Secondary constructions of bent functions and their enforcement

Claude Carlet; Fengrong Zhang; Yupu Hu

doi:10.3934/amc.2012.6.305

Article Contents

2012, Volume 6, Issue 3: 305-314. Doi: 10.3934/amc.2012.6.305

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Secondary constructions of bent functions and their enforcement

1.
LAGA, Universities of Paris 8 and Paris 13; CNRS, UMR 7539, Department of Mathematics, University of Paris 8, 2 rue de la liberté, 93526 Saint-Denis cedex 02
2.
School of Computer Science and Technology, China University of Mining and Technology, Xuzhou, Jiangsu 221116, China, and ISN, Xidian University, Xi'an, Shannxi 710071
3.
State Key Laboratory of Integrated Services Networks, Xidian university, P.O. Box 95, Taibai Road 2, Xi'an, Shannxi 710071

Received: July 2011

Revised: March 2012

Published: August 2012

Abstract / Introduction Related Papers Cited by

Abstract

Thirty years ago, Rothaus introduced the notion of bent function and presented a secondary construction (building new bent functions from already defined ones), which is now called the Rothaus construction. This construction has a strict requirement for its initial functions. In this paper, we first concentrate on the design of the initial functions in the Rothaus construction. We show how to construct Maiorana-McFarland's (M-M) bent functions, which can then be used as initial functions, from Boolean permutations and orthomorphic permutations. We deduce that at least $(2^n!\times 2^{2^n})(2^{2^n}\times2^{2^{n-1}})^2$ bent functions in $2n+2$ variables can be constructed by using Rothaus' construction. In the second part of the note, we present a new secondary construction of bent functions which generalizes the Rothaus construction. This construction requires initial functions with stronger conditions; we give examples of functions satisfying them. Further, we generalize the new secondary construction of bent functions and illustrate it with examples.

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Mathematics Subject Classification: 06E30, 94A60.

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