Four Measures of Nonlinearity

  • Joan Boyar
  • Magnus Find
  • René Peralta
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7878)


Cryptographic applications, such as hashing, block ciphers and stream ciphers, make use of functions which are simple by some criteria (such as circuit implementations), yet hard to invert almost everywhere. A necessary condition for the latter property is to be “sufficiently distant” from linear, and cryptographers have proposed several measures for this distance. In this paper, we show that four common measures, nonlinearity, algebraic degree, annihilator immunity, 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 μ 1(f 1) > μ 1(f 2) but μ 2(f 1) < μ 2(f 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 Function Block Cipher Stream Cipher Bend Function Algebraic Degree 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

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

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