Paper 2022/1566

Characterisation of Bijectivity Preserving Componentwise Modification of S-Boxes

Abstract

Various systematic modifications of vectorial Boolean functions have been used for finding new previously unknown classes of S-boxes with good or even optimal differential uniformity and nonlinearity. Recently, a new method was proposed for modification a component of a bijective vectorial Boolean function by using a linear function. It was shown that the modified function remains bijective under the assumption that the inverse of the function admits a linear structure. A previously known construction of such a modification based on bijective Gold functions in odd dimension is a special case of this type of modification. In this paper, we show that the existence of a linear structure is necessary. Further, we consider replacement of a component of a bijective vectorial Boolean function in the general case. We prove that a permutation on $\mathbb{F}_2^n$ remains bijective if and only if the replacement is done by composing the permutation with an unbalanced Feistel transformation where the round function is any Boolean function on $\mathbb{F}_2^{n-1}$.