On Various Nonlinearity Measures for Boolean Functions

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, μ 1,μ 2, there exist functions f 1,f 2 with f 1 being more nonlinear than f 2 according to μ 1, but less nonlinear according to μ 2. We also present new connections between two of these measures. Additionally, we give a lower bound on the multiplicative complexity of collision-free functions.