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On Semi-bent Functions and Related Plateaued Functions Over the Galois Field \(\mathbb{F}_{2^{n}}\)

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Open Problems in Mathematics and Computational Science

Abstract

Plateaued functions were introduced in 1999 by Zheng and Zhang as good candidates for designing cryptographic functions since they possess desirable various cryptographic characteristics. They are defined in terms of the Walsh–Hadamard spectrum. Plateaued functions bring together various nonlinear characteristics and include two important classes of Boolean functions defined in even dimension: the well-known bent functions and the semi-bent functions. Bent functions (including their constructions) have been extensively investigated for more than 35 years. Very recently, the study of semi-bent functions has attracted the attention of several researchers. Much progress in the design of such functions has been made. The chapter is devoted to certain plateaued functions. The focus is particularly on semi-bent functions defined over the Galois field \(\mathbb{F}_{2^{n}}\) (n even). We review what is known in this framework and investigate constructions.

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Notes

  1. 1.

    A Boolean function f is said to be homogeneous of degree r if \(f(x) =\sum _{ i=0}^{2^{n}-1 }a_{i}x^{i}\) where a i  = 0 for wt(i) ≠ r, where wt(i) is the Hamming weight of i.

  2. 2.

    We say a point p = (x 0, …, x n ) is on a line L[y 0, …, y n ] if and only if \(x_{0}y_{0} + x_{1}y_{1} + \cdots x_{n}y_{n} = 0\).

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Acknowledgements

The author wishes to thank Claude Carlet for his careful reading and interesting comments.

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Authors and Affiliations

  1. Department of Mathematics, University of Paris VIII, LAGA (Laboratoire Analyse, Géometrie et Applications), UMR 7539, CNRS, Paris, France

    Sihem Mesnager

  2. University of Paris XIII, Sorbonne Paris Cité, 2 rue de la liberté, 93526, Saint-Denis Cedex, France

    Sihem Mesnager

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  1. Sihem Mesnager
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Correspondence to Sihem Mesnager .

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  1. Dept. of Computer Science, University of California, Santa Barbara, Santa Barbara, California, USA

    Çetin Kaya Koç

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Mesnager, S. (2014). On Semi-bent Functions and Related Plateaued Functions Over the Galois Field \(\mathbb{F}_{2^{n}}\) . In: Koç, Ç. (eds) Open Problems in Mathematics and Computational Science. Springer, Cham. https://doi.org/10.1007/978-3-319-10683-0_11

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  • DOI: https://doi.org/10.1007/978-3-319-10683-0_11

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