Lattice-based cryptography aims at harnessing the security of cryptographic primitives in the conjectured hardness of well-identified and well-studied algorithmic problems involving Euclidean lattices. This approach leads to more efficient primitives, increased security (the most common lattice problems are conjectured quantum-hard), and improved cryptographic functionalities (fully homomorphic encryption, functional encryption, program obfuscation, etc). A less common but still recurring family of lattices are the so-called orthogonal lattices where the matrix is often sampled from a Gaussian distribution. When lattices have turned out into a major build block in designing cryptographic primitives, orthogonal lattices were used in various constructions such as cryptographic multilinear maps, traitor-tracing schemes, and inner product functional encryption. In this thesis, we study the cryptographic aspects of orthogonal lattices. First, we consider the successive minima and the smoothing parameter of random orthogonal lattices. The main motivation (and the result) is a generalization of the leftover hash lemma (LHL) to lattices and discrete Gaussian distributions. Our results are an improving probabilistic upper bound on the smoothing parameter and give a probabilistic upper bound on the minimum of the orthogonal lattice. Second, we investigate broadcast encryption with anonymous revocation, in which ciphertexts do not reveal any information on which users have been revoked. In this case, the orthogonal lattices are involved in the security proofs of these protocols. We develop a generic transformation of linear functional encryption toward trace-andrevoke systems with the novelty of achieving anonymity. Finally, a fundamental problem related to lattices is the learning with errors (LWE) problem which is an amazingly versatile basis for cryptographic constructions. We focus on the computational hardness of standard lattice problems restricted to orthogonal lattices and on the building of new cryptographic constructions. More precisely, we introduce a new variant of the LWE problem over the integers, without any modular reduction, then give a construction of an encryption scheme whose security is based on the hardness of the integer-LWE problem.
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