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arXiv 1606.08952cs.ITmath.IT
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有限域上的几类Negabent函数

Several Classes of Negabent Functions over Finite Fields

Gaofei Wu, Nian Li, Yuqing Zhang, Xuefeng Liu

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摘要：

Negabent函数作为一类广义Bent函数，由于其在密码学和编码理论中的应用，近年来引起了人们的广泛关注。本文考虑有限域上负函数的构造。首先，利用某些二项式和三项式置换的合成逆，我们提出了几类形式为$f(x)=\Tr_1^n(λx^{2^k+1})+\Tr_1^n(ux)\Tr_1^n(vx)$的负函数，其中$λ\in \F_{2^n}$，$2\leq k\leq n-1$，$(u,v)\in \F^*_{2^n}\times \F^*_{2^n}$，和$\Tr_1^n(\cdot)$是从$\F_{2^n}$到$\F_{2}$的迹函数。其次，利用Kloosterman和，证明了Zhou和Qu（Cryptogr.通信：出现，DOI 10.1007/s12095-015-0167-0。）做一个负责任的人也是必要的。此外，还给出了指数为Niho型的负单项式的一个猜想.

英文摘要：

Negabent functions as a class of generalized bent functions have attracted a lot of attention recently due to their applications in cryptography and coding theory. In this paper, we consider the constructions of negabent functions over finite fields. First, by using the compositional inverses of certain binomial and trinomial permutations, we present several classes of negabent functions of the form $f(x)=\Tr_1^n(λx^{2^k+1})+\Tr_1^n(ux)\Tr_1^n(vx)$, where $λ\in \F_{2^n}$, $2\leq k\leq n-1$, $(u,v)\in \F^*_{2^n}\times \F^*_{2^n}$, and $\Tr_1^n(\cdot)$ is the trace function from $\F_{2^n}$ to $\F_{2}$. Second, by using Kloosterman sum, we prove that the condition for the cubic monomials given by Zhou and Qu (Cryptogr. Commun., to appear, DOI 10.1007/s12095-015-0167-0.) to be negabent is also necessary. In addition, a conjecture on negabent monomials whose exponents are of Niho type is given.

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