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?
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3$\begingroup$ Draw the points of a lattice in two dimensions. Choose a "good" basis (nearly orthogonal) and a "bad" basis (far from orthogonal). Choose a few points in space and use the rounding technique. See what happens. Then ask again. $\endgroup$– K.G.Commented Jan 9, 2017 at 7:49
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1 Answer
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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)
