Title: Characterisation and enumeration of a class of semi-bent quadratic Boolean functions

Authors: Neşe Koçak; Onur Koçak; Ferruh Özbudak; Zülfükar Saygı

Addresses: Institute of Applied Mathematics, Middle East Technical University, Ankara, Turkey ' Institute of Applied Mathematics, Middle East Technical University, Ankara, Turkey; Türktrust Inc., Ankara, Turkey ' Department of Mathematics and Institute of Applied Mathematics, Middle East Technical University, Ankara, Turkey ' Department of Mathematics, TOBB University of Economics and Technology, Ankara, Turkey

Abstract: In this paper, we consider semi-bentness of quadratic Boolean functions defined for even n and give the characterisation of these functions. Up to our knowledge, semi-bentness of this class has not been investigated before and we proved that semi-bent functions of this form exist only for 6|n. Furthermore, we present a method for enumeration of semi-bent and bent functions in certain classes. Using this method we find the exact number of semi-bent functions of this form. Moreover, we complete some previous partial and incomplete enumeration results for three other classes of semi-bent/bent functions in the literature using this method. We also correct some results on quadratic bent functions stated in Ma et al. (2005).

Keywords: quadratic Boolean functions; semi-bent functions; self-reciprocal polynomials; enumeration; bent functions.

DOI: 10.1504/IJICOT.2015.068696

International Journal of Information and Coding Theory, 2015 Vol.3 No.1, pp.39 - 57

Received: 03 Nov 2014
Accepted: 10 Jan 2015

Published online: 06 Apr 2015 *

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