Cryptography and Communications

, Volume 8, Issue 3, pp 313–330 | Cite as

On various nonlinearity measures for boolean functions

  • Joan Boyar
  • Magnus Gausdal Find
  • René 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. 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.


Boolean functions Nonlinearity Multiplicative complexity Algebraic degree Annihilator immunity Thickness Normality Collision-free 

Mathematics Subject Classification (2010)

94C10 94A60 06E30 



We are grateful to Meltem Sönmez Turan for many discussions on the subject of this work.


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Copyright information

© Springer Science+Business Media New York (outside the USA) 2015

Authors and Affiliations

  • Joan Boyar
    • 1
  • Magnus Gausdal Find
    • 2
  • René Peralta
    • 2
  1. 1.Department of Mathematics and Computer ScienceUniversity of Southern DenmarkOdenseDenmark
  2. 2.Information Technology LaboratoryNational Institute of Standards and TechnologyGaithersburgUSA

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