Evolving bijective S-Boxes using hybrid adaptive genetic algorithm with optimal cryptographic properties

Pratap Kumar Behera, Sugata Gangopadhyay

Abstract: The security of the block cipher largely depends upon the cryptographic strength of the S-Boxes resistance to the existing cryptanalytic attacks. The nonlinearity and differential uniformity of S-Boxes are used as a quantitative measurement to measure its resistance against linear approximation attack, and differential attack, respectively. The S-Box with high nonlinearity, and low differential uniformity is considered as cryptographically secure S-Boxes. However, as the size of S-Boxes increases, finding a cryptographically strong S-Boxes with high nonlinearity and low differential uniformity is computationally hard. We considered the problem of constructing bijective S-Boxes as a combinatorial optimization problem. In this paper, we use the genetic algorithm, hybrid genetic algorithm, adaptive genetic algorithm and adaptive genetic algorithm with the integration of a local search procedure called hybrid adaptive genetic algorithm (HAGA) for constructing highly nonlinear S-Boxes along with other important cryptographic properties. We Construct $$8 \times 8$$ 8 × 8 bijective S-Boxes and compare the results of our produced S-Boxes with the previously published S-Boxes generated by different heuristic and evolutionary techniques. The S-Boxes generated by our HAGA attains the nonlinearity 108, differential uniformity 6, and algebraic degree 7, which outperforms most of the existing heuristic and evolutionary techniques.