Some results on q-ary bent functions

Abstract

Kumar et al. [Generalized bent functions and their properties, J. Comb. Theory Ser. A 40 (1985), pp. 90–107] have extended the notion of classical bent Boolean functions in the generalized setup on . They have provided an analogue of classical Maiorana-McFarland type bent functions. In this paper, we study the cross-correlation of a subclass of such generalized Maiorana-McFarland type bent functions. We provide a characterization of quaternary (q=4) bent functions on n+1 variables in terms of their subfunctions on n-variables. Analogues of sum-of-squares’ indicator and absolute indicator of cross-correlation of Boolean functions are defined in the generalized setup. Further, q-ary functions are studied in terms of these indicators and some upper bounds of these indicators are obtained. Finally, we provide some constructions of balanced quaternary functions with high nonlinearity under Lee metric.

Keywords:

• q-ary bent functions
• Walsh–Hadamard transform
• Parseval's identity
• generalized Maiorana-McFarland type bent functions
• cross-correlation

2010 AMS Subject Classifications :

• 94A60
• 94C10
• 06E30

Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments and suggestions that have helped in improving the presentation of the paper. The work of the first author is supported by NBHM (DAE), India.