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 Optical Society of America
The fields of cryptography and cryptanalysis catch people’s attention as the importance of information security in business, communications, military affairs, and other numerous applications continues to be a priority . A cryptosystem is typically constructed using complicated and unique mathematical problems; for instance, the factorization problem the RSA algorithm is a one-way trapdoor function popular in cryptosystems . In contrast, researchers recently have developed ways of hiding information by learning from the nature or manipulating the engineering techniques [3–11]. Famous examples include the DNA microdot techniques  and numerous optical techniques [4–11], which can be divided into three categories. Two of them are well-known and one is the quantum cryptography, using the polarization of each photon for absolute security . The other is related to information security with imaging while the double random-phase encoding and digital holography are common ways [5–7]. The third one takes advantages of optical responses from a macro sample and mainly aims at plaintext messages [8–10]. Several interesting cryptographic schemes were constructed with unique scattering patterns from inhomogeneous media , emittance from quantum dots , and reflectance from silver gratings . Though parts of those works might be imperfect, they clearly demonstrate the promising feasibility of optical cryptographic schemes.
This study proposes an alternative cryptographic scheme of the third category using responses of periodic structures and demonstrates the advantages. For one, the applicable optical response within this proposed scheme could be reflectance, transmittance, absorptance, or emittance at any wavelength. In fact, both directional and hemispherical properties are suitable although only directional reflectance and transmittance will be addressed later for conciseness. The capacity and flexibility of this scheme are thus enlarged and its coding techniques are modified accordingly to enhance confidentiality. Secondly, the surface profiles of periodic structures can be sufficiently defined with few dimensions and identically reproduced without ambiguity. Acquiring optical responses with numerical algorithms during encryption and decryption becomes much easier and trustable. Thirdly, systematically generating a critical key is achieved with optimization methods, a much better approach than the general trial-and-error attempted by a previous study . Last but not least, compatible materials discussed here can serve as a signature of a particular user to fulfill imperative functions of an ideal scheme including non-repudiation and message authenticity . It is thus possible to build co-existing but non-interfering networks for multiple parties using the same scheme.
The following illustration will begin with major components of the proposed cryptographic scheme. In addition to the optical responses of periodic structures, this illustration will include mutually-agreed rules and two optimization methods. These rules contain the regulations and inputs for all scheme users. On the other hand, the methods are the genetic algorithm (GA)  and a hybrid method incorporating GA with the grid search method , where the latter one is potential to enhance searching efficiency. Next, working principles of the scheme will be explained with two characters experiencing encryption and decryption steps. A critical key, i.e., a satisfactory periodic structure, will be efficiently generated with programs using rigorous coupled-wave analysis (RCWA) and either of the two methods of optimization. The generation robustness will be performed for various materials of the structure, incidence wavelengths, and target optical responses. In the end, this study will show how a plaintext message delivered with the scheme and provide a short cryptanalysis.
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 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 . 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  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 . 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 . 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 .
For many materials at room temperature, optical constants are complicated functions although tabulated data  or numerical models  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  with appropriate interpolation was adopted for Al and SiO2. On the other hand, the Ag dielectric function uses the Drude model below :10]. Note that optical constants of Ag are listed in Ref . 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 . Figure 1 demonstrates the way of measuring the specular reflectance from a binary grating. 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 at θ1 and the angle-dependent spectrum at wavelength λ can be plotted and used throughout the cryptographic scheme at a later time. On the other hand, the directional transmittance can be defined and obtained in a similar fashion. Numerically securing the optical responses here is fulfilled using programs based on the RCWA algorithm  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 or transmittance 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 . 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
Another important component of the scheme is an optimization method, which generates Key_2 together with RCWA. Two alternative methods are proposed here: the first is the real-valued genetic algorithm (RVGA)  and the other is the hybrid method combining grid search method with the RVGA. The GA was inspired by evolutionary biology and its search processes were performed in a stochastic way to increase its robustness and promise for wide applications . The applications of this also include the design of periodic structures aimed at various other applications .
Figure 2(a) shows the flowchart of a typical GA whose first step is to generate an initial population. For RVGA, each individual in the population should be a vector of three real-valued components determining the profiles of the periodic structures (i.e. ). The subscript i is a positive integer between 1 and N pop, which is the pre-determined population size. All individuals are randomly-generated although the range of each component may be limited, 0 < fi < 1, for example. The second step is to evaluate the fitness function of each individual. The fitness function here is related to the difference between the desired optical response and that of the selected individual. Hence, its ideal result is zero and usually a small positive constant is picked as the stop criterion. If any result after evaluation is lower than the criterion, the optical responses of the found structure are close enough to those of ideal structures. Key_2 is thus found and the whole process stops. Otherwise, four main operators for GAs are necessary to be subsequently performed.
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 and as two randomly-selected vectors, their crossover is below:12]. 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.
The RVGA can also be incorporated with the grid search method  as a hybrid method and an alternative optimization method. The first step of the grid search method is to construct equally-spaced grids in the applicable dimension ranges of the periodic structures, which are Λ, 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
The scheme can be divided into encryption and decryption steps as shown in Fig. 3(a) 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 and the rules state that the digits 0, 1, and 2 are between and respectively. The ideal at θ = 5°, 25°, and 45° are then assigned respectively as and The fitness function (E) is suggested below: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 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 range value as θ1. Note that the digit “0” can convert into either θ1 or θ4 because the corresponding value of 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 “” and “” 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.
The modified encryption/decryption steps in the proposed scheme are superior to those in previous study  in several ways. For one, the difficulty of searching for satisfactory structures is significantly reduced because the employment of one wavelength is enough for the current scheme. Additionally, the employment of single wavelength renders optical responses much easier to be measured during decryption. Second, searching periodic structures becomes more efficient and systematic because optimization methods are employed while only general guidelines are suggested in previous study. Third, it is much easier to enlarge the capacity of the codebook by either splitting optical response values into more ranges or adding another digit. The former way is generally not applicable to the prior scheme because some codes may never exist at selected wavelengths.
3.2 Generation robustness of Key_2
One of the proposed scheme advancements is the utilization of optimization methods to efficiently search for periodic structures with pre-determined optical responses. Hence, the robustness of an optimization method will be demonstrated below with previously-given fitness function and stop criterion is that the total number of trial structures is limited to 3000. In terms of comprehensiveness, this demonstration will consider and , three wavelengths, both polarization modes, and three materials. Moreover, the angle-dependent spectra are presented with a sketch of suitable structures, a table listing structural dimensions, and the fitness function value of the suitable structures. We first employ the RVGA method and its population size of one generation is set to thirty. Ranges of Λ and 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.
Figure 4 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  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 :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 and are replaced with and in Eq. (3) in Fig. 4(d). Since the oscillation becomes severe at large θ, angles exceeding 50° are not suggested for use. Interestingly, 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
The hybrid optimization method can also help searching satisfactory periodic structures and the method efficiency is compared with that of RVGA here. The first part of the hybrid method is to employ the grid search method. Though the dimensions defining periodic structures should be three (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.
Figure 5(b) shows the minimum fitness function of suitable structures to compare the efficiency between the two methods of optimization. To ensure fair comparison, the ranges of available dimensions, incidence polarization and wavelength, structural material, and the optical response are identically assigned. Method I uses RVGA only and each generation has a population of 30. The number of generations is 100 such a total of 3000 fitness functions are evaluated. On the other hand, Method II is a hybrid method composed of two parts. The first part is the grid search method and a total of 900 fitness functions are evaluated at this part. The second part is the RVGA with the same size population, except the dimensions for searching are significantly narrowed. Both methods are attempted for five times and the history of the minimum fitness function in one of them is plotted. At the 900th evaluation, the Method I outperforms Method II slightly, showing that the heuristic search of GA could be better than an exhaustive approach such as the grid search method. However, this superiority diminishes as the evaluation increases. The 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
Figure 6(a) demonstrates the delivery of a simple message “NANO” using the proposed cryptosystem. Alice (the message sender) delivers three decryption keys and her signature to Bob (the message receiver) via a secure channel, which is inappropriate for plaintext messages possibly due to the limit of capacity or efficiency. The plaintext must thus be encrypted into the ciphertext and is transmitted through an unsecure channel. Each character in the plaintext is represented using three digits according to the aforementioned codebook. Next, the encryption step is completed when each digit is correlated with a polar angle following previously-explained rules. The ciphertext is then a string of numbers and can be delivered easily and promptly. Once Bob received the ciphertext, he decrypts the string of numbers using the three keys and the signature of Alice based on the mutually agreed upon rules. The information related to these keys is listed in Fig. 4(a) and the signature of Alice is the optical constants of Ag.
Two coding strategies are adopted to enhance the safety of the proposed scheme. For one, dummy texts including numbers between 90 and 99 are inserted in the ciphertext to serve two purposes. The first is to remedy the confusion resulting from the loss of a number in the ciphertext during transmission because the received message will be incorrect if numbers are shifted. The existence of a dummy text can split each character clearly such that partial loss of the ciphertext will not ruin the decryption of all messages and thus the data integrity can be achieved. The second objective is to mislead Eve (an eavesdropper), who may not know the difference between the true ciphertext and the dummy text. This way, the ciphertext I in Fig. 6(a) is protected from eavesdroppers and the receiver will remove the dummy text before decryption. The other coding strategy, the employment of multiple wavelengths, can be employed simultaneously to further raise the confidentiality of the scheme. Note that the using multiple wavelengths is much easier here than in prior studies  because the search for periodic structures is aimed at a single wavelength only. Moreover, the utilization of multiple wavelengths here is to provide flexibility and to increase the complexity of the ciphertext.
Figure 6(b) shows the directional reflectance of the selected Ag gratings at three wavelengths for both ciphertext I and ciphertext II. Note that optimization method in the encryption aims only at λ = 405 nm. In the ciphertext I, each number corresponds to the angle of incidence which is at λ = 405 nm or a dummy text. For example, 05, 25, and 45 each correspond to the digits 2, 0, and 1, respectively. On the other hand, the numbers 93, 95, and 97 are dummy texts. In the ciphertext II, another digit is added to the front of those numbers to specify the corresponding wavelength such that the number 145 links to at λ = 405 nm and θ = 45°. The optical responses for λ = 660 nm and λ = 785 nm are also plotted with the structure found when λ = 405 nm. They can be used to enlarge the capacity with the digit “2” and “3” added in front of the angle number, respectively. As a result, numbers 125, 245, and 255 all represent the digit 0 because the associated directional reflectance is below 0.4 as shown in the figure. Alice therefore can send the message “NANO” with the cryptographic scheme to Bob using either ciphertext.
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.
An optical cryptographic scheme utilizing the optical responses of periodic structures for plaintext message was developed by this study. The encryption and decryption steps were divided into stages and explained using a mock scenario. The advantages of this scheme include its high degree of efficiency, robust generation of keys, large capacity, and high degree of flexibility. Two optimization methods benefit the search efficiency of satisfactory structure dimensions and their robustness was demonstrated under various limits. Such robustness actually offers the possibility for multiple users within scheme which are differentiated by their signatures. The employment of dummy texts and multiple wavelengths further assures the great flexibility of the scheme. Compared to most known schemes, the ability of proposed schemes to resist attack becomes much stronger at little cost. We believe that the advancement of the developed cryptographic scheme ensures the security of information, expands the application of optics, and provides an efficient way in searching periodic structures using desired optical responses.
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.
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