A GMM type construction for resilient S-boxes with higher-dimensional vectorial outputs and strictly almost optimal nonlinearity

Resilient substitution boxes (S-boxes) with high nonlinearity are important cryptographic primitives in the design of certain encryption algorithms. There are several trade-offs between the most important cryptographic parameters and their simultaneous optimization is regarded as a difficult task. In this paper we provide a construction technique to obtain resilient S-boxes with so-called strictly almost optimal (SAO) nonlinearity for a larger number of output bits m than previously known. This is the first time that the nonlinearity bound 2n−1− 2 of resilient (n,m) S-boxes, where n and m denote the number of the input and output bits respectively, has been exceeded for m > bn4 c. Thus, resilient S-boxes with extremely high nonlinearity and a larger output space compared to other design methods have been obtained.