Lattice Reduction in Cryptology: An Update

Lattices are regular arrangements of points in space, whose study appeared in the 19th century in both number theory and crystallography. The goal of lattice reduction is to nd useful representations of lattices. A major breakthrough in that eld occurred twenty years ago, with the appearance of Lovv asz's reduction algorithm, also known as LLL or L 3. Lattice reduction algorithms have since proved invaluable in many areas of mathematics and computer science, especially in al-gorithmic number theory and cryptology. In this paper, we survey some applications of lattices to cryptology. We focus on recent developments of lattice reduction both in cryptography and cryptanalysis, which followed seminal works of Ajtai and Coppersmith.