基于特征和的周期序列与线性码研究

Since periodic sequences and linear codes have wide applications in CDMA communications, radar systems and data storage systems, they have drawn much attention all over the world, and many results have been obtained. In this project, based on exponential sums, we are devoted to studying the security analysis of periodic sequences, constructions and analysis of optimal linear codes. The concrete content can be illustrated in the following three aspects. Firstly, we construct periodic sequences with good cryptographic properties, find the explicit formulas or efficient algorithms of k-error linear complexity, 2-adic complexity and correlation distribution. Secondly, properly choose defining sets to design new classes of linear codes with desirable parameters, determine the weight distributions of these codes, and then consider their applications in secrete sharing schemes. Finally, we study linear codes through sequences theory, determine the dimensions and minimum distance of some linear codes by analyzing security indicators of the corresponding sequences. The expected output of this project will not only enhance the cryptography and coding theory, but also promote the development of the related branches of mathematics.

周期序列与线性码因在CDMA通信、雷达系统及数据存储中有着大量的应用而吸引了国内外广泛的关注，并且取得了大量的研究成果。本项目中，我们将基于指数和理论，主要研究周期序列的安全性指标、最优线性码的构造及分析。具体内容包括以下三个方面，首先利用分圆理论、2-adic分析及有限域理论构造密码学性质良好的周期序列，给出一些周期序列的k-错线性复杂度、2-adic复杂度及相关值分布的显示公式或者有效算法；其次，利用bent函数及指数和中的结论，适当的选取定义集，构造出一系列新的性能良好的线性码，确定这类线性码的重量分布，探讨它们在秘密共享方案中的应用；最后，我们基于序列理论研究线性码，利用序列的安全性指标研究对应的线性码的维数、极小距离。本项目的预期研究成果将充实密码编码理论，也有望促进相关数学学科的发展。

线性反馈移位寄存器(LFSR)序列和带进位反馈移位寄存器(FCSR)序列广泛应用于扩频通信、密码学和编码技术等许多工程领域。在这些应用中，通常要求所采用的密钥流序列具有很好的密码学性质以抵抗各种密码攻击。本项目主要围绕（1）构造安全性良好的周期序列；（2）研究周期序列的密码学性质，利用指数和理论等方法计算周期序列的自相关值、线性复杂度、2-adic复杂度等安全性指标；（3）研究了与周期序列密切相关的线性码，构造出几类参数最优或几乎最优的线性码并确定了它们的重量分布；（4）研究了几类经典密码函数的ambiguity与deficiency值。

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