Given a closest vector problem (CVP):
Let $\mathbf{B}=[\mathbf{b}_1, \mathbf{b}_2, \dots, \mathbf{b}_n]$ be the basis matrix of the lattice $\mathcal{L}(\mathbf{B})=\{\mathbf{Bx}|\mathbf{x} \in \mathbb{Z}^n\}$. Find $\mathbf{b} \in \mathcal{L}(\mathbf{B})$ such that the distance to the target vector $\mathbf{t}$ $$||\mathbf{b} - \mathbf{t}||$$ is minimized.
How can we estimate the order of magnitude of $$\min_{\mathbf{b} \in \mathcal{L}(\mathbf{B})} ||\mathbf{b} - \mathbf{t}||$$?
If we are given concrete numbers for the CVP, I know we can just run Babai's Nearest Plane algorithm on the problem and use the obtained solution to bound the desired quantity. However, I would like a solution that is more symbolic, in terms of $\mathbf{B}$ and $\mathbf{t}$.
Specifically, I am working with problems of the form $$\mathbf{B}=\begin{pmatrix} a_1 & 0 & \cdots & 0 \\ 0 & a_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & a_n \\ b_1 & b_2 & \cdots & b_n \end{pmatrix}, \, \mathbf{t} = \begin{pmatrix} 0 \\ 0 \\ \vdots \\ 0 \\ c \\ \end{pmatrix}$$, if this makes any difference.