SECTION I

Optical encryption [1], [2], [3], [4] has been widely studied for data storage and communications. The unique advantages for optical encryption include parallel processing and multiple degrees of freedom or multidimensional characteristics, such as amplitude, phase, wavelength, and polarization [2], [3], [4]. A number of optical encoding systems have been developed (see the examples [2], [3], [4], [5], [6]), and security analyses [7], [8], [9] have also been carried out. Over the past decade, phase retrieval algorithms (such as used for computer-generated hologram) have also been developed for optical encryption [10], [11], [12], [13]. Phase retrieval algorithms provide a tool to determine object parameters from the recorded intensity patterns in the optical imaging [14]. The phase retrieval is usually implemented by using an iterative operation between real and reciprocal spaces. However, the same security keys (such as wavelength and polarization) are widely used in conventional phase-retrieval-based optical security systems [10], [11], [12], [13], which will result in system security deficiency for the long-term repeated applications.

In this paper, we develop a phase-modulated optical system with sparse representation for information encoding and authentication. Inspired by authentication strategy introduced in [15], it can be believed that, when sparse data are used in the phase-modulated cryptosystems, optical authentication method can be employed to verify the decrypted images for achieving higher security. It is also desirable in some applications that decrypted data can be authenticated in optical security system but without the direct information observation. The proposed method can provide a new and effective alternative for phase-retrieval-based optical encoding.

SECTION II

Fig. 1 shows a schematic setup for the proposed optical security system. A collimated monochromatic plane wave is generated for the illumination. The objective of optical encryption is to encode the real-valued input image (i.e., plaintext) into the two cascaded phase-only masks M1 and M2. In our analysis, we model the wave propagation processes in a compact operator form by means of the fractional Fourier transform $(FrFT)$ [5], [16]. The optical setup parameters (geometry and lens focal lengths) are represented by the $FrFT$ parameter (i.e., function order $\alpha$) [16].

During optical encryption, a phase retrieval algorithm is applied to iteratively encode the input image (i.e., plaintext) into phase-only masks M1 and M2. For simplicity, in Fig. 1, the plaintext is simultaneously encrypted into only two phase-only masks; however, it is straightforward to encode the input image into more cascaded phase-only masks. The optical encryption is conducted to find the correct or approximate phase-only masks M1 and M2 under the given constraints, such as the plaintext. In the initial iteration, phase-only masks M1 and M2 are initialized to be randomly distributed in the range of $[0, 2\pi]$. Let $M_{1}^{(n - 1)} (x, y)$ and $M_{2}^{(n - 1)} (\mu, \nu)$ respectively denote phase-only masks M1 and M2. The complex-valued wavefront in the plaintext plane can be described by TeX Source $$O^{(n - 1)}(\xi, \eta) = FrFT_{\alpha_{2}}\left(\left\{FrFT_{\alpha_{1}}\left[M_{1}^{(n - 1)}(x, y)\right]\right\}M_{2}^{(n - 1)}(\mu, \nu)\right)\eqno{\hbox{(1)}}$$ where $\alpha_{1}$, $\alpha_{2}$ denote $FrFT$ function orders, and $(x, y)$, $(\mu, \nu)$, and $(\xi, \eta)$ denote the coordinates of mask (M1) plane, mask (M2) plane, and the plaintext plane, respectively.

Subsequently, a constraint is applied, i.e., setting the amplitude in the plaintext (CCD) plane to be the square root of the real-valued input image TeX Source $$\mathhat{O}^{(n)}(\xi, \eta) = \left[P(\xi, \eta)\right]^{1/2}O^{(n - 1)}(\xi, \eta)/\left\vert O^{(n - 1)}(\xi, \eta)\right\vert\eqno{\hbox{(2)}}$$ where $\mathhat{O}^{(n)} (\xi, \eta)$ denotes the updated complex-valued wavefront obtained in the plaintext plane, $\vert\cdot\vert$ denotes the modulus operation, and $P (\xi, \eta)$ is the desired output (i.e., the plaintext). Hence, phase-only masks M2 and M1 can be respectively updated as TeX Source $$\eqalignno{\mathhat{M}_{2}^{(n)}(\mu, \nu) =&\, {\left.\left\{{FrFT_{-\alpha_{2}}\left[\mathhat{O}^{(n)}(\xi, \eta)\right] \over FrFT_{\alpha_{1}} \left[M_{1}^{(n-1)}(x, y)\right]}\right\}\right/\left\vert{FrFT_{-\alpha_{2}}\left[\mathhat{O}^{(n)}(\xi, \eta) \right] \over FrFT_{\alpha_{1}}\left[M_{1}^{(n-1)}(x, y)\right]}\right\vert}&\hbox{(3)}\cr \mathhat{M}_{1}^{(n)}(x, y) =&\, {FrFT_{-\alpha_{1}}\left(\left\{FrFT_{-\alpha_{2}}\left[\mathhat{O}^{(n)}(\xi, \eta)\right]\right\}\left[\mathhat{M}_{2}^{(n)}(\mu, \nu)\right]^{\ast}\right) \over \left\vert FrFT_{-\alpha_{1}} \left(\left\{FrFT_{-\alpha_{2}}\left[\mathhat{O}^{(n)}(\xi, \eta)\right]\right\}\left[\mathhat{M}_{2}^{(n)}(\mu, \nu)\right]^{\ast}\right)\right\vert}&\hbox{(4)}}$$ where $FrFT_{-\alpha_{1}}$ and $FrFT_{-\alpha_{2}}$ are inverse $FrFT$, and asterisk denotes complex conjugate. The correlation coefficient between the calculated and desired outputs is calculated to monitor the iterative process. If the correlation coefficient is smaller than a preset threshold, the updated phase-only masks $\mathhat{M}_{1}^{(n)} (x, y)$ and $\mathhat{M}_{2}^{(n)} (\mu, \nu)$ are further used for the next iteration. Once the preset condition is satisfied, $\mathhat{M}_{1}^{(n)} (x, y)$ and $\mathhat{M}_{2}^{(n)} (\mu, \nu)$ can be considered as phase-only masks M1 and M2 (i.e., ciphertexts), which can be respectively described by TeX Source $$\eqalignno{M_{1}(x, y) =&\, \mathhat{M}_{1}^{(n)}(x, y)& \hbox{(5)}\cr M_{2}(\mu, \nu) =&\, \mathhat{M}_{2}^{(n)}(\mu, \nu).&\hbox{(6)}}$$

The same set of security keys (such as wavelength, distance, and $FrFT$ function orders) is usually used for encoding the different plaintexts in conventional phase-retrieval-based optical security systems [10], [11], [12], [13], which could result in system security deficiency for the long-term repeated applications. In this paper, we perform optical authentication by introducing sparsity strategies to provide an additional security layer for the phase-modulated-based optical encoding systems. Two simple strategies are developed to generate sparse data, i.e., 1) sparse data randomly generated from the extracted phase-only masks and 2) sparse data randomly generated from the plaintext. In the first case, the plaintext is first encoded into the phase-only masks M1 and M2 based on (1)–(4), and only small parts (such as 10%) of one or two extracted phase-only masks are randomly selected [i.e., sparse data $M_{1c} (x, y)$ and $M_{2c} (\mu, \nu)$] and used for the decryption. Note that sparse data are randomly selected pixel by pixel in this paper. In the second case, small parts (such as 5%) of the plaintext are selected before the encryption, and the two phase-only masks $M_{1p} (x, y)$ and $M_{2p} (\mu, \nu)$ are determined under the given constraints based on (1)–(4). The phase-only masks generated in these ways are used as ciphertexts, and the images obtained during optical decryption do not visually render information about the plaintext. Hence, optical authentication method [15], [17] should be further applied to verify the decrypted images.

During optical decryption, a plane wave is generated for the illumination (see Fig. 1), and a decrypted image can be extracted by using a CCD camera. The optical decryption processes for the two sparsity cases aforementioned can be respectively described by TeX Source $$\eqalignno{P_{1}(\xi, \eta) =&\, \left\vert FrFT_{\alpha_{2}} \left(\left\{FrFT_{\alpha_{1}}\left[M_{1c}(x, y)\right]\right\}M_{2c}(\mu, \nu)\right)\right\vert^{2}&\hbox{(7)} \cr P_{2}(\xi, \eta) =&\, \left\vert FrFT_{\alpha_{2}}\left(\left\{FrFT_{\alpha_{1}}\left[M_{1p}(x, y)\right] \right\}M_{2p}(\mu, \nu)\right)\right\vert^{2}&\hbox{(8)}}$$ where $P_{1} (\xi, \eta)$ and $P_{2} (\xi, \eta)$ are decrypted images corresponding to the two different sparsity cases, respectively. The correlation coefficient [18] is also calculated to evaluate the similarity between the plaintext $P (\xi, \eta)$ and the final decrypted images [$P_{1} (\xi, \eta)$ and $P_{2} (\xi, \eta)$]. The maximum correlation coefficient is 1, which means the plaintext being fully extracted.

Since sparse data are employed, the decrypted images [i.e., $P_{1} (\xi, \eta)$ or $P_{2} (\xi, \eta)$] do not visually render plaintext information, and optical authentication is further applied to verify the decrypted images. In this paper, nonlinear correlation algorithm [15], [17] is applied to compare the decrypted images with the plaintext (i.e., original image used as reference data). First, both the decrypted image and the input image are Fourier transformed. Subsequently, the spectral information obtained is multiplied in the frequency domain, and a coefficient is used to modify the amplitude part. Finally, an inverse Fourier transform is applied, and nonlinear correlation distribution can be determined correspondingly [15], [17]. The optical authentication method aforementioned can be described for the two different sparsity cases as follows: TeX Source $$NC_{h}(\xi, \eta) = \left\vert IFT\left(\left\vert \left\{FT\left[P_{h}(\xi, \eta)\right]\right\}\left\{FT\left[P(\xi, \eta)\right] \right\}\right\vert^{w - 1}\left\{FT\left[P_{h}(\xi, \eta)\right]\right\}\left\{FT\left[P(\xi, \eta)\right]\right\} \right)\right\vert ^{2}\eqno{\hbox{(9)}}$$ where $h = 1, 2, NC_{h} (\xi, \eta)$ denotes nonlinear correlation distribution, $FT$ and $IFT$ respectively denote 2-D Fourier transform and inverse Fourier transform, and $w$ denotes the strength of the applied nonlinearity [15], [17]. The decrypted image is compared with the plaintext through some operations (such as threshold) in nonlinear correlation method to generate good correlation outputs with low sidelobes. Evaluation of the correlation outputs is usually implemented by using peak-to-correlation and discrimination ratio [15].

SECTION III

A simulated experimental setup is shown in Fig. 1 for illustrating validity of the proposed optical security system. A collimated plane wave is generated for the illumination, and the wavelength is 600 nm. The CCD camera (pixel size of 4.65 $\mu\hbox{m}$ and pixel number of 512 × 512) is used for optical decryption. The $FrFT$ orders $\alpha_{1}$ and $\alpha_{2}$ are set as 0.4 and 0.8, respectively. Other $FrFT$ function orders can also be applicable, and other transform domains may also be used. In the proposed optical security system, the encryption is conducted to find the correct or approximate solutions for phase-only masks M1 and M2 (i.e., ciphertexts) under the given constraints, such as unity-amplitude constraints and the plaintext. The threshold is preset as 0.9997 in the proposed phase retrieval algorithm. A digital approach should be applied for the encryption, and either a digital or optical method can be employed for the decryption.

Fig. 2(a) shows the plaintext with 512 × 512 pixels. Fig. 2(b) and (c) show phase-only masks M1 and M2 extracted by using phase retrieval algorithm after 19 iterations during the encryption, respectively. It can be seen in Fig. 2(b) and (c) that the noise-like phase-only masks are generated as ciphertexts and the input image cannot be observed after the encryption. To illustrate the iterative encryption process, a relationship between the number of iterations and correlation coefficients is shown in Fig. 3. It can be seen in Fig. 3 that, when the iterative phase retrieval algorithm is applied during the encryption, correlation coefficients can rapidly increase to a satisfactory point.

Sparse data are generated by randomly selecting small parts of the extracted phase-only masks M1 and M2 for optical decryption. The nonselected pixels are set as zero. Hence, the decrypted images do not visually render any information about the input image, and optical authentication method is further applied to verify the decrypted images. It should be emphasized again that the main purpose is to carry out optical authentication-aided encoding rather than to directly extract the plaintext as those in conventional optical encoding systems. Fig. 4(a) and (b) shows the sparse phase-only masks M1 and M2, which respectively contain only 10% of each extracted phase-only mask [i.e., Fig. 2(b) and (c)]. Note that sparse data are randomly selected pixel by pixel, and the total summation of selected pixels is equivalent to 10% of each originally extracted phase-only mask in Fig. 2(b) and (c). Fig. 4(c) shows a decrypted image, when these sparse phase-only masks are used during optical decryption. The correlation coefficient for Fig. 4(c) is 0.0427. It can be seen in Fig. 4(c) that, when we randomly choose small parts of the extracted phase-only masks for optical decryption, no information about the input image can be extracted even with correct security keys. Fig. 4(d) and (e) shows the corresponding nonlinear correlation outputs, when parameter $w$ is set as 0.3 and 0.5, respectively. It can be seen in Fig. 4(d) and (e) that a remarkable peak is generated, which can help us authenticate the information without the direct visualization of hidden information. Hence, in addition to security guaranteed by the original encoding system, optical authentication operation with sparsity strategy can provide an additional security layer and can reduce system deficiency due to the frequent usage of the same security keys (such as wavelength and $FrFT$ function orders). Since sparse data are directly generated from the extracted phase-only masks, optical authentication with the first sparsity strategy is more suitable for processing binary images.

When only one extracted phase-only mask is sparse during optical decryption, it is also feasible to authenticate the decrypted image, which does not visually render the plaintext information. Fig. 5(a) shows the sparse phase-only mask M2, which contains only 2% of the extracted phase-only mask M2 [see Fig. 2(c)]. In this case, the extracted phase-only mask M1 shown in Fig. 2(b) is directly applied. Fig. 5(b) shows a decrypted image, when the extracted phase-only mask M1 and the sparse phase-only mask M2 are used during optical decryption. The correlation coefficient for Fig. 5(b) is 0.0776. It can be seen in Fig. 5(b) that, when only one extracted phase-only mask is sparse, the decrypted image also cannot visually render the plaintext. Fig. 5(c) and (d) shows the corresponding nonlinear correlation distributions, when parameter $w$ is set as 0.3 and 0.5, respectively. It can be seen in Fig. 5(c) and (d) that a remarkable peak can also be obtained in the correlation outputs for authenticating the information.

During data storage or transmission, the sparse phase-only masks may be contaminated, and robustness against mask contaminations (such as noise) is further tested. Fig. 6(a) shows decrypted image, when the sparse phase-only masks M1 and M2 in Fig. 4(a) and (b) are simultaneously contaminated by additive white noise (zero mean noise with 0.06 variance). It is worth noting that the pixels with zero values are not considered during mask contaminations, since the contaminations are assumed only during data transmission or storage. The correlation coefficient for Fig. 6(a) is 0.0378. Fig. 6(b) and (c) shows the corresponding nonlinear correlation distributions, when parameter $w$ is set as 0.3 and 0.5, respectively. It can be seen in Fig. 6(b) and (c) that a remarkable peak is also obtained, and the proposed method possesses high robustness against mask contaminations. In the optical cryptosystem, some experimental parameters, such as $FrFT$ function orders, can be considered as security keys. Before the optical authentication is implemented, optical decryption should be conducted at first. If the experimental parameters are incorrect, the decision of “false” will be made based on the optical authentication method. Fig. 7(a) shows a decrypted image, when only the $FrFT$ function order $\alpha_{1}$ is wrong (error of 0.02) during optical decryption. In this case, the sparse phase-only masks M1 and M2 in Fig. 4(a) and (b) are used for optical decryption, and other security keys are correct. The correlation coefficient for Fig. 7(a) is 0.0006. Fig. 7(b) and (c) shows the corresponding nonlinear correlation distributions, when parameter $w$ is set as 0.3 and 0.5, respectively. It can be seen in Fig. 7(b) and (c) that, when the security key is wrong during optical decryption, only the noisy correlation distributions can be obtained and the proposed method possesses high key sensitivity. When a similar image shown in Fig. 8(a) is used during the encryption, sparse phase-only masks M1 and M2 [also 10% of each originally extracted mask] are generated as shown in Fig. 8(b) and (c), respectively. Fig. 8(d) shows a decrypted image, when these sparse phase-only masks are used for optical decryption. The correlation coefficient for Fig. 8(d) is 0.0128. Fig. 8(e) and (f) shows the corresponding nonlinear correlation distributions, when parameter $w$ is set as 0.3 and 0.5, respectively. It can be seen in Fig. 8(e) and (f) that, when a similar image is used for optical encryption and decryption, only the noisy correlation distributions can be generated [obtained by a comparison with the original input image in Fig. 2(a)]. Although another plaintext is similar to the original plaintext, its corresponding phase-only masks (i.e., M1 and M2) are generated differently from those generated by using the original plaintext during the iterations (encryption process). Hence, when sparse phase-only masks are used for optical decryption, the decrypted image [based on the similar but false plaintext] will not contain many useful and authentic data, which are further compared with the original plaintext through the nonlinear correlation algorithm.

Different from the sparsity case described in Section 3.1, sparse data can also be directly generated from the plaintext. In this case, experimental parameters are the same as those described in Section 3.1. Fig. 9(a) shows the sparse plaintext, which contains only 5% of the whole input image in Fig. 2(a). It is worth noting that sparse data (i.e., 5%) are generated from the whole input image (i.e., 512 × 512 pixels) in Fig. 2(a) for illustrating the generalized case rather than just restricted to the region that only contains the plaintext information. In practical applications, the generation process may be restricted to the region, which only contains the plaintext information. In this paper, 5% of the whole input image is randomly selected pixel by pixel, and only 4.58% of the plaintext information in Fig. 2(a) has been successfully selected as shown in Fig. 9(a). Fig. 9(b) and (c) shows the phase-only masks M1 and M2 extracted by using the phase retrieval algorithm during the encryption, respectively. It can be seen in Fig. 9(b) and (c) that the sparse plaintext has been fully encrypted, since noise-like distributions (i.e., phase-only masks) are generated as ciphertexts. Two iterations are sufficient for satisfying the preset threshold (i.e., correlation coefficient of 0.9997) in the phase retrieval algorithm during the encryption, since only small parts of the plaintext information should be encoded into phase-only masks M1 and M2. Different from the strategy described in Section 3.1, sparse data are randomly generated from the input image rather than the extracted phase-only masks. Fig. 10(a) shows a decrypted image, when all security keys and the extracted phase-only masks are correct. The correlation coefficient for Fig. 10(a) is 0.2079. It can be seen in Fig. 10(a) that no information about the plaintext can be observed. Fig. 10(b) and (c) shows the corresponding nonlinear correlation distributions, when parameter $w$ is set as 0.3 and 0.5, respectively. It can be seen in Fig. 10(b) and (c) that a remarkable peak can be obviously observed in the correlation distributions for authenticating the decrypted image.

The robustness against mask contaminations (such as noise) is also tested. Fig. 11(a) shows a decrypted image, when the extracted phase-only masks M1 and M2 [see Fig. 9(b) and (c)] are simultaneously contaminated by additive white noise (zero mean noise with 0.5 variance). The correlation coefficient for Fig. 11(a) is 0.2061. Fig. 11(b) and (c) shows the corresponding nonlinear correlation distributions, when parameter $w$ is set as 0.3 and 0.5, respectively. It can be seen in Fig. 11(b) and (c) that the second sparsity strategy can show higher robustness against mask contaminations compared with the first sparsity strategy since the second sparsity strategy can still generate one remarkable peak in the higher-level noisy environment. Fig. 12(a) shows a decrypted image, when only the $FrFT$ function order $\alpha_{1}$ is wrong (error of 0.05) during optical decryption. In this case, the phase-only masks M1 and M2 in Fig. 9(b) and (c) are used for optical decryption, and other security keys are correct. The correlation coefficient for Fig. 12(a) is 0.0044. Fig. 12(b) and (c) shows the corresponding nonlinear correlation outputs, when parameter $w$ is set as 0.3 and 0.5, respectively. It can be seen in Fig. 12(b) and (c) that, when security key is wrong during optical decryption, only the noisy correlation distributions can be obtained. The discrimination capability of the second sparsity case is further studied. When a sparsely similar image [only 5% of the input image in Fig. 8(a)] is used during the encryption, phase-only masks M1 and M2 are generated as shown in Fig. 13(a) and (b), respectively. Fig. 13(c) shows a decrypted image, when all security keys and the extracted phase-only masks are correct. The correlation coefficient for Fig. 13(c) is 0.0679. Fig. 13(d) and (e) shows the corresponding nonlinear correlation distributions, when parameter $w$ is set as 0.3 and 0.5, respectively. It can be seen in Fig. 13(d) and (e) that the noisy correlation distributions with many peaks are obtained, and high discrimination capability can also be achieved in the second sparsity strategy.

Table 1 shows characteristics comparisons between the two sparsifying strategies aforementioned. Both the proposed strategies can work for optical authentication-aided encoding, and the word “Relatively” has been used for the comparisons in Table 1.

SECTION IV

We have proposed the phase-modulated optical system with sparse representation for information encoding and authentication. The optical cryptosystem has been developed with cascaded phase-only masks, and the plaintext has been encoded into the cascaded phase-only masks based on an iterative phase retrieval algorithm during the encryption. Two strategies have been developed to generate sparse (partial) data, and optical authentication method is further applied to verify the decrypted images. The results have demonstrated that the proposed methods are feasible and effective. The analysis has also shown that optical authentication method can provide an additional security layer for the optical security system, and high robustness and high discrimination capability can be achieved.

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