An encryption algorithm for multiple medical images based on a novel chaotic system and an odd-even separation strategy

Abstract

The continuous evolution of information technology underscores the growing emphasis on data security. In the realm of medical imaging, various diagnostic images represent the privacy of individuals, and the potential repercussions of their unauthorized disclosure are substantial. Therefore, this study introduces a novel chaotic system (TLCMCML) and employs it to propose a multi-image medical image encryption algorithm. To simultaneously augment security and optimize encryption efficiency, we undertake a dual-pronged approach. Firstly, we identify the regions of interest (ROIs) within individual medical images, subsequently applying an independent scrambling technique based on an odd-even interleaving configuration. Secondly, we integrate all medical images through horizontal concatenation, forming a comprehensive large-scale image, upon which we implement a synchronized bit-level permutation-diffusion encryption mechanism. Following extensive testing, enhancements were verified across various metrics including information entropy analysis, adjacent pixel correlation examination, differential attack simulations, and robustness assessments, thereby attesting to the exceptional encryption efficacy of the proposed algorithm.

Introduction

With the continuous advancement of information science and technology, the security of data transmission and storage has become increasingly precarious. Images, serving as carriers of diverse data types, warrant heightened attention in terms of security measures. Within medical systems, medical images hold pivotal significance, facilitating precise assessment of patient conditions and enhancing the intuitive discussion of treatment strategies by healthcare professionals. However, these images inherently contain highly sensitive information. Unauthorized access or misuse could compromise patient privacy, potentially precipitating severe medical consequences.

To tackle the aforementioned challenges, researchers have undertaken extensive endeavors in the field of image encryption by integrating various methodologies, including deep learning¹, watermarking techniques², neuronal models³, and chaotic theory etc., to encrypt images and consequently elevate their security levels. The application of chaos theory in the field of image encryption is not only one of the most commonly used methods but also one of the most effective. The theory of chaos is distinguished by its characteristics of pseudo-randomness, unpredictability, and heightened sensitivity to initial conditions and parameters⁴, which is particularly suitable for image encryption. Different individuals have proposed their own image encryption schemes by integrating various techniques. Long et al.⁵ proposed a one-dimensional exponential Chebyshev (1-DEC) chaotic system for image encryption, achieving promising results. Gabr et al. Alexan et al.⁶ utilized the Arnold’s cat map, alongside Langton’s Ant and S-box, for encrypting images⁷. proposed to utilize the unpredictable behavior of the Chua’s system and the hyperchaotic characteristics of the Chen system to resize, rotate, and randomize the target image, thereby breaking through the boundaries of high-security image encryption methods. However, low-dimensional chaotic systems have certain limitations, resulting in insufficient key space to guarantee the security of information. Therefore, researchers have proposed high-dimensional chaotic systems as an alternative. Li et al.⁸ proposed cross coupled map lattice to address the issue of insufficient key space. We proposed Tent-logistic cross mixed coupled map lattice based on cross coupled map lattice. This system possesses superior dynamical characteristics, making it more suitable for image encryption algorithms.

In recent times, there has been a growing focus on encryption methodologies specifically tailored for medical images. El-Damak et al.⁹ proposed to use Fibonacci matrices for processing medical images and then completes the encryption by replacing the pixels in the R, G, and B channels with an S-box. Liu et al.¹⁰ proposed a novel chaotic system employing a hyperbolic sine function as its nonlinearity, and put forward an encryption scheme for medical images based on this newly proposed chaotic system. Singh et al.¹¹ proposed a method that utilizes You Only Look Once v3 (YOLOv3) to extract key features of an image before proceeding with the encryption process. These algorithms are all designed to encrypt individual medical images. However, in diagnosing conditions, doctors often need to supplement and differentially diagnose multiple medical images. Therefore, encrypting multiple images is essential. Various studies have explored the encryption of multiple images¹². Wu et al.¹³ proposed an algorithm that involves overlaying all images to produce a composite image. Initially, an initial key is generated using SHA-256 technology, which serves as the starting value for a linear congruential algorithm to establish an encryption key sequence. Subsequently, the composite image is encrypted using this encryption key and DNA encoding, resulting in a cipher image. Liu et al.¹⁴ proposed an encryption scheme to protect medical multi-images. Initially, three bits of binary information within medical images are randomly altered, and the binary information before and after blurring is concatenated into new pixels. SHA-512 is employed to design and generate random keys. The 3D-Fisher method is designed to achieve random inter-plane scrambling. Next, novel DNA operations and asymmetric DNA encoding/decoding rules are designed for diffusion. Finally, new pixels are generated using the information from the bit planes before blurring. However, these encryption algorithms are still far from adequate for application in the field of encrypting multiple medical images.

The main contributions and innovations of this study are as follows: (1) We propose an encryption algorithm capable of encrypting multiple medical images of arbitrary quantity and size, thereby facilitating the encryption process for multiple medical images. (2) We propose a new chaos system Tent-logistic cross mixed coupled map lattice based on cross coupled map lattice. It possesses good chaotic characteristics, ergodicity, and a relatively wide chaotic range. (3) In the process of image encryption, there are two steps. The first step involves using the odd-even interleaving permutation method to individually scramble the Regions of Interest (ROI) in each medical image, effectively disrupting the medical features within the ROI. In the second step, a synchronous scrambling and diffusion algorithm is applied. After reinserting the regions of interest into the image, an odd-even separation is conducted. Subsequently, simultaneous scrambling and diffusion algorithms are applied to each separated image. This process effectively enhances the randomness of pixel values. Experimental testing has shown that the algorithm proposed in this paper exhibits excellent encryption effectiveness and possesses higher security.

The paper is structured as follows: Sect. 2 introduces the chaotic system TLCMCML. Section 3 describes the whole encryption scheme. Section 4 gives simulation results and performance analysis of medical images. The conclusion is provided in Sect. 5.

Tent-logistic cross mixed coupled map lattice

Trajectory

Cross coupled map lattice (CCML) is defined as Eq. (1)⁸.

1

where the parameter e represents the degree of coupling between different lattice, with its value ranging from 0 to 1, n stands for time, i denotes the index of the lattice, with its range constrained between 1 and L. The i+1 lattice and the i-1 lattice are respectively the adjacent grid points of the i-th lattice. The boundary conditions have been set as follows:

2

The function is logistic map, where is the parameter of the logistic map.

In the iterative process of CCML, the values of even lattice depend solely on the values of the previous iteration, while the values of odd lattice are determined by both the values of the previous iteration and the current iteration.

Tent-dynamic cross coupled map lattice (TDCCML) is defined as¹⁵.

3

4

where parameter , and represents the dynamic coupling coefficient.

According to TDCCML system, we propose tent-logistic cross mixed coupled map lattice (TLCMCML), which is defined as Eq. (5). In the TLCMCML chaotic system, we substitute the parameter from TDCCML with the Logistic-Tent system¹⁶, known for its enhanced stochastic properties. Furthermore, we systematically vary the values of p and q to augment the stochastic characteristics of TLCMCML.

5

6

7

where a, b, c and d satisfy the Fibonacci sequence.

Figure 1 displays the return mapping phase diagrams of CCML, TDCCML, and TLCMCML. Return map phase diagrams are employed to visualize the relationship between adjacent lattice points. As depicted in Fig. 1, TLCMCML demonstrates a wider distribution range compared to both CCML and TDCCML systems. This broader distribution signifies increased stochasticity in TLCMCML, resulting in more diverse output patterns. Consequently, TLCMCML is deemed more suitable for applications in encryption and secure communication.

Fig. 1.

Return mapping phase diagram: (a) CCML with parameter ,; (b) TDCCML with parameter ,; (c)TLCMCML with parameters ,

Lyapunov exponent

Chaotic systems exhibit various dynamical characteristics, with the Lyapunov exponent (LE) being among the most commonly utilized. The Lyapunov exponent quantifies the sensitivity of chaotic systems to initial conditions. A positive Lyapunov exponent indicates chaotic behavior, with a higher value suggesting increased sensitivity to initial conditions and thereby indicating superior performance of the chaotic system. LE is defined as Eq. (8)¹⁷.

8

As shown in Fig. 2, at least half of the LE values of the CCML chaotic system and TDCCML chaotic system are in a state of less than 0 when the parameters satisfy and , showing no chaotic characteristics. When the parameters are and , the TLCMCML is greater than 0 except for the very few LE values, which shows good chaotic characteristics. Therefore, the analysis of LE values shows that TLCMCML is a better chaotic system.

Fig. 2.

Analysis of Shannon entropy: (a) CCML system; (b) TDCCML system; (c) TLCMCML system.

Shannon entropy

Shannon entropy (SE) can be used to represent the randomness of chaotic sequences, and the higher the SE value, the stronger the disorder of chaotic sequences. SE is defined as Eq. (9)¹⁸.

9

where represents the probability of occurrence. Assuming that the CCML, TDCCML and TLCMCML systems have N = 256 states, the theoretical value of SE should be . As shown in Fig. 3, it is clear that at least half of the SE values of the CCML and TDCCML system have a value of 0 when the parameters satisfy and , indicating that the system is only random in certain cases. However, the SE value of TLCMML is close to 8 when the parameter and , which reflects better randomness.

Fig. 3.

Analysis of Shannon entropy: (a) CCML system; (b) TDCCML system; (c) TLCMCML system.

Approximate entropy

The complexity of a time series can be quantified using Approximate Entropy (AE)¹⁹. For a given data time series , this sequence is reconstructed as:

10

where represents an m-dimensional vector. The distance between and is computed as

11

.

Given a threshold , let K represent the count satisfying the condition , where SD denotes the standard deviation of the sequence. Subsequently, we define

12

,

where , then the AE can be calculated as .

As shown in Fig. 4, the horizontal axis denotes precision values ,…, while the vertical axis represents Approximate Entropy (AE) values. It is apparent that our TLCMCML model exhibits clear superiority over CCML and TDCCML.

Fig. 4.

Approximate entropy analysis of CCML, TDCCML and TLCMCML.

Correlation test

Correlation serves as a pivotal metric in assessing pseudo-random sequences. The auto-correlation function characterizes the correlation values of a random signal at various distinct time points, thereby quantifying the similarity of binary sequences following conditional modifications. In general, the autocorrelation function of an ideal pseudo-random sequence tends to approximate the Dirac function. Where Dirac function is

13

where and .

Figure 5 depicts the results of autocorrelation tests conducted on sequences generated by the TDCMCML chaotic system. Three randomly selected sequences were examined, showing autocorrelation functions that closely resemble the Dirac function. This observation suggests that these sequences exhibit a high level of randomness.

Fig. 5.

Analysis of Auto-correlation: (a) Auto-correlation of ; (b) Auto-correlation of ; (c) Auto-correlation of

NIST-800-22 statistical tests

The NIST-800-22 tests are employed to evaluate the suitability of chaotic binary sequences for cryptographic algorithms²⁰. A sequence is deemed random if . According to Table 1, TDCMCML has successfully passed all randomized tests.

Table 1.

The NIST test of chaotic binary sequence.

Test Item	p-Value	Result
ApproximateEntropy	0.31995	Passed
BlockFrequency	0.51386	Passed
CumulativeSums	0.70372	Passed
FFT	0.88327	Passed
Frequency	0.69948	Passed
LinearComplexity	0.82474	Passed
LongestRun	0.34849	Passed
NonOverlappingTemplate	0.96147	Passed
OverlappingTemplate	0.64124	Passed
RandomExcursions	0.49174	Passed
RandomExcursionsVariant	0.93775	Passed
Rank	0.78889	Passed
Runs	0.48941	Passed
Serial	0.67658	Passed
Universal	0.53125	Passed

Encryption process

The encryption algorithm consists of three main stages: preprocessing of images, key and chaotic sequence management, and image encryption. Within the image encryption stage, it involves two distinct sub-stages: the permutation of regions of interest (ROI) within individual medical images using the odd-even interleaving method, and the concurrent permutation and diffusion across multiple medical images using the Simultaneous Permutation and Diffusion (SPD) algorithm. The progression of encryption is depicted in Fig. 6.

Fig. 6.

Encryption process.

Preprocessing of images

Image preprocessing fulfills dual objectives: firstly, standardizing the dimensions of all images to by padding images of varying sizes; secondly, extracting the Region of Interest (ROI) from each image to facilitate subsequent processing of these specific areas.

Step 1

Selecting w medical images and sequentially checking their sizes, choose the largest image, and pad the remaining images with black color, ensuring that all plaintext images have the same dimensions.

Step 2

The process involves extracting the Region of Interest (ROI) from each medical image individually. Typically located centrally within medical images, the ROI is surrounded by black regions. To identify the ROI, an appropriate pixel threshold is selected based on this characteristic. Pixels exceeding this threshold are assigned a value of “1”, while those below it is assigned “0”. This results in a binary image comprising pixels of “0” and “1”. A rectangle encompassing all pixels assigned “1” is then selected, representing the desired ROI. The detailed procedure is illustrated in Algorithm 1.

The effect of extracting the ROI is as shown in the Fig. 7 a, b.

Fig. 7.

(a) Original image, (b) The image with the ROI being outlined, (c) The image after encrypting the ROI regions.

Algorithm 1.

Region of Interest Localization Algorithm.

Handling of keys and chaotic sequences

The key is a vector consisting of components.

Step 3

Select the w padded medical images and concatenate them into a single large image denoted as P. Following the procedural steps outlined in Algorithm 2, the cryptographic key can be derived.

Algorithm 2.

The calculation process of key.

The function hex2dec() converts hexadecimal to decimal, dec2bin() converts decimal to binary, and bin2dec() converts binary to decimal. L represents the number of map lattice.

Step 4

The key is used to generate initial values and parameters of TLCMCML system, as Eq. (14)

14

where the sub-keys assign initial values to each lattice and the parameters and .

Image encryption

During the image encryption stage, two distinct phases are employed: initially, the regions of interest within individual medical images are scrambled using the odd-even interleaving permutation method. Subsequently, the second phase entails the simultaneous scrambling and diffusion across multiple medical images utilizing the SPD algorithm.

In this stage, we will use the odd-even interleaving permutation method to scramble each extracted ROI separately. The permutation illustration will be presented in Fig. 8. The scrambling results are as shown in the Fig. 7c.

Fig. 8.

The odd-even interleaving permutation method.

Step 5

Extract the ROI from the original medical image and save it as the image , with dimensions .

Step 6

Split the image into two equal parts, top T and bottom G.

15

,

16

.

Step 7

Keep the odd positions of T and fill the even positions of Twith the odd positions of G.

Keep the even positions of Gand fill the odd positions of Gwith the even positions of T.

Step 8

Iterate the TLCMCML system times, discard the first computation results, and obtain a matrix with size .

Step 9

Generate two chaotic sequences, AX and BX, each of length , and then sort both AX and BX.

17

,

18

.

where sort() function is used to arrange elements in a sequence in ascending order.

Step 10

Permute T and G using AI and BI respectively. Afterwards, we obtain the scrambled ROI.

Step 11

By integrating the scrambled ROI image back into the original image, we derive the resultant image after the initial round of scrambling.

After scrambling and filling back the ROI of each medical image, horizontally concatenate these w images into a large matrix . Then, separate into odd-positioned and even-positioned pixels, performing SPD algorithm on each part separately, which further enhances the randomness of the image.

Step 12

Iterate the TLCMCML system times, discard the first computation results, and obtain a matrix with size .

Step 13

Acquire two length chaotic serials and .

Step 14

Use the Eq. (19) and Eq. (20) to process the chaotic sequences x and y separately.

19

,

20

.

where gives the minimum integer larger than or equal to x, and denotes the modular operation.

Step 15

Divide sequenceX into two parts, AXand BX, each of size .Similarly, divide sequence Yinto two parts Y1 and Y2.

Step 16

Sort sequencesandseparately to obtain the indices and , respectively.

Step 17

Extract the elements at the odd positions of sequence to form sequence , and extract the elements at the even positions to form sequence . Apply the SPD algorithm separately to both and .

21

22

where , and denotes the bit-level XOR operation.

In the process of simultaneous arrangement and diffusion, the ith value is related to the value of the original image, the value of the chaotic sequence , and the previous encrypted value. As a result, the position and pixel values of the original image can be modified at the same time. Figure 9 shows a schematic of the SPD algorithm.

Fig. 9.

Example of the SPD Algorithm.

Step 18

Recombine and into C, then divide the image C into the final w encrypted images.

23

.

where reshape() function is utilized to reshape the image to a specified size.

Decryption process

The decryption algorithm functions as the inverse operation of the encryption algorithm.

Step 1

Obtain the ciphertext image and secret key.

Step 2

Substitute the secret key into the TLCMCML system to generate chaotic sequences.

Step 3

Obtain sequence Y and matrices V according to Step 12–16.

Step 4

The inverse process of using the SPD algorithm can obtain the decrypted large matrix image.

24

25

where , and denotes the bit-level XOR operation.

Step 5

Divide the large image into w small images according to Eq. (26).

26

Step 6

Extract the encrypted parts of each image separately, reverse the scrambling to obtain the original image of the ROI, and finally fill the decrypted ROI back into the original image.

Analysis of performance and simulation

In this section, we evaluate the security of the cryptographic algorithm under study. The evaluations were performed on a personal computer equipped with an Intel Core i5-12400 H CPU running at 2.50 GHz and 16 GB of RAM, using MATLAB R2016a software. The grayscale medical images used in the evaluation are from the Radiopaedia database (radiopaedia. org) and Dermatology published by the People’s Medical Publishing House³⁹. The color medical images are provided by the Dermatology Department of Qinhuangdao Maternal and Child Health Hospital.

Simulation results

We selected four medical images of varying sizes, one color medical image, and one binary image, then resized them to 512 × 512 dimensions. The proposed algorithm was applied for both encryption and decryption processes, resulting in encrypted images, decrypted images, and their respective histograms as depicted in Fig. 10. Simulation results indicate a significant disparity between the encrypted and original images, rendering it impracticable to retrieve the original image information from its encrypted counterpart. Histogram analysis reveals the pixel value distribution in the original image follows a distinct pattern, thereby posing vulnerability to decryption. In contrast, the pixel distribution in the encrypted image appears notably uniform. These experiments underscore the favorable attributes of the proposed encryption algorithm. Encryption of non-medical images can also be achieved by processing the entire image as a Region of Interest (ROI).

Fig. 10.

The encryption and decryption simulation results and histograms of images. (a) The medical image W1, (b) The medical image W2, (c) The medical image W3, (d) The medical image W4, (e) The color medical image W5, (f) The image ruler.

Key space and sensitivity analysis

Key sensitivity is a critical consideration in image encryption. Firstly, the key space should be sufficiently extensive to mitigate the risk of brute-force attacks. Secondly, encrypted images can only be successfully decrypted with the exact key; even minor modifications to the key will prevent recovery of the correct decrypted image. The initial key for the algorithm is generated by computing the SHA-512 hash value of the plaintext image, resulting in a 512-bit hash. This establishes its key space as , far exceeding the required key space of ²¹. As shown in Fig. 11, it is evident that even slight modifications in the subkey result in substantial variations in the encrypted image. Consequently, these alterations pose a significant obstacle to successfully decrypting the original ciphertext image. This highlights the algorithm’s heightened sensitivity to variations in the encryption and decryption procedures, particularly concerning the key utilized.

Fig. 11.

Key sensitivity tests in decryption. (a) Use correct key in decryption, (b) Use in decryption, (c) Use in decryption, (d) Use in decryption..

Information entropy

Information entropy serves as a quantitative measure to assess the degree of complexity within an image, wherein elevated entropy values signify increased levels of intricacy or disorder. The calculation methodology is as follows:

27

For grayscale medical images consisting of 256 intensity levels, the theoretical entropy limit is 8. Test results presented in Table 2 indicate that the entropy values of the four encrypted medical images all exceed 7.9993, averaging 7.9994, closely approaching the theoretical threshold of 8. Table 3 provides a comparative analysis of entropy among images encrypted using various algorithms, exclusively focusing on 512 512 medical images for fairness. This comparative study reveals that the proposed encryption algorithm achieves higher entropy levels, thereby demonstrating superior algorithmic performance.

Table 2.

The test results of information entropy.

Image	W1	W2	W3	W4
Plaintext	3.1128	4.1601	3.9084	3.7635
Ciphertext	7.9993	7.9992	7.9994	7.9994

Table 3.

The comparison of information entropy.

Algorithm	Ciphertext(avg)
Proposed	7.9994
Kamal’s22	7.9993
Gao’s23	7.9993
Enayatifar’s24	7.9992
Hua’s25	7.9973
Xiong’s26	7.9984

Correlation of adjacent pixels

In plaintext images, there often exists significant similarity between neighboring pixels, which leads to discernible patterns. To mitigate this issue, encrypted images should ideally exhibit minimal correlation. To evaluate the correlation properties of the proposed algorithm, exemplified by images denoted as , , and , we conducted a comparative analysis. Specifically, we examined 10,000 pairs of adjacent pixels in horizontal, vertical, and diagonal directions both before and after encryption. As illustrated in Fig. 12, the encryption process effectively alters the pixel distribution of the original image, resulting in a uniform and irregular distribution of pixels, thereby reducing inter-pixel correlation. Additionally, correlation coefficients were computed for four medical images before and after encryption, with detailed results presented in Table 4. Furthermore, Table 5 presents a comparative analysis of correlation tests conducted using alternative encryption algorithms, focusing on the absolute values of the test data and averaging them for comparison. The findings underscore the enhanced security of the proposed algorithm. Its calculation is defined by Eq. (28).

28

Fig. 12.

Correlation analysis. (a) Horizontal correlation of original image, (b) Horizontal correlation of encrypted image, (c) Vertical correlation of original image, (d) Vertical correlation of encrypted image, (e) Diagonal correlation of original image, (f) Diagonal correlation of encrypted image.

Table 4.

The correlation coefficients of adjacent pixels.

Image	Plaintext			Ciphertext
	Horizontal	Vertical	Diagonal	Horizontal	Vertical	Diagonal
W1	0.9917	0.9929	0.9845	0.0056	0.0092	0.0075
W2	0.9893	0.9867	0.9771	-0.0033	0.0069	0.0026
W3	0.9904	0.9819	0.9642	0.0072	0.0073	-0.0048
W4	0.9945	0.9940	0.9901	-0.0071	-0.0016	0.0047

Table 5.

The comparison of correlation coefficients.

Algorithm	Direction
	Horizontal	Vertical	Diagonal
Proposed	0.0058	0.0062	0.0024
Kamal’s22	0.0146	0.0114	0.0085
Farah’s27	0.0693	0.0610	0.0242
Zhang’s28	0.0239	0.0033	0.0046
Wang’s29	0.0085	0.0054	0.0049
Wu’s12	0.0164	0.0056	0.0289

where

29

30

.

Differential attack

The differential attack stands as a frequently employed method to compromise encrypted images. Consequently, effective resistance against such attacks becomes a crucial criterion for evaluating encryption algorithms. Typically, this attack involves making minor adjustments to pixel values in a plaintext image, resulting in an unchanged ciphertext image and a modified ciphertext image . Two key indicators for assessing resistance against differential attacks include:

31

32

,

33

Four medical images, each sized 512 512 pixels, were selected for testing purposes. Minor adjustments were applied to the pixel values of these medical images prior to simultaneous encryption. Subsequently, Non-Probabilistic Cryptographic Randomness (NPCR) and Unified Average Changing Intensity (UACI) tests were conducted on the resulting encrypted images. Tables 6 and 7 depict the outcomes of four medical images subjected to four rounds of NPCR and UACI testing, with subsequent averaging showing results within expected parameters. Table 8 compares these findings with data from other encryption algorithms, highlighting the encryption algorithm’ s better performance.

Table 6.

The test results of NPCR.

Image	Test1	Test2	Test3	Test4	Avg
W1	99.6237	99.6104	99.5944	99.6026	99.6077
W2	99.6173	99.6114	99.5962	99.6253	99.6125
W3	99.6095	99.6118	99.6245	99.5884	99.6085
W4	99.6136	99.6112	99.5879	99.6046	99.6043

Table 7.

The test results of UACI.

Image	Test1	Test2	Test3	Test4	Avg
W1	33.4342	33.4525	33.4436	33.4463	33.4441
W2	33.4551	33.4657	33.4972	33.4751	33.4732
W3	33.3283	33.4527	33.4474	33.4748	33.4258
W4	33.5274	33.4533	33.4792	33.4924	33.4881

Table 8.

The comparison of average of NPCR and NACI.

Algorithm	NPCR	UACI
Proposed	99.6082	33.4578
Kamal’s29	99.6010	33.4389
Gao’s23	99.5893	33.4635
Farah’s27	99.5677	33.4353
Chai’s30	99.6080	33.4240
Wang’s31	99.6093	33.4635

test

test is employed to quantify the randomness of an image, defined as³²:

34

where represents the frequency of pixel value i occurring within the image, while denotes the average frequency. The term denotes the size of the image. A higher value of indicates more pronounced deviations of pixel values from the average level, thereby reflecting a less uniform distribution of pixels³³.

According to Table 9, it is evident that our proposed algorithm has significantly better value compared to other algorithms and exhibits good randomness in encrypted images.

Table 9.

The results of test.

Plain images	Size
Proposed (Lena)	512*512	226
Shengtao’s33 (Lena)	512*512	229
Wang’s34 (Lena)	512*512	285
Alghafis’s35 (Lena)	512*512	230
Kanafchian’s36 (Lena)	512*512	246

Time complexity analysis

In response to the rapid increase in image data volume, it is imperative that image encryption algorithms exhibit high efficiency. Assuming w images P each of size , the time complexity during the region of interest (ROI) scrambling phase for each image is ensured to be less than . Furthermore, during the concurrent processes of scrambling and diffusion across w images, the aggregate time complexity is . Consequently, the proposed encryption algorithm operates with a time complexity of .

Due to the varying performance capabilities of individual computers, our analysis focuses solely on the algorithmic time complexity. As depicted in Table 10, we compare the time complexities of different algorithms.

Table 10.

Time complexity analysis.

Algorithm	Time complexity
Liu’s14	O(n×M×N) (n represents the number of images)
Xiong’s26	O(8×M×N) (one image)
Chen’s37	O(M×N) (one image)
Zhang’s 38	O(M×N) (one image)
proposed	O(n×M×N) (n represents the number of images)

Robustness analysis

During the process of transmitting and receiving encrypted images, they are vulnerable to external interference, which may lead to partial data loss or modification. Therefore, an exemplary encryption algorithm should exhibit robustness by preserving as much original information as possible in the presence of external disturbances, thereby maintaining the integrity of the encrypted images. Figures 13 and 14 depict the decryption results of four medical images subjected to different levels of cropping attacks, salt-and-pepper noise, gaussian noise and speckle noise. The research findings clearly demonstrate that despite varying degrees of attack, a significant amount of image information can still be recovered, underscoring the algorithm’s resilience.

Fig. 13.

Robustness analysis. (a) Encrypted image after 256 256 pixel cutting, (b) Encrypted image after 128 1024 pixel cutting, (c) Encrypted image after 0.1 salt and pepper noise, (d) Encrypted image after 0.0005 speckle noise, (e) Encrypted image after 0.0001 gaussian noise.

Fig. 14.

Robustness analysis. (a) Decrypted image after 256 256 pixel cutting, (b) Decrypted image after 128 1024 pixel cutting, (c) Decrypted image after 0.1 salt and pepper noise, (d) Decrypted image after 0.0005 speckle noise, (e) Decrypted image after 0.0001 gaussian noise.

Conclusion

This paper introduces an encryption algorithm designed to handle diverse image types, including multiple medical images of arbitrary sizes. Initially, medical images of varying dimensions are standardized to a uniform size and horizontally concatenated into a single large image. A key for this composite image is generated using a hash algorithm. Subsequently, parameters and initial values of a chaotic system are computed based on this key, facilitating the iteration of the proposed TLCMCML chaotic system to generate requisite chaotic sequences for subsequent encryption steps. The encryption process consists of two stages: a permutation stage and a simultaneous permutation and diffusion stage. The first stage employs an odd-even interleaving permutation method to permute regions of interest (ROIs) extracted from the image. In the second step, a synchronous scrambling and diffusion algorithm is applied to the entire large image. Through testing and analysis, this encryption algorithm has demonstrated robust encryption results and shown strong performance across various evaluation metrics. Encryption of non-medical images can also be achieved by processing the entire image as a Region of Interest (ROI). Due to constraints in computational performance, our encryption algorithm may not achieve optimal efficiency in terms of time. Future research will explore applying this algorithm to three-dimensional volumetric medical images, enabling selective encryption of ROIs within volumetric data and secondary encryption based on their significance, thereby enhancing the privacy protection of medical images.

Author contributions

Xu. proposed innovative ideas. Shang. wrote the main manuscript text. Yang. and Zou. prepared figures and tables. All authors reviewed the manuscript.

Data availability

The medical images used in this study are publicly available from the (radiopaedia.org) database.

Declarations

Competing interests

The authors declare no competing interests.