发表机构
Department of Computer Science, Technion（计算机系，海法理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究已知阶有限阿贝尔群的同构判定与基构造，提出随机化同构判定算法达到 $\tilde O(n^{1/4})$ 复杂度，并证明基构造需 $\Omega(\sqrt n)$ 加法，从而分离两者的最优复杂度。
AI 中文摘要
我们研究已知阶为 $n$ 的有限阿贝尔群的同构判定与基构造问题。我们以群内加法运算次数和总运行时间来衡量复杂度。我们给出一个随机化同构判定算法，该算法执行 $\tilde O(n^{1/4})$ 次群加法，运行时间为 $\tilde O(n^{1/4})$。这改进了 Chen 和 Fu 的 $\tilde O(\sqrt n)$ 上界，并在多对数因子内匹配 Bshouty 的 $\Omega(n^{1/4})$ 下界。我们还证明，以至少 $2/3$ 的成功概率找到基在最坏情况下需要 $\Omega(\sqrt n)$ 次群加法。这又在多对数因子内匹配 Chen 和 Fu 的 $\tilde O(\sqrt n)$ 上界。这些结果将有限阿贝尔群中的同构判定与基构造区分开来：它们的最优最坏情况复杂度分别为 $\tilde\Theta(n^{1/4})$ 和 $\tilde\Theta(\sqrt n)$。
英文摘要
We study isomorphism decision and basis construction for finite Abelian groups of known order $n$. We measure complexity by the number of additions performed in the groups and by the total running time. We give a randomized isomorphism decision algorithm that performs $\tilde O(n^{1/4})$ group additions and runs in $\tilde O(n^{1/4})$ time. This improves the $\tilde O(\sqrt n)$ upper bound of Chen and Fu and matches Bshouty $Ω(n^{1/4})$ lower bound up to polylogarithmic factors. We also prove that finding a basis with success probability at least $2/3$ requires $Ω(\sqrt n)$ group additions in the worst case. This matches the $\tilde O(\sqrt n)$ upper bound of Chen and Fu up to polylogarithmic factors. These results separate isomorphism decision from basis construction in finite Abelian groups: their optimal worst-case complexities are $\tildeΘ(n^{1/4})$ and $\tildeΘ(\sqrt n)$, respectivel
