Optimal lower bounds for the Korkine-Zolotareff parameters of a lattice and for Schnorr's algorithm for the shortest vector problem

Abstract

Schnorr’s algorithm for finding an approximation for the shortest nonzero vector in an n-dimensional lattice depends on a parameter k. He proved that for a fixed k ≤ n his algorithm (block 2k-reduction) provides a lattice vector whose length is greater than the length of a shortest nonzero vector in the lattice by at most a factor of (2k)2n/k. (The time required by the algorithm depends on k.) We show that if k = o(n), this bound on the performance of Schnorr’s algorithm cannot be improved (apart from a constant factor in the exponent). Namely, we prove the existence of a basis in Rn which is KZ-reduced on all k-segments and where the ratio ‖b1‖/shortest(L) is at least kcn/k. Noting that such a basis renders all versions of Schnorr’s algorithm idle (output = input), it follows that the quantity kcn/k is a lower bound on the approximation ratio any version of Schnorr’s algorithm can achieve on the shortest vector problem. This proves that Schnorr’s analysis of ∗A preliminary version of this paper has appeared in the Proc. 35th ACM Symp. on Theory of Computing [2]. ACM Classification: F.2.2, G.2 AMS Classification: 68Q25, 68W40, 68W25, 11H55, 11H99, 52C07, 60D05