| 摘要: |
设m是正整数,F23m是含有23m个元素的有限域.本文研究了有限域F23m上的如下方程
x22m+ax2m+bx=0,
其中a,b∈F*23m.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,并刻画了方程有相应解数时系数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 |
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ZHONG Ke-xin, XIA Yong-bo
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College of Mathematics and Statistics, South-Central Minzu University, Wuhan 430074, China
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| Abstract: |
Let m be a positive integer and F23m the flnite fleld with 23m elements. We investigates the following equation over the flnite fleld F23m
x22m+ax2m+bx=0,
where a,b∈F*23m. 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 difierential properties of cryptographic functions. |
| Key words: Finite fleld linearized polynomial permutation polynomial equation of higher degree |
