A construction method of balanced rotation symmetric Boolean functions on arbitrary even number of variables with optima
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A construction method of balanced rotation symmetric Boolean functions on arbitrary even number of variables with optimal algebraic immunity Sihem Mesnager1,2 · Sihong Su3,4
· Hui Zhang3
Received: 1 February 2020 / Revised: 22 September 2020 / Accepted: 22 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Rotation symmetric Boolean functions incorporate a super-class of symmetric functions which represent an attractive corpus for computer investigation. These functions have been investigated from the viewpoints of bentness and correlation immunity and have also played a role in the study of nonlinearity. In the literature, many constructions of balanced oddvariable rotation symmetric Boolean functions with optimal algebraic immunity have been derived. While it seems that the construction of balanced even-variable rotation symmetric Boolean functions with optimal algebraic immunity is very hard work to breakthrough. In this paper, we present for the first time a construction of balanced rotation symmetric Boolean functions on an arbitrary even number of variables with optimal algebraic immunity by modifying the support of the majority function. The nonlinearity of the newly constructed rotation symmetric Boolean functions is also derived. Keywords Rotation symmetric Boolean function · Balancedness · Algebraic immunity · Nonlinearity Mathematics Subject Classification 94C10 · 14G50 · 94A60 · 94B27 · 94B40
Communicated by J. D. Key.
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Sihong Su [email protected] Sihem Mesnager [email protected] Hui Zhang [email protected]
1
Department of Mathematics University of Paris VIII F-93526 Saint-Denis, Laboratory Geometry, Analysis and Applications, LAGA, CNRS, University Sorbonne Paris Nord, CNRS, UMR 7539, 93430 Villetaneuse, France
2
Telecom Paris, 91120 Palaiseau, France
3
School of Mathematics and Statistics, Henan University, Kaifeng 475004, China
4
The Department of Mathematics, University of Paris VIII, 93526 Saint-Denis, France
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S. Mesnager et al.
1 Introduction Boolean functions play an important role in the system of symmetric cryptography, especially in stream ciphers based on linear feedback shift registers (LFSRs). In symmetric cryptosystems, efficient implementations of Boolean functions with a large number of variables combined with some important cryptographic properties are very desirable [1]. If a Boolean function is invariant under the action of the cyclic group on the inputs, it is called a rotation symmetric (RS) Boolean function. RS Boolean functions represent a super-class of symmetric functions that are closely related to idempotent functions (a notion introduced by Filiol and Fontaine [9,10]). Such a class of Boolean functions is of great interest since they can be stored in less space and allow faster computation of the Walsh–Hadamard transform. Carlet and Gao [3] have studied idempotent functions in bivariate and univariate representations and derived constructions of RS Boolean functions. The cryptographic parameters of RS Boolea
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