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Cryptographic algorithms are used to ensure confidentiality, integrity and authenticity of data in information systems. One of the important areas of modern cryptography is that of symmetric key ciphers. They convert the input plaintext into ciphertext, representing it as a random sequence of characters. S-boxes are designed to complicate the input–output relationship of the cipher. In other words, S-boxes introduce nonlinearity into the encryption process, complicating the use of different methods of cryptanalysis (linear, differential, statistical, correlation, etc.). In addition, S-boxes must be random. This property means that nonlinear substitution cannot be represented as simple algebraic constructions. Random S-boxes are designed to protect against algebraic methods of cryptanalysis. Thus, generation of random S-boxes is an important area of research directly related to the design of modern cryptographically strong symmetric ciphers. This problem has been solved in many related works, including some using the simulated annealing (SA) algorithm. Some works managed to generate 8-bit bijective S-boxes with a nonlinearity index of 104. However, this required enormous computational resources. This paper presents the results of our optimization of SA via various parameters. We were able to significantly reduce the computational complexity of substitution generation with SA. In addition, we also significantly increased the probability of generating the target S-boxes with a nonlinearity score of 104.

In the realm of smart communication systems, where the ubiquity of 5G/6G networks and IoT applications demands robust data confidentiality, the cryptographic integrity of block and stream cipher mechanisms plays a pivotal role. This paper focuses on the enhancement of cryptographic strength in these systems through an innovative approach to generating substitution boxes (S-boxes), which are integral in achieving confusion and diffusion properties in substitution–permutation networks. These properties are critical in thwarting statistical, differential, linear, and other forms of cryptanalysis, and are equally vital in pseudorandom number generation and cryptographic hashing algorithms. The paper addresses the challenge of rapidly producing random S-boxes with desired cryptographic attributes, a task notably arduous given the complexity of existing generation algorithms. We delve into the hill climbing algorithm, exploring various cost functions and their impact on computational complexity for generating S-boxes with a target nonlinearity of 104. Our contribution lies in proposing a new cost function that markedly reduces the generation complexity, bringing down the iteration count to under 50,000 for achieving the desired S-box. This advancement is particularly significant in the context of smart communication environments, where the balance between security and performance is paramount.

This paper introduces the hybrid population-based hill-climbing (HPHC) algorithm, a novel approach for generating cryptographically strong S-boxes that combines the efficiency of hill climbing with the exploration capabilities of population-based methods. The algorithm achieves consistent generation of 8-bit S-boxes with a nonlinearity of 104, a critical threshold for cryptographic applications. Our approach demonstrates remarkable efficiency, requiring only 49,277 evaluations on average to generate such S-boxes, representing a 600-fold improvement over traditional simulated annealing methods and a 15-fold improvement over recent genetic algorithm variants. We present comprehensive experimental results from extensive parameter space exploration, revealing that minimal populations (often single-individual) combined with moderate mutation rates achieve optimal performance. This paper provides detailed analysis of algorithm behavior, parameter sensitivity, and performance characteristics, supported by rigorous statistical evaluation. We demonstrate that population size should approximate available thread count for optimal parallel execution despite smaller populations being theoretically more efficient. The HPHC algorithm maintains high reliability across diverse parameter settings while requiring minimal computational resources, making it particularly suitable for practical cryptographic applications.

S-boxes, the key nonlinear component in numerous cryptographic systems, play a crucial role in ensuring security. The quest for random highly nonlinear S-boxes, a desirable attribute for better diffusion and confusion properties, is, therefore, a critical endeavor in cryptographic research. However, generating such nonlinear S-boxes often involves significant computational effort, presenting a major challenge for researchers. This paper addresses this gap by proposing an optimized version of the Simulated Annealing (SA) algorithm specifically tailored for efficient generation of highly nonlinear S-boxes. Our work introduces a multithreaded implementation of the SA algorithm, a heuristic search method known for its proficiency in combinatorial optimization. The multithreading feature enhances computational efficiency, making our approach more suitable for large-scale cryptographic applications. We further optimize the algorithm by incorporating additional exit criteria for both internal and external loops, which significantly reduces the computational complexity associated with the nonlinear substitution generation process. Furthermore, we undertake comprehensive experiments to identify the optimal parameters of the SA algorithm, aiming to maximize the probability of generating target S-boxes while minimizing the number of iterations. This optimization step provides a clear pathway to improve the success rate of the generation process. The results of our study demonstrate a significant improvement over previous works, showing a 30–40% enhancement in the generation of nonlinear S-boxes.

The nonlinear feedback shift registers of the second order inare considered, because based on them it can be developed a generator of stream ciphers with enhanced cryptographic strength. Feasibility of nonlinear feedback shift register search is analyzed. These registers form a maximal length sequence, using programmable logic devices. Performance evaluation of programmable logic devices in the generation of pseudo-random sequence by nonlinear feedback shift registers is given. Recommendations to increase this performance are given. The dependence of the maximum generation rate (clock frequency), programmable logic devices on the number of concurrent nonlinear registers is analyzed. A comparison of the generation rate of the sequences that are generated by nonlinear feedback shift registers is done using hardware and software. The author suggests, describes and explores the search method of nonlinear feedback shift registers, generating a sequence with a maximum period. As the main result are found non-linear 26, 27, 28 and 29 degrees polynomials.

Nonlinear substitutions or S-boxes are important cryptographic primitives of modern symmetric ciphers. They are designed to complicate the plaintext-ciphertext dependency. According to modern ideas, the S-box should be bijective, have high nonlinearity and algebraic immunity, low delta uniformity, and linear redundancy. These criteria directly affect the cryptographic strength of ciphers, providing resistance to statistical, linear, algebraic, differential, and other cryptanalysis techniques. Many researchers have used various heuristic search algorithms to generate random S-boxes with high nonlinearity; however, the complexity of this task is still high. For example, the best-known algorithm to generate a random 8-bit bijective S-box with nonlinearity 104 requires high computational effort—more than 65,000 intermediate estimates or search iterations. In this article, we explore a hill-climbing algorithm and optimize the heuristic search parameters. We show that the complexity of generating S-boxes can be significantly reduced. To search for a random bijective S-box with nonlinearity 104, only about 50,000 intermediate search iterations are required. In addition, we generate cryptographically strong S-Boxes for which additional criteria are provided. We present estimates of the complexity of the search and estimates of the probabilities of generating substitutions with various cryptographic indicators. The extracted results demonstrate a significant improvement in our approach compared to the state of the art in terms of providing linear non-redundancy, nonlinearity, algebraic immunity, and delta uniformity.

The efficiency of heuristic search algorithms is a critical factor in the realm of cryptographic primitive construction, particularly in the generation of highly nonlinear bijective permutations, known as substitution boxes (S-boxes). The vast search space of 256! (256 factorial) permutations for 8-bit sequences poses a significant challenge in isolating S-boxes with optimal nonlinearity, a crucial property for enhancing the resilience of symmetric ciphers against cryptanalytic attacks. Existing approaches to this problem suffer from high computational costs and limited success rates, necessitating the development of more efficient and effective methods. This study introduces a novel approach that addresses these limitations by dynamically adjusting the cost function parameters within the hill-climbing heuristic search algorithm. By incorporating principles from dynamic programming, our methodology leverages feedback from previous iterations to adaptively refine the search trajectory, leading to a significant reduction in the number of iterations required to converge on optimal solutions. Through extensive comparative analyses with state-of-the-art techniques, we demonstrate that our approach achieves a remarkable 100% success rate in locating 8-bit bijective S-boxes with maximal nonlinearity, while requiring only 50,000 iterations on average—a substantial improvement over existing methods. The proposed dynamic parameter adaptation mechanism not only enhances the computational efficiency of the search process, but also showcases the potential for interdisciplinary collaboration between the fields of heuristic optimization and cryptography. The practical implications of our findings are significant, as the ability to efficiently generate highly nonlinear S-boxes directly contributes to the development of more secure and robust symmetric encryption systems. Furthermore, the dynamic parameter adaptation concept introduced in this study opens up new avenues for future research in the broader context of heuristic optimization and its applications across various domains.

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 8x8 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 suggest 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.