| 摘要: |
| 设$m$是正整数, $\mathbb{F}_{2^{3m}}$是含有$2^{3m}$个元素的有限域. 本文研究了有限域$\mathbb{F}_{2^{3m}}$上的如下方程
\begin{equation*}
x^{2^{2m}}+ax^{2^{m}}+bx=0,
\end{equation*}
其中$a,b\in\mathbb{F}^{\ast}_{2^{3m}}$. 利用Bluher论文有关定理及线性化多项式、置换多项式的相关结论, 本文确定了方程解的个数, 并刻画了相应解数时$a,b$所要满足的条件. 所得结果在序列的相关性研究、编码的构造研究及密码函数的差分性质研究中有潜在的应用. |
| 关键词: 有限域 线性化多项式 置换多项式 高次方程 |
| DOI： |
| 分类号:O156.1 |
| 基金项目:国家自然科学基金资助项目(62171479)；中南民族大学中央高校基本科研业务费专项资金项目(CZZ25008) |
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| Solutions for a Higher-Degree Equation over Finite Fields |
|
zhongkexin, xiayongbo
|
|
South-Central Minzu University
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| Abstract: |
| Let $m$ be a positive integer and $\mathbb{F}_{2^{3m}}$ the finite field with $2^{3m}$ elements. We investigates the following equation over the finite field $\mathbb{F}_{2^{3m}}$
\begin{equation*}
x^{2^{2m}}+ax^{2^{m}}+bx=0,
\end{equation*}
where $a,b\in\mathbb{F}^{\ast}_{2^{3m}}$.
By applying relevant theorems from Bluher's work and properties of linearized polynomials and permutation polynomials, we determine the number of solutions to the above equation and characterize the conditions on $a$ and $b$ for each possible solution count. The result may have potential applications in the study of the correlation properties of certain sequences, code constructions, and differential properties of cryptographic functions. |
| Key words: Finite field linearized polynomial permutation polynomial equation of higher degree |
