I just started work on lattice-based cryptography and I could not understand the concept of worst-case to average-case reduction.
We generally say, Average Case Hardness: Random instance of a problem is hard to solve. Worst-Case Hardness: Hard to solve every instance of the problem (even if most instances are easy)
So far, so good
But I could not establish the relationship between lattices and worst-case hardness concept. Let's say our problem is "Shortest Vector Problem (SVP)". I know it is a hard problem and we can prove the security of lattice-based cryptosystems based on hardness of this -or other related problems- problem.
My question is: What is the instances of this problem? Lattices? Given bases? or shortest vector of that lattice? Firstly, I thought "worst" related with "shortest vector" but I was wrong. Could you explain this?