具有良好性质的序列与密码函数的研究

Nowadays the problem of information security is more and more important with the wide employment of communication and computer networks. Cryptographic technique is the key technique of information security. The analysis and design of sequences and cryptographic functions has been a hot research topic for the past several decades, and the analysis and design of cryptographic schemes and protocols could take much advantage from such results. In this project, we want to carry out deep research on the following two issues: 1) cryptographic functions and exponential sums: using Gauss sums and Stickelberger's theorem, we could lift the study of cryptographic functions from finite fields to cyclotomic fields, understand problems better which are not easy to be settled in finite fields, and try to settle them using powerful tools in cyclotomic fields; 2) autocorrelation of sequences and exponential sums: similarly, we could lift the study of autocorrelation from finite fields to cyclotomic fields via Gauss sums and Stickelberger's theorem, simplify certain problems in finite fields, and try to settle some problems which are not easy for former methods via finite fields. Via this project, we will try our best to settle several difficult problems, and to achieve some top level results in the world.

随着通信与计算机网络的广泛应用，信息的安全问题越来越重要，而密码技术是信息安全的核心技术。具有良好性质的序列与密码函数是密码技术中不可或缺的构件。几十年以来，序列与密码函数的设计与分析一直是国际上的研究热点，相关成果可以为密码算法和协议的设计与分析提供重要参考。在本项目中，我们准备深入研究如下两个问题：1）密码函数与指数和（exponential sum）：利用高斯和（Gauss sum）与Stickelberger定理，将对密码函数的研究从有限域提升到分圆数域中去，深刻理解那些在有限域中不容易解决的问题，使用分圆数域中的深刻工具解决之；2）序列自相关与指数和：同样地，利用高斯和与Stickelberger定理，将序列自相关的研究从有限域提升到分圆数域中去，简化有限域中的问题，并解决用纯有限域方法不容易解决的问题。通过本项目的支持，我们力争解决几个困难问题，取得一些国际一流的成果。

本项目围绕密码学中的序列和密码函数开展工作，以国际重要公开问题为导向，圆满完成各项预定任务，为未来的研究工作打下了更扎实的基础。最具有代表性的工作是：1）成功证明了1998年提出的H.A.Lin猜想，解决了这个长达15年的公开问题；2）利用Stickelberger定理，发展出新的组合技巧，找到了新的bent函数；3）给出了理想二值自相关序列2-adic复杂度的非常简单的证明。依托本项目，发表10多篇高质量论文，其中包括4篇IEEE Transactions on Information Theory论文。