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Lattice-based Cryptography

Lattice-based CryptographyDaniele Micciancio Oded Regev July 22, 20081 IntroductionIn this chapter we describe some of the recent progress inlattice- based Cryptography . Lattice-based cryp-tographic constructions hold a great promise for post-quantum Cryptography , as they enjoy very strongsecurity proofs based on worst-case hardness, relatively efficient implementations, as well as great addition, Lattice-based Cryptography is believed to be secure against quantum computers. Our focus herewill be mainly on the practical aspects of Lattice-based Cryptography and less on the methods used to es-tablish their security. For other surveys on the topic of Lattice-based Cryptography , see, , [60, 36, 72, 51]and the lecture notes [50, 68]. The survey by Nguyen and Stern[60] also describes some applications oflattices in cryptanalysis, an important topic that we do notdiscuss here. Another useful resource is thebook by Micciancio and Goldwasser [53], which also containsa wealth of information on the computationalcomplexity aspects of lattice 1: A two-dimensional lattice and two possible what is a lattice ?

complexity aspects of lattice problems. Figure 1: A two-dimensional lattice and two possible bases. So what is a lattice? A lattice is a set of points in n-dimensional space with a periodic structure, such as the one illustrated in Figure 1.

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