The Two Faces of Lattices in CryptologyPhong

Lattices are regular arrangements of points in n-dimensional space, whose study appeared in the 19th century in both number theory and crystallography. Since the appearance of the celebrated Lenstra-Lenstra-Lovv asz lattice basis reduction algorithm twenty years ago, lattices have had surprising applications in cryptology. Until recently, the applications of lattices to cryptology were only negative, as lattices were used to break various cryptographic schemes. Paradoxically, several positive cryptographic applications of lattices have emerged in the past ve years: there now exist public-key cryptosystems based on the hardness of lattice problems, and lattices play a crucial r^ ole in a few security proofs. We survey the main examples of the two faces of lattices in cryptology.