Classification of cubic (n-4)-resilient Boolean functions

Carlet and Charpin classified the set of cubic (n-4)-resilient Boolean functions into four different types with respect to the Walsh spectrum and the dimension of the linear space. Based on the classification of RM(3,6)/RM(1,6), we have completed this classification of cubic (n-4)-resilient Boolean functions by deriving the corresponding algebraic normal form (ANF) and autocorrelation spectrum for each of the four types. At the same time, we have solved an open problem by proving that all plateaued cubic (n-4)-resilient Boolean functions have dimension of the linear space equal either to n-5 or n-6.

I. Introduction

The properties of quadratic Boolean functions (i.e., the second order Reed–Muller code ) are well studied, (e.g., the weight distribution [1], the affine equivalence classes [1], the classification of resilient functions [2] and functions satisfying propagation characteristics [3], etc.) However, it is not trivial to extend these results for functions of higher degrees and even for cubic functions. It is important to understand how the properties behave for the different degrees of functions.