一类PS_(ap)向量bent函数的存在性

On existence of vectorial bent functions from the PS_(ap) class

Bent函数是非常特殊的组合对象,在序列、差集、编码和密码等领域都有重要应用.近年来,形如Trnk(P(x))的bent函数吸引了大量目光,其中k=1或k=n/2且P(x)∈F2n[x].本文在前人研究的基础上进一步研究二项式函数F(x)=Tr_k^n(x^(2^k-1)+ax(r(2~k-1)))(k=n/2≥2)的向量bent性,其中r为奇数.对于r|2~k+1的情形,本文得到了F满足向量bent性的一个充要条件,从而,对所有n和a∈F_(2~n)~*都完全确定了F的向量bent性.而对于r2k+1的情形,Muratovi-Ribi等(2014)曾提出过不存在此类向量bent函数的猜想.通过引入Lucas公式,对r分别等于5、7、9及所有的n和a∈F_(2~n)~*,本文也完全得到了F的向量bent性.特别地,本文找到了一些反例,否定了Muratovi-Ribi等(2014)提出的猜想.

Bent functions are extremal combinatorial objects with wide applications. Recently, the research of bent functions of the form Trnk(P(x)) with k = 1 or k = n/2 and P(x) ∈ F_2~n [x] has attracted much attention.In this paper, based on the existing results, we further study the vectorial bent property of the binomial trace functions F(x)=Tr_k^n(x^(2^k-1)+ax(r(2~k-1))) with k = n/2≥2, r odd and a ∈ F_(2~n)~*. For the case r | 2k+ 1, we obtain a necessary and sufficient condition for F to be vectorial bent and thus completely determine the vectorial bent property of F for all n and all a ∈ a∈F_(2~n)~*. While for the case r 2k+ 1, a conjecture on the nonexistence of binomial trace bent functions has been proposed by Muratovi-Ribi et al.(2014). By employing the powerful Lucas formula, we completely determine the vectorial bent property of F for all n, all a ∈ a∈F_(2~n)~*, when r equals 5,7 and 9. In particular, we find a few counter examples to the conjecture.