## Abstract

Optical image encryption technique has become extremely important in these years. However, most of the proposed multiple-image encryption systems are illuminated with coherent light source. Here we present a multiple-image double-encryption method with spatially incoherent illumination. The first-encryption of multiple images is based on the speckle rotation decorrelation property, and the second-encryption of images’ order is based on the speckle shift decorrelation out of the angular memory-effect range. The double-encryption via two-dimensional rotations of the random phase mask enhances the security and keeps the simplicity of the cryptosystem. The capacity of the ciphertext is greatly increased by multiplexing, and further increased after crosstalk noise removal. The use of incoherent light source reduces the requirements for experimental conditions, and makes the cryptosystem easy to implement in various application scenarios.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

As the development of the computer science and communication technology, information security has become extremely important. Since double random phase encoding (DRPE) was proposed by Refregier and Javidi [1], optical image encryption technique has attracted a lot of attention owing to its inherent ability to make use of multiple parameters and parallel operation. Afterwards, this technique was extended into the fractional Fourier domain [2,3], the Fresnel domain [4], gyrator domain [5] and fractional Mellin domain [6]. Meanwhile, since Situ et al. proposed multiple-image encryption methods based on wavelength and position multiplexing [7,8], various multiple-image encryption techniques have been proposed based on interference principle [9–11], phase retrieval algorithm [12,13], computational ghost imaging [14], and other theories [15–19]. Moreover, these techniques can be used for compression operations and multiple-image hiding simultaneously [20]. However, most of the cryptosystems mentioned above utilize coherent illumination, and thus suffer from high sensitivity to misalignment and coherent artifact noise.

To avoid the limitations of using coherent illumination, an optical encryption system with totally incoherent illumination was proposed [21]. To simplify the above system, Zang et al. presented an image encryption system by using a spatially incoherent light source and a simple optical diffuser as the random phase mask (RPM) [22]. However, this cryptosystem was attacked based on the speckle correlation method [23]. To enhance the security, Sahoo et al. presented an incoherent optical cryptosystem based on an ultra-broadband illumination and a position-multiplexing technique [24]. Recently, an incoherent image encryption system based on a polarized encoding method combined with an incoherent imaging technique is proposed and numerically simulated [25]. These above-mentioned incoherent cryptosystems were very simple, which greatly reduced the complexity of the optical implementation caused by the coherent illumination. Besides, they also avoided information expansion by manipulating the intensity information instead of the complex-valued signals used in the coherent cryptosystems.

The incoherent cryptosystem can be seen as an imaging system. The plaintext can be regarded as the object to be imaged, the ciphertext is the camera image, and the decryption key is the point spread function (PSF) determined by the RPM and the geometry of the imaging system. Moreover, the memory-effect of a scattering medium, which is originally used for imaging through scattering media [26–29], can also be introduced into the cryptosystems. Each point on the object located within the memory-effect range generates a shift-invariant random speckle pattern, i.e. the PSF, on the other side of the RPM. Thus, the camera image is the convolution of the object and the PSF. According to this, the autocorrelation of the object and the camera image are approximately equal, since the autocorrelation of the PSF is a delta function [27,28]. In this way, the incoherent cryptosystem becomes easy to be attacked with speckle correlation method [22,23]. Hence, Sahoo at al. utilized an ultra-broadband illumination and a size reduced ciphertext to avoid the speckle correlation attack [24]. Moreover, this ingenious work uses the position multiplexing concept, so that a complete ciphertext can only be decrypted with unique keys corresponding to various spatial positions. However, the plaintext within the memory-effect region has to be very sparse to avoid the crosstalk between the spatially adjacent keys. Therefore, the information capacity of a ciphertext is limited.

Here we present a multiple-image double-encryption method with an extremely simple system composed of a spatially incoherent light source, a RPM, and a camera. The first-encryption of multiple images is based on the speckle rotation decorrelation (SRD) property [30] via rotations of the RPM around the optical axis. The second encryption of the images’ order is based on the speckle shift decorrelation (SSD) out of the angular memory-effect range (AMER) via a rotation of the RPM around the vertical axis. The plaintext can only be entirely decrypt with both two decryption keys. Based on cross-correlation deconvolution, the first key disorderly decrypts the multiple images, and the second key decrypts the images’ order information. The double-encryption method enhances the security of the cryptosystem and is hard to be attacked by speckle correlation, because the autocorrelation of the ciphertext is the superposition of all the rotated images’ autocorrelations, which is too complicated and blurred to resolve. The capacity of the ciphertext is greatly increased by multiplexing, and further increased after crosstalk noise removal (CTNR). The encryption process is fast and easy to implement owing to the incoherent illumination and the simple system, which is meaningful in practical applications.

## 2. Principle

#### 2.1 Encryption

The principle of the experiments for multiple-image double-encryption via two-dimensional (2D) rotation of a single RPM is presented in Fig. 1. A plaintext is illuminated by a spatially incoherent source and hidden behind the RPM, which is a scattering medium here. Since the plaintext is located within the memory-effect range, each point on the plaintext generates a shift-invariant random speckle pattern on the camera, which can be regarded as the PSF of the imaging system. Thus, the speckle image detected by the camera is the convolution of the PSF and the plaintext.

To realize the first-encryption of multiple images, the spatially incoherent light from various plaintexts or different parts of a complex plaintext are successively transmitted through the RPM. Then, each output speckle is imaged on the camera and recorded in turn. The PSF stays the same during these detections as the system remains unchanged. Hence, each recorded speckle image *I _{i}* can be expressed as:

*P*is the

_{i}*i*th plaintext, and

*S*

^{I}is the PSF of this system. Then, all the recorded speckle images are multiplexed into a single image based on the SRD property we proposed before [30]. The left curve in Fig. 2 is the correlations between speckles before and after rotating the RPM (Newport, 10DKIT-C3-40°) around the optical axis with various angles. The correlations are vanished when the rotational angle is bigger than the rotational decorrelation angle (RDA), which is 0.19 degree here. The speckle rotation can be implemented in three ways: rotating the RPM around the optical axis, rotating the camera around the same axis, and numerically rotating the recorded speckle images around the image center. These three ways are the same since they are relative motions. As shown in Fig. 2, the left three correlation curves corresponding to these three ways are overlapped. Since the first two ways make the system complex, numerical rotation is chosen in our method. The rotated speckles superposed together with interval angles bigger than RDA can keep their independent information unaffected. Based on this, the speckle images are numerically rotated in turn according to an angle sequence. The originally recorded speckle image is regarded as the origin of the angle orientation. And each rotation angle is oriented by rotating the speckle image from the origin to a specific angle in clockwise direction. Then, the rotated speckle images are multiplexed into an image

*I*

^{I}:

*I*(

_{i}*θ*) is the speckle image

_{i}*I*with a rotation of

_{i}*θ*. Notably, the angle intervals between each two rotated images are bigger than the RDA.

_{i}The second-encryption of the images’ order, which is determined by the angle sequence, is based on the property of SSD out of the AMER. As we know, the angle change between the incident light and the scattering medium leads to a speckle shift on the imaging plane [26]. Therefore, the speckle shift can be produced by rotating the RPM around the vertical axis, as shown in Fig. 1. The correlations between the speckles before and after the shift are vanished when the angle change is out of the AMER, which is 3.6degree here (right curve in Fig. 2). This property can be called the SSD out of the AMER. Based on this, another plaintext *P*^{II} recording the angle sequence is encrypted into a speckle image *I*^{II} after rotating the RPM around the vertical axis with an angle change *α* out of the AMER:

*S*

^{II}is uncorrelated with

*S*

^{I}. Then, the speckle images

*I*

^{I}and

*I*

^{II}can be superposed into a ciphertext

*I*

^{C}:

*I*

^{II}(

*θ*

^{II}) is the image

*I*

^{II}with a numerical rotation of

*θ*

^{II}around image center. This is an easy numerical operation for further security, which makes

*P*

^{II}hard to be decrypted without knowing

*θ*

^{II}.

#### 2.2 Decryption

Two keys are needed to decrypt the ciphertext. One key *Key*^{I} is the PSF *S*^{I}, and the other key *Key*^{II} is *S*^{II}. The plaintexts are decrypted through cross-correlation deconvolution method. It is mainly based on the property that the cross-correlation of two correlated PSFs is a strongly peaked function, but the cross-correlation of uncorrelated PSFs is close to zero [30].

The ciphertext can be decrypted in two steps. Firstly, to decrypt the angle sequence in *P*^{II}, *Key*^{II} is numerically rotated with *θ*^{II} around the image center and cross-correlated with *I*^{C}:

*P*

^{II}containing the angle sequence is decrypted through cross-correlation deconvolution. The resolved

*P*

^{II}is rotated with

*θ*

^{II}due to the rotation operation in the encryption process. This effect is eliminated by numerical contrarotation of the recovered

*P*

^{II}. Secondly, based on the angle sequence resolved above and the same angle orientation definition as in encryption,

*Key*

^{I}is numerically rotated around the image center with corresponding angles and then cross-correlated with the ciphertext

*I*

^{C}in sequence:

## 3. Experiments

#### 3.1 Multiple-image double-encryption

In this experiment, the plaintext “534404095” (Fig. 3(i)) is generated from a spatial light modulator (SLM, Daheng Optics, GCI-770102, 1024 × 768, 26μm pixel size). The SLM is placed at a distance of 594mm behind the RPM (Thorlabs, DG20-220) and illuminated by a 532nm spatially incoherent pseudothermal source [31]. The CMOS camera (Point Grey, GS3-U3-91S6C-C, 3376 × 2704 pixels, 3.69μm pixel size) is placed at a distance of 105mm in front of the RPM. In this system condition, the RDA is around 0.15degree, and the AMER is about 2.3degree.

The encryption is completed in three steps. Firstly, the plaintext is broken down into five parts: “53”, “44”, “04”, “0”, “95”, and the five corresponding speckle images output from the RPM are imaged on the camera and recorded in turn. Then, the PSF *S*^{I} is detected by substituting a point for the object on the SLM (Fig. 3(a)). Afterwards, the speckle images are successively rotated around the image center with an angle sequence of 16, 55, 2, 89, and 9degree in MATLAB. To avoid the lack of edge information caused by rotation, 1913 × 1913 pixels in the middle of the images are retained. The rotated speckle images are multiplexed into *I*^{I} according to Eq. (2). Secondly, to implement the second-encryption, the RPM is rotated around the vertical axis with an angle change α = 20degree. Then, the incoherent light of the second plaintext *P*^{II} containing the angle sequence is transmitted through the RPM and the output speckle image *I*^{II} is detected by the camera. Afterwards, the PSF *S*^{II} is detected, as shown in Fig. 3(b). Notably, any *α* bigger than the AMER is effective. Multiple-encryption can be realized by increasing the number of the angle changes, which can integrally encrypt the plaintext and the order information. By the way, *α* is not necessary to be known for decryption. Thirdly, *I*^{II} is numerically rotated around the image center with *θ*^{II} = 28degree. Then, *I*^{I} and *I*^{II} are superposed into a ciphertext *I*^{C} (Fig. 3(c)).

The decryption needs two steps. Firstly, *Key*^{II} is numerically rotated with *θ*^{II} = 28degree and cross-correlated with *I*^{C} according to Eq. (5). Figures 3(d) and 3(e) are the angle sequences recovered with and without numerically rotating *Key*^{II}. It shows that a simple numerical rotation increases the difficulty of the unjustified attack so as to increase the security of the system. Secondly, *Key*^{I} is numerically rotated according to the recovered angle sequence in Fig. 3(d), and then cross-correlated with the ciphertext *I*^{C} in turn. The decrypted plaintext is shown in Fig. 3(g). As a contrast, when *Key*^{II} is not known, a plaintext is recovered in a wrong order through a time-consuming rotation of *Key*^{I} from 0 to 360degree, which turns to “049553440” as shown in Fig. 3(h). Hence, the security is enhanced through the second-encryption. Moreover, the importance of *Key*^{II} increases with the growth of the encrypted images’ number, since a larger number leads to more possibilities of images’ order. Therefore, the second-encryption is a helpful and effective security method in multiple-image encryption techniques. Meanwhile, the first-encryption is also powerful. It is hard to attack without *Key*^{I}, because the autocorrelation of the ciphertext is the superposition of all the rotated speckle images’ autocorrelations, as shown in Fig. 3(f). Obviously, the autocorrelation is complicated and blurred, so that the speckle correlation attack is hard to succeed [23]. Hence, the multiple-image double-encryption system has a high security. Furthermore, the capacity of the ciphertext is greatly increased by multiplexing. Meanwhile, the decryption process is easy and fast owing to the cross-correlation deconvolution method.

#### 3.2 Dynamic multiple-image double-encryption

The double-encryption system can also encrypt dynamic multiple images, for example, the video of moving objects. In this experiment, the system parameters are the same as above. The video of the object “**+**”, which has a motion trail like an “L”, is generated on the SLM. The movement velocity of the object is 130 μm/s. Figures 4(a) and 4(b) are respectively *Key*^{I} and *Key*^{II}. Fourteen frames of the video are encrypted into 14 speckle images and multiplexed based on SRD. The angle sequence is set as 0-60 degree with 10degree intervals for 1st-7th images and 5-65 degree with 10degree intervals for 8th-14th images. Then, the angle sequence and the movement velocity are encrypted into the second plaintext *P*^{II} after rotating the RPM around the vertical axis. The ciphertext (Fig. 4(c)) is obtained from the superposition of the two images according to Eq. (6). Since the image number is big enough and the speckles are similar, the speckles in Fig. 4(c) exhibits an obvious rotation profile.

Figure 4(d) is the plaintext *P*^{II} decrypted with *Key*^{II}. On the basis of the recovered angle sequence, the image frames are successively resolved with *Key*^{I}, and then the video of the moving object is reconstructed. The video is shown in Visualization 1, and 6 frames of the video are shown in Fig. 4(f). When *Key*^{II} is not known, the video is recovered by time-consuming rotation of *Key*^{I} from 0 to 360 degree, but the motion trail of the object is totally wrong, as shown in Visualization 2. When *Key*^{I} is not known, the ciphertext is still hard to attack by the speckle correlation method, although the autocorrelations of each frame are the same. That is because the rotation of the speckle images around the image centers make the superposed autocorrelation totally blurred that cannot recover image any more (Fig. 4(e)).

The image can be decrypted only if *Key*^{I} is rotated to an extremely narrow range around the correct decryption angle. As shown in Fig. 4(g), the images from left to right are decrypted by rotating *Key*^{I} from 65 to 69 degree with 0.5 degree intervals. The image turns misty with 0.5 degree offset and totally blurred with 1 degree or bigger offsets. That is, the image quality quickly decreases as the angle offset increases. As shown in Fig. 4(h), the PSNRs of images decrypted with rotation angles change from 65 to 70 degree are calculated. The PSNR decreases very fast when the rotation angle goes away from 65 degree, i.e. the correct encryption angle.

## 4. Crosstalk noise removal

Most of the multiple-image encryption techniques meet the problem of crosstalk noise [7,11–16], and our method is no exception. The crosstalk noise limits the capacity of the images’ number in a ciphertext. To reduce the crosstalk noise, the Gaussian low-pass filtering is used in many encryption techniques [7,14], which provides a certain optimization. Fortunately, in our method, the crosstalk noise can be removed in principle instead of imprecise filtering.

According to Eq. (6), the crosstalk noise is the background term C produced by other uncorrelated speckle images. When one of the plaintexts *P*_{1} is resolved, the crosstalk noise C turns to:

The crosstalk noise C is estimated and removed in four steps. Firstly, *P _{i}*

_{≠1}and

*P*

^{II}are successively recovered with

*Key*

^{I}and

*Key*

^{II}. Secondly, valid plaintext information is extracted from the recovered images through threshold-value filtering. Thirdly, according to Eq. (7), each extracted plaintext is convoluted and correlated with the keys and then superposed into an image, which is approximately equal to C. Lastly, the denoised plaintext is resolved by subtracting the estimated C from

*P*

_{1}.

Figure 5 is an experimental example of this CTNR method. The plaintexts are respectively decrypted from one speckle image (Fig. 5(a)), one of three multiplexed speckle images (Fig. 5(b)), and one of three multiplexed speckle images after CTNR (Fig. 5(c)). The image quality of Fig. 5(c) is much higher than Fig. 5(b), and nearly the same with Fig. 5(a), which demonstrates the robust denoising performance of the CTNR. To show the effect of the CTNR on the improvement of the capacity of a ciphertext, the PSNRs of each image recovered from the multiplexed speckle images of various image numbers are calculated in sequence. The red solid curve in Fig. 5(d) shows that, before CTNR, the PSNR decreases to half of the maximum (15dB) when the image number increases to 16. As a contrast, after CTNR, the PSNR keeps higher than 15dB even when the image number increases to 35 (blue dotted curve in Fig. 5(d)). If we define the half of the maximum of the red curve as the capacity, then the capacity is more than doubled with CTNR.

In theory, the PSNR should go down very slowly after CTNR. However, as shown in Fig. 5(a), there are a lot of system noises in practical experiments besides the crosstalk noise, such as the irrepressible background light around the plaintext produced by the SLM, and the environment light noises. These noises also increase with the growth of the image number, which results in a decrease of the PSNR. Therefore, the capacity can be further improved by suppressing these system noises.

## 5. Conclusion

We have shown that a multiple-image double-encryption system is realized by 2D rotations of the RPM with spatially incoherent illumination. Based on the SRD property, the first-encryption of multiple images is realized by numerically rotating the speckle image around the image center, which is the same as rotating the RPM around the optical axis. Based on the property of SSD out of the AMER, the second-decryption of images’ order is realized by rotating the RPM around the vertical axis. The cryptosystem keeps extremely simple since the multiplexing and the double-encryption are realized by 2D rotation of the RPM without any additional components. As a contrast, if the second-encryption is realized by changing: the wavelength of the source, the distance between the SLM and the RPM, or the RPM, the system will become much more complex to implement, and even have a risk of attack according to the techniques in [32,33]. Besides, the cryptosystem has a high security. The plaintext will be decrypted in a totally wrong order without *Key*^{II}, and will never be decrypted without *Key*^{I}. The autocorrelation of the ciphertext is the superposition of the autocorrelations of the rotated speckle images, which is too complicated and blurred to be useful for speckle correlation attack. In addition, this method is secure against the cut-and-paste/shear attacks owing to the speckle shift-invariant property and the large pixel number of the ciphertext. The security can be further increased by encrypting the plaintexts into the second or higher encryptions according to specific purpose and application scenarios. Moreover, the capacity of the ciphertext is greatly increased by multiplexing, and further increased after CTNR. The simple cryptosystem has low experimental requirements especially the light source, leading to an easy implementation in practice. This double-encryption method can also be used for multiple-image hidden, compression operations, and other important application scenarios.

## Funding

Graduate Innovation Foundation of Jiangsu (KYCX17_0247); National Natural Foundation of China (Grant No.61675095).

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