Shortest Integer Vectors

Let A be a fixed integer matrix of size m by n and consider all b for which the body K_b = {x: Ax ≤ b} is full dimensional. We examine the set of shortest nonzero integral vectors with respect to the family of norms whose unit balls are given by (K_b − K_b). We show that the number of such shortest vectors is polynomial in the bit size of A, for fixed n. We also show the existence, for any n, of a family of matrices M for which the number of shortest vectors has as a lower bound a polynomial in the bit size of M of the same degree as the polynomial bound.