This study explores the application of genetic algorithms in generating highly nonlinear substitution boxes (S-boxes) for symmetric key cryptography. We present a novel implementation that combines a genetic algorithm with the Walsh-Hadamard Spectrum (WHS) cost function to produce 8×8 S-boxes with a nonlinearity of 104. Our approach achieves performance parity with the best-known methods, requiring an average of 49,399 iterations with a 100% success rate. The study demonstrates significant improvements over earlier genetic algorithm implementations in this field, reducing iteration counts by orders of magnitude. By achieving equivalent performance through a different algorithmic approach, our work expands the toolkit available to cryptographers and highlights the potential of genetic methods in cryptographic primitive generation. The adaptability and parallelization potential of genetic algorithms suggests promising avenues for future research in S-box generation, potentially leading to more robust, efficient, and innovative cryptographic systems. Our findings contribute to the ongoing evolution of symmetric key cryptography, offering new perspectives on optimizing critical components of secure communication systems.
The realm of digital security is in a constant state of evolution, with symmetric key cryptography serving as a fundamental pillar in the architecture of secure communication systems [1][2][3]. At the core of many symmetric encryption algorithms lie Substitution boxes (Sboxes) [4], which play a pivotal role in establishing the nonlinear components essential for robust encryption [5,6]. These S-boxes are critical in creating the confusion and diffusion properties that Claude Shannon identified as crucial for secure ciphers [7,8].
The cryptographic strength of an S-box is multifaceted, encompassing several key indicators [9]. Nonlinearity, which quantifies an S-box's resistance to linear cryptanalysis, stands as a primary measure. For 8×8 S-boxes, commonly employed in modern ciphers, achieving a nonlinearity of 104 represents a significant benchmark [10][11][12]. However, other properties such as differential uniformity, algebraic degree, and algebraic immunity also play crucial roles in determining an S-box's overall cryptographic efficacy [13,14].
While algebraically constructed S-boxes, such as the one used in the Advanced Encryption Standard (AES) with its optimal nonlinearity of 112 [15], might seem ideal, they are not without vulnerabilities. The presence of inherent algebraic structures in such S-boxes can create potential weaknesses, making them susceptible to algebraic cryptanalysis [16][17][18]. This vulnerability underscores the need for randomly generated S-boxes that lack hidden algebraic structures, thereby enhancing resistance against sophisticated cryptanalytic techniques [19][20][21].
The generation of cryptographically robust S-boxes presents a significant computational challenge. The vast search space of possible configurations for 8×8 S-boxes is estimated at 2 8 ! (approximately 10 506 ), renders exhaustive search methods impractical. This complexity has driven research towards heuristic approaches for S-box generation [22][23][24]. Methods such as simulated annealing, hill climbing, and genetic algorithms have shown promise in navigating this expansive solution space efficiently [25].
Recent advances in heuristic S-box generation have made significant advances. Researchers have studied various cost functions, including the Walsh-Hadamard spectrum (WHS) function [23,26], the Picek cost function (PCF) [10], improved Walsh-Hadamard spectrum-based cost functions (WCF) [10,27], and two new extended cost functions (ECF and WCFS) [6,28,29] in conjunction with different search algorithms [10,22]. These efforts have progressively reduced the computational cost of generating highly nonlinear S-boxes, with some methods achieving the target nonlinearity of 104 in fewer than 100,000 iterations.
Despite these advancements, there remains a gap in understanding the full potential of genetic algorithms in this domain. While genetic approaches have been applied to Sbox generation, their performance in comparison to other heuristic methods, particularly in terms of consistency and efficiency in generating S-boxes with optimal cryptographic properties, remains an area ripe for exploration.
Our study aims to address this gap by presenting a comprehensive investigation into the application of genetic algorithms for generating 8×8 S-boxes with a nonlinearity of 104. We explore the synergy between genetic algorithms and the WHS cost function, aiming to match or surpass the efficiency of existing methods while leveraging the inherent advantages of evolutionary approaches, such as adaptability and the potential for parallelization.
The remainder of this paper is structured as follows: Section 2 provides a comprehensive review of the literature, detailing the evolution of S-box generation techniques and the current state of the art. Section 3 offers a background on S-boxes, their cryptographic properties, and the theoretical foundations underpinning their design. Section 4 delineates our methodology and experimental setup, including the specifics of our genetic algorithm implementation and evaluation criteria. Section 5 presents our results and a detailed discussion, comparing our findings with existing methods and analyzing their implications. Finally, Section 6 concludes the paper, summarizing our key findings and outlining promising directions for future research in this critical area of cryptographic system design.
The design and generation of cryptographically strong Sboxes have been subjects of intensive research in the field of symmetric key cryptography. This section provides a comprehensive review of the existing literature, focusing on various approaches to S-box generation and their cryptographic properties.
Algebraic constructions of S-boxes, such as those based on finite field inversion used in the Advanced Encryption Standard (AES) [15,30,31], have been widely studied. However, as Bard (2009) [16] and Courtois and Bard (2007) [17] point out, these constructions may be vulnerable to algebraic attacks due to their inherent mathematical structure. This vulnerability has led to increased interest in generating S-boxes with more complex algebraic structures [32].
Heuristic approaches have gained significant traction in recent years. Clark et al. (2005) [26] introduced a simulated annealing approach for S-box generation [23], demonstrating its effectiveness in producing S-boxes with high nonlinearity. Building on this work, Souravlias et al. (2017) [33] proposed an algorithm portfolio approach combining simulated annealing and tabu search, showing improved results under limited time budgets.
Genetic algorithms have also been explored for S-box generation [24,34]. Tesar (2010) [35] combined a genetic algorithm with a tree search method, generating 8×8 Sboxes with nonlinearity up to 104. Picek et al. (2016) [11] presented a novel cost function for evolving S-boxes, achieving significant improvements in both speed and quality of results compared to previous approaches. Ivanov et al. (2016aIvanov et al. ( , 2016b) ) [36,37] introduced an innovative approach using a modified immune algorithm combined with hill climbing, rapidly generating large sets of highly nonlinear bijective S-boxes. Their work demonstrated the potential of hybrid approaches in S-box generation.
Recent advancements have focused on improving specific cryptographic properties. Rodinko et al. (2017) [38] optimized a method for generating high nonlinear S-boxes, achieving nonlinearity of 104, algebraic immunity of 3, and 8-uniformity within reasonable computational time. Freyre Echevarría and Martínez Díaz (2020) [27] proposed a new cost function specifically designed to improve the nonlinearity of bijective S-boxes.
The importance of multiple cryptographic criteria has been emphasized in recent literature. Freyre-Echevarría et al. (2020) [10] introduced an external parameterindependent cost function for evolving bijective S-boxes, considering both nonlinearity and other important properties. Their work highlighted the need for balanced optimization across multiple cryptographic criteria.
More recent studies have explored novel approaches to S-box generation. Artuğer and Özkaynak (2024) [39] proposed a post-processing approach to improve the nonlinearity of chaos-based S-boxes, addressing a longstanding challenge in this area. Haider et al. (2024) [40] introduced an S-box generator based on elliptic curves, offering a balance between randomization and optimization with minimal computation time.
The application of S-boxes in specific cryptographic contexts has also been a focus of recent research. Jamal et al. (2024) [41] developed a region of interest-based medical image encryption technique using chaotic S-boxes, demonstrating the practical applications of advanced S-box designs in specialized domains.
Emerging threats and the need for enhanced security have led to new considerations in S-box design. Fahd et al. (2024) [42] examined the reality of backdoored S-boxes, highlighting the importance of thorough cryptanalysis and the potential vulnerabilities in S-box structures.
In conclusion, the literature reveals a trend towards more sophisticated, multi-criteria optimization approaches in S-box generation. While significant progress has been made in achieving high nonlinearity and other desirable properties, there remains a need for methods that can consistently produce S-boxes with optimal cryptographic characteristics while balancing computational efficiency and resistance to emerging cryptanalytic techniques.
Symmetric cryptography forms the backbone of secure communication in the digital age. At the heart of many symmetric ciphers lie Substitution boxes (S-boxes), nonlinear components crucial for ensuring the security and robustness of these cryptographic systems. This section provides a comprehensive overview of S-boxes, their role in symmetric cryptography, and the application of genetic algorithms in their optimization.
Substitution boxes (S-boxes) are fundamental components in symmetric-key algorithms, serving as the primary source of nonlinearity [7,8]. An S-box is essentially a vectorial Boolean function that maps a fixed number of input bits to a fixed number of output bits. Formally, an n×m S-box can be defined as [9]:
where 𝐹 and 𝐹 are vector spaces over the Galois field GF(2) with dimensions n and m, respectively.
The cryptographic strength of an S-box is determined by several critical properties [10]:
1) Nonlinearity: A measure of the distance between the S-box and the set of all affine functions. For an n×n S-box, the nonlinearity is defined as:
, where denotes the dot product and ⊕ represents bitwise XOR.
2) Differential uniformity: Quantifies the uniformity of output differences when the input is changed. The differential uniformity δ is given by:
3) Algebraic degree: The highest degree among the component Boolean functions of S. For an n×m S-box, the algebraic degree is:
) Balancedness: An S-box is balanced if each output occurs with equal probability when the input is uniformly distributed.
5) Algebraic Immunity [43]: A measure of resistance against algebraic attacks. For an S-box 𝑆: 𝐹 → 𝐹 , the algebraic immunity is defined as:
, where I(S) is the ideal generated by the polynomials representing the S-box:
( , ,..., ), ( , ,..., ), ( ) ..., ( , ,..., )
The algebraic immunity can be computed by constructing the minimal reduced Gröbner basis of the ideal I(S) using the degree reverse lexicographic (degrevlex) ordering, and finding the polynomial of minimum degree in this basis. These properties collectively contribute to the S-box's ability to resist various cryptanalytic attacks, including differential, linear, and algebraic cryptanalysis. The concept of algebraic immunity for S-boxes, as introduced by Faugère and Perret, provides a crucial measure of resistance against algebraic attacks, which attempt to express the cipher as a system of low-degree multivariate polynomial equations.
The relationship between the algebraic immunity of an S-box and that of Boolean functions can be established through the following construction. Consider a Boolean function 𝑓 : 𝐹 → 𝐹 defined as [44,45]:
( , ,..., , , ,..., )
1, if , : ( , ,..., ) 0, if , : ( , ,..., ) ;
.
The algebraic immunity of the S-box S is then equivalent to the minimum degree of non-zero polynomials in the annihilator of fS:
. This formulation provides a bridge between the algebraic immunity of vectorial Boolean functions (S-boxes) and that of single Boolean functions, unifying the concept across different cryptographic primitives.
S-boxes play a pivotal role in ensuring the security of symmetric ciphers by introducing nonlinearity and complexity into the encryption process [7]. They are employed in widely-used algorithms such as the Advanced Encryption Standard (AES) [15], where the SubBytes operation relies on a carefully designed 8×8 S-box. However, the increasing sophistication of cryptanalytic techniques has necessitated a reevaluation of traditional Sbox design methods. While algebraically constructed S-boxes, such as those used in AES (based on finite field inverses) [30,31], offer certain advantages in terms of implementation efficiency and some cryptographic properties, they may fall short in terms of algebraic immunity [43]. The structured nature of these S-boxes can potentially lead to vulnerabilities against algebraic attacks, which have gained significant attention in recent years [16,17].
Algebraic attacks exploit the possibility of expressing the cipher as a system of low-degree multivariate polynomial equations [17,18]. The complexity of solving such systems is closely related to the algebraic immunity of the S-box [43]. A low algebraic immunity allows for a simpler representation of the cipher, potentially reducing the computational effort required for cryptanalysis [44,45]. This vulnerability has prompted researchers to explore alternative methods for S-box generation that prioritize high algebraic immunity alongside other critical properties.
To address these concerns, there is growing interest in the cryptographic community in random or pseudo-random Sboxes [24,46,47]. These S-boxes, generated through heuristic methods, offer several advantages:
Higher algebraic immunity: Random S-boxes are less likely to exhibit algebraic structures that can be exploited in attacks, potentially leading to higher algebraic immunity values.
Resistance to specialized attacks: Algebraically constructed S-boxes might be vulnerable to attacks tailored to their specific structure. Random Sboxes, lacking such predictable structures, can offer better protection against these targeted attacks.
Flexibility in design: Heuristic methods allow for the optimization of multiple cryptographic criteria simultaneously, enabling a more balanced approach to S-box design.
Adaptability to evolving threat models: As new cryptanalytic techniques emerge, the criteria for S-box generation can be adjusted more easily with heuristic methods compared to algebraic constructions.
Various heuristic approaches have been proposed for generating high-quality random S-boxes, including:
Simulated Annealing [23,26,33,48]: This method mimics the physical process of annealing in metallurgy, gradually "cooling" the system to find an optimal configuration. It has shown promise in generating S-boxes with good cryptographic properties.
Hill Climbing [6,10,36,49,50]: A local search algorithm that iteratively makes small improvements to a candidate solution. This approach can be effective in fine-tuning S-box properties.
Genetic Algorithms [10,35,37,51]: Evolutionary approaches that mimic natural selection to evolve a population of S-boxes towards desired properties. These algorithms have demonstrated the ability to generate S-boxes with excellent cryptographic characteristics, including high algebraic immunity.
In this work, we focus on genetic algorithms due to their ability to efficiently explore large search spaces and handle multi-objective optimization problems. Genetic algorithms offer a promising approach to generating S-boxes that balance multiple cryptographic criteria, including high algebraic immunity, nonlinearity, and differential uniformity.
Genetic Algorithms (GAs) are stochastic optimization techniques inspired by the principles of natural selection and evolution [52,53]. They operate on a population of potential solutions, evolving them over successive generations to improve their fitness concerning a defined objective function. In the context of S-box generation, GAs offer a powerful and flexible approach to optimizing multiple cryptographic properties simultaneously [10,37,54].
The fundamental principle of GAs is to emulate the process of natural selection, where the fittest individuals are more likely to survive and reproduce, passing their beneficial traits to future generations [52,53]. In the case of S-box generation, an "individual" represents a candidate Sbox, and its "fitness" is determined by how well it satisfies the desired cryptographic properties.
The basic structure of a GA includes the following components [54,55]: Select parents for reproduction using tournament selection 5.
Create new population P' through crossover and mutation: 6.
For i = 1 to N/2 do 7.
Select two parents p1 and p2 from P 8.
If random (0,1) < pc then 9.
(c1, c2) = Crossover(p1, p2) 10.
Else 11.
(c1, c2) = (p1, p2) 12.
End If 13.
Mutate c1 and c2 with probability pm 14.
Add c1 and c2 to P' 15.
End For The fitness function is crucial in guiding the evolutionary process towards S-boxes with desired cryptographic properties.
The selection mechanism, often implemented as tournament selection, ensures that fitter individuals have a higher chance of being chosen for reproduction. This process mimics natural selection, where more adapted individuals are more likely to pass on their genes.
Crossover operators for S-boxes must be carefully designed to preserve the bijective property. One approach is to use a permutation-based crossover, where segments of the S-box permutation are exchanged between parents. For example, given two parent S-boxes P1 and P2, a two-point crossover might produce offspring C1 and C2 as follows: . Mutation operators introduce small random changes to maintain genetic diversity and prevent premature convergence. For S-boxes, this might involve swapping two randomly chosen elements or applying a random permutation to a subset of elements.
Our research focuses on developing and implementing a modified genetic algorithm for generating cryptographically strong S-boxes. This section details our approach, the algorithm's structure, and the experimental setup used to evaluate its performance.
We have developed a modified genetic algorithm that incorporates elements of hill climbing, enhancing its ability to navigate the complex search space of S-box configurations. This approach allows for a more targeted exploration of promising regions while maintaining the population-based nature of genetic algorithms.
The core idea of our algorithm is to maintain a population of S-boxes, subject them to controlled mutations, evaluate their cryptographic properties, and selectively propagate the best specimens to subsequent generations. This process is iterated until either an S-box meeting the desired criteria is found or a predefined computational limit is reached. The pseudocode for our modified genetic algorithm is: The elite selection function performs a crucial role in our algorithm. It ranks the S-boxes based on their nonlinearity and objective function values, prioritizing higher nonlinearity and lower objective function values. This function ensures that only the top Kpop S-boxes survive to the next generation, maintaining a high-quality population.
Our mutation operator is designed to preserve the bijectivity of the S-box while introducing controlled randomness. It operates by swapping two randomly selected (distinct) elements within the S-box. This approach ensures that the fundamental property of bijectivity is maintained throughout the evolutionary process.
Formally, the mutation can be described as:
, where 𝑖, 𝑗 ∈ 0,1, … , 255, 𝑖 ≠ 𝑗 and all other elements remain unchanged.
The choice of objective function is critical in guiding the evolutionary process towards cryptographically strong Sboxes. We employ the WHS function proposed by Clark et al. [26], which has shown effectiveness in generating highquality S-boxes. The WHS function is defined as [26]:
, where WHT[b,i] represents the Walsh-Hadamard transform coefficients; i iterate over all component functions and their linear combinations; b iterates over all linear functions; X and R are real-valued parameters.
Based on empirical studies, we set R = 12 and X = 0, which has been shown to yield optimal results in generating bijective S-boxes with high nonlinearity [56,57].
The primary criteria for evaluating the generated S-boxes are: The evaluate function in our algorithm computes these properties for each generated S-box, allowing us to assess its cryptographic strength comprehensively.
Our experiments were conducted on a high-performance computing cluster to handle the computational intensity of the S-box generation process. The implementation was done in C++ for efficiency, with parallelization to utilize multiple cores.
Given that the calculation of the objective function is the most computationally expensive operation in terms of processor time, the complexity of the entire search algorithm can be considered proportional to the number of times the objective function is calculated. This corresponds to the number of S-boxes that were generated and evaluated. We denote this quantity as KSbox.
To accelerate the algorithm's performance, we implemented parallel computation of the new population using Nthread = 8 threads within each iteration. This parallelization significantly reduced the overall execution time of the algorithm.
We conducted a comprehensive parameter sweep to analyze the impact of population size and mutation rate on the quality of the generated S-boxes and the algorithm's convergence rate. Specifically:
Population size (Kpop) was varied from 1 to 21 with a step size of 2.
The mutation rate (Kmut) was varied from 1 to 31 with a step size of 3.
For each combination of Kpop and Kmut, we performed 100 independent runs of the search algorithm to ensure statistical significance. This resulted in a total of 11×11×100 = 12,100 experimental runs.
The algorithm was set to terminate upon finding an Sbox with nonlinearity ≥ 104 or reaching the maximum iteration limit of 150,000. For each run, we recorded the number of S-boxes generated and evaluated (KSbox), which serves as our primary metric for computational efficiency.
This section presents the results of our comprehensive experimental study on the modified genetic algorithm for Sbox generation. We analyze the performance of the algorithm across various parameter configurations and discuss the implications of our findings.
Our primary metric for evaluating the algorithm's efficiency is KSbox, which represents the number of S-boxes generated and evaluated before finding an S-box with the desired nonlinearity of 104. Table 1 presents the average KSbox values for different combinations of population size (Kpop) and mutation rate (Kmut).
One of the most striking observations from our results is the superior performance of the algorithm when Kpop = 1. This configuration consistently yielded the lowest KSbox values across all mutation rates, with averages ranging from 49,277 to 58,213. This finding is somewhat counterintuitive, as genetic algorithms typically benefit from larger population sizes that provide greater genetic diversity.
The effectiveness of a single-individual population suggests that our algorithm's behavior in this configuration closely resembles that of a stochastic hill-climbing method. This approach appears to be particularly well-suited to the S-box optimization problem, possibly due to the following factors:
Landscape structure: The fitness landscape of S-box configurations may have numerous local optima that are relatively close in quality to the global optimum. In such a scenario, an aggressive local search can be highly effective.
Mutation operator efficiency: Our swap-based mutation operator appears to be sufficiently powerful to navigate the search space effectively, even without the diversity typically provided by a larger population.
Reduced computational overhead: With Kpop = 1, the algorithm avoids the computational cost associated with managing and evaluating a large population, allowing for more iterations within the same computational budget.
While the population size shows a clear trend, the impact of the mutation rate (Kmut) is more nuanced. For Kpop = 1, we observe that:
The lowest KSbox (49,277) was achieved with Kmut = 7. Performance generally degraded with higher mutation rates, with KSbox increasing to 58,213 at Kmut = 1. This pattern suggests that there exists an optimal balance between exploration and exploitation in the search process. Lower mutation rates may lead to premature convergence, while higher rates may disrupt good solutions too frequently.
As Kpop increases, we observe a general trend of increasing KSbox values, indicating reduced computational efficiency. This scaling behavior can be attributed to:
Increased evaluation overhead: Larger populations require more objective function evaluations per generation.
Slower convergence: Diversity maintenance in larger populations may slow down the convergence to high-quality solutions.
However, it's worth noting that larger populations might offer benefits not captured by the KSbox metric alone, such as increased robustness or the ability to find a more diverse set of high-quality S-boxes.
Our implementation of parallel computation using 8 threads (Nthread = 8) has proven to be effective in accelerating the search process. This parallelization strategy is particularly beneficial for configurations with larger Kpop and Kmut values, where the workload can be more evenly distributed across threads.
The average number of S-boxes generated (KSbox) before finding an S-box with Nf = 104 Kmut Kpop
The best-performing configuration of our algorithm (Kpop = 1, Kmut = 7) achieves an average KSbox of 49,277. To contextualize our findings within the broader landscape of S-box generation research, we conducted a comprehensive comparison of our genetic algorithm approach with existing methods. Table 2 presents this comparative analysis, encompassing various techniques and cost functions employed in the field. Our genetic algorithm implementation, utilizing the WHS cost function, achieves results that are on par with the bestknown methods in the field. Specifically, our approach generates S-boxes with a nonlinearity of 104 in an average of 49,399 iterations, with a 100% success rate. This performance is comparable to our previous works using hill climbing [6,28], which required 50,000 iterations on average.
Several key observations emerge from this comparative analysis:
Parity in Performance: Our genetic algorithm achieves results that are statistically equivalent to the best-known methods, particularly our earlier hill-climbing approach. This parity is significant, as it demonstrates the versatility and potential of genetic algorithms in this domain. Algorithmic Diversity: By achieving comparable results through a different algorithmic approach, we have expanded the toolkit available to cryptographers and security researchers. This diversity in high-performing methods enhances the robustness of S-box generation techniques.
Consistency and Reliability: Like our previous best results, the genetic algorithm maintains a 100% success rate in generating target S-boxes with nonlinearity 104. This level of reliability is crucial for practical applications in cryptographic system design.
Efficiency Across Methods: The similarity in performance between our genetic algorithm and hill climbing approaches (49,399 vs. 50,000 iterations) suggests that we may be approaching theoretical limits of efficiency for generating Sboxes with these properties using heuristic methods.
Progress from Earlier Genetic Approaches: Compared to earlier genetic algorithm implementations [11,35], our method shows substantial improvement, reducing the required iterations by orders of magnitude while achieving higher nonlinearity.
The achievement of parity with the best-known results using a genetic algorithm is particularly noteworthy and underscores the potential of evolutionary approaches in cryptographic primitive generation.
The superior performance of the Kpop = 1 configuration has important implications for the practical application of our algorithm:
Resource efficiency: The algorithm can be effectively run on systems with limited computational resources, as it doesn't require maintaining a large population. Simplicity: The simplified population management makes the algorithm easier to implement and tune. Adaptability: The algorithm's efficiency makes it suitable for scenarios where S-boxes need to be generated or updated frequently. In conclusion, our modified genetic algorithm demonstrates exceptional efficiency in generating cryptographically strong S-boxes, particularly in its hillclimbing-like configuration. These findings contribute valuable insights to the field of cryptographic primitive design and offer a powerful tool for the development of secure symmetric encryption systems.
This study presents a significant advancement in the field of S-box generation for symmetric key cryptography, focusing on the application of genetic algorithms to produce highly nonlinear substitutions. Our research demonstrates that genetic algorithms, when properly optimized and combined with the Walsh-Hadamard Spectrum (WHS) cost function, can achieve performance parity with the best-known methods in generating 8×8 S-boxes with a nonlinearity of 104.
Key findings of our work include:
The genetic algorithm approach achieves an average of 49,399 iterations to generate target S-boxes, comparable to the best results of 50,000 iterations using hill-climbing methods. A 100% success rate in producing S-boxes with the desired nonlinearity, matching the reliability of top-performing techniques. A significant improvement over earlier genetic algorithm implementations, reducing iteration counts by orders of magnitude.
The achievement of performance parity using a different algorithmic approach expands the toolkit available to cryptographers and highlights the versatility of genetic methods in cryptographic primitive generation. This diversity in high-performing techniques enhances the robustness of S-box generation methodologies. Furthermore, our results underscore the potential of genetic algorithms in this domain, particularly their adaptability to evolving cryptographic criteria and their inherent parallelization capabilities. These characteristics position genetic approaches as promising avenues for future research, potentially leading to more efficient, flexible, and innovative S-box generation techniques.
In conclusion, while not surpassing existing methods in raw performance, our genetic algorithm approach offers a valuable alternative that matches the best-known results. This equivalence, coupled with the unique advantages of genetic algorithms, opens new perspectives in cryptographic research and development. Future work should focus on exploiting these advantages, potentially through hybridization with other heuristic methods or by leveraging parallel computing architectures to further enhance S-box generation efficiency.
1 58,213 65,942 72,830 86,642 101,726 111,990 112,718 125,113 132,806 140,336 149,339 4 56,067 64,863 75,069 89,598 94,726 105,925 122,364 137,003 136,740 151,874 163,291 7 49,277 67,198 77,848 88,353 103,154 109,618 122,382 130,901 142,463 144,601 165,918 10 54,636 65,723 82,198 92,542 102,797 114,163 129,411 137,442 147,416 161,020 165,672 13 56,042 62,660 83,216 94,538 101,073 117,611 124,466 135,244 152,048 158,696 171,756 16 56,010 68,711 79,645 93,134 107,371 120,567 125,274 140,817 150,494 155,049 169,462 19 56,532 65,910 82,883 92,911 105,144 117,877 129,718 142,017 155,463 164,902 175,531 22 54,775 67,236 77,663 92,874 105,559 120,992 131,029 140,772 156,224 162,808 176,669 25 50,066 70,394 79,596 98,967 115,462 118,406 135,294 144,708 157,321 177,621 183,087 28 54,203 70,453 82,200 91,841 108,783 121,683 133,665 152,984 159,887 176,751 181,781 31 53,709 71,581 91,827 101,536 109,625 126,616 143,233 156,987 160,573 183,069 192,493