C OMPARATIVE S TUDY OF C HAOTIC S YSTEM FOR E NCRYPTION

- Chaotic systems leverage their inherent complexity and unpredictability to generate cryptographic keys, enhancing the security of encryption algorithms. This paper presents a comparative study of 13 chaotic keymaps. Several evaluation metrics, including key space size, dimensions, entropy, statistical properties, sensitivity to initial conditions, security level, practical implementation, and adaptability to cloud computing, are utilized to compare the keymaps. Keymaps such as Logistic, Lorenz, and Henon demonstrate robustness and high security levels, offering large key space sizes and resistance to attacks. Their efficient implementation in a cloud computing environment further validates their suitability for real-world encryption scenarios. The context of the study focuses on the role of the key in the encryption and provides a brief specification of each map to assess the effectiveness, security, and suitability of the popular chaotic keymaps for encryption applications. The study also discusses the security assessment of resistance to the popular cryptographic attacks: brute force, known plaintext, chosen plaintext, and side channel. The findings of this comparison reveal the Lorenz Map is the best for the cloud environment based on a specific scenario.


I. INTRODUCTION
In today's era, the abundance of information transferred and stored electronically has raised substantial concerns about data security.Protecting data from access, interception, and manipulation is important for individuals, businesses, and governments, and encryption is a tool for ensuring data security.Encryption involves the conversion of data into text using cryptographic techniques and an encryption key.The resulting scrambled text appears as a sequence of characters, preventing anyone without the correct decryption key from understanding its meaning.This transformation ensures that even if an unauthorized individual accesses the encrypted data, the contents cannot be comprehended without the relevant decryption key [1].
Encryption plays a pivotal role in securing data during transmission over untrusted networks, such as the Internet.As cloud computing becomes more prevalent, encryption becomes essential for securing data stored and processed in cloud environments.Encryption is as an indispensable technology for ensuring data security, confidentiality, and integrity in the digital age.By incorporating encryption into various data processing and communication workflows, organizations and individuals can remarkably reduce the risks associated with data breaches and unauthorized access [2].To classify the importance of encryption in cloud security, a percentage can be assigned to each criterion, which can vary depending on the specific use case and the organization's priorities, as shown in Figure 1.
Encryption keys play a crucial role in the encryption process, serving as the cornerstone of data security.Briefly, the role of encryption keys can be summarized as follows [3], [4]: • Key Generation: Encryption keys are randomly generated using specialized algorithms.The strength and randomness of the key directly affect the security of the encryption.
• Key Distribution: The encryption key needs to be securely distributed to authorized parties who are allowed to encrypt or decrypt the data.Key distribution mechanisms should protect against interception or unauthorized access to the key.• Encryption: During encryption, the plaintext data are combined with the encryption key using a cryptographic algorithm.The encryption key serves as the parameter to control the transformation, producing the ciphertext as the output.• Decryption: The correct decryption key is used to retrieve the original plaintext from the ciphertext in the decryption, which is mathematically related to the encryption key.The decryption key enables inverse transformation, reverting the ciphertext to its original plaintext form.The utilization of chaotic maps in the field of cryptography has garnered substantial interest owing to their crucial role in encryption based on their inherent characteristics such as sensitivity to initial conditions, unpredictability, and complex dynamics, to furnish a secure, efficient approach for the generation of encryption keys [5].The literature on chaotic keymaps and their application in encryption reveals a diverse range of studies exploring the theoretical foundations, cryptographic properties, and practical implementations of chaotic systems in cryptography.While chaotic keymaps demonstrate promising features for secure key generation, further research is needed to address challenges related to key distribution, system parameters, and resistance to cryptanalytic attacks.Future studies could explore hybrid encryption schemes, combining the strengths of chaotic keymaps with traditional cryptographic algorithms, to improve security and efficiency in data encryption.

Logistic Map
• Specialties: 2D noninvertible discrete piecewise map, used in pseudorandom number generation and chaotic cryptography 12. Tinkerbell Map [28]: • Mathematical Equation: x{n+1} = xn 2yn 2 + axn + byn, y{n+1} = 2xnyn + cxn + dyn • Specialties: 2D quadratic chaotic map, exhibits various shapes when iterated 13.Gingerbread-man Map [32]: • Mathematical Equation: x{n+1} = 1yn + |xn|, y{n+1} = xn • Specialties: 2D chaotic map with a self-replicating structure, used in image encryption and pattern generation These properties play a crucial role in selecting the appropriate chaotic map for encryption applications based on the desired security level, computational efficiency, and specific requirements of the encryption system.The selection should consider the trade-offs between complexity, security, and resource requirements to achieve a suitable balance for the encryption application.

Keymaps' Suitability for Encryption Applications
To assess the effectiveness, security, and suitability of three popular chaotic keymaps, namely, the Logistic Map, the Lorenz Map, and the Henon Map, for encryption applications, the evaluation is based on findings from the studies cited in references [31], [32], and [33].
• Randomness: A suitable keymap should produce key sequences that appear random and exhibit a high degree of entropy.o The Logistic Map demonstrates chaotic behavior and can generate key sequences possessing substantial randomness as a result of its sensitivity to initial conditions.o The Henon Map is renowned for its irregular, unpredictable behavior, rendering it appropriate for the generation of random key sequences.o The Lorenz Map, due to its chaotic nature, can produce key sequences with elevated randomness, particularly when parameters are selected suitably.
• • Key Generation Efficiency: This metric measures the proportion of computational time to system resources.
o Logistic Map: The Logistic Map presents a computationally efficient approach for key generation owing to its 1D nature and direct iteration formula.o Lorenz Map: The Lorenz Map requires more computational resources for key generation due to its continuous-time equations and 3D dynamics.o Henon Map: The Henon Map is computationally efficient for key generation, similar to the Logistic Map, due to its simple 2D equations.
• Security and Sensitivity: Chaotic keymaps should exhibit high sensitivity to initial conditions.Minor changes in initial values should result in markedly distinct key sequences.Additionally, they should possess resistance against cryptographic attacks.o Logistic Map: The Logistic Map manifests a remarkable sensitivity to initial conditions, thereby conferring a heightened degree of security.Nevertheless, its 1D nature renders it susceptible to certain cryptanalytic attacks.o Lorenz Map: The generation of keys for the Lorenz Map demands a substantial allocation of computational resources due to the intricate continuous-time equations and 3D dynamics that operate within its system.o Henon Map: The Henon Map is renowned for its computational efficiency in generating keys and analogous to the logistic map owing to its uncomplicated 2D equations.
• Key Space Size: A larger chaotic key space enhances security by making some attacks more computationally infeasible.o Logistic Map: The Logistic Map's key space is confined to the interval [0, 1] for variable "x", which may result in a comparatively diminished key space in relation to chaotic systems of higher dimensions.o Lorenz Map: The Lorenz Map is particularly adept at facilitating image encryption and other scenarios necessitating the generation of high entropy keys.
o Henon Map: The Henon mapping exhibits a 2D domain of keys, thereby presenting a superior key space in contrast to the logistic mapping.
• Applicability in Encryption: Each map works better in some environments based on its specifications.o Logistic Map: The Logistic Map is a fitting choice for straightforward encryption techniques and stream ciphers owing to its efficiency and susceptibility to initial conditions.o Lorenz Map: The Lorenz Map is particularly suitable for image encryption and other applications requiring high-entropy key generation.o Henon Map: The Henon Map exhibits suitability for implementation in lightweight encryption scenarios and image encryption endeavors owing to its dynamic properties in two dimensions.
The comparative study highlights the strengths and weaknesses of the three chaotic keymaps, namely, Logistic, Lorenz, and Henon for encryption purposes.The Logistic Map excels in computational efficiency and simplicity but may have limitations in key space size and security.The Lorenz Map provides high security and sensitivity but requires more computational resources.The Henon Map offers a balance between computational efficiency and security with its 2D dynamics.The choice of the most appropriate keymap is contingent upon the particular encryption criteria, the desired level of security, and the computational resources accessible within a given application.The graphical representation in Figure 2 exhibits the level of randomness inherent in the Lorenz keymap, while its specifications contribute to a heightened level of security when implemented in conjunction with my RC6 encryption algorithm.Comparative Analysis Criteria To provide a comprehensive comparison [33] [34], the following test criteria are used: • Dimensions: The dimensionality of the keymap is evaluated: 1D maps involve a single variable, whereas 2D maps have two variables, and so on.Higher-dimensional maps can provide more complex encryption, but they may require more computational resources.
• Mathematical Formulation: Understanding the mathematical equations that define the keymap involves studying the iterative formulas and parameter values used to generate the chaotic sequence.
• Key Space Size: The size of the key space generated by the chaotic map is determined.A larger key space provides higher security against brute-force attacks, making guessing the key more difficult for adversaries.

• Computational
Efficiency: Measuring the computational efficiency of the considers the time taken to generate key sequences.Faster maps can lead to quicker encryption and decryption.To select the six best tests to apply to chaotic keymaps, those crucial for evaluating the security, randomness, and practicality of the keymaps in an encryption context are considered.The recommended tests based on [23] [35] are key space size, entropy, statistical properties, sensitivity to initial conditions, security level, and practical implementation, as shown in Table 1.  1 compares the specified chaotic keymaps based on the six tests.The experiment examines the efficacy of a 256-bit key, which has the potential to be longer.To gauge the unpredictability of the key, the entropy test is utilized.A key with a high level of entropy is more resilient against statistical attacks due to its increased randomness, and its value is constrained within the range of 0 to 1.One noteworthy characteristic of chaotic systems is their susceptibility to initial conditions.Even a slight alteration in the initial condition can result in a substantially distinct key.This attribute can enhance security by rendering the key highly unpredictable, even when one possesses knowledge of the system.To obtain the statistical characteristics, three examinations are employed.The initial examination is the Monobits test, which evaluates the proportion of 0s and 1s.Ideally, a random sequence should possess an approximately equal quantity of both digits.The subsequent examination is the Run test, which identifies a continuous sequence of identical bits, such as 0000 or 111.This examination determines if the frequency of runs of various lengths aligns with the expected occurrence in a random sequence.The final examination is the autocorrelation test, which investigates patterns by comparing a sequence with shifted versions of itself.Lastly, the level of security for each keymap can be adjusted based on the perceived significance of each factor.

VI. Keymap Suitability for Cloud Environment
Choosing the best chaotic keymap for encryption in a cloud environment is a multifaceted decision, considering factors such as security, performance, and adaptability to cloud computing.Commonly considered options include the Logistic, Lorenz, Henon, Standard, and Gauss maps.After the previous comparative analysis, the Lorenz Map emerges as the prime choice.It boasts a generous key space size, ensuring a formidable defense against brute-force attacks.The Lorenz Map's sequences exhibit high entropy, guaranteeing the generation of robust, unpredictable keys.It also demonstrates good statistical properties, resulting in keys resembling true randomness.Furthermore, it is sensitive to initial conditions, enhancing key security.In terms of security levels, the Lorenz Map ranks first, with resistance to known cryptographic attacks.It is practically implementable, facilitating efficient encryption processes.Lastly, its adaptability to cloud computing environments seals its suitability.Considering these factors and the results in Table 1, the Lorenz Map is the optimal choice for securing a cloud-based system [36].

VII. Security Assessment
This study provides a general overview of the security assessment considerations for the three keymaps (Logistic, Lorenz, and Henon) and their resistance to cryptographic attacks as mentioned in [37], [38], [39] [40].The assessments include brute-force attack resistance, known-plaintext and chosen-plaintext attacks, differential cryptanalysis, meet-inthe-middle attacks, side-channel attacks, and security analysis standards.The assessment relevant for the cloud computing environment is selected to evaluate the strength of encryption schemes using chaotic keymaps: • Brute-Force Attack Resistance: Brute-force attacks involve trying all possible combinations of keys to decrypt the ciphertext.The strength of the keymap depends on the size of the key space [41], [42].
o The Logistic Map has a limited key space size, making it susceptible to brute-force attacks, especially with modern computing capabilities.
o The Lorenz Map and the Henon Map, with larger continuous parameter spaces, generally offer higher resistance to brute-force attacks.
• Known-and Chosen-Plaintext Attacks: Knownplaintext attacks involve an attacker having access to plaintext and corresponding ciphertext pairs.Chosenplaintext attacks involve the attacker choosing plaintexts and observing the corresponding ciphertexts [43].
o The Lorenz Map's chaotic behavior and sensitivity to initial conditions can provide some resistance against these attacks as extracting information about the key becomes challenging.o The Henon Map and the Logistic Map might be less resistant to these attacks due to their potentially weaker chaotic properties.
• Side-Channel Attacks: Side-channel attacks exploit information leaked by a system during the encryption, such as power consumption or timing [44].
o The complexity of the Lorenz Map and its chaotic behavior may make side-channel attacks more challenging.o The Henon Map and the Logistic Map might be more susceptible to certain side-channel attacks due to their simpler mathematical formulations.
Professional cryptanalysts typically evaluate these maps using various cryptographic attack techniques to assess their security levels and their suitability and security in a cloud computing environment.More secure encryption schemes usually involve combining multiple chaotic keymaps or using them as part of hybrid encryption systems to enhance their resistance to attacks.

VIII. Conclusion
The comparative study of chaotic keymaps for encryption in a cloud computing environment yielded several remarkable findings: • Security and Robustness: o The Lorenz Map exhibits the highest level of security and robustness among the keymaps due to its strong chaotic behavior and sensitivity to initial conditions.o The Henon Map provides a reasonable level of security, making it suitable for certain encryption scenarios, although it is not as robust as the Lorenz Map.o The Logistic Map's security is limited due to its potentially weaker chaotic behavior and smaller key space.
• Computational Efficiency and Scalability: o The Logistic Map stands out among the keymaps in terms of computational efficiency and scalability owing to its simple, straightforward mathematical formulation.o The utilization of the Lorenz Map necessitates additional computational resources due to its differential equation system, which may affect performance during cloud deployments on a large scale.o The implementation of the Henon Map displays a moderate level of efficiency, with a notable advantage in terms of scalability when compared to the Lorenz Map.
• Sensitivity and Randomness: • Limitations and Future Research: o Some keymaps, such as the Logistic Map, display limitations with regard to their security and the magnitude of their key space.As a result, additional research and prospective enhancements are necessary.o Future studies could potentially investigate further keymaps, hybrid encryption algorithms, and optimizations to enhance the efficiency and security of chaotic encryption within cloud-based settings.
The comparative analysis reveals the Lorenz Map exhibits robust security and sensitivity features and emerges as a potential contender for employment in applications that necessitate high-security levels and data integrity in a cloud computing environment.

Fig. 1 :
Fig. 1: Importance of encryption in securing data Mathematical Formulation: This formulation is used to determine if it is dynamic or continuous to compute its efficiency.o Logistic Map: The Logistic Map is a 1D map defined by the recurrence relation x(n+1) = r * xn * (1 − xn), where r is the control parameter.It exhibits simple dynamics and can be efficiently computed.o Lorenz Map: The Lorenz Map is a continuoustime 3D chaotic system with the equations dx/dt = σ * (y − x), dy/dt = x * (ρ − z) − y, dz/dt = x * y − β * z, where σ, ρ, and β are system parameters.It involves complex dynamics and continuoustime equations.o Henon Map: The Henon Map is a 2D map defined by the equations x(n+1) = yn + 1 − a * xn 2 and y(n+1) = b * xn, where a and b are control parameters.It displays elementary dynamics and is characterized by computational efficiency.

Fig. 2
Fig.2 Randomness of Lorenz chaotic key sequence

in Cloud Environment: The
These assessments consist of randomness analysis, sensitivity analysis, entropy effect evaluation, statistical properties, and others.E. Security Assessment: The study conducts a security assessment to identify the resistance of keymaps to cryptographic attacks.F. Applicability

Dimensions of Chaos Maps:
F. Security Analysis and Vulnerabilities: o Linqing Huang et al. (2018) examined the security of chaotic encryption algorithms, vulnerabilities to known plaintext attacks, chosen plaintext attacks, and related-key attacks [15].o Mihir Bellare et al. (2011) addressed the design of robust encryption algorithms that are resistant to related-key attacks [16].G. Performance Evaluation: o Research focused on evaluating the performance of chaotic keymaps in terms of key generation speed, entropy generation, and computational overhead, as mentioned in [17] [18] [19].o Extensive studies compared the performance of different chaotic maps in generating keys for practical encryption applications [20] [21].

TABLE I .
RECOMMENDED TESTS ON KEYMAPS

o
The sensitivity of the Lorenz Map to initial conditions constitutes an additional layer of data integrity and security, rendering it appropriate for scenarios where the preservation of chaotic behavior is essential.o The Henon Map exhibits a moderate degree of sensitivity, making it a suitable choice for ensuring a desirable level of robustness in specific use cases of cloud encryption.o The overall robustness of the Logistic Map is influenced by its limited sensitivity.The computational efficiency and simplicity of the Logistic Map render it highly appropriate for parallel computing and real-time encryption in cloud environments characterized by high-throughput demands.The security and robustness of the Lorenz Map render it an appropriate selection for cloud-based applications, where data integrity and resistance to cryptographic attacks are of utmost importance.o The Henon Map's equilibrium between safeguarding and effectiveness renders it appropriate in specific cloud encryption scenarios.
• Applicability in Cloud Environments: o o