Complexity Analysis of a Parallel Lattice Basis Reduction Algorithm

Abstract

Lattice basis reduction is an important problem in geometry of numbers with applications in combinatorial optimization, computer algebra, and cryptography. The well-known sequential LLL algorithm finds a short vector in O(n4 log B) arithmetic operations on integers having binary length O(n log B), where n denotes the dimension of the lattice and B denotes the maximum L2 norm of the initial basis vectors. In this paper a new analysis of the parallel algorithm of Roch and Villard is presented. It is shown that on an n x n mesh it needs O(n2 log B) arithmetic operations on integers having binary length O(n log B). This improves the previous analysis and shows that an asymptotical speedup of n2 is possible using n2 processors.