Chris Peikert – Research Statement

My research is dedicated to developing new, stronger mathematical foundations for cryptography, with a particular focus on geometric objects called lattices. Informally, a lattice is a periodic ‘grid’ of points in n-dimensional real space Rn. Lattices have been studied since the early 1800s, and their apparent simplicity belies many deep connections and applications across mathematics, physics, and computer science. In cryptography, the security of most systems necessarily relies upon computational problems that are conjectured to be intractable, i.e., infeasible to solve with any realistic amount of computational resources. Over the past three decades, the most useful candidate hard problems have come from an area of mathematics called number theory. For instance, a commonly made conjecture is that it is infeasible to compute the prime factors of huge random integers. However, the relatively high computational cost, and largely sequential nature, of operating on such enormous numbers inherently limits the efficiency and applicability of numbertheoretic cryptography. Even more worrisome is that quantum algorithms, which work in a model of computation that exploits quantum mechanics to dramatically speed up certain kinds of computations, can efficiently solve all the number-theoretic problems commonly used in cryptography! Therefore, the future development of a practical, large-scale quantum computer would be devastating to the security of today’s cryptographic systems. Alternative foundations are therefore sorely needed. The seminal works of Ajtai [Ajt96] and Ajtai-Dwork [AD97] in the mid-1990s used conjectured hard problems on lattices as a basis for cryptography. Since then, it has been broadly recognized that lattices have the potential to yield cryptographic schemes with unique and attractive security guarantees—including “worst-case” hardness (explained in the next section) and resistance to quantum attacks—and high levels of asymptotic efficiency and parallelism. However, until 2007 only a few very basic lattice-based objects (having limited applicability) were known, and in practice they were very inefficient and so mainly of theoretical interest. Over the past several years, my research has contributed ground-breaking progress toward realizing the full potential of lattices in cryptography, by (1) strengthening the theoretical foundations of the area, (2) designing new cryptographic constructions that enjoy rich functionality and strong security properties, and (3) making lattice cryptography efficient and practical via new design paradigms, fast algorithms, and optimized implementations.