Vectorial functions and covering sequences

Abstract

The design of large classes of highly nonlinear resilient vectorial functions (mappings from double-struck F sign2n into double-struck F sign2m, also called S-boxes) is needed for iterated block ciphers and for pseudo-random generators with multiple output. In this paper, we recall the diverse known constructions of such S-boxes, and we show that those which provide good candidate functions are, in fact, all in the same class. This class corresponds to a generalization of a well known construction due to Maiorana and MacFarland. We study in detail this construction and we specify it to obtain good S-boxes. In a second part, we generalize to S-boxes the notion of covering sequence. We show that this generalization has the same properties as for Boolean functions, and that it has nice additional properties of stability. We study how this notion can be used to design attacks, and we explain why some functions, including the elements of the new class, cannot be involved in the construction of iterated block ciphers. © Springer-Verlag Berlin Heidelberg 2004.