Modifications of Bijective S-Boxes with Linear Structures

. 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. In this paper, a new general modiﬁcation method is given that preserves the bijectivity property of the function in case 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 the new method.