最坏情况的Hermite-Korkine-Zolotarev约化格基
Worst-Case Hermite-Korkine-Zolotarev Reduced Lattice Bases
摘要：
Hermite-Korkine-Zolotarev约化在强格约化算法中起着核心作用。通过建立在Ajtai介绍的技术，我们证明了厄米-Korkine-Zolotarev减少基地的存在，可以说是最少的减少。我们证明，对于这样的基地，Kannan的算法解决最短的格向量问题需要$d^{\frac{d}{2\e}(1+o(1))}$位操作的尺寸$d$。这与该算法已知的最佳复杂度上界相匹配。这些基还提供了Schnorr常数$α_d$和$β_d$的下界，它们基本上等于最佳上界。最后，我们还表明存在特别坏的基础Schnorr的层次结构的减少。
英文摘要：
The Hermite-Korkine-Zolotarev reduction plays a central role in strong lattice reduction algorithms. By building upon a technique introduced by Ajtai, we show the existence of Hermite-Korkine-Zolotarev reduced bases that are arguably least reduced. We prove that for such bases, Kannan's algorithm solving the shortest lattice vector problem requires $d^{\frac{d}{2\e}(1+o(1))}$ bit operations in dimension $d$. This matches the best complexity upper bound known for this algorithm. These bases also provide lower bounds on Schnorr's constants $α_d$ and $β_d$ that are essentially equal to the best upper bounds. Finally, we also show the existence of particularly bad bases for Schnorr's hierarchy of reductions.
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