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：整数格是所有向量分量为整数且线性组合系数均为整数的向量集合的所有线性组合。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数线性组合仍能生成整个原始格，同时该子集的欧几里得范数被约化。本文提出一种简单高效的迭代格约化算法，该算法保证在每次迭代中单调递减基向量（即子集中的向量）的欧几里得范数。每次迭代通过选择某个基向量，以整数系数沿该向量投影其他基向量的分量，使得约化后基向量的欧几里得范数最小化。每次迭代沿所选基向量投影分量后，算法高效更新所有必要信息，供下次迭代选择最优基向量并执行相关投影操作。