A new class of hyper-bent Boolean functions in binomial forms

Abstract: Bent functions, which are maximally nonlinear Boolean functions with even numbers of variables and whose Hamming distance to the set of all affine functions equals $2^{n-1}\pm 2<sup>{\frac{n}{2}-1}$, were introduced by Rothaus in 1976 when he considered problems in combinatorics. Bent functions have been extensively studied due to their applications in cryptography, such as S-box, block cipher and stream cipher. Further, they have been applied to coding theory, spread spectrum and combinatorial design. Hyper-bent functions, as a special class of bent functions, were introduced by Youssef and Gong in 2001, which have stronger properties and rarer elements. Many research focus on the construction of bent and hyper-bent functions. In this paper, we consider functions defined over $\mathbb{F}<em>{2<sup>n}$ by $$f</em>{a,b}:=\mathrm{Tr}<em>{1}^{{n}(ax^{{(2^{m-1)})+\mathrm{Tr}}}}</em>{1}<sup>{4}(bx<sup>{\frac{2<sup>n-1}{5}})$$, where $n=2m$, $m\equiv 2\pmod 4$, $a\in \mathbb{F}<em>{2<sup>m}$ and $b\in\mathbb{F}</em>{16}$. When $a\in \mathbb{F}<em>{2<sup>m}$ and $(b+1)(b<sup>4+b+1)=0$, with the help of Kloosterman sums and the factorization of $x<sup>5+x+a<sup>{-1}$, we present a characterization of hyper-bentness of $f</em>{a,b}$. Further, we use generalized Ramanujan-Nagell equations to characterize hyper-bent functions of $f_{a,b}$ in the case $a\in\mathbb{F}_{2<sup>{\frac{m}{2}}}$.