Integrating Fractional-Order Hopfield Neural Network with Differentiated Encryption: Achieving High-Performance Privacy Protection for Medical Images
Abstract
1. Introduction
- By leveraging a novel 4D fractional-order HNN with remarkable chaotic performance, we propose a high-performance medical IE scheme named MIES-FHNN-DE. The proposed MIES-FHNN-DE approach addresses the shortcomings of existing schemes, offering enhanced privacy protection for medical images.
- MIES-FHNN-DE creatively combines a 4D fractional-order HNN with a 2D discrete hyperchaotic map to collaboratively generate chaotic sequences. While exploiting the 4D fractional-order HNN’s superior chaotic performance, this design effectively overcomes the efficiency issues that undermine existing schemes.
- MIES-FHNN-DE employs the secret key to generate chaotic sequences and only utilizes hash values to control keystream transformation. Most existing medical IE schemes directly use hash values for chaotic sequence generation, which leads to significant issues with key management and the reusability of sequences. Our design effectively addresses these issues.
- MIES-FHNN-DE presents a novel plaintext-dependent dynamic partitioning mechanism and applies efficient vector-level operations in the diffusion, scrambling, and substitution steps of the encryption process. This innovative design not only introduces plaintext-related dynamics but also achieves outstanding encryption efficiency that far surpasses existing schemes.
- Our in-depth evaluation and comparative study demonstrate that MIES-FHNN-DE offers a formidable combination of outstanding security and efficiency. Its ultra-high encryption rate far exceeds the latest IE schemes, making it a superior choice for diverse emerging medical applications.
2. Fractional-Order Calculus and 4D Fractional-Order HNN
2.1. Fractional-Order Calculus
- : The Caputo fractional derivative operator. The subscript a is the lower limit of integration, the superscript C denotes “Caputo”, and is the non-integer order of the derivative.
- : The function for which the fractional derivative is being calculated, with t as the independent variable.
- : A normalization constant. is the Euler gamma function, and n is an integer such that .
- : The definite integral with respect to from a to t, used to combine the contributions of the function and its derivatives over the interval.
- : The kernel of the integral, a power function of with exponent , determining the weighting of the contributions.
- : The n-th derivative of with respect to , where n is related to the order of the fractional derivative.
- : The differential of the integration variable .
2.2. Hopfield Neural Networks
2.3. The 4D Fractional-Order HNN
3. Proposed MIES-FHNN-DE
3.1. Fusion of Chaotic Sequences
- Step 1: Input the first five components of K into the 4D fractional-order HNN to generate a chaotic sequence of length . Note that represents the size of the medical images to be encrypted, and denotes the number of these images (or image channels).
- Step 2: Initialize a sequence of length where all elements are zero, and let .
- Step 3: Create a 2D matrix by executing matrix multiplication on the remaining elements of . Specifically, .
- Step 4: Reshape into a 1D vector and let .
- Step 5: Input the last four components () of K into 2D-ELMM to obtain a chaotic sequence of length .
- Step 6: Perform vector addition on and to finalize the fusion of the chaotic sequences produced by the 4D fractional-order HNN and 2D-ELMM: .
3.2. Creation of Two Pixel Bit Matrices
- Step 1: Reshape into a 4D pixel matrix of size . Note that , , represents the size of the medical images, and denotes the number of these images (or image channels). Let extract the H3B part of each pixel in .
- Step 2: Initialize a 3D pixel matrix of size with all elements set to zero. This matrix will be used to store the weighted accumulation results of the pixel H3B parts.
- Step 3: Conduct weighted accumulation over the fourth dimension of , indexed by i (), and let .
- Step 4: Let , and then reshape into the 2D H3B pixel matrix of size .
- Step 5: Reshape into a 4D pixel matrix of size , and let to extract the L5B part of each pixel in .
- Step 6: Initialize a 3D pixel matrix of size with all elements set to zero. This matrix will be used to store the weighted accumulation results of the pixel L5B parts.
- Step 7: Conduct weighted accumulation over the 4th dimension of , indexed by j (), and let .
- Step 8: Let , and then reshape into the 2D L5B pixel matrix of size .
3.3. Determination of Control Parameters
- Step 1: Perform a modular addition operation in vector form to integrate the hash value of and the hash value of into a hash value byte vector of length 32: .
- Step 2: Initialize a zero-valued variable , which will be employed to accumulate the results of subsequent operations.
- Step 3: Employ as the index to perform four rounds of successive multiplication and accumulation operations on the 32 byte elements of :
- Step 4: Perform modular addition operations on to produce three dynamic control parameters , , and that will be used in the subsequent encryption procedures. Specifically, , , and .
3.4. Generation of Keystreams
- Step 1: Perform a truncation operation on by redefining it as , thereby discarding the first elements.
- Step 2: Set . Here, , and the keystream will be used in the first round of vector-level scrambling operations for the H3B pixel matrix .
- Step 3: Set . The keystream will be used in the second round of vector-level scrambling operations for .
- Step 4: Set . The keystream will be used in the vector-level scrambling operations for the L5B pixel matrix .
- Step 5: Set and then reshape into a 2D form of size . Here, means rounding the operand down, and the keystream will be used in the diffusion and substitution operations for .
- Step 6: Set and then reshape into a 2D form of size . The keystream will be used in the diffusion and substitution operations for .
3.5. Dynamic Partitioned Alternating Diffusion
- Step 1: Initialize two zero-valued intermediate ciphertext matrices, and , of identical size to , and then determine the first two partitioning indices:
- Step 2: Perform an XOR-based diffusion operation on the first row of . Specifically, .
- Step 3: For each integer , perform a modular addition diffusion operation on the i-th row of . Specifically, .
- Step 4: For each integer , perform an XOR-based diffusion operation on the j-th row of . Specifically, .
- Step 5: For each integer , perform a modular addition diffusion operation on the k-th row of . Specifically, .
- Step 6: Determine the last two partitioning indices:
- Step 7: Perform an XOR-based diffusion operation on the first column of . Specifically, .
- Step 8: For each integer , perform a modular addition diffusion operation on the i-th column of . Specifically, .
- Step 9: For each integer , perform an XOR-based diffusion operation on the j-th column of . Specifically, .
- Step 10: For each integer , perform a modular addition diffusion on the k-th row of . Specifically, .
3.6. Dynamic Partitioned Vector-Level Scrambling
- Step 1: Initialize two zero-valued intermediate ciphertext matrices, and , of identical size to , and then determine the first two partitioning indices:
- Step 2: Sort the first elements of the keystream to obtain the scrambling index vector . Then, perform quick row scrambling on the first columns of with :
- Step 3: Sort the next elements of to obtain the scrambling index vector . Then, perform quick row scrambling on the columns to of with :
- Step 4: Sort the next elements of to obtain the scrambling index vector . Then, perform quick row scrambling on the columns to of with :
- Step 5: Determine the last two partitioning indices:
- Step 6: Sort the next elements of to obtain the scrambling index vector . Then, perform quick column scrambling on the first rows of with :
- Step 7: Sort the next elements of to obtain the scrambling index vector . Then, perform quick column scrambling on the rows to of with :
- Step 8: Sort the last elements of to obtain the scrambling index vector . Then, perform quick column scrambling on the rows to of with :
3.7. Dynamic Partitioned Dual-Operation Substitution
- Step 1: Initialize one zero-valued intermediate ciphertext matrix of identical size to , and then determine the two partitioning indices:
- Step 2: Apply modular addition substitution to the first columns of using the keystream . Namely, .
- Step 3: Perform XOR substitution on the columns to of with . Specifically, .
- Step 4: Apply modular addition substitution on the columns to of using . Namely, .
3.8. Complete Process of MIES-FHNN-DE
- Step 1: Input the nine components of the secret key K into the 4D fractional-order HNN and 2D-ELMM based on the size and quantity of the medical images requiring encryption. Generate the chaotic sequences and . Transform into and fuse it with to obtain the fused chaotic sequence . For more details, see Section 3.1.
- Step 2: Perform splitting and weighted accumulation operations on the 3D pixel matrix composed of the input images to obtain the 2D pixel matrices and . Here, corresponds to the H3B parts of all pixels, while corresponds to the L5B parts. For more details, see Section 3.2.
- Step 3: Generate three dynamic control parameters , , and using the hash values and of and . For more details, see Section 3.3.
- Step 4: Generate the keystreams , , , , and from for encrypting and by applying . For more details, see Section 3.4.
- Step 5: Perform the “Dynamic Partitioned Alternating Diffusion” on the H3B pixel matrix with and , resulting in the intermediate ciphertext matrix . For more details, see Section 3.5.
- Step 6: Apply the “Dynamic Partitioned Vector-level Scrambling” on using and to obtain the intermediate ciphertext matrix . For more details, see Section 3.6.
- Step 7: Execute the “Dynamic Partitioned Dual-operation Substitution” on with and to obtain the intermediate ciphertext matrix . For more details, see Section 3.7.
- Step 8: Perform the second round of alternating diffusion on with and to obtain the intermediate ciphertext matrix .
- Step 9: Apply the second round of vector-level scrambling on using and to obtain the intermediate ciphertext matrix .
- Step 10: Perform one round of alternating diffusion on the L5B pixel matrix with and to obtain the intermediate ciphertext matrix .
- Step 11: Apply one round of vector-level scrambling on using and to obtain the intermediate ciphertext matrix .
- Step 12: Execute one round of dual-operation substitution on with and to obtain the intermediate ciphertext matrix .
- Step 13: Transform and into the final encrypted medical images via “Reconstruction of Images” (the mirror inverse step of “Creation of Two Pixel Bit Matrices”).
4. Security and Efficiency Evaluations
4.1. Visual Evaluation
4.2. Key Space
4.3. Key Sensitivity
4.4. Pixel Correlation
4.5. Entropy Analysis
4.6. Pixel Distribution
4.7. Differential Attacks
4.8. Robustness Evaluation
4.9. Efficiency Evaluation
4.10. Overall Comparison with Eight Latest Schemes
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
IE | Image encryption |
HNN | Hopfield neural network |
MIES-FHNN-DE | Medical IE scheme based on 4D fractional-order HNN and differentiated encryption |
NPCR | Number of pixels change rate |
UACI | Unified average changing intensity |
ANN | Artificial neural networks |
AI | Artificial intelligence |
2D-ELMM | Two-dimensional enhanced logistic modular map |
H3B | High 3-bit |
L5B | Low 5-bit |
SAC | Strict avalanche criterion |
CC | Correlation coefficient |
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Medical Image | Plaintext (Horiz., Vert., Diag.) | Ciphertext (Horiz., Vert., Diag.) | ||||
---|---|---|---|---|---|---|
Xray hand | 0.9926 | 0.9936 | 0.9860 | 0.0015 | 0.0016 | 0.0011 |
X-ray chest | 0.9891 | 0.9927 | 0.9833 | 0.0010 | 0.0006 | 0.0017 |
CT chest | 0.9426 | 0.9643 | 0.9200 | 0.0001 | 0.0006 | 0.0006 |
MRI head | 0.9303 | 0.9126 | 0.8656 | 0.0012 | 0.0015 | 0.0003 |
US abdomen (R) | 0.9621 | 0.9310 | 0.9179 | 0.0014 | 0.0006 | 0.0015 |
US abdomen (G) | 0.9587 | 0.9250 | 0.9235 | 0.0009 | 0.0013 | 0.0002 |
US abdomen (B) | 0.9668 | 0.9439 | 0.9237 | 0.0010 | 0.0004 | 0.0002 |
Skin scar (R) | 0.9842 | 0.9825 | 0.9682 | 0.0012 | 0.0010 | 0.0019 |
Skin scar (G) | 0.9869 | 0.9853 | 0.9740 | 0.0020 | 0.0018 | 0.0008 |
Skin scar (B) | 0.9867 | 0.9836 | 0.9694 | 0.0014 | 0.0006 | 0.0009 |
Medical Image | Plaintext | Ciphertext |
---|---|---|
X-ray hand | 6.3627 | 7.9994 |
X-ray chest | 7.4003 | 7.9993 |
CT chest | 6.4875 | 7.9994 |
MRI head | 5.0947 | 7.9993 |
US abdomen (R) | 4.5534 | 7.9994 |
US abdomen (G) | 4.5405 | 7.9993 |
US abdomen (B) | 4.5860 | 7.9994 |
Skin scar (R) | 6.4920 | 7.9993 |
Skin scar (G) | 6.6131 | 7.9994 |
Skin scar (B) | 6.7231 | 7.9994 |
Medical Image | NPCR | UACI |
---|---|---|
X-ray hand | 99.6105% | 33.4366% |
X-ray chest | 99.6052% | 33.4871% |
CT chest | 99.6147% | 33.4329% |
MRI head | 99.6118% | 33.4277% |
US abdomen (R) | 99.6116% | 33.4594% |
US abdomen (G) | 99.6096% | 33.5037% |
US abdomen (B) | 99.6059% | 33.4819% |
Skin scar (R) | 99.6091% | 33.4857% |
Skin scar (G) | 99.6048% | 33.4591% |
Skin scar (B) | 99.6127% | 33.4638% |
Average | 99.6096% | 33.4638% |
Optimal value | 99.6094% | 33.4635% |
Scheme | Input Scale | ||||
---|---|---|---|---|---|
Ours | 0.0046 | 0.0144 | 0.0196 | 0.0591 | 0.0830 |
[46] | 0.1524 | 0.4497 | 0.6313 | 1.8927 | 2.5712 |
[47] | 0.0768 | 0.2197 | 0.3213 | 0.9417 | 1.3806 |
[48] | 0.0407 | 0.1248 | 0.1893 | 0.5649 | 0.8892 |
[49] | 0.0324 | 0.0975 | 0.1638 | 0.4987 | 0.9118 |
[35] | 0.0203 | 0.0629 | 0.0878 | 0.2766 | 0.3755 |
Scheme | Year | Protection Target | Key Space | NPCR/UACI (%) | Entropy (bits) | Throughput (Mbps) |
---|---|---|---|---|---|---|
[47] | 2021 | Ordinary Image | 99.6088/33.4564 | 7.9993 | 6.3457 | |
[48] | 2021 | Ordinary Image | 99.6101/33.4668 | 7.9993 | 10.8975 | |
[49] | 2022 | Ordinary Image | 99.6101/33.4992 | 7.9993 | 12.7664 | |
[35] | 2023 | Ordinary Image | 99.6072/33.4572 | 7.9993 | 22.8508 | |
[50] | 2024 | Ordinary Image | 99.6120/33.4880 | 7.9993 | 3.2664 | |
[24] | 2024 | Medical Image | 99.6079/33.4534 | 7.9991 | 0.52323 | |
[22] | 2025 | Medical Image | 99.6588/33.4882 | 7.9022 | 8.6038 | |
[23] | 2025 | Medical Image | 99.6476/33.5953 | 7.9994 | 25.1848 | |
Ours | 2025 | Medical Image | 99.6096/33.4638 | 7.9994 | 102.5623 |
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Feng, W.; Zhang, K.; Zhang, J.; Zhao, X.; Chen, Y.; Cai, B.; Zhu, Z.; Wen, H.; Ye, C. Integrating Fractional-Order Hopfield Neural Network with Differentiated Encryption: Achieving High-Performance Privacy Protection for Medical Images. Fractal Fract. 2025, 9, 426. https://doi.org/10.3390/fractalfract9070426
Feng W, Zhang K, Zhang J, Zhao X, Chen Y, Cai B, Zhu Z, Wen H, Ye C. Integrating Fractional-Order Hopfield Neural Network with Differentiated Encryption: Achieving High-Performance Privacy Protection for Medical Images. Fractal and Fractional. 2025; 9(7):426. https://doi.org/10.3390/fractalfract9070426
Chicago/Turabian StyleFeng, Wei, Keyuan Zhang, Jing Zhang, Xiangyu Zhao, Yao Chen, Bo Cai, Zhengguo Zhu, Heping Wen, and Conghuan Ye. 2025. "Integrating Fractional-Order Hopfield Neural Network with Differentiated Encryption: Achieving High-Performance Privacy Protection for Medical Images" Fractal and Fractional 9, no. 7: 426. https://doi.org/10.3390/fractalfract9070426
APA StyleFeng, W., Zhang, K., Zhang, J., Zhao, X., Chen, Y., Cai, B., Zhu, Z., Wen, H., & Ye, C. (2025). Integrating Fractional-Order Hopfield Neural Network with Differentiated Encryption: Achieving High-Performance Privacy Protection for Medical Images. Fractal and Fractional, 9(7), 426. https://doi.org/10.3390/fractalfract9070426