One doubt on an NP hard problem It seems that Shortest Vector Problem in a lattice  is NP hard. Then how Ajtai-Kumar-Sivakumar (AKS) algorithm solves it? I mean, what is the witness?
First of all, SVP is not known to be NP-complete. There are some hardness results for SVP, but just considering very specific scenarios (for instance, SVP is NP-Hard in the $\ell_{\infty}$ norm) or under other types of reductions (such as randomized or non-uniform reductions).

Even so, the AKS algorithm runs in exponential time, so, it doesn't give us any information about the hardness of SVP.

When we say that a problem is NP-hard, we expected that there is no polynomial-time algorithm to solve the problem, but there is nothing strange with the fact that we know exponential-time algorithms for the problem (and we usually know trivial exponential-time algorithms to solve hard problems...).