Lattice-Based Cryptography, the Picture Way

When working on lattice-based cryptography, two main formalisms co-exist. One, let us call it the matrix formalism, where the problems are described with matrices, usually formulated as finding a short solution of a linear system. And another one, let us call it the geometric formalism, where the problems are defined with lattices, and the goal is usually to find short non-zero vectors in these lattices. The matrix formalism is mostly used when constructing cryptographic protocols, whereas the geometric formalism is often used in cryptanalysis. In this talk, I plan to describe a few results, usually formulated with the matrix formalism, in the geometric formalism. In addition to providing the opportunity to draw nice pictures, we will see that keeping in mind the geometric formalism may help us understand the motivations behind some cryptographic constructions, and we will also gain intuition about the hardness of some algorithmic problems.