
==== Front
iScience
iScience
iScience
2589-0042
Elsevier

S2589-0042(24)01799-1
10.1016/j.isci.2024.110574
110574
Article
Robust optical multi-image encryption with lossless decryption Recovery Based on phase recombination and vector decomposition
Guo Yuan guoyuan171@126.com
1∗
Li Wenpeng leewenpeng@126.com
24∗∗
Wu Lanlan 3
Zhai Ping 2
1 Heilongjiang University, No. 74 Xuefu Road, Harbin 150080, Heilongjiang, China
2 Qiqihar University, No. 42 Wenhua Street, Qiqihar 161006, Heilongjiang, China
3 Anhui Wenda University of Information Engineering, No. 3 Forest Avenue, Hefei 231201, Anhui, China
∗ Corresponding author guoyuan171@126.com
∗∗ Corresponding author leewenpeng@126.com
4 Lead contact

25 7 2024
20 9 2024
25 7 2024
27 9 1105741 5 2024
10 7 2024
22 7 2024
© 2024 The Author(s)
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
Summary

Image encryption is crucial for protecting image privacy and ensuring security. Encrypting large batches of images of different types and sizes simultaneously with losslessly decryption is often necessary. This paper proposes an optical asymmetric multi-image encryption algorithm to meet these demands. First, plaintext images are converted into one-dimensional pixels and blocked. Image information, image count, and pixels are stored in corresponding areas and reassembled. Unit equal-modulus vector decomposition (UEMD) and phase truncation generate the ciphertext image and keys. The decrypted image is reconstructed from the ciphertext’s information and quantity areas. Asymmetric encryption with different keys for encryption and decryption enhances security, while UEMD ensures lossless recovery and robustness. Experiments demonstrate the proposed algorithm’s efficiency in encrypting multiple grayscale and color images of varying sizes, providing high security, and lossless recovery. This technology offers superior protection for sensitive image data, enhancing encryption system practicality and digital security.

Graphical abstract

Highlights

• An optical asymmetric encryption algorithm was proposed for large batches of images of different types and sizes simultaneously

• The ciphertext images can be recovered losslessly

• Strong robustness can achieve high decryption quality with a variance of 0.015 Gaussian noise and 1/2 shear

• The algorithm has a large encryption capacity and strong anti-attack ability

Applied sciences; Data encryption

Subject areas

applied sciences
data encryption
Published: July 25, 2024
==== Body
pmcIntroduction

With the emergence and rapid development of meta-universe, blockchain, cloud technology, and digital twins, images, and videos have become an important way of recording information and interacting and are widely used.1 When digital images are transmitted over the network or stored in the cloud, there is a danger of information loss and leakage due to interception and destruction by attackers. It is necessary to take certain protection measures, and image encryption is one of the most important and effective methods for ensuring image data security.2,3

With the rapid growth of network information transmission data and the enhancement of transmission capacity, single-image encryption has been difficult to adapt to the needs of the era of big data, so multi-image encryption algorithms have been the attention of experts in the field of image security and research. In the military field, image encryption is crucial. Military images often contain sensitive information such as battlefield maps, satellite images, and reconnaissance photos. These images may have different resolutions and sizes.4,5 Multi-image encryption can handle images of various sizes simultaneously, meeting practical needs. In the medical field, image encryption protects patient privacy and sensitive data. Medical images, including X-rays, CT scans, and MRIs, have varying resolutions and sizes. Multi-image encryption ensures these images’ secure transmission and storage while maintaining their quality, preventing unauthorized access.6 In legal and law enforcement, images handled by police and judicial agencies may come from surveillance cameras, mobile phones, and other devices with different resolutions and sizes. Multi-image encryption ensures the security of this evidence during transmission and storage, preventing unauthorized access, and tampering.7 An optical information system has the advantages of high speed and large bandwidth, which allows it to process images in parallel and realize fast encryption and decryption.8,9,10 It is suitable for multi-image encryption, and various methods have been proposed recently. These methods mainly scramble image pixels through optical transformations such as interference, diffraction, polarization, phase modulation, imaging, etc., and encrypt a plaintext image into a noise-like ciphertext image. Many research results have been achieved in multi-image encryption.11,12,13 Yong et al.14 proposed a multi-image optical encryption method with rotational multiplexing of the key in the Fresnel domain. Deepan et al.15 used spatial multiplexing of compressed sensing and double random phase coding to realize multi-image encryption. Tang et al.16 encrypted multiple images based on bit-plane decomposition and chaotic mapping algorithm. Li et al.17 used cascaded fractional Fourier transform to superimpose multiple images into a single image to realize multi-image encryption. Ye et al.18 proposed an optical multi-image compression encryption algorithm based on quaternionic discrete fractional Hartley transform and improved pixel adaptive diffusion. This algorithm requires high computational storage accuracy. Liu et al.19 proposed an optical interferometric multi-image encryption and authentication algorithm based on sparse and spatial multiplexing; the encryption quality of this method is affected by image overlapping. Su et al.20 proposed a security-enhanced multi-image encryption algorithm based on an improved structured phase mask iterative phase restoration algorithm in the Fresnel domain.

However, most of the current optical multi-image algorithms generally have three problems21,22,23: (1) limited practicality: most algorithms are unable to encrypt any number of different sizes, different types of color, and grayscale images at the same time, and the optical components are high-cost and large, which are not easy to deal with, and the components also need to be precisely arranged and cumbersome; (2) weak security: most of the optical encryption is a linear system, which is poor in anti-attack ability; (3) poor image recovery: the decrypted image is susceptible to multi-image crosstalk noise.

To address the previous problems, this paper proposes an optical asymmetric encryption algorithm that simultaneously encrypts multiple images of different sizes and types. Furthermore, the ciphertext image can be recovered losslessly by adopting the unit equal mode vector decomposition. This algorithm can realize simultaneous asymmetric encryption of grayscale and color images, which has a large encryption capacity and strong anti-attack ability. Moreover, digital information technology can realize this optical technology on the computer. It is more suitable for medical data, drawing design, confidential documents, identity information, and other applications with many images, high confidentiality requirements, and good-quality decrypted images.

Results and discussion

Principles

Multi-image recombination

In this paper, the multiple images are reorganized to achieve different types and sizes of multi-image encryption, and the principle is shown in Figure 1.Figure 1 Principle of multi-image recombination

The multi-image reorganization process follows: (1) convert n images into one-dimensional arrays sequentially and then combine each one-dimensional array into a whole pixel area. (2) Store each image type, length, and width in the information area sequentially. (3) Divide the pixel area into n blocks to form a pixel block when there is not enough to fill in the end with zeroes. (4) Add the information area of different images after the divided n blocks of pixels to get the data area. (5) Place the quantity area in front of the data area to obtain a new data area data, and the width of the image reorganization is obtained according to Equation 1. (6) Fill the data with zeroes so that the image can be evenly divided. After obtaining the height of the reorganized image from the filled data0 according to Equation 2, pixel blocks are reshaped into n equal-sized grayscale images.(Equation 1) width=ceil{floor{len(data)}}

(Equation 2) height=len(data0)÷width

The one-dimensional array after image reorganization is mainly composed of three parts: quantity area, pixel area, and information area, shown in Figure 2. The kk value is a very important parameter representing the storage capacity used to save data in the quantity and information areas. Specifically, kk = 1 means using 1 unit of storage capacity to save the information; when kk = 100, it means using 100 units of storage capacity. The kk value is closely related to shear resistance. In 4.2.3, we conducted a detailed experimental analysis of the variation of kk size with the image noise resistance and shear resistance. In Figure 2A is the quantity area, which consists of a one-dimensional array of 10×kk size, the first 2×kk size stores the number of images N, and the last 8×kk size stores the total number of pixels in the pixel area block S; the pixel area stores the total number of pixels in n images; Figure 2B is the information area, which consists of a one-dimensional array of 9×kk size, the first kk size stores the image type T (color image, grayscale image), the last 4×kk size stores the length L of the image, and the last 4×kk size stores the width W of the image.Figure 2 Results of image recombination

(A) Quantity area.

(B) Information area.

(C) Type-converted range.

(D) The converted range of pixels in the quantity area information area.

To resist the noise attack, in this paper, the pixel storage of the information area of the size area is shown in Figures 2C and 2D. There are two main types of images: color image T=3, grayscale image, and binary image T=1. Figure 2C converts T to a number in the interval [0,255], T′=191 when T=3, and T′=63 when T=1. Since N and S in Figure 2A, L, and W in Figure 2B may be more than 256, the four variables are converted to octal number x first. Then, each bit of the octal x is converted according to Equation 3 to obtain eight corresponding numbers x′ between [0,256]. Figure 2D shows the values corresponding to each converted octal number.(Equation 3) x′=16+32x

Principle of UEMD

Figure 3 shows the unit equal-modulus vector decomposition principle. The two circles are unit circles, and f1 and f2 are unit vectors with modulus 1. The vector quantities of f, f1, and f2 satisfy the vector synthesis and decomposition principle. Therefore two phase templates f1 and f2 are obtained by vector decomposition of complex number f as follows:(Equation 4) f=f1+f2

(Equation 5) f1=exp(i·ϕ1)

(Equation 6) f2=exp(i·ϕ2)

Figure 3 Principle of isometric vector decomposition

This can be seen in Figure 3, ϕ1=θ+φ and ϕ2=θ−φ.

According to the Cosine law, we can get:(Equation 7) φ=arccosA2+1−12A·1=arccosA2

ϕ1, ϕ1 take the values as the Equations 8 and 9:(Equation 8) ϕ1=θ+φ={2π+θ+arccosA2,θ+φ≤−πθ+arccosA2,−π<θ+φ<πθ+arccosA2−2π,θ+φ≥π

(Equation 9) ϕ2=θ−φ={2π+θ−arccosA2,θ−φ≤−πθ−arccosA2,−π<θ−φ<πθ−arccosA2−2π,θ−φ≥π

Encryption

The encryption principle is shown in Figure 4, where {f1(x,y),f2(x,y)……fn(x,y)} denotes the n images to be encrypted, φ(α,β) and ϕ(x,y) denote two independent white noise sequences uniformly distributed on [0,1].Figure 4 Phase truncation encryption based on UEMD

The first image f1(x,y) is subjected in the spatial domain to a random phase template R1(x,y)=exp[j2πφ(α,β)] to be modulated. Then by Fourier transform to frequency domain, it undergoes unit equal mode vector decomposition to get two unit modes E1 and E2 , which are stored as private keys. E1 and the second image f2(x,y) is multiplied, and then equal-mode vector decomposition is performed to obtain two unit modes, one of which is saved as a private key and the other is then manipulated with the other images. Until the nth image, the vector decomposition yields two unit modes E2n−1 and E2n. E2n−1 is modulated by a random phase template R2(α,β)=exp[j2πϕ(x,y)], and finally undergoes a Fourier inverse transformation in the output plane to obtain the ciphertext image C(x,y).

Decryption

The decryption schematic diagram is shown in Figure 5. The ciphertext image is placed at the input of the phase truncation encryption system of the cascade UEMD after Fourier transforms and multiplied with the complex conjugate R2∗=exp[−j2πϕ(x,y)] of R2. By phase intercepting, we can get E2n−1. With adding the preserved private key E2n, we take the amplitude to get the nth image fn(x,y). Then with the phase and the private key E2(n−1), we take the amplitude to get the (n−1)th image fn−1(x,y), and take the phase to continue to operate until the final phase E1 is obtained. Adding E1 to the private key E2, and then carrying out the Fourier inverse transform, we get the product of the first image f1(x,y) and R1.Figure 5 Phase truncation decryption based on UEMD

If f1(x,y) is a positive real-valued function, it is only necessary to know R2∗ and take the mode after the Fourier inverse transform to recover the first image, which can be recorded with a CCD intensity detector. If f1(x,y) is a complex function, then in addition to knowing R2∗, R1∗=exp[−j2πφ(α,β)] must be also known to recover the first image correctly.

Multi-image reconstruction

Multi-image reconstruction is the inverse process of multi-image reorganization, the specific principle is shown in Figure 6.Figure 6 Principle of multi-image reconstruction

Multi-image reconstruction should firstly restore the pixel storage method octal to decimal and then reconstruct, the specific reconstruction process is as follows: (1) convert the restructured images into one-dimensional arrays in turn, and then the one-dimensional arrays to form a whole data area. (2) In front of the data area, take the size of 10×kk to get the quantity area, divide the value of the quantity area by 32 to get the integer of [0,7], take the first 2×kk size of the number of non-repeating numbers to arrange into octal numbers, and then converted to decimal numbers to get the number of the converted image N, and get the total number of pixels after blocking of converted pixel area with the same size of the remaining 8×kk operation S. (3) Based on the sum of the obtained and the extracted data area, the pixel area, and the information area. Pixel area N and S information area. (4) Extract the image type T, length L and width W from the information area. (5) Reconstruct the pixels in the pixel area into N original images based on the information obtained from the information area.

Algorithm description

Encryption process

The n images will be encrypted according to the phase truncation encryption system of cascaded unit equal mode vector decomposition for multi-image encryption. The algorithm encryption flow is shown in Figure 7, the specific steps are as follows.Figure 7 Flowchart of multi-image encryption algorithm

Step1: select n plaintext images A={f1(x,y),f2(x,y),f3(x,y),…,fn(x,y)} of different sizes and types, and according to the principle of multi-image reorganization, reorganize the n plaintext images into n grayscale images B={f1′(x,y),f2′(x,y),f3′(x,y),…,fn′(x,y)} of the same size in sequence.

Step 2: the first grayscale image f1′(x,y) from B is first Fourier transformed after modulation by the phase template R1(x,y) and then the unit equal-mode vector decomposition is performed to obtain two unit modes E1 and E2.(Equation 10) E1,E2=UEMD{FT{f1′(x,y)·R1(x,y)}}

Where f1′(x,y) is the first grayscale image in B, R1(x,y)=exp[j2πφ(α,β)] is the phase template, FT{} denotes the Fourier transform, and UEMD{} denotes the unit equal mode vector decomposition.

Step 3: E2 is saved as private key 1, E1 and the second grayscale image f2′(x,y) are multiplied and then the unit equal mode vector decomposition is performed to obtain two unit modes E3 and E4.(Equation 11) E3,E4=UEMD{E1·f2′(x,y)}

Step 4: repeat operation (3) until E2n−1 and E2n are obtained by multiplying the last grayscale image fn′(x,y) in B with E2n−3 and then performing unit equal mode vector decomposition, at which time a total of n private keys are saved, respectively, E2,E4,…,E2(n−1),E2n.

Step 5: the unit mode E2n−1 takes the phase angle and is modulated by the second phase template R2(α,β) after the Fourier inverse transforms, the phase intercept gets P(x,y) as another private key to save, and the amplitude is taken to get the final cipher image C(x,y).(Equation 12) C(x,y)=PT{IFT{ang{E2n−1}·R2(α,β)}}

(Equation 13) P(x,y)=PR{IFT{ang{E2n−1}·R2(α,β)}}

Where R2(α,β)=exp[j2πϕ(x,y)] is the phase template, PT{} denotes taking the amplitude, PR{} denotes phase intercept, IFT{} denotes Fourier inverse transform, and ang{} denotes taking the phase angle.

Decryption process

The decryption process is shown in Figure 8 with the following steps.Figure 8 Flowchart of multi-image decryption

Step 1: the ciphertext image C(x,y) is combined with the private key P(x,y) and then multiplied by R2∗(α,β) after Fourier transform to get t1, the real part of the complex t1 is taken, and then the phase form is taken to get the recovered unit module E2n−1.(Equation 14) E2n−1=exp(j×REAL{FT{C(x,y)·P(x,y)}·R2∗(α,β)})

Where R2∗(α,β) is the complex conjugate of the phase template R2(α,β) and REAL{} means taking the real part of the complex number.

Step 2: E2n−1 is summed with the saved private key E2n and the amplitude is taken to get the nth grayscale image fn′(x,y) and the phase is intercepted to get the unit mode E2n−3.(Equation 15) fn′(x,y)=PT{E2n−1+E2n}

(Equation 16) E2n−3=PR{E2n−1+E2n}

Where PT{} denotes amplitude truncation operation and PR{} denotes phase truncation.

Step 3: repeat the operation in step (2) until the penultimate grayscale image f2′(x,y) is taken out and the unit mode E1 is obtained.

Step 4: E1 is added with the last remaining private key E2 and after Fourier inverse transformation, the amplitude is intercepted to get the last grayscale image f1′(x,y). So that all the reorganized n grayscale images B={f1′(x,y),f2′(x,y),f3′(x,y),…,fn′(x,y)} have been taken out. The f1′ can be gotten as:(Equation 17) f1′(x,y)=PT{IFT{E1+E2}}

Step 5: all reorganized n grayscale images B={f1′(x,y),f2′(x,y),f3′(x,y),…,fn′(x,y)} were subjected to multi-image reconstruction to obtain A={f1(x,y),f2(x,y),f3(x,y),…,fn(x,y)}, which are recovered plaintext images of different sizes and types.

Experiments

Results of encryption and decryption

To verify the feasibility and effectiveness of the multi-image encryption system in this paper, the color images baboon (512×512×3) and bird (256×344×3), the grayscale images barbara (256×256), CT (657×500), and woman (668×500) and the binary map dragon (269×500) are selected for experiments in the Python 3.7 and Pycharm environments for the experiments. The number of images in multi-image reorganization and reconstruction is 6, and the kk value is 100. The results are shown in Figure 9.Figure 9 Encryption and decryption results

(A) Plaintext image.

(B) Recombination image.

(C) Encrypted image.

(D) Decrypted image.

(E) Plaintext image histogram.

(F) Reassemble the image histogram.

(G) Ciphertext image histogram.

(H) Correlation distribution of adjacent pixels in ciphertext image.

Figure 9A shows that six images of different types and sizes are finally encrypted into one ciphertext image Figure 9C, which shows that the encrypted image completely masks the contour information of any of the plaintext images. Figure 9D shows that the six decrypted images have no difference between the naked eye and the plaintext images. The grayscale histogram and adjacent pixel correlation features measure the security level of the encryption system. The more uniform the grayscale histogram of the ciphertext image is and the more dispersed the adjacent pixel correlation distribution map is, the more secure the encryption system is. Figure 9E shows the six plaintext image grayscale histograms, which show that the distribution is not uniform and shows a clear distribution pattern. Figure 9F shows the reorganization of the image grayscale histogram, which can still be seen as a certain distribution pattern. Figure 9G shows the encrypted image grayscale histogram, which cannot be seen in the distribution pattern and completely mask the information of the plaintext image. Figure 9H shows the encrypted image adjacent to the pixel correlation distribution map. The adjoining pixel correlation of the encrypted image was destructed thoroughly, indicating that the encryption effect is good.

Robustness analysis

The robustness analysis of image encryption is aimed at whether the decryption quality remains good after transmitting or storing. Robustness is an important evaluation index of the performance of encryption algorithms and is an important guarantee of image security. Strong robustness will be manifested when the error between the decrypted image and the plaintext image is still very small after the noise, shear, and other attacks are for the ciphertext image. This paper analyzes the robustness of the proposed algorithm in detail. It is verified that the security guarantee of strong robustness of images can be realized.

Noise analysis

To test the anti-Gaussian noise ability, the experiments of analysis of anti-noise of reorganized image and encryption system were done, respectively.

In the reorganization method test, the experiments were done without changing the information area pixel storage and different numbers of image reorganization.

Analysis of noise in recombined images

After adding Gaussian noise with variance 0.015 to the reorganized image, the reorganized image is shown in Figure 10 and peak signal noise ratio (PSNR) is shown in Table 1.Figure 10 Reconstructed images after adding Gaussian noise

Table 1 PSNR of plaintext image and decrypted image

image	baboon	barbara	bird	CT	dargon	woman	
PSNR/dB	31.0682	33.1038	18.6775	8.3721	4.8838	30.9981	

The general multi-image reorganization method cannot reconstruct the image because the information area stores the image-related information, and adding noise will change the image information, resulting in the image being unable to be reconstructed. In this paper, Figure 10 shows that the reconstructed image is still visible after adding Gaussian noise with a variance of 0.015 to all six reconstructed images. The information area pixels were stored according to our restructuring algorithm’s Equation 3. Figure 10 and Table 1 show that the color and grayscale images have a strong noise immunity. Binary map and medical image noise immunity is relatively weak but still has clear recovery states. Overall, the multi-image reorganization algorithm in this paper has strong noise immunity.

Analysis of noise in ciphertext image

To test the anti-Gaussian noise attack ability of the multi-image encryption system in this paper, we added Gaussian noise to the ciphertext with a mean value of 0 and a variance of [0,0.15]. The results of the PSNR and the average mean square error (MSE) are shown in Figure 11. We selected “barbara”, which is one of the image decryption, as the example.Figure 11 The Relationship between noise variance and PSNR/MSE of decrypted images

(A) PSNR.

(B) MSE.

(C) V = 0.005.

(D) V = 0.1.

(E) V = 0.15.

(F) V = 0.2.

As shown in Figure 11, the more noise is added, the smaller the PSNR, the larger the MSE, the more blurred the decrypted image is. The decrypted image has a clear contour when the variance is 0.2 and retains most of the original information of the plaintext image. This suggests that this paper’s asymmetric encryption algorithm has a certain degree of effectiveness in resisting noise damage.

Cutting analysis

Analysis of cutting in recombined images

If the image information is cut off in the process of cutting the image, it is impossible to reconstruct the multi-image; this paper, for this problem, will be related to the information set to kk values, that is, a certain degree of anti-shearing ability. To test the proposed multi-image restructuring algorithm’s ability to resist shear attacks, this paper gives the restructured image sub-case cut of different sizes after multi-image reconstruction. The effect is shown in Figure 12.Figure 12 Composite and corresponding reconstructed image after cropping

(A) cut 99∗99.

(B) Reconstructed image.

(C) cut 300∗300.

(D) Reconstructed image.

In Figure 12A, each reconfigured image in the information area is after clipping 99×99 pixels. Figure 12B shows that each reconfigured image can still clearly identify the original plaintext image. And Table 2 show that between the plaintext image and the reconfigured image, PSNR values are around 20dB, indicating that the reconfiguration process of the information area and the number of areas of the information is not lost in the process of clipping. In this paper, the value of kk is defined as 100, and any shear no more than 99 size can reconstruct multiple images, and the kk value determines the clipping area. In Figure 12C, 300×300 pixels were cut, avoiding the information and data areas. Figure 12D shows the reconstructed image, which can still identify the original plaintext image outline, although partially obscured, indicating the algorithm has a certain shear attack ability.Table 2 PSNR of reconstructed Image after Cropping (dB)

Cut size	baboon	barbara	bird	CT	dargon	woman	
99×99	19.5965	18.8018	26.9166	22.1909	100	17.7184	
300×300	11.0285	8.7156	13.6119	12.7160	12.3596	7.9459	

Analysis of ciphertext image cropping

To test the encryption system’s ability to resist shear attack, this paper gives the ciphertext image cut to different sizes for testing. By selecting six images for encryption and the “cameraman” image, for example, the specific effect of decrypting the image is shown in Figure 13.Figure 13 Decrypted image after cropping

(A) cut 1/16.

(B) cut 1/16.

(C) cut 1/16.

(D) cut 1/4.

(E) cut 1/2.

As can be seen in Figures 13A–13C show the ciphertext image and decrypted image with 1/16 cuts and different positions. Figures 13D and 13E show the ciphertext image with 1/4 and 1/2 cuts, respectively. Although the quality of the decrypted image is degraded, the contour of the plaintext image is still clearly visible, and most of the information in the plaintext image is retained. This shows that the encryption algorithm in this paper is strongly resistant to shear attacks.

The relationship between the value of kk and the robustness

This paper determines image restructuring and shear resistance by the kk value. When there is no cutting, the image can be decrypted and restored lossless. The larger the kk value, the larger the available image shear area. This section analyzes the variation of kk size with the image noise resistance and shear resistance.

As shown in Table 3, when kk is 100, the maximum value of anti-Gaussian variance is 0.015. When kk is 1, the maximum value of anti-Gaussian variance can be up to 0.022. In short, the larger the value of kk is, the larger the value of PSNR is, and the weaker the anti-Gaussian noise ability is, the larger the area can be clipped. In the table, the image cannot be cut when the kk value is 1. Otherwise, the data lost in the information area cannot be reconstructed. And the larger the kk value is, the larger the area available for image cutting. By comprehensive analysis, this paper selects the kk value of 100.Table 3 Analysis of kk values

Value of kk	maximum Gaussian noise	image	PSNR/dB	maximum shear area	image	PSNR/dB	
1	0.022	baboon	31.0682	no	baboon	–	
barbara	33.1038		barbara	–	
dragon	4.8838		dragon	–	
100	0.015	baboon	32.7496	99×99	baboon	19.5965	
barbara	35.3109		barbara	18.8018	
dragon	5.0654		dragon	100	
300	0.010	baboon	33.6425	299×299	baboon	11.0284	
barbara	36.3820		barbara	8.7156	
dragon	5.1708		dragon	12.3596	

Security analysis

The main objective of cryptographic algorithm security analysis is to determine whether the algorithm has vulnerabilities and whether an attacker can exploit these vulnerabilities to attack it. The most important aspects include both key and attack resistance. This paper analyzes the algorithm’s robustness in the previous section, and the key and attack resistance are interpreted as follows.

Key sensitivity analysis

In this paper, the key generation of the encryption scheme is determined by the image, which is vectorially decomposed to produce two vectors, one of which is used as the private key. Therefore, the number of images is the number of keys. To test the effect of the error of one of the keys on the quality of the entire decrypted image, the Gaussian noise [0,0.2] is added to each of the six keys. The first encrypted image, “barbara,” is used as an example to evaluate the decryption quality using PSNR, structural similarity (SSIM), correlation coefficient (CC), and MSE.

As shown in Figures 14A–14C, shows PSNR, SSIM, and CC with the noise added larger, the value decreases. In Figure 14D, as the noise is larger, the value of MSE is larger, indicating that the gap between the decrypted image and the plaintext image is larger, i.e., the quality of the decrypted image is worse than that of the plaintext image. Among the six keys, with the addition of the same noise size, the difference between the decrypted image and plaintext image changes more when adding noise to the third key, P3, indicating that key P3 is more sensitive. Overall, the multi-image encryption key to this paper is highly sensitive.Figure 14 Key sensitivity analysis

(A) PSNR.

(B) SSIM.

(C) CC.

(D) MSE.

Analysis of chosen-plaintext attack

There are many types of cryptographic attacks, the chosen plaintext attack being the most aggressive. If a cryptosystem can resist the chosen plaintext attack, it must be able to resist the ciphertext-only attack and the known plaintext attack. In this paper, experiments of chosen plaintext attacks have been done to test the algorithm’s security. Three plaintext images are to be encrypted, and the encryption algorithm is known. Each of the three plaintext images is changed by only 1 pixel. The encryption system outputs two ciphertext images, as shown in Figure 15.Figure 15 Results of chosen-plaintext attack

(A) Plaintext image.

(B) Clear text image after changing one pixel.

(C) Ciphertext image.

(D) Ciphertext image after pixel change.

(E) The difference between two ciphertext images.

Figure 15A show three plaintext images of 256×256 with all 0 pixels, and a ciphertext image of Figure 15C is finally obtained by the encryption algorithm of this paper. Figure 15B shows another three plaintext images after the first-pixel value 0 has been changed to 1. And the ciphertext image obtained after the encryption algorithm is Figures 15C–15E is the image obtained by subtracting the pixel values of the ciphertext image in Figure 15D from the pixel values of the ciphertext image in Figure 15C. Figure 15E is a disorderly white noise sequence, which is completely out of order. It shows that the algorithm in this paper can resist the selection of plaintext attacks.

Analysis of deep learning attack

Zhao et al.24 proposed a PTFT attack method based on deep learning. The neural network based on the residual network ResNet was used to train 10,000 pairs of plaintext and ciphertext to learn the matching process from ciphertext to plaintext automatically. Using this method attacks the proposed algorithm, the results are shown in Figure 16.Figure 16 Deep learning attack

(A) Training Loss and Test Loss.

(B) Plaintext Images.

(C) Decryption Images.

Figure 16B shows the plaintext image. Figure 16C shows the decrypted image of the proposed algorithm after a deep learning attack. It can be seen that the decrypted images after the deep learning attack are very different from the plaintext images. The information of plaintext images cannot be seen in the decrypted images, which shows that this paper’s deep learning attack method cannot break the proposed encryption system.

KeySpace analysis

Taking the key P as an example, we calculate its keyspace to analyze its resistance to brute-force attacks. During the attack, all other keys and the encryption algorithm are assumed to be known. Figure 17A shows that when P[0][0] changes by 10−3, the decryption results clearly reveal the original information. As shown in Figure 17B, when P[0][0] changes by 10−2, the original content in the decrypted image is indistinguishable. To perform a brute-force attack, an attacker must enumerate all possible values of each element in the matrix P. The size of matrix P is related to the recombination of multiple images, and since each element requires at least 102 potential values, the computational complexity becomes quite large. If the size of the recombined image is 256×256, then it is necessary to search exhaustively for 102×256×256 possible values. Therefore, this method demonstrates strong resistance to brute-force attacks.Figure 17 KeySpace analysis

(A) ΔP[0][0]=10−3.

(B) ΔP[0][0]=10−2.

Impact of encryption quantity on results

This paper analyzed and compared the relationship between the number of encrypted images and the encryption effect in CC, PSNR values and information entropy (IE) values.

CC analysis

As can be seen in Figure 18, compared with the literature of Li,25 Kang,26 and Guo,27 when encrypting 2 to 12 images, the CC of this paper are all close to 1, which indicates that the decrypted images of this paper can be recovered lossless and the system has large encrypted image capacity. The CC value tends to decrease with the increase in the number of images. The more encrypted images are encrypted, the worse the decryption quality. So, the encrypted image capacity is limited.Figure 18 Correlation coefficient (CC) of decrypted images under different number of images

PSNR analysis

In Table 4, it can be seen that the more the number of encrypted images, the smaller the PSNR value, and the worse the recovery quality. For other methods, when only one image is encrypted, the PSNR values are 47.7521dB, 52.2011dB, 50.3369dB, 47.1033dB, 51.5011dB, and 47.5138dB, respectively, indicating that the decrypted image still has some slight differences from the original image. However, when the number of encrypted images reaches 16, the PSNR values decrease rapidly, indicating poor decryption quality and limited encryption capacity. In contrast, the PSNR value of this paper’s encryption algorithm remains infinite even when encrypting up to 16 images, demonstrating that this paper’s encryption algorithm has lossless decryption capability and high-capacity encryption ability.Table 4 Comparison of PSNR(dB)

Number of images	Li’s paper25	Kang’s paper26	Guo’s paper27	Wu’s paper28	Guo’s paper29	Ping’s paper30	Ours	
1	47.7521	50.3369	52.2011	47.1033	51.5011	47.5138	inf	
2	44.2200	46.8849	49.3335	43.7159	48.9871	44.0985	inf	
4	40.9966	45.5522	47.6134	40.0265	47.4987	39.5139	inf	
8	35.1021	40.5583	48.9971	34.3996	47.0162	34.9198	inf	
16	27.1056	33.8313	46.8691	25.9865	46.6982	27.0956	inf	

IE analysis

In image encryption, information entropy is an important metric to evaluate the randomness and complexity of pixel values in an encrypted image. High information entropy means that the pixel values of the encrypted image are more uniformly distributed and have higher randomness, making the encrypted image more difficult to decipher. The maximum value of information entropy is 8. Table 5 compares the information entropy after encryption in the literature25,26,27,28,29 and the method proposed in this paper. The information entropy of the algorithm proposed in this paper is closest to 8, indicating that the ciphertext values are most uniformly distributed and the encryption performance is better.Table 5 Information entropy(IE) of different encryption methods

	Method	
Information	Wu’s paper28	Li’s paper25	Guo’s paper27	Kang’s paper26	Guo’s paper29	Ping’s paper30	Ours	
Entropy(IE)	7.3584	7.5220	7.3271	7.4982	7.4440	7.3469	7.9320	

Runtime analysis and comparison

Encryption and decryption time are important indicators of the feasibility of the algorithm. We did the following experiments to verify the feasibility and efficiency of this paper’s algorithm.

Runtime analysis

This paper’s total running time of the encryption system contains multi-image reorganization time, encryption time, decryption time, and reconstruction time. Table 6 shows the running time of the encryption system at various stages with different numbers (containing different types and sizes) of images.Table 6 Runtime analysis

Number of images	recombination time/s	encryption time/s	decryption time/s	reconstruction time/s	total time/s	
1	0.128656	0.429177	0.281248	0.084773	0.923854	
2	0.149600	0.506646	0.377988	0.097740	1.131974	
4	0.224365	0.594444	0.526559	0.165558	1.510926	
8	0.450785	0.742055	0.645801	0.294181	2.132822	
16	0.818111	1.168060	1.117648	0.526637	3.630455	

Table 6 shows that as the number of encrypted images increases, the running time of each stage increases. When 16 images were encrypted, the image reorganization and reconstruction time is 0.818111s and 0.526637s, and the encryption and decryption time is 1.168060s and 1.117648s, which is in line with practical needs. This shows that the algorithm in this paper has short system runtime and high encryption and decryption efficiency.

Runtime comparison

We selected literature of Wu, Guo and we selected literature of Wu,28 Guo,29 and Ping30 to do a comparison of run time in the various stages of each system under the conditions of 8 images; the results are shown in Table 7.Table 7 Runtime comparison

Different algorithm	recombination time/s	encryption time/s	decryption time/s	reconstruction time/s	total time/s	
Wu’s paper28	0.8497	0.4957	2.9969	2.9301	7.2724	
Guo’s paper29	0.1322	0.6548	1.4456	0.6400	2.8726	
Ping’s paper30	0.9914	0.7302	1.9988	1.8864	5.6068	
Li’s paper25	0.5762	0.6536	3.7261	2.7324	7.6889	
Kang’s paper26	0.5013	0.7439	2.8973	2.0376	6.1801	
Guo’s paper27	0.4598	0.7016	1.1374	0.6357	2.9345	
Ours	0.4507	0.6420	0.6458	0.2941	2.0389	

As can be seen from Table 7, when the same number of images were encrypted, the recombination and encryption times of our method were slightly higher than the recombination time of Guo’s paper29 (0.1322 s) and the encryption time of Wu’s paper28 (0.4957 s). This is because our recombination step is more complex. Our method’s decryption and reconstruction times are significantly shorter than those of other methods, being 0.6458 s and 0.2941 s, respectively. The total time of our method is the shortest, at 2.0389 s. This indicates that our algorithm is highly efficient and feasible.

Conclusion

This paper proposes a multi-image asymmetric encryption system which can encrypt different grayscale and color images in batches simultaneously. The ciphertext can be recovered losslessly by the decryption system and image reconstruction. A large number of experiments have been done to prove the feasibility and safety of this algorithm. The results and comparative analysis show that it takes only 1.168 s to encrypt 16 images of different sizes and types. Moreover, compared with other multi-image encryption methods, the speed of decrypting different numbers of images is the fastest, and the decrypted images can be recovered lossless. Security experiments show that the proposed algorithm is robust and can achieve high decryption quality with a variance of 0.015 Gaussian noise and 1/2 shear. This method can also resist many kinds of attacks, such as select plaintext attacks, deep learning attacks, and others. It is suitable for multi-image encryption protection, which requires high transmission and storage security and good decryption image quality. The aforementioned characteristics bring this proposed a wide application prospect.

Limitations of the study

Although our method can encrypt multiple images and achieve lossless recovery, the time spent on reassembly and reconstruction phases increases for images with higher resolutions, which leads to a reduction in the efficiency of encryption and decryption. In future work, our focus will be on developing fast encryption and decryption methods for multiple high-resolution images.

STAR★Methods

Key resources table

REAGENT or RESOURCE	SOURCE	IDENTIFIER	
Deposited data	
	
Data	This paper	https://osf.io/9xkcn	
Original Code	This paper	https://osf.io/9xkcn	
	
Software and algorithms	
	
Python (version 3.8)	Python Software Foundation	https://www.python.org/	
Numpy(version1.24.3)	Python package	https://numpy.org	
PyTorch (version 1.13.1)	Python package	https://pytorch.org/	
DGL Cuda11.6(version 0.9.1)	Python package	https://docs.dgl.ai/en/latest/install/	
opencv-python (4.4.0.46)	Python package	https://github.com/skvark/opencv-python	

Resource availability

Lead contact

Further information and requests for resources and reagents should be directed to and will be fulfilled by the lead contact, Wenpeng Li (leewenpeng@126.com).

Materials availability

This study did not generate new unique reagents.

Data and code availability

• Data of the main experiment: https://osf.io/9xkcn.

• TATA code to reproduce the statistical analyses and figures: https://osf.io/9xkcn.

• Any additional information required to reanalyze the data reported in this paper is available from the lead contact upon request.

Experimental model and study participant details

Focusing on the actual demands of many kinds and numbers of images to be encrypted, high decryption quality and high security of transmission and storage, we proposed a optical asymmetric multi-image encryption algorithm. Firstly, the plaintext images of different types and sizes were converted into one-dimensional pixel, and each pixel area was obtained by blocking. The image information, the number of images and all the pixels after blocking were stored in the corresponding area in order and the reassembled images were obtained by merging. Then, a ciphertext image and different private keys are obtained by unit equal-modulus vector decomposition (UEMD) and phase truncation. The final decrypted image can be obtained by multi-image reconstruction of the pixel information converted from the information area and the quantity area of the ciphertext. Because of completely different keys of the encryption and decryption, the asymmetric multi-image encryption is realized to improve the anti-attack ability of the system. The UEMD makes the ciphertext image lossless recovery. The image is reorganized and the pixel storage mode of information area and quantity area are changed to enhance the robustness of the system. Experiments and comparative analysis show that the proposed algorithm can quickly encrypt multiple grayscale and color images of different sizes at the same time with high efficiency, fine security performance and lossless recovery of decrypted images, which greatly improves the practicability of the encryption system. We believe this technology can provide a higher level of protection for highly sensitive image information and data, making a greater contribution to the security and convenience of the digital world.

Method details

Encryption process

Step1: Select n plaintext images A={f1(x,y),f2(x,y),f3(x,y),…,fn(x,y)} of different sizes and types, and according to the principle of multi-image reorganization, reorganize the n plaintext images into n grayscale images B={f1′(x,y),f2′(x,y),f3′(x,y),…,fn′(x,y)} of the same size in sequence.

Step 2: The first grayscale image f1′(x,y) from B is first Fourier transformed after modulation by the phase template R1(x,y) and then the unit equal-mode vector decomposition is performed to obtain two unit modes E1 and E2.(Equation 18) E1,E2=UEMD{FT{f1′(x,y)·R1(x,y)}}

Where f1′(x,y) is the first grayscale image in B, R1(x,y)=exp[j2πφ(α,β)] is the phase template, FT{} denotes the Fourier transform, and UEMD{} denotes the unit equal mode vector decomposition.

Step 3: E2 is saved as private key 1, E1 and the second grayscale image f2′(x,y) are multiplied and then the unit equal mode vector decomposition is performed to obtain two unit modes E3 and E4.(Equation 19) E3,E4=UEMD{E1·f2′(x,y)}

Step 4: Repeat operation (3) until E2n−1 and E2n are obtained by multiplying the last grayscale image fn′(x,y) in B with E2n−3 and then performing unit equal mode vector decomposition, at which time a total of n private keys are saved, respectively E2,E4,…,E2(n−1),E2n.

Step 5: The unit mode E2n−1 takes the phase angle and is modulated by the second phase template R2(α,β) after the Fourier inverse transforms, the phase intercept gets P(x,y) as another private key to save, and the amplitude is taken to get the final cipher image C(x,y).(Equation 20) C(x,y)=PT{IFT{ang{E2n−1}·R2(α,β)}}

(Equation 21) P(x,y)=PR{IFT{ang{E2n−1}·R2(α,β)}}

Where R2(α,β)=exp[j2πϕ(x,y)] is the phase template, PT{} denotes taking the amplitude, PR{} denotes phase intercept, IFT{} denotes Fourier inverse transform, and ang{} denotes taking the phase angle.

Decyption process

Step 1: The ciphertext image C(x,y) is combined with the private key P(x,y) and then multiplied by R2∗(α,β) after Fourier transform to get t1, the real part of the complex t1 is taken, and then the phase form is taken to get the recovered unit module E2n−1.(Equation 22) E2n−1=exp(j×REAL{FT{C(x,y)·P(x,y)}·R2∗(α,β)})

Where R2∗(α,β) is the complex conjugate of the phase template R2(α,β) and REAL{} means taking the real part of the complex number.

Step 2: E2n−1 is summed with the saved private key E2n and the amplitude is taken to get the nth grayscale image fn′(x,y) and the phase is intercepted to get the unit mode E2n−3.(Equation 23) fn′(x,y)=PT{E2n−1+E2n}

(Equation 24) E2n−3=PR{E2n−1+E2n}

Where PT{} denotes amplitude truncation operation and PR{} denotes phase truncation.

Step 3: Repeat the operation in step (2) until the penultimate grayscale image f2′(x,y) is taken out and the unit mode E1 is obtained.

Step 4: E1 is added with the last remaining private key E2 and after Fourier inverse transformation, the amplitude is intercepted to get the last grayscale image f1′(x,y). So that all the reorganized n grayscale images B={f1′(x,y),f2′(x,y),f3′(x,y),…,fn′(x,y)} have been taken out. The f1′ can be gotten as:(Equation 25) f1′(x,y)=PT{IFT{E1+E2}}

Step 5: All reorganized n grayscale images B={f1′(x,y),f2′(x,y),f3′(x,y),…,fn′(x,y)} were subjected to multi-image reconstruction to obtain A={f1(x,y),f2(x,y),f3(x,y),…,fn(x,y)}, which are recovered plaintext images of different sizes and types.

Quantification and statistical analysis

All statistical details and sample sizes are provided. The exact statistical tests and variables used are described in the text and the legends of the tables and figures.

Acknowledgments

This work was supported partly by 10.13039/501100001809 National Natural Science Foundation of China (61872204 ), 10.13039/501100005046 Heilongjiang Province Natural Science Foundation (LH2021F056 ), 10.13039/501100003851 Heilongjiang Provincial Education Department , grant/award(135509113 ), and Graduate Innovation Research Project of 10.13039/501100010616 Qiqihar University (authorization number QUZLTS_CX2023007).

Author contributions

Conceptualization, Y.G., W.L., P.Z., and L.W.; methodology, Y.G., W.L., P.Z., and L.W.; investigation Y.G., W.l., and L.W.; data curation, Y.G. and W.l.; visualization, P.Z. and L.W.; formal analysis, Y.G. and W.L.; writing—original draft, Y.G. and W.L.; writing—review and editing, Y.G. and W.L.; funding acquisition Y.G.; supervision, W.L., P.Z., and L.W.

Declaration of interests

The authors declare no competing interests.
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References

1 Al-Husainy M.A.F. A novel encryption method for image security Int. J. Secur. Appl. 6 2012 1 8
2 Pareek N.K. Patidar V. Sud K.K. Diffusion–substitution based gray image encryption scheme Digit. Signal Process. 23 2013 894 901 10.1016/j.dsp.2013.01.005
3 Li H. Wang Y. Yan H. Li L. Li Q. Zhao X. Double-image encryption by using chaos-based local pixel scrambling technique and gyrator transform Opt Laser. Eng. 51 2013 1327 1331 10.1016/j.optlaseng.2013.05.011
4 Unnikrishnan G. Joseph J. Singh K. Optical encryption by double-random phase encoding in the fractional fourier domain Opt. Lett. 25 2000 887 889 10.1364/OL.25.000887 18064216
5 Gong Q. Wang H. Qin Y. Wang Z. Modified diffractive-imaging-based image encryption Opt Laser. Eng. 121 2019 66 73 10.1016/j.optlaseng.2019.03.013
6 Lin C. Shen X. Tang R. Zou X. Multiple images encryption based on fourier transform hologram Opt Commun. 285 2012 1023 1028 10.1016/j.optcom.2011.10.046
7 He X. Tao H. Liu C. Zhu J. Single-shot color image encryption based on mixed state diffractive imaging Opt Laser. Eng. 107 2018 112 118 10.1016/j.optlaseng.2018.03.018
8 Liu Z. Zhang Y. Zhao H. Ahmad M.A. Liu S. Optical multi-image encryption based on frequency shift Optik 122 2011 1010 1013 10.1016/j.ijleo.2010.06.039
9 Pan S.M. Wen R.H. Zhou Z.H. Zhou N.R. Optical multi-image encryption scheme based on discrete cosine transform and nonlinear fractional mellin transform Multimed. Tools Appl. 76 2017 2933 2953 10.1007/s11042-015-3209-x
10 Xiao M. Tan R. Ye H. Gong L. Zhu Z. Double-color-image compression-encryption algorithm based on quaternion multiple parameter dfrat and feature fusion with preferable restoration quality Entropy 24 2022 941 10.3390/e24070941 35885163
11 Sahasrabuddhe A. Laiphrakpam D.S. Multiple images encryption based on 3d scrambling and hyper-chaotic system Inf. Sci. 550 2021 252 267 10.1016/j.ins.2020.10.031
12 ul Haq T. Shah T. 4d mixed chaotic system and its application to rgb image encryption using substitution-diffusion J. Inf. Secur. Appl. 61 2021 102931 10.1016/j.jisa.2021.102931
13 Yavuz E. A new parallel processing architecture for accelerating image encryption based on chaos J. Inf. Secur. Appl. 63 2021 103056 10.1016/j.jisa.2021.103056
14 Yong-Liang X. Su X. Li S. Liu X. Zeng S. Key rotation multiplexing for multiple-image optical encryption in the fresnel domain Opt Laser. Technol. 43 2011 889 894 10.1016/j.optlastec.2010.10.003
15 Deepan B. Quan C. Wang Y. Tay C.J. Multiple-image encryption by space multiplexing based on compressive sensing and the double-random phase-encoding technique Appl. Opt. 53 2014 4539 4547 10.1364/AO.53.004539 25090076
16 Tang Z. Song J. Zhang X. Sun R. Multiple-image encryption with bit-plane decomposition and chaotic maps Opt Laser. Eng. 80 2016 1 11 10.1016/j.optlaseng.2015.12.004
17 Li Y. Zhang F. Li Y. Tao R. Asymmetric multiple-image encryption based on the cascaded fractional fourier transform Opt Laser. Eng. 72 2015 18 25 10.1016/j.optlaseng.2015.03.027
18 Ye H.-S. Zhou N.-R. Gong L.-H. Multi-image compression-encryption scheme based on quaternion discrete fractional hartley transform and improved pixel adaptive diffusion Signal Process. 175 2020 107652 10.1016/j.sigpro.2020.107652
19 Liu L. Shan M. Zhong Z. Liu B. Multiple-image encryption and authentication based on optical interference by sparsification and space multiplexing Opt Laser. Technol. 122 2020 105858 10.1016/j.optlastec.2019.105858
20 Su Y. Wang X. Wang Z. Liu C. Li J. Xu K. Li S. Cai Z. Wan W. Security-enhanced multiple-image encryption based on modified iterative phase retrieval algorithm with structured phase mask in fresnel domain Optik 254 2022 168649 10.1016/j.ijleo.2022.168649
21 Chu R. Zhang S. Mou J. A multi-image compression and encryption scheme based on fractional chaotic map Phys. Scr. 98 2023 075213 10.1088/1402-4896/acdb01
22 Gao X. Mou J. Banerjee S. Cao Y. Xiong L. Chen X. An effective multiple-image encryption algorithm based on 3d cube and hyperchaotic map J. King Saud Univ. Comput. Inf. Sci. 34 2022 1535 1551 10.1016/j.jksuci.2022.01.017
23 Abdi H. Hacene I.B. A multiple and robust image watermarking approach based on lwt and fwht for medical data security J. Mech. Med. Biol. 23 2023 2350011 10.1142/S0219519423500112
24 Zhao X. Xin Z. Xing B. Cong L. Jie C. Yang N. Attacking asymmetric cryptosystem based on phase truncated fourier fransform by deep learning Acta Phys. Sin. 70 2021 144202 10.7498/aps.70.20202075
25 Li X. Meng X. Yang X. Wang Y. Yin Y. Peng X. He W. Dong G. Chen H. Multiple-image encryption via lifting wavelet transform and xor operation based on compressive ghost imaging scheme Opt Laser. Eng. 102 2018 106 111 10.1016/j.optlaseng.2017.10.023
26 Kang Y. Zhang L. Ye H. Zhang D. Zhuang S. Ghost imaging-based optical cryptosystem for multiple images using integral property of the fourier transform Chin. Phys. B 30 2021 124207 10.1088/1674-1056/ac0815
27 GUO Y. WANG X. WANG C. JIANG J. Ciphertext multi-image reversible information hiding based on datagram reorganization and optical interference Acta Photonica Sin. 50 2021 1210002 10.3788/gzxb20215012.1210002
28 Wu J. Xie Z. Liu Z. Liu W. Zhang Y. Liu S. Multiple-image encryption based on computational ghost imaging Opt Commun. 359 2016 38 43 10.1016/j.optcom.2015.09.039
29 Guo Y. Zhou Y.Y. Jing S.W. Multiple-image encryption based on image recombination and bit scrambling Acta Photonica Sin. 49 2020 410002 10.3788/gzxb20204904.0410002
30 Ping P. Jianhua L. Yingchi M. Rongzhi Q. Image encryption algorithm based on chaotic maps and bit reconstruction J. Image Graph. 22 2017 1348 1355
