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的所有整数线性组合构成的集合，其中向量的所有分量均为整数，且线性组合的系数也为整数。格约简问题是指在给定格中寻找一组向量，使得该子集的所有整数线性组合仍能生成整个原始格，同时子集中向量的欧几里得范数尽可能减小。本文提出了一种简单高效的迭代式格约简算法，该算法保证在每次迭代中单调降低基向量（子集中的向量）的欧几里得范数。每次迭代选择这样一个基向量：沿该向量方向（以整数系数）投影去除其他基向量的分量，使得约简基向量的欧几里得范数最小化。随后，每次迭代沿所选基向量去除分量投影，并高效更新下一次迭代所需的所有信息，以便选取最优基向量并执行对应投影操作。