Lattices are beautiful high-dimensional geometric objects that arise naturally in many areas of mathematics and computer science—from algebraic number theory to sphere packing. And, over the past decade, lattices have taken on a central role in cryptography. In particular, lattice-based cryptography is by far the most mature form of public-key cryptography that is thought to be secure against quantum computers, and it is therefore quite likely that most internet traffic will be encrypted using lattice-based techniques in the not-too-distant future.
We will study the mathematical and computational theory of lattices using a particularly elegant tool called the discrete Gaussian measure. Using this magical tool, we will take a quick tour of the theory and practice of lattices—proving foundational geometric results such as transference theorems, building a cryptographic scheme and proving its security, and even proving some of the cutting-edge results concerning the discrete Gaussian.