Parallel Complexitiy of Lattice Basis Reduction and a Floating-Point Parallel Algorithm

Lattice basis reduction is an important problem in the areas of computer algebra and geometry of numbers There are several e cient sequential algorithms for lattice basis reduction e g the well known LLL algorithm and a variant of Schnorr and Euchner which uses fast oating point arithmetic Recently parallel algorithms were developed but they use slow exact integer arithmetic and until now a formal proof for the e ciency of parallel algorithms with n processors is omitted where n denotes the dimension of the lattice In this paper a variant of the well known basis reduction algorithms is presented that is well suited for the computation with fast oating point arithmetic and for the implementation on a mesh connected array of n processors In addition an error analysis the parallel implementation and a theorem about the parallel e ciency is provided