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On various nonlinearity measures for boolean functions



Joan Boyar, Magnus G. Find, Rene Peralta


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. 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, mu_1,mu_2, there exist functions f_1,f_2 with f_1 being more nonlinear than f_2 according to \mu_1, but less nonlinear according to \mu_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.
Cryptography and Communication


nonlinearity, cryptography, boolean functions, algebraic degree, annihilator immunity, collision-free, thickness, normality


Boyar, J. , Find, M. and Peralta, R. (2016), On various nonlinearity measures for boolean functions, Cryptography and Communication, [online],, (Accessed June 16, 2024)


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Created May 18, 2016, Updated October 12, 2021