因子分解多项式的逐次子格约化和一种新的复杂度
Gradual sub-lattice reduction and a new complexity for factoring polynomials
摘要：
我们提出了一个专门为一些经典的应用程序格减少格算法。应用程序是一个广义背包型结构的格基，其中的目标向量是boundarily短。对于这样的应用程序，算法的复杂性，改善了传统的格约简取代一些依赖于输入向量的位长度的依赖于输出向量的边界。如果目标向量的位长与输入向量的位长无关，则该算法仅与输入向量的位长成线性关系，这是对二次复杂度浮点LLL算法的改进。为了说明该算法的实用性，我们表明，直接应用因式分解一元多项式的整数导致自1984年以来的第一个复杂性界的改善。第二个应用是代数数重建，其中一个新的复杂性界也得到了。
英文摘要：
We present a lattice algorithm specifically designed for some classical applications of lattice reduction. The applications are for lattice bases with a generalized knapsack-type structure, where the target vectors are boundably short. For such applications, the complexity of the algorithm improves traditional lattice reduction by replacing some dependence on the bit-length of the input vectors by some dependence on the bound for the output vectors. If the bit-length of the target vectors is unrelated to the bit-length of the input, then our algorithm is only linear in the bit-length of the input entries, which is an improvement over the quadratic complexity floating-point LLL algorithms. To illustrate the usefulness of this algorithm we show that a direct application to factoring univariate polynomials over the integers leads to the first complexity bound improvement since 1984. A second application is algebraic number reconstruction, where a new complexity bound is obtained as well.
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