  A lattice of integers is the collection of all linear combinations of a set
of vectors for which all entries of the vectors are integers and all
coefficients in the linear combinations are also integers. Lattice reduction
refers to the problem of finding a set of vectors in a given lattice such that
the collection of all integer linear combinations of this subset is still the
entire original lattice and so that the Euclidean norms of the subset are
reduced. The present paper proposes simple, efficient iterations for lattice
reduction which are guaranteed to reduce the Euclidean norms of the basis
vectors (the vectors in the subset) monotonically during every iteration. Each
iteration selects the basis vector for which projecting off (with integer
coefficients) the components of the other basis vectors along the selected
vector minimizes the Euclidean norms of the reduced basis vectors. Each
iteration projects off the components along the selected basis vector and
efficiently updates all information required for the next iteration to select
its best basis vector and perform the associated projections.
