当前位置: X-MOL 学术Des. Codes Cryptogr. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
Vectorial bent functions and partial difference sets
Designs, Codes and Cryptography ( IF 1.4 ) Pub Date : 2021-08-14 , DOI: 10.1007/s10623-021-00919-y
Ayça Çeşmelioğlu 1 , Wilfried Meidl 2 , Isabel Pirsic 2, 3
Affiliation  

The objective of this article is to broaden the understanding of the connections between bent functions and partial difference sets. Recently, the first two authors showed that the elements which a vectorial dual-bent function with certain additional properties maps to 0, form a partial difference set, which generalizes the connection between Boolean bent functions and Hadamard difference sets, and some later established connections between p-ary bent functions and partial difference sets to vectorial bent functions. We discuss the effects of coordinate transformations. As all currently known vectorial dual-bent functions \(F:{\mathbb {F}}_{p^n}\rightarrow {\mathbb {F}}_{p^s}\) are linear equivalent to l-forms, i.e., to functions satisfying \(F(\beta x) = \beta ^lF(x)\) for all \(\beta \in {\mathbb {F}}_{p^s}\), we investigate properties of partial difference sets obtained from l-forms. We show that they are unions of cosets of \({\mathbb {F}}_{p^s}^*\), which also can be seen as certain cyclotomic classes. We draw connections to known results on partial difference sets from cyclotomy. Motivated by experimental results, for a class of vectorial dual-bent functions from \({\mathbb {F}}_{p^n}\) to \({\mathbb {F}}_{p^s}\), we show that the preimage set of the squares of \({\mathbb {F}}_{p^s}\) forms a partial difference set. This extends earlier results on p-ary bent functions.



中文翻译:

矢量弯曲函数和偏差分集

本文的目的是拓宽对弯曲函数和偏差分集之间联系的理解。最近,前两位作者表明,具有某些附加性质的向量对偶弯曲函数映射到 0 的元素形成偏差分集,概括了布尔弯曲函数与 Hadamard 差分集之间的联系,以及一些后来建立的联系p 元弯曲函数和偏差分集到矢量弯曲函数。我们讨论坐标变换的影响。由于所有目前已知的向量双弯曲函数\(F:{\mathbb {F}}_{p^n}\rightarrow {\mathbb {F}}_{p^s}\)线性等价于l -forms , 即函数满足\(F(\beta x) = \beta ^lF(x)\)对于所有\(\beta \in {\mathbb {F}}_{p^s}\),我们研究了获得的部分差异集的性质从l -形式。我们证明它们是\({\mathbb {F}}_{p^s}^*\)的陪集的并集,也可以看作是某些分圆类。我们在分圆法的部分差异集上绘制与已知结果的联系。受实验结果启发,对于一类从\({\mathbb {F}}_{p^n}\)\({\mathbb {F}}_{p^s}\),我们证明\({\mathbb {F}}_{p^s}\)的平方的原像集形成了一个部分差分集。这扩展了p 元弯曲函数的早期结果。

更新日期:2021-08-19
down
wechat
bug