## Information hiding based on double random-phase encoding and public-key cryptography

Optics Express, Vol. 17, Issue 5, pp. 3270-3284 (2009)

http://dx.doi.org/10.1364/OE.17.003270

Acrobat PDF (4164 KB)

### Abstract

A novel information hiding method based on double random-phase encoding (DRPE) and Rivest-Shamir-Adleman (RSA) public-key cryptosystem is proposed. In the proposed technique, the inherent diffusion property of DRPE is cleverly utilized to make up the diffusion insufficiency of RSA public-key cryptography, while the RSA cryptosystem is utilized for simultaneous transmission of the cipher text and the two phase-masks, which is not possible under the DRPE technique. This technique combines the complementary advantages of the DPRE and RSA encryption techniques and brings security and convenience for efficient information transmission. Extensive numerical simulation results are presented to verify the performance of the proposed technique.

© 2009 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. G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption by double-random phase encoding in the fractional Fourier domain,” Opt. Lett. **25**, 887–889 (2000). [CrossRef]

7. S. Kishk and B. Javidi, “Information hiding technique with double phase encoding,” Appl. Opt. **41**, 5462–5470 (2002). [CrossRef] [PubMed]

## 2. RSA public-key cryptography

- Select two large prime numbers
*p*and*q*randomly; - Calculate
*n*=*p*×*q*,*φ*(*n*) = (*p*-1)(*q*-1), where*φ*(*n*) is the Euler function of*n*; - Select an integer
*e*, such that 1 <*e*<*φ*(*n*) and gcd(*φ*(*n*),*e*) = 1, where gcd(·) implies that*φ*(*n*) and*e*are prime to each other; - The decryption key
*d*is calculated by*d*·*e*≠ 1 mod*φ*(*n*), where mod denotes modular arithmetic operation; - {
*e*,*n*} denotes the public key, and {*d*,*n*} denotes the private key.

*n*. Then perform the encryption operation on each plaintext group

*m*such that

## 3. The secret image hiding and extraction methods

### 3.1 Phase retrieval algorithm based on DRPE system

*θ*

_{0}(

*x*,

*y*) and

*φ*

_{0}(

*u*,

*v*) are the two phase-functions inserted in the input plane and Fourier plane respectively, and their values are randomly distributed over the interval [0,1]. The transform process from the input image

*f*(

*x*,

*y*) to the output image

*g*(

*x*,

*y*) can be expressed as [1

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]

*FT*and

*FT*

^{-1}denote the Fourier transform and the inverse Fourier transform operations, respectively.

### 3.2 Specific method for information hiding

*f*(

*x*,

*y*) can be transformed into the secret image

*g*(

*x*,

*y*) by the modulation of the two phase-masks

*θ*(

*x*,

*y*) and

*φ*(

*u*,

*v*). Thus, hiding the secret image

*g*(

*x*,

*y*) can be substituted by hiding the input image

*f*(

*x*,

*y*) and the two phase-masks

*θ*(

*x*,

*y*) and

*φ*(

*u*,

*v*) (i.e. the two phase-functions

*θ*

_{0}(

*x*,

*y*) and

*θ*

_{0}(

*u*,

*v*)). The process of the information hiding is shown in Fig. 2, and the specific steps are described below:

*f*(

*x*,

*y*) with only eight gray levels is divided into three bit-planes

*f*

_{1}(

*x*,

*y*),

*f*

_{2}(

*x*,

*y*), and

*f*

_{3}(

*x*,

*y*), respectively. Then the three bit-planes are encrypted by RSA public-key cryptography according to Eq. (1). Thus the three encrypted binary images

*f*'

_{1}(

*x*,

*y*),

*f*'

_{2}(

*x*,

*y*), and

*f*'

_{3}(

*x*,

*y*) are obtained;

*h*(

*x*,

*y*) with same size as the input image. Then replace the lowest three bit-planes of the host image with the three binary images obtained in step 1). As a result, the image

*h*'(

*x*,

*y*) is gotten;

*h*'(

*x*,

*y*) with

*M*×

*N*pixels into the image

*H*(

*x*,

*y*) with 2

*M*× 2

*N*pixels. The enlargement rules are as follows [12

12. X. Zhou and J. G. Chen, “Information hiding based on double random phase encoding technology,” J. Mod. Optics. **53**, 1777–1783 (2006). [CrossRef]

13. H. Zhang, L. Z. Cai, X. F. Meng, X. F. Xu, X. L. Yang, X. X. Shen, and G. Y. Dong, “Image watermarking based on an iterative phase retrieval algorithm and sine-cosine modulation in the discrete-cosine-transform domain,” Opt. Commun. **278**, 257–263 (2007). [CrossRef]

*θ*

_{0}(

*x*,

*y*) and

*φ*

_{0}(

*u*,

*v*) into the enlarged image

*H*(

*x*,

*y*) according to the following equations:

*H*'(.,.) represents the combined image, and

*α*represents a constant superposition parameter.

### 3.3 Extraction and decryption of the hidden information

*A*, given by

*θ*

_{0}(

*x*,

*y*) and the superposition parameter

*α*as

*angle*(·) represents the angle of the argument, and

*abs*(·) represents the modulus of the argument. Using Eqs. (12) and (15), we can obtain the Fourier plane phase-function, expressed as

*h*'(

*x*,

*y*) can also be obtained from the combined image as

*h*', i.e., the three encrypted binary images

*f*' can be extracted by using the RSA algorithm. After these binary images are decrypted by using the private RSA key according to Eq. (2), the input image

*f*(

*x*,

*y*) can be retrieved by assembling the three decrypted bit-planes in order. The input image

*f*(

*x*,

*y*) and the phase-functions

*θ*

_{0}(

*x*,

*y*) and

*φ*

_{0}(

*u*,

*v*) are then introduced in the DRPE system as shown in Fig. 1, in order to generate the secret image

*g*(

*x*,

*y*).

### 3.4 Numerical simulation and discussion

*n*=

*p*×

*q*= 91593 × 77041 = 5515596313 (Actually,

*p*and

*q*are often chosen as big primes which are more than a decimal number 10

^{100}in practical application),

*e*= 1757316971, and

*d*= 2674607171. The phase-masks PM1 and PM2 were obtained by performing 100 iterations of the CIFT algorithm. Here, the value of the parameter

*α*in Eq. (9) was selected as 20. To measure the performance of the proposed technique, the peak signal-to-noise ratio (PSNR) metric [7

7. S. Kishk and B. Javidi, “Information hiding technique with double phase encoding,” Appl. Opt. **41**, 5462–5470 (2002). [CrossRef] [PubMed]

*M*×

*N*is the size of image,

*k*represents the gray level number,

*g*and

*g*' represent the original image and the obtained image, respectively. The PSNR is an indicator of image quality. It is based on the sum of the squared differences between corresponding pixels of two images, and decreases as the difference between

*g*and

*g*' increases. The PSNR of the combined image has been found to be 27.2 dB, and of the decrypted image is 31.2 dB.

*α*affects the PSNR for both the combined and decrypted images in Fig. 3. There is no optimum value for

*α*, but the choice of

*α*depends on following conditions. Firstly, perceptible deterioration cannot be induced to the host image by the superposed phase-masks. Secondly, the secret image can be decrypted by the DRPE system as an acceptable image, because the values of the phase functions must be rounded to the nearest integers before they are added into the digital host image, and their error values aroused by rounding are determined by the value of

*α*. Thirdly, the proposed method should be robust enough to resist some kinds of attacks, such as adding noise, compression, filtering and so on. In this paper, the gray values of the host image are distributed in the interval [0,255], while the values of cos(2

*πθ*

_{0}(

*m*,

*n*)), sin(2

*πθ*

_{0}(

*m*,

*n*)) and

*φ*

_{0}(

*m*,

*n*) in Eq.(9) are distributed in the interval [0,1]. From Fig. 4, a comparably good result will be achieved if the value of

*α*is set to 20.

12. X. Zhou and J. G. Chen, “Information hiding based on double random phase encoding technology,” J. Mod. Optics. **53**, 1777–1783 (2006). [CrossRef]

13. H. Zhang, L. Z. Cai, X. F. Meng, X. F. Xu, X. L. Yang, X. X. Shen, and G. Y. Dong, “Image watermarking based on an iterative phase retrieval algorithm and sine-cosine modulation in the discrete-cosine-transform domain,” Opt. Commun. **278**, 257–263 (2007). [CrossRef]

*f*(

*x*,

*y*) can be regarded as the ciphertext of the secret image

*g*(

*x*,

*y*), thus avoiding the separate delivery of keys in the information transmission process.

12. X. Zhou and J. G. Chen, “Information hiding based on double random phase encoding technology,” J. Mod. Optics. **53**, 1777–1783 (2006). [CrossRef]

13. H. Zhang, L. Z. Cai, X. F. Meng, X. F. Xu, X. L. Yang, X. X. Shen, and G. Y. Dong, “Image watermarking based on an iterative phase retrieval algorithm and sine-cosine modulation in the discrete-cosine-transform domain,” Opt. Commun. **278**, 257–263 (2007). [CrossRef]

**53**, 1777–1783 (2006). [CrossRef]

**278**, 257–263 (2007). [CrossRef]

## 4. Security analysis

### 4.1. Complementary Advantages of DRPE and RSA public-key cryptography

*f*system with two phase-masks has better diffusion quality than that with one phase-mask. It makes the combined image robust to resist occlusion attack. The decrypted image is shown in Fig. 6(e) (using a single phase-mask,

*φ*) when 25% of the combined images’ pixels are occluded. Comparing Figs. 6(d) with 6(e), we can see that the decrypted image using two phase-masks is clearer than that using one. Therefore, we employ two phase-masks to diffuse and confuse information in our method.

### 4.2 Robustness of the Proposed Technique

*α*is set to 20 for all simulations tests conducted.

## 5. Conclusion

## Appendix

4. B. Javidi, A. Sergent, G. Zhang, and L. Guibert, “Fault tolerance properties of a double phase encoding encryption technique,” Opt. Eng. **36**, 992–998 (1997). [CrossRef]

7. S. Kishk and B. Javidi, “Information hiding technique with double phase encoding,” Appl. Opt. **41**, 5462–5470 (2002). [CrossRef] [PubMed]

*f*(

*x*,

*y*)). We occlude 1/2, 3/4, and 7/8 of the input image pixels, and the occluded images are displayed in Figs. A1(a), A1(b), and A1(c), respectively. When the occluded input images and the two phase masks are introduced into the DRPE system as shown in Fig. 1, the corresponding decrypted images are obtained, and shown in Figs. A1(d), A1(e), and A1(f), respectively.

*f*(

*x*,

*y*), the attacker even need not to decode the RSA-encoded input image

*f*'(

*x*,

*y*). Because a binary image only has two kinds gray values (0 and 1), a random binary image, which is arbitrarily chosen, will have about 1/2 of its pixels which have the same gray value as the corresponding pixels of the input random binary image (as shown in Fig. A2(a) for example). Especially, an image, with 0 or 1 gray value for all pixels, has about 1/2 of its pixels which have the same gray value as the corresponding pixels of the input random binary image. For example, Fig. A2(b) is an image with 1 gray value for all pixels. Those pixels with 1 gray value in Fig. A2(a) are the same as the corresponding pixels of Fig. A2(b). The number of pixels with 1 gray value in Fig. A2(a) is about 1/2 of its total pixels, so Fig. A2(b) have about 1/2 of its pixels which are the same as the corresponding pixels of Fig. A2(a). Therefore, when the binary image (Fig. A2(a) for example) is taken as the input image, even he does not know how to decode the RSA-encoded input image, the attacker can decrypt the secret image by arbitrarily choosing a random binary image, or more simply, by choosing an image with 0 or 1 gray value for all pixels as the input image. Fig. A2(c) shows the decrypted image when Fig. A2(b) is used as the input image. So a binary image cannot be used as the input image in this method.

^{n}gray-scales,

*n*=1,2,3⋯. From Fig. A1(f), we can see that 1/8 of the input image pixels can not recover the secret image, so 8 gray levels should be the minimum number for this method. That is to say, in this condition, the secret information will remain safe if the attacker cannot decode the RSA-encoded input image. Therefore, we take an image with only 8 gray levels in this paper as the input image, whose gray values are uniformly distributed from 0 to 7, shown in Fig. A3(a).

*M*×

*M*is the size of image.

*M*×

*M*things taken

*n*at a time.

*a*(

*n*) = 7

^{n}

*C*

^{n}_{65536}, so

*a*(

*n*-1) = 7

^{n-1}

*C*

_{65536}

^{n-1}. Thus, from

*n*< 57345 can be obtained. That is, when

*n*< 57345,

*P*<5.5×10

^{-1640}.

## References and links

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

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

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

4. | B. Javidi, A. Sergent, G. Zhang, and L. Guibert, “Fault tolerance properties of a double phase encoding encryption technique,” Opt. Eng. |

5. | B. Javidi and T. Nomura, “Securing information by use of digital holography,” Opt. Lett. |

6. | X. Zhou, S. Yuan, S. W. Wang, and J. Xie, “Affine cryptosystem of double-random-phase encryption based on the fractional Fourier transform,” Appl. Opt. |

7. | S. Kishk and B. Javidi, “Information hiding technique with double phase encoding,” Appl. Opt. |

8. | J. Rosen and B. Javidi, “Hidden images in halftone pictures,” Appl. Opt. |

9. | Y. S. Shi, G. H. Situ, and J. J. Zhang, “Multiple-image hiding in the Fresnel domain,” Opt. Lett. |

10. | K. T. Kim, J. H. Kim, and E. S. Kim, “Multiple information hiding technique using random sequence and Hadamard matrix,” Opt. Eng. |

11. | J. J. Kim, J. H. Choi, and E. S. Kim, “Optodigital implementation of multiple information hiding and extraction system,” Opt. Eng. |

12. | X. Zhou and J. G. Chen, “Information hiding based on double random phase encoding technology,” J. Mod. Optics. |

13. | H. Zhang, L. Z. Cai, X. F. Meng, X. F. Xu, X. L. Yang, X. X. Shen, and G. Y. Dong, “Image watermarking based on an iterative phase retrieval algorithm and sine-cosine modulation in the discrete-cosine-transform domain,” Opt. Commun. |

14. | D. R. Stinson, |

15. | B. Yang, |

16. | S. Yuan, X. Zhou, D. H. Li, and D. F. Zhou, “Simultaneous transmission for an encrypted image and a double random-phase encryption key,” Appl. Opt. |

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

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

19. | G. H. Situ and J. J. Zhang, “A cascaded iterative Fourier transform algorithm for optical security applications,” Optik. |

20. | G. H. Situ, J. J. Zhang, Y. Zhang, and Z. S. Zhao, “A cascaded-phase retrieval algorithm for optical image encryption,” J. Optoelectron. Laser. |

**OCIS Codes**

(100.2960) Image processing : Image analysis

(100.3010) Image processing : Image reconstruction techniques

(100.5070) Image processing : Phase retrieval

**ToC Category:**

Image Processing

**History**

Original Manuscript: November 12, 2008

Revised Manuscript: December 28, 2008

Manuscript Accepted: January 19, 2009

Published: February 17, 2009

**Citation**

Yuan Sheng, Zhou Xin, Mohammed S. Alam, Lu Xi, and Li Xiao-feng, "Information hiding based on double random-phase encoding and public-key
cryptography," Opt. Express **17**, 3270-3284 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-5-3270

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### 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]
- G. Unnikrishnan, J. Joseph, and K. Singh, "Optical encryption by double-random phase encoding in the fractional Fourier domain," Opt. Lett. 25, 887-889 (2000). [CrossRef]
- O. Matoba and B. Javidi, "Encrypted optical memory system using three-dimensional keys in the Fresnel domain," Opt. Lett. 24, 762-764 (1999). [CrossRef]
- B. Javidi, A. Sergent, G. Zhang, and L. Guibert, "Fault tolerance properties of a double phase encoding encryption technique," Opt. Eng. 36, 992-998 (1997). [CrossRef]
- B. Javidi and T. Nomura, "Securing information by use of digital holography," Opt. Lett. 25, 28-30 (2000). [CrossRef]
- X. Zhou, S. Yuan, S. W. Wang, and J. Xie, "Affine cryptosystem of double-random-phase encryption based on the fractional Fourier transform," Appl. Opt. 45, 8434-8439 (2006). [CrossRef]
- S. Kishk and B. Javidi, "Information hiding technique with double phase encoding," Appl. Opt. 41, 5462-5470 (2002). [CrossRef] [PubMed]
- J. Rosen and B. Javidi, "Hidden images in halftone pictures," Appl. Opt. 40, 3346-3353 (2001). [CrossRef]
- Y. S. Shi, G. H. Situ, and J. J. Zhang, "Multiple-image hiding in the Fresnel domain," Opt. Lett. 32, 1914-1916 (2007). [CrossRef] [PubMed]
- K. T. Kim and J. H. Kim, and E. S. Kim, "Multiple information hiding technique using random sequence and Hadamard matrix," Opt. Eng. 40, 2489-2494, (2001). [CrossRef]
- J. J. Kim, J. H. Choi, and E. S. Kim, "Optodigital implementation of multiple information hiding and extraction system," Opt. Eng. 43, 113-125 (2004). [CrossRef]
- X. Zhou and J. G. Chen, "Information hiding based on double random phase encoding technology," J. Mod. Optics. 53, 1777-1783 (2006). [CrossRef]
- H. Zhang, L. Z. Cai, X. F. Meng, X. F. Xu, X. L. Yang, X. X. Shen, and G. Y. Dong, "Image watermarking based on an iterative phase retrieval algorithm and sine-cosine modulation in the discrete-cosine-transform domain," Opt. Commun. 278, 257-263 (2007). [CrossRef]
- D. R. Stinson, Cryptography: Theory and Practice (CRC Press 2002).
- B. Yang, Modern Cryptography (Tsinghua University Press 2007) (in Chinese).
- S. Yuan, X. Zhou, D. H. Li, and D. F. Zhou, "Simultaneous transmission for an encrypted image and a double random-phase encryption key," Appl. Opt. 46, 3747-3753 (2007). [CrossRef] [PubMed]
- Q1. R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik. 35, 237-246 (1972).
- J. R. Fienup, "Phase retrieval algorithms: a comparison," Appl. Opt. 21, 2758-2769 (1982). [CrossRef] [PubMed]
- Q2. G. H. Situ and J. J. Zhang, "A cascaded iterative Fourier transform algorithm for optical security applications," Optik. 114, 473-477 (2003). [CrossRef]
- Q3. G. H. Situ, J. J. Zhang, Y. Zhang, and Z. S. Zhao, "A cascaded-phase retrieval algorithm for optical image encryption," J. Optoelectron. Laser. 15, 341-343 (2004) (in Chinese).

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