Collaborative Research: AF: SaTC: Medium: Theoretical Foundations of Lattice-Based Cryptography

格是一种几何对象，在计算机科学中有许多应用，特别是在安全密码设计中。这种基于格的加密技术有许多吸引人的特性，包括它明显的安全性，甚至可以对抗配备量子计算机的对手（“后量子”），以及它在构建高级原语方面的实用性，包括完全同态加密（FHE），它允许“在加密数据上进行计算”。基于此，美国国家标准与技术研究所（NIST）最近选择了几个基于格的密码系统进行标准化，作为其长达数年的后量子密码标准化过程的一部分。由于基于格的密码系统将在不久的将来得到广泛使用，因此了解其背后问题的复杂性（安全性）尤为迫切。这个项目有三个主要的研究目标。首先，该项目旨在更好地理解晶格问题的细粒度复杂性，即解决基于晶格密码系统的计算问题所需的精确运行时间。这项工作将理想地导致更好地理解这些密码系统的实际安全性。其次，该项目将研究格和纠错码之间的联系，它们与格有许多相似之处，并且本身就是重要的和被充分研究的对象。第三，本项目将研究代数结构格上问题的复杂性。基于这些格的密码系统——包括大多数实用的密码系统，包括那些最近被NIST选择用于标准化的密码系统——通常效率要高得多，但人们对其背后问题的复杂性知之甚少。除了这些主要的研究目标之外，研究人员还将编写一本关于计算机科学中的格子的全面的、免费的教科书。特别是，本书将详细介绍格的算法、复杂性理论、密码学和几何方面。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Lattices are geometric objects that have many applications in computer science, and especially to the design of secure cryptography. Such lattice-based cryptography has many attractive properties including its apparent security even against adversaries equipped with quantum computers (being "post-quantum") and its usefulness in constructing advanced primitives, including Fully Homomorphic Encryption (FHE), which allows for "computing on encrypted data." Based on this, the National Institute of Standards and Technology (NIST) recently selected several lattice-based cryptosystems for standardization as part of their years-long post-quantum cryptography standardization process. As lattice-based cryptosystems will be in widespread use in the near future, it is especially urgent to understand the complexity (security) of the problems that underlie them.This project has three primary research goals. First, the project seeks to better understand the fine-grained complexity of lattice problems, i.e., the precise running time necessary to solve the computational problems underlying lattice-based cryptosystems. This work will ideally lead to a better understanding of the practical security of these cryptosystems. Second, this project will study connections between lattices and error-correcting codes, which have many similarities to lattices and are important and well-studied objects in their own right. Third, this project will study the complexity of problems on algebraically structured lattices. Cryptosystems based on these lattices---which include most practical cryptosystems, including those recently selected for standardization by NIST---are generally much more efficient, but much less is known about the complexity of the problems that underlie them. In addition to these main research goals, the investigators will write a comprehensive, freely available textbook about lattices in computer science. In particular, this book will cover algorithmic, complexity-theoretic, cryptographic, and geometric aspects of lattices in detail.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.