关于KZ约化
On the KZ Reduction
摘要：
Korkine-Zolotareff（KZ）约简是格译码中常用的约简策略之一。本文首先研究了KZ约化矩阵的一些重要性质。具体地说，我们提出了一个线性上限的Hermit常数，这是约$\frac{7}{8}$倍的现有的尖锐的线性上限，和上限的KZ常数是{ \em多项式}小于现有的尖锐的。我们还提出了KZ约化矩阵的列长的上界，以及分别比提升KZ约化矩阵的列长小{ \em多项式和指数}的KZ约化矩阵的正交性亏损的上界.然后，我们得到的最短向量问题（SVP）的任何解决方案时，其基矩阵是LLL减少的项目的幅度上界。这些上界对于分析KZ约简算法中基展开的复杂性和理解其数值稳定性是有用的。最后，通过改进求解SVP问题常用的Schnorr-Euchner搜索策略和Zhang { \em et al.}仿真结果表明，新的KZ约简算法比Zhang { \em et al.}提出的KZ约简算法速度快，数值可靠性高，特别是当基矩阵是病态的时。
英文摘要：
The Korkine-Zolotareff (KZ) reduction is one of the often used reduction strategies for lattice decoding. In this paper, we first investigate some important properties of KZ reduced matrices. Specifically, we present a linear upper bound on the Hermit constant which is around $\frac{7}{8}$ times of the existing sharpest linear upper bound, and an upper bound on the KZ constant which is {\em polynomially} smaller than the existing sharpest one. We also propose upper bounds on the lengths of the columns of KZ reduced matrices, and an upper bound on the orthogonality defect of KZ reduced matrices which are even {\em polynomially and exponentially} smaller than those of boosted KZ reduced matrices, respectively. Then, we derive upper bounds on the magnitudes of the entries of any solution of a shortest vector problem (SVP) when its basis matrix is LLL reduced. These upper bounds are useful for analyzing the complexity and understanding numerical stability of the basis expansion in a KZ reduction algorithm. Finally, we propose a new KZ reduction algorithm by modifying the commonly used Schnorr-Euchner search strategy for solving SVPs and the basis expansion method proposed by Zhang {\em et al.} Simulation results show that the new KZ reduction algorithm is much faster and more numerically reliable than the KZ reduction algorithm proposed by Zhang {\em et al.}, especially when the basis matrix is ill conditioned.
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