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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 9 — Apr. 25, 2011
  • pp: 8187–8199
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Cryptographic scheme using genetic algorithm and optical responses of periodic structures

Jia-Shiang Chen, Yu-Bin Chen, Pei-feng Hsu, Nghia Nguyen-Huu, and Yu-Lung Lo  »View Author Affiliations


Optics Express, Vol. 19, Issue 9, pp. 8187-8199 (2011)
http://dx.doi.org/10.1364/OE.19.008187


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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

2. Scheme components

2.1 One-Dimensional periodic structures

The simplest type of periodic structures is one-dimensional (1-D) binary gratings, which are shown in Fig. 1
Fig. 1 One-dimensional periodic structures at a plane wave incidence. The optical responses are employed within the proposed cryptographic scheme and the dimensions are determined by the groove depth d, lateral filling ratio f, and grating period Λ, respectively. S 1, S 2, and S 3 are the reflected intensity when the angle of incidence is θ1, θ2, and θ3, respectively.
and employed to demonstrate the capability of the proposed scheme. Other types are also applicable, but they are omitted here for simplicity. The surface profiles of the gratings are identified by the groove depth 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

One critical component of the proposed scheme is the optical responses of periodic structures, whose complexity provides sufficient capacity and security. For periodic structures, these responses become highly anisotropic and depend on individual diffraction efficiency, which should be bi-directionally determined [14

14. E. G. Loewen and E. Popov, Diffraction Gratings and Applications (M. Dekker, 1997).

]. Summations of transmitted and reflected diffraction efficiency are hemispherical transmittance and reflectance, respectively. The energy conservation can determine the absorptance and the Kirchhoff’s law [15

15. F. P. Incropera, D. P. DeWitt, T. L. Bergman, and A. S. Lavain, Fundamentals of Heat and Mass Transfer (John Wiley, 2007).

] can then decide the emittance. All responses are functions of multiple variables and can be classified into three groups in order to play different roles in the proposed scheme.

The first group involves incoming/outgoing light waves, including their orientation, wavelength, and polarization. In this work, the polar angle of incidence (θ) is the core of the ciphertext while the wavelength and polarization serve as an encryption/decryption key (Key_1) for the scheme. The second group comprises structure dimensions, whose slight change may lead to significant variation in optical responses. For example, a binary grating surface profile and its responses vary with d, f, w, and Λ [16

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. 5, 201–213 (2008).

]. These dimensions are employed as a second key (Key_2) and its generation is a quite challenge due to the difficulty in searching for structures with arbitrarily specified optical responses, especially at those cases that without efficient algorithms. In contrast, acquiring quantitative optical responses from periodic structures numerically is much easier thanks to algorithms capable of solving Maxwell’s equations [17

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]

]. The third group is the relative dielectric function ε or optical constants (i.e. the refractive index n and extinction coefficient κ). They can be correlated with each other by ε = (n + iκ)2, where i is the square root of (−1). Since they vary with the material, wavelength, and temperature, this work takes advantages of these as the signature of the ciphertext sender in assuring message authenticity and non-repudiation [1

1. A. J. Menezes, P. C. Van Oorschot, and S. A. Vanstone, Handbook of Applied Cryptography (CRC Press, 1997).

].

For many materials at room temperature, optical constants are complicated functions although tabulated data [18

18. E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1998).

] or numerical models [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]

] are available. Silver (Ag), aluminum (Al), and silicon dioxide (SiO2) are selected as the representative metals and dielectric for demonstrating the scheme here. The tabulated data in [18

18. E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1998).

] with appropriate interpolation was adopted for Al and SiO2. 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]

]:
ε(ω)=εωp2ω2+iωγ
(1)
where ω, ωp, and γ denote the angular frequency, plasma frequency, and damping constant, respectively. The parameter ε becomes one in the high frequency range while ωp = 1.29×1016 rad/s and γ = 1.14×1014 rad/s [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]

]. Note that optical constants of Ag are listed in Ref [18

18. E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1998).

]. as well, but the applicability of the Drude model enlarges the number of allowable users of a given scheme. Users’ signatures can be replaced with different ε, ω, ωp, and γ for a virtual material in the numerical implementation of the proposed cryptographic scheme.

Though optical responses are mainly numerically obtained throughout this work, they can also be measured experimentally [19

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]

]. Figure 1 demonstrates the way of measuring the specular reflectance (R) from a binary grating. R is the reflected zeroth order diffraction efficiency (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 R at θ1 and the angle-dependent R spectrum at wavelength λ can be plotted and used throughout the cryptographic scheme at a later time. On the other hand, the directional transmittance T can be defined and obtained in a similar fashion. Numerically securing the optical responses here is fulfilled using programs based on the RCWA algorithm [17

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]

] because of its high computational efficiency. In RCWA modeling, the electromagnetic fields of either region can be solved and thus yield Poynting vectors and efficiency of each diffraction order.

2.3 Mutually-Agreed rules

Mutually-agreed rules are regulations and inputs acknowledged by both the message sender and receiver. These rules may vary to provide uniqueness to each scheme; however, all users in a scheme should follow the same rules. Three of such rules are emphasized below while others are omitted and are relatively trivial. First, very few polar angles (θ = 5°, 15°, 25°, …, and 85°) are chosen and their difference is no less than 10°. Using few angles simplifies the scheme demonstration and their separation significantly reduces the uncertainty in encryption and decryption both in modeling and experiments. The second rule is the correlation between the code and the target optical response. For example, the digit 0 is assigned when the directional reflectance (R) or transmittance (T) is between 0 and 0.4. Other digits can be assigned similarly such that the target optical response at any polar angle is linked to a digit. The correlation can be refined to enlarge the number of possible digit combinations for numerous characters. The third rule names the utilized wavelengths and their order, which include 405 nm (λ1), 660 nm (λ2), and 785 nm (λ3) 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]

]. Though a single wavelength can work well within the scheme, multiple wavelengths can further strengthen the security of the scheme. This will be explained in more detail later.

2.4 Optimization methods

The operator selection and reproduction of good individuals are conducted with Roulette-wheel selection following the results of their fitness function evaluation. The crossover operator is achieved using a real-valued approach with a crossover rate (p c) for randomly-selected vectors from the pool of parents. Taking u1 and u2 as two randomly-selected vectors, their crossover is below:
u1=αu1+(1α)u2
(2a)
u2=(1α)u1+αu2
(2b)
where u1 and u2 are produced offspring vectors, and α is a random number between 0 and 2. The offspring individuals become closer than parents when α is between 0 and 1, but they separate further when α is between 1 and 2. A jump mutation operator is subsequently introduced to circumvent the risk that the solutions stick at a local optimum. Operator Elitism is then performed to ensure that the most fit candidate solution can be retained through generations [12

12. D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, 1989).

]. Once a new population is generated, a new loop the fitness function of each individual is evaluated as the beginning of a new loop. The process ends when an individual satisfies the criterion or the loop number is too big to bear.

3. Working principles

3.1 Encryption and decryption

The scheme can be divided into encryption and decryption steps as shown in Fig. 3(a)
Fig. 3 Working principles of the proposed cryptographic scheme are demonstrated using the characters “E” and “F.” (a) Encryption process; (b) Decryption process.
and 3(b), respectively. An overview of them will be given in the description below. On the other hand, a detailed process of describing the generation of each key generation and an example will be given later. Both the encryption and decryption require three commonly shared keys and the signature of the message sender. At the encryption stage, a character in the intended plaintext message is encrypted into the ciphertext after experiencing coding and transformation stages. A codebook (Key_3) is employed at the coding stage such that the character is converted into a triplet code. An example of codebook for the English alphabet is given in Fig. 3(a) and will be referred to hereafter. Next, the triplet code is transformed into the ciphertext with the information from Key_1, Key_2, and the sender’s signature. Even for long messages, the ciphertext consists only of serial numbers and is able to be delivered easily and quickly. Though eavesdroppers may copy the ciphertext, they are unable to decrypt them with little information and improper decryption stages. In fact, the ciphertext should experience calculation, transformation, and decoding stages for decryption as shown in Fig. 3(b). Though the decryption step is almost the inverse of encryption, one more stage is added and functions of other stages are not the same as those in encryption. The main difference is the employment of optimization methods and the sequence of using keys.

As shown in Fig. 3(a), an optimization method is necessary only during encryption and this paragraph will discuss its operation. First, the fitness function and the stop criterion should be determined based on the mutually-agreed rules, the target’s optical responses and their ideal values. Their determination should ensure that each code can be represented with at least one angle of incidence. For example, the target optical response is R and the rules state that the digits 0, 1, and 2 are between 0R<0.4, 0.4R<0.7, and 0.7R1.0, respectively. The ideal R at θ = 5°, 25°, and 45° are then assigned respectively as Rideal,1=1.0, Rideal,2=0.0, and Rideal,3=0.55. The fitness function (E) is suggested below:
E=i=13(Rideal,iRi)2
(3)
where R1, R2, and R3 represent the directional reflectance of the found structures at θ = 5°, 25°, and 45°, respectively. The stop criterion is assigned as zero hereafter although such a demanding criterion is unnecessary for the scheme to function. The role of R can be replaced with T easily for transmittance through slits. Next, the dimensions of the periodic structures are generated. Those dimensions together with the information contained within Key_1, the signature of the sender and mutually agreed upon rules, are then placed into programs to determine their optical responses. If any evaluation results in meeting the criterion of the fitness function, the satisfactory Key_2 is found and the searching process stops. Otherwise, the searching process continues with the newly generated population. The robustness of Key_2 will be shown later while two plaintext characters “F” and “G” in Fig. 3 are selected to vividly illustrate the encryption and decryption below.

At the first stage of encryption, characters “F” and “G” are respectively replaced by codes 012 and 020 following the codebook. After identifying the satisfactory R of the satisfactory periodic structure, both triplet codes are transformed into the ciphertext. Code 012 is converted into the ciphertext θ1θ2θ3 while codes 020 is converted into θ1θ3θ4. θ4 is another polar angle that provides the same R range value as θ1. Note that the digit “0” can convert into either θ1 or θ4 because the corresponding value of R is both between 0 and 0.4. Actually, angles can be used interchangeably as long as their correlated optical responses are within the same range. This way, an eavesdropper may become easily confused as the same digit can be represented with different numbers in the ciphertext. In contrast to encryption, the receiver of the ciphertext initiates the decryption by obtaining “R1R2R3” and “R1R3R4” with Key_1, Key_2, and numerical programs. Then, the receiver can decode these angles into the characters “F” and “G” after referring to the mutually agreed upon rules and Key_3.

3.2 Generation robustness of Key_2

Figure 4
Fig. 4 Satisfactory optical responses of suitable structures for the demonstration of the robustness of the optimization methods: (a) The directional reflectance from a Ag grating at the TM wave incidence; (b) The directional reflectance from a Ag grating at the TE wave incidence; (c) The directional reflectance from a Al grating at the TM wave incidence; (d) The directional transmittance through a Ag slit array at the TE wave incidence. The ideal values of optical responses are marked with red solid circles.
shows the directional reflectance from gratings made of Ag and Al. Red solid circles marked in each figure represent ideal values of optical responses while the ranges of each digit are labeled on the right side of figures. Figure 4(a) and 4(b) show the directional reflectance spectra from Ag gratings at the TM wave and TE wave incidences, respectively. Any spectrum can work well and is sufficient for the proposed scheme. For instance, the dashed line in Fig. 4(a) is the spectrum at λ = 405 nm from a grating of Λ = 433 nm, 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 SiO2 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.

Most wavy features in Fig. 4 can be explained with Wood’s anomaly [16

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. 5, 201–213 (2008).

] as the energy redistribution occurs when a diffracted wave propagates at the grazing angle (θ = 90°). The occurrence of Wood’s anomaly can be predicted from the well-known grating equation [14

14. E. G. Loewen and E. Popov, Diffraction Gratings and Applications (M. Dekker, 1997).

]:
sinθj=sinθ+jλΛ
(4)
where θj is the angle of incidence of jth diffraction wave and j is the diffraction order. Wood’s anomaly occurs at θj = ±90° for any jth 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 Rideal,iand Ri are replaced with Tideal,i and Ti in Eq. (3) in Fig. 4(d). Since the oscillation becomes severe at large θ, angles exceeding 50° are not suggested for use. Interestingly, T or 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

4. Example demonstration and enhanced coding strategies

A brief cryptanalysis of the scheme is provided here for the ciphertext-only attack but it is applicable to other attacks also. Assume that Eve overhears the ciphertext without any other information and she can only analyze the frequency of those numbers. Fortunately, the frequency of numbers in the ciphertext does not reflect the frequency of the plaintext characters at all. First, a character is represented with three numbers rather than a single digit. Any number is not unique and can be replaced with other numbers so long as the associated optical response is consistent. Second, the involvement of dummy texts and multiple wavelengths largely reduces the possibility of repeating numbers. In fact, Alice can manipulate the numbers shown in the ciphertext such that the same message can be easily delivered with dissimilar ciphertexts. Moreover, the choice of available angles can be significantly increased if the resolution changes from 10° to 1°. In short, the complexity and aforementioned strategies indeed guarantee the safety of the scheme at the cost of length in the ciphertext.

5. Conclusions

Acknowledgments

Authors appreciate financial supports from the National Science Council (NSC) in Taiwan under grants NSC-99-2120-M-006-001 and NSC-99-2628-E-006-009.

References and links

1.

A. J. Menezes, P. C. Van Oorschot, and S. A. Vanstone, Handbook of Applied Cryptography (CRC Press, 1997).

2.

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]

3.

C. T. Clelland, V. Risca, and C. Bancroft, “Hiding messages in DNA microdots,” Nature 399(6736), 533–534 (1999). [CrossRef] [PubMed]

4.

N. Gisin, G. G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002). [CrossRef]

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.

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

7.

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]

8.

R. Pappu, B. Recht, J. Taylor, and N. Gershenfeld, “Physical one-way functions,” Science 297(5589), 2026–2030 (2002). [CrossRef] [PubMed]

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 12(13), 2925–2931 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-13-2925. [CrossRef] [PubMed]

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]

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 438 (7066), 343–346 (2005). [CrossRef] [PubMed]

12.

D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, 1989).

13.

S. S. Rao, Engineering Optimization: Theory and Practice (John Wiley, 2009).

14.

E. G. Loewen and E. Popov, Diffraction Gratings and Applications (M. Dekker, 1997).

15.

F. P. Incropera, D. P. DeWitt, T. L. Bergman, and A. S. Lavain, Fundamentals of Heat and Mass Transfer (John Wiley, 2007).

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. 5, 201–213 (2008).

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]

18.

E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1998).

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]

20.

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

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]

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

  1. A. J. Menezes, P. C. Van Oorschot, and S. A. Vanstone, Handbook of Applied Cryptography (CRC Press, 1997).
  2. 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]
  3. C. T. Clelland, V. Risca, and C. Bancroft, “Hiding messages in DNA microdots,” Nature 399(6736), 533–534 (1999). [CrossRef] [PubMed]
  4. N. Gisin, G. G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002). [CrossRef]
  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. B. Javidi and T. Nomura, “Securing information by use of digital holography,” Opt. Lett. 25(1), 28–30 (2000). [CrossRef]
  7. 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]
  8. R. Pappu, B. Recht, J. Taylor, and N. Gershenfeld, “Physical one-way functions,” Science 297(5589), 2026–2030 (2002). [CrossRef] [PubMed]
  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 12(13), 2925–2931 (2004), http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-13-2925 . [CrossRef] [PubMed]
  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]
  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 438 (7066), 343–346 (2005). [CrossRef] [PubMed]
  12. D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning (Addison-Wesley, 1989).
  13. S. S. Rao, Engineering Optimization: Theory and Practice (John Wiley, 2009).
  14. E. G. Loewen and E. Popov, Diffraction Gratings and Applications (M. Dekker, 1997).
  15. F. P. Incropera, D. P. DeWitt, T. L. Bergman, and A. S. Lavain, Fundamentals of Heat and Mass Transfer (John Wiley, 2007).
  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. 5, 201–213 (2008).
  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]
  18. E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1998).
  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]
  20. R. Katayama and Y. Komatsu, “Blue/DVD/CD compatible optical head,” Appl. Opt. 47(22), 4045–4054 (2008). [CrossRef] [PubMed]
  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]

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