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：整数格是所有整数向量线性组合的集合，其中每个向量元素均为整数，且线性组合中的系数也均为整数。格约化是指在一个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构成整个原始格，同时子集向量的欧几里得范数得到缩减。本文提出了一种简单高效的格约化迭代算法，该算法保证在每次迭代中单调递减基向量（子集中的向量）的欧几里得范数。每次迭代选取一个基向量，通过以整数系数投影移除其他基向量沿该选定向量的分量，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，并高效更新下一次迭代所需的所有信息，以便选择最优基向量并执行相应投影。