On lower bounds of second-order nonlinearities of cubic bent functions constructed by concatenating Gold functions

Abstract

We consider the problem of identifying the classes of Boolean functions having high second-order nonlinearities. In this paper, we demonstrate that the cubic bent functions obtained by Leander and McGuire (J. Combin. Theory Ser. A, 116 (2009), pp. 960–970), which are concatenations of the quadratic Gold functions, possess high second-order nonlinearities.

Keywords:

• Boolean functions
• cubic functions
• derivatives
• second-order nonlinearity
• low-order approximations

2010 AMS Subject Classifications :

• 06E30
• 12Y05
• 94A60
• 94C10

Acknowledgements

The authors like to thank the anonymous referees for many valuable comments that improved the technical as well as the editorial qualities of this paper. The authors thank Goutam Kumar Paul for carefully going through the paper and suggesting many changes that improved the presentation of this paper.