特征理论在若干编码密码问题中的应用

The connection between information theory and number theory is becoming increasing close recently. Character theory is not only the important object of algebraic number theory, but also has close relationship with many practical topics in coding theory and cryptography, and continually presents new applications, for example, cyclotomic numbers can be used to obtain optimal algebraic manipulation detection codes through constructions of strong external difference families, and the computation of the generalized Hamming weights of irreducible cyclic codes can be transformed to the computation of Gauss sums, etc. In this project, we study the following questions with the character theory as a main research tool: Firstly, we study the constructions of strong external difference families and generalized strong external difference families by using Jacobi sums, cyclotomic numbers and direct sum decomposition of groups. Secondly, by using Gauss sums and geometry etc., we determine the generalized Hamming weights of linear codes. Thirdly, we analyze the pseudo randomness of period sequences with Jacobi sums, cyclotomic numbers and the Weil Theorem. Finally, we combine the study of strong external difference families, sequences and linear codes, and investigate the construction of generalized strong external difference families by using the sequences with low autocorrelation, and compute the generalized Hamming weights of cyclotomic linear codes . This research project is expected to promote the.development of synchronous communication and secret sharing etc.

近年来，信息技术与数论的联系日益密切。特征理论不仅是代数数论的重要研究对象，它还与编码和密码中许多应用问题都有密切联系，并不断呈现出新的应用，如：分圆数可用于研究强外差组的构造，进而得到最优代数操作检测码，不可约循环码的广义汉明重量的计算可转化为高斯和的计算等。本项目以特征理论为主要工具，研究如下问题：首先，利用雅可比和、分圆数以及群的直和分解研究强外差组、广义强外差组的构造；其次，利用高斯和、几何等方法研究线性码的广义汉明重量的计算；再次，利用雅可比和、分圆数及Weil定理分析分圆序列的伪随机性；最后，将强外差组、序列、线性码三者的研究相结合，利用低自相关序列来研究广义强外差组等的构造，并计算分圆线性码的广义汉明重量。本项目的研究有望促进同步通信、密钥共享等领域的发展。

序列密码广泛应用于密码与通信等领域，在这些应用中，通常要求所采用的密钥流序列具有低自相关性、较大线性复杂度、较大N-adic复杂度等伪随机性指标以抵抗各种密码攻击。本项目以特征理论为主要工具，研究了以下问题：1. 利用广义分圆类构造了一类大周期序列，并证明这类序列的线性复杂度达到了最大，足以抵抗Berlekamp-Massey 算法的攻击。 2. 研究了Sidel’nikov子序列在素域F_d上的 k-错线性复杂度，所得结果表明这类序列可抵抗近似算法的攻击。3. 利用类高斯和工具给出计算序列2-adic复杂度的新方法，并利用此方法计算了几类最优自相关序列的2-adic复杂度，所得结果表明我们所研究的这几类序列可以抵抗有理逼近算的攻击 4. 研究了De Bruijn序列与零差分平衡函数的构造。