Division of Mathematical Sciences (DMS)
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Submitted by Akshay Venkatesh on Tue, 02/06/2018 - 6:08pm
As was realized by Descartes, the solution of algebraic equations can be realized geometrically. This observation was the start of a rich interaction between algebra and geometry. This project will study two topics in number theory. The first concerns the shape of arithmetic manifolds -- i.e., certain geometries defined by their number theoretic symmetries. The PI has conjectured the existence of new structures that govern their shape (mathematically speaking their topology), which he will investigate in more detail. The second topic relates to lattices of high dimension.
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Submitted by Katherine Stange on Thu, 02/01/2018 - 5:09pm
This award supports the principal investigator's research in number theory and its cryptographic applications. Number theory serves as the basis for modern cryptography and internet security. The underlying mathematical theories, of elliptic curves and integer factorization, have been studied for centuries. This project includes components of basic research, concerning the relationship between geometry and number theory, as well as foundational research in cryptographic applications, with an eye toward the advent of quantum computers.
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Submitted by Jintai Ding on Tue, 11/14/2017 - 10:46am
Cyber security is considered one of the most important aspects of our information technology based society. Key Exchange(KE) is a fundamental cryptographic primitive, and authenticated KE (AKE) is one of the most used cryptographic tools in secure communication protocols (e.g. SSL/TLS, IPSec, SSH) over the Internet. In light of the threat that quantum computers pose to cryptosystems such as RSA and ECC, this project is devoted to the development of secure and efficient AKE alternatives for the post-quantum computer world, which is now considered of a high priority by the US government.
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Submitted by Paul Gunnells on Wed, 10/11/2017 - 12:08pm
This research project will investigate and develop new mathematical tools in public-key cryptography. Such tools have been introduced in recent years as suitable for implementation on low-cost platforms with constrained computational resources; it is expected that such platforms will become more and prevalent as consumer devices become "smart" and connect to large networks with the emergence of the "Internet of Things." Public-key cryptography is used by each person hundreds and perhaps thousands of times daily, as it is the main security used in cellular, ATM, and other computer networks.
