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.

翻译：整数格是由一组向量所有整系数线性组合构成的集合，其中向量所有分量为整数，且线性组合系数亦为整数。格约化问题是指在给定格中寻找一组向量，使得该子集的所有整系数线性组合仍构成整个原始格，并降低该子集的欧几里得范数。本文提出简单高效的格约化迭代方法，确保每次迭代单调降低基向量（子集中的向量）的欧几里得范数。每次迭代选择特定基向量，通过整数系数投影去除其他基向量沿该向量的分量，从而最小化约化后基向量的欧几里得范数。每次迭代执行沿选定基向量的分量投影，并高效更新下一迭代所需的所有信息，以便选择最佳基向量并执行相应投影。