Shortest Vector Problem ( 1982 ; Lenstra , Lenstra , Lovasz )

of n linearly independent vectors b1, . . . ,bn ∈ Rm in m-dimensional Euclidean space. For computational purposes, the lattice vectors b1, . . . ,bn are often assumed to have integer (or rational) entries, so that the lattice can be represented by an integer matrix B = [b1, . . . ,bn] ∈ Zm×n (called basis) having the generating vectors as columns. Using matrix notation, lattice points in L(B) can be conveniently represented as Bx where x is an integer vector. The integers m and n are called the dimension and rank of the lattice respectively. Notice that any lattice admits multiple bases, but they all have the same rank and dimension. The main computational problems on lattices are the Shortest Vector Problem, which asks to find the shortest nonzero vector in a given lattice, and the Closest Vector Problem, which asks to find the lattice point closest to a given target. Both problems can be defined with respect to any norm, but the Euclidean norm ‖v‖ = √