Comment on 'Particle Swarm Optimization Based Highly Nonlinear Substitution-Boxes Generation for Security Applications'

This report presents a comment on the article 'Particle Swarm Optimization Based Highly Nonlinear Substitution-Boxes Generation for Security Applications', that was recently published in IEEE Access (Volume 8, 2020, date of publication June 23, 2020). This article explores a new s-box generation method, which based on the particle swarm optimization. The authors claim, that their proposed technique they surpasses the known results. In particular, they assert that the s-boxes, generated by them, (the article lists three examples in Tables 1-3) are comparable in nonlinearity to the best-known results, or even outperform them. In addition, the authors argue that the new s-boxes are able to offer better resistance against linear cryptanalysis than the most recent optimization-based s-boxes. We show that this is not the case. Indeed, the resistance against linear cryptanalysis is expressed with the indexes of the s-box nonlinearity. The nonlinearity is estimated as the minimum nonlinearity of the component Boolean functions, that describes the certain s-box. However, the authors of the article erroneously estimate the nonlinearity of only the coordinate Boolean functions. Any component Boolean function has to be highly nonlinear, and we show that in the examples given by the authors, there are functions with very low value of the nonlinearity. Therefore, the authors' general conclusion about the better resistance against linear cryptanalysis is also not confirmed. Nevertheless, it should be noted that s-box generation method, based on the particle swarm optimization, is the promising direction that needs the further researches. However, it is necessary to revise the analysis of the performance of this technique (see Section 4) and to clarify the obtained dependences (see Fig. 2-4) and the gained insights.