论文标题
进一步研究特征两个特征功能
Further Study of Planar Functions in Characteristic Two
论文作者
论文摘要
平面功能在类似DES的迭代密码，错误校正代码，信号集和数学领域的构造中非常重要。它们是在最初由奇数特征的有限字段定义的，并由Y. khou \ cite {zhou}概括为什至特征。在2016年，L。Qu \ cite {q}提出了一种新方法，用于在$ \ f_ {2^n} $上构建二次平面函数。最近，D。Bartoli和M. Timpanella \ Cite {Bartoli}的特征是系数$ a，b $的条件，使函数$ f_ {a，b}（x）= ax^{2m} {2m} {2m} +1} +1}+bx^+bx^{2^m+1}}} $ \ f_ {2^{3M}} $由Hasse-Weil Bound。 在本文中，使用lang-weil绑定，hasse-weil绑定的概括以及在\ cite {q}中引入的新方法，我们完全表征了$ \ f_ {q^k} $的四类平面函数系数的必要和充分条件，其中$ q = 2^m $与$ q = 2^m $ aff $ m $ groud foreem ref foreem（请参阅theem pofe to the theem（reem）。它们的第一类和最后一类分别超过$ \ f_ {q^2} $和$ \ f_ {q^4} $，而其他两个类别超过$ \ f_ {q^3} $。 $ \ f_ {q^3} $上的一个类是$ f_ {a，b}（x）$的扩展名，在\ cite {bartoli}中研究了，而我们的证明似乎更简单。此外，尽管我们结果的$ \ f_ {q^2} $以上的平面二项式终于是已知的平面单元，但我们也同时回答了必要性，并部分解决了\ cite {q}中提出的二项式案例的开放问题。
Planar functions are of great importance in the constructions of DES-like iterated ciphers, error-correcting codes, signal sets and the area of mathematics. They are defined over finite fields of odd characteristic originally and generalized by Y. Zhou \cite{Zhou} in even characteristic. In 2016, L. Qu \cite{Q} proposed a new approach to constructing quadratic planar functions over $\F_{2^n}$. Very recently, D. Bartoli and M. Timpanella \cite{Bartoli} characterized the condition on coefficients $a,b$ such that the function $f_{a,b}(x)=ax^{2^{2m}+1}+bx^{2^m+1} \in\F_{2^{3m}}[x]$ is a planar function over $\F_{2^{3m}}$ by the Hasse-Weil bound. In this paper, using the Lang-Weil bound, a generalization of the Hasse-Weil bound, and the new approach introduced in \cite{Q}, we completely characterize the necessary and sufficient conditions on coefficients of four classes of planar functions over $\F_{q^k}$, where $q=2^m$ with $m$ sufficiently large (see Theorem \ref{main}). The first and last classes of them are over $\F_{q^2}$ and $\F_{q^4}$ respectively, while the other two classes are over $\F_{q^3}$. One class over $\F_{q^3}$ is an extension of $f_{a,b}(x)$ investigated in \cite{Bartoli}, while our proofs seem to be much simpler. In addition, although the planar binomial over $\F_{q^2}$ of our results is finally a known planar monomial, we also answer the necessity at the same time and solve partially an open problem for the binomial case proposed in \cite{Q}.
