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

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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.