## Optically-induced-potential-based image encryption |

Optics Express, Vol. 19, Issue 23, pp. 22619-22627 (2011)

http://dx.doi.org/10.1364/OE.19.022619

Acrobat PDF (4930 KB)

### Abstract

We present a technique of nonlinear image encryption by use of virtual optics. The image to be encrypted is superposed on a random intensity image. And this superposed image propagates through a nonlinear medium and a 4-f system with single phase key. The image is encrypted to a stationary white noise. The decryption process is sensitive to the parameters of the encryption system and the phase key in 4-f system. This sensitivity makes attackers hard to access the phase key. In nonlinear medium, optically-induced potentials, which depend on intensity of optical wave, make the superposition principle frustrated. This nonlinearity based on optically induced potentials highly improves the secrecy level of image encryption. Resistance against attacks based on the phase retrieval technique proves that it has the high secrecy level. This nonlinear image encryption based on optically induced potentials is proposed and demonstrated for the first time.

© 2011 OSA

## 1. Introduction

1. C. Barsi and J. W. Fleischer, “Digital reconstruction of optically-induced potentials,” Opt. Express **17**(25), 23338–23343 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-25-23338. [CrossRef] [PubMed]

2. C. Barsi, W. Wan, and J. W. Fleischer, “Imaging through nonlinear media using digital holography,” Nat. Photonics **3**(4), 211–215 (2009). [CrossRef]

3. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. **22**(16), 1268–1270 (1997). [CrossRef] [PubMed]

4. 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**(10), 1414–1416 (2006). [CrossRef] [PubMed]

5. 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]

9. M. He, Q. Tan, L. Cao, Q. He, and G. Jin, “Security enhanced optical encryption system by random phase key and permutation key,” Opt. Express **17**(25), 22462–22473 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-25-22462. [CrossRef] [PubMed]

5. 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]

6. G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption by double-random phase encoding in the fractional Fourier domain,” Opt. Lett. **25**(12), 887–889 (2000). [CrossRef] [PubMed]

7. G. Situ and J. Zhang, “Double random-phase encoding in the Fresnel domain,” Opt. Lett. **29**(14), 1584–1586 (2004). [CrossRef] [PubMed]

10. X. Peng, L. Yu, and L. Cai, “Double-lock for image encryption with virtual optical wavelength,” Opt. Express **10**(1), 41–45 (2002), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-1-41. [PubMed]

12. X. Wang, D. Zhao, F. Jing, and X. Wei, “Information synthesis (complex amplitude addition and subtraction) and encryption with digital holography and virtual optics,” Opt. Express **14**(4), 1476–1486 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-4-1476. [CrossRef] [PubMed]

13. 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]

14. X. Peng, P. Zhang, H. Wei, and B. Yu, “Known-plaintext attack on optical encryption based on double random phase keys,” Opt. Lett. **31**(8), 1044–1046 (2006). [CrossRef] [PubMed]

15. 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), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-16-102532. [CrossRef] [PubMed]

16. W. Liu, G. Yang, and H. Xie, “A hybrid heuristic algorithm to improve known-plaintext attack on Fourier plane encryption,” Opt. Express **17**(16), 13928–13938 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-13928. [CrossRef] [PubMed]

14. X. Peng, P. Zhang, H. Wei, and B. Yu, “Known-plaintext attack on optical encryption based on double random phase keys,” Opt. Lett. **31**(8), 1044–1046 (2006). [CrossRef] [PubMed]

15. 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), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-16-102532. [CrossRef] [PubMed]

## 2. Principle of optically-induced-potential image encryption

*r*(

*x, y*), where

*r*(

*x, y*) is an independent random function. The treated plaintext image is uploaded on the spatial light modulator (SLM) in plane P

_{0}. The SLM is illuminated by a polarized plane wave and the input field propagates through the nonlinear medium. The field exiting out of the nonlinear medium propagates through a 4-f system with single phase key. Two lenses L

_{1}and L

_{2}with same focal length form the 4-f system. Random phase masks (RPM) M is placed in plane P

_{2}. Complex transmittances of the phase mask can be expressed as exp(i2

*πp*(

*x, y*)), where

*p*(

*x, y*) is an independent random function that uniformly distributed in [0, 1

1. C. Barsi and J. W. Fleischer, “Digital reconstruction of optically-induced potentials,” Opt. Express **17**(25), 23338–23343 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-25-23338. [CrossRef] [PubMed]

_{3}is recorded by phase-shifting holography.

1. C. Barsi and J. W. Fleischer, “Digital reconstruction of optically-induced potentials,” Opt. Express **17**(25), 23338–23343 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-25-23338. [CrossRef] [PubMed]

2. C. Barsi, W. Wan, and J. W. Fleischer, “Imaging through nonlinear media using digital holography,” Nat. Photonics **3**(4), 211–215 (2009). [CrossRef]

*ψ*is the slow varying envelope of the optical field,

*k*= 2π/

*λ*,

*λ*is the wavelength in free space,

*n*is the medium’s base refractive index,

*D*and

*N*stand for the diffraction and nonlinear operators, respectively. Photorefractive medium lithium niobate crystal doped with iron Fe whose crystalline

*c*axis is perpendicular to the propagation direction is used as the nonlinear medium. For open-circuit condition and neglecting diffusion effect, refractive index change caused by

*e*-polarized light can be expressed as [17

17. N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals 1. Steady-state,” Ferroelectrics **22**(1), 949–960 (1978). [CrossRef]

18. W. L. She, C. C. Xu, B. Guo, and W. K. Lee, “Formation of photovoltaic bright spatial soliton in photorefractive LiNbO3 crystal by a defocused laser beam induced by a background laser beam,” J. Opt. Soc. Am. B **23**(10), 2121–2126 (2006). [CrossRef]

*I*= |

_{s}*ψ*|

^{2}is the intensity normalized with the background intensity,

*γ*is the effective electro-optic coefficient and

_{eff}*E*is the photovoltaic field. Field evolution in nonlinear medium can be calculated numerically using Fourier split step method [1

_{pv}**17**(25), 23338–23343 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-25-23338. [CrossRef] [PubMed]

2. C. Barsi, W. Wan, and J. W. Fleischer, “Imaging through nonlinear media using digital holography,” Nat. Photonics **3**(4), 211–215 (2009). [CrossRef]

*dz*. In fact, the diffraction process and the nonlinear process happen simultaneously. But when the propagation distance

*dz*is very small, this approximation will have very small error.

_{0}should be as small as possible. And in numerical calculation with Fourier split step method, the propagating field of the nonlinear medium should be padded with zeros to enlarge the calculation window to prevent diffractive light escaping from the calculation window because of the periodic boundary condition in numerical Fourier calculation.

_{3}back-propagates through the system.where IFT denotes inverse Fourier transform. And for nonlinear medium, backward propagation can be calculated by:

_{0}can be obtained by:

_{0}. With a correct intensity key, correct plaintext could be obtained from the input field. During propagation in nonlinear medium, the wavefront of the intensity key and the plaintext would interact with each other. The output field of nonlinear medium in plane P

_{1}is not the sum of individual output field of the intensity key and the plaintext because of this interaction. This property makes attackers hard to retrieve intensity key.

## 3. Results and discussions

_{0}is 10μm*10μm. In this situation, paraxial approximation could be satisfied. The Lena image with 256×256 pixels shown in Fig. 2(a) is used as plaintext. Zeros is padded to extend the size of the image to 512×512 and a random intensity image is superimposed on the plaintext. The length of the photorefractive crystal Fe:LiNbO3 is 20cm and parameters of the crystal are

*E*=−27×10

_{pv}^{5}V/m,

*n*=2.20,

_{e}*γ*=30.9pV/m, at the wavelength of

_{eff}*λ*=514.5nm. The intensity of the treated Lena image is normalized with the background intensity. The optical wave in the crystal is

*e*-polarized. The step length

*dz*is 200μm. After the propagation of the nonlinear crystal, the real-part/image-part of complex-amplitude of the field exiting out of the nonlinear medium in plane P

_{1}is shown in Fig. 2(b)/ (c). The real-part/image part of the ciphertext is shown in Fig. 2(d)/ (e). Decrypted image with correct keys and correct parameters is shown in Fig. 2(f). The correlation coefficient (CC) between the plaintext and the decrypted image is 1.0000 meaning that the image can be decrypted without any distortion. The correlation coefficient is defined as CC(

*x,y*)=cov(

*x,y*)/(σ

*x*σ

*y*), where cov(

*x,y*) is the cross-covariance of two variables and σ

*x*σ

*y*is the product of two variables’ standard deviations.

_{2}and the intensity key in plane P

_{0}are exactly known. In decryption procedure, the parameter of nonlinear crystal should be exactly correct, or the optically-induced potentials would go wrong which makes the decrypted result incorrect. When

*γ*increases by 0.0001%, the decrypted image is shown in Fig. 3(a) . The decrypted image is added with noise. When

_{eff}*γ*increased by 0.0005%, the decrypted image is shown in Fig. 3(b), the useful information is totally blurring. When

_{eff}*γ*increased by 0.001%, the decrypted image is shown in Fig. 3(c), the decrypted image becomes a random noise. The decrypted result is sensitive to nonlinear parameter

_{eff}*γ*.

_{eff}*λ*increases by 0.0001% in linear process while the nonlinear parameters are kept exactly correct, the decrypted image is shown in Fig. 3(d). The decrypted image is added with noise. When

*λ*increased by 0.0005% in linear process, the decrypted image is shown in Fig. 3(e), the useful information is totally blurring. When

*λ*increased by 0.001%, the decrypted image is shown in Fig. 3(f), the decrypted image becomes a random noise. The linear diffraction parameter is also critical to the decrypted result.

_{2}and the intensity key in plane P

_{0}are exactly known, we calculate the dependence of CC on the nonlinear coefficient and wavelength

*λ*in linear diffraction process. We define nonlinear coefficient as:

*δγ/γ*and the dependence of CC on

*δλ/λ*are shown in Fig. 4(a) and Fig. 4(b), respectively. When CC<0.2, the decrypted image cannot be distinguishable. When the deviation of

*γ*is larger than 0.001%, the decrypted image would be undistinguishable. The insets show the zoomed figures. It’s clear that the dependence of CC on nonlinear coefficient

*γ*or linear wavelength

*λ*is highly sensitive.

_{2}. The random noise can be expressed as

*A*exp(i2

_{m}*πq*(

*x, y*)), where

*q*(

*x, y*) is an independent random function that uniformly distributed in [0, 1],

*A*is the amplitude of the noise. A random noise whose amplitude

_{m}*A*is 0.00001% to the amplitude of the phase key (the amplitude of the phase key is 1) is superimposed on the phase key in plane P

_{m}_{2}. The decrypted image with this noise-added phase key is shown in Fig. 5(a) .the decrypted result is added with noise. When the amplitude of the noise increases to 0.0000005,the quality of the decrypted image decreased dramatically(see Fig. 5(b)). When the amplitude of the noise increases to 0.000001, the decrypted image cannot be distinguishable (see Fig. 5(c)). When the amplitude of the noise increases to 0.000005, the decrypted image becomes a noise (see Fig. 5(d)). We also investigate decrypted results in the absence of the nonlinear medium when a random noise is superimposed on the phase key in plane P

_{2}. While the parameter of

*γ*equals zeros (in the absence of nonlinear medium), the encryption system becomes a linear system. Figure 5(e), (f), (g), and (h) respectively show the decrypted images in the absence of the nonlinear medium while random noises with different amplitude of 0.01, 0.05, 0.1, and 0.5 are superimposed on the phase mask M. When the amplitude of the noise increases to 0.1, the decrypted image is still distinguishable (see Fig. 5(g)). The comparison of the decrypted results with noise-added phase key shows that phase encoding in the presence of the nonlinear medium is more sensitive to phase changes than in the absence of the nonlinear medium.

_{eff}14. X. Peng, P. Zhang, H. Wei, and B. Yu, “Known-plaintext attack on optical encryption based on double random phase keys,” Opt. Lett. **31**(8), 1044–1046 (2006). [CrossRef] [PubMed]

**31**(8), 1044–1046 (2006). [CrossRef] [PubMed]

_{1}. In our system, the complex-amplitude of the field in plane P

_{1}is the output field of the nonlinear crystal. The attacker cannot obtain the distribution of complex-amplitude in plane P

_{1}because of the existence of the intensity key and the nonlinear medium. Phase retrieval technique which works in the linear transform systems doesn’t work in nonlinear transform systems. In our nonlinear encryption system, the wavefront would be distorted by the intensity of field. The wavefront of the intensity key and the plaintext interact with each other during propagation in nonlinear medium. Phase retrieval technique to recover the phase key will not work. All the attacks based on phase retrieval technique will fail.

## 4. Conclusion

## Acknowledgment

## References and links

1. | C. Barsi and J. W. Fleischer, “Digital reconstruction of optically-induced potentials,” Opt. Express |

2. | C. Barsi, W. Wan, and J. W. Fleischer, “Imaging through nonlinear media using digital holography,” Nat. Photonics |

3. | I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. |

4. | 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. |

5. | P. Refregier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. |

6. | G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption by double-random phase encoding in the fractional Fourier domain,” Opt. Lett. |

7. | G. Situ and J. Zhang, “Double random-phase encoding in the Fresnel domain,” Opt. Lett. |

8. | X. C. Cheng, L. Z. Cai, Y. R. Wang, X. F. Meng, H. Zhang, X. F. Xu, X. X. Shen, and G. Y. Dong, “Security enhancement of double-random phase encryption by amplitude modulation,” Opt. Lett. |

9. | M. He, Q. Tan, L. Cao, Q. He, and G. Jin, “Security enhanced optical encryption system by random phase key and permutation key,” Opt. Express |

10. | X. Peng, L. Yu, and L. Cai, “Double-lock for image encryption with virtual optical wavelength,” Opt. Express |

11. | H. Kim, D. H. Kim, and Y. Lee, “Encryption of digital hologram of 3-D object by virtual optics,” Opt. Express |

12. | X. Wang, D. Zhao, F. Jing, and X. Wei, “Information synthesis (complex amplitude addition and subtraction) and encryption with digital holography and virtual optics,” Opt. Express |

13. | 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. |

14. | X. Peng, P. Zhang, H. Wei, and B. Yu, “Known-plaintext attack on optical encryption based on double random phase keys,” Opt. Lett. |

15. | Y. Frauel, A. Castro, T. J. Naughton, and B. Javidi, “Resistance of the double random phase encryption against various attacks,” Opt. Express |

16. | W. Liu, G. Yang, and H. Xie, “A hybrid heuristic algorithm to improve known-plaintext attack on Fourier plane encryption,” Opt. Express |

17. | N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals 1. Steady-state,” Ferroelectrics |

18. | W. L. She, C. C. Xu, B. Guo, and W. K. Lee, “Formation of photovoltaic bright spatial soliton in photorefractive LiNbO3 crystal by a defocused laser beam induced by a background laser beam,” J. Opt. Soc. Am. B |

**OCIS Codes**

(070.0070) Fourier optics and signal processing : Fourier optics and signal processing

(070.4340) Fourier optics and signal processing : Nonlinear optical signal processing

(100.0100) Image processing : Image processing

**ToC Category:**

Image Processing

**History**

Original Manuscript: July 26, 2011

Revised Manuscript: October 4, 2011

Manuscript Accepted: October 10, 2011

Published: October 25, 2011

**Citation**

Bing-Chu Chen and He-Zhou Wang, "Optically-induced-potential-based image encryption," Opt. Express **19**, 22619-22627 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-23-22619

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### References

- C. Barsi and J. W. Fleischer, “Digital reconstruction of optically-induced potentials,” Opt. Express17(25), 23338–23343 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-25-23338 . [CrossRef] [PubMed]
- C. Barsi, W. Wan, and J. W. Fleischer, “Imaging through nonlinear media using digital holography,” Nat. Photonics3(4), 211–215 (2009). [CrossRef]
- I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett.22(16), 1268–1270 (1997). [CrossRef] [PubMed]
- 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(10), 1414–1416 (2006). [CrossRef] [PubMed]
- 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]
- G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption by double-random phase encoding in the fractional Fourier domain,” Opt. Lett.25(12), 887–889 (2000). [CrossRef] [PubMed]
- G. Situ and J. Zhang, “Double random-phase encoding in the Fresnel domain,” Opt. Lett.29(14), 1584–1586 (2004). [CrossRef] [PubMed]
- X. C. Cheng, L. Z. Cai, Y. R. Wang, X. F. Meng, H. Zhang, X. F. Xu, X. X. Shen, and G. Y. Dong, “Security enhancement of double-random phase encryption by amplitude modulation,” Opt. Lett.33(14), 1575–1577 (2008). [CrossRef] [PubMed]
- M. He, Q. Tan, L. Cao, Q. He, and G. Jin, “Security enhanced optical encryption system by random phase key and permutation key,” Opt. Express17(25), 22462–22473 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-25-22462 . [CrossRef] [PubMed]
- X. Peng, L. Yu, and L. Cai, “Double-lock for image encryption with virtual optical wavelength,” Opt. Express10(1), 41–45 (2002), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-1-41 . [PubMed]
- H. Kim, D. H. Kim, and Y. Lee, “Encryption of digital hologram of 3-D object by virtual optics,” Opt. Express12(20), 4912–4921 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-20-4912 . [CrossRef] [PubMed]
- X. Wang, D. Zhao, F. Jing, and X. Wei, “Information synthesis (complex amplitude addition and subtraction) and encryption with digital holography and virtual optics,” Opt. Express14(4), 1476–1486 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-4-1476 . [CrossRef] [PubMed]
- 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]
- X. Peng, P. Zhang, H. Wei, and B. Yu, “Known-plaintext attack on optical encryption based on double random phase keys,” Opt. Lett.31(8), 1044–1046 (2006). [CrossRef] [PubMed]
- Y. Frauel, A. Castro, T. J. Naughton, and B. Javidi, “Resistance of the double random phase encryption against various attacks,” Opt. Express15(16), 10253–10265 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-16-102532 . [CrossRef] [PubMed]
- W. Liu, G. Yang, and H. Xie, “A hybrid heuristic algorithm to improve known-plaintext attack on Fourier plane encryption,” Opt. Express17(16), 13928–13938 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-13928 . [CrossRef] [PubMed]
- N. V. Kukhtarev, V. B. Markov, S. G. Odulov, M. S. Soskin, and V. L. Vinetskii, “Holographic storage in electrooptic crystals 1. Steady-state,” Ferroelectrics22(1), 949–960 (1978). [CrossRef]
- W. L. She, C. C. Xu, B. Guo, and W. K. Lee, “Formation of photovoltaic bright spatial soliton in photorefractive LiNbO3 crystal by a defocused laser beam induced by a background laser beam,” J. Opt. Soc. Am. B23(10), 2121–2126 (2006). [CrossRef]

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