why babai rounding technique does not work for finding closest vector using bad basis Given, a basis of the lattice such that the basis not nearly orthogonal(bad basis) babai rounding technique to find the closest vector does not give the desired result. Can any one explain why?
The rounding algorithm, given $\mathbf{t=v+e}$ will output
\begin{equation}
\lfloor\mathbf{t}^t\cdot \mathbf{S}\rceil\cdot \mathbf{S}^{-1}=
\lfloor(\mathbf{v}^t + \mathbf{e}^t) \cdot \mathbf{S}\rceil\cdot \mathbf{S}^{-1}=
\mathbf{v}^t +\lfloor\mathbf{e}^t\cdot \mathbf{S}\rceil\cdot \mathbf{S}^{-1}
\end{equation}
where $\mathbf{v}$ is in the dual lattice. Now, the algorithm works correctly if $\lfloor\mathbf{e}^t\cdot \mathbf{S}\rceil$ becomes zero, which holds as long as both $\mathbf{e}$ and $\mathbf{s}_i$ are short enough (i.e., the basis is good enough)