Fredrik Heed Elvegård: Gitter: Från detaljerad teori till kryptering

Abstract: In this paper we give detailed proofs of fundamental theorems regarding lattices and discuss various crypto-systems associated with lattices. The aim is to allow undergraduate mathematic students to quickly interpret the proofs and gain a strong understanding of lattices and how they apply to cryptography. We prove a number of lattice properties that eventually allows us to prove Minkowskis theorem. This theorem helps us show that certain sets that may look like lattices can never be lattices and it also lets us determine properties of the shortest non-zero vector in a lattice. The lattice theory has a meaningful role in the crypto-systems GGH and NTRU that are then explained in details. Finally there is a discussion about why the private key in NTRU probably is the shortest nonzero vector in the associated lattice.