用GRH交换代数：分解多项式和相关结构的算法
Trading GRH for algebra: algorithms for factoring polynomials and related structures
摘要：
在本文中，我们开发的技术，消除了需要的广义黎曼假设（GRH）从各种（几乎所有）已知的结果确定性多项式分解有限域。我们的主要结果表明，给定有限域k上的n次多项式f（x），我们可以在确定性poly（n^{ \log n}，\log |k|）时间“中找到”f（x）的一个非平凡因子“或”k[x]/（f（x））的一个n阶非平凡自同构“.这个主要工具导致了各种新的GRH自由的结果，其中最引人注目的是：（1）给定有限域上的非交换代数，我们可以找到一个零因子在确定性次指数时间。(2)给定一个正整数r，使得|r或r至少有两个不同的奇素因子。有一个确定性的多项式时间算法来寻找有限域上的r次分圆多项式的非平凡因子。在Lenstra（1991）关于构造有限域之间的同构的开创性工作之后，本文进一步将经典的Galois理论构造如分圆扩张、Kummer扩张、Teichmuller子群推广到具有自同构的交换半单代数的情形.这些一般化的结构有助于消除对GRH的依赖。
英文摘要：
In this paper we develop techniques that eliminate the need of the Generalized Riemann Hypothesis (GRH) from various (almost all) known results about deterministic polynomial factoring over finite fields. Our main result shows that given a polynomial f(x) of degree n over a finite field k, we can find in deterministic poly(n^{\log n},\log |k|) time "either" a nontrivial factor of f(x) "or" a nontrivial automorphism of k[x]/(f(x)) of order n. This main tool leads to various new GRH-free results, most striking of which are: (1) Given a noncommutative algebra over a finite field, we can find a zero divisor in deterministic subexponential time. (2) Given a positive integer r such that either 8|r or r has at least two distinct odd prime factors. There is a deterministic polynomial time algorithm to find a nontrivial factor of the r-th cyclotomic polynomial over a finite field. In this paper, following the seminal work of Lenstra (1991) on constructing isomorphisms between finite fields, we further generalize classical Galois theory constructs like cyclotomic extensions, Kummer extensions, Teichmuller subgroups, to the case of commutative semisimple algebras with automorphisms. These generalized constructs help eliminate the dependence on GRH.
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