Lattices in Computer Science Lecture 3 CVP Algorithm

In this lecture, we describe an approximation algorithm to the Closest Vector Problem (CVP). This algorithm, known as the Nearest Plane Algorithm, was developed by L. Babai in 1986. It obtains a 2( 2 √ 3 ) approximation ratio, where n is the rank of the lattice. In many applications, this algorithm is applied for a constant n; in such cases, we obtain a constant approximation factor. One can define approximate-CVP as a search problem, as an optimization problem, or as a decision problem (where the latter is often known as a gap problem). In the following definitions, γ ≥ 1 is the approximation factor. By setting γ = 1 we obtain the exact version of the problems. DEFINITION 1 (CV Pγ , SEARCH) Given a basis B ∈ Zm×n and a point t ∈ Zm, find a point x ∈ L(B) such that ∀y ∈ L(B), ‖x− t‖ ≤ γ‖y − t‖. DEFINITION 2 (CV Pγ , OPTIMIZATION) Given a basis B ∈ Zm×n and a point t ∈ Zm, find r ∈ Q such that dist(t,L(B)) ≤ r ≤ γ · dist(t,L(B)). DEFINITION 3 (CV Pγ , DECISION) Given a basis B ∈ Zm×n, a point t ∈ Zm and r ∈ Q, decide if dist(t,L(B)) ≤ r or dist(t,L(B)) > γ · r.