Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 653 https://internationalpubls.com Advanced Nonlinear Analysis Technique in Modern Transposition Ciphers 1G.M.Karpura Dheepan, 2P.M. Sithar Selvam, 3L.Kartheesan, 4S. Abhirami, 5Vaishali Joshi, 6Raja Ambethkar M 1Associate Professor, Sathyabama Institute of Science and Technology, Chennai, India. mail2dheeps@gmail.com 2Professor of Mathematics, KCG College of Technology, Karapakkam, Chennai, India. sithar.maths@kcgcollege.com 3Assistant Professor (Senior), Department of Computer Science & Engineering, School of Computing, Vel Tech Rangarajan Dr.Sagunthala R&D Institute of Science and Technology, Avadi, Chennai, Tamil Nadu, India. drkartheesanl@veltech.edu.in 4Assistant Professor, Department of Mathematics, Sona College of Technology, Salem, India. abhiramis@sonatech.ac.in 5Associate Professor, School of Computer Science and Information Technology (SCSIT), Symbiosis University of Applied Sciences (SUAS), Indore, India. vaishali.joshi22@gmail.com 6Associate Professor, Department of Engineering English, Koneru Lakshmaiah Education Foundation, KL University, Green Fields, Vaddeswaram, Guntur, Andhra Pradesh, India. rajaambethkar@kluniversity.in Article History: Received: 17-06-2024 Revised: 26-07-2024 Accepted: 10-08-2024 Abstract: Particularly with modern transposition ciphers, the evolving field of cryptography requires advances in safe encryption methods. The key problem fixed is the absence of robust security against advanced cryptanalytic methods provided by traditional transposition ciphers. This work uses chaos theory to automate the production of substitution boxes (S- boxes), which are fundamental to transposition ciphers, so addressing this constraint. Strong nonlinearity and resistance to differential and linear cryptanalysis are produced by the proposed method by combining chaos-driven algorithms. The experimental data emphasizes the relative efficiency of the chaos-driven S-box to traditional transposition ciphers. In nonlinearity and differential uniformity, the chaos-driven S-box considerably exceeds the performance of the Columnar Transposition Cipher, Rail Fence Cipher, and Permutation Cipher achieving values of 65.00 to 78.00 and 2 respectively. considerable resistance to both linear and differential cryptanalysis indicated by this low differential uniformity and considerable nonlinearity greatly enhances its security profile. Keywords: cryptography, S-box generation, chaos theory, transposition ciphers, nonlinear analysis 1. Introduction Maintaining confidentiality and information integrity is quite important in the field of cryptography [1]. For many years, conventional encryption methods—including many transposition ciphers like the Columnar Transposition Cipher, Rail Fence Cipher, and Permutation Cipher—have been basic in protecting correspondence [2]-[6]. The key problem fixed is the absence of robust security against advanced cryptanalytic methods provided by traditional transposition ciphers [7]-[8]. Particularly, traditional methods rely on robustness against differential attacks, linear approximations, and other complex cryptanalytic techniques, therefore failing the degree of security required for modern usage [9]. This difference demands the development of a new cryptographic technique that not only enhances Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 654 https://internationalpubls.com security but also maintains computational performance to be pragmatic for real-world applications [10]. The objectives of this research are threefold: 1. To design a new S-box generating method combining chaotic systems to provide improved security characteristics. Including less differential homogeneity and more nonlinearity helps to raise resistance to both linear and differential cryptanalysis. 2. To offer execution times to be pragmatic for real-world uses; the new method ensures security improvements in step with computational capability. 3. To compare with standard transposition ciphers, the proposed approach performs in terms of cryptographic measures and general security efficacy. The proposed approach offers a novel approach to generate S-boxes using nonlinear chaos theory. Using chaotic maps, the chaos-driven S-box builds complex substitution boxes demonstrating considerably superior cryptographic properties than current methods dependent on basic permutations. Attackers find this approach difficult to exploit nonlinearity and erratic behavior. Relatively new and a significant departure from conventional techniques, the use of chaotic systems in cryptographic S- box generation offers a fresh strategy of enhancing encryption security. This research makes several significant contributions to the field of cryptography: 1. One major advancement in cryptography techniques is the development of an S-box generating technique motivated by chaos. Including chaotic sequences into S-box generating helps the method attain lower differential uniformity and higher degrees of nonlinearity than conventional ciphers. 2. The second provides performance evaluations showing that the chaos-driven S-box far outperforms traditional transposition ciphers in major security parameters including nonlinearity and differential homogeneity. 2. Literature Review Combining the classical columnar transposition cipher, double columnar transposition cipher, changeable columnar transposition cipher, route transposition cipher, the review in [11] will present a complete view of the several forms of columnar transposition ciphers. The paper will also cover modern cryptographic dangers and advanced cryptanalysis techniques including machine learning, brute force attack, differential cryptanalysis, chosen plaint-text attack etc.). Designed to enhance columnar transposition cipher security will be fresh algorithmic innovations. Moreover discussed will be a better form of the columnar transposition cipher approach covering its mathematical background, theoretical framework, and predicted security enhancements. This [12] proposed a New Lightweight Cryptographic Algorithm for Enhancing Data Security to be deployed on cloud computing to secure apps. The method is a 16 byte (128-bit block cipher) key search for which encrypts the data. Ideas in Feistal and substitution permutation architecture inspire me to raise the complexity of the encryption. Using XOR, XNOR, shifting, swapping, the method follows Shannon's theory of diffusion and confusion. It also lets vary the length of the secret key and the rotational count. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 655 https://internationalpubls.com Combining the [13] proposed an designs to enhance columnar transposition cipher security will be fresh algorithmic innovations. Among these advancements are dynamic key generation, column permutation, interaction with other cryptographic primitives, and key scheduling methods. Moreover discussed will be a better form of the columnar transposition cipher approach covering its mathematical basis, theoretical background, and predicted security enhancement. Table 1: Reference Method Methodology Outcome [11] Variants of Columnar Transposition Ciphers Comprehensive review of classical and modern columnar transposition ciphers, including classical columnar, double columnar, mutable columnar, and route transposition ciphers. Analysis of vulnerabilities, limitations, and cryptographic threats. Development of new algorithmic advancements such as dynamic key generation, column permutation, and integration with other cryptographic primitives. Identifies key vulnerabilities and limitations of traditional ciphers. Proposes advancements that enhance security through improved key management and cryptographic integration. Theoretical structure and anticipated security upgrades are detailed. [12] New Lightweight Cryptographic Algorithm Design of a 128-bit block cipher with a 128-bit key, inspired by Feistel and substitution- permutation methods. Implements logical operations (XOR, XNOR, shifting, swapping) to achieve Shannon's principles of diffusion and confusion. Achieves strong security with significant improvements in execution time and security measures compared to existing cloud computing cryptographic systems. Demonstrates flexibility in key length and rounds. [13] Enhanced Columnar Transposition Cipher Development of advanced columnar transposition ciphers with dynamic key generation, column permutation, and integration with other cryptographic primitives. Detailed analysis of the mathematical basis and theoretical structure. Proposes enhancements that address the limitations of traditional columnar transposition ciphers. Focuses on improving security through algorithmic advancements and key scheduling. Emphasizing the advancements and contributions to cryptographic procedures, the table 1 provides a thorough overview of the methods, approaches, and outcomes from every cited work. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 656 https://internationalpubls.com Proposed Nonlinear Chaos-Driven S-Box Generation For Transposition Ciphers In this section, application a nonlinear chaos-driven S-box generating method for transposition ciphers using these careful guidelines. Proposed Method 1. Selection of Chaotic Map: The research select an appropriate chaotic map distinguished by complexity of behavior and sensitivity to starting conditions. Typical decisions call for the tent map, logistic map, or henon map. Excellent nonlinearity in the generated S-boxes will be guaranteed in part by strong chaotic aspects of the map. 2. Parameter Initialization: Here set the parameters of the chosen chaotic map. This includes configuring control parameters and beginning conditions influencing the behavior of the map. Make sure these settings support anarchy and avoid following trends. 3. Chaotic Sequence Generation: Use the chaotic map to produce an ordered series of numbers. This sequence will cause the values of the S-box to be permuted or mixed. The length of the sequence should follow the mandated S-box size. 4. Normalization: Standardize the disorganized series such that it falls inside the specified S-box value range. Usually, this means mapping the chaotic sequence to a set of numbers suitable for the S- box. 5. S-box Construction: Create the S-box using a square matrix configuration of the normalized chaotic values. The size of this matrix determines the specific S-box needs—that is, 8x8 for 64-bit encryption. Verify that the S-box satisfies among other cryptographic criteria uniform distribution and nonlinearity. 6. Validation: Validate the generated S-box's cryptographic properties. Common validations include testing nonlinearity, differential uniformity, resistance to linear and differential cryptanalysis, and differential consistency. Pseudocode // Step 1: Selection of Chaotic Map chaotic_map = ChooseChaoticMap(“Logistic”) // Example choice // Step 2: Parameter Initialization initial_condition = 0.5 control_parameter = 3.99 sequence_length = 256 // Size of the S-box // Step 3: Chaotic Sequence Generation chaotic_sequence = GenerateChaoticSequence(chaotic_map, initial_condition, control_parameter, sequence_length) // Step 4: Normalization Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 657 https://internationalpubls.com normalized_sequence = NormalizeChaoticSequence(chaotic_sequence, min_value=0, max_value=255) // Step 5: S-box Construction s_box_size = sqrt(sequence_length) // Determine the S-box size s_box = ConstructSBox(normalized_sequence, s_box_size) // Step 6: Validation is_valid = ValidateSBox(s_box) if is_valid: Print(“S-box is valid and cryptographically secure.”) else: Print(“S-box failed validation.”) 3.1. Transposition Ciphers in cryptography Transposition ciphers are a subset of cryptographic methods whereby information in a plaintext is carefully rearranged to encrypt. Unlike substitution ciphers—which replace characters with others— transposition ciphers permuate the characters depending on a certain algorithm. From this reordering, the encrypted text appears scrambled, but the value of the original characters remains constant. The plaintext comes first arranged in a transposition cipher into a matrix or grid shape. One chooses the character arrangement of this matrix depending on a given key or algorithm. For a simple columnar transposition cipher, for example, the plaintext is typed out in rows of a given length then permuted in line with a key. Reading the produced ciphertext column by column in the specified permutation sequence returns The key thus determines the rearranging pattern of the columns, hence providing a unique ciphertext for every key. Strongest in their ability to conceal the structure of the plaintext without altering the characters themselves, are transposition ciphers Combining numerous rounds of transpositions or using more complex permutation techniques will help to enhance this obfuscation. Advanced transposition ciphers may make use of numerous matrices with varying row and column sizes or dynamic keys varying with every encryption round. By leveraging the capabilities of both transpositions and substitutions, these hybrid methods provide increasingly robust encryption systems. By means of their design and analysis, these hybrid ciphers seek to ensure that they possess desired cryptographic characteristics, such diffusion and confusion, thereby permitting sufficient protection of data against many attack vectors. 3.1.1. Columnar Transposition Cipher 1. Matrix Construction: One could argue that one should write the plaintext in a matrix form with a designated column count n. M stands for matrix; P is plaintext. The matrix is constructed as follows: ,i j i n jM P += Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 658 https://internationalpubls.com where i - row index, and j - column index. 2. Column Permutation: Sort the columns according to a key K that outlines the reading column sequence to follow. Assume the key provides a permutation π from which column j should be understood as column π(j). Building the ciphertext C by reading the matrix column-wise in the sequence provided by π: helps one , ( )i i jC M = where Ci is the i-th character of the ciphertext. 3.1.2. Rail Fence Cipher 1. Fence Construction: Write the plaintext in a zigzag pattern on r rails. Let rrr be the number of rails and PPP be the plaintext. The characters are placed in a zigzag manner, such that: , length of rowi j i jM P += where i - rail index and j - position within the rail. 2. Reading Rails: Read the characters from each rail sequentially to form the ciphertext C: 0,: 1,: 1,:concat( , , , )rC M M M −=  where concat denotes concatenation of rows. 3.1.3. Permutation Cipher 1. Permutation Function: Let P be the plaintext and K be the permutation key. The key specifies a permutation π that reorders the positions of characters. Each character Pi is mapped to Cπ(i): ( )i iC P = where C - ciphertext. Nonlinear chaos-driven S-box generation The proposed nonlinear chaos-driven S-box generation method generates substitution boxes (S-boxes) with high cryptographic strength for application in modern encryption systems using chaos theory. This approach satisfies the need for advanced and non-linear S-boxes able to significantly raise cryptographic system security. Chaos theory provides a basis for developing sequences with significant sensitivity to initial conditions and parameters that produce quite surprising and complex behavior. Chaos theory is applied in the framework of S-box generation to produce sequences of values Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 659 https://internationalpubls.com subsequently used to form S-boxes. Perfect for creating S-boxes with robust cryptographic capabilities, these sequences are naturally random and difficult to forecast. 1. Selection of Chaotic Map: The approach begins with selecting a good chaotic map, such the logistic map, henon map, or tent map. These maps are chosen for their ability to produce complex and apparently random sequences. The chaotic map lays the foundation for building the sequences meant for the S-box. 2. Parameter Initialization: The second is define control parameters and starting conditions for the chaotic map. These criteria should be chosen to maximize the chaotic elements of the map since they define the behavior of the turbulent sequence. Correct starting point determines the suitable unpredictability and complexity in the generated sequence. 3. Chaotic Sequence Generation: Using the selected chaotic map and initised parameters generates a chaotic sequence of values. Here a sequence of pseudo-random numbers will generate the S-box. The length of the sequence is chosen to correspond with the required S-box size—256 values for an 8x8 S-box. 4. Normalization: By means of normalizing the turbulent series, the S-box values' required range is better met. This phase scales the chaotic values such that they lie inside the integer range needed for the S-box—for an 8-bit S-box, 0 to 255. 5. S-box Construction: The S-box arises from a matrix's ordered, normalized chaotic values. The matrix dimensions depend on the S-box's 8x8, for example, size. Plaintext values are changed using this matrix during encryption; each matrix point provides a unique substitution value. 6. Validation: The generated S-box is verified to meet cryptographic standards. We analyze basic properties like differential uniformity, nonlinearity, and resistance to both linear and differential cryptanalysis. These properties are quite necessary to ensure that the S-box enhances the security of the encryption technique. Logistic Map: 1 (1 )n n nx r x x+ =   − where: xn - current value in the sequence. xn+1 - next value. r - control parameter (typically between 3.57 and 4 for chaotic behavior). xn - initialized with x0, the initial condition. Henon Map: 2 1 1 1n n n n n n x a x y x y b x + +  = −  + =  =  Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 660 https://internationalpubls.com where: a and b - parameters (commonly a=1.4 and b=0.3). xn and yn are the current values in the sequence. Tent Map: 1 if 0 1 if 1 1 n n n n n x x x x x     +    =  −    − where: α - bifurcation parameter (typically between 0.5 and 1). Normalization To map the chaotic sequence values into a desired range (e.g., 0 to 255 for an 8-bit S-box): ( min ) ( 1) ma n x mi i x i x x x N M − =  − − where: xi - i-th value in the chaotic sequence. Minx and maxx - minimum and maximum values of the chaotic sequence, respectively. M - number of possible values for the S-box (e.g., 256 for an 8-bit S-box). Ni - normalized value used in the S-box. S-box Construction Arrange the normalized values into an M×M matrix (for an 8x8 S-box, M=8): ,i j i M jS N  += where: Si,j - value at position (i,j) in the S-box matrix. Ni×M+j - normalized chaotic value. S-box Validation Evaluate the cryptographic properties of the generated S-box: Nonlinearity: 2 ( ) max W( ( ) W( ))nx NL S S x x  =  F where: W - Walsh-Hadamard transform. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 661 https://internationalpubls.com ⊕ - XOR operation. S(x) - output of the S-box for input xxx. Differential Uniformity: 2 ( ) max { ( ) ( ) 0}nx DU S S x S x  =     F ∣ where: Δ - difference vector. The term counts the number of distinct output differences. Chaotic sequences in S-box manufacture introduce a large degree of nonlinearity and unpredictability, which raises the S-box resistance against various cryptographic attacks. Since the innate randomness of chaotic sequences helps to achieve desired cryptographic properties, the S-box is more resistant to assaults including differential and linear cryptanalysis. Pseudocode // Function to generate a chaotic sequence Function GenerateChaoticSequence(map_type, initial_condition, control_parameter, sequence_length): Initialize x = initial_condition chaotic_sequence = [] For i from 1 to sequence_length: x = ApplyChaoticMap(map_type, x, control_parameter) Append x to chaotic_sequence Return chaotic_sequence // Function to apply a specific chaotic map Function ApplyChaoticMap(map_type, x, control_parameter): If map_type is “Logistic”: Return control_parameter * x * (1 - x) ElseIf map_type is “Henon”: Return (1 - control_parameter[0] * x^2 + control_parameter[1] * y) ElseIf map_type is “Tent”: If x < control_parameter: Return x / control_parameter Else: Return (1 - x) / (1 - control_parameter) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 662 https://internationalpubls.com Else: Throw Error(“Unsupported chaotic map type”) // Function to normalize chaotic sequence Function NormalizeChaoticSequence(chaotic_sequence, min_value, max_value): min_x = Minimum(chaotic_sequence) max_x = Maximum(chaotic_sequence) normalized_sequence = [] For each x in chaotic_sequence: normalized_value = Floor(((x - min_x) / (max_x - min_x)) * (max_value - min_value)) Append normalized_value to normalized_sequence Return normalized_sequence // Function to construct S-box from normalized sequence Function ConstructSBox(normalized_sequence, s_box_size): s_box = Matrix(s_box_size, s_box_size) index = 0 For i from 0 to s_box_size - 1: For j from 0 to s_box_size - 1: s_box[i, j] = normalized_sequence[index] index = index + 1 Return s_box // Function to validate the S-box Function ValidateSBox(s_box): nonlinearity = ComputeNonlinearity(s_box) differential_uniformity = ComputeDifferentialUniformity(s_box) If nonlinearity > threshold_nonlinearity AND differential_uniformity <= threshold_differential_uniformity: Return True Else: Return False // Main procedure Function Main(): Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 663 https://internationalpubls.com map_type = “Logistic” // Example choice initial_condition = 0.5 control_parameter = 3.99 // For Logistic Map sequence_length = 256 // Example for 8x8 S-box chaotic_sequence = GenerateChaoticSequence(map_type, initial_condition, control_parameter, sequence_length) normalized_sequence = NormalizeChaoticSequence(chaotic_sequence, 0, 255) s_box_size = Sqrt(sequence_length) s_box = ConstructSBox(normalized_sequence, s_box_size) Explanation: 1. GenerateChaoticSequence: o Generates a sequence using a chosen chaotic map and parameters. o Calls ApplyChaoticMap to compute each value in the sequence. 2. ApplyChaoticMap: o Applies the chaotic map function based on the selected map type (Logistic, Henon, Tent). 3. NormalizeChaoticSequence: o Normalizes the chaotic values to a specified range (e.g., 0 to 255 for an 8-bit S-box). 4. ConstructSBox: o Arranges the normalized sequence into an M×MM \times MM×M matrix to form the S-box. 5. ValidateSBox: o Checks the S-box properties like nonlinearity and differential uniformity to ensure it meets cryptographic standards. 6. Main: o Coordinates the overall process: generating the chaotic sequence, normalizing it, constructing the S-box, and validating it. 4. Results and Discussion In the experimental setup for evaluating the nonlinear chaos-driven S-box generation method, simulations were conducted using MATLAB for its extensive support of matrix operations and chaotic system implementations. The testing were conducted on a high-performance computer cluster built with Intel Xeon CPUs and 64 GB of RAM in order to guarantee sufficient computational resources. Fit for producing strong nonlinearity, the chaotic sequences were produced by the logistic map with an initial condition of 0.5 and a control parameter of 3.99. Among the transposition cipher systems Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 664 https://internationalpubls.com now in use against which the 8x8 (256 value) S-boxes were evaluated were the Columnar Transposition Cipher, Rail Fence Cipher, and Permutation Cipher. Table 2: Experimental Setup Parameter Value Simulation Tool MATLAB Computer Hardware Intel Xeon, 64 GB RAM Chaotic Map Logistic Map Initial Condition 0.5 Control Parameter 3.99 Sequence Length 256 S-box Size 8x8 (256 values) Columnar Transposition Cipher Key Length 8 Rail Fence Cipher Rails 2 Permutation Cipher Key Length 4 Nonlinearity Threshold 100.00 Differential Uniformity Threshold 2 Testing Dataset Size 1000 samples Number of Encryption Rounds 1 Performance Metrics 1. Nonlinearity: o Calculates the S-box's divergence from a linear function, hence determining its resistance to linear approximations. Greater nonlinearity points to better resistance. One can compute nonlinearity by use of Walsh-Hadamard transform. Relative to a linear function, it is the transform of the S-box output with maximum value. 2. Differential Uniformity: o Counts the most solutions to a differential equation whereby a set output difference results from an input difference. Lower values imply more robust resistance to different kinds of attack. Count the output variations for each input variation then determine the maximum. 3. Resistance to Linear Cryptanalysis: o Examines S-box resilience to assaults using linear approximations of the encryption function. It studied S-box against linear cryptanalytic techniques and with success rate of approximation. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 665 https://internationalpubls.com 4. Resistance to Differential Cryptanalysis: o Tests the S-box's robustness against assaults using differences between plaintexts and corresponding ciphertexts. 5. Permutation Effectiveness: o Assesses the quality utilized in the transposition cypher methods. 6. Execution Time: o Calculates the S-box producing and encryption process processing efficiency. Table 3: Performance Assessment Metric Method 56 bits 128 bits 192 bits 256 bits 448 bits Nonlinearity Columnar Transposition 20.00 25.00 28.00 30.00 32.00 Rail Fence 18.00 22.00 26.00 28.00 30.00 Permutation Cipher 22.00 27.00 31.00 34.00 36.00 Chaos-driven S- box 60.00 70.00 75.00 80.00 85.00 Differential Uniformity Columnar Transposition 8 7 6 5 4 Rail Fence 9 8 7 6 5 Permutation Cipher 6 5 4 3 2 Chaos-driven S- box 2 2 2 2 2 Resistance to Linear Cryptanalysis Columnar Transposition Medium Low Low Low Low Rail Fence Low Low Low Low Low Permutation Cipher Medium Medium High High High Chaos-driven S- box High High High Very High Very High Resistance to Differential Cryptanalysis Columnar Transposition Low Low Low Low Low Rail Fence Low Low Low Low Low Permutation Cipher Medium Medium High High High Chaos-driven S- box Very High Very High Very High Very High Very High Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 666 https://internationalpubls.com Permutation Effectiveness Columnar Transposition Moderate High High High High Rail Fence Low Moderate Moderate High High Permutation Cipher High High High Very High Very High Chaos-driven S- box Very High Very High Very High Very High Very High Execution Time (ms) Columnar Transposition 20 40 60 80 120 Rail Fence 30 50 70 90 130 Permutation Cipher 25 45 65 85 125 Chaos-driven S- box 50 80 110 150 200 Regarding cryptographic security aspects, the experimental results indisputably reveal that the chaos- driven S-box has a clear advantage over conventional transposition ciphers. • Nonlinearity: The chaos-driven S-box generates values spanning from 60.00 to 85.00 over several bit sizes, far more than the range of conventional approaches between 20.00 and 36.00. This exhibit improved resilience to linear approximations, so the S-box offers more protection against linear cryptanalysis. • Differential Uniformity: Comparatively to the 4 to 9 range of traditional ciphers, the chaos- driven S-box keeps a low differential uniformity of 2 across all bit sizes. This indicates amazing resistance to different kinds of attack. • Resistance to Linear Cryptanalysis: Comparatively to “Medium” to “High,” the chaos- driven S-box provides fairly strong resistance, categorized as “Very High” across all bit sizes. Resistance to Linear Cryptanalysis This shows how resistant it is to attacks applying linear relationships. • Resistance to Differential Cryptanalysis: Comparatively, the chaos-driven S-box is scored as “Very High” for resistance, whereas conventional techniques are classified as “Low,” to “High,” hence stressing its increased security against differential cryptanalysis. • Permutation Effectiveness: The chaos-driven S-box shines with a “Very High” rating, above the “Moderate” to “Very High” ratings of traditional ciphers, therefore displaying improved scrambling of plaintext. • Execution Time: The chaos-driven S-box has higher execution times ranging from 50 to 200 milliseconds than the 20 to 130 milliseconds of conventional approaches. Its improved security features support the justification of this compromise. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 667 https://internationalpubls.com Performance over various block size Metric Method 64 bits 128 bits 256 bits Nonlinearity Columnar Transposition 22.00 25.00 27.00 Rail Fence 20.00 22.00 24.00 Permutation Cipher 26.00 29.00 31.00 Chaos-driven S-box 65.00 72.00 78.00 Differential Uniformity Columnar Transposition 7 6 5 Rail Fence 8 7 6 Permutation Cipher 5 4 3 Chaos-driven S-box 2 2 2 Resistance to Linear Cryptanalysis Columnar Transposition Medium Medium Medium Rail Fence Low Low Low Permutation Cipher High High High Chaos-driven S-box Very High Very High Very High Resistance to Differential Cryptanalysis Columnar Transposition Low Low Low Rail Fence Low Low Low Permutation Cipher High High High Chaos-driven S-box Very High Very High Very High Permutation Effectiveness Columnar Transposition Moderate High High Rail Fence Low Moderate High Permutation Cipher High Very High Very High Chaos-driven S-box Very High Very High Very High Execution Time (ms) Columnar Transposition 30 45 80 Rail Fence 40 55 90 Permutation Cipher 35 50 85 Chaos-driven S-box 60 90 150 The chaos-driven S-box exhibits stronger cryptographic security metrics over many block sizes than traditional ciphers. • Nonlinearity: Much above the 22.00 to 31.00 range for traditional approaches, the chaos- driven S-box reveals degrees of nonlinearity between 65.00 and 78.00. This guarantees that outputs do not match expectedly with input patterns, hence strengthening resistance to linear cryptanalysis. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 31 No. 6s (2024) 668 https://internationalpubls.com • Differential Uniformity: The chaos-driven S-box much surpasses conventional methods covering from 3 to 7 with a constant value of 2. This low differential uniformity reflects its resistance against differential attacks, therefore making it more difficult for attackers to exploit plaintext variants. • Resistance to Linear and Differential Cryptanalysis: Though traditional procedures vary from “Low” to “High,” the chaos-driven S-box is rated as “Very High” for both types of attacks, therefore giving better security against both linear and differential cryptanalytic techniques. • Permutation Effectiveness: The chaos-driven S-box rates “Very High,” above the “Moderate” to “Very High” grades of conventional ciphers. This suggests it provides better permutation efficiency, hence enhancing scrambling and concealing of plaintext. • Execution Time: Although safer, the chaos-driven S-box consumes more processing time (60 to 150 milliseconds) than traditional methods (30 to 90 milliseconds). This compromise underlines how harmonious better security is with computational economy. Conclusion The experimental data emphasizes the relative efficiency of the chaos-driven S-box to traditional transposition ciphers. In nonlinearity and differential uniformity, the chaos-driven S-box considerably exceeds the performance of the Columnar Transposition Cipher, Rail Fence Cipher, and Permutation Cipher achieving values of 65.00 to 78.00 and 2 respectively. considerable resistance to both linear and differential cryptanalysis indicated by this low differential uniformity and considerable nonlinearity greatly enhances its security profile. About resistance to cryptanalytic attacks, the chaos- driven S-box continuously scores at a “Very High” level, very unlike the “Low” to “High” ratings of conventional techniques. This shows its great enhanced ability to stop intricate attacks. Though this enhanced security results in a trade-off with execution speed. Conventional ciphers require 30 to 90 milliseconds; the chaos-driven S-box requires 60 to 150 milliseconds for encryption. References [1] Mahboob, A., Asif, M., Zulqarnain, R. M., Siddique, I., Ahmad, H., Askar, S. S., & Pau, G. (2023). An Innovative Technique for Constructing Highly Non-Linear Components of Block Cipher for Data Security against Cyber Attacks. Comput. Syst. Sci. Eng., 47(2), 2547-2562. [2] Salami, Y., Khajevand, V., & Zeinali, E. (2023). Cryptographic algorithms: a review of the literature, weaknesses and open challenges. J. Comput. Robot, 16(2), 46-56. [3] Vithalkar, P. N. (2024). 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