This project is about using recent advances in mathematics to improve lattice based cryptography. Lattice based cryptography is a foundation for advanced cryptographic schemes that are resistant to attacks by quantum computers. Secure cryptography is essential for digital communication, for example for ensuring the safe transfer of sensitive financial data. The new mathematical advance behind this project is the efficient construction of lattices that have both addition and multiplication operations and that are more densely packed than the ones now typically used in cryptography. The central goals of the project are to improve these constructions, to develop faster algorithms to operate on these lattices, and to build more efficient cryptographic applications using them. The technical advances in this project concern number fields with small root discriminants. This project will study the computational complexity of constructing such number fields and of performing arithmetic operations in them. Prior work on infinite families of number fields with small root discriminants has focused on existence theorems. This project will build on work of two of the P.I.s on efficient explicit constructions of such families. The goal is to improve these constructions using a variety of mathematical techniques including Galois cohomology, explicit Chebotarev theorems and recent advances on Hilbert's 12th problem via p-adic methods and modular forms. Another goal is to develop fast Four