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

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 analy...

• Type: Journal
• Year: 2008
• Where: TOC
• Authors: Miklós Ajtai
• Keywords: Algorithm, Shortest Nonzero Vector, Shortest Vector Problem, TOC 2008