Improved Construction of Nonlinear Resilient S-Boxes

Abstract

Abstract. We provide two new construction methods for nonlinear resilient functions. The first method is a simple modification of an elegant construction due to Zhang and Zheng and constructs n-input, m-output resilient S-boxes with degree d > m. We prove by an application of the Griesmer bound for linear error correcting codes that the modified Zhang-Zheng construction is superior to the previous method of Cheon in Crypto 2001. Our second construction uses a sharpened version of the Maiorana-McFarland technique to construct nonlinear resilient functions. The nonlinearity obtained by our second construction is better than previously known construction methods.

Highly Nonlinear Resilient Functions Through Disjoint Codes in Projective Spaces

“…In a recent work [22] the known construction methods have been merged into a general construction technique that uses the existence of a single linear code with certain parameters. This technique has been later modified in [14] slightly improving the results in [22] by utilizing all the 2 k − 1 nonzero codewords of a linear [u, k, t + 1] code rather than only 2 k−1 .…”

“…The construction of (36, 8, t) functions of high nonlinearity has been discussed a lot in the literature [14,15,22]. Here we only consider the case t = 2.…”

“…Some of these functions and the corresponding codes are listed in the Table 2, where the nonlinearity of our functions is compared to those presented in [14].…”

“…It is enough to consider the currently best known method in [14]. Applying this method one uses a code of shortest length, i.e.…”

“…a [7,3,4] code, together with an highly nonlinear function G ∈ M 3 9 . As given in [14,Theorem 8], the nonlinearity of the resulting (16, 3, 3) function is…”