Many (though not all) lattice-based cryptographic constructions are known to be secure if SVP is in fact hard in this regime. In linear algebra, a lattice L ⊂ R n, and even with a quantum computer. For example, in 2009, Craig Gentry introduced the first fully homomorphic encryption scheme, which was based on a lattice problem. Much more work has been devoted to constructing additional cryptographic primitives based on LWE and related problems. Since then, much follow-up work has focused on improving Regev's security proof and improving the efficiency of the original scheme. The first lattice-based public-key encryption scheme whose security was proven under worst-case hardness assumptions was introduced by Oded Regev in 2005, together with the Learning with Errors problem (LWE). However, their scheme is not known to be at least as hard as solving a worst-case lattice problem. Silverman introduced a lattice-based public-key encryption scheme, known as NTRU. In 1998, Jeffrey Hoffstein, Jill Pipher, and Joseph H. She then showed a cryptographic hash function whose security is equivalent to the computational hardness of SIS. In 1996, Miklós Ajtai introduced the first lattice-based cryptographic construction whose security could be based on the hardness of well-studied lattice problems, and Cynthia Dwork showed that a certain average-case lattice problem, known as Short Integer Solutions (SIS), is at least as hard to solve as a worst-case lattice problem. Furthermore, many lattice-based constructions are considered to be secure under the assumption that certain well-studied computational lattice problems cannot be solved efficiently. Unlike more widely used and known public-key schemes such as the RSA, Diffie-Hellman or elliptic-curve cryptosystems - which could, theoretically, be defeated using Shor's algorithm on a quantum computer - some lattice-based constructions appear to be resistant to attack by both classical and quantum computers. Lattice-based constructions are currently important candidates for post-quantum cryptography. Lattice-based cryptography is the generic term for constructions of cryptographic primitives that involve lattices, either in the construction itself or in the security proof. Constructions of cryptographic primitives that involve lattices Of a bounded lattice-ordered set, we say that is complemented in if there exists an element such that and. For a bounded lattice-ordered set, the upperīound is frequently denoted 1 and the lower bound is frequently denoted 0. From a universal algebraist's point of view, however, a lattice is different from a lattice-ordered set because lattices are algebraic structures that form an equational class or variety, but lattice-ordered sets are not algebraic structures, and therefore do not form a variety.Ī lattice-ordered set is bounded provided that it is a bounded poset, i.e., if it has an upper bound and a lower bound. Lattice-ordered sets abound in mathematics and its applications, and many authors do not distinguish between them and lattices. (In other words, one may prove that for any lattice,Īnd for any two members and of, if and only if. One obtains the same lattice-ordered set from the given lattice by setting in if and only if. Also, from a lattice, one may obtain a lattice-ordered set by setting in if and only if. In fact, a lattice is obtained from a lattice-ordered poset by defining and for any. There is a natural relationship between lattice-ordered A lattice-ordered set is a poset in which each two-element subset has an infimum, denoted, and a supremum, denoted.
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