有限域上多项式分解的拍指数3/2等价代数问题
Algebraic Problems Equivalent to Beating Exponent 3/2 for Polynomial Factorization over Finite Fields
摘要：
在有限域上分解一元多项式的最快已知算法是Kaltofen-Shoup算法的Kedlaya-Umans（快速模复合）实现。它是随机的，并且花费$\widetilde{O}(n^{3/2}\log q + n \log^2 q)$时间来分解具有$q$元素的有限域$\mathbb{F}_q$上的次数为$n$的多项式。一个重要的未决问题是$3/2$指数是否可以改进。我们研究了一系列代数问题，并在它们之间建立了一个减少的网络。结果是，对于这些问题中的任何一个，具有比$3/2$更好的指数的算法将产生具有比$3/2$更好的指数的多项式分解算法。
英文摘要：
The fastest known algorithm for factoring univariate polynomials over finite fields is the Kedlaya-Umans (fast modular composition) implementation of the Kaltofen-Shoup algorithm. It is randomized and takes $\widetilde{O}(n^{3/2}\log q + n \log^2 q)$ time to factor polynomials of degree $n$ over the finite field $\mathbb{F}_q$ with $q$ elements. A significant open problem is if the $3/2$ exponent can be improved. We study a collection of algebraic problems and establish a web of reductions between them. A consequence is that an algorithm for any one of these problems with exponent better than $3/2$ would yield an algorithm for polynomial factorization with exponent better than $3/2$.
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