On July 5, 2022, the National Institute of Standards and Technology announced four possible post-quantum cryptography standards, three of them are based on lattice theory and the other one is based on Hash function. It is well-known that the security of the lattice cryptography relies on the hardness of the shortest vector problem (SVP) and the closest vector problem (CVP). In fact, the SVP is a sphere packing problem and the CVP is a sphere covering problem. Furthermore, both SVP and CVP are equivalent to arithmetic problems of positive definite quadratic forms. This paper will briefly introduce the post-quantum cryptography and show its connections with sphere packing, sphere covering, and positive definite quadratic forms.

翻译：2022年7月5日，美国国家标准与技术研究院公布了四种可能的后量子密码学标准，其中三种基于格理论，另一种基于哈希函数。众所周知，格密码的安全性依赖于最短向量问题（SVP）和最近向量问题（CVP）的困难性。实际上，SVP是一个球堆积问题，而CVP是一个球覆盖问题。此外，SVP和CVP均等价于正定二次型的算术问题。本文将简要介绍后量子密码学，并展示其与球堆积、球覆盖及正定二次型之间的联系。