Lattice problem

In computer science, lattice problems are a class of optimization problems on lattices. The conjectured intractability of such problems is central to construction of secure lattice-based cryptosystems. For applications in such cryptosystems, lattices over vector spaces (often ${\displaystyle \mathbb {Q} ^{n}}$) or free modules (often ${\displaystyle \mathbb {Z} ^{n}}$) 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 L². However, other norms (such as L^p) are also considered and show up in a variety of results. Let ${\displaystyle \lambda (L)}$ denote the length of the shortest non-zero vector in the lattice L, that is,

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