TWC: Small: Algorithms for Number-Theoretic Problems Arising in Cryptography

这个项目研究了几个可以应用于密码学的问题。其中一个目标是开发对量子计算机安全的经典密码系统。特别是，该项目探索了最近提出的一些基于格子的系统的安全性。另一个目标是使目前正在使用的系统更有效率。该项目旨在改进一些构造可用于密码系统的曲线的算法。这个项目将对理解现在或将来应该使用哪些密码系统产生影响。最近，一些基于格的密码体制被量子算法破解。研究人员正在探索其他基于晶格的系统，例如基于Ring-LWE的系统，是否也可以被量子计算机打破。Ring-LWE问题很重要，因为许多密码构造，包括全同态加密，都可以基于它。PI和co-PI还在研究其他后量子密钥交换、签名和加密方案。在不同的方向上，研究人员致力于更有效的算法来构建曲线，这些曲线可以用于基于离散对数的密码系统。研究人员让他们的研究生参与这些研究项目的许多部分。通过她参与各种辅导活动，该协会的目标是增加从事科学或数学职业的妇女人数。她还教初中生和高中生有关密码学的知识。

This project studies several questions that have applications to cryptography. One goal is to develop classical cryptosystems that are secure against quantum computers. In particular, the project explores the security of some of the recently proposed lattice-based systems. Another goal is to make systems that are currently being used more efficient. The project aims to improve some of the algorithms for constructing curves that can be used in cryptosystems. This project will have implications for understanding which cryptosystems should be used now or in the future. Some lattice-based cryptosystems were recently broken by use of quantum algorithms. The investigators are exploring whether other lattice-based systems, such as systems based on Ring-LWE, can be broken by quantum computers as well. The Ring-LWE problem is important because a number of cryptographic constructions, including fully homomorphic encryption, can be based on it. The PI and the co-PI are also investigating other post-quantum key-exchange, signature and encryption schemes. In a different direction, the investigators work on more efficient algorithms for constructing curves that can be used in discrete-log based cryptosystems. The investigators involve their graduate students in many parts of these research projects. Through her involvement in various mentoring activities the PI aims to increase the number of women who will pursue a career in science or mathematics. She also teaches middle school and high school students about cryptography.