信息可靠存储与传输及安全保护中几类编码研究

This project is devoted to investigating several forefront coding problems in information processing and information security, which have deep connection with the storage of big data, digital and quantum communication, and the security protection for digital multimedia. These problems include: Regenerating Codes, Cyclic Codes, Quantum Codes, and Secure Codes etc. Distributed storage is a new technique to store big data that is based on the technology of network. Regenerating codes are a class of distributed storage codes used to ensure the durability and availability of big data. All of cyclic codes, quantum MDS codes and quantum codes for synchronization have a good algebraic structure and a fast decoding algorithm. They have important applications in digital or quantum communications. Secure codes have played a crucial role in the copyright protection for multimedia products. We will describe more meticulously the mathematical models for these problems originated from information science, and analyze thoroughly their combinatorial properties. Combining with the theory and methods in Algebra, Number Theory, Combinatorics, Algebraic Geometry, Finite Geometry, Probability and Statistics, we aim to make significant progress in the specific problems as follows: The evaluation of lower and upper bounds for the size of regenerating codes in distributed storage, the construction of optimal codes and the improvement of modeling; The evaluation for the weight distribution and the generalized Hamming weight of cyclic codes; The constructions of quantum MDS codes, quantum codes for synchronization and quantum codes with nice parameters; The analysis of collusion model for digital fingerprinting, the construction and detection for fingerprinting codes, the evaluation of lower and upper bounds for the size of codes with traceability, and the design of tracing algorithms for such codes.

本项目拟研究信息处理与信息安全中与海量数据存储、数字与量子通信及数字媒体安全保护密切相关的几类编码问题，包括：再生码、循环码、量子码、安全码等前沿课题。分布式存储以网络技术为基础，实现对数据的海量存储，而再生码用来保证分布式存储中数据的耐久性和可用性。循环码、量子MDS码、量子同步码皆具有良好的代数结构及快速的译码算法，在数字或量子通信中有重要的应用。安全码在多媒体产品版权保护中起着至关重要的作用。本项目拟更精确地刻画这些源于信息科学问题的数学模型，深入剖析其组合特性，结合代数、数论、组合学、代数几何、有限几何和概率统计的理论方法，在以下具体问题上取得重要进展：分布式存储中再生码容量的上下界估计、最优码的构造和模型的改进；循环码的重量分布与广义汉明重量的计算；量子MDS码、量子同步码及具有良好参数的量子码的构造；数字指纹中合谋模式分析、指纹的构造与检测、追踪码容量的上下界估计及其追踪算法。

本项目以代数、数论、代数几何、有限几何、概率论为数学工具，对信息处理与信息安全中几类与组合数学密切相关的前沿课题进行了系统的研究，包括：背驮码、分离哈希函数族、防诬陷码、父代识别码和追踪码、量子纠错码、线性纠错码、BCH码、字符结对码与b-字符结对码、闪存编码、置换码、集中式缓存方案、Lee 码、完美分解集、隐私保护信息检索、多层常重码、子空间码、光正交签名样式码、压缩感知、大数据的分布式存储编码方案、置换多项式、Cayley图的度数-直径问题以及有限域上的无直角集等前沿课题。2016年至2019年四年期间，先后在重要国际专业刊物《IEEE Transactions on Information Theory》、《SIAM Journal on Discrete Mathematics》、《Science China Mathematics》、《Designs, Codes and Cryptography》、《Journal of Algebraic Combinatorics》、《Journal of Combinatorial Designs》、《Finite Fields and Their Applications》和《Discrete Mathematics》上发表论文45篇，它们都已被SCI收录。其中，17篇发表在信息学领域顶尖期刊《IEEE Transactions on Information Theory》上。项目主持人荣获“北京学者”称号。