Polynomial-time computation of the annular lattice-vector count

Determine whether there exists a polynomial-time algorithm that, given an n-dimensional lattice whose shortest nonzero vector has length 2 and a positive number α, determines the cardinality of X(α,Λ) = {v ∈ Λ : 2 ≤ ‖v‖ ≤ 2 + α}.

Background

This problem strengthens the preceding algorithmic question by imposing a complexity requirement. The quantity X(α,Λ) counts lattice vectors in a prescribed Euclidean annulus, and the paper does not resolve whether that count can be computed in time polynomial in an appropriate representation of the lattice and the parameters.

The question is relevant to the paper’s discussion of lattice-based cryptography, where the number of lattice vectors within a multiple of the shortest-vector length is an important structural quantity.