## Optical stream-cipher-like system for image encryption based on Michelson interferometer |

Optics Express, Vol. 19, Issue 3, pp. 2634-2642 (2011)

http://dx.doi.org/10.1364/OE.19.002634

Acrobat PDF (1231 KB)

### Abstract

A novel optical image encryption scheme based on interference is proposed. The original image is digitally encoded into one phase-only mask by employing an improved Gerchberg-Saxton phase retrieval algorithm together with another predefined random phase mask which serves as the encryption key. The decryption process can be implemented optically based on Michelson interferometer by using the same key. The scheme can be regarded as a stream-cipher-like encryption system, the encryption and decryption keys are the same, however the operations are different. The position coordinates and light wavelength can also be used as additional keys during the decryption. Numerical simulations have demonstrated the validity and robustness of the proposed method.

© 2011 Optical Society of America

## 1. Introduction

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

2. D. S. Monaghan, U. Gopinathan, T. J. Naughton, and J. T. Sheridan, “Key-space analysis of double random phase encryption technique,” Appl. Opt. **46**, 6641–6647 (2007). [CrossRef] [PubMed]

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

5. 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**, 1644–1646 (2005). [CrossRef] [PubMed]

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

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

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

8. Y. Zhang, CH. Zheng, and N. Tanno, “Optical encryption based on iterative fractional Fourier transform,” Opt. Commun. **202**, 277–285 (2002). [CrossRef]

9. X. Wang and D. Zhao, “Image encryption based on anamorphic fractional Fourier transform and three-step phase-shifting interferometry,” Opt. Commun. **268**, 240–244 (2006). [CrossRef]

13. Z. Liu and S. Liu, “Randomization of the Fourier transform,” Opt. Lett. **32**, 478–480 (2007). [CrossRef] [PubMed]

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

19. Z. Liu, Q. Li, J. Dai, X. Sun, S. Liu, and M. A. Ahmad, “A new kind of double image encryption by using a cutting spectrum in the 1-D fractional Fourier transform domains,” Opt. Commun. **282**, 1536–1540 (2009). [CrossRef]

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

21. M. Madjarova, M. Kakuta, M. Yamaguchi, and N. Ohyama, “Optical implementation of the stream cipher based on the irreversible cellular automata algorithm,” Opt. Lett. **22**, 1624–1626 (1997). [CrossRef]

25. T. Sasaki, H. Togo, J. Tanida, and Y. Ichioka, “Stream cipher based on pseudorandom number generation with optical affine transformation,” Appl. Opt. **39**, 2340–2346 (2000). [CrossRef]

## 2. Encryption and decryption

### 2.1. Encryption algorithm

*f*(

*x,y*) and the given random phase distribution POM1, we digitally determine the phase distribution of POM2.

*ϕ*

_{1}(

*x*

_{1},

*y*

_{1}) and

*ϕ*

_{2}(

*x*

_{2},

*y*

_{2}) denote the phase distributions of POM1 and POM2,

*f*(

*x,y*) exp [i

*φ*(

*x,y*)] denotes the complex amplitude distribution at the output plane, where

*f*(

*x,y*) is the original image and

*φ*(

*x,y*) is an arbitrary phase, respectively. Now, we pay attention to the mathematical description computing the two masks POM1 and POM2.

*f*(

*x,y*) is the magnitude distribution at the output plane, which is proportional to the square root of the intensity of the image as The complex field of the interference beam at the output plane is

*f*

_{1}(

*x, y*) exp[i

*φ*

_{1}(

*x, y*)] and

*f*

_{2}(

*x, y*) exp[i

*φ*

_{2}(

*x, y*)] are the Fresnel domain functions of POM1 and POM2, respectively. The symbol ‘⊗’ expresses the convolution operation. The point pulse function

*h*(

*x,y,l*) is equal to where

*l*is the distance of the center points of object planes and image plane,

*λ*is the wavelength of the incident light, respectively. Therefore where the symbols ‘ℱ’ and ‘ℱ

^{−1}’ express the Fourier transform and the inverse Fourier transform, respectively. The phase function of POM1 is created absolutely independent of the primary image by random number generator on a computer. Thereafter, the original image is encoded into corresponding POM2 only. The advantage of our method is obvious that mask POM1 can be used to encrypted different images and more degrees of freedom of the optical system are used. In Eq. (2), the magnitude

*f*(

*x,y*) and the complex amplitude

*f*

_{1}(

*x, y*) exp[i

*φ*

_{1}(x, y)] are known. The numerical solution of the mask POM2, exp[i

*ϕ*

_{2}(x

_{2}, y

_{2})] can be determined by employing a phase retrieval algorithm via Eq. (4).

27. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. **21**, 2758–2769 (1982). [CrossRef] [PubMed]

*f*

_{1}(

*x, y*) exp[i

*φ*

_{1}(

*x, y*)]) in the output function.

*ϕ*

_{1}(

*x*

_{1},

*y*

_{1}) and

*φ*(

*x,y*) can be generated between [0,

*π*] randomly,

*f*(

*x,y*) is the magnitude of the original image. Other parameters of optical system will be pre-defined. Eq. (5) show the

*k*th iteration loop of the phase retrieval process (for the sake of simplicity, the coordinates notations are neglected).

*ϕ*

_{2}(x

_{2}, y

_{2})], a magnitude

*g*(

*x*

_{2},

*y*

_{2}) is acquired in every iteration. (c) Remove the modulus of the previous result,

*i.e.*let

*g*

^{(k)}(

*x*

_{2},

*y*

_{2}) = 1, and compute the phase function only to obtain an estimate of POM2, then Fresnel transform this phase only estimate

*φ*

^{(k+1)}(

*x,y*)] after a simple addition (Eq. (5d)) which is corresponding to the optical interference process and replace the modulus

*f*

^{(k+1)}(

*x,y*) with the image modulus

*f*(

*x,y*). The next (

*k*+ 1)-th loop is then carried out to form a new estimate of POM2. This phase retrieval algorithm will give an approximate phase distribution

*g*(

*x*

_{2},

*y*

_{2}) → 1. The convergence of our algorithm can be proved using the method similar to [26] or [27

27. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. **21**, 2758–2769 (1982). [CrossRef] [PubMed]

*f*

_{1}(

*x, y*) exp[i

*φ*

_{1}(

*x, y*)] in our algorithm would not disturb the convergence property. Finally, the original image is encrypted into the mask POM2 digitally. In the following decryption process, the encrypted information of POM2 can be retrieved by interfering it with the POM1 optically using a Michelson interferometer.

### 2.2. Decryption process

*l*

_{1}and

*l*

_{2}, respectively. The key mask POM1 is a random phase image statistically used as the key, the other mask POM2 is the encoding mask which contains the entire information of the encryption image. A coherent plane wave is incident on a beam splitter(BS) with the energy ratio 50% : 50%. Then the two divided beams are modulated by masks POM1 and POM2, respectively. After the return beams are combined by the BS, the interference pattern can be obtained at the output plane. This Michelson interferometer system finally achieve the interference decoding progress optically.

*a*into a ciphertext

*c*via the Exclusive OR (XOR) operation with a key steam

*b*, i.e.

*c*=

*a*⊕

*b*. Performing another XOR operation on the ciphertext

*c*with the same key stream

*b*will retrieve the original information

*a*(

*a*=

*c*⊕

*b*). Such stream cipher systems can be implemented optically, with several random key bit stream generator achieved through optical XOR operations [22

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

21. M. Madjarova, M. Kakuta, M. Yamaguchi, and N. Ohyama, “Optical implementation of the stream cipher based on the irreversible cellular automata algorithm,” Opt. Lett. **22**, 1624–1626 (1997). [CrossRef]

25. T. Sasaki, H. Togo, J. Tanida, and Y. Ichioka, “Stream cipher based on pseudorandom number generation with optical affine transformation,” Appl. Opt. **39**, 2340–2346 (2000). [CrossRef]

## 3. Numerical simulations

*l*

_{1}= 10cm and

*l*

_{2}= 12cm, respectively. The reconstructed result with the correct masks and right parameters of the system is displayed in Fig. 4(b). The phases distribution of mask POM1 and POM2 are simultaneously shown in Fig. 4(c) and 4(d). It is clear that the initial image has been encrypted into the mask POM2, in which the phase distribution has the same statistically independent qualities as white noise.

*N*×

*N*is the size of the image.

*f*(

*m, n*) and

*f*

^{(k)}(

*m, n*) denote the amplitude value of the original and retrieved image at the position (

*m, n*), respectively. The convergence speed of the phase retrieval algorithm is shown in Table 1. In the begin stage, the NMSE curve drops sharply. Whereafter the algorithm converges slowly and tends to be steady, which means that the algorithm has a satisfying convergence property and the almost entire information of the original image can be retrieved. The decrypted image of Fig. 4(b) is recovered from the mask POM2 with 100 retrieval times, which will be used in the following analysis.

*l*

_{2}and

*λ*are the correct values, the NMSE can reach to a small difference compared to right value. Otherwise a white noise image will be received at the output plane. As shown in Fig. 5, the sensitivity of distance and wavelength are about 0.1

*μ*m and 1nm, respectively. The high sensitivity will cause great difficulty in copying the decryption system.

## 4. Conclusion

## Acknowledgments

## References and links

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

2. | D. S. Monaghan, U. Gopinathan, T. J. Naughton, and J. T. Sheridan, “Key-space analysis of double random phase encryption technique,” Appl. Opt. |

3. | B. Javidi, G. Zhang, and J. Li, “Encrypted optical memory using double-random phase encoding,” Opt. Eng. |

4. | O. Matoba and B. Javidi, “Encrypted optical memory system using three-dimensional keys in the Fresnel domain” Opt. Lett. |

5. | 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. |

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

7. | S. Liu, Q. Mi, and B. Zhu, “Optical image encryption with multistage and multichannel fractional Fourier-domain filtering,” Opt. Lett. |

8. | Y. Zhang, CH. Zheng, and N. Tanno, “Optical encryption based on iterative fractional Fourier transform,” Opt. Commun. |

9. | X. Wang and D. Zhao, “Image encryption based on anamorphic fractional Fourier transform and three-step phase-shifting interferometry,” Opt. Commun. |

10. | Z. Liu and S. Liu, “Random fractional Fourier transform,” Opt. Lett. |

11. | L. Chen and D. Zhao, “Optical image encryption with Hartley transforms,” Opt. Lett. |

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

13. | Z. Liu and S. Liu, “Randomization of the Fourier transform,” Opt. Lett. |

14. | Z. Liu and S. Liu, “Double image encryption based on iterative fractional Fourier transform,” Opt. Commun. |

15. | G. Situ and J. Zhang, “Multiple-image encryption by wavelength multiplexing,” Opt. Lett. |

16. | H. Li and Y. Wang, “Double-image encryption based on iterative gyrator transform,” Opt. Commun. |

17. | M. Z. He, L. Z. Cai, Q. Liu, X. C. Wang, and X. F. Meng, “Multiple image encryption and watermarking by random phase matching,” Opt. Commun. |

18. | Z. Liu, J. Dai, X. Sun, and S. Liu, “Triple image encryption scheme in fractional Fourier transform domains,” Opt. Commun. |

19. | Z. Liu, Q. Li, J. Dai, X. Sun, S. Liu, and M. A. Ahmad, “A new kind of double image encryption by using a cutting spectrum in the 1-D fractional Fourier transform domains,” Opt. Commun. |

20. | Y. Zhang and B. Wang, “Optical image encryption based on interference,” Opt. Lett. |

21. | M. Madjarova, M. Kakuta, M. Yamaguchi, and N. Ohyama, “Optical implementation of the stream cipher based on the irreversible cellular automata algorithm,” Opt. Lett. |

22. | J. W. Han, C. S. Park, D. H. Ryu, and E. S. Kim, “Optical image encryption based on XOR operations,” Opt. Eng. |

23. | J. W. Han, S. Lee, and E. S. Kim, “Optical key bit stream generator,” Opt. Eng. |

24. | S. Zhang and R. Karim, “High-security optical integrated stream ciphers,” Opt. Eng. |

25. | T. Sasaki, H. Togo, J. Tanida, and Y. Ichioka, “Stream cipher based on pseudorandom number generation with optical affine transformation,” Appl. Opt. |

26. | R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik |

27. | J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. |

**OCIS Codes**

(100.0100) Image processing : Image processing

(100.4998) Image processing : Pattern recognition, optical security and encryption

**ToC Category:**

Image Processing

**History**

Original Manuscript: November 30, 2010

Revised Manuscript: January 14, 2011

Manuscript Accepted: January 23, 2011

Published: January 27, 2011

**Citation**

Bing Yang, Zhengjun Liu, Bo Wang, Yan Zhang, and Shutian Liu, "Optical stream-cipher-like system for image encryption based on Michelson
interferometer," Opt. Express **19**, 2634-2642 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-3-2634

Sort: Year | Journal | Reset

### References

- P. Refregier, and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20, 767–769 (1995). [CrossRef] [PubMed]
- D. S. Monaghan, U. Gopinathan, T. J. Naughton, and J. T. Sheridan, “Key-space analysis of double random phase encryption technique,” Appl. Opt. 46, 6641–6647 (2007). [CrossRef] [PubMed]
- B. Javidi, G. Zhang, and J. Li, “Encrypted optical memory using double-random phase encoding,” Opt. Eng. 36, 1054–1058 (1997).
- O. Matoba, and B. Javidi, “Encrypted optical memory system using three-dimensional keys in the Fresnel domain,” Opt. Lett. 24, 762–764 (1999). [CrossRef]
- 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, 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, 1044–1046 (2006). [CrossRef] [PubMed]
- S. Liu, Q. Mi, and B. Zhu, “Optical image encryption with multistage and multichannel fractional Fourier-domain filtering,” Opt. Lett. 26, 1242–1244 (2001). [CrossRef]
- Y. Zhang, C. H. Zheng, and N. Tanno, “Optical encryption based on iterative fractional Fourier transform,” Opt. Commun. 202, 277–285 (2002). [CrossRef]
- X. Wang, and D. Zhao, “Image encryption based on anamorphic fractional Fourier transform and three-step phase-shifting interferometry,” Opt. Commun. 268, 240–244 (2006). [CrossRef]
- Z. Liu, and S. Liu, “Random fractional Fourier transform,” Opt. Lett. 32, 2088–2090 (2007). [CrossRef] [PubMed]
- L. Chen, and D. Zhao, “Optical image encryption with Hartley transforms,” Opt. Lett. 31, 3438–3440 (2006). [CrossRef] [PubMed]
- H. Kim, D. H. Kim, and Y. Lee, “Encryption of digital hologram of 3-d object by virtual optics,” Opt. Express 12,Q14912–4921 (2004). [CrossRef] [PubMed]
- Z. Liu, and S. Liu, “Randomization of the Fourier transform,” Opt. Lett. 32, 478–480 (2007). [CrossRef] [PubMed]
- Z. Liu, and S. Liu, “Double image encryption based on iterative fractional Fourier transform,” Opt. Commun. 275, 324–329 (2007). [CrossRef]
- G. Situ, and J. Zhang, “Multiple-image encryption by wavelength multiplexing,” Opt. Lett. 30, 1306–1308 (2005). [CrossRef] [PubMed]
- H. Li, and Y. Wang, “Double-image encryption based on iterative gyrator transform,” Opt. Commun. 281, 5745–5749 (2008). [CrossRef]
- M. Z. He, L. Z. Cai, Q. Liu, X. C. Wang, and X. F. Meng, “Multiple image encryption and watermarking by random phase matching,” Opt. Commun. 247, 29–37 (2005). [CrossRef]
- Z. Liu, J. Dai, X. Sun, and S. Liu, “Triple image encryption scheme in fractional Fourier transform domains,” Opt. Commun. 282, 518–522 (2009). [CrossRef]
- Z. Liu, Q. Li, J. Dai, X. Sun, S. Liu, and M. A. Ahmad, “A new kind of double image encryption by using a cutting spectrum in the 1-D fractional Fourier transform domains,” Opt. Commun. 282, 1536–1540 (2009). [CrossRef]
- Y. Zhang, and B. Wang, “Optical image encryption based on interference,” Opt. Lett. 33, 2443–2445 (2008). [CrossRef] [PubMed]
- M. Madjarova, M. Kakuta, M. Yamaguchi, and N. Ohyama, “Optical implementation of the stream cipher based on the irreversible cellular automata algorithm,” Opt. Lett. 22, 1624–1626 (1997). [CrossRef]
- J. W. Han, C. S. Park, D. H. Ryu, and E. S. Kim, “Optical image encryption based on XOR operations,” Opt. Eng. 38, 47–54 (1999). [CrossRef]
- J. W. Han, S. Lee, and E. S. Kim, “Optical key bit stream generator,” Opt. Eng. 38, 33–38 (1999). [CrossRef]
- S. Zhang, and R. Karim, “High-security optical integrated stream ciphers,” Opt. Eng. 38, 20–24 (1999). [CrossRef]
- T. Sasaki, H. Togo, J. Tanida, and Y. Ichioka, “Stream cipher based on pseudorandom number generation with optical affine transformation,” Appl. Opt. 39, 2340–2346 (2000). [CrossRef]
- R. W. Gerchberg, and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–246 (1972).
- J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982). [CrossRef] [PubMed]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.