On Propagation Criteria of Some Classes of Boolean Functions

Abstract

The denition of the propagation criterion of Boolean functions was introduced by Bart Preneel and co-authors. This concept represent a set of vectors, for which the corresponding derivatives of a Boolean function are balanced.  It characterizes the statistical properties of a family of Boolean function  derivatives that play an important role in the cryptosystem analysis and  synthesis. For some classes of Boolean functions, the propagation criterion  determines their extreme properties. For example, the propagation criterion of bent functions determines their maximum nonlinearity. However, the main disadvantage of bent functions is the lack of balancedness, which means that  such functions do not have a uniform output distribution. The construction of  balanced Boolean functions having a high nonlinearity and a large number of  vectors satisfying the propagation criterion is still an open problem in  cryptography. In this paper we obtain exact values and estimates of the number  of vectors satisfying the propagation criterion of Boolean functions from well- known cryptographic classes, such as plateaued functions, Maiorana-McFarland  functions, quadratic functions, algebraic degenerate functions and multiane  functions. We also show that the number of vectors satisfying the propagation  criterion is an invariant for the extension of the general affine group of the first  degree.