Construction of nonlinear component based on bent Boolean functions

Abstract

The substitution box (S-box) is the core component of any block cipher that creates confusion in the ciphertext. This research contributes to creating a substitution box based on bent Boolean functions. Bent Boolean functions are constructed using the Maiorana–McFarland method and used as the coordinate functions of the substitution box. They are maximal nonlinear and inherently imbalanced, making them unsuitable for direct use in constructing a confusion component. The imbalance nature of the coordinate Boolean function directly impacts the bijective property of the nonlinear component. To create a bijective confusion component, we focus on mitigating the imbalance of each coordinate Boolean function. The methodology adopted is simple and efficient as compared to other heuristic techniques. By defining an empty matrix and putting bent Boolean functions side by side as matrix elements, we shape the initial $2^{n} \times k$ sized substitution box, $k\, \in (2,...,n)$. We limit the number of occurrences of each possible outcome of initial $2^{n} \times k$ substitution box to make a bijective $n \times n$ S-box. These small initial S-boxes lay a good foundation for constructing strong subsequent S-boxes. Experimental results based on the generation of $8 \times 8$ substitution box indicate that the suggested algorithm outperforms other competing heuristic approaches in nonlinearity, while maintaining sufficient performance in the strict avalanche criterion, bit-independence, linear and differential probability properties. The suggested nonlinear component is tested using well-known images from the literature, and the histogram analysis of the confused images demonstrates a high degree of randomness.