On Generalized Bent and Negabent Functions

Abstract

From the last few years, generalized bent functions gain a lot of attention in research as they have many applications in various fields such as combinatorial design, sequence design theory, cryptography, CDMA communication, etc. A deep and broad study of generalized bent functions with their properties is done in literature. Kumar et al.[11] first gave the concept of generalized bent function. Many researchers studied the properties and characterizations of generalized bent functions. In [2] authors introduced the concept of generalized (<img src=image/13426756_03.gif>-ary) negabent functions and studied some properties of generalized (<img src=image/13426756_03.gif>-ary) negabent functions. In this paper, we study the generalized (<img src=image/13426756_03.gif>-ary) bent functions <img src=image/13426756_01.gif>, where <img src=image/13426756_02.gif> is the ring of integers with mod <img src=image/13426756_03.gif>, <img src=image/13426756_04.gif> is the vector space of dimension <img src=image/13426756_05.gif> over <img src=image/13426756_02.gif> and <img src=image/13426756_03.gif>≥2 is any positive integer. We discuss several properties of generalized (<img src=image/13426756_03.gif>-ary) bent functions with respect to their nega-Hadamard transform. We also study the relation between generalized nega-Hadamard transforms and generalized nega-autocorrelations. Furthermore, we prove the necessary and sufficient conditions for the bentness and negabentness of generalized (<img src=image/13426756_03.gif>-ary) bent function generated by the secondary construction for <img src=image/13426756_04.gif>, where <img src=image/13426756_06.gif>.