Univariate Niho Bent Functions from o-Polynomials

TL;DR

This paper characterizes univariate Niho bent functions as sums of Leander-Kholosha functions with specific coefficients, enabling explicit construction from o-polynomials and analysis of their algebraic degree.

Contribution

It provides a new representation of univariate Niho bent functions in terms of Leander-Kholosha functions and derives explicit forms for quadratic and cubic cases.

Findings

Univariate Niho bent functions are sums of Leander-Kholosha functions.

Explicit forms are derived for quadratic and cubic o-polynomial cases.

The algebraic degree of these bent functions is calculated.

Abstract

In this paper, we discover that any univariate Niho bent function is a sum of functions having the form of Leander-Kholosha bent functions with extra coefficients of the power terms. This allows immediately, knowing the terms of an o-polynomial, to obtain the powers of the additive terms in the polynomial representing corresponding bent function. However, the coefficients are calculated ambiguously. The explicit form is given for the bent functions obtained from quadratic and cubic o-polynomials. We also calculate the algebraic degree of any bent function in the Leander-Kholosha class.