On Various Nonlinearity Measures for Boolean Functions

Abstract

A necessary condition for the security of cryptographic functions is to be “sufficiently distant” from linear, and cryptographers have proposed several measures for this distance. In this paper, we show that six common measures, nonlinearity, algebraic degree, annihilator immunity, algebraic thickness, normality, and multiplicative complexity, are incomparable in the sense that for each pair of measures, µ1, µ2, there exist functions f1, f2 with f1 being more nonlinear than f2 according to µ1, but less nonlinear according to µ2. We also present new connections between two of these measures. Additionally, we give a lower bound on the multiplicative com-plexity of collision-free functions