Constructions of p-ary Quadratic Bent Functions

Abstract

Bent functions have many applications in the fields of coding theory, communications and cryptography. This paper studies the constructions of bent functions having the form $\sum_{i=1}^{(n-1)/2}c_{i}{\mathrm{tr}}_{1}^{n}(x^{p^{i}+1})$ for odd n and $\sum_{i=1}^{n/2-1}c_{i}{\mathrm{tr}}_{1}^{n}(x^{p^{i}+1})+c_{n/2}{\mathrm{tr}}_{1}^{n/2}(x^{p^{n/2}+1})$ for even n, over the finite field $\mathbb{F}_{p^{n}}$ of odd characteristic p, where $c_{i}\in \mathbb{F}_{p}$ . Based on the irreducibility of some polynomials on $\mathbb{F}_{p}$ , we focus on characterizing the bent functions for n=p ^v q ^r and n=2p ^v q ^r, where $v\geq0,\;r\geq1,\;q$ is an odd prime and p a primitive root modulo q ². Moreover, the enumerations of those functions are also considered.