Abstract
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-box es created by the Feistel structure have a poor property of boomerang uniformity. The genetic algorithm is introduced to improve the properties of the S-box es 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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\times $ </tex-math></inline-formula> 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.
Highlights
The boomerang attack [1] is a variant of the differential attack
In this paper, a genetic algorithm is used to improve the properties of the S-boxes created by the Feistel structure
New genetic operators are designed for the genetic algorithm to develop 8 × 8 S-boxes with low differential uniformity, high nonlinearity, and low boomerang uniformity
Summary
The boomerang attack [1] is a variant of the differential attack. For ciphers that the probabilities of the differential characteristics decrease exponentially with respect to the growth of rounds, the boomerang attack can concatenate two short characteristics to form a longer characteristic with a better probability. Wang: New Genetic Operators for Developing S-Boxes With Low Boomerang Uniformity and a class of 4-uniform BCT permutations over F2n were obtained. A new genetic algorithm is introduced to improve the properties of the S-boxes created by the Feistel structure. The new genetic algorithm generates 8 × 8 bijective S-boxes with low differential uniformity, high nonlinearity, and low boomerang uniformity. Benefiting from the full use of the advantages of gene exchange and gene mutation, the new genetic algorithm in this paper dramatically improves the properties of the S-boxes created by the Feistel structure. The new genetic algorithm generates the best S-box with differential uniformity 6, nonlinearity 108, and boomerang uniformity 10, whereas the Feistel structure creates the best S-box with differential uniformity 16, nonlinearity 96, and boomerang uniformity 52.
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