In computer science, lattice problems are a class of optimization problems related to mathematical objects called lattices. The conjectured intractability of such problems is central to the construction of secure lattice-based cryptosystems: lattice problems are an example of NP-hard problems which have been shown to be average-case hard, providing a test case for the security of cryptographic algorithms. In addition, some lattice problems which are worst-case hard can be used as a basis for extremely secure cryptographic schemes. The use of worst-case hardness in such schemes makes them among the very few schemes that are very likely secure even against quantum computers. For applications in such cryptosystems, lattices over vector spaces (often ) or free modules (often ) are generally considered.
For all the problems below, assume that we are given (in addition to other more specific inputs) a basis for the vector space V and a norm N. The norm usually considered is the Euclidean norm L2. However, other norms (such as Lp) are also considered and show up in a variety of results.[1]
Throughout this article, let denote the length of the shortest non-zero vector in the lattice L: that is,
^Khot, Subhash (2005). "Hardness of approximating the shortest vector problem in lattices". J. ACM. 52 (5): 789–808. doi:10.1145/1089023.1089027. S2CID 13438130.
In computer science, latticeproblems are a class of optimization problems related to mathematical objects called lattices. The conjectured intractability...
congruence latticeproblem asks whether every algebraic distributive lattice is isomorphic to the congruence lattice of some other lattice. The problem was posed...
solution (SIS) and ring-SIS problems are two average-case problems that are used in lattice-based cryptography constructions. Lattice-based cryptography began...
random packing of equal spheres generally has a density around 63.5%. A lattice arrangement (commonly called a regular arrangement) is one in which the...
finite lattice representation problem, or finite congruence latticeproblem, asks whether every finite lattice is isomorphic to the congruence lattice of...
Leech lattice is an even unimodular lattice Λ24 in 24-dimensional Euclidean space, which is one of the best models for the kissing number problem. It was...
assumption that the shortest vector problem (SVP) is hard in these ideal lattices. In general terms, ideal lattices are lattices corresponding to ideals in rings...
assumptions used in cryptography (including RSA, discrete log, and some latticeproblems) can be based on worst-case assumptions via worst-case-to-average-case...
In mathematics, the Gauss circle problem is the problem of determining how many integer lattice points there are in a circle centered at the origin and...
asymmetric cryptosystem uses a variant of the learning with errors latticeproblem as its basic trapdoor function. It won the NIST competition for the...
mathematics, the E8 lattice is a special lattice in R8. It can be characterized as the unique positive-definite, even, unimodular lattice of rank 8. The name...
mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). A lattice that satisfies at...
In geometry and crystallography, a Bravais lattice, named after Auguste Bravais (1850), is an infinite array of discrete points generated by a set of...
learning problem. Regev showed that the LWE problem is as hard to solve as several worst-case latticeproblems. Subsequently, the LWE problem has been...
Lattice QCD is a well-established non-perturbative approach to solving the quantum chromodynamics (QCD) theory of quarks and gluons. It is a lattice gauge...
Computational geometry – see Closest pair of points problem Cryptanalysis – for latticeproblem Databases – e.g. content-based image retrieval Coding...
errors problem is the fact that the solution to the RLWE problem can be used to solve a version of the shortest vector problem (SVP) in a lattice (a polynomial-time...
nine possible definable packings. The 8-dimensional E8 lattice and 24-dimensional Leech lattice have also been proven to be optimal in their respective...
one-dimensional (1D) latticeproblem. In 1944 Onsager was able to get an exact solution to a two-dimensional (2D) latticeproblem at the critical density...
existence of highly symmetrical lattices: the E8 lattice and the Leech lattice. If arrangements are restricted to lattice arrangements, in which the centres...
the lattice of stable matchings is a distributive lattice whose elements are stable matchings. For a given instance of the stable matching problem, this...
to each problem. The Clay Mathematics Institute officially designated the title Millennium Problem for the seven unsolved mathematical problems, the Birch...
generalization of the previously mentioned problems, as well as graph isomorphism and certain latticeproblems. Efficient quantum algorithms are known for...
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