Decomposing Bent Function


Anne Canteaut and Pascale Charpin
INRIA, projet CODES
BP 105
78153 Le Chesnay Cedex, France
{Anne.Canteaut,Pascale.Charpin}@inria.fr

Regular paper in IEEE Trans. Inform. Theory , 49(8), August 2003.

presented at ISIT 2002, Lausanne, Switzerland.

Abstract

In a recent paper, it is shown that the restrictions of bent functions to subspaces of codimension 1 and 2 are highly nonlinear. Here, we present an extensive study of the restrictions of bent functions to affine subspaces. We propose several methods which are mainly based on properties of the derivatives and of the dual of a given bent function. We solve an open problem due to Hou (1990). We especially describe the connection, for a bent function, between the Fourier spectra of its restrictions and the decompositions of its dual. Most notably, we show that the Fourier spectra of the restrictions of a bent function to the subspaces of codimension 2 can be explicitly derived from the Hamming weights of the second derivatives of the dual function. The last part of the paper is devoted to some infinite classes of bent functions which cannot be decomposed into four bent functions.

Keywords

Bent functions, Boolean functions, derivatives of Boolean functions, Reed-Muller codes, restrictions of Boolean functions.