The boomerang uniformity measures the resistance of block ciphers to boomerang attacks and has become an essential criterion of the substitution box (S-box). However, the S-boxes created by the Feistel structure have a poor property of boomerang uniformity. The genetic algorithm is introduced to improve the properties of the S-boxes created by the Feistel structure. New genetic operators are designed for the genetic algorithm to improve its searchability. The new genetic algorithm generates some 8 × 8 bijective S-boxes with differential uniformity 6, nonlinearity 108, and boomerang uniformity 10, which has dramatically improved the properties of the S-boxes created by the Feistel structure. Furthermore, the new genetic algorithm also improves the properties of the S-box population created by the Feistel structure as a whole. We compare the S-boxes generated by the new genetic algorithm with those generated by the traditional one. The comparison results show that the S-boxes generated by the new genetic algorithm have better properties than the S-boxes generated by the traditional genetic algorithm, demonstrating the new genetic algorithm's effectiveness and superiority in developing S-boxes.INDEX TERMS Boomerang uniformity, S-box, genetic algorithm, genetic operator.
Genetic algorithms are widely adopted to solve optimization problems in robotic applications. In such safety-critical systems, it is vitally important to formally prove the correctness when genetic algorithms are applied. This paper focuses on formal modeling of crossover operations that are one of most important operations in genetic algorithms. Specially, we for the first time formalize crossover operations with higher-order logic based on HOL4 that is easy to be deployed with its user-friendly programing environment. With correctness-guaranteed formalized crossover operations, we can safely apply them in robotic applications. We implement our technique to solve a path planning problem using a genetic algorithm with our formalized crossover operations, and the results show the effectiveness of our technique.
We propose some properties for a fuzzy correlation test by reduced-spread fuzzy variance for sample fuzzy data. First, we define the condition of fuzzy data for repeatedly observed data or that which includes error term data. By using the average of spreads for fuzzy numbers, we reduce the spread of fuzzy variance and define the agreement index for the degree of acceptance and rejection. Given a non-normal random fuzzy sample, we have bivariate normal distribution by apply Box-Cox power fuzzy transformation and test the fuzzy correlation for independence between the variables provided by the agreement index.
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