The best known asymptotic bit complexity bound for factoring univariate polynomials over finite fields grows with \(d^{1.5 + o (1)}\) for input polynomials of degree d, and with the square of the bit size of the ground field. It relies on a variant of the Cantor–Zassenhaus algorithm which exploits fast modular composition. Using techniques by Kaltofen and Shoup, we prove a refinement of this bound when the finite field has a large extension degree over its prime field. We also present fast practical algorithms for the case when the extension degree is smooth.

中文翻译：

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大扩展度有限域上的单变量多项式分解

开往有限域上因式分解单变量多项式最好已知渐近位复杂性的增长与\（d ^ {1.5 + O（1）} \）为度的输入多项式d，并与地面字段的比特大小的平方。它依赖于利用快速模块化组合的 Cantor-Zassenhaus 算法的变体。使用 Kaltofen 和 Shoup 的技术，我们证明了当有限域在其素域上具有大的扩展度时对这个界的改进。我们还针对扩展度平滑的情况提出了快速实用的算法。