Gaussian Sampling in Lattice-Based Cryptography

Abstract

Although rather recent, lattice-based cryptography has stood out on numerous points, be it by the variety of constructions that it allows, by its expected resistance to quantum computers, of by its efficiency when instantiated on some classes of lattices. One of the most powerful tools of lattice-based cryptography is Gaussian sampling. At a high level, it allows to prove the knowledge of a particular lattice basis without disclosing any information about this basis. It allows to realize a wide array of cryptosystems. Somewhat surprisingly, few practical instantiations of such schemes are realized, and the algorithms which perform Gaussian sampling are seldom studied. The goal of this thesis is to fill the gap between the theory and practice of Gaussian sampling. First, we study and improve the existing algorithms, byboth a statistical analysis and a geometrical approach. We then exploit the structures underlying many classes of lattices and apply the ideas of the fast Fourier transform to a Gaussian sampler, allowing us to reach a quasilinearcomplexity instead of quadratic. Finally, we use Gaussian sampling in practice to instantiate a signature scheme and an identity-based encryption scheme. The first one yields signatures that are the most compact currently obtained in lattice-based cryptography, and the second one allows encryption and decryption that are about one thousand times faster than those obtained with a pairing-based counterpart on elliptic curves.