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A Theory of Highly Nonlinear Functions

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Applied Algebra, Algebraic Algorithms and Error-Correcting Codes (AAECC 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3857))

Abstract

Highly nonlinear functions are important as sources of low-correlation sequences, high-distance codes and cryptographic primitives, as well as for applications in combinatorics and finite geometry.

We argue that the theory of such functions is best seen in terms of splitting factor pairs. This introduces an extra degree of freedom, through the pairing of a normalised function φ : G → N between groups with a homomorphism \(\varrho : {G} \rightarrow Aut{(N)}\).

From this perspective we introduce a new definition of equivalence for functions, relative to \(\varrho\), and show it preserves their difference distributions. When \(\varrho \equiv 1\) it includes CCZ and generalised linear equivalence, as well as planar and linear equivalence.

More generally, we use splitting factor pairs to relate several important measures of nonlinearity. We propose approaches to both linear approximation theory and bent functions, and to difference distribution theory and perfect nonlinear functions, which encompass the current approaches.

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Authors and Affiliations

  1. Royal Melbourne Institute of Technology, Melbourne, VIC, 3001, Australia

    K. J. Horadam

Authors
  1. K. J. Horadam
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Editors and Affiliations

  1. Department of Electrical Engineering, University of Hawaii, Honolulu, USA

    Marc P. C. Fossorier

  2. National Institute of Advanced Industrial Science and Technology (AIST), Research Center for Information Security (RCIS),  

    Hideki Imai

  3. Department of Electrical and Computer Engineering, University of California, CA 95616, Davis,  

    Shu Lin

  4. University Paul Sabatier, AAECC/IRIT, 118 route de Narbonne, 31300, Toulouse cedex, France

    Alain Poli

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© 2006 Springer-Verlag Berlin Heidelberg

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Horadam, K.J. (2006). A Theory of Highly Nonlinear Functions. In: Fossorier, M.P.C., Imai, H., Lin, S., Poli, A. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 2006. Lecture Notes in Computer Science, vol 3857. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11617983_8

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  • DOI: https://doi.org/10.1007/11617983_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-31423-3

  • Online ISBN: 978-3-540-31424-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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