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Four Measures of Nonlinearity

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Book cover Algorithms and Complexity (CIAC 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7878))

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Abstract

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.

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Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Southern Denmark, Denmark

    Joan Boyar & Magnus Find

  2. Information Technology Laboratory, National Institute of Standards and Technology, USA

    René Peralta

Authors
  1. Joan Boyar
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  2. Magnus Find
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  3. René Peralta
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Editor information

Editors and Affiliations

  1. School of Engineering, Department of Computer Engineering and Informatics, University of Patras, 265 00, Patras, Greece

    Paul G. Spirakis

  2. LSI Department, Edif Omega, Universitat Politècnica de Catalunya, Campus Nord, Jordi Girona 1-3, 08034, Barcelona, Spain

    Maria Serna

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Boyar, J., Find, M., Peralta, R. (2013). Four Measures of Nonlinearity. In: Spirakis, P.G., Serna, M. (eds) Algorithms and Complexity. CIAC 2013. Lecture Notes in Computer Science, vol 7878. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38233-8_6

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  • DOI: https://doi.org/10.1007/978-3-642-38233-8_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-38232-1

  • Online ISBN: 978-3-642-38233-8

  • eBook Packages: Computer ScienceComputer Science (R0)

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