On the Relationship Between Resilient Boolean Functions and Linear Branch Number of S-Boxes

Abstract

Differential branch number and linear branch number are critical for the security of symmetric ciphers. The recent trend in the designs like PRESENT block cipher, ASCON authenticated encryption shows that applying S-boxes that have nontrivial differential and linear branch number can significantly reduce the number of rounds. As we see in the literature that the class of $4\times 4$ S-boxes have been well-analysed, however, a little is known about the $n \times n$ S-boxes for $n \ge 5$. For instance, the complete classification of $5 \times 5$ affine equivalent S-boxes is still unknown. Therefore, it is challenging to obtain “the best” S-boxes with dimension $\ge$5 that can be used in symmetric cipher designs. In this article, we present a novel approach to construct S-boxes that identifies classes of $n \times n$ S-boxes ($n = 5, 6$) with differential branch number 3 and linear branch number 3, and ensures other cryptographic properties. To the best of our knowledge, we are the first to report $6\times 6$ S-boxes with linear branch number 3, differential branch number 3, and with other good cryptographic properties such as nonlinearity 24 and differential uniformity 4.