## Abstract

Scattering media are generally regarded as an obstacle in optical imaging. However, the scattering of a diffuser can be exactly taken as an advantage to act as random phase masks in the field of optical encryption to enhance information security. Here, we propose and demonstrate a dynamic diffuser based optical encryption method, which increases the ciphering strength by exploiting the uncorrelated characteristics of the dynamic diffuser as well as randomly sampling the plaintext multiple times. The light emitted from a randomly sampled plaintext passing through the dynamic diffuser generates noise-like speckles, and then SNR of the recorded speckles is further reduced for obtaining the ciphertexts, which makes COA using PRA almost impossible. The specific uncorrelated characteristics of the dynamic diffuser make the ciphertexts and the PSF keys of the optical encryption unique. Therefore, only authorized users who mastered the keys can decrypt the plaintext. The proposed method is very simple and flexible since it can also achieve the encryption offline by performing convolutions on partial-plaintexts with pre-recorded uncorrelated PSFs to generate speckle patterns and then reducing their SNR to obtain the ciphertexts. This type of encryption technique has a promising prospect in applications involving images and/or videos information encryption owing to its simplicity and flexibility.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Nowadays, images and videos have gradually become the main media of information transmission in people's daily life and work. Therefore, it is a significant task to encrypt them with great security. Among various encryptions, optical encryptions [1–9] have aroused great interests as they provide the possibility of high-speed parallel processing of 2D images carrying information in many different dimensions [10].

Random phase masks (RPMs) based optical encryptions transform the original plaintexts to noise-like speckles by inserting physical RPMs into the encryption systems. According to the number of RPMs utilized in the encryption systems, they can be divided into two categories: single random phase mask (SRPM) based encryptions and double random phase masks (DRPM) based encryptions. DRPM based encryptions can encrypt images both in spatial and frequency domains using two RPMs [2,11]. Fractional Fourier transforms [12–18] and Fresnel transforms [19–22] have been utilized in the encryptions since they can introduce various degree-of-freedoms, such as polarization, wavelength, and three-dimensional positions of the RPM in Fresnel and/or Fourier domain as extra keys to strengthening the security of the encryption systems. However, in most reported DRPM based encryptions, the illuminations are coherent, which causes them to become susceptible to misalignment and coherent noise. In contrast, the SRPM based optical encryptions can effectively avoid this problem by utilizing an incoherent illumination and an optical diffuser [23,24]. Nevertheless, their weakness is that within the memory effect (ME) region, the autocorrelation of the plaintext and that of the ciphertext are approximately equal [25,26], which results in a potential of decryption in ciphertext-only attack (COA) [27] when using the phase retrieval algorithm (PRA). To solve this problem, two ingenious encryption methods are proposed and demonstrated by Sahoo et al. [28] and Shi et al. [29]. Whereas, the plaintext within the ME region has to be very sparse to avoid the crosstalk between the spatially adjacent keys, so the information capacity of a ciphertext is limited [28]. Moreover, the plaintext could be obtained by performing deconvolutions on the ciphertext with the reshaped point spread functions (PSFs) obtained by rotating the ciphertext [29,30].

Here, we propose and demonstrate a dynamic diffuser based optical encryption method, which increases the ciphering strength by exploiting the uncorrelated characteristics of the dynamic diffuser as well as randomly sampling the plaintext multiple times. The light emitted from a randomly sampled plaintext (i.e. a random partial-plaintext as shown in 1(b), passing through the dynamic diffuser, generates noise-like speckles, and then the signal-to-noise ratio (SNR) of the recorded speckles can be further reduced for obtaining the ciphertexts as shown in 1(c), which makes COA using PRA almost impossible. The specific uncorrelated characteristics of the dynamic diffuser make the ciphertexts and the PSF keys of the optical encryption unique. Therefore, only authorized users who mastered the keys can decrypt the plaintext. The crucial factor is the PSF key consisted of the uncorrelated PSFs. After the uncorrelated PSFs are recorded, the encryption not only can be performed using the optical encryption setup but also can be offline conducted by convoluting the partial-plaintexts with the pre-recorded uncorrelated PSFs, which makes the proposed method more simple, flexible, and practical. This type of encryption technique has a promising prospect in applications involving images and/or videos information encryption due to its flexibility and simplicity.

## 2. Principle

The proposed optical encryption method is shown in Fig. 1. The plaintext to be encrypted can be a two-dimensional image as shown in Fig. 1(a). Unlike the existing approaches of inputting the plaintext directly, the proposed encryption method randomly samples the plaintext multiple times without repetition, which converts the plaintext to be a series of different sparse samplings of it i.e. the partial-plaintexts as shown in Fig. 1(b). Each of them only containing partial information of the plaintext is input at the input plane, light emitted from which will then pass through a dynamic diffuser to form the corresponding noise-like ciphertext at the output plane as shown in Fig. 1(c).

Thanks to the uncorrelated characteristics of the dynamic diffuser, when the diffuser dynamically changes, the point PSF of the optical encryption system changes accordingly, which generates the uncorrelated PSFs called the PSF key. Therefore, light emitted from the partial-plaintexts passing through the dynamic diffuser with these uncorrelated PSFs generates the noise-like speckle patterns, which can be considered as the convolution results of the partial-plaintexts and the corresponding uncorrelated PSFs since the optical encryption system is incoherent and linear. Accordingly, the one-to-one correspondence between the uncorrelated PSFs and the noise-like speckle patterns is called the order key.

To further enhance the security of the encryption for more effectively resisting the COA using PRA, we can reduce the SNR of the ciphertexts by adding noise or using a broadband illumination in the optical encryption system, because the PRA is not only more sensitive to the complexity of the object [31,32] but also more sensitive to noise as compared to deconvolution.

The uncorrelated PSFs can be gained by exploiting the spatial, spectral, or structural uncorrelated characteristics of a dynamic diffuser, which have recently been studied and utilized in optical imaging [31,33–34]. Here, for the first time to our knowledge, we propose and demonstrate the uncorrelated PSFs, obtained by simply shifting or rotating a static diffuser, can be used for information encryption. The generated uncorrelated PSFs can be expressed as,

After the encryption, we envisage that the encrypted ciphertexts can be transmitted through a channel open to the public, the order key can be sent to the receiver through an authenticated channel, and the PSF key can be passed to the receiver in a physical form.

For decryption, as shown in Fig. 2, the ciphertexts are first reordered according to the order key and then the partial-plaintexts are decrypted by deconvoluting the ciphertexts in the right order with the uncorrelated PSFs. Finally, the reconstructed partial-plaintexts are superposed to obtain the plaintext. Thanks to the PSF key, each PSF can only decrypt its corresponding ciphertext, thus the decryption almost cannot be performed when the PSF key is not available. Meanwhile, the order key makes the decryption computationally difficult. Furthermore, the SNR reduction of the ciphertexts by adding noise or using a broadband illumination makes the reconstruction of each partial-plaintext using PRA fail, which guarantees it is almost impossible to make the decryption by COA using PRA. Also, as a result of the intrinsic stochasticity of the position and orientation of the object reconstructed by PRA [31,35], assuming even if each partial-plaintext can be reconstructed using PRA, it is computationally difficult to decrypt the plaintext.

## 3. Experimental setup

As shown in Fig. 3, (a) very simple experimental setup can be established for demonstrating the proposed optical encryption method. The dynamic diffuser in the setup can be a shifting or rotating ground glass, a ground glass illuminated by a light source with changing wavelengths, etc.. Here, a ground glass (Thorlabs DG10-120) mounted on a rotator is utilized as the dynamic diffuser. The plaintext to be encrypted can be a two-dimensional image. It is randomly sampled without repetition to be multiple partial-plaintexts, which then can be displayed on the input plane by a projector (Acer X1210 K). For controlling the illumination wavelength, a filter is placed between the dynamic diffuser and the input plane. The light emitted from a partial-plaintext passes through the filter, a unique region of the rotating diffuser, and a 3 mm iris almost touching the diffuser, which generates the noise-like speckle pattern recorded by a camera (Andor Zyla 4.2 plus with a resolution of 2048×2048 pixels, a pixel size of 6.5µm, and a 16bit AD) placed at the output plane. The iris is used for controlling the aperture of the optical system, which determines the grain size of the speckles of the recorded pattern, and guarantees the generated speckle patterns can be sampled by the camera appropriately. Then the recorded patterns are chopped down to images with a resolution of 400 × 400 pixels, which are then processed by MATLAB on a normal PC (Lenovo Thinkcentre M8400t, Intel Core i7, 4 GB memory). To record a PSF, one pixel in the center of the input plane can be lighted up by the projector to simulate a point source. The light emitted from which passes through a unique region of the rotating diffuser, and then the PSF corresponding to the unique region can be recorded by the camera. By appropriately rotating the diffuser, the uncorrelated PSFs can be recorded.

## 4. Results and discussion

#### 4.1 Encryption using the spatially uncorrelated characteristic of a diffuser

First of all, a simple plaintext “T” was randomly sampled to be a series of non-repetitive partial-plaintexts, parts of which are shown in Fig. 4(a)-(c). The light emitted from those partial-plaintexts passes through the rotating diffuser, which generates the corresponding ciphertexts as shown in Fig. 4(d)-(f). The original plaintext “T” is encrypted into a series of ciphertexts.

Figure 5(a)-(d) give the decryption process of the plaintext “T” correctly using the PSF key, in which an authorized user first utilizes the order key to reorder the received ciphertexts, and then exploits the PSFs to specifically decrypt the corresponding ciphertexts, and eventually superposes all the partial-plaintexts to decrypt the plaintext “T” as shown in Fig. 5(d).

To investigate the security performance of the optical encryption method, an imaginary attacker is first assumed to decrypt the ciphertexts through COA using PRA. For each ciphertext, PRA is utilized to do the reconstruction, and all the reconstructions are superposed and shown in Fig. 5(e). Due to the intrinsic stochasticity of the positions and orientations of the reconstructions using PRA, it is computationally difficult for the attacker to make the correct decryption. The more complex the plaintext is, the more difficult the decryption using PRA will be. Then, the attacker is assumed to master the PSF key but not the order key, using a wrong PSF to decrypt a ciphertext leads to a failure of the decryption as expected as shown in Fig. 5(f), which suggests the order key can make the decryption computationally difficult. Therefore, during the decryption, the PSF key needs to be used in conjunction with the order key, which enhances the security of the encryption. Finally, if the PSF key is not available to the attacker, the decryption cannot be performed, since the uncorrelated characteristics of the dynamic diffuser make the ciphertexts and the uncorrelated PSFs of the optical encryption unique, thus each PSF can only decrypt its corresponding ciphertext.

Then, as mentioned above, more complexity of the plaintext means more random samplings which will make the decryption by COA using PRA more difficult. To verify this, as shown in Fig. 6(a), (a) more complex plaintext i.e. a ‘puppy’ image with a resolution of 50×50 pixels was randomly sampled to generate 147 different partial-plaintexts, some of which are shown in Fig. 6(b). Accordingly, 147 uncorrelated PSFs were recorded, and some of them are shown in Fig. 6(c) and the corresponding ciphertexts are shown in Fig. 6(d).

Figure 7(a)&(c) depict the decryption process of the ‘puppy’ image correctly using the PSF key. Figure 7(b)&(d) give the decryption using PRA. By comparing Fig. 7(a) with Fig. 7(b), the decrypted partial-plaintexts using the PSF key are significantly better than that using PRA, this is because the PRA is more sensitive to the complexity of an object than the deconvolution does [31,32]. Figure 7(b) shows some of the partial-plaintexts can be roughly decrypted using PRA, however, the positions and orientations of them are uncertain, and most of the partial-plaintexts decrypted using PRA are noise-like patterns. As a result, the decryption of the plaintext using PRA is failed which is also a noise-like pattern as shown in Fig. 7(d). The results suggest that the more complex the plaintext it is, the more partial-plaintexts there are, the tougher it is to make the decryption through PRA, which also can be detailedly noticed in the first row of Visualization 1.

During the above experiments, we found that the decryption using PRA is also more sensitive to noise than that using the PSF key, which suggests PRA is not only more sensitive to the complexity of an object but also more sensitive to noise as compared to deconvolution. This means when the SNR of the ciphertexts is low, the partial-plaintexts cannot be successfully decrypted by PRA but can be decrypted through the deconvolution using the PSF key. Consequently, noise can be added into the ciphertexts to make the decryption using PRA completely unsuccessful, however, the decryption using the deconvolution with the PSF key works.

Two simple approaches are proposed to reduce the SNR of the ciphertexts. First, Gaussian noise was added into the ciphertexts, and the SNR of them was 4.1 dB. Figure 8 depicts the decryption process of the Gaussian-noise-added ciphertexts. As can be seen in the zoom-in insert of Fig. 8(d), the decryption of a noise-added ciphertext using PRA always gives one central bright spot surrounded by noise. This can be also noticed in Fig. 8(f) as well as the second row of Visualization 1 which detailedly demonstrates the encryption and decryption of a more complex plaintext i.e. the ‘puppy’ image using the same method. Figure 8(e) & (f) give the decryption results of the noise-added ciphertexts using the PSF key and PRA, in which the plaintext ‘T’ was randomly sampled to be five partial-plaintexts, and then encrypted to be five noise-added ciphertexts. It can be seen that the plaintext ‘T’ can still be successfully decrypted using the deconvolution with the PSF key, whereas PRA fails to make the decryption as expected as shown in Fig. 8(f). This suggests the generated ciphertexts with artificially added noise have a stronger resistance to the COA using PRA.

Second, the SNR of the ciphertexts can be reduced by using a broadband illumination in the optical encryption setup. As can be seen in Fig. 9, the plaintext ‘T’ can be successfully decrypted using deconvolution with the PSF key as shown in Fig. 9(d), whereas PRA fails to decrypt the plaintext ‘T’ as shown in Fig. 9(e). This suggests the generated broadband ciphertexts enhance their resistance to COA using PRA.

In brief, the proposed encryption method is capable of resisting COA using PRA due to the inevitable ambiguities of phase retrieval of PRA, especially for the encryption of complex plaintexts. Taking the advantage of PRA's characteristic of being more susceptive to noise, it can make COA using PRA almost impossible by reducing the SNR of the ciphertexts through artificially adding noise or using broadband illumination, even when the plaintext is not complex.

#### 4.2 Encryption using the spectrally uncorrelated characteristic of a diffuser

It should be noted that the proposed encryption method can also exploit the spectrally uncorrelated characteristic of a diffuser. This means light from a point source with different wavelengths passes through the diffuser to generate spectrally uncorrelated PSFs. The ciphertexts equal to the convolution of the partial-plaintexts with the spectrally uncorrelated PSFs, which can be expressed as,

where $P{P_\lambda }$ are the partial-plaintexts, ${\otimes}$ is the convolution calculation, and $PS{F_\lambda }$ is the generated PSF when the illumination wavelength is $\lambda$, which is spectrally uncorrelated to other PSFs. Similar to the encryption using the spatially uncorrelated characteristic, the encryption using the spectrally uncorrelated characteristic also has the PSF key and the order key. The PSF key consists of the spectrally uncorrelated PSFs, and the order key describes a one-to-one correspondence between the generated ciphertexts and the spectrally uncorrelated PSFs, i.e. the wavelength-dependent order key.To verify the effectiveness of the encryption using the spectrally uncorrelated PSFs, proof-of-concept experiments are conducted. Three narrowband filters with a center wavelength of 405 nm, 532 nm, and 635 nm were used to generate the uncorrelated PSFs as shown in Fig. 10(d)-(f). By the encryption using the PSFs recorded, the ciphertexts can be generated as shown in Fig. 10(g)-(i), corresponding to the partial-plaintexts as shown in Fig. 10(a)-(c) obtained by randomly sampling the plaintext ‘T’.

As can be seen from Fig. 11, the wavelength-dependent ciphertexts can only be decrypted by the PSF with the same wavelength, which is consistent with the prediction of Eq. (3). The result verifies the effectiveness of the proposed method. Therefore, thanks to the spectrally uncorrelated characteristic, the wavelength of the illumination can be exploited as another degree of freedom to enhance the strength of the ciphering of the proposed encryption method.

Apart from the spatially and spectrally uncorrelated characteristic, the structural uncorrelated characteristic of a diffuser, which is always considered as an impediment in imaging [33], could be also exploited for optical encryption. Some biological materials (such as eggshell membranes, chicken breasts, etc.) with unique and non-reproducible dynamic structural characteristics could be the candidates for further increasing the security of the ciphering.

Optical encryptions utilizing coherent illumination are highly sensitive to the optical misalignment and unavoidable coherent artifact noise, however, the proposed encryption method gets rid of these problems by utilizing incoherent illumination. Meanwhile, it utilizes a simple dynamic optical diffuser, which greatly reduces the complexity of the system and decreases the errors generated from the coherent artifact noise.

Additionally, as compared with other diffuser-based optical encryption systems, the proposed method can effectively resist COA using PRA by making full use of the uncorrelated characteristics of the diffuser and the spatially random sampling. Further taking the advantage of PRA's characteristic of being more susceptive to noise, it can make COA using PRA almost impossible by reducing the SNR of the ciphertexts through artificially adding noise or using broadband illumination, even when the plaintext is not complex. In the experiments, the realization of the spatially random sampling of a plaintext relies on the random number matrix generated by MATLAB, which is essentially pseudo-random. Advanced random number generation technologies can be introduced into the proposed method to generate a random number matrix with stronger randomness, thereby further reinforcing its security. It should be noted that the spatially random sampling of plaintexts in the proposed method is not necessary when the speed of the encryption is preferred. In this case, the plaintexts can be directly encrypted by the uncorrelated PSFs, and then the noise can be added to the ciphertexts for making the COA using PRA almost impossible, while the decryptions using deconvolutions with the uncorrelated PSFs still work, the detailed results can be seen in Visualization 2.

In our method, the crucial factor is the uncorrelated PSFs obtained from the uncorrelated characteristics of dynamic scattering media. After the uncorrelated PSFs are recorded, the ciphertexts not only can be recorded using the optical encryption setup in Fig. 3 but also can be obtained offline by the calculation of convoluting the partial-plaintexts with the uncorrelated PSFs. Thus, the encryption can be simply realized by an algorithm, which makes our proposed method more simple, flexible, and practical. Especially, when the dynamic scattering media are the unique natural scattering materials e.g. biological tissues, the later solution is preferred, since the unique dynamic scatterings of them can give the unique uncorrelated PSFs, which is almost unrepeatable.

## 5. Conclusion

To conclude, making use of the spatial and spectral uncorrelated characteristics of a dynamic diffuser, spatially random sampling of a plaintext, and PRA's characteristic of being more susceptive to noise, a safer diffuser-based encryption method is proposed and demonstrated. Proof-of-concept experiments, for demonstrating the proposed method, are successfully implemented. Experimental results suggest that the proposed method can make COA using PRA almost impossible, which enhances the security of the diffuser-based type of optical encryptions. After the uncorrelated PSFs are recorded, the encryption not only can be performed using the encryption setup, but also can be offline conducted by convoluting the partial-plaintexts with the pre-recorded uncorrelated PSFs, which makes the proposed method more simple, flexible, and practical. This type of encryption technique has a promising prospect in applications involving images and/or videos information encryption owing to its simplicity and flexibility.

## Funding

National Natural Science Foundation of China (11404237, 61805167); Taiyuan University of Technology (TYUTRC-2019).

## Acknowledgment

We thank the support of the National Natural Science Foundation of China (61805167, 11404237) as well as the Qualified Personnel Foundation of Taiyuan University of Technology (NO. TYUTRC-2019). We thank the Editors and Reviewers a lot for their efforts to help us improve the manuscript during this difficult time due to COVID-19.

## Disclosures

The authors declare no conflicts of interest.

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