On the smoothing parameter and last minimum of random orthogonal lattices

Let X∈ Z^{n × m}, with each entry independently and identically distributed from an integer Gaussian distribution. We consider the orthogonal lattice Λ^⊥(X) of X, i.e., the set of vectors v∈ Z^m such that Xv= 0. In this work, we prove probabilistic upper bounds on the smoothing parameter and the (m- n) -th minimum of Λ^⊥(X). These bounds improve and the techniques build upon prior works of Agrawal et al. (Adv Cryptol 2013:97–116, 2013), and of Aggarwal and Regev (Chic J Theor Comput Sci 7:1–11, 2016).