Theory of 3-rotation symmetric cubic Boolean functions

Abstract

Rotation symmetric Boolean functions have been extensively studied in the last 15 years or so because of their importance in cryptography and coding theory. Until recently, very little was known about such basic questions as when two such functions are affine equivalent. This question in important in applications, because almost all important properties of Boolean functions (such as Hamming weight, nonlinearity, etc.) are affine invariants, so when searching a set for functions with useful properties, it suffices to consider just one function in each equivalence class. This can greatly reduce computation time. Even for quadratic functions, the analysis of affine equivalence was only completed in 2009. The much more complicated case of cubic functions was completed in the special case of affine equivalence under permutations for monomial rotation symmetric functions in two papers from 2011 and 2014. There has also been recent progress for some special cases for functions of degree >3. In 2007 it was found that functions satisfying a new notion of k-rotation symmetry for k > 1 (where the case k = 1 is ordinary rotation symmetry) were of substantial interest in cryptography and coding theory. Since then several researchers have used these functions for k = 2 and 3 to study such topics as construction of bent functions, nonlinearity and covering radii of various codes. In this paper we develop a detailed theory for the monomial 3-rotation symmetric cubic functions, extending earlier work for the case k = 2 of these functions.