I am currently reading about lattice cryptography and am interested in the cryptosystems based on the LWE problem. I understand the reductions from lattice problems to dLWE. Then we base our belief in the cryptosystem (say Regev or dual Regev PKC) on dLWE. But this is however asymptotic. From what I understand, the way this works is we believe breaking SIVP or whatever lattice problem takes time a very large function of n, and from the reduction we get a SIVP solver, so any algorithm which breaks the cryptosystem also is necessarily is a very large function of n.
What I don't understand however is how does one give parameters to these cryptosystems (for example, if I want 128 bits of security, how large should the LWE instance be)? I have read a couple of papers on this topic and have not really understood it. My comprehension is that we solve LWE like an instance of BDD. But these papers discuss basis reduction and ways of solving SVP and I do not know how to link these things together.
Sorry, my English is really bad.