1.

INTRODUCTION

In the last 20 years, a large number of light-based techniques for security and encryption problems have been suggested [1-4]. Very recently, nanostructured devices have been proposed for producing physical unclonable functions. Among other techniques, the information is encoded by using metal nanoparticles or thin layer structures. In practical applications, random phase masks attached to the image to be secured make the information inaccessible once the light interacting with the sample is propagated. These devices display a distinctive behavior under polarized light and consequently, the use of illuminating sources with different polarizations increase the degrees of freedom of the cryptosystem resulting in gadget which is more complex and difficult to attack [5-8].

3D optical random phase masks were proposed some years ago [9]. These ensembles are produced by combining glass and diffuser materials. They are simple to produce and very difficult to copy. As in nanoparticle-based devices, the coded information is not accessible once the light interacting with the sample is propagated. The irradiance of the propagated signal is usually a speckle-like noise distribution and consequently, statistical analyses of the recorded light can be performed. Note that small differences in the noise pattern enable to distinguish the same encoded message but attached to a different random code. For this kind of problems, classification methods based on machine learning techniques can be used [10]. In this way, it is possible to discriminate among messages with the correct encoding (true class) and counterfeit information (false class).

In this communication we present a simplified numerical model that approximately describes the behavior of 3D optical random phase codes. In particular, we show that encoded messages can be distinguished using machine learning algorithms under certain propagation conditions.

2.

A NUMERICAL MODEL FOR 3D RANDOM PHASE CODES

In this section we present a computer simulation to model the behavior of 3D random phase masks [9]. This calculation provides more insight on the results presented in reference [11]. In that paper, we used 3D phase masks to encode information in such a way that it cannot be read after propagation. Despite the speckle noise embeds and hides the original data, these signals are statistically labelled so they can be successfully classified using machine learning algorithms. In [11], Random Forests Classification [12] was used for discriminating among samples classes using experimentally obtained data.

In order to produce a correct model of such devices, light propagation through a diffuser should be taken into account. However, the analysis of the transmitted light though an inhomogeneous system is very complex. Here, we propose a simplified model based on combining phase masks, glass slides and blurring, that can be considered as a first-order approximation.

Fig. 1(a) depicts the three 3D random codes designs used as numerical tests. The image used (a QR code) is shown in Fig. 1(b). Code 1 is generated by attaching a phase-only random mask ϕ₁ to Q(x,y). For convenience, the complementary image of the QR was used. Code 2 was been simulated by combining random phase masks ϕ₂ and ϕ₃ and glass: light is propagated through a 2 mm-wide microscope slide with index of refraction n=1.5195, corresponding to a BK7 glass illuminated at λ =532 nm. Code 3 is calculated in an equivalent way according to the corresponding diagram in Fig. 1(a). According to the Fresnel Laws, the amplitude transmission of the ensemble is

Figure 1.

(a) Sketch of the three 3D codes masks used in the simulation; (b) test image: the QR encodes the message 0034934021143.

and therefore, the transmittance for codes 2 and 3 is t₂ =0.9575 and t₃ =0.9168 respectively. Blurring produced by the diffuser is simulated by convolving the modulus of the propagated beam by an NxN constant kernel. Propagation is calculated by means of the scalar plane wave power spectrum formula [13],

where FT stands for the Fourier transform operator, u and v are the spatial frequencies, E(x,y, 0) is the scalar electric field at the reference plane, E(x,y,z) is the propagated beam and H(u,v,z) is the transference function. Note that the wavelength is λ/n when the beam is propagated in a medium with refractive index n. Table 1 displays a summary of variables and values used in the simulation.

Table 1:

Propagation variables

Calculations are performed as follows: light propagates through one of the three encoding devices [Fig. 1(a)]. Then, the beams are propagated a distance z=250 mm and the light irradiance is recorded by means of an 8-bit camera. The generated distributions are shown in Figs 2(a-c). Note that the irradiance displays a speckle noise pattern and the information encoded in the QR is no accessible by simple visual inspection [see Fig2. 2(d-f)]. Figures 2(g-i) show the amplitude histograms (the square root of the irradiance). In order to provide enough cases, 30 different codes for each class have been calculated. Note that histograms present subtle differences that can be used to distinguish among the three classes.

Figure 2.

First row: Encoded light distributions trough (a) Code 1, (b) Code 2, (c) Code 3. Second row: Propagated light distributions at z = 250 mm (d) Code 1, (e) Code 2, (f) Code 3. Third row: Histograms of the propagated amplitudes: (g) Code 1, (h) Code 2, (i) Code 3.

Using a different propagation distance (z=400 mm), a second set of calculations have been carried out. The recorded distributions and the corresponding histograms are shown in Figs. 3(a-c) and 3(d-f), respectively. Note that in this case, the profile of the histograms clearly follows a Gamma probability distribution [14-16], i.e.

Figure 3.

First row: Propagated light distributions z = 400 mm (a) Code 1, (b) Code 2, (c) Code 3. Second row: Histograms of the propagated amplitudes (d) Code 1, (e) Code 2, (f) Code 3.

where I, ‹I› and σ are the intensity data points, its average and the corresponding standard deviation respectively; Parameter n₀ is defined as n₀ = (‹I› / σ)² and Γ() stands for the Gamma special function.

3.

CLASSIFICATION

Using the information provided by the histograms, it is possible to distinguish among the three different classes of codes. As explained above, 30 different codes for each class (Code 1, 2, and 3) have been produced. After propagation, the corresponding amplitude histograms are used to feed a machine learning classification system. In the present work two algorithms have been considered: k-Nearest Neighbors (kNN) with k=1 [17] and Random Forest (RF) with 100 trees [12]. Classification is performed by splitting the dataset into training and test subsets. Validation is carried out 10 times. In each trial, the samples used in the training or test sets are randomly selected. Classification accuracy is calculated by means of the ratio of the number of correctly classified samples, divided by the total number of test samples. Calculations were carried out using the scikit-learn library [18].

Table 2 summarizes the results obtained: for z = 250 mm, both kNN and RF produce very good results. In particular kNN is able to produce an error-free classification. Nevertheless, classification accuracy is very low for z = 400 mm. An alternative approach for classification has been considered. Instead of using the 256 values of the histogram, several statistical parameters have been obtained from the histogram distribution: mean, standard deviation, kurtosis and skewness. On top of that, the entropy of the irradiance has also been considered. These five statistics are used to train a system using kNN (k=1) and RF (100 trees). As in the previous case, each class is composed of 30 samples. Validation is performed 10 times using a hold-out strategy. In this case, RF provides almost error-free classification for z = 250 mm. When irradiance is recorded at z = 400 mm, both kNN and RF fails and a bad classification accuracy is obtained.

Table 2:

Classification results

At first sight, it seems that histograms for the z = 250 mm case are somewhat different whereas the histograms generated for the z = 400 mm case are almost undistinguishable.

4.

CONCLUDING REMARKS

In this communication, we proposed a simplified numerical model for understanding the behavior of optical 3D codes based on diffusers. These structures encode the image in a unique way and after propagation, the irradiance presents a speckle-like noise pattern. As a result, the original information is no longer accessible. The statistics of these distributions is analyzed and the data provided is used to feed machine learning classification methods. We demonstrated that using kNN and RF is possible to distinguish among the signals produced by different classes of codes. We found that classification is possible for a certain range of distances.