2011/12/01 by Chunming Tang, Yanfeng Qi, Tang, Chunming +7
Computer Science · Engineering · #Coding theory and cryptography #Cryptographic Implementations and Security #FOS: Computer and information sciences #Information Theory (cs.IT) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.1112.0062
openalex publication_date 2011/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
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 2n-1± 2(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 \mathbbF2n by fa,b:=Tr1n(ax(2m-1))+Tr14(bx(2n-1)/(5)), where n=2m, m≡ 2\pmod 4, a∈ \mathbbF2m and b∈\mathbbF16. When a∈ \mathbbF2m and (b+1)(b4+b+1)=0, with the help of Kloosterman sums and the factorization of x5+x+a-1, 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∈\mathbbF2(m)/(2).