On the smoothing parameter and last minimum of random orthogonal lattices

Let \(X \in {{\mathbb Z}}^{n \times m}\), with each entry independently and identically distributed from an integer Gaussian distribution. We consider the orthogonal lattice \(\varLambda ^\perp (X)\) of X, i.e., the set of vectors \(\mathbf {v}\in {{\mathbb Z}}^m\) such that \(X \mathbf {v}= \mathbf {0}\). In this work, we prove probabilistic upper bounds on the smoothing parameter and the \((m-n)\)-th minimum of \(\varLambda ^\perp (X)\). 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).