Worst-Case hardness of lattice problems
An instance of the (approximate) Shortest Vector Problem is simply a basis of some lattice. The desired output is any (approximately) shortest nonzero vector of that lattice. To solve this problem in the worst case, an algorithm must produce the desired output given any basis of any arbitrary lattice. (Other lattice problems have similar instances, usually including a basis and sometimes other information, like an arbitrary target point in the case of the Closest Vector Problem.)