Highly nonlinear plateaued functions

Abstract

The authors describe a method for producing Boolean functions of degree d ≥ 3 in n = 2dk − 1 (k = 1,  2,  …) variables, such that the functions are plateaued and balanced, have high nonlinearity and have no linear structures. The nonlinearity is 2 n−1 − 2(n−1)/2, which is the same as the largest possible nonlinearity for a quadratic function in n (odd) variables (the so-called ‘quadratic bound’). Their theorem uses some new ideas to generalise a theorem, which gave the case d = 3, in a 2009 paper by Fengrong Zhang et al. They discuss the cryptographic properties and applications for the functions.