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Title:
A generalization of the class of hyper-bent Boolean functions in binomial forms
Authors: Chunming Tang, Yu Lou, Yanfeng Qi, Baocheng Wang, 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 f^{(r)}_{a,b}:=\mathrm{Tr}_{1}^{n}(ax^{r(2^m-1)}) +\mathrm{Tr}_{1}^{4}(bx^{\frac{2^n-1}{5}}), where n=2m, m\equiv 2\pmod 4, a\in \mathbb{F}_{2^m} and b\in\mathbb{F}_{16}. When r\equiv 0\pmod 5, we characterize the hyper-bentness of f^{(r)}_{a,b}. When r\not \equiv 0\pmod 5, a\in mathbb{F}_{2^m} and (b+1)(b^4+b+1)=0, with the help of Kloosterman sums and the factorization of x^5+x+a^{-1}, we present a characterization of hyper-bentness of f^{(r)}_{a,b}. Further, we give all the hyper-bent functions of f^{(r)}_{a,b} in the case a\in\mathbb{F}_{2^{\frac{m}{2}}}.
ePrint: https://eprint.iacr.org/2011/698
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