Integral lattices and their theta series

格子算法在解决代数、数论、组合学、最优化和密码学等领域的问题中起着非常重要的作用。我们提出的工作将填补当前使用晶格计算能力的一个主要漏洞。枚举短向量的增强能力将在丢番图方程、线性最优化以及基于寻找最短向量的困难的基础上理解密码系统的安全性方面具有重要应用。关于theta级数的工作应该会产生这个问题的第一个算法。这也将发现许多应用，包括球形码的构造和设计。

Lattice algorithms play a very important role in solving problems in algebra, number theory, combinatorics, optimisation and cryptography. Our proposed work will fill a major hole in current capabilities for computing with lattices. The enhanced ability to enumerate short vectors will have important applications to Diophantine equations, linear optimisation and also to understanding the security of cryptosystems based on the difficulty of finding a shortest vector. The work on theta series should result in the first algorithm for this problem. This will also find many applications including to the construction of spherical codes and designs.