Improved cryptographic hash functions with worst-case/average-case connection

(MATH) We define a new family of collision resistant hash functions whose security is based on the worst case hardness of approximating the covering radius of a lattice within a factor O(πn²log n), where π is a value between 1 and √ \over n that depends on the solution of the closest vector problem in certain "almost perfect" lattices. Even for π = √ \over n, this improves the smallest (worst-case) inapproximability factor for lattice problems known to imply the existence of one-way functions. (Previously known best factor was O(n^3+ε) for the shortest independent vector problem, due to Cai and Nerurkar, based on work of Ajtai.) Using standard transference theorems from the geometry of numbers, our result immediately gives a connection between the worst-case and average-case complexity of the shortest vector problem with connection factor O(πn³}log n), improving the best previously known connection factor O(n^4+ε), also due to Ajtai, Cai and Nerurkar.