随着量子计算机研制进程的加快，能抵抗量子算法攻击的格密码方案受到了极大关注。为了分析基于理想格的密码方案的安全密钥长度，研究它们的解码算法尤为关键。本项目主要研究二次域的格规约算法的基础理论和密码应用，具体内容包括：1）模格在分圆域的理想上的构造；2）二次域上格的基本性质；3）二次域上BKZ规约算法研究；4）NTRU格和R-LWE问题的解码攻击。本项目的研究成果将为格密码方案的实用化提供相关的理论支持，在后量子时代为国家信息系统的安全运行提供参考。

With the rapid development of quantum computers, lattice-based cryptography which can resist attacks from quantum algorithms has attracted great attention. For the sake of analyzing the bit-level security of ideal lattice-based cryptographic schemes, it is of vital importance to investigate the decoding of ideal lattices. This project focuses on the fundamental theory and cryptographic applications of lattice reduction over quadratic fields, including: 1) construction of module-lattices over ideals of cyclotomic fields; 2) basic properties of lattices over quadratic fields; 3) construction of a Block Korkine Zolotarev (BKZ) algorithm over quadratic fields; 4) decoding attacks of NTRU and Ring Learning With Error (R-LWE) problems. The research fruits of the project will provide related theoretical support to facilitate the practical deployment of lattice-based cryptography, and offer reference for the secure operation of the national information system in the post-quantum era.