Paper 2011/698

A generalization of the class of hyper-bent Boolean functions in binomial forms

Chunming Tang, Yu Lou, Yanfeng Qi, Baocheng Wang, and Yixian Yang

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 $2^{n-1}\pm 2^{\frac{n}{2}-1}$, 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 ${\mathbb{F}}_{2^n}$ by $$, where $$, $$, $$ and $$. When $$, we characterize the hyper-bentness of $$. When $$, $$ and $$, with the help of Kloosterman sums and the factorization of $$, we present a characterization of hyper-bentness of $$. Further, we give all the hyper-bent functions of $$ in the case $$.