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

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 ± 2n2 , 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 F2n by fa,b := Tr n 1 (ax (2m−1)) + Tr1(bx 2n−1 5 ), where n = 2m, m ≡ 2 (mod 4), a ∈ F2m and b ∈ F16. When a ∈ F2m and (b + 1)(b + b + 1) = 0, with the help of Kloosterman sums and the factorization of x + x + a, we present a characterization of hyper-bentness of fa,b. Further, we use generalized Ramanujan-Nagell equations to characterize hyper-bent functions of fa,b in the case a ∈ 2m2 . Chungming Tang, Yanfeng Qi and Maozhi Xu acknowledge support from the Natural Science Foundation of China (Grant No.10990011 & No.60763009). Baocheng Wang and Yixian Yang acknowledge support from National Science Foundation of China Innovative Grant (70921061), the CAS/SAFEA International Partnership Program for Creative Research Teams. Chunming Tang, Yanfeng Qi and Maozhi Xu Laboratory of Mathematics and Applied Mathematics, School of Mathematical Sciences, Peking University, Beijing, 100871, China Chunming Tang’s e-mail: [email protected] Baocheng Wang and Yixian Yang Information Security Center, Beijing University of Posts and Telecommunications and Research Center on fictitious Economy and Data Science, Chinese Academy of Sciences, Beijing, 100871, China 2 Chunming Tang et al.