## Cryptographic scheme using genetic algorithm and optical responses of periodic structures |

Optics Express, Vol. 19, Issue 9, pp. 8187-8199 (2011)

http://dx.doi.org/10.1364/OE.19.008187

Acrobat PDF (1578 KB)

### Abstract

This study employed the optical responses of periodic structures, multiple-variable functions with sufficient complexity, to develop a cryptographic scheme. The characteristics of structures could be delivered easily with the ciphertext, a series of numbers containing plaintext messages. Two optimization methods utilizing a genetic algorithm were adopted to generate the periodic structure profile as a critical encryption/decryption key. The robustness of methods was further confirmed under various limits. The ciphertext could only be decrypted by referring to the codebook after acquiring the pre-determined optical response. The confidentiality and large capacity of the scheme revealed the enhanced coding strategies here while the success of the scheme was demonstrated with the delivery of an example message.

© 2011 OSA

## 1. Introduction

10. Y.-B. Chen and J. S. Chen, “Cryptosystem for plaintext messages utilizing optical properties of gratings,” Appl. Opt. **49**(11), 2041–2046 (2010). [CrossRef] [PubMed]

## 2. Scheme components

### 2.1 One-Dimensional periodic structures

*d*, lateral filling ratio of the strip

*f*, groove width

*w*, and grating period Λ. The ridges extend infinitely along the

*y*-direction and the plane of incidence is the

*x*-

*z*plane, determined by the incidence and the substrate normal. The space is vertically divided into three regions, and Region II is the grating region composed of ridges and grooves. Region I contains the incidence and reflected diffractions, while transmitted diffractions are in Region III. Region III is assigned either the free space or an opaque substrate of the same material as grating ridges without losing generality. Note that the material in Region III can be different from the two aforementioned cases. The

*z*-axis is parallel to the normal of the gratings, whose periodicity is along the

*x*-axis. The azimuthal angle is fixed such that the angle of incidence is simply defined with the polar angle θ, the angle between the

*z*-axis and the incident plane wave. The Optical responses will be studied with two incidence polarizations: the transverse magnetic (TM) mode and transverse electric (TE) mode.

### 2*.*2 Optical responses of periodic structures

10. Y.-B. Chen and J. S. Chen, “Cryptosystem for plaintext messages utilizing optical properties of gratings,” Appl. Opt. **49**(11), 2041–2046 (2010). [CrossRef] [PubMed]

_{2}) are selected as the representative metals and dielectric for demonstrating the scheme here. The tabulated data in [18] with appropriate interpolation was adopted for Al and SiO

_{2}. On the other hand, the Ag dielectric function uses the Drude model below [10

10. Y.-B. Chen and J. S. Chen, “Cryptosystem for plaintext messages utilizing optical properties of gratings,” Appl. Opt. **49**(11), 2041–2046 (2010). [CrossRef] [PubMed]

*, and γ denote the angular frequency, plasma frequency, and damping constant, respectively. The parameter ε*

_{p}_{∞}becomes one in the high frequency range while ω

*= 1.29×10*

_{p}^{16}rad/s and γ = 1.14×10

^{14}rad/s [10

**49**(11), 2041–2046 (2010). [CrossRef] [PubMed]

_{∞}, ω, ω

*, and γ for a virtual material in the numerical implementation of the proposed cryptographic scheme.*

_{p}19. Y. J. Shen, Q. Z. Zhu, and Z. M. Zhang, “A scatterometer for measuring the bidirectional reflectance and transmittance of semiconductor wafers with rough surfaces,” Rev. Sci. Instrum. **74**(11), 4885–4892 (2003). [CrossRef]

*R*

_{0}) and will be called directional reflectance hereafter. This directional reflectance is the intensity ratio of the reflected light to that of the incidence light at the same polar angle. For example,

*S*

_{1}/

*S*

_{in}is

_{1}and the angle-dependent

17. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. **71**(7), 811–818 (1981). [CrossRef]

### 2.3 Mutually-Agreed rules

_{1}), 660 nm (λ

_{2}), and 785 nm (λ

_{3}) from three commonly used laser diodes [20

20. R. Katayama and Y. Komatsu, “Blue/DVD/CD compatible optical head,” Appl. Opt. **47**(22), 4045–4054 (2008). [CrossRef] [PubMed]

### 2.4 Optimization methods

21. H. C. Cheng and Y. L. Lo, “Arbitrary strain distribution measurement using a genetic algorithm approach and two fiber Bragg grating intensity spectra,” Opt. Commun. **239**(4-6), 323–332 (2004). [CrossRef]

22. E. G. Johnson and M. A. G. Abushagur, “Microgenetic-algorithm optimization methods applied to dielectric gratings,” J. Opt. Soc. Am. A **12**(5), 1152–1160 (1995). [CrossRef]

*p*

_{c}) for randomly-selected vectors from the pool of parents. Taking

*d*, and

*f*. They may correspond to the

*x*,

*y*, and

*z*coordinates as shown in Fig. 2(b). For example, each range is divided into three, five, and four parts equally such that a total of 4×6×5 = 120 grids are formed. This way, the grids are uniformly distributed within the available space and each grid represents a unique periodic structure. The next step is to evaluate the fitness of those structures represented by all grids and then map the results with respect to dimensions. If any result satisfies the criterion, the appropriate structure (Key_2) is found and no further steps are necessary. Otherwise, the minima shown on the map locate the greatest possible dimension ranges and these reduced ranges can be employed in the following RVGA. Certainly the computational cost increases greatly with finer grids. Both RVGA and the hybrid method will be utilized later, but the discussion of the best grid spacing will not be further investigated for clarity.

## 3. Working principles

### 3.1 Encryption and decryption

*E*) is suggested below:where

**49**(11), 2041–2046 (2010). [CrossRef] [PubMed]

### 3.2 Generation robustness of Key_2

*d*are constrained within 150 nm and 4500 nm while their resolution is 1 nm. On the other hand,

*f*is allowed to vary from 0 to 1 with a resolution of three digits after the decimal. Such a high resolution is not a problem with numerical modeling and it can be lowered in real situations without losing the generality the proposed scheme.

*d*= 605 nm, and

*f*= 0.117. Actually, if any of dimensions is 5% enlarged, reflectance spectra from those gratings of modified profiles still satisfy the assigned ranges (not shown here). The satisfaction facilitates the usage of proposed scheme in reality by allowing microfabrication tolerance. Either mentioned grating above is able to serve as Key_2 and contains the information required in the sender’s signature. Since the fitness function is less than 0.01, its optical response is almost the same as the ideal values and thus the robustness of the optimization methods are confirmed. The claim can be further supported with spectra at other wavelengths and associated gratings, such as λ = 660 nm (Λ = 779 nm,

*d*= 834 nm, and

*f*= 0.124) and λ = 785 nm (Λ = 959 nm,

*d*= 177 nm, and

*f*= 0.155) as shown in Fig. 4(a). Ag gratings can also function well at the TE wave incidence of either wavelength as shown in Fig. 4(b). Structural dimensions and detail of the fitness function are also tabulated within the figure. When the material is changed or another optical response is adopted, the optimization methods can still work well in searching for a satisfactory structural dimension. A good proof is the directional reflectance from the Al gratings at the TM wave incidence shown in Fig. 4(c) shows and the directional transmittance spectra through SiO

_{2}periodic slits at the TE wave incidence shown in Fig. 4(d). In short, the generation of Key_2 can be efficient and robust with a proposed optimization method regardless of incident wavelength, polarization, material, and selected optical responses.

*is the angle of incidence of*

_{j}*j*th diffraction wave and

*j*is the diffraction order. Wood’s anomaly occurs at θ

*= ±90° for any*

_{j}*j*th diffraction order while the magnitude of the anomaly becomes trivial for large values of

*j*. The peaks in spectra of Fig. 4(a) and 4(b) mostly correlate with the anomaly that takes place at

*j*= −2. The peaks and dip near θ = 45° for λ = 405 nm in Fig. 4(c) are due to

*j*= −1. On the other hand, the spectra in Fig. 4(d) are severely deteriorated with oscillations although their optical responses are satisfactory. Note that

*T*

_{0}is almost zero through the transparent slits at θ = 25° because other transmitted diffraction waves dominate. In fact, the efficiency summation of all transmitted diffraction waves (hemispherical transmittance,

*T*

_{hem}) is larger than 0.9 as specified in the figure. However, both the +1 and −1 order transmitted diffractions have the diffraction efficiency near to 0.4.

### 3.3 Comparison between two optimization methods

*d*,

*f*, and Λ),

*f*is fixed to 0.1 here for the sake of simplicity. Ranges of both

*d*and Λ are maintained between 150 nm and 4500 nm while each are divided into 30 equally spaced grids. Al is chosen as the structural material and the wavelength of the TM wave incidence is 405 nm. Figure 5(a) shows the map of the fitness function values with respect to the dimensions

*d*and Λ. The values are interpreted between those grids and the minimum is 0.1924 (

*d*= 600 nm and Λ = 450 nm) among the 900 grids. The next part of the method is to employ the RVGA for determining the dimensions of a smaller area in the map which is marked with a rectangle. The range of

*d*is thus between 450 nm and 1050 nm while the range of Λ is between 150 nm and 600 nm. This way, the area of possible dimensions is less than 2% from the initial one, making the RVGA more efficient.

*E*

_{min}average for five trials is 0.1130 when using Method I and 0.0829 when using Method II. This can be attributed to the nature of GA such that Method I attempts to explore the entire area continuously, even those areas with unfavorable fitness. In comparison, Method II is much more precise and its exploitation is within a more narrow or favorable range. Though Method II could be better, it should be noted that the choice of grid size is crucial. The size should be determined according to the map of function values. In summary, both methods can be employed interchangeably in the scheme dependent on the case.

## 4. Example demonstration and enhanced coding strategies

**49**(11), 2041–2046 (2010). [CrossRef] [PubMed]

## 5. Conclusions

## Acknowledgments

## References and links

1. | A. J. Menezes, P. C. Van Oorschot, and S. A. Vanstone, |

2. | R. L. Rivest, A. Shamir, and L. Adleman, “Method for obtaining digital signatures and public-key cryptosystems,” Commun. ACM |

3. | C. T. Clelland, V. Risca, and C. Bancroft, “Hiding messages in DNA microdots,” Nature |

4. | N. Gisin, G. G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. |

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

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

7. | W. Chen, X. D. Chen, and C. J. R. Sheppard, “Optical image encryption based on diffractive imaging,” Opt. Lett. |

8. | R. Pappu, B. Recht, J. Taylor, and N. Gershenfeld, “Physical one-way functions,” Science |

9. | M. Zhou, S. D. Chang, and C. P. Grover, “Cryptography based on the absorption/emission features of multicolor semiconductor nanocrystal quantum dots,” Opt. Express |

10. | Y.-B. Chen and J. S. Chen, “Cryptosystem for plaintext messages utilizing optical properties of gratings,” Appl. Opt. |

11. | A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature |

12. | D. E. Goldberg, |

13. | S. S. Rao, |

14. | E. G. Loewen and E. Popov, |

15. | F. P. Incropera, D. P. DeWitt, T. L. Bergman, and A. S. Lavain, |

16. | B. J. Lee, Y.-B. Chen, and Z. M. Zhang, “Transmission enhancement through nanoscale metallic slit arrays from the visible to mid-infrared,” J. Comput. Theor. Nanosci. |

17. | M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. |

18. | E. D. Palik, |

19. | Y. J. Shen, Q. Z. Zhu, and Z. M. Zhang, “A scatterometer for measuring the bidirectional reflectance and transmittance of semiconductor wafers with rough surfaces,” Rev. Sci. Instrum. |

20. | R. Katayama and Y. Komatsu, “Blue/DVD/CD compatible optical head,” Appl. Opt. |

21. | H. C. Cheng and Y. L. Lo, “Arbitrary strain distribution measurement using a genetic algorithm approach and two fiber Bragg grating intensity spectra,” Opt. Commun. |

22. | E. G. Johnson and M. A. G. Abushagur, “Microgenetic-algorithm optimization methods applied to dielectric gratings,” J. Opt. Soc. Am. A |

**OCIS Codes**

(050.2770) Diffraction and gratings : Gratings

(160.4760) Materials : Optical properties

(060.4785) Fiber optics and optical communications : Optical security and encryption

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: January 13, 2011

Revised Manuscript: April 7, 2011

Manuscript Accepted: April 8, 2011

Published: April 14, 2011

**Citation**

Jia-Shiang Chen, Yu-Bin Chen, Pei-feng Hsu, Nghia Nguyen-Huu, and Yu-Lung Lo, "Cryptographic scheme using genetic algorithm and optical responses of periodic structures," Opt. Express **19**, 8187-8199 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-9-8187

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### References

- A. J. Menezes, P. C. Van Oorschot, and S. A. Vanstone, Handbook of Applied Cryptography (CRC Press, 1997).
- R. L. Rivest, A. Shamir, and L. Adleman, “Method for obtaining digital signatures and public-key cryptosystems,” Commun. ACM 21(2), 120–126 (1978). [CrossRef]
- C. T. Clelland, V. Risca, and C. Bancroft, “Hiding messages in DNA microdots,” Nature 399(6736), 533–534 (1999). [CrossRef] [PubMed]
- N. Gisin, G. G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002). [CrossRef]
- 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]
- B. Javidi and T. Nomura, “Securing information by use of digital holography,” Opt. Lett. 25(1), 28–30 (2000). [CrossRef]
- W. Chen, X. D. Chen, and C. J. R. Sheppard, “Optical image encryption based on diffractive imaging,” Opt. Lett. 35(22), 3817–3819 (2010). [CrossRef] [PubMed]
- R. Pappu, B. Recht, J. Taylor, and N. Gershenfeld, “Physical one-way functions,” Science 297(5589), 2026–2030 (2002). [CrossRef] [PubMed]
- M. Zhou, S. D. Chang, and C. P. Grover, “Cryptography based on the absorption/emission features of multicolor semiconductor nanocrystal quantum dots,” Opt. Express 12(13), 2925–2931 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-13-2925 . [CrossRef] [PubMed]
- Y.-B. Chen and J. S. Chen, “Cryptosystem for plaintext messages utilizing optical properties of gratings,” Appl. Opt. 49(11), 2041–2046 (2010). [CrossRef] [PubMed]
- A. Argyris, D. Syvridis, L. Larger, V. Annovazzi-Lodi, P. Colet, I. Fischer, J. Garcia-Ojalvo, C. R. Mirasso, L. Pesquera, and K. A. Shore, “Chaos-based communications at high bit rates using commercial fibre-optic links,” Nature 438 (7066), 343–346 (2005). [CrossRef] [PubMed]
- D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, 1989).
- S. S. Rao, Engineering Optimization: Theory and Practice (John Wiley, 2009).
- E. G. Loewen and E. Popov, Diffraction Gratings and Applications (M. Dekker, 1997).
- F. P. Incropera, D. P. DeWitt, T. L. Bergman, and A. S. Lavain, Fundamentals of Heat and Mass Transfer (John Wiley, 2007).
- B. J. Lee, Y.-B. Chen, and Z. M. Zhang, “Transmission enhancement through nanoscale metallic slit arrays from the visible to mid-infrared,” J. Comput. Theor. Nanosci. 5, 201–213 (2008).
- M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71(7), 811–818 (1981). [CrossRef]
- E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1998).
- Y. J. Shen, Q. Z. Zhu, and Z. M. Zhang, “A scatterometer for measuring the bidirectional reflectance and transmittance of semiconductor wafers with rough surfaces,” Rev. Sci. Instrum. 74(11), 4885–4892 (2003). [CrossRef]
- R. Katayama and Y. Komatsu, “Blue/DVD/CD compatible optical head,” Appl. Opt. 47(22), 4045–4054 (2008). [CrossRef] [PubMed]
- H. C. Cheng and Y. L. Lo, “Arbitrary strain distribution measurement using a genetic algorithm approach and two fiber Bragg grating intensity spectra,” Opt. Commun. 239(4-6), 323–332 (2004). [CrossRef]
- E. G. Johnson and M. A. G. Abushagur, “Microgenetic-algorithm optimization methods applied to dielectric gratings,” J. Opt. Soc. Am. A 12(5), 1152–1160 (1995). [CrossRef]

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