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Substitution boxes are the main nonlinear component of block ciphers. The security of these ciphers against linear, differential, or side-channel attacks is dependent on the design of such component and their intrinsic properties. There are several methods that aim to cryptographically define, generate, or search for strong substitution boxes. The application of combinatorial optimization algorithms is one of the most useful methodologies in this research area. In this article, we present a novel hybrid method based on the Leaders and Followers and hill-climbing over Hamming Weight Classes metaheuristics, coupled with a new trade-off fitness function that generates 8-bit bijective substitution boxes with good resisting properties towards classical cryptanalysis and side-channel attacks by power consumption. We address the best Pareto optimal solutions for the multi-objective optimization of non-linearity and confusion coefficient variance.

The search of bijective n × n S-boxes resilient to power attacks in the space of dimension (2 n) ! is a controversial topic in the cryptology community nowadays. This paper proposes partitioning the space of (2 n) ! S-boxes into equivalence classes using the hypothetical power leakage according to the Hamming weights model, which ensures a homogeneous theoretical resistance within the class against power attacks. We developed a fast algorithm to generate these S-boxes by class. It was mathematically demonstrated that the theoretical metric confusion coefficient variance takes constant values within each class. A new search strategy—jumping over the class space—is justified to find S-boxes with high confusion coefficient variance in the space partitioned by Hamming weight classes. In addition, a decision criterion is proposed to move quickly between or within classes. The number of classes and the number of S-boxes within each class are calculated, showing that, as n increases, the class space dimension is an ever-smaller fraction of the space of S-boxes, which significantly reduces the space of search of S-boxes resilient to power attacks, when the search is performed from class to class. [ABSTRACT FROM AUTHOR]

The search of bijective n×n S-boxes resilient to power attacks in the space of dimension (2n)! is a controversial topic in the cryptology community nowadays. This paper proposes partitioning the space of (2n)! S-boxes into equivalence classes using the hypothetical power leakage according to the Hamming weights model, which ensures a homogeneous theoretical resistance within the class against power attacks. We developed a fast algorithm to generate these S-boxes by class. It was mathematically demonstrated that the theoretical metric confusion coefficient variance takes constant values within each class. A new search strategy—jumping over the class space—is justified to find S-boxes with high confusion coefficient variance in the space partitioned by Hamming weight classes. In addition, a decision criterion is proposed to move quickly between or within classes. The number of classes and the number of S-boxes within each class are calculated, showing that, as n increases, the class space dimension is an ever-smaller fraction of the space of S-boxes, which significantly reduces the space of search of S-boxes resilient to power attacks, when the search is performed from class to class.

A local search method is developed and applied to create trajectories over S-box spaces in this work. These trajectories show a strong linear correlation between confusion coefficient variance, transparency order, modified transparency order, and revised transparency order under the Hamming weight model. When the confusion coefficient variance values increase, the values of transparency order, modified transparency order beta zero, and revised transparency order beta zero decrease, reflecting the same theoretical resistance against side-channel attacks by power consumption. From the best of our knowledge, it is the first time that Local Search trajectories are used to discover relations between cryptography properties. The experimental results and comparisons provide evidence of the capabilities of the proposal to handle the cryptography problems. • The Local Search is used to create Trajectories over s-box Space. • The CCV has linear correlation with MTO and RTO. • Trajectories are used to study the correlations between S-boxes properties. • Linear correlations between CCV, MTO and RTO are found. [ABSTRACT FROM AUTHOR]