分圆相关的一些问题及其应用研究

项目名称： 分圆相关的一些问题及其应用研究

项目编号： No.11471178

项目类型： 面上项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 杨晶

作者单位： 清华大学

项目金额： 70万元

中文摘要： 分圆问题是一个古老的几何问题，它在数论与代数组合中发展延伸为许多重要的课题，如：分圆域理论，分圆函数域理论，分圆周期，分圆差集，分圆数等；这些课题与编码密码学中很多应用问题有密切关系，如：不可约循环码的重量分布本质上相当于对应分圆周期的计算，分圆方法可用于构造的密码序列，其线性复杂度与自相关值可归结于相关分圆数的计算等。本项目中，我们以分圆域理论及指数和理论为主要的数学工具，研究以下三个方面的问题： ⑴ 计算若干分圆周期，进而求解相关循环码的重量分布； ⑵ 计算若干分圆数，进而分析相关分圆序列的线性复杂度和自相关值； ⑶ 研究和构作Galois环上低乘法复杂度的正规基。

中文关键词： 有限域；伽罗华环；分圆周期；分圆数；指数和

英文摘要： Cyclotomic problem is an ancient problem in geometry, which has developed and extended into many important subjects in number theory and algebraic combinatorics, such as theory of cyclotomic fields, theory of cyclotomic function fields, cyclotomic period, cyclotomic different set, cyclotomic number etc. These subjects have close relationship with many practical topics in coding theory and cryptography, such as, the weight distribution of irreducible cyclic code is essentially equivalent to the computation of the corresponding cyclotomic periods, and by cyclotomic method, some sequences can be constructed, whose linear complexity and autocorrelation values can be calculated with the relevant cyclotomic numbers, etc. In this project, exploiting the theory of cyclotomic fields and exponential sums, we will consider three problems below: (1) To calculate certain cyclotomic periods, and then to compute the weight distribution of relevant cyclic code. (2) To calculate certain cyclotomic numbers, and then to analyzed the linear complexity and autocorrelation values of relevant cyclotomic sequences. (3) To research and construct the normal basis of Galois rings with low multiplication complexity.

英文关键词： finite field;Galois ring;cyclotomic period;cyclotomic number;exponential sum