

15. X. F. Meng, X. Peng, L. Z. Cai, A. M. Li, Z. Gao, and Y. R. Wang, “Cryptosystem based on two-step phase-shifting interferometry and the RSA public-key encryption algorithm,” J. Opt. A, Pure Appl. Opt. 11(8), 085402 (2009). [CrossRef]  

16. Y. Zhang and B. Wang, “Optical image encryption based on interference,” Opt. Lett. 33(21), 2443–2445 (2008). [CrossRef]   [PubMed]  

References

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  1. P. Refregier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20(7), 767–769 (1995).
    [Crossref] [PubMed]
  2. O. Matoba and B. Javidi, “Encrypted optical memory system using three-dimensional keys in the Fresnel domain,” Opt. Lett. 24(11), 762–764 (1999).
    [Crossref]
  3. J. W. Han, C. S. Park, D. H. Ryu, and E. S. Kin, “Optical image encryption based on XOR operations,” Opt. Eng. (Bellingham) 38(1), 47–54 (1999).
    [Crossref]
  4. J. W. Han, S. H. Lee, and E. S. Kin, “Optical key bit stream generator,” Opt. Eng. (Bellingham) 38(1), 33–38 (1999).
    [Crossref]
  5. S. T. Liu, Q. L. Mi, and B. H. Zhu, “Optical image encryption with multistage and multichannel fractional Fourier-domain filtering,” Opt. Lett. 26(16), 1242–1244 (2001).
    [Crossref]
  6. Y. Zhang, C. H. Zheng, and N. Tanno, “Optical encryption based on iterative fractional Fourier transform,” Opt. Commun. 202(4-6), 277–285 (2002).
    [Crossref]
  7. A. Carnicer, M. Montes-Usategui, S. Arcos, and I. Juvells, “Vulnerability to chosen-cyphertext attacks of optical encryption schemes based on double random phase keys,” Opt. Lett. 30(13), 1644–1646 (2005).
    [Crossref] [PubMed]
  8. U. Gopinathan, D. S. Monaghan, T. J. Naughton, and J. T. Sheridan, “A known-plaintext heuristic attack on the Fourier plane encryption algorithm,” Opt. Express 14(8), 3181–3186 (2006).
    [Crossref] [PubMed]
  9. Y. Frauel, A. Castro, T. J. Naughton, and B. Javidi, “Resistance of the double random phase encryption against various attacks,” Opt. Express 15(16), 10253–10265 (2007).
    [Crossref] [PubMed]
  10. P. K. Wang, L. A. Watson, and C. Chatwin, “Random phase encoding for optical security,” Opt. Eng. (Bellingham) 35(9), 2464–2469 (1996).
    [Crossref]
  11. Y. Li, K. Kreske, and J. Rosen, “Security and encryption optical systems based on a correlator with significant output images,” Appl. Opt. 39(29), 5295–5301 (2000).
    [Crossref]
  12. Z. J. Liu and S. T. Liu, “Double image encryption based on iterative fractional Fourier transform,” Opt. Commun. 275(2), 324–329 (2007).
    [Crossref]
  13. B. Javidi and T. Nomura, “Securing information by use of digital holography,” Opt. Lett. 25(1), 28–30 (2000).
    [Crossref]
  14. N. K. Nishchal, J. Joseph, and K. Singh, “Securing information using fractional Fourier transform in digital holography,” Opt. Commun. 235(4-6), 253–259 (2004).
    [Crossref]
  15. X. F. Meng, X. Peng, L. Z. Cai, A. M. Li, Z. Gao, and Y. R. Wang, “Cryptosystem based on two-step phase-shifting interferometry and the RSA public-key encryption algorithm,” J. Opt. A, Pure Appl. Opt. 11(8), 085402 (2009).
    [Crossref]
  16. Y. Zhang and B. Wang, “Optical image encryption based on interference,” Opt. Lett. 33(21), 2443–2445 (2008).
    [Crossref] [PubMed]

2009 (1)

X. F. Meng, X. Peng, L. Z. Cai, A. M. Li, Z. Gao, and Y. R. Wang, “Cryptosystem based on two-step phase-shifting interferometry and the RSA public-key encryption algorithm,” J. Opt. A, Pure Appl. Opt. 11(8), 085402 (2009).
[Crossref]

2008 (1)

2007 (2)

Z. J. Liu and S. T. Liu, “Double image encryption based on iterative fractional Fourier transform,” Opt. Commun. 275(2), 324–329 (2007).
[Crossref]

Y. Frauel, A. Castro, T. J. Naughton, and B. Javidi, “Resistance of the double random phase encryption against various attacks,” Opt. Express 15(16), 10253–10265 (2007).
[Crossref] [PubMed]

2006 (1)

2005 (1)

2004 (1)

N. K. Nishchal, J. Joseph, and K. Singh, “Securing information using fractional Fourier transform in digital holography,” Opt. Commun. 235(4-6), 253–259 (2004).
[Crossref]

2002 (1)

Y. Zhang, C. H. Zheng, and N. Tanno, “Optical encryption based on iterative fractional Fourier transform,” Opt. Commun. 202(4-6), 277–285 (2002).
[Crossref]

2001 (1)

2000 (2)

1999 (3)

O. Matoba and B. Javidi, “Encrypted optical memory system using three-dimensional keys in the Fresnel domain,” Opt. Lett. 24(11), 762–764 (1999).
[Crossref]

J. W. Han, C. S. Park, D. H. Ryu, and E. S. Kin, “Optical image encryption based on XOR operations,” Opt. Eng. (Bellingham) 38(1), 47–54 (1999).
[Crossref]

J. W. Han, S. H. Lee, and E. S. Kin, “Optical key bit stream generator,” Opt. Eng. (Bellingham) 38(1), 33–38 (1999).
[Crossref]

1996 (1)

P. K. Wang, L. A. Watson, and C. Chatwin, “Random phase encoding for optical security,” Opt. Eng. (Bellingham) 35(9), 2464–2469 (1996).
[Crossref]

1995 (1)

Arcos, S.

Cai, L. Z.

X. F. Meng, X. Peng, L. Z. Cai, A. M. Li, Z. Gao, and Y. R. Wang, “Cryptosystem based on two-step phase-shifting interferometry and the RSA public-key encryption algorithm,” J. Opt. A, Pure Appl. Opt. 11(8), 085402 (2009).
[Crossref]

Carnicer, A.

Castro, A.

Chatwin, C.

P. K. Wang, L. A. Watson, and C. Chatwin, “Random phase encoding for optical security,” Opt. Eng. (Bellingham) 35(9), 2464–2469 (1996).
[Crossref]

Frauel, Y.

Gao, Z.

X. F. Meng, X. Peng, L. Z. Cai, A. M. Li, Z. Gao, and Y. R. Wang, “Cryptosystem based on two-step phase-shifting interferometry and the RSA public-key encryption algorithm,” J. Opt. A, Pure Appl. Opt. 11(8), 085402 (2009).
[Crossref]

Gopinathan, U.

Han, J. W.

J. W. Han, C. S. Park, D. H. Ryu, and E. S. Kin, “Optical image encryption based on XOR operations,” Opt. Eng. (Bellingham) 38(1), 47–54 (1999).
[Crossref]

J. W. Han, S. H. Lee, and E. S. Kin, “Optical key bit stream generator,” Opt. Eng. (Bellingham) 38(1), 33–38 (1999).
[Crossref]

Javidi, B.

Joseph, J.

N. K. Nishchal, J. Joseph, and K. Singh, “Securing information using fractional Fourier transform in digital holography,” Opt. Commun. 235(4-6), 253–259 (2004).
[Crossref]

Juvells, I.

Kin, E. S.

J. W. Han, S. H. Lee, and E. S. Kin, “Optical key bit stream generator,” Opt. Eng. (Bellingham) 38(1), 33–38 (1999).
[Crossref]

J. W. Han, C. S. Park, D. H. Ryu, and E. S. Kin, “Optical image encryption based on XOR operations,” Opt. Eng. (Bellingham) 38(1), 47–54 (1999).
[Crossref]

Kreske, K.

Lee, S. H.

J. W. Han, S. H. Lee, and E. S. Kin, “Optical key bit stream generator,” Opt. Eng. (Bellingham) 38(1), 33–38 (1999).
[Crossref]

Li, A. M.

X. F. Meng, X. Peng, L. Z. Cai, A. M. Li, Z. Gao, and Y. R. Wang, “Cryptosystem based on two-step phase-shifting interferometry and the RSA public-key encryption algorithm,” J. Opt. A, Pure Appl. Opt. 11(8), 085402 (2009).
[Crossref]

Li, Y.

Liu, S. T.

Z. J. Liu and S. T. Liu, “Double image encryption based on iterative fractional Fourier transform,” Opt. Commun. 275(2), 324–329 (2007).
[Crossref]

S. T. Liu, Q. L. Mi, and B. H. Zhu, “Optical image encryption with multistage and multichannel fractional Fourier-domain filtering,” Opt. Lett. 26(16), 1242–1244 (2001).
[Crossref]

Liu, Z. J.

Z. J. Liu and S. T. Liu, “Double image encryption based on iterative fractional Fourier transform,” Opt. Commun. 275(2), 324–329 (2007).
[Crossref]

Matoba, O.

Meng, X. F.

X. F. Meng, X. Peng, L. Z. Cai, A. M. Li, Z. Gao, and Y. R. Wang, “Cryptosystem based on two-step phase-shifting interferometry and the RSA public-key encryption algorithm,” J. Opt. A, Pure Appl. Opt. 11(8), 085402 (2009).
[Crossref]

Mi, Q. L.

Monaghan, D. S.

Montes-Usategui, M.

Naughton, T. J.

Nishchal, N. K.

N. K. Nishchal, J. Joseph, and K. Singh, “Securing information using fractional Fourier transform in digital holography,” Opt. Commun. 235(4-6), 253–259 (2004).
[Crossref]

Nomura, T.

Park, C. S.

J. W. Han, C. S. Park, D. H. Ryu, and E. S. Kin, “Optical image encryption based on XOR operations,” Opt. Eng. (Bellingham) 38(1), 47–54 (1999).
[Crossref]

Peng, X.

X. F. Meng, X. Peng, L. Z. Cai, A. M. Li, Z. Gao, and Y. R. Wang, “Cryptosystem based on two-step phase-shifting interferometry and the RSA public-key encryption algorithm,” J. Opt. A, Pure Appl. Opt. 11(8), 085402 (2009).
[Crossref]

Refregier, P.

Rosen, J.

Ryu, D. H.

J. W. Han, C. S. Park, D. H. Ryu, and E. S. Kin, “Optical image encryption based on XOR operations,” Opt. Eng. (Bellingham) 38(1), 47–54 (1999).
[Crossref]

Sheridan, J. T.

Singh, K.

N. K. Nishchal, J. Joseph, and K. Singh, “Securing information using fractional Fourier transform in digital holography,” Opt. Commun. 235(4-6), 253–259 (2004).
[Crossref]

Tanno, N.

Y. Zhang, C. H. Zheng, and N. Tanno, “Optical encryption based on iterative fractional Fourier transform,” Opt. Commun. 202(4-6), 277–285 (2002).
[Crossref]

Wang, B.

Wang, P. K.

P. K. Wang, L. A. Watson, and C. Chatwin, “Random phase encoding for optical security,” Opt. Eng. (Bellingham) 35(9), 2464–2469 (1996).
[Crossref]

Wang, Y. R.

X. F. Meng, X. Peng, L. Z. Cai, A. M. Li, Z. Gao, and Y. R. Wang, “Cryptosystem based on two-step phase-shifting interferometry and the RSA public-key encryption algorithm,” J. Opt. A, Pure Appl. Opt. 11(8), 085402 (2009).
[Crossref]

Watson, L. A.

P. K. Wang, L. A. Watson, and C. Chatwin, “Random phase encoding for optical security,” Opt. Eng. (Bellingham) 35(9), 2464–2469 (1996).
[Crossref]

Zhang, Y.

Y. Zhang and B. Wang, “Optical image encryption based on interference,” Opt. Lett. 33(21), 2443–2445 (2008).
[Crossref] [PubMed]

Y. Zhang, C. H. Zheng, and N. Tanno, “Optical encryption based on iterative fractional Fourier transform,” Opt. Commun. 202(4-6), 277–285 (2002).
[Crossref]

Zheng, C. H.

Y. Zhang, C. H. Zheng, and N. Tanno, “Optical encryption based on iterative fractional Fourier transform,” Opt. Commun. 202(4-6), 277–285 (2002).
[Crossref]

Zhu, B. H.

Appl. Opt. (1)

J. Opt. A, Pure Appl. Opt. (1)

X. F. Meng, X. Peng, L. Z. Cai, A. M. Li, Z. Gao, and Y. R. Wang, “Cryptosystem based on two-step phase-shifting interferometry and the RSA public-key encryption algorithm,” J. Opt. A, Pure Appl. Opt. 11(8), 085402 (2009).
[Crossref]

Opt. Commun. (3)

Z. J. Liu and S. T. Liu, “Double image encryption based on iterative fractional Fourier transform,” Opt. Commun. 275(2), 324–329 (2007).
[Crossref]

N. K. Nishchal, J. Joseph, and K. Singh, “Securing information using fractional Fourier transform in digital holography,” Opt. Commun. 235(4-6), 253–259 (2004).
[Crossref]

Y. Zhang, C. H. Zheng, and N. Tanno, “Optical encryption based on iterative fractional Fourier transform,” Opt. Commun. 202(4-6), 277–285 (2002).
[Crossref]

Opt. Eng. (Bellingham) (3)

J. W. Han, C. S. Park, D. H. Ryu, and E. S. Kin, “Optical image encryption based on XOR operations,” Opt. Eng. (Bellingham) 38(1), 47–54 (1999).
[Crossref]

J. W. Han, S. H. Lee, and E. S. Kin, “Optical key bit stream generator,” Opt. Eng. (Bellingham) 38(1), 33–38 (1999).
[Crossref]

P. K. Wang, L. A. Watson, and C. Chatwin, “Random phase encoding for optical security,” Opt. Eng. (Bellingham) 35(9), 2464–2469 (1996).
[Crossref]

Opt. Express (2)

Opt. Lett. (6)

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Figures (5)

Fig. 1
Fig. 1 Schematic of the verification optical system
Fig. 2
Fig. 2 Encryption results. (a) The first original image for encryption, (b) the second original image for encryption, (c) and (d) two reconstructed images with correct encryption keys, (e) and (f) two reconstructed images with incorrect encryption keys.
Fig. 3
Fig. 3 Height distributions of three encrypted DPEs
Fig. 4
Fig. 4 (a) Dependence of RE on wavelength difference Δ λ 1 and Δ λ 2 , (b) decryption result with Δ λ 1 = 1 × 10 4 n m , (c) decryption result with Δ λ 2 = 1 × 10 4 n m .
Fig. 5
Fig. 5 (a) Dependence of RE on distance difference Δ l and Δ l 3 , (b) decryption result with Δ l = 2 n m , (c) decryption result with Δ l 3 = 2 n m .

Equations (13)

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o λ 1 ' ( m , n ) = o λ 1 ( m , n ) exp [ i 2 π r a n d ( m , n ) ] ,
o λ 1 ' ( m , n ) = exp ( i Φ 1 λ 1 ) h ( x , y , λ 1 , l 1 ) + exp ( i Φ 2 λ 1 ) h ( x , y , λ 1 , l 2 ) + exp ( i Φ 3 λ 1 ) h ( x , y , λ 1 , l 3 ) ,
h ( x , y , λ , l ) = exp ( i 2 π l / λ ) i l λ exp [ i π l λ ( x 2 + y 2 ) ]
exp ( i Φ 1 λ 1 ) + exp ( i Φ 2 λ 1 ) = F 1 { F { o λ 1 ' ( m , n ) } F { exp ( i Φ 3 λ 1 ) } F { h ( x , y , λ 1 , l 3 ) } F { h ( x , y , λ 1 , l ) } } ,
D = F 1 { F { o λ 1 ' ( m , n ) } F { exp ( i Φ 3 λ 1 ) } F { h ( x , y , λ 1 , l 3 ) } F { h ( x , y , λ 1 , l ) } } ,
exp ( i Φ 2 λ 1 ) = D exp ( i Φ 1 λ 1 ) .
| D exp ( i Φ 1 λ 1 ) | 2 = [ D exp ( i Φ 1 λ 1 ) ] [ D * exp ( i Φ 1 λ 1 ) ] = 1.
Φ 1 λ 1 = arg ( D ) arc cos ( a b s ( D ) / 2 ) ,
Φ 2 λ 1 = arg ( D exp ( i Φ 1 λ 1 ) ) .
Φ 1 λ 1 = 2 π λ 1 ( n ( λ 1 ) 1 ) h 1 ,
Φ 1 λ 2 = 2 π λ 2 ( n ( λ 2 ) 1 ) h 1
h 1 ( x , y ) = Φ 1 λ 1 ( x , y ) + 2 P ( x , y ) π 2 π ( n ( λ 1 ) 1 ) λ 1 Φ 1 λ 2 ( x , y ) + 2 Q ( x , y ) π 2 π ( n ( λ 2 ) 1 ) λ 2
R E = m = 1 N n = 1 N | | r λ 1 ( m , n ) | | o λ 1 ' ( m , n ) | | 2 + m = 1 N n = 1 N | | r λ 2 ( m , n ) | | o λ 2 ' ( m , n ) | | 2 m = 1 N n = 1 N | o λ 1 ' ( m , n ) | 2 + m = 1 N n = 1 N | o λ 2 ' ( m , n ) | 2 ,

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