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On the Higher Order Nonlinearities of Boolean Functions and S-Boxes, and Their Generalizations

(Invited Paper)

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Sequences and Their Applications - SETA 2008 (SETA 2008)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5203))

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Abstract

The r-th order nonlinearity of a Boolean function \(f:F_2^n\to F_2\) is its minimum Hamming distance to all functions of algebraic degrees at most r, where r is a positive integer. The r-th order nonlinearity of an S-box \(F:F_2^n\to F_2^m\) is the minimum r-th order nonlinearity of its component functions v·F, \(v\in F_2^m\setminus \{0\}\). The role of this cryptographic criterion against attacks on stream and block ciphers has been illustrated by several papers. Its study is also interesting for coding theory and is related to the covering radius of Reed-Muller codes (i.e. the maximum multiplicity of errors that have to be corrected when maximum likelihood decoding is used on a binary symmetric channel). We give a survey of what is known on this parameter, including the bounds involving the algebraic immunity of the function, the bounds involving the higher order nonlinearities of its derivatives, and the resulting bounds on the higher order nonlinearities of the multiplicative inverse functions (used in the S-boxes of the AES). We show an improvement, when we consider an S-box instead of a Boolean function, of the bounds on the higher order nonlinearity expressed by means of the algebraic immunity. We study a generalization (for S-boxes) of the notion and we give new results on it.

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Authors and Affiliations

  1. Department of Mathematics (MAATICAH), University of Paris 8, 2 rue de la liberté, 93526, Saint-Denis Cedex, France

    Claude Carlet

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  1. Claude Carlet
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Solomon W. Golomb Matthew G. Parker Alexander Pott Arne Winterhof

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Carlet, C. (2008). On the Higher Order Nonlinearities of Boolean Functions and S-Boxes, and Their Generalizations. In: Golomb, S.W., Parker, M.G., Pott, A., Winterhof, A. (eds) Sequences and Their Applications - SETA 2008. SETA 2008. Lecture Notes in Computer Science, vol 5203. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-85912-3_31

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  • DOI: https://doi.org/10.1007/978-3-540-85912-3_31

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-85911-6

  • Online ISBN: 978-3-540-85912-3

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