A novel color image encryption algorithm based on fractional-order conservative memristive hyperchaotic system and extended zig-zag transform
Open accessAbstract
Color images are a valuable information carrier. This paper presents a novel image encryption scheme based on fractional-order conservative memristive hyperchaotic system (FCMHS) and extended Zig-Zag transform to protect images from being attacked. First, the dynamic characteristics of FCMHS were systematically studied and analyzed, and its potential application value in the field of information security is verified. The pseudo-random sequences generated by FCMHS passed all the tests of the National Institute of Standards and Technology (NIST). Second, a new extended Zig-Zag confusing method is designed, which realizes the scrambling of pixels in different channels of a color image. Finally, the performance of the designed encryption algorithm is analyzed using key space, information entropy, histogram, differential attack, correlation coefficient and robustness analysis. The average number of pixels change rate (NPCR) and uniform average change intensity (UACI) values of the encryption scheme are 99.61% and 33.46%, respectively. Additionally, the encryption time is only 0.12533 s for a color image of size $$256\times 256$$ . The experimental results demonstrate that our algorithm has better encryption performance and security.
AI snapshot
Written from the paper’s full text.
TL;DR
The paper proposes a color image encryption scheme combining a fractional-order conservative memristive hyperchaotic system (FCMHS) with an extended 3D Zig-Zag transform. It reports strong security metrics, including average NPCR 99.61%, UACI 33.46%, and 0.12533 s encryption time for a 256×256 color image.
Key findings
- The FCMHS-generated pseudo-random sequences passed all 15 NIST-800-22 subtests, with average p-values within [0.01, 1].
- The encryption scheme achieved average NPCR and UACI values of 99.61% and 33.46%, respectively, close to theoretical ideals (99.6094% and 33.4635%).
- Key space is reported as 10^(9×15) ≈ 2^448, exceeding the 2^100 threshold; key sensitivity tests with 10^-15 perturbations produced pixel change rates above 99.6%.
- Encrypted images had information entropy close to 8 (e.g., Lena R/G/B: 7.9976, 7.9977, 7.9971) and near-zero correlation coefficients, while encryption time was 0.12533 s for 256×256 and 0.53289 s for 512×512.
Methods
- Design: A fractional-order conservative memristive hyperchaotic system is built from a 4D conservative chaotic system and a fractional-order multistable memristor; an extended Zig-Zag transform extends 2D Zig-Zag scrambling to 3D RGB channel rearrangement.
- Encryption: Color images are divided into blocks, scrambled via chaotic-sequence-selected Zig-Zag transforms, permuted, diffused using a Fibonacci Q-matrix, and row/column substituted; decryption reverses these steps.
- Data and analysis: Tests used Lena (256×256), Baboon (512×512), and Airplane (512×512) from the USC-SIPI database; analyses included phase/bifurcation/Lyapunov/Kaplan-Yorke diagrams, NIST tests, key space, key sensitivity, histogram variance, correlation, NPCR/UACI, information entropy, cropping, noise, and execution time.
Limitations
- The paper does not state an explicit limitations section; it only lists future research directions such as combining multiple fractional-order chaotic systems and exploring quantum computing for large images.
- Evident from the text: the execution-time comparison in Table 9 uses different computer specifications across schemes, so the reported speed advantage is not hardware-controlled.
- Evident from the text: security validation is empirical (statistical and robustness tests) and does not include formal cryptanalytic proofs or side-channel/chosen-ciphertext attack analysis.
Model-written summary. Check it against the paper before relying on it.
Integrity signals
Registry
Cites
Statistics
Claims the paper makes
0 of 8 cited sources checked · 0 supported · 0 flagged
- Findingown data
The chaotic sequence generated by the proposed system passed all the test items of NIST-800-22.
“It can be seen from the test results in Table 1 that the chaotic sequence passed all the test items.”
Quote in the paperSubstantial claim - Findingown data
The FCMHS exhibits strongly nonperiodic orbits in the spaces x∼y, x∼z, y∼z, x∼w, x∼w∼u and x∼y∼z.
“spaces x ∼ y , x ∼ z , y ∼ z , x ∼ w , x ∼ w ∼ u and x ∼ y ∼ z .”
Quote in the paperSubstantial claim - Findingown data
The Lyapunov exponent is widely positive in the initial value definition domain of FCMHS.
“the Lyapunov exponent is widely positive in the initial value definition domain of FCMHS”
Quote in the paperSubstantial claim - Findingown data
The Kaplan-Yorke fractal dimension values of the FCMHS are always greater than 4.
“the values of the Kaplan-Yorke fractal dimension are always greater than 4”
Quote in the paperSubstantial claim - Findingown data
The FCMHS has complex dynamic behavior changes within the parameter interval.
“The bifurcation diagram shows that the system has complex dynamic behavior changes within the parameter interval”
Quote in the paperSubstantial claim
- Findingown data
The Lyapunov exponent diagram of the FCMHS has at least one exponent greater than 0.
“The Lyapunov exponential diagram has at least one exponent greater than 0”
Quote in the paperSubstantial claim - Findingown data
The phase trajectories of the FCMHS have no attractors in each phase space.
“The phase trajectories have no attractors in each phase space”
Quote in the paperSubstantial claim - Findingnone given
The proposed algorithm's key space is 2^448.
“Proposed | 2 448”
Quote in the paperSubstantial claim - Findingown data
The proposed algorithm has strong key sensitivity.
“the experimental results show that the proposed algorithm has a large key space, strong key sensitivity and high encryption efficiency.”
Quote in the paperSubstantial claim - Findingown data
The proposed algorithm has good encryption efficiency.
“The results show that the proposed algorithm has good encryption efficiency.”
- Also stated in: § 6 Conclusion · boosted
“the experimental results show that the proposed algorithm has a large key space, strong key sensitivity and high encryption efficiency.”
Quote in the paperSubstantial claim - Also stated in: § 6 Conclusion · boosted
- Findingnone given
The proposed algorithm has time complexity Θ(M × N).
“The time complexity of the proposed algorithm in this paper is Θ(M × N) .”
Quote in the paperSubstantial claim - Findingown data
The proposed algorithm's average information entropy result is superior to that of other algorithms.
“and its average result is superior to that of other algorithms.”
Quote in the paperSubstantial claim - Findingown data
A tiny initial value perturbation leads to a pixel change rate of the ciphertext image of more than 99.6%.
“a tiny initial value perturbation leads to the pixel change rate of the ciphertext image more than 99.6%”
Quote in the paperSubstantial claim - Findingown data
For Lena encrypted image with salt-and-pepper noise densities 0.01, 0.05, and 0.1, most of the original image information can be decrypted despite varying degrees of effect.
“adding different densities of SPN in the encrypted image, although the decrypted image is affected to varying degrees, most of the information in the original image can be decrypted”
Quote in the paperSubstantial claim - Findingown data
The tested NPCR and UACI values are close to the ideal values.
“The tested values are close to the ideal values as shown in Table 6”
Quote in the paperSubstantial claim - Findingown data
The correct plaintext image cannot be decrypted even if the key is slightly changed.
“the correct plaintext image cannot be decrypted even if the key is slightly changed”
Quote in the paperSubstantial claim - Findingown data
The variance value of the image before and after encryption changes significantly.
“The experimental data demonstrates that the variance value of the image before and after encryption changes significantly.”
Quote in the paperSubstantial claim - Findingown data
The proposed algorithm has a large key space.
“the experimental results show that the proposed algorithm has a large key space, strong key sensitivity and high encryption efficiency.”
Quote in the paperSubstantial claim - Findingown data
Encryption and decryption time increases as the size of the image increases.
“It is evident that the encryption and decryption time increases as the size of the picture increases.”
Quote in the paperNot a substantial claim - Findingown data
The encryption time is 0.12533 s for a color image of size 256 × 256.
“Additionally, the encryption time is only 0.12533 s for a color image of size 256 × 256 .”
Quote in the paperNot a substantial claim - Findingown data
The FCMHS chaotic sequences pass the NIST-800-22 randomness tests.
“Table 1 NIST-800-22 test result”
- Also stated in: § 6 Conclusion · plain
“Frequency | 0.5558 | Passed”
Quote in the paperSubstantial claim - Also stated in: § 6 Conclusion · plain
- Findingown data
The proposed algorithm produces ciphertext images with information entropy close to 8.
“By comparing the data in Table 7, the encryption algorithm designed in this paper produces ciphertext images with an information entropy close to 8”
Quote in the paperSubstantial claim - Findingown data
Original color images (Lena, Baboon, Airplane) have high correlation between neighboring pixels in horizontal, vertical, and diagonal directions.
“Figure 13 shows that the neighbouring pixels of the original image are arranged diagonally in different directions, indicating high correlation.”
- Also stated in: § 5.4 Correlation analysis · hedged
“The presence of image pixel points closed to 1 in three directions before encryption and close to 0 in the encrypted image indicates that the proposed encryption algorithm can effectively break the correlation between pixels.”
Quote in the paperSubstantial claim - Also stated in: § 5.4 Correlation analysis · hedged
- Premisecitation
Li et al. (2023) proposed a new fractional-order multistable memristor model.
“Li et al. (2023) proposes a new fractional-order multistable memristor model, which can be expressed as:”
Quote in the paperSubstantial claim0 of 1 sources checked- cites Li et al. (2023)
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- Interpretationreasoning
The phase space trajectories of the FCMHS under different combinations of variables exhibit high complexity and unpredictability.
“indicating that the phase space trajectories of the FCMHS under different combinations of variables exhibit high complexity and unpredictability.”
Quote in the paperSubstantial claim - Interpretationnone given
The extended Zig-Zag transform breaks the spatial neighbourhood of the image so that pixels that were originally neighbouring are mapped away from each other.
“Extended Zig-Zag transform breaks the spatial neighbourhood of the image so that pixels that were originally neighbouring are mapped away from each other.”
Quote in the paperNot a substantial claim - Premisecitation
Fractional-order calculus describes the complex dynamic behaviour of chaotic systems more accurately than integer-order calculus because of its long-term memory effect and non-local dependence property.
“Fractional order calculus is able to describe the complex dynamic behaviour of chaotic systems more accurately than integer order calculus with its long-term memory effect and non-local dependence property (Liu et al., 2022;Meng et al., 2024).”
Quote in the paperSubstantial claim0 of 2 sources checked- cites Liu et al., 2022
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
- cites Meng et al., 2024 source
- Interpretationown data
The chaotic sequences generated by the proposed system have good randomness.
“Therefore, the chaotic sequences generated by the proposed system have good randomness.”
Quote in the paperSubstantial claim - Interpretationreasoning
In three-dimensional space, the extended Zig-Zag transform disrupts the coupling information between RGB channels.
“In three-dimensional space, the coupling information between RGB channels is disrupted”
Quote in the paperSubstantial claim - Interpretationown data
The FCMHS is a chaotic system with highly sensitive initial conditions.
“which verifies that FCMHS is a chaotic system with highly sensitive initial conditions.”
- Also stated in: § 5.1 Key space analysis · plain
“The FCMHS is a nonlinear chaotic system that is extremely sensitive to initial conditions.”
Quote in the paperSubstantial claim - Also stated in: § 5.1 Key space analysis · plain
- Interpretationown data
The FCMHS is highly sensitive to its parameters.
“reflecting the high sensitivity of the system to the parameters.”
Quote in the paperSubstantial claim - Interpretationown data
The proposed encryption algorithm has better resistance to statistical attacks.
“The analysis of the experimental results in Fig. 12 and Table 4 indicates that the proposed encryption algorithm has better resistance to statistical attacks.”
Quote in the paperSubstantial claim - Interpretationown data
The proposed encryption algorithm can effectively break the correlation between neighboring pixels in ciphertext images.
“The presence of image pixel points closed to 1 in three directions before encryption and close to 0 in the encrypted image indicates that the proposed encryption algorithm can effectively break the correlation between pixels.”
Quote in the paperSubstantial claim - Interpretationown data
The encryption process can alter the distribution of the pixel information in the plaintext image so that the plaintext information can be hidden as much as possible in the ciphertext image.
“This shows that the encryption process can alter the distribution of the pixel information in the plaintext image so that the plaintext information can be hidden as much as possible in the ciphertext image.”
Quote in the paperSubstantial claim - Interpretationown data
The proposed algorithm can resist statistical attacks.
“It is able to resist statistical statistics, differential attacks, cropping attacks and Noise attacks.”
Quote in the paperSubstantial claim - Interpretationown data
The proposed encryption algorithm has strong resistance to cropping attacks.
“Therefore, the proposed encryption algorithm based on the fractional-order conservative memristive hyperchaotic system and extended Zig-Zag transform has a strong resistance to cropping attacks.”
- Also stated in: § 6 Conclusion · boosted
“It is able to resist statistical statistics, differential attacks, cropping attacks and Noise attacks.”
Quote in the paperSubstantial claim - Also stated in: § 6 Conclusion · boosted
- Interpretationown data
The proposed encryption algorithm can effectively guarantee the security of transmitted image information.
“Therefore, the encryption algorithm designed in this paper can effectively guarantee the security of transmitted image information.”
Quote in the paperSubstantial claim - Interpretationown data
The encryption algorithm based on FCMHS and extended Zig-Zag transform has good sensitivity to plaintext.
“indicating that the encryption algorithm based on FCMHS and extended Zig-Zag transform has good sensitivity to plaintext and can resist differential and selective plaintext attacks.”
Quote in the paperSubstantial claim - Interpretationnone given
The chaotic sequences required for decryption are deterministically regenerated by FCMHS using the identical cryptographic key used in encryption, enabling precise reconstruction of the original data.
“The chaotic sequences required for decryption are deterministically regenerated by FCMHS through deployment of the identical cryptographic key utilized in the encryption stage, thereby enabling precise reconstruction of the original data.”
Quote in the paperSubstantial claim - Interpretationown data
The encryption algorithm is extremely sensitive to the key.
“From Table 3, it can be seen that the encryption algorithm is extremely sensitive to the key”
Quote in the paperSubstantial claim
- May combine two claims. 16
- The quote only partly states this claim. 2
- Classified as probably not a substantial claim. 4
May combine two claims.
Findingown dataThe histogram of the original images does not demonstrate a uniform distribution, whereas the histogram of the encrypted images displays a uniform distribution.
“As can be seen in Fig. 12 (a-c), the histogram of the original images do not demonstrate a uniform distribution.Conversely, in Fig. 12 (d-f) the histogram of the encrypted images display a uniform distribution.”
Quote in the paperSubstantial claimMay combine two claims.
Findingown dataThe average NPCR and UACI values of the encryption scheme are 99.61% and 33.46%, respectively.
“The average number of pixels change rate (NPCR) and uniform average change intensity (UACI) values of the encryption scheme are 99.61% and 33.46%, respectively.”
Quote in the paperSubstantial claimThe quote only partly states this claim.
Findingown dataFor encrypted Lena, Baboon, and Airplane images, correlation coefficients between neighboring pixels in horizontal, vertical, and diagonal directions are close to 0.
“The presence of image pixel points closed to 1 in three directions before encryption and close to 0 in the encrypted image indicates that the proposed encryption algorithm can effectively break the correlation between pixels.”
Quote in the paperSubstantial claimThe quote only partly states this claim.
Findingown dataThe proposed scheme has a very large key space compared with recent literature.
“It is clear that the proposed scheme has a very large key space.”
Quote in the paperSubstantial claimMay combine two claims.
Findingown dataFor Lena ciphertext cropped at rates 1/16, 1/8, and 1/4, the decrypted images are somewhat blurred but the image content can still be roughly seen.
“The resulting figure shows that the decrypted image is somewhat blurred and the image content can still be roughly seen”
Quote in the paperSubstantial claimMay combine two claims.
FindingcitationAssuming a system accuracy of 10^-15, the total key space of the proposed scheme is 10^9×15 ≈ 2^448, which far exceeds the minimal threshold of 2^100.
“Assuming a system accuracy of 10 -15 , the total key space is 10 9×15 ≈ 2 448 , which far exceeds the minimal thresh- old outlined in 2 100 (Koçak et al. 2024).”
Quote in the paperSubstantial claim0 of 1 sources checked- cites Koçak et al. 2024
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
May combine two claims.
Findingown dataThe encrypted image bears no correlation with the original image, while the reconstructed image is visually indistinguishable from the original.
“Figure 9 illustrates that the encrypted image bears no correlation with the original, while the reconstructed image is visually indistinguishable from the original.”
Quote in the paperSubstantial claimMay combine two claims.
InterpretationreasoningThe extended Zig-Zag transform enhances the complexity of the disarrangement algorithm and is more capable of combating statistical attacks than the traditional Zig-Zag transform method.
“which enhances the complexity of the disarrangement algorithm, and the Extended Zig-Zag transform is more capable of combating statistical attacks than the traditional Zig-Zag transform method.”
Quote in the paperSubstantial claimClassified as probably not a substantial claim.
PremisecitationChaotic systems have high sensitivity and unpredictability to initial conditions and system parameters.
“Chaotic systems have properties such as high sensitivity and unpredictability to initial conditions and system parameters (Youssef et al. 2024).”
Quote in the paperSubstantial claim0 of 1 sources checked- cites Youssef et al. 2024 source
Classified as probably not a substantial claim.
PremisecitationCang et al. (2017) presented a Hamiltonian energy-conserving 4D conservative chaotic system with complex dynamical properties.
“Cang et al. (2017) presented a Hamiltonian energy-conserving 4D conservative chaotic system with complex dynamical properties.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites Cang et al. (2017) source
May combine two claims.
Interpretationown dataThe widely positive Lyapunov exponent of FCMHS is conducive to achieving strong diffusivity and high obfuscation of the global pixels of an image, and thus effectively resisting statistical attacks and differential attacks.
“which is conducive to achieving strong diffusivity and high obfuscation of the global pixels of the image, and thus effectively resisting statistical attacks and differential attacks.”
Quote in the paperSubstantial claimMay combine two claims.
PremisecitationUnlike dissipative chaotic systems, conservative chaotic systems usually do not have an attractor structure but exhibit stronger pseudo-random characteristics.
“Unlike DCSs, CCSs usually do not have the attractor structure (Ji'e et al. 2022) but exhibit stronger pseudo-random characteristics.”
Quote in the paperSubstantial claim0 of 1 sources checked- cites Ji'e et al. 2022
Couldn’t be matched to a paper in Crossref, so it can’t be checked.
Classified as probably not a substantial claim.
PremisecitationImage encryption technology can effectively prevent sensitive images from being stolen or tampered with during transmission and storage.
“Image encryption technology can effectively prevent sensitive images from being stolen or tampered with during transmission and storage (Kaur et al. 2022).”
Quote in the paperSubstantial claim0 of 1 sources checked- cites Kaur et al. 2022 source
May combine two claims.
InterpretationreasoningThe extended Zig-Zag transform method achieves effective pixel rearrangement in both spatial and channel dimensions, which significantly enhances the disambiguation degree and encryption strength of the image and shows a good comprehensive performance.
“The method achieves effective pixel rearrangement in both spatial and channel dimensions, which significantly enhances the disambiguation degree and encryption strength of the image and shows a good comprehensive performance.”
Quote in the paperNot a substantial claimMay combine two claims.
Interpretationown dataThe proposed encryption scheme is effective and yields satisfactory performance.
“These results confirm that the proposed encryption scheme is effective and yields satisfactory performance.”
Quote in the paperSubstantial claimMay combine two claims.
Interpretationown dataThe proposed algorithm has better encryption performance and security.
“The experimental results demonstrate that our algorithm has better encryption performance and security.”
Quote in the paperSubstantial claim- Rests on an assumption: Combining Zig-Zag transform with a chaotic system enhances the complexity and security of image encryption and improves resistance to statistical and analytical attacks.
- Rests on an assumption: The security of image encryption methods using only chaotic systems depends on the complexity of chaotic systems and should be explored in combination with other theories to improve security.
May combine two claims.
Interpretationown dataThe encryption algorithm based on FCMHS and extended Zig-Zag transform can resist differential and selective plaintext attacks.
“indicating that the encryption algorithm based on FCMHS and extended Zig-Zag transform has good sensitivity to plaintext and can resist differential and selective plaintext attacks.”
- Also stated in: § 6 Conclusion · plain
“It is able to resist statistical statistics, differential attacks, cropping attacks and Noise attacks.”
Quote in the paperSubstantial claim- Also stated in: § 6 Conclusion · plain
May combine two claims.
Interpretationown dataThe proposed encryption algorithm has very good robustness and can resist noise attacks.
“indicating that the encryption algorithm designed in this paper has a very good robustness able to resist noise attacks.”
- Also stated in: § 6 Conclusion · plain
“It is able to resist statistical statistics, differential attacks, cropping attacks and Noise attacks.”
Quote in the paperSubstantial claim- Also stated in: § 6 Conclusion · plain
May combine two claims.
Interpretationown dataThe proposed algorithm avoids the effect of data loss to a certain extent and recovers the original image to a large extent.
“demonstrating that the algorithm avoids the effect of data loss to a certain extent and recovers the original image to a large extent.”
Quote in the paperSubstantial claimMay combine two claims.
InterpretationreasoningThe key generation mechanism has good randomness and a highly complex state space, which effectively avoids key guessing and exhaustive attacks.
“Therefore, the key generation mechanism has good randomness and highly complex state space, which effectively avoids key guessing and exhaustive attacks.”
Quote in the paperSubstantial claim- Rests on an assumption: The security of image encryption methods using only chaotic systems depends on the complexity of chaotic systems and should be explored in combination with other theories to improve security.
May combine two claims.
Interpretationown dataThe proposed image encryption algorithm has good key sensitivity and security.
“which reflects the system's good key sensitivity, and significantly enhances the resistance to similar key attack and exhaustive attack.”
- Also stated in: § 5.2 Key sensitivity analysis · hedged
“which indicates that the proposed image encryption algorithm has good key sensitivity and security.”
Quote in the paperSubstantial claim- Also stated in: § 5.2 Key sensitivity analysis · hedged
Probably not a substantial claim: may not pass S4 (based).
Interpretationnone givenThe extended Zig-Zag transform method proposed in this paper makes full use of the information redundancy and correlation between dimensions.
“The extended Zig-Zag transform method proposed in this paper makes full use of the information redundancy and correlation between dimensions.”
Quote in the paperNot a substantial claim
The paper relies on this without support.
The nonlinear characteristics of memristors can cause a system to exhibit more complex chaotic properties.
“The nonlinear characteristics of memristors can cause the system to exhibit more complex chaotic properties.”
The paper relies on this without support.
The introduction of a memristor makes a chaotic system more sensitive to initial conditions, which can improve the security performance of an image encryption algorithm based on that system.
“Moreover, the introduction of the memristor makes the system more sensitive to the initial conditions, which can improve the security performance of the image encryption algorithm based on the memristor chaotic system.”
The paper relies on this without support.
Conservative chaotic systems have not been widely studied.
“In contrast, conservative chaotic systems (CCSs) have not been widely studied.”
The paper relies on this without support.
Image data has a larger volume and stronger inter-pixel correlation than traditional text data, so image encryption requires higher efficiency and stronger security.
“Compared with traditional text encryption, image data has a larger volume and stronger inter-pixel correlation, which determines that its encryption processing requires higher efficiency and stronger security in design.”
The paper relies on this without support.
Combining Zig-Zag transform with a chaotic system enhances the complexity and security of image encryption and improves resistance to statistical and analytical attacks.
“The combination of this transform with a chaotic system enhances the complexity and security of image encryption and improves the resistance to statistical and analytical attacks.”
The paper relies on this without support.
The security of image encryption methods using only chaotic systems depends on the complexity of chaotic systems and should be explored in combination with other theories to improve security.
“The security of image encryption methods using only chaotic systems depends on the complexity of chaotic systems and should be explored in combination with other theories to improve the security of image encryption.”
The paper relies on this without support.
The properties of memristor have more advantages in constructing conservative chaotic system.
“The properties of memristor have more advantages in constructing conservative chaotic system.”
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
Noise densities disagree with Fig. 15 caption
MinorThe text says the three salt-and-pepper noise densities are 0.01, 0.05, and 0.1, but the figure caption labels them 0.01, 0.1, and 0.2.
“Figure 15 shows the Lena image after the SPN attack with noise densities of 0.01, 0.05, and 0.1, respectively.”
“Fig. 15 SPN attack on Lena image with different noise rates.(a) SPN 0.01, (b) SPN 0.1, (c) SPN 0.2, (d)-(f) are the corresponding decryption images”
Pixel vector loses the color-image dimension
MinorThe input contains 3MN values, but Step 6 creates a permutation of only MN positions and Step 8 specifies an M × N matrix. The stated steps do not account for the other two color channels while claiming to encrypt and decrypt a color image.
“Step 1: Input an original color plaintext image C of dimen- sion M × N× 3 and divide it into n equal image blocks C n .”
“Step 8: Convert the sequence PCZ′ into the matrix PCZ of size M × N.”
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“The secret key is constituted of five initial values x 0 , y 0 , z 0 , w 0 , u 0 and four parameters b , q , c , k of FCMHS. Assuming a system accuracy of 10 –15 , the total key space is 10 9 × 15 ≈ 2 448 , which far exceeds the minimal threshold outlined in 2 100 (Koçak et al. 2024).”
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Self-citationAn image encryption algorithm based on dynamic row scrambling and Zigzag transformation
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“…systems have been playing an increasingly important role in a variety of fields such as image encryption (Moysis et al. 2025; Shi et al. 2024; Deng et al. 2025; Zhang et al., 2024), secure communication (Alexan et al. 2024a; Yu et al. 2025a, b) and neural network construction (Wang et al.…”
A novel color image encryption method based on new three-dimensional chaotic mapping and DNA coding
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“Qi et al. (2020) analyzed in detail the mechanisms of periodic to quasi-periodic, chaotic and strongly chaotic Hamiltonian conserved chaotic systems.”
Yu, F., Lin, Y., Yao, W., Cai, S., Lin, H., & Li, Y. (2024) · Neural Networks · doi:10.1016/j.neunet.2024.106904 · Cited by 86 · cited 1× · see citing works
“…been playing an increasingly important role in a variety of fields such as image encryption (Moysis et al. 2025; Shi et al. 2024; Deng et al. 2025; Zhang et al., 2024), secure communication (Alexan et al. 2024a; Yu et al. 2025a, b) and neural network construction (Wang et al. 2024; Li et al. 2024).”
Four-dimensional autonomous dynamical systems with conservative flows: two-case study
Cang, S., Wu, A., Wang, Z., & Chen, Z. (2017) · Nonlinear Dynamics · doi:10.1007/s11071-017-3599-6 · Cited by 81 · cited 1× · see citing works
“2.1 Mathematical model Cang et al. (2017) presented a Hamiltonian energy-conserving 4D conservative chaotic system with complex dynamical properties.”
InfluentialA new fast multiple color image encryption algorithm
Alexan, W., Hosny, K., & Gabr, M. (2025) · Cluster Computing · doi:10.1007/s10586-024-04919-0 · Cited by 79 · cited 1× · see citing works
“Subsequently, they designed a multiple color image encryption algorithm based on the hyperchaotic system (Alexan et al. 2025a, 2025b).”
Kaur, G., Agarwal, R., & Patidar, V. (2022) · The Visual Computer · doi:10.1007/s00371-021-02066-w · Cited by 78 · see citing works
Symmetric multi-double-scroll attractors in Hopfield neural network under pulse controlled memristor
Li, J., Wang, C., & Deng, Q. (2024) · Nonlinear Dynamics · doi:10.1007/s11071-024-09791-6 · Cited by 75 · cited 1× · see citing works
“…been playing an increasingly important role in a variety of fields such as image encryption (Moysis et al. 2025; Shi et al. 2024; Deng et al. 2025; Zhang et al., 2024), secure communication (Alexan et al. 2024a; Yu et al. 2025a, b) and neural network construction (Wang et al. 2024; Li et al. 2024).”
Fibonacci Q-Matrix, Hyperchaos, and Galois Field (2⁸) for Augmented Medical Image Encryption
El-Damak, D., Alexan, W., Mamdouh, E., El-Aasser, M., Fathy, A., & Gabr, M. (2024) · IEEE Access · doi:10.1109/access.2024.3433499 · Cited by 75 · cited 1× · see citing works
“Damak et al. (2024) and Hosny et al. (2021) proposed an image encryption scheme based on Fibo-nacci Q-Matrix and hyperchaotic system, respectively.”
Youssef, M., Gabr, M., Alexan, W., Mansour, M. B. M., Kamal, K., Hosny, H., & El-Damak, D. (2024) · IEEE Access · doi:10.1109/access.2024.3454512 · Cited by 74 · cited 1× · see citing works
“Chaotic systems have properties such as high sensitivity and unpredictability to initial conditions and system parameters (Youssef et al. 2024).”
Chaos-based medical image encryption scheme using special nonlinear filtering function based LFSR
Deb, S., & Bhuyan, B. (2021) · Multimedia Tools and Applications · doi:10.1007/s11042-020-10308-7 · Cited by 65 · cited 1× · see citing works
“Deb et al. (2021) proposed a medical image encryption scheme based on a Logistic-Tent system and Arnold’s cat map.”
Wang, C., Luo, D., Deng, Q., & Yang, G. (2024) · Chaos Solitons & Fractals · doi:10.1016/j.chaos.2024.115471 · Cited by 63 · cited 1× · see citing works
“…been playing an increasingly important role in a variety of fields such as image encryption (Moysis et al. 2025; Shi et al. 2024; Deng et al. 2025; Zhang et al., 2024), secure communication (Alexan et al. 2024a; Yu et al. 2025a, b) and neural network construction (Wang et al. 2024; Li et al. 2024).”
Alexan, W., Elabyad, N., Khaled, M., Osama, R., El-Damak, D., Abd El Ghany, M. A., Korayem, Y., Mamdouh, E., & Gabr, M. (2024) · IEEE Access · doi:10.1109/access.2024.3446733 · Cited by 53 · see citing works
A Simple Method for Constructing a Family of Hamiltonian Conservative Chaotic Systems
Ji’e, M., Yan, D., Sun, S., et al. (2022) · IEEE Transactions on Circuits and Systems I Regular Papers · doi:10.1109/tcsi.2022.3172313 · Cited by 52 · see citing works
An efficient image encryption scheme using elementary cellular automata with novel permutation box
Kumar, A., & Raghava, N. S. (2021) · Multimedia Tools and Applications · doi:10.1007/s11042-021-10750-1 · Cited by 48 · see citing works
A New Four-Dimensional Non-Hamiltonian Conservative Hyperchaotic System
Gu, S., Du, B., & Wan, Y. (2020) · International Journal of Bifurcation and Chaos · doi:10.1142/s0218127420502429 · Cited by 44 · cited 1× · see citing works
“Gu et al. (2020) presented a new four-dimensional non-Hamiltonian conserved hyperchaotic system and analyses its dynamical properties.”
Yu, F., Tan, B., He, T., et al. (2025) · Mathematics · doi:10.3390/math13050726 · Cited by 43 · cited 1× · see citing works
“…been playing an increasingly important role in a variety of fields such as image encryption (Moysis et al. 2025; Shi et al. 2024; Deng et al. 2025; Zhang et al., 2024), secure communication (Alexan et al. 2024a; Yu et al. 2025a, b) and neural network construction (Wang et al. 2024; Li et al. 2024).”
Yuan, Y., Yu, F., Tan, B., Huang, Y., Yao, W., Cai, S., & Lin, H. (2025) · Chaos An Interdisciplinary Journal of Nonlinear Science · doi:10.1063/5.0238893 · Cited by 42 · cited 1× · see citing works
“Therefore, many scholars have turned their attention to the field of conserved chaos (Yuan et al. 2025), and they have proposed a number of construction methods for CCSs. Dong et al. (2019) proposed a general method for designing CCS based on Hamilton system.”
Image encryption based on novel Hill Cipher variant and 2D-IGSCM hyper-chaotic map
Zheng, Y., Huang, Q., Cai, S., et al. (2024) · Nonlinear Dynamics · doi:10.1007/s11071-024-10324-4 · Cited by 41 · see citing works
Image encryption algorithm based on the dynamic RNA computing and a new chaotic map
Zhou, S., Wei, Y., Zhang, Y., Iu, H. H., & Zhang, H. (2024) · Integration · doi:10.1016/j.vlsi.2024.102336 · Cited by 39 · cited 1× · see citing works
“Zhou et al. (2025) presented an image encryption algorithm using the dynamic RNA computing and chaotic map.”
A new image encryption algorithm based on cubic fractal matrix and L-LCCML system
Zhao, H., Wang, S., & Fu, Z. (2024) · Chaos Solitons & Fractals · doi:10.1016/j.chaos.2024.115076 · Cited by 36 · cited 1× · see citing works
“Zhao et al. (2024) investigated a new image encryption algorithm based on L-LCCML system.”
Meng, F., & Gu, Z. (2023) · Fractal and Fractional · doi:10.3390/fractalfract7110795 · Cited by 34 · cited 1× · see citing works
“Meng et al. (2023) designed a color image encryption algo-rithm using extended DNA coding and Zig-Zag transform.”
Self-citationExploiting Circular Shifts for Efficient Chaotic Image Encryption
Moysis, L., Lawnik, M., Alexan, W., Goudos, S. K., Baptista, M. S., & Fragulis, G. F. (2025) · IEEE Access · doi:10.1109/access.2025.3572589 · Cited by 33 · cited 1× · see citing works
“In recent years, chaotic systems have been playing an increasingly important role in a variety of fields such as image encryption (Moysis et al. 2025; Shi et al. 2024; Deng et al. 2025; Zhang et al., 2024), secure communication (Alexan et al. 2024a; Yu et al. 2025a, b) and neural network…”
Improved reverse zigzag transform and DNA diffusion chaotic image encryption method
Guo, Z., & Sun, P. (2022) · Multimedia Tools and Applications · doi:10.1007/s11042-022-12269-5 · Cited by 29 · cited 1× · see citing works
“Guo et al. (2022) proposed an image encryption algorithm using the reverse Zig-Zag transformation method.”
A new controllable multi-wing chaotic system: applications in high-security color image encryption
Ding, P., Geng, P., & Hu, W. (2024) · The Journal of Supercomputing · doi:10.1007/s11227-024-06635-3 · Cited by 27 · see citing works
A class of 5D Hamiltonian conservative hyperchaotic systems with symmetry and multistability
Dong, Q., Zhou, S., Zhang, Q., et al. (2022) · Nonlinear Dynamics · doi:10.1007/s11071-022-07735-6 · Cited by 24 · cited 1× · see citing works
“Dong et al. (2022) investigated five-dimensional Hamiltonian CCS with symmetry and multistability.”
Pan, H., Li, G., Gu, Y., et al. (2024) · Nonlinear Dynamics · doi:10.1007/s11071-024-09880-6 · Cited by 20 · cited 1× · see citing works
“Pan et al. (2024) constructed a class n-dimensional CCS and verified its effectiveness by circuit simulation.”
Firing activities in a fractional-order Hindmarsh–Rose neuron with multistable memristor as autapse
Li, Z., Xie, W.-Q., Zeng, J., et al. (2022) · Chinese Physics B · doi:10.1088/1674-1056/ac65f7 · Cited by 19 · cited 2× · see citing works
“Li et al. (2023) proposed in a novel fractional multistable locally active memristor.”
A new 5D fractional-order conservative hyperchaos system
Tian, B., Peng, Q., Leng, X., et al. (2022) · Physica Scripta · doi:10.1088/1402-4896/aca568 · Cited by 18 · see citing works
A novel infinitely coexisting attractor and its application in image encryption
Shi, Q., An, X., Yang, F., & Li, Z. (2024) · Multimedia Tools and Applications · doi:10.1007/s11042-023-17999-8 · Cited by 15 · cited 1× · see citing works
“…recent years, chaotic systems have been playing an increasingly important role in a variety of fields such as image encryption (Moysis et al. 2025; Shi et al. 2024; Deng et al. 2025; Zhang et al., 2024), secure communication (Alexan et al. 2024a; Yu et al. 2025a, b) and neural network…”
Color Image Encryption Based on LSS-Type Coupled Mapped Lattice
Zhang, F., & Wang, X. (2024) · IEEE Access · doi:10.1109/access.2024.3378276 · Cited by 15 · cited 1× · see citing works
“…been playing an increasingly important role in a variety of fields such as image encryption (Moysis et al. 2025; Shi et al. 2024; Deng et al. 2025; Zhang et al., 2024), secure communication (Alexan et al. 2024a; Yu et al. 2025a, b) and neural network construction (Wang et al. 2024; Li et al.…”
Leng, X., Du, B., Gu, S., et al. (2022) · Nonlinear Dynamics · doi:10.1007/s11071-022-07498-0 · Cited by 14 · cited 1× · see citing works
“Leng et al. (2022) investigated the dynamical behaviours of fractional-order CCS and its DSP implementation.”
Yan, M., & Li, S. (2025) · Soft Computing · doi:10.1007/s00500-025-10443-z · Cited by 6 · cited 1× · see citing works
“Yan et al. (2025) designed an image encryption algorithm based on five-dimensional Hamiltonian CCS. Yu et al. (2025a, b) investigated the dynamics of tunable five-dimensional conserved amnesia hyperchaotic systems with transient quasi-periodic properties and designed an image encryption algorithm…”
Sarfraz, M., Zhou, J., Islam, M., et al. (2025) · Fractal and Fractional · doi:10.3390/fractalfract9020076 · Cited by 2 · cited 1× · see citing works
“The Kaplan-Yorke dimension (Sarfraz et al. 2025) is a fractional-dimensional (non-integer) metric used to estimate how many free dimensions of a system are active in phase space.”
A class of 5D Hamiltonian conservative hyperchaotic systems with symmetry and multistabilityPreprint
Dong, Q., Zhou, S., Zhang, Q., et al. (2022) · Research Square · doi:10.21203/rs.3.rs-1632303/v1 · Cited by 1 · see citing works
Meng, F. (2023) · International Conference on Artificial Intelligence and Computer Science (AICS) · doi:10.1117/12.3009385
Self-citationA New Controllable Multi-Wing Chaotic System: Applications in High- Security Color Image EncryptionPreprint
Ding, P., Geng, P., & Hu, W. (2024) · Research Square · doi:10.21203/rs.3.rs-5008632/v1
Image Encryption based on novel Hill Cipher Variant and 2D-IGSCM Hyper-chaotic Map
Zheng, Y., Huang, Q., Cai, S., et al. (2024) · doi:10.21203/rs.3.rs-4441118/v1
Flagged first, then by citations
Cited by
Sorted by citations
Nabil, H., & Tayeb, H. (2025) · The European Physical Journal Plus · doi:10.1140/epjp/s13360-025-06953-2 · Cited by 8 · see citing works
“[58] (0.5328 s ), despite the latter two utilizing a significantly more powerful CPU (Intel Ultra 7, 3.80 GHz) and 32 GB RAM, while being moderately slower than Ref.”
InfluentialHu, C., Zhang, L., Gu, Q., Kang, C., Ning, B., Huang, Y., & Chen, R. S. (2026) · Journal of King Saud University - Computer and Information Sciences · doi:10.1007/s44443-026-00581-1 · Cited by 4 · see citing works
“Nevertheless, many existing chaos-based encryption schemes are mainly tailored for single-image scenarios, which limits their applicability in large-scale medical data protection (Gao et al. 2025; Meng et al. 2025).”
Secure color image encryption using a fractional-order chaotic PRNG and DNA-level operations
Yang, C., Liu, Y., Sun, X., Han, Q., & Zhou, D. (2026) · Scientific Reports · doi:10.1038/s41598-026-48822-7 · Cited by 5 · see citing works
Zhang, W., Chen, J., & Kang, Z. (2026) · Physica Scripta · doi:10.1088/1402-4896/ae32d7 · Cited by 2 · see citing works
Alsubaei, F. S., Eltoukhy, M. M., Darwish, M. M., & Hosny, K. M. (2026) · AIMS Mathematics · doi:10.3934/math.2026704 · Cited by 2 · see citing works
Image encryption optimization algorithm based on GA-PSO and DNA convolutional code
Tian, J., Liu, M., Yang, S., & Shi, D. (2026) · Journal of King Saud University - Computer and Information Sciences · doi:10.1007/s44443-026-00728-0 · Cited by 1 · see citing works
ÖZDEMİR, B., & Doğan, N. (2026) · Fractal and Fractional · doi:10.3390/fractalfract10070484 · Cited by 1 · see citing works
“The closer the entropy value of the encryption scheme is to the ideal value of 8, the more difficult it becomes for attackers to extract information from the encrypted image [37,39,43,46,61].”
Enhancing image encryption security with a 2D SF-SIMM hyperchaotic map
Al–Jamal, M. F., Al‐Talib, M., & Baniabedalruhman, A. (2026) · Multimedia Tools and Applications · doi:10.1007/s11042-026-21413-4
“Meng et al. [18] proposed a fractional-order conservative mem-ristive hyperchaotic system, and Kumar [19] combined a 4D Lorenz system with timestamp hashing and wavelet packet diffusion.”
Li, Y., Li, M., & Wang, Y. (2026) · The European Physical Journal Plus · doi:10.1140/epjp/s13360-026-07835-x
Long, X., Tang, M., Xiong, L., & Bian, H. (2026) · The European Physical Journal Plus · doi:10.1140/epjp/s13360-026-08088-4
Zhai, S., Cheng, J., Li, G., et al. (2026) · Journal of King Saud University - Computer and Information Sciences · doi:10.1007/s44443-026-01060-3
“Fractional calculus extends classical integer-order calculus to non-integer orders, providing a powerful mathematical framework for describing the inherent memory and non-local interactions found in many materials and physical processes (Gokul and Said 2025; Meng et al. 2025; Li et al. 2025).”
A novel 4D conservative memristive hyperchaotic system and its application in image encryption
Xiao, H., Wang, Y., & Jin, J. (2026) · Journal of King Saud University - Computer and Information Sciences · doi:10.1007/s44443-026-01158-8
“Unlike dissipative systems that possess negative divergence (Deng et al. 2026; Li et al. 2026b), conservative systems exhibit zero divergence, indicating zero dissipation across all state variables (Meng et al. 2025; Zhang et al. 2023).”
Bian, H., Zhou, S., Xiong, L., et al. (2026) · Physica Scripta · doi:10.1088/1402-4896/ae9ceb
“Fractional-order hyperchaos, four-wing chaotic systems, and DeepFace-assisted encryption further improved color-image and content-aware protection [34–36].”
Secure Chaotic Image Encryption Implemented in FPGA
Sarkar, P., Das, A. K., Saha, A., et al. (2026) · doi:10.1109/ciacon70148.2026.11689361
- Authors
- Meng, F., Wu, G., & Zhang, J.
- Affiliations
- Yancheng Teachers University (CN)
- Nanjing Normal University of Special Education (CN)
- Corresponding author
- Meng, Fanqi
- Zhang, Juxiao
- Venue
- Journal of King Saud University Computer and Information Sciences
- Published
- 2025-08
- Volume
- 37
- Issue
- 6
- ISSN
- 1319-1578 (ISSN-L)
- 2213-1248
- OpenAlex
- W4412992206
- Semantic Scholar
- 77a9da6a8c0f5e4223ca21ec1c5bc142a3d78742
- Cited by
- 14
- References
- 43
- Citations by year
- 2026: 13 · 2025: 1
- FWCI
- 5.25
- Type
- journal article
- Language
- en
- Open access
- gold
- doi.org
- Licence
- Locations
- Journal of King Saud University - Computer and Information Sciences (publishedVersion, cc-by-nc-nd)
- DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
- Funders
- Natural Science Research of Jiangsu Higher Education Institutions of China (23KJA120004)
- Topics
- Physical Sciences › Computer Science › Computer Vision and Pattern Recognition › Chaos-based Image/Signal Encryption
- Physical Sciences › Computer Science › Artificial Intelligence › Computer Science and Engineering
- Physical Sciences › Computer Science › Computer Vision and Pattern Recognition › Advanced Steganography and Watermarking Techniques
- Keywords
- color image encryption, conservative chaotic systems, pixel scrambling, pseudorandom sequence generation, key space analysis, NPCR and UACI, differential attack resistance, image scrambling
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