Several Classes of Negabent Functions over Finite Fields

TL;DR

This paper introduces new classes of negabent functions over finite fields, utilizing permutation inverses and Kloosterman sums, and explores conditions for negabent monomials, advancing cryptographic function design.

Contribution

It presents novel constructions of negabent functions using permutation inverses and Kloosterman sums, and establishes necessary conditions for negabent monomials, including a conjecture on Niho exponents.

Findings

Constructed negabent functions using permutation inverses.

Proved the necessity of conditions for cubic negabent monomials.

Proposed a conjecture on negabent monomials with Niho exponents.

Abstract

Negabent functions as a class of generalized bent functions have attracted a lot of attention recently due to their applications in cryptography and coding theory. In this paper, we consider the constructions of negabent functions over finite fields. First, by using the compositional inverses of certain binomial and trinomial permutations, we present several classes of negabent functions of the form $f(x)=\Tr_1^n(\lambda x^{2^k+1})+\Tr_1^n(ux)\Tr_1^n(vx)$, where $\lambda\in \F_{2^n}$, $2\leq k\leq n-1$, $(u,v)\in \F^*_{2^n}\times \F^*_{2^n}$, and $\Tr_1^n(\cdot)$ is the trace function from $\F_{2^n}$ to $\F_{2}$. Second, by using Kloosterman sum, we prove that the condition for the cubic monomials given by Zhou and Qu (Cryptogr. Commun., to appear, DOI 10.1007/s12095-015-0167-0.) to be negabent is also necessary. In addition, a conjecture on negabent monomials whose exponents are of Niho type is given.