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

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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.