Chapter 6 - Special Types of Boolean Functions

Abstract

This chapter discusses specific types of Boolean functions, with emphasis on the properties that are relevant to cryptography applications. The first topic is symmetric functions, since these are so special that extensive analysis of their properties is possible. The more general class of rotation symmetric functions turns out to be much richer in useful cryptographic functions, and much of the chapter is devoted to the theory of these functions, which were introduced under that name in 1999. In the following decade, various results were proved which led to cryptographic and coding theory applications for these functions. In 2008, a conjecture (still unproved) that there are no homogeneous rotation symmetric bent functions of degree larger than 2 was made. After 2009, an extensive theory about the affine equivalence classes of rotation symmetric functions, with consequences for their weight and nonlinearity, was developed. The main results for this theory so far are given in the chapter, with references to the literature for detailed proofs.