I've been reading How to estimate the hardness of SIS instances? and following some of its sources, and I want to confirm a few things.
LLL algorithm runs in polynomial time, but isn't capable of producing arbitrarily small basis, and is therefore not very relevant in cryptanalysis in most lattice-based schemes. Is that correct?
BKZ algorithm invokes a SVP (Shortest Vector Problem) oracle polynomial number of times, and the lower-bounded of the norm of the basis it produces is inversely proportional to the block size the SVP oracle operates on, regardless of the norm of the input basis; and the SVP oracle it uses runs in time exponential to the block size. Is that correct?