This may be too broad question but it is not. I have been studying lattices for few months now, more specifically I studied:
- Lattice problems ($SVP$, $CVP$ and etc.)
- Lattice cryptography in post quantum computers
- Worst-case hardness in Lattice cryptography
- Public key encryption using Lattice
- Lattice reduction algorithms (LLL and its running time)
However, I could not figure out the relationship with reducing a lattice basis and lattice problems. Essentially, why we should reduce a lattice basis? I know the goal of reducing lattice basis is to get short and nearly orthogonal basis vectors, but why? what is the bigger picture?
To be fair, during past few months of reading papers, I found only few only sentences (here, and there):
Reduced bases allow to solve the following important lattice problems (SVP, CVP), either exactly or approximately
It would be greatly appreciated if I could get a direction or hint or something to help me figure out the concept.