Thickness distribution of Boolean functions in 4 and 5 variables and a comparison with other cryptographic properties

This paper explores the distribution of algebraic thickness of Boolean functions (that is, the minimum number of terms in the ANF of the functions in the orbit of a Boolean function, through all affine transformations), in four and five variables, and the complete distribution is presented. Additionally, a complete analysis of some complexity properties (e.g., nonlinearity, balancedness, etc.) of all relevant orbits of Boolean functions is presented. Some properties of our notion of rigid function (which enabled us to reduce significantly the computation) are shown and some open questions are proposed, providing some further explanation of one of these questions.